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20 Effective Math Strategies To Approach ProblemSolving
Katie Keeton
Math strategies for problemsolving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.
Problemsolving skills are essential to math in the general classroom and reallife. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.
This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in reallife situations.
What are problemsolving strategies?
Problemsolving strategies in math are methods students can use to figure out solutions to math problems. Some problemsolving strategies:
 Draw a model
 Use different approaches
 Check the inverse to make sure the answer is correct
Students need to have a toolkit of math problemsolving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better.
Strategies can help guide students to the solution when it is difficult ot know when to start.
The ultimate guide to problem solving techniques
Download these readytogo problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.
20 Math Strategies For ProblemSolving
Different problemsolving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem.
Here are 20 strategies to help students develop their problemsolving skills.
Strategies to understand the problem
Strategies that help students understand the problem before solving it helps ensure they understand:
 The context
 What the key information is
 How to form a plan to solve it
Following these steps leads students to the correct solution and makes the math word problem easier .
Here are five strategies to help students understand the content of the problem and identify key information.
1. Read the problem aloud
Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.
2. Highlight keywords
When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed. For example, if the word problem asks how many are left, the problem likely requires subtraction. Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.
3. Summarize the information
Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary. Summaries should include only the important information and be in simple terms that help contextualize the problem.
4. Determine the unknown
A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer. Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.
5. Make a plan
Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it. The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer. Encourage students to make a list of each step they need to take to solve the problem before getting started.
Strategies for solving the problem
1. draw a model or diagram.
Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process. It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.
Similarly, you could draw a model to represent the objects in the problem:
2. Act it out
This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives . When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts. The examples below show how 1stgrade students could “act out” an addition and subtraction problem:
The problem  How to act out the problem 
Gia has 6 apples. Jordan has 3 apples. How many apples do they have altogether?  Two students use counters to represent the apples. One student has 6 counters and the other student takes 3. Then, they can combine their “apples” and count the total. 
Michael has 7 pencils. He gives 2 pencils to Sarah. How many pencils does Michael have now?  One student (“Michael”) holds 7 pencils, the other (“Sarah”) holds 2 pencils. The student playing Michael gives 2 pencils to the student playing Sarah. Then the students count how many pencils Michael is left holding. 
3. Work backwards
Working backwards is a popular problemsolving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution. This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.
For example,
To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71. Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.
4. Write a number sentence
When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved. It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.
5. Use a formula
Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve. Make sure students know the important mathematical formulas they will need in tests and reallife. It can help to display these around the classroom or, for those who need more support, on students’ desks.
Strategies for checking the solution
Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense.
There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.
Here are five strategies to help students check their solutions.
1. Use the Inverse Operation
For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work.
2. Estimate to check for reasonableness
Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable. Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten. For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10. When the estimate is clear the two numbers are close. This means your answer is reasonable.
3. PlugIn Method
This method is particularly useful for algebraic equations. Specifically when working with variables. To use the plugin method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.
If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓
4. Peer Review
Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly. Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixedability partners or similarability partners. In mixedability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills. If partners have comparable ability levels and problemsolving skills, they may find that they approach problems differently or have unique insights to offer each other about the problemsolving process.
5. Use a Calculator
A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to realworld situations.
Stepbystep problemsolving processes for your classroom
In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4step process to solve problems.
Polya’s 4 steps include:
 Understand the problem
 Devise a plan
 Carry out the plan
Today, in the style of George Polya, many problemsolving strategies use various acronyms and steps to help students recall.
Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom.
Here are 5 problemsolving strategies to introduce to students and use in the classroom.
How Third Space Learning improves problemsolving
Resources .
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Explore the range of problem solving resources for 2nd to 8th grade students.
Oneonone tutoring
Third Space Learning offers oneonone math tutoring to help students improve their math skills. Highly qualified tutors deliver highquality lessons aligned to state standards.
Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problemsolving to independent practice.
Throughout each lesson, tutors ask higherlevel thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problemsolving skills.
Problemsolving
Educators can use many different strategies to teach problemsolving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra.
Teaching students how to choose and implement problemsolving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to reallife problemsolving.
READ MORE : 8 Common Core math examples
There are many different strategies for problemsolving; Here are 5 problemsolving strategies: • draw a model • act it out • work backwards • write a number sentence • use a formula
Here are 10 strategies of problemsolving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model • Act it out • Work backwards • Write a number sentence • Use a formula
1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back
Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.
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4 Best Steps To Problem Solving in Math That Lead to Results
Eastern Shore Math Teacher
What does problem solving in math mean, and how to develop these skills in students? Problem solving involves tasks that are challenging and make students think. In teaching through problem solving, learning takes place while trying to solve problems with specific concepts and skills. Therefore, teachers need to provide safe learning spaces that foster a growth mindset in math in order for students to take risks to solve problems. In addition, providing students with problem solving steps in math builds success in solving problems.
By providing rich mathematical tasks and engaging puzzles, students improve their number sense and mindset about mathematics. Click Here to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students.
Students who feel successful in math class are happier and more engaged in learning. Check out The Bonus Guide for Creating a Growth Mindset Classroom and Students Who Love Math for ideas, lessons, and mindset surveys for students to use in your classroom to cultivate a positive classroom community in mathematics. You can also sign up for other freebies from me Here at Easternshoremathteacher.com .
Have you ever given students a word problem or rich task, and they froze? They have no idea how to tackle the problem, even if it is a concept they are successful with. This is because they need problem solving strategies. I started to incorporate more problem solving tasks into my teaching in addition to making the 4 steps for problem solving a schoolwide initiative and saw results.
What is Problem Solving in Math?
When educators use the term problem solving , they are referring to mathematical tasks that are challenging and require students to think. Such tasks or problems can promote students’ conceptual understanding, foster their ability to reason and communicate mathematically, and capture their interests and curiosity (Hiebert & Wearne, 1993; Marcus & Fey, 2003; NCTM, 1991; van de Walle, 2003).
How Should Problem Solving For Math Be Taught?
Problem solving should not be done in isolation. In the past, we would teach the concepts and procedures and then assign onestep “story” problems designed to provide practice on the content. Next, we would teach problem solving as a collection of strategies such as “draw a picture” or “guess and check.” Eventually, students would be given problems to apply the skills and strategies. Instead, we need to make problem solving an integral part of mathematics learning.
In teaching through problem solving, learning takes place while trying to solve problems with specific concepts and skills. As students solve problems, they can use any strategy. Then, they justify their solutions with their classmates and learn new ways to solve problems.
Students do not need every task to involve problem solving. Sometimes the goal is to just learn a skill or strategy.
Criteria for Problem Solving Math
Lappan and Phillips (1998) developed a set of criteria for a good problem that they used to develop their middle school mathematics curriculum (Connected Mathematics). The problem:
 has important, useful mathematics embedded in it.
 requires higherlevel thinking and problem solving.
 contributes to the conceptual development of students.
 creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
 can be approached by students in multiple ways using different solution strategies.
 has various solutions or allows different decisions or positions to be taken and defended.
 encourages student engagement and discourse.
 connects to other important mathematical ideas.
 promotes the skillful use of mathematics.
 provides an opportunity to practice important skills.
Of course, not every problem will include all of the above. However, the first four are essential. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.
The real value of these criteria is that they provide teachers with guidelines for making decisions about how to make problem solving a central aspect of their instruction. Read more at NCTM .
Problem Solving Teaching Methods
Teaching students these 4 steps for solving problems allows them to have a process for unpacking difficult problems.
As you teach, model the process of using these 4 steps to solve problems. Then, encourage students to use these steps as they solve problems. Click here for Posters, Bookmarks, and Labels to use in your classroom to promote the use of the problem solving steps in math.
How Problem Solving Skills Develop
Problem solving skills are developed over time and are improved with effective teaching practices. In addition, teachers need to select rich tasks that focus on the math concepts the teacher wants their students to explore.
Problem Solving 4 Steps
Understand the problem.
Read & Think
 Circle the needed information and underline the question.
 Write an answer STEM sentence. There are_____ pages left to read.
Plan Out How to Solve the Problem
Make a Plan
 Use a strategy. (Draw a Picture, Work Backwards, Look for a Pattern, Create a Table, Bar Model)
 Use math tools.
Do the Problem
Solve the Problem
 Show your work to solve the problem. This could include an equation.
Check Your Work on the Problem
Answer & Check
 Write the answer into the answer stem.
 Does your answer make sense?
 Check your work using a different strategy.
Check out these Printables for Problem Solving Steps in Math .
Teaching Problem Solving Strategies
A problem solving strategy is a plan used to find a solution. Understanding how a variety of problem solving strategies work is important because different problems require you to approach them in different ways to find the best solution. By mastering several problemsolving strategies, you can select the right plan for solving a problem. Here are a few strategies to use with students:
 Draw a Picture
 Work Backwards
 Look for a Pattern
 Create a Table
Why is Using Problem Solving Steps For Math Important?
Problem solving allows students to develop an understanding of concepts rather than just memorizing a set of procedures to solve a problem. In addition, it fosters collaboration and communication when students explain the processes they used to arrive at a solution. Through problemsolving, students develop a deeper understanding of mathematical concepts, become more engaged, and see the importance of mathematics in their lives.
NCTM Process Standards
In 2011 the Common Core State Standards incorporated the NCTM Process Standards of problemsolving, reasoning and proof, communication, representation, and connections into the Standards for Mathematical Practice. With these process standards, the focus became more on mathematics through problem solving. Students could no longer just develop procedural fluency, they needed to develop conceptual understanding in order to solve new problems and make connections between mathematical ideas.
Engaging Students to Learn in Mathematics Class
Engaging students to learn in math class will help students to love math. Children develop a dislike of math early on and end up resenting it into adult life. Even in the real world, students will likely have to do some form of mathematics in their personal or working life. So how can teachers make math more interesting to engage students in the subject? Read more at 5 Best Strategies for Engaging Students to Learn in Mathematics Class
Teachers can promote number sense by providing rich mathematical tasks and encouraging students to make connections to their own experiences and previous learning.
Sign up on my webpage to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students. Providing opportunities to do math puzzles daily is one way to help students develop their number sense. CLICK Here to sign up for 71 Math Number Puzzles and check out my website.
Promoting a Growth Mindset
Research shows that there is a link between a growth mindset and success. In addition, kids who have a growth mindset about their abilities perform better and are more engaged in the classroom. Students need to be able to preserve and make mistakes when problem solving.
Read more … 5 Powerful and Easy Lessons Teaching Students How to Get a Growth Mindset
Here are some Resources to Use to Grow a Growth Mindset
 Free Mindset Survey
 Growth Mindset Classroom Display Free
 Growth Mindset Lessons
Using Word Problems
Story Problems and word problems are one way to promote problem solving. In addition, they provide great practice in using the 4 steps of solving problems. Then, students are ready for more challenging problems.
For Kindergarten
 Subtraction within 5
For First Grade
 Word Problems to 20
 Word Problems of Subtraction
For Second Grade
 Two Step Word Problems with Addition and Subtraction
 Grade 2 Addition and Subtraction Word Problems
 Word Problems with Subtraction
For Third Grade
 Word Problems Division and Multiplication
 Multiplication Word Problems
For Fourth Grade
 Multiplication Area Model
 Multiplicative Comparison Word Problems
Resources for Problem Solving
 3 Act Tasks
 What’s the Best Proven Way to Teach Word Problems with Two Step Equations?
 5 Powerful and Easy Lessons Teaching Students How to Get a Growth Mindset
 5 Powerful Ideas to Help Students Develop a Growth Mindset in Mathematics
Problem Solving Steps For Math
In mathematics, problem solving is one of the most important topics to teach. Learning to problem solve helps students apply mathematics to realworld situations. In addition, it is used for a deeper understanding of mathematical concepts.
By providing rich mathematical tasks and engaging puzzles, students improve their number sense and mindset about mathematics. Click Here to get this Freebie of 71 Math Number Puzzles delivered to your inbox to use with your students.
Check out The Free Ultimate Guide for Creating a Growth Mindset Classroom and Students Who Love Math for ideas, lessons, and mindset surveys to use to cultivate a growth mindset classroom.
Start by modeling using the problem solving steps in math and allowing opportunities for students to use the steps to solve problems. As students become more comfortable with using the steps and have some strategies to use, provide more challenging tasks. Then, students will begin to see the importance of problem solving in math and connecting their learning to realworld situations.
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Problem Solving in Mathematics Education
 Open Access
 First Online: 28 June 2016
Cite this chapter
You have full access to this open access chapter
 Peter Liljedahl 6 ,
 Manuel SantosTrigo 7 ,
 Uldarico Malaspina 8 &
 Regina Bruder 9
Part of the book series: ICME13 Topical Surveys ((ICME13TS))
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Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students’ development of mathematical knowledge and problem solving competencies. The accumulated knowledge and field developments include conceptual frameworks to characterize learners’ success in problem solving activities, cognitive, metacognitive, social and affective analysis, curriculum proposals, and ways to foster problem solving approaches. In the survey, four interrelated areas are reviewed: (i) the relevance of heuristics in problem solving approaches—why are they important and what research tells us about their use? (ii) the need to characterize and foster creative problem solving approaches—what type of heuristics helps learners think of and practice creative solutions? (iii) the importance for learners to formulate and pursue their own problems; and (iv) the role played by the use of both multiple purpose and ad hoc mathematical action types of technologies in problem solving activities—what ways of reasoning do learners construct when they rely on the use of digital technologies and how technology and technology approaches can be reconciled?
You have full access to this open access chapter, Download chapter PDF
 Mathematical Problem
 Prospective Teacher
 Creative Process
 Digital Technology
 Mathematical Task
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Mathematical problem solving has long been seen as an important aspect of mathematics, the teaching of mathematics, and the learning of mathematics. It has infused mathematics curricula around the world with calls for the teaching of problem solving as well as the teaching of mathematics through problem solving. And as such, it has been of interest to mathematics education researchers for as long as our field has existed. More relevant, mathematical problem solving has played a part in every ICME conference, from 1969 until the forthcoming meeting in Hamburg, wherein mathematical problem solving will reside most centrally within the work of Topic Study 19: Problem Solving in Mathematics Education. This booklet is being published on the occasion of this Topic Study Group.
To this end, we have assembled four summaries looking at four distinct, yet interrelated, dimensions of mathematical problem solving. The first summary, by Regina Bruder, is a nuanced look at heuristics for problem solving. This notion of heuristics is carried into Peter Liljedahl’s summary, which looks specifically at a progression of heuristics leading towards more and more creative aspects of problem solving. This is followed by Luz Manuel Santos Trigo’s summary introducing us to problem solving in and with digital technologies. The last summary, by Uldarico Malaspina Jurado, documents the rise of problem posing within the field of mathematics education in general and the problem solving literature in particular.
Each of these summaries references in some critical and central fashion the works of George Pólya or Alan Schoenfeld. To the initiated researchers, this is no surprise. The seminal work of these researchers lie at the roots of mathematical problem solving. What is interesting, though, is the diverse ways in which each of the four aforementioned contributions draw on, and position, these works so as to fit into the larger scheme of their respective summaries. This speaks to not only the depth and breadth of these influential works, but also the diversity with which they can be interpreted and utilized in extending our thinking about problem solving.
Taken together, what follows is a topical survey of ideas representing the diversity of views and tensions inherent in a field of research that is both a means to an end and an end onto itself and is unanimously seen as central to the activities of mathematics.
1 Survey on the StateoftheArt
1.1 role of heuristics for problem solving—regina bruder.
The origin of the word heuristic dates back to the time of Archimedes and is said to have come out of one of the famous stories told about this great mathematician and inventor. The King of Syracuse asked Archimedes to check whether his new wreath was really made of pure gold. Archimedes struggled with this task and it was not until he was at the bathhouse that he came up with the solution. As he entered the tub he noticed that he had displaced a certain amount of water. Brilliant as he was, he transferred this insight to the issue with the wreath and knew he had solved the problem. According to the legend, he jumped out of the tub and ran from the bathhouse naked screaming, “Eureka, eureka!”. Eureka and heuristic have the same root in the ancient Greek language and so it has been claimed that this is how the academic discipline of “heuristics” dealing with effective approaches to problem solving (socalled heurisms) was given its name. Pólya ( 1964 ) describes this discipline as follows:
Heuristics deals with solving tasks. Its specific goals include highlighting in general terms the reasons for selecting those moments in a problem the examination of which could help us find a solution. (p. 5)
This discipline has grown, in part, from examining the approaches to certain problems more in detail and comparing them with each other in order to abstract similarities in approach, or socalled heurisms. Pólya ( 1949 ), but also, inter alia, Engel ( 1998 ), König ( 1984 ) and Sewerin ( 1979 ) have formulated such heurisms for mathematical problem tasks. The problem tasks examined by the authors mentioned are predominantly found in the area of talent programmes, that is, they often go back to mathematics competitions.
In 1983 Zimmermann provided an overview of heuristic approaches and tools in American literature which also offered suggestions for mathematics classes. In the Germanspeaking countries, an approach has established itself, going back to Sewerin ( 1979 ) and König ( 1984 ), which divides schoolrelevant heuristic procedures into heuristic tools, strategies and principles, see also Bruder and Collet ( 2011 ).
Below is a review of the conceptual background of heuristics, followed by a description of the effect mechanisms of heurisms in problemsolving processes.
1.1.1 Research Review on the Promotion of Problem Solving
In the 20th century, there has been an advancement of research on mathematical problem solving and findings about possibilities to promote problem solving with varying priorities (c.f. Pehkonen 1991 ). Based on a model by Pólya ( 1949 ), in a first phase of research on problem solving, particularly in the 1960s and the 1970s, a series of studies on problemsolving processes placing emphasis on the importance of heuristic strategies (heurisms) in problem solving has been carried out. It was assumed that teaching and learning heuristic strategies, principles and tools would provide students with an orientation in problem situations and that this could thus improve students’ problemsolving abilities (c.f. for instance, Schoenfeld 1979 ). This approach, mostly researched within the scope of talent programmes for problem solving, was rather successful (c.f. for instance, Sewerin 1979 ). In the 1980s, requests for promotional opportunities in everyday teaching were given more and more consideration: “ problem solving must be the focus of school mathematics in the 1980s ” (NCTM 1980 ). For the teaching and learning of problem solving in regular mathematics classes, the current view according to which cognitive, heuristic aspects were paramount, was expanded by certain studentspecific aspects, such as attitudes, emotions and selfregulated behaviour (c.f. Kretschmer 1983 ; Schoenfeld 1985 , 1987 , 1992 ). Kilpatrick ( 1985 ) divided the promotional approaches described in the literature into five methods which can also be combined with each other.
Osmosis : actionoriented and implicit imparting of problemsolving techniques in a beneficial learning environment
Memorisation : formation of special techniques for particular types of problem and of the relevant questioning when problem solving
Imitation : acquisition of problemsolving abilities through imitation of an expert
Cooperation : cooperative learning of problemsolving abilities in small groups
Reflection : problemsolving abilities are acquired in an actionoriented manner and through reflection on approaches to problem solving.
Kilpatrick ( 1985 ) views as success when heuristic approaches are explained to students, clarified by means of examples and trained through the presentation of problems. The need of making students aware of heuristic approaches is by now largely accepted in didactic discussions. Differences in varying approaches to promoting problemsolving abilities rather refer to deciding which problemsolving strategies or heuristics are to imparted to students and in which way, and not whether these should be imparted at all or not.
1.1.2 Heurisms as an Expression of Mental Agility
The activity theory, particularly in its advancement by Lompscher ( 1975 , 1985 ), offers a wellsuited and manageable model to describe learning activities and differences between learners with regard to processes and outcomes in problem solving (c.f. Perels et al. 2005 ). Mental activity starts with a goal and the motive of a person to perform such activity. Lompscher divides actual mental activity into content and process. Whilst the content in mathematical problemsolving consists of certain concepts, connections and procedures, the process describes the psychological processes that occur when solving a problem. This course of action is described in Lompscher by various qualities, such as systematic planning, independence, accuracy, activity and agility. Along with differences in motivation and the availability of expertise, it appears that intuitive problem solvers possess a particularly high mental agility, at least with regard to certain contents areas.
According to Lompscher, “flexibility of thought” expresses itself
… by the capacity to change more or less easily from one aspect of viewing to another one or to embed one circumstance or component into different correlations, to understand the relativity of circumstances and statements. It allows to reverse relations, to more or less easily or quickly attune to new conditions of mental activity or to simultaneously mind several objects or aspects of a given activity (Lompscher 1975 , p. 36).
These typical manifestations of mental agility can be focused on in problem solving by mathematical means and can be related to the heurisms known from the analyses of approaches by Pólya et al. (c.f. also Bruder 2000 ):
Reduction : Successful problem solvers will intuitively reduce a problem to its essentials in a sensible manner. To achieve such abstraction, they often use visualisation and structuring aids, such as informative figures, tables, solution graphs or even terms. These heuristic tools are also very well suited to document in retrospect the approach adopted by the intuitive problem solvers in a way that is comprehensible for all.
Reversibility : Successful problem solvers are able to reverse trains of thought or reproduce these in reverse. They will do this in appropriate situations automatically, for instance, when looking for a key they have mislaid. A corresponding general heuristic strategy is working in reverse.
Minding of aspects : Successful problem solvers will mind several aspects of a given problem at the same time or easily recognise any dependence on things and vary them in a targeted manner. Sometimes, this is also a matter of removing barriers in favour of an idea that appears to be sustainable, that is, by simply “hanging on” to a certain train of thought even against resistance. Corresponding heurisms are, for instance, the principle of invariance, the principle of symmetry (Engel 1998 ), the breaking down or complementing of geometric figures to calculate surface areas, or certain terms used in binomial formulas.
Change of aspects : Successful problem solvers will possibly change their assumptions, criteria or aspects minded in order to find a solution. Various aspects of a given problem will be considered intuitively or the problem be viewed from a different perspective, which will prevent “getting stuck” and allow for new insights and approaches. For instance, many elementary geometric propositions can also be proved in an elegant vectorial manner.
Transferring : Successful problem solvers will be able more easily than others to transfer a wellknown procedure to another, sometimes even very different context. They recognise more easily the “framework” or pattern of a given task. Here, this is about own constructions of analogies and continual tracing back from the unknown to the known.
Intuitive, that is, untrained good problem solvers, are, however, often unable to access these flexibility qualities consciously. This is why they are also often unable to explain how they actually solved a given problem.
To be able to solve problems successfully, a certain mental agility is thus required. If this is less well pronounced in a certain area, learning how to solve problems means compensating by acquiring heurisms. In this case, insufficient mental agility is partly “offset” through the application of knowledge acquired by means of heurisms. Mathematical problemsolving competences are thus acquired through the promotion of manifestations of mental agility (reduction, reversibility, minding of aspects and change of aspects). This can be achieved by designing subactions of problem solving in connection with a (temporarily) conscious application of suitable heurisms. Empirical evidence for the success of the active principle of heurisms has been provided by Collet ( 2009 ).
Against such background, learning how to solve problems can be established as a longterm teaching and learning process which basically encompasses four phases (Bruder and Collet 2011 ):
Intuitive familiarisation with heuristic methods and techniques.
Making aware of special heurisms by means of prominent examples (explicit strategy acquisition).
Short conscious practice phase to use the newly acquired heurisms with differentiated task difficulties.
Expanding the context of the strategies applied.
In the first phase, students are familiarised with heurisms intuitively by means of targeted aid impulses and questions (what helped us solve this problem?) which in the following phase are substantiated on the basis of model tasks, are given names and are thus made aware of their existence. The third phase serves the purpose of a certain familiarisation with the new heurisms and the experience of competence through individualised practising at different requirement levels, including in the form of homework over longer periods. A fourth and delayed fourth phase aims at more flexibility through the transfer to other contents and contexts and the increasingly intuitive use of the newly acquired heurisms, so that students can enrich their own problemsolving models in a gradual manner. The second and third phases build upon each other in close chronological order, whilst the first phase should be used in class at all times.
All heurisms can basically be described in an actionoriented manner by means of asking the right questions. The way of asking questions can thus also establish a certain kind of personal relation. Even if the teacher presents and suggests the line of basic questions with a prototypical wording each time, students should always be given the opportunity to find “their” wording for the respective heurism and take a note of it for themselves. A possible key question for the use of a heuristic tool would be: How to illustrate and structure the problem or how to present it in a different way?
Unfortunately, for many students, applying heuristic approaches to problem solving will not ensue automatically but will require appropriate early and longterm promoting. The results of current studies, where promotion approaches to problem solving are connected with selfregulation and metacognitive aspects, demonstrate certain positive effects of such combination on students. This field of research includes, for instance, studies by Lester et al. ( 1989 ), Verschaffel et al. ( 1999 ), the studies on teaching method IMPROVE by Mevarech and Kramarski ( 1997 , 2003 ) and also the evaluation of a teaching concept on learning how to solve problems by the gradual conscious acquisition of heurisms by Collet and Bruder ( 2008 ).
1.2 Creative Problem Solving—Peter Liljedahl
There is a tension between the aforementioned story of Archimedes and the heuristics presented in the previous section. Archimedes, when submersing himself in the tub and suddenly seeing the solution to his problem, wasn’t relying on osmosis, memorisation, imitation, cooperation, or reflection (Kilpatrick 1985 ). He wasn’t drawing on reduction, reversibility, minding of aspects, change of aspect, or transfer (Bruder 2000 ). Archimedes was stuck and it was only, in fact, through insight and sudden illumination that he managed to solve his problem. In short, Archimedes was faced with a problem that the aforementioned heuristics, and their kind, would not help him to solve.
According to some, such a scenario is the definition of a problem. For example, Resnick and Glaser ( 1976 ) define a problem as being something that you do not have the experience to solve. Mathematicians, in general, agree with this (Liljedahl 2008 ).
Any problem in which you can see how to attack it by deliberate effort, is a routine problem, and cannot be an important discover. You must try and fail by deliberate efforts, and then rely on a sudden inspiration or intuition or if you prefer to call it luck. (Dan Kleitman, participant cited in Liljedahl 2008 , p. 19).
Problems, then, are tasks that cannot be solved by direct effort and will require some creative insight to solve (Liljedahl 2008 ; Mason et al. 1982 ; Pólya 1965 ).
1.2.1 A History of Creativity in Mathematics Education
In 1902, the first half of what eventually came to be a 30 question survey was published in the pages of L’Enseignement Mathématique , the journal of the French Mathematical Society. The authors, Édouard Claparède and Théodore Flournoy, were two Swiss psychologists who were deeply interested in the topics of mathematical discovery, creativity and invention. Their hope was that a widespread appeal to mathematicians at large would incite enough responses for them to begin to formulate some theories about this topic. The first half of the survey centered on the reasons for becoming a mathematician (family history, educational influences, social environment, etc.), attitudes about everyday life, and hobbies. This was eventually followed, in 1904, by the publication of the second half of the survey pertaining, in particular, to mental images during periods of creative work. The responses were sorted according to nationality and published in 1908.
During this same period Henri Poincaré (1854–1912), one of the most noteworthy mathematicians of the time, had already laid much of the groundwork for his own pursuit of this same topic and in 1908 gave a presentation to the French Psychological Society in Paris entitled L’Invention mathématique —often mistranslated to Mathematical Creativity Footnote 1 (c.f. Poincaré 1952 ). At the time of the presentation Poincaré stated that he was aware of Claparède and Flournoy’s work, as well as their results, but expressed that they would only confirm his own findings. Poincaré’s presentation, as well as the essay it spawned, stands to this day as one of the most insightful, and thorough treatments of the topic of mathematical discovery, creativity, and invention.
Just at this time, I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incident of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuschian functions were identical with those of nonEuclidean geometry. I did not verify the idea; I should not have had the time, as, upon taking my seat in the omnibus, I went on with the conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the results at my leisure. (Poincaré 1952 , p. 53)
So powerful was his presentation, and so deep were his insights into his acts of invention and discovery that it could be said that he not so much described the characteristics of mathematical creativity, as defined them. From that point forth mathematical creativity, or even creativity in general, has not been discussed seriously without mention of Poincaré’s name.
Inspired by this presentation, Jacques Hadamard (1865–1963), a contemporary and a friend of Poincaré’s, began his own empirical investigation into this fascinating phenomenon. Hadamard had been critical of Claparède and Flournoy’s work in that they had not adequately treated the topic on two fronts. As exhaustive as the survey appeared to be, Hadamard felt that it failed to ask some key questions—the most important of which was with regard to the reason for failures in the creation of mathematics. This seemingly innocuous oversight, however, led directly to his second and “most important criticism” (Hadamard 1945 ). He felt that only “firstrate men would dare to speak of” (p. 10) such failures. So, inspired by his good friend Poincaré’s treatment of the subject Hadamard retooled the survey and gave it to friends of his for consideration—mathematicians such as Henri Poincaré and Albert Einstein, whose prominence were beyond reproach. Ironically, the new survey did not contain any questions that explicitly dealt with failure. In 1943 Hadamard gave a series of lectures on mathematical invention at the École Libre des Hautes Études in New York City. These talks were subsequently published as The Psychology of Mathematical Invention in the Mathematical Field (Hadameard 1945 ).
Hadamard’s classic work treats the subject of invention at the crossroads of mathematics and psychology. It provides not only an entertaining look at the eccentric nature of mathematicians and their rituals, but also outlines the beliefs of mid twentiethcentury mathematicians about the means by which they arrive at new mathematics. It is an extensive exploration and extended argument for the existence of unconscious mental processes. In essence, Hadamard took the ideas that Poincaré had posed and, borrowing a conceptual framework for the characterization of the creative process from the Gestaltists of the time (Wallas 1926 ), turned them into a stage theory. This theory still stands as the most viable and reasonable description of the process of mathematical creativity.
1.2.2 Defining Mathematical Creativity
The phenomena of mathematical creativity, although marked by sudden illumination, actually consist of four separate stages stretched out over time, of which illumination is but one stage. These stages are initiation, incubation, illumination, and verification (Hadamard 1945 ). The first of these stages, the initiation phase, consists of deliberate and conscious work. This would constitute a person’s voluntary, and seemingly fruitless, engagement with a problem and be characterized by an attempt to solve the problem by trolling through a repertoire of past experiences. This is an important part of the inventive process because it creates the tension of unresolved effort that sets up the conditions necessary for the ensuing emotional release at the moment of illumination (Hadamard 1945 ; Poincaré 1952 ).
Following the initiation stage the solver, unable to come up with a solution stops working on the problem at a conscious level and begins to work on it at an unconscious level (Hadamard 1945 ; Poincaré 1952 ). This is referred to as the incubation stage of the inventive process and can last anywhere from several minutes to several years. After the period of incubation a rapid coming to mind of a solution, referred to as illumination , may occur. This is accompanied by a feeling of certainty and positive emotions (Poincaré 1952 ). Although the processes of incubation and illumination are shrouded behind the veil of the unconscious there are a number of things that can be deduced about them. First and foremost is the fact that unconscious work does, indeed, occur. Poincaré ( 1952 ), as well as Hadamard ( 1945 ), use the very real experience of illumination, a phenomenon that cannot be denied, as evidence of unconscious work, the fruits of which appear in the flash of illumination. No other theory seems viable in explaining the sudden appearance of solution during a walk, a shower, a conversation, upon waking, or at the instance of turning the conscious mind back to the problem after a period of rest (Poincaré 1952 ). Also deducible is that unconscious work is inextricably linked to the conscious and intentional effort that precedes it.
There is another remark to be made about the conditions of this unconscious work: it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work. These sudden inspirations never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come … (Poincaré 1952 , p. 56)
Hence, the fruitless efforts of the initiation phase are only seemingly so. They not only set up the aforementioned tension responsible for the emotional release at the time of illumination, but also create the conditions necessary for the process to enter into the incubation phase.
Illumination is the manifestation of a bridging that occurs between the unconscious mind and the conscious mind (Poincaré 1952 ), a coming to (conscious) mind of an idea or solution. What brings the idea forward to consciousness is unclear, however. There are theories of the aesthetic qualities of the idea, effective surprise/shock of recognition, fluency of processing, or breaking functional fixedness. For reasons of brevity I will only expand on the first of these.
Poincaré proposed that ideas that were stimulated during initiation remained stimulated during incubation. However, freed from the constraints of conscious thought and deliberate calculation, these ideas would begin to come together in rapid and random unions so that “their mutual impacts may produce new combinations” (Poincaré 1952 ). These new combinations, or ideas, would then be evaluated for viability using an aesthetic sieve, which allows through to the conscious mind only the “right combinations” (Poincaré 1952 ). It is important to note, however, that good or aesthetic does not necessarily mean correct. Correctness is evaluated during the verification stage.
The purpose of verification is not only to check for correctness. It is also a method by which the solver reengages with the problem at the level of details. That is, during the unconscious work the problem is engaged with at the level of ideas and concepts. During verification the solver can examine these ideas in closer details. Poincaré succinctly describes both of these purposes.
As for the calculations, themselves, they must be made in the second period of conscious work, that which follows the inspiration, that in which one verifies the results of this inspiration and deduces their consequences. (Poincaré 1952 , p. 62)
Aside from presenting this aforementioned theory on invention, Hadamard also engaged in a farreaching discussion on a number of interesting, and sometimes quirky, aspects of invention and discovery that he had culled from the results of his empirical study, as well as from pertinent literature. This discussion was nicely summarized by Newman ( 2000 ) in his commentary on the elusiveness of invention.
The celebrated phrenologist Gall said mathematical ability showed itself in a bump on the head, the location of which he specified. The psychologist Souriau, we are told, maintained that invention occurs by “pure chance”, a valuable theory. It is often suggested that creative ideas are conjured up in “mathematical dreams”, but this attractive hypothesis has not been verified. Hadamard reports that mathematicians were asked whether “noises” or “meteorological circumstances” helped or hindered research [..] Claude Bernard, the great physiologist, said that in order to invent “one must think aside”. Hadamard says this is a profound insight; he also considers whether scientific invention may perhaps be improved by standing or sitting or by taking two baths in a row. Helmholtz and Poincaré worked sitting at a table; Hadamard’s practice is to pace the room (“Legs are the wheels of thought”, said Emile Angier); the chemist J. Teeple was the twobath man. (p. 2039)
1.2.3 Discourses on Creativity
Creativity is a term that can be used both loosely and precisely. That is, while there exists a common usage of the term there also exists a tradition of academic discourse on the subject. A common usage of creative refers to a process or a person whose products are original, novel, unusual, or even abnormal (Csíkszentmihályi 1996 ). In such a usage, creativity is assessed on the basis of the external and observable products of the process, the process by which the product comes to be, or on the character traits of the person doing the ‘creating’. Each of these usages—product, process, person—is the roots of the discourses (Liljedahl and Allan 2014 ) that I summarize here, the first of which concerns products.
Consider a mother who states that her daughter is creative because she drew an original picture. The basis of such a statement can lie either in the fact that the picture is unlike any the mother has ever seen or unlike any her daughter has ever drawn before. This mother is assessing creativity on the basis of what her daughter has produced. However, the standards that form the basis of her assessment are neither consistent nor stringent. There does not exist a universal agreement as to what she is comparing the picture to (pictures by other children or other pictures by the same child). Likewise, there is no standard by which the actual quality of the picture is measured. The academic discourse that concerns assessment of products, on the other hand, is both consistent and stringent (Csíkszentmihályi 1996 ). This discourse concerns itself more with a fifth, and as yet unmentioned, stage of the creative process; elaboration . Elaboration is where inspiration becomes perspiration (Csíkszentmihályi 1996 ). It is the act of turning a good idea into a finished product, and the finished product is ultimately what determines the creativity of the process that spawned it—that is, it cannot be a creative process if nothing is created. In particular, this discourse demands that the product be assessed against other products within its field, by the members of that field, to determine if it is original AND useful (Csíkszentmihályi 1996 ; Bailin 1994 ). If it is, then the product is deemed to be creative. Note that such a use of assessment of end product pays very little attention to the actual process that brings this product forth.
The second discourse concerns the creative process. The literature pertaining to this can be separated into two categories—a prescriptive discussion of the creativity process and a descriptive discussion of the creativity process. Although both of these discussions have their roots in the four stages that Wallas ( 1926 ) proposed makes up the creative process, they make use of these stages in very different ways. The prescriptive discussion of the creative process is primarily focused on the first of the four stages, initiation , and is best summarized as a cause  and  effect discussion of creativity, where the thinking processes during the initiation stage are the cause and the creative outcome are the effects (Ghiselin 1952 ). Some of the literature claims that the seeds of creativity lie in being able to think about a problem or situation analogically. Other literature claims that utilizing specific thinking tools such as imagination, empathy, and embodiment will lead to creative products. In all of these cases, the underlying theory is that the eventual presentation of a creative idea will be precipitated by the conscious and deliberate efforts during the initiation stage. On the other hand, the literature pertaining to a descriptive discussion of the creative process is inclusive of all four stages (Kneller 1965 ; Koestler 1964 ). For example, Csíkszentmihályi ( 1996 ), in his work on flow attends to each of the stages, with much attention paid to the fluid area between conscious and unconscious work, or initiation and incubation. His claim is that the creative process is intimately connected to the enjoyment that exists during times of sincere and consuming engagement with a situation, the conditions of which he describes in great detail.
The third, and final, discourse on creativity pertains to the person. This discourse is space dominated by two distinct characteristics, habit and genius. Habit has to do with the personal habits as well as the habits of mind of people that have been deemed to be creative. However, creative people are most easily identified through their reputation for genius. Consequently, this discourse is often dominated by the analyses of the habits of geniuses as is seen in the work of Ghiselin ( 1952 ), Koestler ( 1964 ), and Kneller ( 1965 ) who draw on historical personalities such as Albert Einstein, Henri Poincaré, Vincent Van Gogh, D.H. Lawrence, Samuel Taylor Coleridge, Igor Stravinsky, and Wolfgang Amadeus Mozart to name a few. The result of this sort of treatment is that creative acts are viewed as rare mental feats, which are produced by extraordinary individuals who use extraordinary thought processes.
These different discourses on creativity can be summed up in a tension between absolutist and relativist perspectives on creativity (Liljedahl and Sriraman 2006 ). An absolutist perspective assumes that creative processes are the domain of genius and are present only as precursors to the creation of remarkably useful and universally novel products. The relativist perspective, on the other hand, allows for every individual to have moments of creativity that may, or may not, result in the creation of a product that may, or may not, be either useful or novel.
Between the work of a student who tries to solve a problem in geometry or algebra and a work of invention, one can say there is only a difference of degree. (Hadamard 1945 , p. 104).
Regardless of discourse, however, creativity is not “part of the theories of logical forms” (Dewey 1938 ). That is, creativity is not representative of the lockstep logic and deductive reasoning that mathematical problem solving is often presumed to embody (Bibby 2002 ; Burton 1999 ). Couple this with the aforementioned demanding constraints as to what constitutes a problem, where then does that leave problem solving heuristics? More specifically, are there creative problem solving heuristics that will allow us to resolve problems that require illumination to solve? The short answer to this question is yes—there does exist such problem solving heuristics. To understand these, however, we must first understand the routine problem solving heuristics they are built upon. In what follows, I walk through the work of key authors and researchers whose work offers us insights into progressively more creative problem solving heuristics for solving true problems.
1.2.4 Problem Solving by Design
In a general sense, design is defined as the algorithmic and deductive approach to solving a problem (Rusbult 2000 ). This process begins with a clearly defined goal or objective after which there is a great reliance on relevant past experience, referred to as repertoire (Bruner 1964 ; Schön 1987 ), to produce possible options that will lead towards a solution of the problem (Poincaré 1952 ). These options are then examined through a process of conscious evaluations (Dewey 1933 ) to determine their suitability for advancing the problem towards the final goal. In very simple terms, problem solving by design is the process of deducing the solution from that which is already known.
Mayer ( 1982 ), Schoenfeld ( 1982 ), and Silver ( 1982 ) state that prior knowledge is a key element in the problem solving process. Prior knowledge influences the problem solver’s understanding of the problem as well as the choice of strategies that will be called upon in trying to solve the problem. In fact, prior knowledge and prior experiences is all that a solver has to draw on when first attacking a problem. As a result, all problem solving heuristics incorporate this resource of past experiences and prior knowledge into their initial attack on a problem. Some heuristics refine these ideas, and some heuristics extend them (c.f. Kilpatrick 1985 ; Bruder 2000 ). Of the heuristics that refine, none is more influential than the one created by George Pólya (1887–1985).
1.2.5 George Pólya: How to Solve It
In his book How to Solve It (1949) Pólya lays out a problem solving heuristic that relies heavily on a repertoire of past experience. He summarizes the fourstep process of his heuristic as follows:
Understanding the Problem
First. You have to understand the problem.
What is the unknown? What are the data? What is the condition?
Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
Draw a figure. Introduce suitable notation.
Separate the various parts of the condition. Can you write them down?
Devising a Plan
Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
Have you seen it before? Or have you seen the same problem in a slightly different form?
Do you know a related problem? Do you know a theorem that could be useful?
Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
Could you restate the problem? Could you restate it still differently? Go back to definitions.
If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
Carrying Out the Plan
Third. Carry out your plan.
Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
Looking Back
Fourth. Examine the solution obtained.
Can you check the result? Can you check the argument?
Can you derive the solution differently? Can you see it at a glance?
Can you use the result, or the method, for some other problem?
The emphasis on auxiliary problems, related problems, and analogous problems that are, in themselves, also familiar problems is an explicit manifestation of relying on a repertoire of past experience. This use of familiar problems also requires an ability to deduce from these related problems a recognizable and relevant attribute that will transfer to the problem at hand. The mechanism that allows for this transfer of knowledge between analogous problems is known as analogical reasoning (English 1997 , 1998 ; Novick 1988 , 1990 , 1995 ; Novick and Holyoak 1991 ) and has been shown to be an effective, but not always accessible, thinking strategy.
Step four in Pólya’s heuristic, looking back, is also a manifestation of utilizing prior knowledge to solve problems, albeit an implicit one. Looking back makes connections “in memory to previously acquired knowledge [..] and further establishes knowledge in longterm memory that may be elaborated in later problemsolving encounters” (Silver 1982 , p. 20). That is, looking back is a forwardlooking investment into future problem solving encounters, it sets up connections that may later be needed.
Pólya’s heuristic is a refinement on the principles of problem solving by design. It not only makes explicit the focus on past experiences and prior knowledge, but also presents these ideas in a very succinct, digestible, and teachable manner. This heuristic has become a popular, if not the most popular, mechanism by which problem solving is taught and learned.
1.2.6 Alan Schoenfeld: Mathematical Problem Solving
The work of Alan Schoenfeld is also a refinement on the principles of problem solving by design. However, unlike Pólya ( 1949 ) who refined these principles at a theoretical level, Schoenfeld has refined them at a practical and empirical level. In addition to studying taught problem solving strategies he has also managed to identify and classify a variety of strategies, mostly ineffectual, that students invoke naturally (Schoenfeld 1985 , 1992 ). In so doing, he has created a better understanding of how students solve problems, as well as a better understanding of how problems should be solved and how problem solving should be taught.
For Schoenfeld, the problem solving process is ultimately a dialogue between the problem solver’s prior knowledge, his attempts, and his thoughts along the way (Schoenfeld 1982 ). As such, the solution path of a problem is an emerging and contextually dependent process. This is a departure from the predefined and contextually independent processes of Pólya’s ( 1949 ) heuristics. This can be seen in Schoenfeld’s ( 1982 ) description of a good problem solver.
To examine what accounts for expertise in problem solving, you would have to give the expert a problem for which he does not have access to a solution schema. His behavior in such circumstances is radically different from what you would see when he works on routine or familiar “nonroutine” problems. On the surface his performance is no longer proficient; it may even seem clumsy. Without access to a solution schema, he has no clear indication of how to start. He may not fully understand the problem, and may simply “explore it for a while until he feels comfortable with it. He will probably try to “match” it to familiar problems, in the hope it can be transformed into a (nearly) schemadriven solution. He will bring up a variety of plausible things: related facts, related problems, tentative approaches, etc. All of these will have to be juggled and balanced. He may make an attempt solving it in a particular way, and then back off. He may try two or three things for a couple of minutes and then decide which to pursue. In the midst of pursuing one direction he may go back and say “that’s harder than it should be” and try something else. Or, after the comment, he may continue in the same direction. With luck, after some aborted attempts, he will solve the problem. (p. 3233)
Aside from demonstrating the emergent nature of the problem solving process, this passage also brings forth two consequences of Schoenfeld’s work. The first of these is the existence of problems for which the solver does not have “access to a solution schema”. Unlike Pólya ( 1949 ), who’s heuristic is a ‘one size fits all (problems)’ heuristic, Schoenfeld acknowledges that problem solving heuristics are, in fact, personal entities that are dependent on the solver’s prior knowledge as well as their understanding of the problem at hand. Hence, the problems that a person can solve through his or her personal heuristic are finite and limited.
The second consequence that emerges from the above passage is that if a person lacks the solution schema to solve a given problem s/he may still solve the problem with the help of luck . This is an acknowledgement, if only indirectly so, of the difference between problem solving in an intentional and mechanical fashion verses problem solving in a more creative fashion, which is neither intentional nor mechanical (Pehkonen 1997 ).
1.2.7 David Perkins: Breakthrough Thinking
As mentioned, many consider a problem that can be solved by intentional and mechanical means to not be worthy of the title ‘problem’. As such, a repertoire of past experiences sufficient for dealing with such a ‘problem’ would disqualify it from the ranks of ‘problems’ and relegate it to that of ‘exercises’. For a problem to be classified as a ‘problem’, then, it must be ‘problematic’. Although such an argument is circular it is also effective in expressing the ontology of mathematical ‘problems’.
Perkins ( 2000 ) also requires problems to be problematic. His book Archimedes’ Bathtub: The Art and Logic of Breakthrough Thinking (2000) deals with situations in which the solver has gotten stuck and no amount of intentional or mechanical adherence to the principles of past experience and prior knowledge is going to get them unstuck. That is, he deals with problems that, by definition, cannot be solved through a process of design [or through the heuristics proposed by Pólya ( 1949 ) and Schoenfeld ( 1985 )]. Instead, the solver must rely on the extralogical process of what Perkins ( 2000 ) calls breakthrough thinking .
Perkins ( 2000 ) begins by distinguishing between reasonable and unreasonable problems. Although both are solvable, only reasonable problems are solvable through reasoning. Unreasonable problems require a breakthrough in order to solve them. The problem, however, is itself inert. It is neither reasonable nor unreasonable. That quality is brought to the problem by the solver. That is, if a student cannot solve a problem by direct effort then that problem is deemed to be unreasonable for that student. Perkins ( 2000 ) also acknowledges that what is an unreasonable problem for one person is a perfectly reasonable problem for another person; reasonableness is dependent on the person.
This is not to say that, once found, the solution cannot be seen as accessible through reason. During the actual process of solving, however, direct and deductive reasoning does not work. Perkins ( 2000 ) uses several classic examples to demonstrate this, the most famous being the problem of connecting nine dots in a 3 × 3 array with four straight lines without removing pencil from paper, the solution to which is presented in Fig. 1 .
Nine dots—four lines problem and solution
To solve this problem, Perkins ( 2000 ) claims that the solver must recognize that the constraint of staying within the square created by the 3 × 3 array is a selfimposed constraint. He further claims that until this is recognized no amount of reasoning is going to solve the problem. That is, at this point in the problem solving process the problem is unreasonable. However, once this selfimposed constraint is recognized the problem, and the solution, are perfectly reasonable. Thus, the solution of an, initially, unreasonable problem is reasonable.
The problem solving heuristic that Perkins ( 2000 ) has constructed to deal with solvable, but unreasonable, problems revolves around the idea of breakthrough thinking and what he calls breakthrough problems . A breakthrough problem is a solvable problem in which the solver has gotten stuck and will require an AHA! to get unstuck and solve the problem. Perkins ( 2000 ) poses that there are only four types of solvable unreasonable problems, which he has named wilderness of possibilities , the clueless plateau , narrow canyon of exploration , and oasis of false promise . The names for the first three of these types of problems are related to the Klondike gold rush in Alaska, a time and place in which gold was found more by luck than by direct and systematic searching.
The wilderness of possibilities is a term given to a problem that has many tempting directions but few actual solutions. This is akin to a prospector searching for gold in the Klondike. There is a great wilderness in which to search, but very little gold to be found. The clueless plateau is given to problems that present the solver with few, if any, clues as to how to solve it. The narrow canyon of exploration is used to describe a problem that has become constrained in such a way that no solution now exists. The ninedot problem presented above is such a problem. The imposed constraint that the lines must lie within the square created by the array makes a solution impossible. This is identical to the metaphor of a prospector searching for gold within a canyon where no gold exists. The final type of problem gets its name from the desert. An oasis of false promise is a problem that allows the solver to quickly get a solution that is close to the desired outcome; thereby tempting them to remain fixed on the strategy that they used to get this almostanswer. The problem is, that like the canyon, the solution does not exist at the oasis; the solution strategy that produced an almostanswer is incapable of producing a complete answer. Likewise, a desert oasis is a false promise in that it is only a reprieve from the desolation of the dessert and not a final destination.
Believing that there are only four ways to get stuck, Perkins ( 2000 ) has designed a problem solving heuristic that will “up the chances” of getting unstuck. This heuristic is based on what he refers to as “the logic of lucking out” (p. 44) and is built on the idea of introspection. By first recognizing that they are stuck, and then recognizing that the reason they are stuck can only be attributed to one of four reasons, the solver can access four strategies for getting unstuck, one each for the type of problem they are dealing with. If the reason they are stuck is because they are faced with a wilderness of possibilities they are to begin roaming far, wide, and systematically in the hope of reducing the possible solution space to one that is more manageable. If they find themselves on a clueless plateau they are to begin looking for clues, often in the wording of the problem. When stuck in a narrow canyon of possibilities they need to reexamine the problem and see if they have imposed any constraints. Finally, when in an oasis of false promise they need to reattack the problem in such a way that they stay away from the oasis.
Of course, there are nuances and details associated with each of these types of problems and the strategies for dealing with them. However, nowhere within these details is there mention of the main difficulty inherent in introspection; that it is much easier for the solver to get stuck than it is for them to recognize that they are stuck. Once recognized, however, the details of Perkins’ ( 2000 ) heuristic offer the solver some ways for recognizing why they are stuck.
1.2.8 John Mason, Leone Burton, and Kaye Stacey: Thinking Mathematically
The work of Mason et al. in their book Thinking Mathematically ( 1982 ) also recognizes the fact that for each individual there exists problems that will not yield to their intentional and mechanical attack. The heuristic that they present for dealing with this has two main processes with a number of smaller phases, rubrics, and states. The main processes are what they refer to as specializing and generalizing. Specializing is the process of getting to know the problem and how it behaves through the examination of special instances of the problem. This process is synonymous with problem solving by design and involves the repeated oscillation between the entry and attack phases of Mason et al. ( 1982 ) heuristic. The entry phase is comprised of ‘getting started’ and ‘getting involved’ with the problem by using what is immediately known about it. Attacking the problem involves conjecturing and testing a number of hypotheses in an attempt to gain greater understanding of the problem and to move towards a solution.
At some point within this process of oscillating between entry and attack the solver will get stuck, which Mason et al. ( 1982 ) refer to as “an honourable and positive state, from which much can be learned” (p. 55). The authors dedicate an entire chapter to this state in which they acknowledge that getting stuck occurs long before an awareness of being stuck develops. They proposes that the first step to dealing with being stuck is the simple act of writing STUCK!
The act of expressing my feelings helps to distance me from my state of being stuck. It frees me from incapacitating emotions and reminds me of actions that I can take. (p. 56)
The next step is to reengage the problem by examining the details of what is known, what is wanted, what can be introduced into the problem, and what has been introduced into the problem (imposed assumptions). This process is engaged in until an AHA!, which advances the problem towards a solution, is encountered. If, at this point, the problem is not completely solved the oscillation is then resumed.
At some point in this process an attack on the problem will yield a solution and generalizing can begin. Generalizing is the process by which the specifics of a solution are examined and questions as to why it worked are investigated. This process is synonymous with the verification and elaboration stages of invention and creativity. Generalization may also include a phase of review that is similar to Pólya’s ( 1949 ) looking back.
1.2.9 Gestalt: The Psychology of Problem Solving
The Gestalt psychology of learning believes that all learning is based on insights (Koestler 1964 ). This psychology emerged as a response to behaviourism, which claimed that all learning was a response to external stimuli. Gestalt psychologists, on the other hand, believed that there was a cognitive process involved in learning as well. With regards to problem solving, the Gestalt school stands firm on the belief that problem solving, like learning, is a product of insight and as such, cannot be taught. In fact, the theory is that not only can problem solving not be taught, but also that attempting to adhere to any sort of heuristic will impede the working out of a correct solution (Krutestkii 1976 ). Thus, there exists no Gestalt problem solving heuristic. Instead, the practice is to focus on the problem and the solution rather than on the process of coming up with a solution. Problems are solved by turning them over and over in the mind until an insight, a viable avenue of attack, presents itself. At the same time, however, there is a great reliance on prior knowledge and past experiences. The Gestalt method of problem solving, then, is at the same time very different and very similar to the process of design.
Gestalt psychology has not fared well during the evolution of cognitive psychology. Although it honours the work of the unconscious mind it does so at the expense of practicality. If learning is, indeed, entirely based on insight then there is little point in continuing to study learning. “When one begins by assuming that the most important cognitive phenomena are inaccessible, there really is not much left to talk about” (Schoenfeld 1985 , p. 273). However, of interest here is the Gestalt psychologists’ claim that focus on problem solving methods creates functional fixedness (Ashcraft 1989 ). Mason et al. ( 1982 ), as well as Perkins ( 2000 ) deal with this in their work on getting unstuck.
1.2.10 Final Comments
Mathematics has often been characterized as the most precise of all sciences. Lost in such a misconception is the fact that mathematics often has its roots in the fires of creativity, being born of the extralogical processes of illumination and intuition. Problem solving heuristics that are based solely on the processes of logical and deductive reasoning distort the true nature of problem solving. Certainly, there are problems in which logical deductive reasoning is sufficient for finding a solution. But these are not true problems. True problems need the extralogical processes of creativity, insight, and illumination, in order to produce solutions.
Fortunately, as elusive as such processes are, there does exist problem solving heuristics that incorporate them into their strategies. Heuristics such as those by Perkins ( 2000 ) and Mason et al. ( 1982 ) have found a way of combining the intentional and mechanical processes of problem solving by design with the extralogical processes of creativity, illumination, and the AHA!. Furthermore, they have managed to do so without having to fully comprehend the inner workings of this mysterious process.
1.3 Digital Technologies and Mathematical Problem Solving—Luz Manuel SantosTrigo
Mathematical problem solving is a field of research that focuses on analysing the extent to which problem solving activities play a crucial role in learners’ understanding and use of mathematical knowledge. Mathematical problems are central in mathematical practice to develop the discipline and to foster students learning (Pólya 1945 ; Halmos 1994 ). Mason and JohnstonWilder ( 2006 ) pointed out that “The purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitized to notice and competent to carry out” (p. 25). Tasks are essential for learners to elicit their ideas and to engage them in mathematical thinking. In a problem solving approach, what matters is the learners’ goals and ways to interact with the tasks. That is, even routine tasks can be a departure point for learners to extend initial conditions and transform them into some challenging activities.
Thus, analysing and characterizing ways in which mathematical problems are formulated (Singer et al. 2015 ) and the process involved in pursuing and solving those problems generate important information to frame and structure learning environments to guide and foster learners’ construction of mathematical concepts and problem solving competences (SantosTrigo 2014 ). Furthermore, mathematicians or discipline practitioners have often been interested in unveiling and sharing their own experience while developing the discipline. As a results, they have provided valuable information to characterize mathematical practices and their relations to what learning processes of the discipline entails. It is recognized that the work of Pólya ( 1945 ) offered not only bases to launch several research programs in problem solving (Schoenfeld 1992 ; Mason et al. 1982 ); but also it became an essential resource for teachers to orient and structure their mathematical lessons (Krulik and Reys 1980 ).
1.3.1 Research Agenda
A salient feature of a problem solving approach to learn mathematics is that teachers and students develop and apply an enquiry or inquisitive method to delve into mathematical concepts and tasks. How are mathematical problems or concepts formulated? What types of problems are important for teachers/learners to discuss and engage in mathematical reasoning? What mathematical processes and ways of reasoning are involved in understanding mathematical concepts and solving problems? What are the features that distinguish an instructional environment that fosters problemsolving activities? How can learners’ problem solving competencies be assessed? How can learners’ problem solving competencies be characterized and explained? How can learners use digital technologies to understand mathematics and to develop problemsolving competencies? What ways of reasoning do learners construct when they use digital technologies in problem solving approaches? These types of questions have been important in the problem solving research agenda and delving into them has led researchers to generate information and results to support and frame curriculum proposals and learning scenarios. The purpose of this section is to present and discuss important themes that emerged in problem solving approaches that rely on the systematic use of several digital technologies.
In the last 40 years, the accumulated knowledge in the problem solving field has shed lights on both a characterization of what mathematical thinking involves and how learners can construct a robust knowledge in problem solving environments (Schoenfeld 1992 ). In this process, the field has contributed to identify what types of transformations traditional learning scenarios might consider when teachers and students incorporate the use of digital technologies in mathematical classrooms. In this context, it is important to briefly review what main themes and developments the field has addressed and achieved during the last 40 years.
1.3.2 Problem Solving Developments
There are traces of mathematical problems and solutions throughout the history of civilization that explain the humankind interest for identifying and exploring mathematical relations (Kline 1972 ). Pólya ( 1945 ) reflects on his own practice as a mathematician to characterize the process of solving mathematical problems through four main phases: Understanding the problem, devising a plan, carrying out the plan, and looking back. Likewise, Pólya ( 1945 ) presents and discusses the role played by heuristic methods throughout all problem solving phases. Schoenfeld ( 1985 ) presents a problem solving research program based on Pólya’s ( 1945 ) ideas to investigate the extent to which problem solving heuristics help university students to solve mathematical problems and to develop a way of thinking that shows consistently features of mathematical practices. As a result, he explains the learners’ success or failure in problem solving activities can be characterized in terms their mathematical resources and ways to access them, cognitive and metacognitive strategies used to represent and explore mathematical tasks, and systems of beliefs about mathematics and solving problems. In addition, Schoenfeld ( 1992 ) documented that heuristics methods as illustrated in Pólya’s ( 1945 ) book are ample and general and do not include clear information and directions about how learners could assimilate, learn, and use them in their problem solving experiences. He suggested that students need to discuss what it means, for example, to think of and examining special cases (one important heuristic) in finding a closed formula for series or sequences, analysing relationships of roots of polynomials, or focusing on regular polygons or equilateral/right triangles to find general relations about these figures. That is, learners need to work on examples that lead them to recognize that the use of a particular heuristic often involves thinking of different type of cases depending on the domain or content involved. Lester and Kehle ( 2003 ) summarize themes and methodological shifts in problem solving research up to 1995. Themes include what makes a problem difficult for students and what it means to be successful problem solvers; studying and contrasting experts and novices’ problem solving approaches; learners’ metacognitive, beliefs systems and the influence of affective behaviours; and the role of context; and social interactions in problem solving environments. Research methods in problem solving studies have gone from emphasizing quantitative or statistical design to the use of cases studies and ethnographic methods (Krutestkii ( 1976 ). Teaching strategies also evolved from being centred on teachers to the active students’ engagement and collaboration approaches (NCTM 2000 ). Lesh and Zawojewski ( 2007 ) propose to extend problem solving approaches beyond class setting and they introduce the construct “model eliciting activities” to delve into the learners’ ideas and thinking as a way to engage them in the development of problem solving experiences. To this end, learners develop and constantly refine problemsolving competencies as a part of a learning community that promotes and values modelling construction activities. Recently, English and Gainsburg ( 2016 ) have discussed the importance of modeling eliciting activities to prepare and develop students’ problem solving experiences for 21st Century challenges and demands.
Törner et al. ( 2007 ) invited mathematics educators worldwide to elaborate on the influence and developments of problem solving in their countries. Their contributions show a close relationship between countries mathematical education traditions and ways to frame and implement problem solving approaches. In Chinese classrooms, for example, three instructional strategies are used to structure problem solving lessons: one problem multiple solutions , multiple problems one solution , and one problem multiple changes . In the Netherlands, the realistic mathematical approach permeates the students’ development of problem solving competencies; while in France, problem solving activities are structured in terms of two influential frameworks: The theory of didactical situations and anthropological theory of didactics.
In general, problem solving frameworks and instructional approaches came from analysing students’ problem solving experiences that involve or rely mainly on the use of paper and pencil work. Thus, there is a need to reexamined principles and frameworks to explain what learners develop in learning environments that incorporate systematically the coordinated use of digital technologies (Hoyles and Lagrange 2010 ). In this perspective, it becomes important to briefly describe and identify what both multiple purpose and ad hoc technologies can offer to the students in terms of extending learning environments and representing and exploring mathematical tasks. Specifically, a task is used to identify features of mathematical reasoning that emerge through the use digital technologies that include both mathematical action and multiple purpose types of technologies.
1.3.3 Background
Digital technologies are omnipresent and their use permeates and shapes several social and academic events. Mobile devices such as tablets or smart phones are transforming the way people communicate, interact and carry out daily activities. Churchill et al. ( 2016 ) pointed out that mobile technologies provide a set of tools and affordances to structure and support learning environments in which learners continuously interact to construct knowledge and solve problems. The tools include resources or online materials, efficient connectivity to collaborate and discuss problems, ways to represent, explore and store information, and analytical and administration tools to management learning activities. Schmidt and Cohen ( 2013 ) stated that nowadays it is difficult to imagine a life without mobile devices, and communication technologies are playing a crucial role in generating both cultural and technical breakthroughs. In education, the use of mobile artefacts and computers offers learners the possibility of continuing and extending peers and groups’ mathematical discussions beyond formal settings. In this process, learners can also consult online materials and interact with experts, peers or more experienced students while working on mathematical tasks. In addition, dynamic geometry systems (GeoGebra) provide learners a set of affordances to represent and explore dynamically mathematical problems. Leung and BoliteFrant ( 2015 ) pointed out that tools help activate an interactive environment in which teachers and students’ mathematical experiences get enriched. Thus, the digital age brings new challenges to the mathematics education community related to the changes that technologies produce to curriculum, learning scenarios, and ways to represent, explore mathematical situations. In particular, it is important to characterize the type of reasoning that learners can develop as a result of using digital technologies in their process of learning concepts and solving mathematical problems.
1.3.4 A Focus on Mathematical Tasks
Mathematical tasks are essential elements for engaging learners in mathematical reasoning which involves representing objects, identifying and exploring their properties in order to detect invariants or relationships and ways to support them. Watson and Ohtani ( 2015 ) stated that task design involves discussions about mathematical content and students’ learning (cognitive perspective), about the students’ experiences to understand the nature of mathematical activities; and about the role that tasks played in teaching practices. In this context, tasks are the vehicle to present and discuss theoretical frameworks for supporting the use of digital technology, to analyse the importance of using digital technologies in extending learners’ mathematical discussions beyond formal settings, and to design ways to foster and assess the use of technologies in learners’ problem solving environments. In addition, it is important to discuss contents, concepts, representations and strategies involved in the process of using digital technologies in approaching the tasks. Similarly, it becomes essential to discuss what types of activities students will do to learn and solve the problems in an environment where the use of technologies fosters and values the participation and collaboration of all students. What digital technologies are important to incorporate in problem solving approaches? Dynamic Geometry Systems can be considered as a milestone in the development of digital technologies. Objects or mathematical situations can be represented dynamically through the use of a Dynamic Geometry System and learners or problem solvers can identify and examine mathematical relations that emerge from moving objects within the dynamic model (MorenoArmella and SantosTrigo 2016 ).
Leung and BoliteFrant ( 2015 ) stated that “dynamic geometry software can be used in task design to cover a large epistemic spectrum from drawing precise robust geometrical figures to exploration of new geometric theorems and development of argumentation discourse” (p. 195). As a result, learners not only need to develop skills and strategies to construct dynamic configuration of problems; but also ways of relying on the tool’s affordances (quantifying parameters or objects attributes, generating loci, graphing objects behaviours, using sliders, or dragging particular elements within the configuration) in order to identify and support mathematical relations. What does it mean to represent and explore an object or mathematical situation dynamically?
A simple task that involves a rhombus and its inscribed circle is used to illustrate how a dynamic representation of these objects and embedded elements can lead learners to identify and examine mathematical properties of those objects in the construction of the configuration. To this end, learners are encouraged to pose and pursue questions to explain the behaviours of parameters or attributes of the family of objects that is generated as a result of moving a particular element within the configuration.
1.3.5 A Task: A Dynamic Rhombus
Figure 2 represents a rhombus APDB and its inscribed circle (O is intersection of diagonals AD and BP and the radius of the inscribed circle is the perpendicular segment from any side of the rhombus to point O), vertex P lies on a circle c centred at point A. Circle c is only a heuristic to generate a family of rhombuses. Thus, point P can be moved along circle c to generate a family of rhombuses. Indeed, based on the symmetry of the circle it is sufficient to move P on the semicircle B’CA to draw such a family of rhombuses.
A dynamic construction of a rhombus
1.3.6 Posing Questions
A goal in constructing a dynamic model or configuration of problems is always to identify and explore mathematical properties and relations that might result from moving objects within the model. How do the areas of both the rhombus and the inscribed circle behave when point P is moved along the arc B’CB? At what position of point P does the area of the rhombus or inscribed circle reach the maximum value? The coordinates of points S and Q (Fig. 3 ) are the x value of point P and as y value the corresponding area values of rhombus ABDP and the inscribed circle respectively. Figure 2 shows the loci of points S and Q when point P is moved along arc B’CB. Here, finding the locus via the use of GeoGebra is another heuristic to graph the area behaviour without making explicit the algebraic model of the area.
Graphic representation of the area variation of the family of rhombuses and inscribed circles generated when P is moved through arc B’CB
The area graphs provide information to visualize that in that family of generated rhombuses the maximum area value of the inscribed circle and rhombus is reached when the rhombus becomes a square (Fig. 4 ). That is, the controlled movement of particular objects is an important strategy to analyse the area variation of the family of rhombuses and their inscribed circles.
Visualizing the rhombus and the inscribed circle with maximum area
It is important to observe the identification of points P and Q in terms of the position of point P and the corresponding areas and the movement of point P was sufficient to generate both area loci. That is, the graph representation of the areas is achieved without having an explicit algebraic expression of the area variation. Clearly, the graphic representations provide information regarding the increasing or decreasing interval of both areas; it is also important to explore what properties both graphic representations hold. The goal is to argue that the area variation of the rhombus represents an ellipse and the area of the inscribed circle represents a parabola. An initial argument might involve selecting five points on each locus and using the tool to draw the corresponding conic section (Fig. 5 ). In this case, the tool affordances play an important role in generating the graphic representation of the areas’ behaviours and in identifying properties of those representations. In this context, the use of the tool can offer learners the opportunity to problematize (SantosTrigo 2007 ) a simple mathematical object (rhombus) as a means to search for mathematical relations and ways to support them.
Drawing the conic section that passes through five points
1.3.7 Looking for Different Solutions Methods
Another line of exploration might involve asking for ways to construct a rhombus and its inscribed circle: Suppose that the side of the rhombus and the circle are given, how can you construct the rhombus that has that circle inscribed? Figure 6 shows the given data, segment A 1 B 1 and circle centred at O and radius OD. The initial goal is to draw the circle tangent to the given segment. To this end, segment AB is congruent to segment A 1 B 1 and on this segment a point P is chosen and a perpendicular to segment AB that passes through point P is drawn. Point C is on this perpendicular and the centre of a circle with radius OD and h is the perpendicular to line PC that passes through point C. Angle ACB changes when point P is moved along segment AB and point E and F are the intersection of line h and the circle with centre M the midpoint of AB and radius MA (Fig. 6 ).
Drawing segment AB tangent to the given circle
Figure 7 a shows the right triangle AFB as the base to construct the rhombus and the inscribed circle and Fig. 7 b shows the second solution based on triangle AEB.
a Drawing the rhombus and the inscribed circle. b Drawing the second solution
Another approach might involve drawing the given circle centred at the origin and the segment as EF with point E on the yaxis. Line OC is perpendicular to segment EF and the locus of point C when point E moves along the yaxis intersects the given circle (Fig. 8 a, b). Both figures show two solutions to draw the rhombus that circumscribe the given circle.
a and b Another solution that involves finding a locus of point C
In this example, the GeoGebra affordances not only are important to construct a dynamic model of the task; but also offer learners and opportunity to explore relations that emerge from moving objects within the model. As a result, learners can rely on different concepts and strategies to solve the tasks. The idea in presenting this rhombus task is to illustrate that the use of a Dynamic Geometry System provides affordances for learners to construct dynamic representation of mathematical objects or problems, to move elements within the representation to pose questions or conjectures to explain invariants or patterns among involved parameters; to search for arguments to support emerging conjectures, and to develop a proper language to communicate results.
1.3.8 Looking Back
Conceptual frameworks used to explain learners’ construction of mathematical knowledge need to capture or take into account the different ways of reasoning that students might develop as a result of using a set of tools during the learning experiences. Figure 9 show some digital technologies that learners can use for specific purpose at the different stages of problem solving activities.
The coordinated use of digital tools to engage learners in problem solving experiences
The use of a dynamic system (GeoGebra) provides a set of affordances for learners to conceptualize and represent mathematical objects and tasks dynamically. In this process, affordances such as moving objects orderly (dragging), finding loci of objects, quantifying objects attributes (lengths, areas, angles, etc.), using sliders to vary parameters, and examining family of objects became important to look for invariance or objects relationships. Likewise, analysing the parameters or objects behaviours within the configuration might lead learners to identify properties to support emerging mathematical relations. Thus, with the use of the tool, learners might conceptualize mathematical tasks as an opportunity for them to engage in mathematical activities that include constructing dynamic models of tasks, formulating conjectures, and always looking for different arguments to support them. Similarly, learners can use an online platform to share their ideas, problem solutions or questions in a digital wall and others students can also share ideas or solution methods and engaged in mathematical discussions that extend mathematical classroom activities.
1.4 Problem Posing: An Overview for Further Progress—Uldarico Malaspina Jurado
Problem posing and problem solving are two essential aspects of the mathematical activity; however, researchers in mathematics education have not emphasized their attention on problem posing as much as problem solving. In that sense, due to its importance in the development of mathematical thinking in students since the first grades, we agree with Ellerton’s statement ( 2013 ): “for too long, successful problem solving has been lauded as the goal; the time has come for problem posing to be given a prominent but natural place in mathematics curricula and classrooms” (pp. 100–101); and due to its importance in teacher training, with AbuElwan’s statement ( 1999 ):
While teacher educators generally recognize that prospective teachers require guidance in mastering the ability to confront and solve problems, what is often overlooked is the critical fact that, as teachers, they must be able to go beyond the role as problem solvers. That is, in order to promote a classroom situation where creative problem solving is the central focus, the practitioner must become skillful in discovering and correctly posing problems that need solutions. (p. 1)
Scientists like Einstein and Infeld ( 1938 ), recognized not only for their notable contributions in the fields they worked, but also for their reflections on the scientific activity, pointed out the importance of problem posing; thus it is worthwhile to highlight their statement once again:
The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old questions from a new angle, requires creative imagination and marks real advance in science. (p. 92)
Certainly, it is also relevant to remember mathematician Halmos’s statement ( 1980 ): “I do believe that problems are the heart of mathematics, and I hope that as teachers (…) we will train our students to be better problem posers and problem solvers than we are” (p. 524).
An important number of researchers in mathematics education has focused on the importance of problem posing, and we currently have numerous, very important publications that deal with different aspects of problem posing related to the mathematics education of students in all educational levels and to teacher training.
1.4.1 A Retrospective Look
Kilpatrick ( 1987 ) marked a historical milestone in research related to problem posing and points out that “problem formulating should be viewed not only as a goal of instruction but also as a means of instruction” (Kilpatrick 1987 , p. 123); and he also emphasizes that, as part of students’ education, all of them should be given opportunities to live the experience of discovering and posing their own problems. Drawing attention to the few systematic studies on problem posing performed until then, Kilpatrick contributes defining some aspects that required studying and investigating as steps prior to a theoretical building, though he warns, “attempts to teach problemformulating skills, of course, need not await a theory” (p. 124).
Kilpatrick refers to the “Source of problems” and points out how virtually all problems students solve have been posed by another person; however, in real life “many problems, if not most, must be created or discovered by the solver, who gives the problem an initial formulation” (p. 124). He also points out that problems are reformulated as they are being solved, and he relates this to investigation, reminding us what Davis ( 1985 ) states that, “what typically happens in a prolonged investigation is that problem formulation and problem solution go hand in hand, each eliciting the other as the investigation progresses” (p. 23). He also relates it to the experiences of software designers, who formulate an appropriate sequence of subproblems to solve a problem. He poses that a subject to be examined by teachers and researchers “is whether, by drawing students’ attention to the reformulating process and given them practice in it, we can improve their problem solving performance” (p. 130). He also points out that problems may be a mathematical formulation as a result of exploring a situation and, in that sense, “school exercises in constructing mathematical models of a situation presented by the teacher are intended to provide students with experiences in formulating problems.” (p. 131).
Another important section of Kilpatrick’s work ( 1987 ) is Processes of Problem Formulating , in which he considers association, analogy, generalization and contradiction. He believes the use of concept maps to represent concept organization, as cognitive scientists Novak and Gowin suggest, might help to comprehend such concepts, stimulate creative thinking about them, and complement the ideas Brown and Walter ( 1983 ) give for problem posing by association. Further, in the section “Understanding and developing problem formulating abilities”, he poses several questions, which have not been completely answered yet, like “Perhaps the central issue from the point of view of cognitive science is what happens when someone formulates the problem? (…) What is the relation between problem formulating, problem solving and structured knowledge base? How rich a knowledge base is needed for problem formulating? (…) How does experience in problem formulating add to knowledge base? (…) What metacognitive processes are needed for problem formulating?”
It is interesting to realize that some of these questions are among the unanswered questions proposed and analyzed by Cai et al. ( 2015 ) in Chap. 1 of the book Mathematical Problem Posing (Singer et al. 2015 ). It is worth stressing the emphasis on the need to know the cognitive processes in problem posing, an aspect that Kilpatrick had already posed in 1987, as we just saw.
1.4.2 Researches and Didactic Experiences
Currently, there are a great number of publications related to problem posing, many of which are research and didactic experiences that gather the questions posed by Kilpatrick, which we just commented. Others came up naturally as reflections raised in the framework of problem solving, facing the natural requirement of having appropriate problems to use results and suggestions of researches on problem solving, or as a response to a thoughtful attitude not to resign to solving and asking students to solve problems that are always created by others. Why not learn and teach mathematics posing one’s own problems?
1.4.3 New Directions of Research
Singer et al. ( 2013 ) provides a broad view about problem posing that links problem posing experiences to general mathematics education; to the development of abilities, attitudes and creativity; and also to its interrelation with problem solving, and studies on when and how problemsolving sessions should take place. Likewise, it provides information about research done regarding ways to pose new problems and about the need for teachers to develop abilities to handle complex situations in problem posing contexts.
Singer et al. ( 2013 ) identify new directions in problem posing research that go from problemposing task design to the development of problemposing frameworks to structure and guide teachers and students’ problem posing experiences. In a chapter of this book, Leikin refers to three different types of problem posing activities, associated with school mathematics research: (a) problem posing through proving; (b) problem posing for investigation; and (c) problem posing through investigation. This classification becomes evident in the problems posed in a course for prospective secondary school mathematics teachers by using a dynamic geometry environment. Prospective teachers posed over 25 new problems, several of which are discussed in the article. The author considers that, by developing this type of problem posing activities, prospective mathematics teachers may pose different problems related to a geometric object, prepare more interesting lessons for their students, and thus gradually develop their mathematical competence and their creativity.
1.4.4 Final Comments
This overview, though incomplete, allows us to see a part of what problem posing experiences involve and the importance of this area in students mathematical learning. An important task is to continue reflecting on the questions posed by Kilpatrick ( 1987 ), as well as on the ones that come up in the different researches aforementioned. To continue progressing in research on problem posing and contribute to a greater consolidation of this research line, it will be really important that all mathematics educators pay more attention to problem posing, seek to integrate approaches and results, and promote joint and interdisciplinary works. As Singer et al. ( 2013 ) say, going back to Kilpatrick’s proposal ( 1987 ),
Problem posing is an old issue. What is new is the awareness that problem posing needs to pervade the education systems around the world, both as a means of instruction (…) and as an object of instruction (…) with important targets in reallife situations. (p. 5)
Although it can be argued that there is a difference between creativity, discovery, and invention (see Liljedahl and Allan 2014 ) for the purposes of this book these will be assumed to be interchangeable.
AbuElwan, R. (1999). The development of mathematical problem posing skills for prospective middle school teachers. In A. Rogerson (Ed.), Proceedings of the International Conference on Mathematical Education into the 21st century: Social Challenges, Issues and Approaches , (Vol. 2, pp. 1–8), Cairo, Egypt.
Google Scholar
Ashcraft, M. (1989). Human memory and cognition . Glenview, Illinois: Scott, Foresman and Company.
Bailin, S. (1994). Achieving extraordinary ends: An essay on creativity . Norwood, NJ: Ablex Publishing Corporation.
Bibby, T. (2002). Creativity and logic in primaryschool mathematics: A view from the classroom. For the Learning of Mathematics, 22 (3), 10–13.
Brown, S., & Walter, M. (1983). The art of problem posing . Philadelphia: Franklin Institute Press.
Bruder, R. (2000). Akzentuierte Aufgaben und heuristische Erfahrungen. In W. Herget & L. Flade (Eds.), Mathematik lehren und lernen nach TIMSS. Anregungen für die Sekundarstufen (pp. 69–78). Berlin: Volk und Wissen.
Bruder, R. (2005). Ein aufgabenbasiertes anwendungsorientiertes Konzept für einen nachhaltigen Mathematikunterricht—am Beispiel des Themas “Mittelwerte”. In G. Kaiser & H. W. Henn (Eds.), Mathematikunterricht im Spannungsfeld von Evolution und Evaluation (pp. 241–250). Hildesheim, Berlin: Franzbecker.
Bruder, R., & Collet, C. (2011). Problemlösen lernen im Mathematikunterricht . Berlin: CornelsenVerlag Scriptor.
Bruner, J. (1964). Bruner on knowing . Cambridge, MA: Harvard University Press.
Burton, L. (1999). Why is intuition so important to mathematicians but missing from mathematics education? For the Learning of Mathematics, 19 (3), 27–32.
Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem posing research in mathematics: Some answered and unanswered questions. In F.M. Singer, N. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp.3–34). Springer.
Churchill, D., Fox, B., & King, M. (2016). Framework for designing mobile learning environments. In D. Churchill, J. Lu, T. K. F. Chiu, & B. Fox (Eds.), Mobile learning design (pp. 20–36)., lecture notes in educational technology NY: Springer.
Chapter Google Scholar
Collet, C. (2009). Problemlösekompetenzen in Verbindung mit Selbstregulation fördern. Wirkungsanalysen von Lehrerfortbildungen. In G. Krummheuer, & A. Heinze (Eds.), Empirische Studien zur Didaktik der Mathematik , Band 2, Münster: Waxmann.
Collet, C., & Bruder, R. (2008). Longtermstudy of an intervention in the learning of problemsolving in connection with selfregulation. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PMENA XXX, (Vol. 2, pp. 353–360).
Csíkszentmihályi, M. (1996). Creativity: Flow and the psychology of discovery and invention . New York: Harper Perennial.
Davis, P. J. (1985). What do I know? A study of mathematical selfawareness. College Mathematics Journal, 16 (1), 22–41.
Article Google Scholar
Dewey, J. (1933). How we think . Boston, MA: D.C. Heath and Company.
Dewey, J. (1938). Logic: The theory of inquiry . New York, NY: Henry Holt and Company.
Einstein, A., & Infeld, L. (1938). The evolution of physics . New York: Simon and Schuster.
Ellerton, N. (2013). Engaging preservice middleschool teachereducation students in mathematical problem posing: Development of an active learning framework. Educational Studies in Math, 83 (1), 87–101.
Engel, A. (1998). Problemsolving strategies . New York, Berlin und Heidelberg: Springer.
English, L. (1997). Children’s reasoning processes in classifying and solving comparison word problems. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 191–220). Mahwah, NJ: Lawrence Erlbaum Associates Inc.
English, L. (1998). Reasoning by analogy in solving comparison problems. Mathematical Cognition, 4 (2), 125–146.
English, L. D. & Gainsburg, J. (2016). Problem solving in a 21st Century mathematics education. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 313–335). NY: Routledge.
Ghiselin, B. (1952). The creative process: Reflections on invention in the arts and sciences . Berkeley, CA: University of California Press.
Hadamard, J. (1945). The psychology of invention in the mathematical field . New York, NY: Dover Publications.
Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87 , 519–524.
Halmos, P. R. (1994). What is teaching? The American Mathematical Monthly, 101 (9), 848–854.
Hoyles, C., & Lagrange, J.B. (Eds.). (2010). Mathematics education and technology–Rethinking the terrain. The 17th ICMI Study . NY: Springer.
Kilpatrick, J. (1985). A retrospective account of the past 25 years of research on teaching mathematical problem solving. In E. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 1–15). Hillsdale, New Jersey: Lawrence Erlbaum.
Kilpatrick, J. (1987). Problem formulating: Where do good problem come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Hillsdale, NJ: Erlbaum.
Kline, M. (1972). Mathematical thought from ancient to modern times . NY: Oxford University Press.
Kneller, G. (1965). The art and science of creativity . New York, NY: Holt, Reinhart, and Winstone Inc.
Koestler, A. (1964). The act of creation . New York, NY: The Macmillan Company.
König, H. (1984). Heuristik beim Lösen problemhafter Aufgaben aus dem außerunterrichtlichen Bereich . Technische Hochschule Chemnitz, Sektion Mathematik.
Kretschmer, I. F. (1983). Problemlösendes Denken im Unterricht. Lehrmethoden und Lernerfolge . Dissertation. Frankfurt a. M.: Peter Lang.
Krulik, S. A., & Reys, R. E. (Eds.). (1980). Problem solving in school mathematics. Yearbook of the national council of teachers of mathematics . Reston VA: NCTM.
Krutestkii, V. A. (1976). The psychology of mathematical abilities in school children . University of Chicago Press.
Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. K. Lester, Jr. (Ed.), The second handbook of research on mathematics teaching and learning (pp. 763–804). National Council of Teachers of Mathematics, Charlotte, NC: Information Age Publishing.
Lester, F., & Kehle, P. E. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 501–518). Mahwah, NJ: Lawrence Erlbaum.
Lester, F. K., Garofalo, J., & Kroll, D. (1989). The role of metacognition in mathematical problem solving: A study of two grade seven classes. Final report to the National Science Foundation, NSF Project No. MDR 8550346. Bloomington: Indiana University, Mathematics Education Development Center.
Leung, A., & BoliteFrant, J. (2015). Designing mathematical tasks: The role of tools. In A. Watson & M. Ohtani (Eds.), Task design in mathematics education (pp. 191–225). New York: Springer.
Liljedahl, P. (2008). The AHA! experience: Mathematical contexts, pedagogical implications . Saarbrücken, Germany: VDM Verlag.
Liljedahl, P., & Allan, D. (2014). Mathematical discovery. In E. Carayannis (Ed.), Encyclopedia of creativity, invention, innovation, and entrepreneurship . New York, NY: Springer.
Liljedahl, P., & Sriraman, B. (2006). Musings on mathematical creativity. For the Learning of Mathematics, 26 (1), 20–23.
Lompscher, J. (1975). Theoretische und experimentelle Untersuchungen zur Entwicklung geistiger Fähigkeiten . Berlin: Volk und Wissen. 2. Auflage.
Lompscher, J. (1985). Die Lerntätigkeit als dominierende Tätigkeit des jüngeren Schulkindes. In L. Irrlitz, W. Jantos, E. Köster, H. Kühn, J. Lompscher, G. Matthes, & G. Witzlack (Eds.), Persönlichkeitsentwicklung in der Lerntätigkeit . Berlin: Volk und Wissen.
Mason, J., & JohnstonWilder, S. (2006). Designing and using mathematical tasks . St. Albans: Tarquin Publications.
Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically . Harlow: Pearson Prentice Hall.
Mayer, R. (1982). The psychology of mathematical problem solving. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research (pp. 1–13). Philadelphia, PA: Franklin Institute Press.
Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A multidimensional method for teaching mathematics in heterogeneous classrooms. American Educational Research Journal, 34 (2), 365–394.
Mevarech, Z. R., & Kramarski, B. (2003). The effects of metacognitive training versus workedout examples on students’ mathematical reasoning. British Journal of Educational Psychology, 73 , 449–471.
MorenoArmella, L., & SantosTrigo, M. (2016). The use of digital technologies in mathematical practices: Reconciling traditional and emerging approaches. In L. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 595–616). New York: Taylor and Francis.
National Council of Teachers of Mathematics (NCTM). (1980). An agenda for action . Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics . Reston, VA: National Council of Teachers of Mathematics.
Newman, J. (2000). The world of mathematics (Vol. 4). New York, NY: Dover Publishing.
Novick, L. (1988). Analogical transfer, problem similarity, and expertise. Journal of Educational Psychology: Learning, Memory, and Cognition, 14 (3), 510–520.
Novick, L. (1990). Representational transfer in problem solving. Psychological Science, 1 (2), 128–132.
Novick, L. (1995). Some determinants of successful analogical transfer in the solution of algebra word problems. Thinking & Reasoning, 1 (1), 5–30.
Novick, L., & Holyoak, K. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology, 17 (3), 398–415.
Pehkonen, E. K. (1991). Developments in the understanding of problem solving. ZDM—The International Journal on Mathematics Education, 23 (2), 46–50.
Pehkonen, E. (1997). The stateofart in mathematical creativity. Analysis, 97 (3), 63–67.
Perels, F., Schmitz, B., & Bruder, R. (2005). Lernstrategien zur Förderung von mathematischer Problemlösekompetenz. In C. Artelt & B. Moschner (Eds.), Lernstrategien und Metakognition. Implikationen für Forschung und Praxis (pp. 153–174). Waxmann education.
Perkins, D. (2000). Archimedes’ bathtub: The art of breakthrough thinking . New York, NY: W.W. Norton and Company.
Poincaré, H. (1952). Science and method . New York, NY: Dover Publications Inc.
Pólya, G. (1945). How to solve It . Princeton NJ: Princeton University.
Pólya, G. (1949). How to solve It . Princeton NJ: Princeton University.
Pólya, G. (1954). Mathematics and plausible reasoning . Princeton: Princeton University Press.
Pólya, G. (1964). Die Heuristik. Versuch einer vernünftigen Zielsetzung. Der Mathematikunterricht , X (1), 5–15.
Pólya, G. (1965). Mathematical discovery: On understanding, learning and teaching problem solving (Vol. 2). New York, NY: Wiley.
Resnick, L., & Glaser, R. (1976). Problem solving and intelligence. In L. B. Resnick (Ed.), The nature of intelligence (pp. 230–295). Hillsdale, NJ: Lawrence Erlbaum Associates.
Rusbult, C. (2000). An introduction to design . http://www.asa3.org/ASA/education/think/intro.htm#process . Accessed January 10, 2016.
SantosTrigo, M. (2007). Mathematical problem solving: An evolving research and practice domain. ZDM—The International Journal on Mathematics Education , 39 (5, 6): 523–536.
SantosTrigo, M. (2014). Problem solving in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 496–501). New York: Springer.
Schmidt, E., & Cohen, J. (2013). The new digital age. Reshaping the future of people nations and business . NY: Alfred A. Knopf.
Schoenfeld, A. H. (1979). Explicit heuristic training as a variable in problemsolving performance. Journal for Research in Mathematics Education, 10 , 173–187.
Schoenfeld, A. H. (1982). Some thoughts on problemsolving research and mathematics education. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research (pp. 27–37). Philadelphia: Franklin Institute Press.
Schoenfeld, A. H. (1985). Mathematical problem solving . Orlando, Florida: Academic Press Inc.
Schoenfeld, A. H. (1987). What’s all the fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–215). Hillsdale, NJ: Lawrence Erlbaum Associates.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York, NY: Simon and Schuster.
Schön, D. (1987). Educating the reflective practitioner . San Fransisco, CA: JosseyBass Publishers.
Sewerin, H. (1979): Mathematische Schülerwettbewerbe: Beschreibungen, Analysen, Aufgaben, Trainingsmethoden mit Ergebnissen . Umfrage zum Bundeswettbewerb Mathematik. München: Manz.
Silver, E. (1982). Knowledge organization and mathematical problem solving. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research (pp. 15–25). Philadelphia: Franklin Institute Press.
Singer, F., Ellerton, N., & Cai, J. (2013). Problem posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83 (1), 9–26.
Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing. From research to practice . NY: Springer.
Törner, G., Schoenfeld, A. H., & Reiss, K. M. (2007). Problem solving around the world: Summing up the state of the art. ZDM—The International Journal on Mathematics Education, 39 (1), 5–6.
Verschaffel, L., de Corte, E., Lasure, S., van Vaerenbergh, G., Bogaerts, H., & Ratinckx, E. (1999). Learning to solve mathematical application problems: A design experiment with fifth graders. Mathematical Thinking and Learning, 1 (3), 195–229.
Wallas, G. (1926). The art of thought . New York: Harcourt Brace.
Watson, A., & Ohtani, M. (2015). Themes and issues in mathematics education concerning task design: Editorial introduction. In A. Watson & M. Ohtani (Eds.), Task design in mathematics education, an ICMI Study 22 (pp. 3–15). NY: Springer.
Zimmermann, B. (1983). Problemlösen als eine Leitidee für den Mathematikunterricht. Ein Bericht über neuere amerikanische Beiträge. Der Mathematikunterricht, 3 (1), 5–45.
Further Reading
Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex, and setting . Buckingham, PA: Open University Press.
Borwein, P., Liljedahl, P., & Zhai, H. (2014). Mathematicians on creativity. Mathematical Association of America.
Burton, L. (1984). Thinking things through . London, UK: Simon & Schuster Education.
Feynman, R. (1999). The pleasure of finding things out . Cambridge, MA: Perseus Publishing.
Gardner, M. (1978). Aha! insight . New York, NY: W. H. Freeman and Company.
Gardner, M. (1982). Aha! gotcha: Paradoxes to puzzle and delight . New York, NY: W. H. Freeman and Company.
Gardner, H. (1993). Creating minds: An anatomy of creativity seen through the lives of Freud, Einstein, Picasso, Stravinsky, Eliot, Graham, and Ghandi . New York, NY: Basic Books.
Glas, E. (2002). Klein’s model of mathematical creativity. Science & Education, 11 (1), 95–104.
Hersh, D. (1997). What is mathematics, really? . New York, NY: Oxford University Press.
RootBernstein, R., & RootBernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people . Boston, MA: Houghton Mifflin Company.
Zeitz, P. (2006). The art and craft of problem solving . New York, NY: Willey.
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Liljedahl, P., SantosTrigo, M., Malaspina, U., Bruder, R. (2016). Problem Solving in Mathematics Education. In: Problem Solving in Mathematics Education. ICME13 Topical Surveys. Springer, Cham. https://doi.org/10.1007/9783319407302_1
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10 Strategies for Problem Solving in Math
Created on May 19, 2022
Updated on January 6, 2024
When faced with problemsolving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them. Therefore, this article discusses math strategies and how your students may use them since instructors often have to lead students through this problemsolving maze.
What Are Problem Solving Strategies in Math?
If you want to fix a problem, you need a solid plan. Math strategies for problem solving are ways of tackling math in a way that guarantees better outcomes. These strategies simplify math for kids so that less time is spent figuring out the problem. Both those new to mathematics and those more knowledgeable about the subject may benefit from these methods.
There are several methods to apply problemsolving procedures in math, and each strategy is different. While none of these methods failsafe, they may help your student become a better problem solver, particularly when paired with practice and examples. The more math problems kids tackle, the more math problem solving skills they acquire, and practice is the key.
Strategies for Problemsolving in Math
Even if a student is not a math wiz, a suitable solution to mathematical problems in math may help them discover answers. There is no one best method for helping students solve arithmetic problems, but the following ten approaches have shown to be very effective.
Understand the Problem
Understanding the nature of math problems is a prerequisite to solving them. They need to specify what kind of issue it is ( fraction problem , word problem, quadratic equation, etc.). Searching for keywords in the math problem, revisiting similar questions, or consulting the internet are all great ways to strengthen their grasp of the material. This step keeps the pupil on track.
1:1 Math Lessons
Want to raise a genius? Start learning Math with Brighterly
Guess and check.
One of the timeintensive strategies for resolving mathematical problems is the guess and check method. In this approach, students keep guessing until they get the answer right.
After assuming how to solve a math issue, students should reintroduce that assumption to check for correctness. While the approach may appear cumbersome, it is typically successful in revealing patterns in a child’s thought process.
Work It Out
Encourage pupils to record their thinking process as they go through a math problem. Since this technique requires an initial comprehension of the topic, it serves as a selfmonitoring method for mathematics students. If they immediately start solving the problem, they risk making mistakes.
Students may keep track of their ideas and fix their math problems as they go along using this method. A youngster may still need you to explain their methods of solving the arithmetic questions on the extra page. This confirmation stage etches the steps they took to solve the problem in their minds.
Work Backwards
In mathematics, a fresh perspective is sometimes the key to a successful solution. Young people need to know that the ability to recreate math problems is valuable in many professional fields, including project management and engineering.
Students may better prepare for difficulties in realworld circumstances by using the “Work Backwards” technique. The end product may be used as a startoff point to identify the underlying issue.
In most cases, a visual representation of a math problem may help youngsters understand it better. Some of the most helpful math tactics for kids include having them play out the issue and picture how to solve it.
One way to visualize a workout is to use a blank piece of paper to draw a picture or make tally marks. Students might also use a marker and a whiteboard to draw as they demonstrate the technique before writing it down.
Find a Pattern
Kids who use pattern recognition techniques can better grasp math concepts and retain formulae. The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition.
Students may use this strategy to spot patterns and fill in the blanks. Over time, this strategy will help kids answer math problems quickly.
When faced with a math word problem, it might be helpful to ask, “What are some possible solutions to this issue?” It encourages you to give the problem more thought, develop creative solutions, and prevent you from being stuck in a rut. So, tell the pupils to think about the math problems and not just go with the first solution that comes to mind.
Draw a Picture or Diagram
Drawing a picture of a math problem can help kids understand how to solve it, just like picturing it can help them see it. Shapes or numbers could be used to show the forms to keep things easy. Kids might learn how to use dots or letters to show the parts of a pattern or graph if you teach them.
Charts and graphs can be useful even when math isn’t involved. Kids can draw pictures of the ideas they read about to help them remember them after they’ve learned them. The plan for how to solve the mathematical problem will help kids understand what the problem is and how to solve it.
Trial and Error Method
The trial and error method may be one of the most common problem solving strategies for kids to figure out how to solve problems. But how well this strategy is used will determine how well it works. Students have a hard time figuring out math questions if they don’t have clear formulas or instructions.
They have a better chance of getting the correct answer, though, if they first make a list of possible answers based on rules they already know and then try each one. Don’t be too quick to tell kids they shouldn’t learn by making mistakes.
Review Answers with Peers
It’s fun to work on your math skills with friends by reviewing the answers to math questions together. If different students have different ideas about how to solve the same problem, get them to share their thoughts with the class.
During class time, kids’ ways of working might be compared. Then, students can make their points stronger by fixing these problems.
Check out the Printable Math Worksheets for Your Kids!
There are different ways to solve problems that can affect how fast and well students do on math tests. That’s why they need to learn the best ways to do things. If students follow the steps in this piece, they will have better experiences with solving math questions.
Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.
As adults, we take numbers for granted, but preschoolers and kindergartners have no idea what these symbols mean. Yet, we often demand instant understanding and flawless performance when we start teaching numbers to our children. If you don’t have a clue about how to teach numbers for kids, browse no more. You will get four […]
May 19, 2022
Teaching children is a complex process because they require more attention than an adult person. You may need to employ different teaching strategies when teaching kids. But what are teaching strategies? Teaching strategies are the methods to ensure your kids or students learn efficiently. But not all strategies yield similarly, and if the one you […]
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Math Problem Solving Strategies That Make Students Say “I Get It!”
Even students who are quick with math facts can get stuck when it comes to problem solving.
As soon as a concept is translated to a word problem, or a simple mathematical sentence contains an unknown, they’re stumped.
That’s because problem solving requires us to consciously choose the strategies most appropriate for the problem at hand . And not all students have this metacognitive ability.
But you can teach these strategies for problem solving. You just need to know what they are.
We’ve compiled them here divided into four categories:
Strategies for understanding a problem
Strategies for solving the problem, strategies for working out, strategies for checking the solution.
Get to know these strategies and then model them explicitly to your students. Next time they dive into a rich problem, they’ll be filling up their working out paper faster than ever!
Before students can solve a problem, they need to know what it’s asking them. This is often the first hurdle with word problems that don’t specify a particular mathematical operation.
Encourage your students to:
Read and reread the question
They say they’ve read it, but have they really ? Sometimes students will skip ahead as soon as they’ve noticed one familiar piece of information or give up trying to understand it if the problem doesn’t make sense at first glance.
Teach students to interpret a question by using selfmonitoring strategies such as:
 Rereading a question more slowly if it doesn’t make sense the first time
 Asking for help
 Highlighting or underlining important pieces of information.
Identify important and extraneous information
John is collecting money for his friend Ari’s birthday. He starts with $5 of his own, then Marcus gives him another $5. How much does he have now?
As adults looking at the above problem, we can instantly look past the names and the birthday scenario to see a simple addition problem. Students, however, can struggle to determine what’s relevant in the information that’s been given to them.
Teach students to sort and sift the information in a problem to find what’s relevant. A good way to do this is to have them swap out pieces of information to see if the solution changes. If changing names, items or scenarios has no impact on the end result, they’ll realize that it doesn’t need to be a point of focus while solving the problem.
Schema approach
This is a math intervention strategy that can make problem solving easier for all students, regardless of ability.
Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:
[Number/Quantity A] with [Number/Quantity B] removed becomes [end result].
This is the underlying procedure or schema students are being asked to use. Once they have a list of schema for different mathematical operations (addition, multiplication and so on), they can take turns to apply them to an unfamiliar word problem and see which one fits.
Struggling students often believe math is something you either do automatically or don’t do at all. But that’s not true. Help your students understand that they have a choice of problemsolving strategies to use, and if one doesn’t work, they can try another.
Here are four common strategies students can use for problem solving.
Visualizing
Visualizing an abstract problem often makes it easier to solve. Students could draw a picture or simply draw tally marks on a piece of working out paper.
Encourage visualization by modeling it on the whiteboard and providing graphic organizers that have space for students to draw before they write down the final number.
Guess and check
Show students how to make an educated guess and then plug this answer back into the original problem. If it doesn’t work, they can adjust their initial guess higher or lower accordingly.
Find a pattern
To find patterns, show students how to extract and list all the relevant facts in a problem so they can be easily compared. If they find a pattern, they’ll be able to locate the missing piece of information.
Work backward
Working backward is useful if students are tasked with finding an unknown number in a problem or mathematical sentence. For example, if the problem is 8 + x = 12, students can find x by:
 Starting with 12
 Taking the 8 from the 12
 Being left with 4
 Checking that 4 works when used instead of x
Now students have understood the problem and formulated a strategy, it’s time to put it into practice. But if they just launch in and do it, they might make it harder for themselves. Show them how to work through a problem effectively by:
Documenting working out
Model the process of writing down every step you take to complete a math problem and provide working out paper when students are solving a problem. This will allow students to keep track of their thoughts and pick up errors before they reach a final solution.
Check along the way
Checking work as you go is another crucial selfmonitoring strategy for math learners. Model it to them with think aloud questions such as:
 Does that last step look right?
 Does this follow on from the step I took before?
 Have I done any ‘smaller’ sums within the bigger problem that need checking?
Students often make the mistake of thinking that speed is everything in math — so they’ll rush to get an answer down and move on without checking.
But checking is important too. It allows them to pinpoint areas of difficulty as they come up, and it enables them to tackle more complex problems that require multiple checks before arriving at a final answer.
Here are some checking strategies you can promote:
Check with a partner
Comparing answers with a peer leads is a more reflective process than just receiving a tick from the teacher. If students have two different answers, encourage them to talk about how they arrived at them and compare working out methods. They’ll figure out exactly where they went wrong, and what they got right.
Reread the problem with your solution
Most of the time, students will be able to tell whether or not their answer is correct by putting it back into the initial problem. If it doesn’t work or it just ‘looks wrong’, it’s time to go back and fix it up.
Fixing mistakes
Show students how to backtrack through their working out to find the exact point where they made a mistake. Emphasize that they can’t do this if they haven’t written down everything in the first place — so a single answer with no working out isn’t as impressive as they might think!
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5 Teaching Mathematics Through Problem Solving
Janet Stramel
In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)
What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving reallife problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.
According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.
There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.
Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a twodigit number by a onedigit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.
Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problemsolving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.
Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.
Consider the following worthwhileproblem criteria developed by Lappan and Phillips (1998):
 The problem has important, useful mathematics embedded in it.
 The problem requires highlevel thinking and problem solving.
 The problem contributes to the conceptual development of students.
 The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
 The problem can be approached by students in multiple ways using different solution strategies.
 The problem has various solutions or allows different decisions or positions to be taken and defended.
 The problem encourages student engagement and discourse.
 The problem connects to other important mathematical ideas.
 The problem promotes the skillful use of mathematics.
 The problem provides an opportunity to practice important skills.
Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.
Key features of a good mathematics problem includes:
 It must begin where the students are mathematically.
 The feature of the problem must be the mathematics that students are to learn.
 It must require justifications and explanations for both answers and methods of solving.
Problem solving is not a neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.
But look at the b ack.
It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.
When you teach through problem solving , your students are focused on ideas and sensemaking and they develop confidence in mathematics!
Mathematics Tasks and Activities that Promote Teaching through Problem Solving
Choosing the Right Task
Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:
 Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
 What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
 Can the activity accomplish your learning objective/goals?
Low Floor High Ceiling Tasks
By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].
The strengths of using Low Floor High Ceiling Tasks:
 Allows students to show what they can do, not what they can’t.
 Provides differentiation to all students.
 Promotes a positive classroom environment.
 Advances a growth mindset in students
 Aligns with the Standards for Mathematical Practice
Examples of some Low Floor High Ceiling Tasks can be found at the following sites:
 YouCubed – under grades choose Low Floor High Ceiling
 NRICH Creating a Low Threshold High Ceiling Classroom
 Inside Mathematics Problems of the Month
Math in 3Acts
Math in 3Acts was developed by Dan Meyer to spark an interest in and engage students in thoughtprovoking mathematical inquiry. Math in 3Acts is a wholegroup mathematics task consisting of three distinct parts:
Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.
In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.
Act Three is the “reveal.” Students share their thinking as well as their solutions.
“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3Acts can be found at the following websites:
 Dan Meyer’s ThreeAct Math Tasks
 Graham Fletcher3Act Tasks ]
 Math in 3Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete
Number Talks
Number talks are brief, 515 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:
 The teacher presents a problem for students to solve mentally.
 Provide adequate “ wait time .”
 The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
 For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
 Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.
“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:
 Inside Mathematics Number Talks
 Number Talks Build Numerical Reasoning
Saying “This is Easy”
“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.
When the teacher says, “this is easy,” students may think,
 “Everyone else understands and I don’t. I can’t do this!”
 Students may just give up and surrender the mathematics to their classmates.
 Students may shut down.
Instead, you and your students could say the following:
 “I think I can do this.”
 “I have an idea I want to try.”
 “I’ve seen this kind of problem before.”
Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.
Using “Worksheets”
Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?
What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use handson materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the handson learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a handson activity.
Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higherorder thinking skills, and worksheets will not allow them to do that.
One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”
You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include handson manipulatives. Using manipulatives can
 Provide your students a bridge between the concrete and abstract
 Serve as models that support students’ thinking
 Provide another representation
 Support student engagement
 Give students ownership of their own learning.
Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.
any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method
should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning
involves teaching a skill so that a student can later solve a story problem
when we teach students how to problem solve
teaching mathematics content through real contexts, problems, situations, and models
a mathematical activity where everyone in the group can begin and then work on at their own level of engagement
20 seconds to 2 minutes for students to make sense of questions
Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.
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The easy 4 step problemsolving process (+ examples)
This is the 4 step problemsolving process that I taught to my students for math problems, but it works for academic and social problems as well.
Every problem may be different, but effective problem solving asks the same four questions and follows the same method.
 What’s the problem? If you don’t know exactly what the problem is, you can’t come up with possible solutions. Something is wrong. What are we going to do about this? This is the foundation and the motivation.
 What do you need to know? This is the most important part of the problem. If you don’t know exactly what the problem is, you can’t come up with possible solutions.
 What do you already know? You already know something related to the problem that will help you solve the problem. It’s not always obvious (especially in the real world), but you know (or can research) something that will help.
 What’s the relationship between the two? Here is where the heavy brainstorming happens. This is where your skills and abilities come into play. The previous steps set you up to find many potential solutions to your problem, regardless of its type.
When I used to tutor kids in math and physics , I would drill this problemsolving process into their heads. This methodology works for any problem, regardless of its complexity or difficulty. In fact, if you look at the various advances in society, you’ll see they all follow some variation of this problemsolving technique.
“The gap between understanding and misunderstanding can best be bridged by thought!” ― Ernest Agyemang Yeboah
Generally speaking, if you can’t solve the problem then your issue is step 3 or step 4; you either don’t know enough or you’re missing the connection.
Good problem solvers always believe step 3 is the issue. In this case, it’s a simple matter of learning more. Less skilled problem solvers believe step 4 is the root cause of their difficulties. In this instance, they simply believe they have limited problemsolving skills.
This is a fixed versus growth mindset and it makes a huge difference in the effort you put forth and the belief you have in yourself to make use of this stepbystep process. These two mindsets make a big difference in your learning because, at its core, learning is problemsolving.
Let’s dig deeper into the 4 steps. In this way, you can better see how to apply them to your learning journey.
Step 1: What’s the problem?
The ability to recognize a specific problem is extremely valuable.
Most people only focus on finding solutions. While a “solutionsoriented” mindset is a good thing, sometimes it pays to focus on the problem. When you focus on the problem, you often make it easier to find a viable solution to it.
When you know the exact nature of the problem, you shorten the time frame needed to find a solution. This reminds me of a story I was once told.
When does the problemsolving process start?
The process starts after you’ve identified the exact nature of the problem.
Homeowners love a wellkept lawn but hate mowing the grass.
Many companies and inventors raced to figure out a more timeefficient way to mow the lawn. Some even tried to design robots that would do the mowing. They all were chasing the solution, but only one inventor took the time to understand the root cause of the problem.
Most people figured that the problem was the labor required to maintain a lawn. The actual problem was just the opposite: maintaining a lawn was laborintensive. The rearrangement seems trivial, but it reveals the true desire: a wellmaintained lawn.
The best solution? Remove maintenance from the equation. A lawn made of artificial grass solved the problem . Hence, an application of Astroturf was discovered.
This way, the law always looked its best. Taking a few moments to apply critical thinking identified the true nature of the problem and yielded a powerful solution.
An example of choosing the right problem to work the problemsolving process on
One thing I’ve learned from tutoring high school students in math : they hate word problems.
This is because they make the student figure out the problem. Finding the solution to a math problem is already stressful. Forcing the student to also figure out what problem needs solving is another level of hell.
Word problems are not always clear about what needs to be solved. They also have the annoying habit of adding extraneous information. An ordinary math problem does not do this. For example, compare the following two problems:
What’s the height of h?
A radio station tower was built in two sections. From a point 87 feet from the base of the tower, the angle of elevation of the top of the first section is 25º, and the angle of elevation of the top of the second section is 40º. To the nearest foot, what is the height of the top section of the tower?
The first is a simple problem. The second is a complex problem. The end goal in both is the same.
The questions require the same knowledge (trigonometric functions), but the second is more difficult for students. Why? The second problem does not make it clear what the exact problem is. Before mathematics can even begin, you must know the problem, or else you risk solving the wrong one.
If you understand the problem, finding the solution is much easier. Understanding this, ironically, is the biggest problem for people.
Problemsolving is a universal language
Speaking of people, this method also helps settle disagreements.
When we disagree, we rarely take the time to figure out the exact issue. This happens for many reasons, but it always results in a misunderstanding. When each party is clear with their intentions, they can generate the best response.
Education systems fail when they don’t consider the problem they’re supposed to solve. Foreign language education in America is one of the best examples.
The problem is that students can’t speak the target language. It seems obvious that the solution is to have students spend most of their time speaking. Unfortunately, language classes spend a ridiculous amount of time learning grammar rules and memorizing vocabulary.
The problem is not that the students don’t know the imperfect past tense verb conjugations in Spanish. The problem is that they can’t use the language to accomplish anything. Every year, kids graduate from American high schools without the ability to speak another language, despite studying one for 4 years.
Well begun is half done
Before you begin to learn something, be sure that you understand the exact nature of the problem. This will make clear what you need to know and what you can discard. When you know the exact problem you’re tasked with solving, you save precious time and energy. Doing this increases the likelihood that you’ll succeed.
Step 2: What do you need to know?
All problems are the result of insufficient knowledge. To solve the problem, you must identify what you need to know. You must understand the cause of the problem. If you get this wrong, you won’t arrive at the correct solution.
Either you’ll solve what you thought was the problem, only to find out this wasn’t the real issue and now you’ve still got trouble or you won’t and you still have trouble. Either way, the problem persists.
If you solve a different problem than the correct one, you’ll get a solution that you can’t use. The only thing that wastes more time than an unsolved problem is solving the wrong one.
Imagine that your car won’t start. You replace the alternator, the starter, and the ignition switch. The car still doesn’t start. You’ve explored all the main solutions, so now you consider some different solutions.
Now you replace the engine, but you still can’t get it to start. Your replacements and repairs solved other problems, but not the main one: the car won’t start.
Then it turns out that all you needed was gas.
This example is a little extreme, but I hope it makes the point. For something more relatable, let’s return to the problem with language learning.
You need basic communication to navigate a foreign country you’re visiting; let’s say Mexico. When you enroll in a Spanish course, they teach you a bunch of unimportant words and phrases. You stick with it, believing it will eventually click.
When you land, you can tell everyone your name and ask for the location of the bathroom. This does not help when you need to ask for directions or tell the driver which airport terminal to drop you off at.
Finding the solution to chess problems works the same way
The book “The Amateur Mind” by IM Jeremy Silman improved my chess by teaching me how to analyze the board.
It’s only with a proper analysis of imbalances that you can make the best move. Though you may not always choose the correct line of play, the book teaches you how to recognize what you need to know . It teaches you how to identify the problem—before you create an action plan to solve it.
The problemsolving method always starts with identifying the problem or asking “What do you need to know?”. It’s only after you brainstorm this that you can move on to the next step.
Learn the method I used to earn a physics degree, learn Spanish, and win a national boxing title
 I was a terrible math student in high school who wrote off mathematics. I eventually overcame my difficulties and went on to earn a B.A. Physics with a minor in math
 I pieced together the best works on the internet to teach myself Spanish as an adult
 *I didn’t start boxing until the very old age of 22, yet I went on to win a national championship, get a highpaying amateur sponsorship, and get signed by Roc Nation Sports as a profession.
I’ve used this method to progress in mentally and physically demanding domains.
While the specifics may differ, I believe that the general methods for learning are the same in all domains.
This free ebook breaks down the most important techniques I’ve used for learning.
Step 3: What do you already know?
The only way to know if you lack knowledge is by gaining some in the first place. All advances and solutions arise from the accumulation and implementation of prior information. You must first consider what it is that you already know in the context of the problem at hand.
Isaac Newton once said, “If I have seen further, it is by standing on the shoulders of giants.” This is Newton’s way of explaining that his advancements in physics and mathematics would be impossible if it were not for previous discoveries.
Mathematics is a great place to see this idea at work. Consider the following problem:
What is the domain and range of y=(x^2)+6?
This simple algebra problem relies on you knowing a few things already. You must know:
 The definition of “domain” and “range”
 That you can never square any real number and get a negative
Once you know those things, this becomes easy to solve. This is also how we learn languages.
An example of the problemsolving process with a foreign language
Anyone interested in serious foreign language study (as opposed to a “crash course” or “survival course”) should learn the infinitive form of verbs in their target language. You can’t make progress without them because they’re the root of all conjugations. It’s only once you have a grasp of the infinitives that you can completely express yourself. Consider the problemsolving steps applied in the following example.
I know that I want to say “I don’t eat eggs” to my Mexican waiter. That’s the problem.
I don’t know how to say that, but last night I told my date “No bebo alcohol” (“I don’t drink alcohol”). I also know the infinitive for “eat” in Spanish (comer). This is what I already know.
Now I can execute the final step of problemsolving.
Step 4: What’s the relationship between the two?
I see the connection. I can use all of my problemsolving strategies and methods to solve my particular problem.
I know the infinitive for the Spanish word “drink” is “beber” . Last night, I changed it to “bebo” to express a similar idea. I should be able to do the same thing to the word for “eat”.
“No como huevos” is a pretty accurate guess.
In the math example, the same process occurs. You don’t know the answer to “What is the domain and range of y=(x^2)+6?” You only know what “domain” and “range” mean and that negatives aren’t possible when you square a real number.
A domain of all real numbers and a range of all numbers equal to and greater than six is the answer.
This is relating what you don’t know to what you already do know. The solutions appear simple, but walking through them is an excellent demonstration of the process of problemsolving.
In most cases, the solution won’t be this simple, but the process or finding it is the same. This may seem trivial, but this is a model for thinking that has served the greatest minds in history.
A recap of the 4 steps of the simple problemsolving process
 What’s the problem? There’s something wrong. There’s something amiss.
 What do you need to know? This is how to fix what’s wrong.
 What do you already know? You already know something useful that will help you find an effective solution.
 What’s the relationship between the previous two? When you use what you know to help figure out what you don’t know, there is no problem that won’t yield.
Learning is simply problemsolving. You’ll learn faster if you view it this way.
What was once complicated will become simple.
What was once convoluted will become clear.
Ed Latimore
I’m a writer, competitive chess player, Army veteran, physicist, and former professional heavyweight boxer. My work focuses on selfdevelopment, realizing your potential, and sobriety—speaking from personal experience, having overcome both poverty and addiction.
Follow me on Twitter.
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Powerful online learning at your pace
What IS ProblemSolving?
Ask teachers about problemsolving strategies, and you’re opening a can of worms! Opinions about the “best” way to teach problemsolving are all over the board. And teachers will usually argue for their process quite passionately.
When I first started teaching math over 25 years ago, it was very common to teach “keywords” to help students determine the operation to use when solving a word problem. For example, if you see the word “total” in the problem, you always add. Rather than help students become better problem solvers, the use of keywords actually resulted in students who don’t even feel the need to read and understand the problem–just look for the keywords, pick out the numbers, and do the operation indicated by the keyword.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.
Another common strategy for teaching problemsolving is the use of acrostics that students can easily remember to perform the “steps” in problemsolving. CUBES is an example. Just as with keywords, however, students often follow the steps with little understanding. As an example, a common step is to underline or highlight the question. But if you ask students why they are underlining or highlighting the question, they often can’t tell you. The question is , in fact, super important, but they’ve not been told why. They’ve been told to underline the question, so they do.
The problem with both keywords and the rotestep strategies is that both methods try to turn something that is inherently messy into an algorithm! It’s way past time that we leave both methods behind.
First, we need to broaden the definition of problemsolving. Somewhere along the line, problemsolving became synonymous with “word problems.” In reality, it’s so much more. Every one of us solves dozens or hundreds of problems every single day, and most of us haven’t solved a word problem in years. Problemsolving is often described as figuring out what to do when you don’t know what to do. My power went out unexpectedly this morning, and I have work to do. That’s a problem that I had to solve. I had to think about what the problem was, what my options were, and formulate a plan to solve the problem. No keywords. No acrostics. I’m using my phone as a hotspot and hoping my laptop battery doesn’t run out. Problem solved. For now.
If you want to get back to what problemsolving really is, you should consult the work of George Polya. His book, How to Solve It , which was first published in 1945, outlined four principles for problemsolving. The four principles are: understand the problem, devise a plan, carry out the plan, and look back. This document from UC Berkeley’s Mathematics department is a great 4page overview of Polya’s process. You can probably see that the keyword and rotesteps strategies were likely based on Polya’s method, but it really got out of hand. We need to help students think , not just follow steps.
I created both primary and intermediate posters based on Polya’s principles. Grab your copies for free here !
I would LOVE to hear your comments about problemsolving!
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Do you tutor teachers?
I do professional development for district and schools, and I have online courses.
You make a great point when you mentioned that teaching students to look for “keywords” is not teaching students to become better problem solvers. I was once guilty of using the CUBES strategy, but have since learned to provide students with opportunity to grapple with solving a problem and not providing them with specified steps to follow.
I think we’ve ALL been there! We learn and we do better. 🙂
Love this article and believe that we can do so much better as math teachers than just teaching key words! Do you have an editable version of this document? We are wanting to use something similar for our school, but would like to tweak it just a bit. Thank you!
I’m sorry, but because of the clip art and fonts I use, I am not able to provide an editable version.
Hi Donna! I am working on my dissertation that focuses on problemsolving. May I use your intermediate poster as a figure, giving credit to you in my citation with your permission, for my section on Polya’s Traditional ProblemSolving Steps? You laid out the process so succinctly with examples that my research could greatly benefit from this image. Thank you in advance!
Absolutely! Good luck with your dissertation!
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Mathematical ProblemSolving: Techniques and Strategies
by Ali  Mar 8, 2023  Blog Post , Blogs  0 comments
Introduction to Mathematical ProblemSolving
Mathematical problemsolving is the process of using logical reasoning and critical thinking to find a solution to a mathematical problem. It is an essential skill that is required in a wide range of academic and professional fields, including science, technology, engineering, and mathematics (STEM).
Importance of Mathematical ProblemSolving Skills
Mathematical problemsolving skills are critical for success in many areas of life, including education, career, and daily life. It helps students to develop analytical and critical thinking skills, enhances their ability to reason logically, and encourages them to persevere when faced with challenges.
The Process of Mathematical ProblemSolving
The process of mathematical problemsolving involves several steps that include identifying the problem, understanding the problem, making a plan, carrying out the plan, and checking the answer.
Techniques and Strategies for Mathematical ProblemSolving
1. identify the problem.
The first step in problemsolving is to identify the problem. It involves reading the problem carefully and determining what the problem is asking.
2. Understand the problem
The next step is to understand the problem by breaking it down into smaller parts, identifying any relevant information, and determining what needs to be solved.
3. Make a plan
After understanding the problem, the next step is to develop a plan to solve it. This may involve identifying a formula or method to use, drawing a diagram or chart, or making a list of steps to follow.
4. Carry out the plan
Once a plan is developed, the next step is to carry out the plan by solving the problem using the chosen method. It is important to show all steps and work neatly to avoid making mistakes.
5. Check the answer
Finally, it is essential to check the answer to ensure it is correct. This can be done by rereading the problem, checking the solution for accuracy, and verifying that it makes sense.
Know About: HOW TO FIND PERFECT MATH TUTOR
Importance of using online calculators while learning math.
Utilizing online calculators can prove to be a beneficial resource for learning mathematics. There are numerous reasons why incorporating them into your studies is a wise choice.
Firstly, online calculators offer the convenience of being easily accessible at any time and from anywhere. No longer do you need to carry a physical calculator with you; you can use them on any device that has internet connectivity.
In addition, online calculators excel in accuracy and can efficiently handle complex calculations that may be difficult to do manually. They can perform arithmetic at a faster speed, saving you time and increasing productivity.
Another advantage is that some online calculators include builtin visualizations such as graphs and charts, which can help students grasp mathematical concepts better.
Furthermore, feedback can be provided by certain online calculators, assisting students in identifying and rectifying errors in their calculations. This feature can be especially useful for students who are new to learning mathematics .
Online calculators have a versatile range of functions beyond basic arithmetic, including algebraic equations, trigonometry, and calculus . This makes them useful for students at all levels of math education.
Overall, online calculators are an invaluable tool for students learning math. They are convenient, accurate, efficient, and versatile, and aid in the understanding of mathematical concepts, making them an essential component of modernday education.
Common Errors in Mathematical ProblemSolving
There are several common errors that can occur in mathematical problemsolving, including misunderstanding the problem, using incorrect formulas or methods, making computational errors, and not checking the answer. To avoid these errors, it is essential to read the problem carefully, use the correct formulas and methods, check all computations, and doublecheck the answer for accuracy.
Improving Mathematical ProblemSolving Skills
There are several ways to improve mathematical problemsolving skills, including practicing regularly, working with others, seeking help from a teacher or tutor, and reviewing past problems. It is also helpful to develop a positive attitude towards problemsolving, persevere through challenges, and learn from mistakes.
Must Know: WHICH IS THE BEST WAY OF LEARNING ONLINE TUTORING OR TRADITIONAL TUTORING
Mathematical problemsolving is a crucial skill that is required for success in many academic and professional fields. By following the process of problemsolving and using the techniques and strategies outlined in this article, individuals can improve their problemsolving skills and achieve success in their academic and professional endeavors.
Frequently Asked Questions
What is mathematical problemsolving.
Mathematical problemsolving is the process of using logical reasoning and critical thinking to find a solution to a mathematical problem.
Why are mathematical problemsolving skills important?
What are the steps involved in the process of mathematical problemsolving, how can online calculators aid in learning mathematics.
Online calculators can aid in learning mathematics by providing convenience, accuracy, and efficiency. They can also help students grasp mathematical concepts better through builtin visualizations and provide feedback to identify and rectify errors in their calculations.
What are common errors to avoid in mathematical problemsolving?
Common errors to avoid in mathematical problemsolving include misunderstanding the problem, using incorrect formulas or methods, making computational errors, and not checking the answer. To avoid these errors, it is essential to read the problem carefully, use the correct formulas and methods, check all computations, and doublecheck the answer for accuracy.
We are committed to help students by one on one online private tutoring to maximize their elearning potential and achieve the best results they can.
For this, we offer a free of cost trial class so that we can satisfy you. There is a free trial class for firsttime students.
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4.9: Strategies for Solving Applications and Equations
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Learning Objectives
By the end of this section, you will be able to:
 Use a problem solving strategy for word problems
 Solve number word problems
 Solve percent applications
 Solve simple interest applications
Before you get started, take this readiness quiz.
 Translate “six less than twice x ” into an algebraic expression. If you missed this problem, review [link] .
 Convert 4.5% to a decimal. If you missed this problem, review [link] .
 Convert 0.6 to a percent. If you missed this problem, review [link] .
Have you ever had any negative experiences in the past with word problems? When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. Realize that your negative experiences with word problems are in your past. To move forward you need to calm your fears and change your negative feelings.
Start with a fresh slate and begin to think positive thoughts. Repeating some of the following statements may be helpful to turn your thoughts positive. Thinking positive thoughts is a first step towards success.
 I think I can! I think I can!
 While word problems were hard in the past, I think I can try them now.
 I am better prepared now—I think I will begin to understand word problems.
 I am able to solve equations because I practiced many problems and I got help when I needed it—I can try that with word problems.
 It may take time, but I can begin to solve word problems.
 You are now well prepared and you are ready to succeed. If you take control and believe you can be successful, you will be able to master word problems.
Use a Problem Solving Strategy for Word Problems
Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.
EXAMPLE \(\PageIndex{1}\)
Normal yearly snowfall at the local ski resort is 12 inches more than twice the amount it received last season. The normal yearly snowfall is 62 inches. What was the snowfall last season at the ski resort?
the problem.  
what you are looking for.  What was the snowfall last season? 
what we are looking for and choose a variable to represent it.  Let \(s=\) the snowfall last season. 
Restate the problem in one sentence with all the important information.  
Translate into an equation.  
the equation.  
Subtract 12 from each side.  
Simplify.  
Divide each side by two.  
Simplify.  
First, is our answer reasonable? Yes, having 25 inches of snow seems OK. The problem says the normal snowfall is twelve inches more than twice the number of last season. Twice 25 is 50 and 12 more than that is 62.  
the question.  The snowfall last season was 25 inches. 
EXAMPLE \(\PageIndex{2}\)
Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than twice the number of notebooks. He bought seven textbooks. How many notebooks did he buy?
He bought two notebooks
EXAMPLE \(\PageIndex{3}\)
Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?
He did seven crosswords puzzles
We summarize an effective strategy for problem solving.
PROBLEM SOLVING STRATEGY FOR WORD PROBLEMS
 Read the problem. Make sure all the words and ideas are understood.
 Identify what you are looking for.
 Name what you are looking for. Choose a variable to represent that quantity.
 Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
 Solve the equation using proper algebra techniques.
 Check the answer in the problem to make sure it makes sense.
 Answer the question with a complete sentence.
Solve Number Word Problems
We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.
EXAMPLE \(\PageIndex{4}\)
The sum of seven times a number and eight is thirtysix. Find the number.
the problem.  

what you are looking for.  the number 
what you are looking for and choose a variable to represent it.  Let = the number. 
Restate the problem as one sentence. Translate into an equation. 

the equation. Subtract eight from each side and simplify. Divide each side by seven and simplify. 

Is the sum of seven times four plus eight equal to 36? \[\begin{align} 7·4+8 & \stackrel{?}{=}36 \\ 28+8 & \stackrel{?}{=}36 \\ 36 & =36✓ \end{align}\]  
the question.  The number is 4. 
Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.
EXAMPLE \(\PageIndex{5}\)
The sum of four times a number and two is fourteen. Find the number.
EXAMPLE \(\PageIndex{6}\)
The sum of three times a number and seven is twentyfive. Find the number.
Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far, we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.
EXAMPLE \(\PageIndex{7}\)
The sum of two numbers is negative fifteen. One number is nine less than the other. Find the numbers.
the problem.  
what you are looking for.  two numbers 
what you are looking for by choosing a variable to represent the first number. “One number is nine less than the other.”  Let \(n=1^{\text{st}}\) number. \(n−9=2^{\text{nd}}\) number 
Write as one sentence. Translate into an equation.  The sum of two numbers is negative fifteen.

the equation. Combine like terms. Add nine to each side and simplify. Simplify. 

Is \(−12\) nine less than \(−3\)? \[\begin{align}−3−9 & \stackrel{?}{=}−12 \\ −12 & =−12✓ \end{align}\] Is their sum \(−15?\) \[\begin{align} −3+(−12) & \stackrel{?}{=}−15 \\ −15 & =−15✓ \end{align}\]  
the question.  The numbers are \(−3\) and \(−12\). 
EXAMPLE \(\PageIndex{8}\)
The sum of two numbers is negative twentythree. One number is seven less than the other. Find the numbers.
\(−15,−8\)
EXAMPLE \(\PageIndex{9}\)
The sum of two numbers is negative eighteen. One number is forty more than the other. Find the numbers.
\(−29,11\)
Consecutive Integers (optional)
Some number problems involve consecutive integers . Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:
\[\begin{array}{rrrr} 1, & 2, & 3, & 4 \\ −10, & −9, & −8, & −7\\ 150, & 151, & 152, & 153 \end{array}\]
Notice that each number is one more than the number preceding it. Therefore, if we define the first integer as n , the next consecutive integer is \(n+1\). The one after that is one more than \(n+1\), so it is \(n+1+1\), which is \(n+2\).
\[\begin{array}{ll} n & 1^{\text{st}} \text{integer} \\ n+1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; & 2^{\text{nd}}\text{consecutive integer} \\ n+2 & 3^{\text{rd}}\text{consecutive integer} \;\;\;\;\;\;\;\; \text{etc.} \end{array}\]
We will use this notation to represent consecutive integers in the next example.
EXAMPLE \(\PageIndex{10}\)
Find three consecutive integers whose sum is \(−54\).
the problem.  
what you are looking for.  three consecutive integers 
each of the three numbers  Let \(n=1^{\text{st}} \text{integer}\). \(n+1=2^{\text{nd}} \text{consecutive integer}\) \(n+2=3^{\text{rd}} \text{consecutive integer}\) 
Restate as one sentence. Translate into an equation.  The sum of the three integers is \(−54\).

the equation. Combine like terms. Subtract three from each side. Divide each side by three.  
\(\begin{align} −19+(−18)+(−17) & =−54 \\ −54 & =−54✓ \end{align}\)  
the question.  The three consecutive integers are −17,−18, and −19. 
EXAMPLE \(\PageIndex{11}\)
Find three consecutive integers whose sum is \(−96\).
\(−33,−32,−31\)
EXAMPLE \(\PageIndex{12}\)
Find three consecutive integers whose sum is \(−36\).
\(−13,−12,−11\)
Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers . Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:
\[24, 26, 28\]
\[−12,−10,−8\]
Notice each integer is two more than the number preceding it. If we call the first one n , then the next one is \(n+2\). The one after that would be \(n+2+2\) or \(n+4\).
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 63, 65, and 67.
\[63, 65, 67\]
\[n,n+2,n+4\]
Does it seem strange to have to add two (an even number) to get the next odd number? Do we get an odd number or an even number when we add 2 to 3? to 11? to 47?
Whether the problem asks for consecutive even numbers or odd numbers, you do not have to do anything different. The pattern is still the same—to get to the next odd or the next even integer, add two.
EXAMPLE \(\PageIndex{13}\)
Find three consecutive even integers whose sum is \(120\).
the problem.  
what you are looking for.  three consecutive even integers 
each of the three numbers  Let \(n = 1^{\text{st}} \text{consecutive even integer}\). \(n + 2 = 2^{\text{nd}} \text{consecutive even integer}\). \(n + 4 = 3^{\text{rd}} \text{consecutive even integer}\). 
Restate as one sentence. Translate into an equation.  The sum of the three even integers is 120 \(n + n + 2 + n + 4 = 120\) 
the equation. Combine like terms. Subtract three from each side. Divide each side by three.  \(n + n + 2 + n + 4 = 120\) \(\begin{aligned} &{3n+6=120} \\ &{3n=114} \\ &{n=38} &{1^\text{st} \text{integer}}\end{aligned}\) \(\begin{aligned} &{n+2} & &{2^\text{nd} \text{integer}}\\ &{38+2} \\ &{40} \end{aligned}\) \(\begin{aligned} &{n+2} & &{3^\text{rd} \text{integer}}\\ &{38+4} \\ &{42} \end{aligned}\) 
\(\begin{align} 38 + 40 + 42 &\overset{?}{=} &120 \nonumber\\ 120 &=& 120 &✓ \nonumber\end{align}\)  
the question.  The three consecutive integers are 38, 40, and 42. 
EXAMPLE \(\PageIndex{14}\)
Find three consecutive even integers whose sum is 102.
\(32, 34, 36\)
EXAMPLE \(\PageIndex{15}\)
Find three consecutive even integers whose sum is \(−24\).
\(−10,−8,−6\)
When a number problem is in a real life context, we still use the same strategies that we used for the previous examples.
EXAMPLE \(\PageIndex{16}\)
A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?
the problem.  
what you are looking for.  How much does the husband earn? 
each of the three numbers  Let \(h=\text{the amount the husband earns}\). 
Restate the problem in one sentence with all the important information. Translate into an equation.  \(2h−16,000=\text{the amount the wife earns}.\) Together the husband and wife earn $110,000. \(h+2h−16,000=110,000\) 
the equation. Combine like terms. Add 16,000 to both sides and simplify. Divide each side by three.  \(h+2h−16,000=110,000\) \(\begin{aligned} &{3h−16,000=110,000} \\ &{3h=126,000} \\ &{h=42,000} &{\text{amount husband earns}} \end{aligned}\) \(\begin{aligned} &{2h−16,000} &{\text{ amount wife earns}} \\ &{2(42,000)−16,000} \\ &{84,000−16,000} \\ &{68,000} \end{aligned}\) 
If the wife earns $68,000 and the husband earns $42,000, is that $110,000? Yes!  
the question.  The husband earns $42,000 a year. 
According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,400. This was $1,600 less than six times the cost in 1975. What was the average cost of a car in 1975?
The average cost was $5,000.
EXAMPLE \(\PageIndex{18}\)
US Census data shows that the median price of new home in the U.S. in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?
The median price was $19,300.
Solve Percent Applications
There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.
EXAMPLE \(\PageIndex{19}\)
Translate and solve:
 What number is 45% of 84?
 8.5% of what amount is $4.76?
 168 is what percent of 112?
Translate into algebra. Let the number.  
Multiply.  
37.8 is 45% of 84. 
Translate. Let the amount.  
Multiply.  
Divide both sides by 0.085 and simplify.  
8.5% of $56 is $4.76 
We are asked to find percent, so we must have our result in percent form.  
Translate into algebra. Let = the percent.  
Multiply.  
Divide both sides by 112 and simplify.  
Convert to percent.  
168 is 150% of 112. 
 What number is 45% of 80?
 7.5% of what amount is $1.95?
 110 is what percent of 88?
ⓐ 36 ⓑ $26 ⓒ \(125 \% \)
EXAMPLE \(\PageIndex{21}\)
 What number is 55% of 60?
 8.5% of what amount is $3.06?
 126 is what percent of 72?
ⓐ 33 ⓑ $36 ⓐ \(175 \% \)
Now that we have a problem solving strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we will solve involve everyday situations, you can rely on your own experience.
EXAMPLE \(\PageIndex{22}\)
The label on Audrey’s yogurt said that one serving provided 12 grams of protein, which is 24% of the recommended daily amount. What is the total recommended daily amount of protein?
What are you asked to find?  What total amount of protein is recommended? 
Choose a variable to represent it.  Let a=a= total amount of protein. 
Write a sentence that gives the information to find it.  
Translate into an equation.  
Solve.  
Check: Does this make sense? Yes, 24% is about \(\frac{1}{4}\) of the total and 12 is about \(\frac{1}{4}\) of 50.  
Write a complete sentence to answer the question.  The amount of protein that is recommended is 50 g. 
EXAMPLE \(\PageIndex{23}\)
One serving of wheat square cereal has 7 grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber?
EXAMPLE \(\PageIndex{24}\)
One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?
Remember to put the answer in the form requested. In the next example we are looking for the percent.
EXAMPLE \(\PageIndex{25}\)
Veronica is planning to make muffins from a mix. The package says each muffin will be 240 calories and 60 calories will be from fat. What percent of the total calories is from fat?
What are you asked to find?  What percent of the total calories is fat? 
Choose a variable to represent it.  Let p=percent of fat. 
Write a sentence that gives the information to find it.  
Translate the sentence into an equation.  
Multiply.  
Divide both sides by 240.  
Put in percent form.  
Check: does this make sense? Yes, \(25 \% \) is onefourth; 60 is onefourth of 240. So, \(25 \%\) makes sense.  
Write a complete sentence to answer the question.  Of the total calories in each muffin, \(25 \%\) is fat. 
EXAMPLE \(\PageIndex{26}\)
Mitzi received some gourmet brownies as a gift. The wrapper said each 28% brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat? Round the answer to the nearest whole percent.
EXAMPLE \(\PageIndex{27}\)
The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat? Round the answer to the nearest whole percent.
It is often important in many fields—business, sciences, pop culture—to talk about how much an amount has increased or decreased over a certain period of time. This increase or decrease is generally expressed as a percent and called the percent change .
To find the percent change, first we find the amount of change, by finding the difference of the new amount and the original amount. Then we find what percent the amount of change is of the original amount.
FIND PERCENT CHANGE
\[\text{change}= \text{new amount}−\text{original amount}\]
change is what percent of the original amount?
EXAMPLE \(\PageIndex{28}\)
Recently, the California governor proposed raising community college fees from $36 a unit to $46 a unit. Find the percent change. (Round to the nearest tenth of a percent.)
Find the amount of change.  \(46−36=10\) 
Find the percent.  Change is what percent of the original amount? 
Let p=p= the percent.  
Translate to an equation.  
Simplify.  \(10=36 p\) 
Divide both sides by 36.  \(0.278 \approx p\) 
Change to percent form; round to the nearest tenth  \(27.8 \% \approx p\) 
Write a complete sentence to answer the question.  The new fees are approximately a \(27.8 \% \) increase over the old fees. 
Remember to round the division to the nearest thousandth in order to round the percent to the nearest tenth. 
EXAMPLE \(\PageIndex{29}\)
Find the percent change. (Round to the nearest tenth of a percent.) In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents.
\(8.8 \% \)
EXAMPLE \(\PageIndex{30}\)
Find the percent change. (Round to the nearest tenth of a percent.) In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was 2.25.
Applications of discount and markup are very common in retail settings.
When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate , usually given as a percent, is used to determine the amount of the discount . To determine the amount of discount, we multiply the discount rate by the original price.
The price a retailer pays for an item is called the original cost . The retailer then adds a markup to the original cost to get the list price , the price he sells the item for. The markup is usually calculated as a percent of the original cost. To determine the amount of markup, multiply the markup rate by the original cost.
\[ \begin{align} \text{amount of discount} &= \text{discount rate}· \text{original price} \\ \text{sale price} &= \text{original amount}– \text{discount price} \end{align}\]
The sale price should always be less than the original price.
\[\begin{align} \text{amount of markup} &= \text{markup rate}·\text{original price} \\ \text{list price} &= \text{original cost}–\text{markup} \end{align}\]
The list price should always be more than the original cost.
EXAMPLE \(\PageIndex{31}\)
Liam’s art gallery bought a painting at an original cost of $750. Liam marked the price up 40%. Find
 the amount of markup and
 the list price of the painting.
Identify what you are asked to find, and choose a variable to represent it.  What is the amount of markup? Let m=m= the amount of markup. 
Write a sentence that gives the information to find it.  
Translate into an equation.  
Solve the equation.  
Write a complete sentence.  The markup on the painting was $300. 
Identify what you are asked to find, and choose a variable to represent it.  What is the list price? Let \(p=\) the list price. 
Write a sentence that gives the information to find it.  
Translate into an equation.  
Solve the equation.  
Check.  Is the list price more than the original cost? Is $1,050 more than $750? Yes. 
Write a complete sentence.  The list price of the painting was $1,050. 
EXAMPLE \(\PageIndex{32}\)
Find ⓐ the amount of markup and ⓑ the list price: Jim’s music store bought a guitar at original cost $1,200. Jim marked the price up 50%.
ⓐ $600 ⓑ $1,800
EXAMPLE \(\PageIndex{33}\)
Find ⓐ the amount of markup and ⓑ the list price: The Auto Resale Store bought Pablo’s Toyota for $8,500. They marked the price up 35%.
ⓐ $2,975 ⓑ $11,475
Solve Simple Interest Applications
Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.
The amount of money you initially deposit into a bank is called the principal , P , and the bank pays you interest, I. When you take out a loan, you pay interest on the amount you borrow, also called the principal.
In either case, the interest is computed as a certain percent of the principal, called the rate of interest , r . The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable t , (for time) represents the number of years the money is saved or borrowed.
Interest is calculated as simple interest or compound interest. Here we will use simple interest.
SIMPLE INTEREST
If an amount of money, P , called the principal, is invested or borrowed for a period of t years at an annual interest rate r , the amount of interest, I , earned or paid is
\[ \begin{array}{ll} I=Prt \; \; \; \; \; \; \; \; \; \; \; \; \text{where} & { \begin{align} I &= \text{interest} \\ P &= \text{principal} \\ r &= \text{rate} \\ t &= \text{time} \end{align}} \end{array}\]
Interest earned or paid according to this formula is called simple interest .
The formula we use to calculate interest is \(I=Prt\). To use the formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information in a chart.
EXAMPLE \(\PageIndex{34}\)
Areli invested a principal of $950 in her bank account that earned simple interest at an interest rate of 3%. How much interest did she earn in five years?
\( \begin{aligned} I & = \; ? \\ P & = \; \$ 950 \\ r & = \; 3 \% \\ t & = \; 5 \text{ years} \end{aligned}\)
\(\begin{array}{ll} \text{Identify what you are asked to find, and choose a} & \text{What is the simple interest?} \\ \text{variable to represent it.} & \text{Let } I= \text{interest.} \\ \text{Write the formula.} & I=Prt \\ \text{Substitute in the given information.} & I=(950)(0.03)(5) \\ \text{Simplify.} & I=142.5 \\ \text{Check.} \\ \text{Is } \$142.50 \text{ a reasonable amount of interest on } \$ \text{ 950?} \; \;\;\;\;\; \;\;\;\;\;\; \\ \text{Yes.} \\ \text{Write a complete sentence.} & \text{The interest is } \$ \text{142.50.} \end{array}\)
EXAMPLE \(\PageIndex{35}\)
Nathaly deposited $12,500 in her bank account where it will earn 4% simple interest. How much interest will Nathaly earn in five years?
He will earn $2,500.
EXAMPLE \(\PageIndex{36}\)
Susana invested a principal of $36,000 in her bank account that earned simple interest at an interest rate of 6.5%.6.5%. How much interest did she earn in three years?
She earned $7,020.
There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we do not know the rate.
EXAMPLE \(\PageIndex{37}\)
Hang borrowed $7,500 from her parents to pay her tuition. In five years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of simple interest?
\( \begin{aligned} I & = \; \$ 1500 \\ P & = \; \$ 7500 \\ r & = \; ? \\ t & = \; 5 \text{ years} \end{aligned}\)
Identify what you are asked to find, and choose What is the rate of simple interest? a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Change to percent form. Let r = rate of interest. I = P r t 1,500 = ( 7,500 ) r ( 5 ) 1,500 = 37,500 r 0.04 = r 4 % = r Check. I = P r t 1,500 = ? ( 7,500 ) ( 0.04 ) ( 5 ) 1,500 = 1,500 ✓ Write a complete sentence. The rate of interest was 4%. Identify what you are asked to find, and choose What is the rate of simple interest? a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Change to percent form. Let r = rate of interest. I = P r t 1 ,500 = ( 7,500 ) r ( 5 ) 1,500 = 37,500 r 0.04 = r 4 % = r Check. I = P r t 1 ,500 = ? ( 7,500 ) ( 0.04 ) ( 5 ) 1,500 = 1, 500 ✓ Write a complete sentence. The rate of interest was 4%.
EXAMPLE \(\PageIndex{38}\)
Jim lent his sister $5,000 to help her buy a house. In three years, she paid him the $5,000, plus $900 interest. What was the rate of simple interest?
The rate of simple interest was 6%.
EXAMPLE \(\PageIndex{39}\)
Loren lent his brother $3,000 to help him buy a car. In four years, his brother paid him back the $3,000 plus $660 in interest. What was the rate of simple interest?
The rate of simple interest was 5.5%.
In the next example, we are asked to find the principal—the amount borrowed.
EXAMPLE \(\PageIndex{40}\)
Sean’s new car loan statement said he would pay $4,866,25 in interest from a simple interest rate of 8.5% over five years. How much did he borrow to buy his new car?
\( \begin{aligned} I & = \; 4,866.25 \\ P & = \; ? \\ r & = \; 8.5 \% \\ t & = \; 5 \text{ years} \end{aligned}\)
Identify what you are asked to find, What is the amount borrowed (the principal)? and choose a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Let P = principal borrowed. I = P r t 4,866.25 = P ( 0.085 ) ( 5 ) 4,866.25 = 0.425 P 11,450 = P Check. I = P r t 4,866.25 = ? ( 11,450 ) ( 0.085 ) ( 5 ) 4,866.25 = 4,866.25 ✓ Write a complete sentence. The principal was $11,450. Identify what you are asked to find, What is the amount borrowed (the principal)? and choose a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Let P = principal borrowed. I = P r t 4 ,866.25 = P ( 0.085 ) ( 5 ) 4,866.25 = 0.425 P 11,450 = P Check. I = P r t 4 ,866.25 = ? ( 11,450 ) ( 0.085 ) ( 5 ) 4,866.25 = 4,866.25 ✓ Write a complete sentence. The principal was $11,450.
EXAMPLE \(\PageIndex{41}\)
Eduardo noticed that his new car loan papers stated that with a 7.5% simple interest rate, he would pay $6,596.25 in interest over five years. How much did he borrow to pay for his car?
He paid $17,590.
EXAMPLE \(\PageIndex{42}\)
In five years, Gloria’s bank account earned $2,400 interest at 5% simple interest. How much had she deposited in the account?
She deposited $9,600.
Access this online resource for additional instruction and practice with using a problem solving strategy.
 Begining Arithmetic Problems
Key Concepts
\(\text{change}=\text{new amount}−\text{original amount}\)
\(\text{change is what percent of the original amount?}\)
 \( \begin{align} \text{amount of discount} &= \text{discount rate}· \text{original price} \\ \text{sale price} &= \text{original amount}– \text{discount price} \end{align}\)
 \(\begin{align} \text{amount of markup} &= \text{markup rate}·\text{original price} \\ \text{list price} &= \text{original cost}–\text{markup} \end{align}\)
 If an amount of money, P , called the principal, is invested or borrowed for a period of t years at an annual interest rate r , the amount of interest, I , earned or paid is: \[\begin{aligned} &{} &{} &{I=interest} \nonumber\\ &{I=Prt} &{\text{where} \space} &{P=principal} \nonumber\\ &{} &{\space} &{r=rate} \nonumber\\ &{} &{\space} &{t=time} \nonumber \end{aligned}\]
Practice Makes Perfect
1. List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Answers will vary.
2. List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.
In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.
3. There are \(16\) girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.
4. There are \(18\) Cub Scouts in Troop 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.
5. Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is \(12\) less than three times the number of hardbacks. Huong had \(162\) paperbacks. How many hardback books were there?
58 hardback books
6. Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are \(42\) adult bicycles. How many children’s bicycles are there?
In the following exercises, solve each number word problem.
7. The difference of a number and \(12\) is three. Find the number.
8. The difference of a number and eight is four. Find the number.
9. The sum of three times a number and eight is \(23\). Find the number.
10. The sum of twice a number and six is \(14\). Find the number.
11 . The difference of twice a number and seven is \(17\). Find the number.
12. The difference of four times a number and seven is \(21\). Find the number.
13. Three times the sum of a number and nine is \(12\). Find the number.
14. Six times the sum of a number and eight is \(30\). Find the number.
15. One number is six more than the other. Their sum is \(42\). Find the numbers.
\(18, \;24\)
16. One number is five more than the other. Their sum is \(33\). Find the numbers.
17. The sum of two numbers is \(20\). One number is four less than the other. Find the numbers.
\(8, \;12\)
18 . The sum of two numbers is \(27\). One number is seven less than the other. Find the numbers.
19. One number is \(14\) less than another. If their sum is increased by seven, the result is \(85\). Find the numbers.
\(32,\; 46\)
20 . One number is \(11\) less than another. If their sum is increased by eight, the result is \(71\). Find the numbers.
21. The sum of two numbers is \(14\). One number is two less than three times the other. Find the numbers.
\(4,\; 10\)
22. The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.
23. The sum of two consecutive integers is \(77\). Find the integers.
\(38,\; 39\)
24. The sum of two consecutive integers is \(89\). Find the integers.
25. The sum of three consecutive integers is \(78\). Find the integers.
\(25,\; 26,\; 27\)
26. The sum of three consecutive integers is \(60\). Find the integers.
27. Find three consecutive integers whose sum is \(−36\).
\(−11,\;−12,\;−13\)
28. Find three consecutive integers whose sum is \(−3\).
29. Find three consecutive even integers whose sum is \(258\).
\(84,\; 86,\; 88\)
30. Find three consecutive even integers whose sum is \(222\).
31. Find three consecutive odd integers whose sum is \(−213\).
\(−69,\;−71,\;−73\)
32. Find three consecutive odd integers whose sum is \(−267\).
33. Philip pays \($1,620\) in rent every month. This amount is \($120\) more than twice what his brother Paul pays for rent. How much does Paul pay for rent?
34. Marc just bought an SUV for \($54,000\). This is \($7,400\) less than twice what his wife paid for her car last year. How much did his wife pay for her car?
35. Laurie has \($46,000\) invested in stocks and bonds. The amount invested in stocks is \($8,000\) less than three times the amount invested in bonds. How much does Laurie have invested in bonds?
\($13,500\)
36. Erica earned a total of \($50,450\) last year from her two jobs. The amount she earned from her job at the store was \($1,250\) more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?
In the following exercises, translate and solve.
37. a. What number is 45% of 120? b. 81 is 75% of what number? c. What percent of 260 is 78?
a. 54 b. 108 c. 30%
38. a. What number is 65% of 100? b. 93 is 75% of what number? c. What percent of 215 is 86?
39. a. 250% of 65 is what number? b. 8.2% of what amount is $2.87? c. 30 is what percent of 20?
a. 162.5 b. $35 c. 150%
40. a. 150% of 90 is what number? b. 6.4% of what amount is $2.88? c. 50 is what percent of 40?
In the following exercises, solve.
41. Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16% of the total bill as a tip. How much should the tip be?
42. When Hiro and his coworkers had lunch at a restaurant near their work, the bill was $90.50. They want to leave 18% of the total bill as a tip. How much should the tip be?
43. One serving of oatmeal has 8 grams of fiber, which is 33% of the recommended daily amount. What is the total recommended daily amount of fiber?
44. One serving of trail mix has 67 grams of carbohydrates, which is 22% of the recommended daily amount. What is the total recommended daily amount of carbohydrates?
45. A bacon cheeseburger at a popular fast food restaurant contains 2070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?
46. A grilled chicken salad at a popular fast food restaurant contains 650 milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount of sodium?
47. The nutrition fact sheet at a fast food restaurant says the fish sandwich has 380 calories, and 171 calories are from fat. What percent of the total calories is from fat?
48. The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has 190 calories, and 114 calories are from fat. What percent of the total calories is from fat?
49. Emma gets paid $3,000 per month. She pays $750 a month for rent. What percent of her monthly pay goes to rent?
50. Dimple gets paid $3,200 per month. She pays $960 a month for rent. What percent of her monthly pay goes to rent?
51. Tamanika received a raise in her hourly pay, from $15.50 to $17.36. Find the percent change.
52. Ayodele received a raise in her hourly pay, from $24.50 to $25.48. Find the percent change.
53. Annual student fees at the University of California rose from about $4,000 in 2000 to about $12,000 in 2010. Find the percent change.
54. The price of a share of one stock rose from $12.50 to $50. Find the percent change.
55. A grocery store reduced the price of a loaf of bread from $2.80 to $2.73. Find the percent change.
−2.5%
56. The price of a share of one stock fell from $8.75 to $8.54. Find the percent change.
57. Hernando’s salary was $49,500 last year. This year his salary was cut to $44,055. Find the percent change.
58. In ten years, the population of Detroit fell from 950,000 to about 712,500. Find the percent change.
In the following exercises, find a. the amount of discount and b. the sale price.
59. Janelle bought a beach chair on sale at 60% off. The original price was $44.95.
a. $26.97 b. $17.98
60. Errol bought a skateboard helmet on sale at 40% off. The original price was $49.95.
In the following exercises, find a. the amount of discount and b. the discount rate (Round to the nearest tenth of a percent if needed.)
61. Larry and Donna bought a sofa at the sale price of $1,344. The original price of the sofa was $1,920.
a. $576 b. 30%
62. Hiroshi bought a lawnmower at the sale price of $240. The original price of the lawnmower is $300.
In the following exercises, find a. the amount of the markup and b. the list price.
63. Daria bought a bracelet at original cost $16 to sell in her handicraft store. She marked the price up 45%. What was the list price of the bracelet?
a. $7.20 b. $23.20
64. Regina bought a handmade quilt at original cost $120 to sell in her quilt store. She marked the price up 55%. What was the list price of the quilt?
65. Tom paid $0.60 a pound for tomatoes to sell at his produce store. He added a 33% markup. What price did he charge his customers for the tomatoes?
a. $0.20 b. $0.80
66. Flora paid her supplier $0.74 a stem for roses to sell at her flower shop. She added an 85% markup. What price did she charge her customers for the roses?
67. Casey deposited $1,450 in a bank account that earned simple interest at an interest rate of 4%. How much interest was earned in two years?
68 . Terrence deposited $5,720 in a bank account that earned simple interest at an interest rate of 6%. How much interest was earned in four years?
69. Robin deposited $31,000 in a bank account that earned simple interest at an interest rate of 5.2%. How much interest was earned in three years?
70. Carleen deposited $16,400 in a bank account that earned simple interest at an interest rate of 3.9% How much interest was earned in eight years?
71. Hilaria borrowed $8,000 from her grandfather to pay for college. Five years later, she paid him back the $8,000, plus $1,200 interest. What was the rate of simple interest?
72. Kenneth lent his niece $1,200 to buy a computer. Two years later, she paid him back the $1,200, plus $96 interest. What was the rate of simple interest?
73. Lebron lent his daughter $20,000 to help her buy a condominium. When she sold the condominium four years later, she paid him the $20,000, plus $3,000 interest. What was the rate of simple interest?
74. Pablo borrowed $50,000 to start a business. Three years later, he repaid the $50,000, plus $9,375 interest. What was the rate of simple interest?
75. In 10 years, a bank account that paid 5.25% simple interest earned $18,375 interest. What was the principal of the account?
76. In 25 years, a bond that paid 4.75% simple interest earned $2,375 interest. What was the principal of the bond?
77. Joshua’s computer loan statement said he would pay $1,244.34 in simple interest for a threeyear loan at 12.4%. How much did Joshua borrow to buy the computer?
78. Margaret’s car loan statement said she would pay $7,683.20 in simple interest for a fiveyear loan at 9.8%. How much did Margaret borrow to buy the car?
Everyday Math
79 . Tipping At the campus coffee cart, a medium coffee costs $1.65. MaryAnne brings $2.00 with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?
80 . Tipping Four friends went out to lunch and the bill came to $53.75 They decided to add enough tip to make a total of $64, so that they could easily split the bill evenly among themselves. What percent tip did they leave?
Writing Exercises
81. What has been your past experience solving word problems? Where do you see yourself moving forward?
82. Without solving the problem “44 is 80% of what number” think about what the solution might be. Should it be a number that is greater than 44 or less than 44? Explain your reasoning.
83. After returning from vacation, Alex said he should have packed 50% fewer shorts and 200% more shirts. Explain what Alex meant.
84. Because of road construction in one city, commuters were advised to plan that their Monday morning commute would take 150% of their usual commuting time. Explain what this means.
a. After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.
b. After reviewing this checklist, what will you do to become confident for all objectives?
COMMENTS
Here are five strategies to help students check their solutions. 1. Use the Inverse Operation. For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7.
Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help!
This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. ... Make sure you use Polya's 4 problemsolving steps. Problem Solving Strategy 2 (Using a variable to find the sum of a sequence.) Gauss's strategy for sequences ...
The very first Mathematical Practice is: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of ...
Therefore, highquality assessment of problem solving in public tests and assessments1 is essential in order to ensure the effective learning and teaching of problem solving throughout primary and secondary education. Although the focus here is on the assessment of problem solving in mathematics, many of the ideas will be directly transferable ...
Problemsolving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.
A Guide to Problem Solving. When confronted with a problem, in which the solution is not clear, you need to be a skilled problemsolver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problemsolving skill is out of your reach, but like any skill, you can improve your problemsolving with practice.
In the past, we would teach the concepts and procedures and then assign onestep "story" problems designed to provide practice on the content. Next, we would teach problem solving as a collection of strategies such as "draw a picture" or "guess and check.". Eventually, students would be given problems to apply the skills and strategies.
1.1 Role of Heuristics for Problem Solving—Regina Bruder. The origin of the word heuristic dates back to the time of Archimedes and is said to have come out of one of the famous stories told about this great mathematician and inventor. The King of Syracuse asked Archimedes to check whether his new wreath was really made of pure gold. Archimedes struggled with this task and it was not until ...
The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition. Students may use this strategy to spot patterns and fill in the blanks.
Schema approach. This is a math intervention strategy that can make problem solving easier for all students, regardless of ability. Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:
Teaching about problem solving begins with suggested strategies to solve a problem. For example, "draw a picture," "make a table," etc. You may see posters in teachers' classrooms of the "Problem Solving Method" such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no ...
Consider the problemsolving steps applied in the following example. I know that I want to say "I don't eat eggs" to my Mexican waiter. That's the problem. I don't know how to say that, but last night I told my date "No bebo alcohol" ("I don't drink alcohol"). I also know the infinitive for "eat" in Spanish (comer).
Step 1: Understanding the problem. We are given in the problem that there are 25 chickens and cows. All together there are 76 feet. Chickens have 2 feet and cows have 4 feet. We are trying to determine how many cows and how many chickens Mr. Jones has on his farm. Step 2: Devise a plan.
1. Link problemsolving to reading. When we can remind students that they already have many comprehension skills and strategies they can easily use in math problemsolving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools ...
Polya's (1957) fourstep process has provided a model for the teaching and assessing. problem solving in mathematics classrooms: understanding the problem, devising a plan, carrying out the plan, and looking back. Other educators have adapted these steps, but the. essence of these adaptations is very similar to what Polya initially developed.
Another common strategy for teaching problemsolving is the use of acrostics that students can easily remember to perform the "steps" in problemsolving. CUBES is an example. Just as with keywords, however, students often follow the steps with little understanding. As an example, a common step is to underline or highlight the question.
The following formula will come in handy for solving example 6: Perimeter of a Rectangle = 2 (length) + 2 (width) Example 6 : In a blueprint of a rectangular room, the length is 1 inch more than 3 times the width. Find the dimensions if the perimeter is to be 26 inches. Step 1: Understand the problem.
1. Identify the problem. The first step in problemsolving is to identify the problem. It involves reading the problem carefully and determining what the problem is asking. 2. Understand the problem. The next step is to understand the problem by breaking it down into smaller parts, identifying any relevant information, and determining what ...
Step 1. Read the problem. Step 2. Identify what you are looking for. the number: Step 3. Name what you are looking for and. choose a variable to represent it. Let n = the number. Step 4. Translate: Restate the problem as one sentence. Translate into an equation. Step 5. Solve the equation. Subtract eight from each side and simplify.
the studies showing how strategies can improve mathematics problem solving, Koichu, Berman, and Moore (2004) aimed to promote heuristic literacy in a regular mathematics classroom. Moreover, Dewey's (1933) "How we think", Polya's (1988) problemsolving methods and the stages of Krulik and
Learn the steps you can follow to solve any math word problem.We hope you are enjoying this video! For more indepth learning, check out Miacademy.co (https:...
solving is the ability to understand issues as well as the steps involved (Mandina and Ochonogor, 2018). Understanding problemsolving is ... regarding mathematics problemsolving understanding, which