problem solving as a method of teaching ppt

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Teaching problem solving.

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Tips and Techniques

Expert vs. novice problem solvers, communicate.

• Have students  identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
• If students are unable to articulate their concerns, determine where they are having trouble by  asking them to identify the specific concepts or principles associated with the problem.
• In a one-on-one tutoring session, ask the student to  work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
• When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)

Encourage Independence

• Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
• Have students work through problems on their own. Ask directing questions or give helpful suggestions, but  provide only minimal assistance and only when needed to overcome obstacles.
• Don’t fear  group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others

Be sensitive

• Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing  positive reinforcement to let students know when they have mastered a new concept or skill.

Encourage Thoroughness and Patience

• Try to communicate that  the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.

Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills,  a teacher should be aware of principles and strategies of good problem solving in his or her discipline .

The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book  How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes  a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

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Teaching Problem-Solving Skills

Many instructors design opportunities for students to solve “problems”. But are their students solving true problems or merely participating in practice exercises? The former stresses critical thinking and decision­ making skills whereas the latter requires only the application of previously learned procedures.

Problem solving is often broadly defined as "the ability to understand the environment, identify complex problems, review related information to develop, evaluate strategies and implement solutions to build the desired outcome" (Fissore, C. et al, 2021). True problem solving is the process of applying a method – not known in advance – to a problem that is subject to a specific set of conditions and that the problem solver has not seen before, in order to obtain a satisfactory solution.

Below you will find some basic principles for teaching problem solving and one model to implement in your classroom teaching.

Principles for teaching problem solving

• Model a useful problem-solving method . Problem solving can be difficult and sometimes tedious. Show students how to be patient and persistent, and how to follow a structured method, such as Woods’ model described below. Articulate your method as you use it so students see the connections.
• Teach within a specific context . Teach problem-solving skills in the context in which they will be used by students (e.g., mole fraction calculations in a chemistry course). Use real-life problems in explanations, examples, and exams. Do not teach problem solving as an independent, abstract skill.
• Help students understand the problem . In order to solve problems, students need to define the end goal. This step is crucial to successful learning of problem-solving skills. If you succeed at helping students answer the questions “what?” and “why?”, finding the answer to “how?” will be easier.
• Take enough time . When planning a lecture/tutorial, budget enough time for: understanding the problem and defining the goal (both individually and as a class); dealing with questions from you and your students; making, finding, and fixing mistakes; and solving entire problems in a single session.
• Ask questions and make suggestions . Ask students to predict “what would happen if …” or explain why something happened. This will help them to develop analytical and deductive thinking skills. Also, ask questions and make suggestions about strategies to encourage students to reflect on the problem-solving strategies that they use.
• Link errors to misconceptions . Use errors as evidence of misconceptions, not carelessness or random guessing. Make an effort to isolate the misconception and correct it, then teach students to do this by themselves. We can all learn from mistakes.

Woods’ problem-solving model

Define the problem.

• The system . Have students identify the system under study (e.g., a metal bridge subject to certain forces) by interpreting the information provided in the problem statement. Drawing a diagram is a great way to do this.
• Known(s) and concepts . List what is known about the problem, and identify the knowledge needed to understand (and eventually) solve it.
• Unknown(s) . Once you have a list of knowns, identifying the unknown(s) becomes simpler. One unknown is generally the answer to the problem, but there may be other unknowns. Be sure that students understand what they are expected to find.
• Units and symbols . One key aspect in problem solving is teaching students how to select, interpret, and use units and symbols. Emphasize the use of units whenever applicable. Develop a habit of using appropriate units and symbols yourself at all times.
• Constraints . All problems have some stated or implied constraints. Teach students to look for the words "only", "must", "neglect", or "assume" to help identify the constraints.
• Criteria for success . Help students consider, from the beginning, what a logical type of answer would be. What characteristics will it possess? For example, a quantitative problem will require an answer in some form of numerical units (e.g., $/kg product, square cm, etc.) while an optimization problem requires an answer in the form of either a numerical maximum or minimum.

Think about it

• “Let it simmer”.  Use this stage to ponder the problem. Ideally, students will develop a mental image of the problem at hand during this stage.
• Identify specific pieces of knowledge . Students need to determine by themselves the required background knowledge from illustrations, examples and problems covered in the course.
• Collect information . Encourage students to collect pertinent information such as conversion factors, constants, and tables needed to solve the problem.

Plan a solution

• Consider possible strategies . Often, the type of solution will be determined by the type of problem. Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards.
• Choose the best strategy . Help students to choose the best strategy by reminding them again what they are required to find or calculate.

Carry out the plan

• Be patient . Most problems are not solved quickly or on the first attempt. In other cases, executing the solution may be the easiest step.
• Be persistent . If a plan does not work immediately, do not let students get discouraged. Encourage them to try a different strategy and keep trying.

Encourage students to reflect. Once a solution has been reached, students should ask themselves the following questions:

• Does the answer make sense?
• Does it fit with the criteria established in step 1?
• Did I answer the question(s)?
• What did I learn by doing this?
• Could I have done the problem another way?

If you would like support applying these tips to your own teaching, CTE staff members are here to help.  View the  CTE Support  page to find the most relevant staff member to contact. 

• Fissore, C., Marchisio, M., Roman, F., & Sacchet, M. (2021). Development of problem solving skills with Maple in higher education. In: Corless, R.M., Gerhard, J., Kotsireas, I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science, vol 1414. Springer, Cham. https://doi.org/10.1007/978-3-030-81698-8_15
• Foshay, R., & Kirkley, J. (1998). Principles for Teaching Problem Solving. TRO Learning Inc., Edina MN.  (PDF) Principles for Teaching Problem Solving (researchgate.net)
• Hayes, J.R. (1989). The Complete Problem Solver. 2nd Edition. Hillsdale, NJ: Lawrence Erlbaum Associates.
• Woods, D.R., Wright, J.D., Hoffman, T.W., Swartman, R.K., Doig, I.D. (1975). Teaching Problem solving Skills.
• Engineering Education. Vol 1, No. 1. p. 238. Washington, DC: The American Society for Engineering Education.

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Problem-Solving Method in Teaching

The problem-solving method is a highly effective teaching strategy that is designed to help students develop critical thinking skills and problem-solving abilities . It involves providing students with real-world problems and challenges that require them to apply their knowledge, skills, and creativity to find solutions. This method encourages active learning, promotes collaboration, and allows students to take ownership of their learning.

Table of Contents

Definition of problem-solving method.

Problem-solving is a process of identifying, analyzing, and resolving problems. The problem-solving method in teaching involves providing students with real-world problems that they must solve through collaboration and critical thinking. This method encourages students to apply their knowledge and creativity to develop solutions that are effective and practical.

Meaning of Problem-Solving Method

The meaning and Definition of problem-solving are given by different Scholars. These are-

Woodworth and Marquis(1948) : Problem-solving behavior occurs in novel or difficult situations in which a solution is not obtainable by the habitual methods of applying concepts and principles derived from past experience in very similar situations.

Skinner (1968): Problem-solving is a process of overcoming difficulties that appear to interfere with the attainment of a goal. It is the procedure of making adjustments in spite of interference

Benefits of Problem-Solving Method

The problem-solving method has several benefits for both students and teachers. These benefits include:

• Encourages active learning: The problem-solving method encourages students to actively participate in their own learning by engaging them in real-world problems that require critical thinking and collaboration
• Promotes collaboration: Problem-solving requires students to work together to find solutions. This promotes teamwork, communication, and cooperation.
• Builds critical thinking skills: The problem-solving method helps students develop critical thinking skills by providing them with opportunities to analyze and evaluate problems
• Increases motivation: When students are engaged in solving real-world problems, they are more motivated to learn and apply their knowledge.
• Enhances creativity: The problem-solving method encourages students to be creative in finding solutions to problems.

Steps in Problem-Solving Method

The problem-solving method involves several steps that teachers can use to guide their students. These steps include

• Identifying the problem: The first step in problem-solving is identifying the problem that needs to be solved. Teachers can present students with a real-world problem or challenge that requires critical thinking and collaboration.
• Analyzing the problem: Once the problem is identified, students should analyze it to determine its scope and underlying causes.
• Generating solutions: After analyzing the problem, students should generate possible solutions. This step requires creativity and critical thinking.
• Evaluating solutions: The next step is to evaluate each solution based on its effectiveness and practicality
• Selecting the best solution: The final step is to select the best solution and implement it.

Verification of the concluded solution or Hypothesis

The solution arrived at or the conclusion drawn must be further verified by utilizing it in solving various other likewise problems. In case, the derived solution helps in solving these problems, then and only then if one is free to agree with his finding regarding the solution. The verified solution may then become a useful product of his problem-solving behavior that can be utilized in solving further problems. The above steps can be utilized in solving various problems thereby fostering creative thinking ability in an individual.

The problem-solving method is an effective teaching strategy that promotes critical thinking, creativity, and collaboration. It provides students with real-world problems that require them to apply their knowledge and skills to find solutions. By using the problem-solving method, teachers can help their students develop the skills they need to succeed in school and in life.

• Jonassen, D. (2011). Learning to solve problems: A handbook for designing problem-solving learning environments. Routledge.
• Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students learn? Educational Psychology Review, 16(3), 235-266.
• Mergendoller, J. R., Maxwell, N. L., & Bellisimo, Y. (2006). The effectiveness of problem-based instruction: A comparative study of instructional methods and student characteristics. Interdisciplinary Journal of Problem-based Learning, 1(2), 49-69.
• Richey, R. C., Klein, J. D., & Tracey, M. W. (2011). The instructional design knowledge base: Theory, research, and practice. Routledge.
• Savery, J. R., & Duffy, T. M. (2001). Problem-based learning: An instructional model and its constructivist framework. CRLT Technical Report No. 16-01, University of Michigan. Wojcikowski, J. (2013). Solving real-world problems through problem-based learning. College Teaching, 61(4), 153-156

Teaching and Learning Mathematics through Problem Solving

Nov 17, 2014

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Teaching and Learning Mathematics through Problem Solving. Facilitator ’ s Handbook A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 (with reference to Volume Two). The Literacy and Numeracy Secretariat Professional Learning Series.

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Teaching and Learning Mathematics through Problem Solving Facilitator’s Handbook A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 (with reference to Volume Two) The Literacy and Numeracy Secretariat Professional Learning Series

Aims of Numeracy Professional Learning • Promote the belief that all students have learned some mathematics through their lived experiences in the world and that the mathematics classroom is one where students bring that thinking to their work • Build teachers’ expertise at setting classroom conditions where students can move from their informal math understandings to generalizations and formal representations of their mathematical thinking • Assist educators working with teachers of students in the junior division to implement student-focused instructional methods referenced in A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 to improve student achievement

Aims continued • Have teachers experience mathematical problem solving as a model of what effective math instruction entails by: • collectively solving problems relevant to students’ lives that reflect the expectations in the Ontario mathematics curriculum • viewing and discussing the thinking and strategies in the solutions • sorting and classifying the responses to a problem to provide a visual image of the range of experience and understanding of the mathematics • analysing the visual continuum of thinking to determine starting points and next steps for instruction

Overall Learning Goals for Problem Solving During this session, participants will: • become familiar with the notion of learning mathematics for teaching as a focus for numeracy professional learning • experience learning mathematics through problem solving • solve problems in different ways • develop strategies for teaching mathematics through problem solving

Effective Mathematics Teaching and Learning • Mathematics classrooms must be challenging and engaging environments for all students, where students learn significant mathematics. • Students are called to engage in solving rich and relevant problems. These problems offer several entry points so that all students can achieve, given sufficient time and support. • Lessons are structured to build on students’ prior knowledge. Agree, Disagree, Not Sure

Effective Mathematics Teaching and Learning continued • Students develop their own varied solutions to problems and thus develop a deeper understanding of the mathematics involved. • Students consolidate their knowledge through shared and independent practice. • Teachers select and/or organize students’ solutions for sharing to highlight the mathematics learning (e.g., bansho, gallery walk, math congress). • Teachers need specific mathematics knowledge and mathematics pedagogy to teach effectively. Agree, Disagree, Not Sure

What Does It Mean to Learn Mathematics for Teaching?

Deborah Loewenberg BallMathematics for Teaching • Expert personal knowledge of subject matter is often, ironically, inadequate for teaching. • It requires the capacity to deconstruct one’s own knowledge into a less polished final form where critical components are accessible and visible. • Teachers must be able to do something perverse: work backward from a mature and compressed understanding of the content to unpack its constituent elements and make mathematical ideas accessible to others. • Teachers must be able to work with content for students while it is in a growing and unfinished state.

What Do Teachers Need to Know and Be Able to Do Mathematically? • Understand the sequence and relationship between math strands within textbook programs and materials within and across grade levels • Know the relationship between mathematical ideas, conceptual models, terms, and symbols • Generate and use strategic examples and different mathematical representations using manipulatives • Develop students’ mathematical communication – description, explanation, and justification • Understand and evaluate the mathematical significance of students’ comments and coordinate discussion for mathematics learning

Why Study Problem Solving?

Why Study Problem Solving? EQAO suggests that • a significant number of Grades 3 and 6 students exhibited difficulty in understanding the demands of open-response problem-solving questions in mathematics • many Grades 3 and 6 students, when answering open-response questions in mathematics, had difficulty explaining their thinking in mathematical terms Excerpted from EQAO. (2006). Summary of Results and Strategies for Teachers: Grade 3 and 6 Assessments of Reading, Writing, and Mathematics, 2005 –2006

An Overview A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 Volume 5:Teaching Basic Facts and Multidigit Computations Volume 1: Foundations of Mathematics Instruction Volume 2: Problem Solving and Communication Volume 4: Assessment and Home Connections Volume 3: Classroom Resources and Management

1. List 2 ideas about problem solving that are familiar. 2. List 2 ideas about problem solving that are unfamiliar. 3. List 2 ideas about problem solving that are puzzling. <put in graphic of Volume 2 – Problem solving> In What Ways Does A Guide to Effective Instruction in Mathematics Describe Problem Solving? Familiar, Unfamiliar, Puzzling

Problem SolvingSession A Activating Prior Knowledge

Learning Goals of the Module Experience learning mathematics through problem solving by: • identifying what problem solving looks, sounds, and feels like • relating aspects of Polya’s problem-solving process to problem-solving experiences • experiencing ways that questioning and prompts provoke our mathematical thinking

Curriculum Connections

Warm Up – Race to Take Up Space Goal: To cover the game board with rectangles Players: 2 (individual) to 4 (teams of 2) Materials: 7x9 square tiles grid game board, 32 same- colour square tiles per player, 2 dice How to Play: Take turns rolling the dice to get 2 numbers. Multiply the 2 dice numbers to calculate the area of a rectangle (e.g., 4, 6  area = 24 square units). Construct a rectangle of the area calculated, using square tiles of the same colour. Place your rectangle on the game board. Lose a turn if the rectangle you constructed cannot be placed on the empty space on the game board. The game ends when no more rectangles can be placed on the game board. Which player is left with the most tiles?

Working on It – Carpet Problem Hello Grade 4 students, The carpet you have been asking for arrives tonight. Please clear a space in your room today that will fit this new carpet. The perimeter of the carpet is 12 m. From your principal • What is the problem to solve? • Why is this problem a problem? • Show two different ways to solve this carpet problem. • How do you know we have all the possible solutions? A Guide to Effective Instruction, Vol. 2 – Problem Solving, pp. 18–25

Read one page from the “Problem Solving Vignette” on pp. 18–25. Mathematical Processes problem solving reasoning and proving reflecting selecting tools and computational strategies connecting representing communicating Look Back – Reflect and Connect How Were the Students Solving the Problem? 1. What mathematics was evident in the students’ development of a solution to the carpet problem? 2. Describe the mathematical processes that the students were using to develop a solution. 3. Provide specific details from the vignette text to justify your description.

Look Back continued Polya’s Problem-Solving Process Understand the Problem Communicate – talk to understand the problem Make a Plan Communicate – discuss ideas with others to clarify strategies Carry Out the Plan Communicate – record your thinking using manipulatives, pictures, words, numbers, and symbols Look Back Communicate – verify, summarize/ generalize, validate, and explain Focus on the one or two pages that you read from the “Problem- Solving Vignette.” 4. What questions does the teacher ask to make the problem-solving process explicit? 5. What strategies does the teacher use to engage all the students in solving this carpet problem?

Next Steps in Our Classroom • Describe two strategies from the “Problem- Solving Vignette” that you use and two strategies that you will begin to use in your classroom to engage students in problem solving. • Keep a written record of the questions that you ask to make the problem-solving process explicit. • Practise noticing the breadth of mathematics that students use in their solutions. • During class discussions, make explicit comments about the mathematics students are showing in their solutions.

Problem SolvingSession B Modelling and Representing Area

Learning Goals of the Module Solving problems in different ways and developing strategies for teaching mathematics through problem solving in order to: • understand the range of students’ mathematical thinking (mathematical constructs) inherent in solutions developed by students in a combined Grade 4 and 5 class • develop strategies for posing questions and providing prompts to provoke a range of mathematical thinking • develop strategies for coordinating students’ mathematical thinking and communication (bansho)

Curriculum Connections Specific Expectations Gr 3 Specific Expectations Gr 4 Specific Expectations Gr 5

Curriculum Connections Specific Expectations

Warm Up – The Size of Things How do the areas of the items compare? How do you know? a five-dollar bill 1. Examine the cards in the envelope on your table. • Order the items on the cards from smallest to largest area. • How do you know that your order is accurate? 3. Compare the order of your cards with that of another group at your table. 4. Discuss any differences you observe. a credit card a cheque an envelope

Working on It – 4 Square Units Problem Show as many polygons as possible that have an area of 4 square units. a) Create your polygons on a geoboard. b) Record them on square dot paper. c) Label the polygons by the number of sides (e.g., triangle, rectangle, quadrilateral, octagon). d) Show how you know that each of your polygons are 4 square units.

a) one square unit b) two square units 2. a) What is the area of this polygon? b) How can you reason about half-square units? Working on It continued • What could a polygon look like that is a) 1 square unit? b) 2 square units?

Working on It continued 3. Show as many polygons as possible that have an area of 4 square units. a) Create your polygons on a geoboard. b) Record them on square dot paper. c) Label the polygons by the number of sides (e.g., triangle, rectangle, quadrilateral, octagon). d) Show how you know that each of your polygons is 4 square units.

Constructing a Collective Thinkpad Bansho as Assessment for Learning Organize student solutions to make explicit the mathematics inherent in this problem. Solutions that show similar mathematical thinking are arranged vertically to look like a concrete bar graph. Polygons (composed of other polygons): hexagons, and so on Squares Rectangles Polygons (composed of rectangles): octagons, hexagons, and so on Parallelograms, and so on Triangles

See Volume 2, Problem Solving and Communication (pp. 32–34). In groups of 2 or 3, share the reading and answer these questions: What is the purpose of carefully questioning and prompting students during and after problem solving? What are a few things that teachers need to keep in mind when preparing questions for a reflect-and-connect part of a lesson? 3. How should teachers use think-alouds to promote learning during a math lesson? Look Back – Reflect and Connect Questioning and Prompting Students to Share Their Mathematical Thinking

Next Steps in Our Classroom • Describe 2 strategies that you use to get students to share their mathematical thinking as they solve problems. • Describe 2 strategies that you will begin to use in your classroom to engage students in communicating their mathematical thinking. • Keep a written record of the prompts you use to unearth the mathematical ideas during problem solving. • Practise noticing the breadth of mathematics that students use in their solutions. • During class discussions, make explicit comments about the mathematics students are showing in their solutions.

Problem SolvingSession C Organizing and Coordinating Student Solutions to Problems Using Criteria

Learning Goals of the Module Develop strategies for teaching mathematics through problem solving by: • recognizing and understanding the range of mathematical thinking (e.g., concepts, strategies) in students’ solutions • organizing student solutions purposefully to make explicit the mathematics • developing strategies for coordinating students’ mathematical thinking and communication (bansho) • describing the teacher’s role in teaching through problem solving

Warm Up – Composite Shape Problem What could a composite shape look like that … • has an area of 4 square units? • is composed of 3 rectangles (Grade 5)? • is composed of at least one rectangle, one triangle, and one parallelogram (Grade 6)? 1. Draw and describe at least one composite shape that meets these criteria. 2. Explain the strategies you used to create each composite shape. 3. Justify how your composite shape meets the criteria listed in the problem. Think-Aloud

4 cm 8 cm 4 cm 6 cm Working on It – L-shaped Problem 1. What is the area of this shape? a) Show at least 2 different solutions. b) Explain the strategies used. 2. For Grade 5: a) Use only rectangles. b) What is the relationship between the side lengths and the area of the rectangle? • For Grade 6: a) Use only triangles. b) What is the relationship between the area of a rectangle and the area of a triangle?

Constructing a Collective Thinkpad Banshoas Assessment for Learning What does the teacher need to do to understand the range of student responses? (See pp. 48–50.) 2. What does the teacher need to know and do to coordinate class discussion so it builds on the mathematical knowledge from student responses? (See pp. 48–50.)

4 cm 4 cm 4 cm 4 cm 4 cm 8 cm 8 cm 8 cm 8 cm 8 cm 8 cm 8 cm 8 cm 4 cm 4 cm 4 cm 4 cm 4 cm 4 cm 4 cm 4 cm 6 cm 6 cm 6 cm 6 cm 6 cm 6 cm 6 cm 6 cm 4 cm 4 cm 4 cm 4 cm 6 cm 4 cm 8 cm 8 cm 4 cm 4 cm 6 cm Understanding Range of Gr 5 Responses Will these strategies work for any size L-shaped figure? Bansho

4 cm 4 cm 4 cm 4 cm 4 cm 8 cm 8 cm 8 cm 8 cm 8 cm 8 cm 4 cm 4 cm 4 cm 4 cm 4 cm 4 cm 6 cm 6 cm 6 cm 6 cm 6 cm 6 cm 4 cm 4 cm 6 cm 4 cm 8 cm 8 cm 4 cm 4 cm 6 cm Understanding Range of Gr 6 Responses What’s the relationship between calculating the area of a rectangle and calculating the area of a triangle? Bansho

Look Back – Reflect and Connect 1. What mathematics is evident in the solutions? • Which problem-solving strategies were used to develop solutions? (See pp. 38–44.) • How are the following mathematical processes evident in the development of the solutions: a) problem solving b) reasoning and proving c) reflecting d) selecting tools and computation strategies e) connecting f) representing g) communicating

Look Back continued 4. What are some ways that the teacher can support student problem solving? (See pp. 30-34.)

Next Steps in Our Classroom 1. Choose 4 student work samples to analyse and describe in terms of: a) the mathematics evident in their work b) the problem-solving strategies used to develop their solutions. 2. Reflect on and apply 2 of the following strategies to support student learning of mathematics through problem solving: a) bansho b) think-aloud c) any teaching strategy from pp. 30–34, 38–44, or 48–50.

Problem SolvingSession D Selecting and Writing Effective Mathematics Problems for Learning

Learning Goals of the Module Develop strategies for teaching mathematics through problem solving by: • identifying the purpose of problems for learning mathematics • analysing the characteristics of effective problems • analysing problems from resource materials according to criteria of effective problems • selecting, adapting, and/or writing problems

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Problem Solving Methods

Published by Hilary Bradford Modified over 9 years ago

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PROBLEM SOLVING STRATEGIES

Subject: Mathematics

Age range: 11-14

Resource type: Other

Last updated

12 January 2015

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If the solutions were included this would be a 5 star resource. It really is an excellent way to demonstrate various problem solving strategies. Using each problem included as a starter activity with able groups we have found all solutions and I&#39;ve just added them on the slide notes.

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Teaching Through Problem Solving - PowerPoint PPT Presentation

Teaching Through Problem Solving

Elementary and middle school mathematics: teaching developmentally, (6th ed. ... begin with the math ... workstations and games ... – powerpoint ppt presentation.

• __________________________________________________ ___________________
• Adapted from
• Van de Walle (2007). Elementary and Middle School Mathematics Teaching Developmentally, (6th Ed.). Pearson Allyn Bacon, pp. 37-70.
• Places students attention on ideas and sense making
• Develops mathematical power
• Develops the belief in students that they are capable of doing mathematics and that is makes sense
• Provides ongoing assessment data that can be used in instructional decisions
• Allows entry point for a wide range of students
• Strategies and Process
• Develop problem analysis skills
• Develop and select strategies
• Justify solutions
• Extend of generalize problems
• Metacognition
• Monitor and regulate actions
• Disposition
• Gain confidence and belief in abilities
• Be willing to try and persevere
• Enjoy doing mathematics
• Upper two levels
• Higher-Order
• Any task or activity for which a student has no prescribed or memorized rules or methods, nor is there a perception by the student that there is a specific correct solution method.
• (Hiebert et al., 1997)
• Must begin where the students are
• Task must take into consideration the current understanding of the students
• Problematic or engaging aspect of the problem must be due to the mathematics the students are to learn
• Mathematics of the problem should be the focus not the context or creation of a product
• Must require justifications and explanations of answersshould be an important part to problem solutions
• The following task might be used in grades 5-6 as part of the development of fraction concepts
• __________________________________________________ ______
• Place an X on the number line about where 11/8 would be. Explain why you put the X where you did. Perhaps you will want to draw and label other points on the line to help explain your answer.
• Process Problems
• Requires solution processes other than computational procedures
• Translation Problems
• Include one- and two-step story problems typically found in textbooks
• Application Problems
• Generally uses computation once data is gathered and a decision about a solution strategy has been made
• U.S. teachers usually spend a small portion of a lesson explaining or reviewing an idea and then have students go through a set of exercises. Lessons in this format follow the explain-then-practice pattern. Teachers find themselves going from desk to desk re-teaching and explaining to individuals. This is in contrast to a lesson built around a single problemwhich is the typical approach for student-centered problem-based lessons.
• __________________________________________________ ____
• Teaching through problem solving suggests a simple three part structure for lessons.
• Before--Introducing
• During--Exploring
• After--Summarizing
• Be sure the task is understood by students
• Establish expectations for how students are to work and what products are to be prepared
• Prepare students mentally for the task by activating student prior knowledge related to the problem
• Let gogive students a chance to work without guidance.
• Observe interactions and listen for group ideas, strategies, and work procedures to use in class discussions
• Offer assistance when needed, either when all members of a group raise their hands or if a group is not working
• Provide an extension for groups that finish more quickly than others
• Engage class in productive discussionencourage student-student dialogue and not just student-teacher dialogue that may exclude the class by having students ask questions also
• Listen actively without evaluationhave students explain methods, solutions, and justifications for answers
• Summarize main ideas and identify problems that could be used for future explorations
• Some Questions to Consider
• How did you organize your work?
• What problems did you encounter?
• What strategies did you use? Which were most helpful?
• How did you arrive at your solution?
• What other strategy might have been used instead?
• Selecting appropriate tasks and materials
• Identifying sources of problems
• Clarifying the teachers role
• Organizing and implementing instruction
• Changing the difficulty of problems by altering the --Problem Context --Problem Structure
• Sources of Effective Tasks
• Standards-Based Curricula
• National Council of Teachers of Mathematics JournalMathematics Teaching in the Middle School
• National Council of Teachers of Mathematics Navigation Series books
• Internet Lesson Plans
• 1. How is the activity done?
• Actually do the activity to determine how students might do the activity, what materials are needed, what is recorded or written down, and what misconceptions may emerge.
• 2. What is the purpose of the activity?
• Determine what mathematical ideas are developed by the activity. Are these ideas concepts or procedural skills? What connections can be made to other topics?
• 3. Will the activity accomplish its purpose?
• 4. What must the teacher do?
• --In the before part of the lesson?
• --In the during part of the lesson?
• --In the after part of the lesson?
• Content and Task Decisions
• Begin with the Math!
• Articulate clearly the ideas you want students to learn as a result of the lesson
• Consider your students
• What do the students already know and understand about the topic
• Decide on task
• Keep it simplegood tasks need not be elaborate
• Predict what will happen
• Think about what your students will likely do with the taskmodify task as needed based on your predictions
• Teaching Actions
• Articulate student responsibilities
• Think about how students will supply information about what they learned
• Plan the Before portion of lesson
• Think about the During portion of the lesson
• Think about the After portion of the lesson
• MinilessonsTasks that do not require the entire class period--think-pair-share strategy is useful
• Workstations and GamesCan be set-up around the room without the need to distribute and collect materials to allow students to work on different tasks and concepts
• Problem-Solving MenuA menu is a collection of activities for a student to do. A menu can provide class work activities for several days, a week, or for a longer period of time. The tasks on the menu or not hierarchical and do not conceptually build upon each other.
• Understand the Problem
• Read and explore
• Devise a strategy to use to solve the problem
• Carry out the Plan
• Solve the problem
• Reflect on answer and justify solution
• Look for Patterns
• Consider all Possibilities
• Make an Organized List
• Draw a Picture
• Guess and Check
• Write an Equation
• Construct a Table or Graph
• Use Objects
• Work Backward
• Solve a Simpler (or similar) Problem
• Three types of information teachers should provide to their students
• Math conventions
• Alternative methods
• Clarification of students methods
• In addition to using a problem-based approach, consider the following to attend to the diversity in the classroom.
• Be sure that problems have multiple entry points
• Plan differentiated tasks
• Use heterogeneous groupings
• Make accommodations and modifications for English-language learners
• Listen carefully to students
• Different problem-based tasks or experiences, spread over numerous class periods, each addressing the same basic ideas
• Repetitive, non-problem-based exercises designed to improve skills or procedures already acquired
• Increased opportunity to develop conceptual ideas and more elaborate and useful connections
• Opportunity to develop alternative and flexible strategies
• Greater chance for all students to understand, not just a few
• Clear message that math is about figuring things out and making sense
• Increased facility with a strategy but only with a strategy already learned
• Focus on a singular method and an exclusion of flexible alternatives
• False appearance of understanding
• Rule-oriented view of what mathematics is about
• An efficient strategy for the skill to be drilled is already in place
• Automaticity with the skill or strategy is a desired outcome
• Practice as Homework
• Allows students to practice at home
• Communicates to parents that math is sense-making and has value
• Drill as Homework
• Keep it short
• Provide a key
• Do not grade based on correctness
• Do not use class time going over drill homework

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