teaching problem solving mathematics

Teaching Problem Solving in Math

  • Freebies , Math , Planning

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

There’s also this visuals strategy wheel practice.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem.   To help students understand the problem, I provided them with sample problems, and together we did five important things:

  • read the problem carefully
  • restated the problem in our own words
  • crossed out unimportant information
  • circled any important information
  • stated the goal or question to be solved

We did this over and over with example problems.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

  • taking our time
  • working the problem out
  • showing all our work
  • estimating the answer
  • using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

  • switch strategies or try a different one
  • rethink the problem
  • think of related content
  • decide if you need to make changes
  • check your work
  • but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Finally, Step 4 – Check It.   This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

  • compare your answer to your estimate
  • check for reasonableness
  • check your calculations
  • add the units
  • restate the question in the answer
  • explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act  – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

You can grab these problem-solving bookmarks for FREE by clicking here .

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it  may work for other grade levels. The practice problems are all for the early third-grade level.

teaching problem solving mathematics

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

Problem Solving

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 real-life 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 two-digit number by a one-digit 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’ problem-solving 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 worthwhile-problem criteria developed by Lappan and Phillips (1998):

  • The problem has important, useful mathematics embedded in it.
  • The problem requires high-level 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.

Needlepoint of cats

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

Back of a needlepoint

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 sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Teacher teaching a math lesson

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?

teaching problem solving mathematics

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 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group 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 3-Acts can be found at the following websites:

  • Dan Meyer’s Three-Act Math Tasks
  • Graham Fletcher3-Act Tasks ]
  • Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 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

Light bulb

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 hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on 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 higher-order 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 hands-on 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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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.

teaching problem solving mathematics

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Making Sense of Mathematics

Making Sense of Mathematics

Teaching Mathematics through Problem Solving- An Upside-Down Approach

By inviting children to solve problems in their own ways, we are initiating them into the community of mathematicians who engage in structuring and modeling their “lived worlds” mathematically.

 Fosnot and Jacob, 2007

Teaching mathematics through problem solving requires you to think about the types of tasks you pose to students, how you facilitate discourse in your classroom, and how you support students use of a variety of representations as tools for problem solving, reasoning, and communication.

This is a different approach from “do-as-I-show-you” approach where the teacher shows all the mathematics, demonstrates strategies to solve a problem, and then students just have to practice that exact same skill/strategy, perhaps using a similar problem.

Teaching mathematics through problem solving means that students solve problems to learn new mathematics through real contexts, problems, situations, and strategies and models that allow them to build concept and make connections on their own.

The main difference between the traditional approach “I-do-you-do” and teaching through problem solving, is that the problem is presented at the beginning of the lesson, and the skills, strategies and ideas emerge when students are working on the problem. The teacher listens to students’ responses and examine their work, determining the moment to extend students’ thinking and providing targeted feedback.

Here are the 4 essential moves in a math lesson using a student-centered approach or problem-solving approach:

  • Number Talk (5-8 min) (Connection)

The mini-lesson starts with a Number Talk. The main purpose of a Number Talk is:

*to build number sense, and 

*to provide opportunities for students to explain their thinking and respond to the mathematical thinking of others.

teaching problem solving mathematics

Please refer to the document Int§roducing Number Talks . Or watch this video with Sherry Parrish to gain understanding about how Number Talks can build fluency with your students.

Here are some videos of Number Talks so you can observe some of the main teaching moves.

The role of the teacher during a number talk is crucial. He/she needs to listen carefully to the way student is explaining his/her reasoning, then use a visual representation of what the student said. Other students also share their strategies, and the teacher represents those strategies as well. Students then can visualize a variety of strategies to solve a problem. They learn how to use numbers flexibly, there is not just one way to solve a problem. When students have a variety if strategies in their math tool box, they can solve any problem, they can make connections with mathematical concepts.

teaching problem solving mathematics

There are a variety of resources that can be used for Math Talks. Note : the main difference between Number Talks and Math Talks, is that one allows students to use numbers flexibly leading them to fluency, develop number sense, and opportunities to communicate and reason with mathematics; the other allows for communicating and reasoning, building arguments to critique the reasoning of others, the use of logical thinking, and the ability to recognize different attributes to shapes and other figures and make sense of the mathematics involved.

  • 2. Using problems to teach (5-8 min) Mini Lesson

teaching problem solving mathematics

Problems that can serve as effective tasks or activities for students to solve have common features. Use the following points as a guide to assess if the problem/task has the potential to be a genuine problem:

*Problem should be appropriate to their current understanding, and yet still find it challenging and interesting.

*The main focus of the problem should allow students to do the mathematics they need to learn, the emphasis should be on making sense of the problem, and developing the understanding of the mathematics. Any context should not overshadow the mathematics to be learned.

*Problems must require justification, students explain why their solution makes sense. It is not enough when the teacher tells them their answer is correct.

*Ideally, a problem/task should have multiple entries. For example “find 3 factors whose product is 108”, instead of just “multiplying 3 numbers. “

The most important part of the mini-lesson is to avoid teaching tricks or shortcuts, or plain algorithms. Our goal is always to help guide students to understand why the math works (conceptual understanding). And most importantly how different mathematical concepts/ideas are connected! “Math is a connected subject”  Jo Boaler’s video

“Students can learn mathematics through exploring and solving contextual and mathematical problems vs. students can learn to apply mathematics only after they have mastered the basic skills.” By Steve Leinwand author of Principles to Action .

  • 3. Active Engagement (20-30 min)

teaching problem solving mathematics

This is the opportunity for students to work with partners or independently on the problem, making connections of what they know, and trying to use the strategy that makes sense to them. Always making sure to represent the problem with a visual representation. It can be any model that helps student understand what the problem is about.

The job of the teacher during this time, is to walk around asking questions to students to guide them in the right direction, but without telling too much. Allowing students to come up with their own solutions and justifications.

  • Teacher can clarify any questions around the problem, not the solution.
  • Teacher emphasizes reasoning to make sense of the problem/task.
  • Teacher encourages student-student dialogue to help build a sense of self.

Some lessons will include a rich task, or a project based learning, or a number problem (find 3 numbers whose product is 108). There are a variety of learning target tasks to choose from, for each grade level on the Assessment Live Binders website created by Erma Anderson and Project AERO.

Again, keep in mind that some lessons will follow a different structure depending on the learning target for that day. Regardless of instructional design, the teacher should not be doing the thinking, reasoning, and connection building; it must be the students who are engaged in these activities

  • 4. Share (8-12 min) (Link)

teaching problem solving mathematics

The most crucial part of the lesson is here. This is where the teaching/learning happens, not only learning from teacher, but learning from peers reaching their unique “zone of proximal development” (Vygotsky, 1978).

We bring back our students to share how they solved their problem. Sometimes they share with a partner first, to make sure they are using the right vocabulary, and to make sure they make sense of their answer. Then a few of them can share with the rest of the class. But sharing with a partner first is helpful so everyone has the opportunity to share.

“Talk to each other and the teacher about ideas – Why did I choose this method? Does it work in other cases? How is the method similar or different to methods other people used?” Jo Boaler’s article “How Students Should Be Taught Mathematics.”

Students make sense of their solution. The teacher listens and makes connections between different strategies that students are sharing. Teacher paraphrases the strategy student described, perhaps linking it with an efficient strategy.

teaching problem solving mathematics

“It is a misperception that student-centered classrooms don’t include any lecturing. At times it’s essential the teacher share his or her expertise with the larger group. Students could drive the discussion and the teacher guides and facilitates the learning.” Trevor MacKenzie

If the target for today’s lesson was to introduce the use a number line, for example, this is where the teacher will share that strategy as another possible way to solve today’s problem!

This could also be a good time for any formative assessment, using See Saw, using exit slips, or any kind of evidence of what they learned today.

References.

“Teaching Student-Centered Mathematics” Table 2.1 page 26 , Van de Walle, Karp, Lovin, Bay-Williams

“Number Talks” , Sherry Parrish

“How Students Should be Taught Mathematics: Reflections from Research and Practice” Jo Boaler

“Erma Anderson, Project AERO Assessments live binders

“Principles to Action” , Steve Leinwand

“ Turning Teaching Upside Down “, by Cathy Seeley

“Four Inquiry Qualities At The Heart of Student-Centered Teaching”

By Trevor MacKenzie

“The Zone of Proximal Development” Vygotsky, 1978

*** Here is a link to my favorite places to plan Math padlet, you will find a variety of resources, videos, articles, etc. By Caty Romero

***One more padlet for many resources to plan, teach, and assess mathematics that make sense: Making Sense of Mathematics Padlet.

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Published by Caty Romero - Math Specialist

Passionate about learning and making sense of mathematics. Teacher, Math Learning Specialist, K-8 Math Consultant, and Instructional Coach. Student-Centered-Learning is my approach! Contact me at [email protected] or follow me on Twitter @catyrmath View all posts by Caty Romero - Math Specialist

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How to Use Real-World Problems to Teach Elementary School Math: 6 Tips

teaching problem solving mathematics

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When you think back on elementary school math, do you have fond memories of the countless worksheets you completed on adding fractions or solving division problems? Probably not.

Researchers and educators have been pushing for years for schools to move away from teaching math through a set of equations with no context around them, and towards an approach that pushes kids to use numerical reasoning to solve real problems, mirroring the way that they’ll encounter the use of math as adults.

The strategy is largely about setting kids up for success in the professional world, and educators can lay the groundwork decades earlier, even in kindergarten .

Here are some tips for using a real world problem-solving approach to teaching math to elementary school students.

1. There’s more than one right answer and more than one right method

A “real world task” can be as simple as asking students to think of equations that will get them to a particular “target” number, say, 14. Students could say 7 plus 7 is 14 or they could say 25 minus 11 is 14. Neither answer is better than the other, and that lesson teaches kids that there are multiple ways to use math to solve problems.

2. Give kids a chance to explain their thinking

The process you use to solve a real world math problem can be just as important as arriving at the correct answer, said Robbi Berry, who teaches 5th grade in Las Cruces, N.M. Her students have learned not to ask her if a particular answer is correct, she said, because she’ll turn the question back on them, asking them to explain how they know that it is right. She also gives her students a chance to explain to one another how they arrived at a particular solution, “We always share our strategies so that the kids can see the different ways” to arrive at an answer, she said. Students get excited, she said, when one of their classmates comes up with an approach they never would have thought of. “Math is creative,” Berry said. “It’s not just learning and memorizing.”

3. Be willing to deal with some off-the-wall answers

Problem solving does not necessarily mean going to the word problems in your textbook, said Latrenda Knighten, a mathematics instructional coach in Baton Rouge, La. For little kids, it can be as simple as showing a group of geometric shapes and asking what they have in common. Students may go off track a bit by talking about things like color, she said, but teachers can steer them towards thinking about things like how a rectangle differs from a triangle.

4. Let your students push themselves

Tackling these richer, real-world problems can be tougher than solving equations on a worksheet. And that is a good thing, said Jo Boaler, a professor at Stanford University and an expert on math education. “It’s really good for your brain to struggle,” she said. “We don’t want kids getting right answers all the time because that’s not giving their brains a really good workout.” These types of problems require collaboration, a skill that many don’t associate with math, but that is key to how math reasoning works beyond the classroom. The complexity and difficulty of the tasks means that students “have to talk to each other and really figure out what to do, what’s a good method?”

5. Celebrate ‘favorite mistakes’ to encourage intellectual risk taking

Wrong answers should be viewed as learning opportunities, Berry said. When one of her students makes an error, she asks if she can share it with the class as a “favorite mistake.” Most of the time, students are comfortable with that, and the class will work together to figure how the misstep happened.

6. Remember there’s no such thing as a being born with a ‘math brain’

Some teachers believe that certain students are just naturally good at math, and others are not, Boaler said. But that’s not true. “Brains are constantly shaping, changing, developing, connecting, and there is no fixed anything,” said Boaler, who often works alongside neuroscientists. What’s more, many elementary school teachers lack confidence in their own math abilities, she said. “They think they can’t do [math],” Boaler said. “And they often pass those ideas on” to their students.

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“Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does,” says Jon R. Star.

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One way is the wrong way to do math. Here’s the right way.

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Research by Ed School psychologist reinforces case for stressing multiple problem-solving paths over memorization

There’s never just one way to solve a math problem, says Jon R. Star , a psychologist and professor of education at the Harvard Graduate School of Education. With researchers from Vanderbilt University, Star found that teaching students multiple ways to solve math problems instead of using a single method improves teaching and learning. In an interview with the Gazette, Star, a former math teacher, outlined the research and explained how anyone, with the right instruction, can develop a knack for numbers.

Jon R. Star

GAZETTE: What is the most common misconception about math learning?

STAR: That you’re either a math person or you’re not a math person — that some people are just born with math smarts, and they can do math, and other people are just not, and there’s not much you can do about it.

GAZETTE: What does science say about the process of learning math?

STAR: One thing we know from psychology about the learning process is that the act of reaching into your brain, grabbing some knowledge, pulling it out, chewing on it, talking about it, and putting it back helps you learn. Psychologists call this elaborative encoding. The more times you can do that process — putting knowledge in, getting it out, elaborating on it, putting it back in — the more you will have learned, remembered, and understood the material. We’re trying to get math teachers to help students engage in that process of elaborative encoding.

GAZETTE: How did you learn math yourself?

STAR: Learning math should involve some sense-making. It’s necessary that we listen to what our teacher tells us about the math and try to make sense of it in our minds. Math learning is not about pouring the words directly from the teacher’s mouth into the students’ ears and brains. That’s not the way it works. I think that’s how I learned math. But that’s not how I hope students learn math and that’s not how I hope teachers think about the teaching of math. Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does.

GAZETTE: Tell us about the teaching method described in the research.

STAR: One of the strategies that some teachers may use when teaching math is to show students how to solve problems and expect that the student is going to end up using the same method that the teacher showed. But there are many ways to solve math problems; there’s never just one way.

The strategy we developed asks that teachers compare two ways for solving a problem, side by side, and that they follow an instructional routine to lead a discussion to help students understand the difference between the two methods. That discussion is really the heart of this routine because it is fundamentally about sharing reasoning: Teachers ask students to explain why a strategy works, and students must dig into their heads and try to say what they understand. And listening to other people’s reasoning reinforces the process of learning.

GAZETTE: Why is this strategy an improvement over just learning a single method?

STAR: We think that learning multiple strategies for solving problems deepens students’ understanding of the content. There is a direct benefit to learning through comparing multiple methods, but there are also other types of benefits to students’ motivation. In this process, students come to see math a little differently — not just as a set of problems, each of which has exactly one way to solve it that you must memorize, but rather, as a terrain where there are always decisions to be made and multiple strategies that one might need to justify or debate. Because that is what math is.

For teachers, this can also be empowering because they are interested in increasing their students’ understanding, and we’ve given them a set of tools that can help them do that and potentially make the class more interesting as well. It’s important to note, too, that this approach is not something that we invented. In this case, what we’re asking teachers to do is something that they do a little bit of already. Every high school math teacher, for certain topics, is teaching students multiple strategies. It’s built into the curriculum. All that we’re saying is, first, you should do it more because it’s a good thing, and second, when you do it, this is a certain way that we found to be especially effective, both in terms of the visual materials and the pedagogy. It’s not a big stretch for most teachers. Conversations around ways to teach math for the past 30 or 40 years, and perhaps longer, have been emphasizing the use of multiple strategies.

GAZETTE: What are the potential challenges for math teachers to put this in practice?

STAR: If we want teachers to introduce students to multiple ways to solve problems, we must recognize that that is a lot of information for students and teachers. There is a concern that there could be information overload, and that’s very legitimate. Also, a well-intentioned teacher might take our strategy too far. A teacher might say something like, “Well, if comparing two strategies is good, then why don’t I compare three or four or five?” Not that that’s impossible to do well. But the visual materials you would have to design to help students manage that information overload are quite challenging. We don’t recommend that.

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1.6: Problem Solving Strategies

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  • Michelle Manes
  • University of Hawaii

Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solve them), you learn strategies and techniques that can be useful. But no single strategy works every time.

How to Solve It

George Pólya was a great champion in the field of teaching  effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, How to Solve it . Pólya died at the age 98 in 1985. [1]

George_Pólya_ca_1973.jpg

George Pólya, circa 1973

  • Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

  • First, you have to understand the problem.
  • After understanding, then make a plan.
  • Carry out the plan.
  • Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. 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. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

  • What if the picture was different?
  • What if the numbers were simpler?
  • What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

A Problem Solving Strategy: Try Something!

If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

Note that being "good at mathematics" is not about doing things right the first time. It is about figuring things out. Practice being okay with having done something incorrectly. Try to avoid using an eraser and just lightly cross out incorrect work (do not black out the entire thing). This way if it turns out that you did something useful, you still have that work to reference! If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what was left after paying Brianna. Finally, Alex saw David and gave him 1/2 of the remaining money. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

A Problem Solving Strategy: 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! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

A Problem Solving Strategy: Make Up Numbers

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

Try this: Assume (that is, pretend) Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person.

Or try working backward: suppose Alex has some specific amount left at the end, say $10. Since he gave David half of what he had before seeing David, that means he had $20 before running into David. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

(Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

Most people want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. Instead of asking the teacher, “Is this right?”, you should be ready to justify it and say, “Here’s my answer, and here is how I got it.”

A Problem Solving Strategy: Try a Simpler Problem

Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said, “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

The ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

A Problem Solving Strategy: Work Systematically

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

A Problem Solving Strategy: Use Manipulatives to Help You Investigate

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

A Problem Solving Strategy: Look for and Explain Patterns

Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

  • Describe all of the patterns you see in the table. If possible, actually describe these to a friend.
  • Explain and justify any of the patterns you see (if possible, actually do this with a friend). If you don't have a partner to work with, imagine they asked you, "How can you be sure the patterns will continue?"
  • Expand this to find what calculation(s) you would perform to find the total number of squares on a 100 × 100 chess board.

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

(Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

index-12_1-300x282-1.png

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What progress have you made?

A Problem Solving Strategy: Find the Math, Remove the Context

Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

  • What is the sum of all the numbers on the clock’s face?
  • Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
  • How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

A Problem Solving Strategy: Check Your Assumptions

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

index-13_1-300x296.png

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

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11 Real World Math Activities That Engage Students

Bridging the gap between abstract math concepts and real life experiences can make the subject accessible and relevant for kids.

During a unit on slope, José Vilson’s students just weren’t getting it, and their frustration was growing. The former middle school math teacher began brainstorming creative ways to illustrate the concept. “I kept thinking, ‘My students already understand how this works—they just don’t know that they know,’” Vilson writes in a recent article for Teacher2Teacher . “How can I activate knowledge they don’t believe they have?”

Then he thought about a hill a couple of blocks from school that his students “walk up every day to get to the subway.” He tacked up paper and began sketching stick figures on the hill. “One was at the top of the hill, one was halfway up, one was near the bottom skating on flat ground, and one was on a cliff,” writes Vilson, now the executive director of EduColor. “Which of these figures will go faster and why?” he asked his students. “That got my kids laughing because, of course, my stick figures weren’t going to hang in the MoMA.” Still, his sketch got them thinking and talking, and it provided a simple stepping stone that “gave that math relevance and belonging in their own lives,” Vilson concludes. 

“It’s not unusual for students to walk into our classrooms thinking that math belongs to people who are smarter, who are older, or who aren’t in their immediate circle,” Vilson writes. “But every time I teach math in a way that’s accessible and real for my students, I’m teaching them: ‘The math is yours.’”

To build on Vilson’s idea, we posted on our social channels asking teachers to share their favorite strategies for connecting math to students’ experiences and lives outside of school. We received hundreds of responses from math educators across grade levels. Here are 11 teacher-tested ideas that get students seeing and interacting with the math that surrounds them each day.

Hunt for clues

Coordinate systems can feel abstract to some students—but using coordinates to navigate a familiar space can solidify the concept in a relevant and fun way. “Before starting a unit on coordinates, I make gridded maps of the school—I make them look old using tea staining —and send my students off on a treasure hunt using the grid references to locate clues,” says Kolbe Burgoyne, an educator in Australia. “It’s meaningful, it’s fun, and definitely gets them engaged.”

Budget a trip

Students enjoy planning and budgeting for imaginary trips, teachers tell us, offering ample opportunities to practice adding, subtracting, and multiplying large numbers. In Miranda Henry’s resource classroom, for example, students are assigned a budget for a fictional spring break trip; then they find flights, hotels, food, and whatever else they’ll need, while staying within budget.

Math teacher Alicia Wimberley has her Texas students plan and budget a hypothetical trip to the Grand Canyon. “They love the real world context of it and start to see the relevance of the digits after the decimal—including how the .00 at the end of a price was relevant when adding.” One of Wimberley’s students, she writes, mixed up his decimals and nearly planned a $25,000 trip, but found his mistake and dialed back his expenses to under $3,000.

Tap into pizza love

Educators in our audience are big fans of “pizza math”—that is, any kind of math problem that involves pizza. “Pizza math was always a favorite when teaching area of a circle,” notes Shane Capps. If a store is selling a 10-inch pizza, for example, and we know that’s referring to its diameter, what is its total area? “Pizza math is a great tool for addition, subtraction, multiplication, word problems, fractions, and geometry,” another educator writes on our Instagram. There are endless pizza-based word problems online. Here’s a simple one to start, from Jump2Math : “The medium pizza had six slices. Mom and Dad each ate one slice. How much pizza is left?”

Break out the measuring cups

Lindsey Allan has her third-grade students break into pairs, find a recipe they like online, and use multiplication to calculate how much of each ingredient they’d need in order to feed the whole class. The class then votes on a favorite recipe, and they write up a shopping list—“which involves more math, because we have to decide, ‘OK, if we need this much butter for the doubled recipe, will we need three or four sticks, and then how much will be left over?’” Allan writes. “And then it turns out students were also doing division without even realizing!” 

Sometimes, a cooking mistake teaches students about proportions the hard way. “Nobody wants a sad chocolate chip cookie where you doubled the dough but not the chocolate chips,” adds teacher Holly Satter.

Heading outdoors is good for kids’ bodies , of course, but it can also be a rich mathematical experience. In second grade, kids can head out to measure perimeters, teacher Jenna McCann suggests—perhaps of the flower boxes in the school garden. If outdoors isn’t an option, there’s plenty of math to be found by walking around inside school—like measuring the perimeter of the tables in the cafeteria or the diameters of circles taped off on the gym floor.

In Maricris Lamigo’s eighth-grade geometry class, “I let [students] roam around the school and take photos of things where congruent triangles were applied,” says Lamigo. “I have students find distances in our indoor courtyard between two stickers that I place on the floor using the Pythagorean theorem,” adds Christopher Morrone, another eighth-grade teacher. In trigonometry, Cathee Cullison sends students outside “with tape measures and homemade clinometers to find heights, lengths, and areas using learned formulas for right and non-right triangles.” Students can make their own clinometers , devices that measure angles of elevation, using protractors and a few other household items.

Plan for adult life

To keep her math lessons both rigorous and engaging, Pamela Kranz runs a monthlong project-based learning activity where her middle school students choose an occupation and receive a salary based on government data. Then they have to budget their earnings to “pay rent, figure out transportation, buy groceries,” and navigate any number of unexpected financial dilemmas, such as medical expenses or car repairs. While learning about personal finance, they develop their mathematical understanding of fractions, decimals, and percents, Kranz writes.

Dig into sports stats

To help students learn how to draw conclusions from data and boost their comfort with decimals and percentages, fourth-grade teacher Kyle Pisselmyer has his students compare the win-loss ratio of the local sports team to that of Pisselmyer’s hometown team. While students can struggle to grasp the relevance of decimals—or to care about how 0.3 differs from 0.305—the details snap into place when they look at baseball players’ stats, educator Maggierose Bennion says.

March Madness is a great source of real world data for students to analyze in math class, says sixth-grade math teacher Jeff Norris. Last March, Norris decorated his classroom like a basketball court, then had his students do basic statistical analysis—like calculating mean, median, and mode—using March Madness data, including individual game scores and the total win rate of each team. “We also did some data collection through our own basketball stations to make it personally relevant,” Norris says; students lined up in teams to shoot paper balls into a basket in a set amount of time, recorded their scores in a worksheet, and then examined the scoring data of the entire class to answer questions about mean, median, mode, range, and outliers.

Go on a (pretend) shopping spree

“My students love any activities that include SHOPPING!” says Jessie, a sixth-grade teacher who creates shopping-related problems using fake (or sometimes real) store ads and receipts. Her students practice solving percentage problems, and the exercise includes opportunities to work with fractions and decimals.

To get students more engaged with the work, math educator Rachel Aleo-Cha zeroes in on objects she knows students are excited about. “I make questions that incorporate items like AirPods, Nike shoes, makeup, etc.,” Aleo-Cha says. She also has students calculate sales tax and prompts them to figure out “what a 50% off plus 20% off discount is—it’s not 70% off.”

Capture math on the fly

Math is everywhere, and whipping out a smartphone when opportunities arise can lead to excellent content for math class. At the foot of Mount Elbert in Colorado, for example, math teacher Ryan Walker recorded a short word problem for his fourth- and fifth-grade students. In the video, he revealed that it was 4:42 a.m., and it would probably take him 249 minutes to reach the summit. What time would he reach the summit, he asked his students—and, assuming it took two-thirds as long to descend, what time would he get back down?

Everyday examples can be especially relatable. At the gas station, “I record a video that tells the size of my gas tank, shows the current price of gas per gallon, and shows how empty my gas tank is,” says Walker. “Students then use a variety of skills (estimation, division, multiplying fractions, multiplying decimals, etc.) to make their estimate on how much money it will cost to fill my tank.”

Connect to social issues

It can be a powerful exercise to connect math to compelling social issues that students care about. In a unit on ratios and proportions, middle school teacher Jennifer Schmerler starts by having students design the “most unfair and unjust city”—where resources and public services like fire departments are distributed extremely unevenly. Using tables and graphs that reflect the distribution of the city’s population and the distribution of its resources, students then design a more equitable city.

Play entrepreneur

Each year, educator Karen Hanson has her fourth- and fifth-grade students brainstorm a list of potential business ideas and survey the school about which venture is most popular. Then the math begins: “We graph the survey results and explore all sorts of questions,” Hanson writes, like whether student preferences vary with age. Winning ideas in the past included selling T-shirts and wallets made of duct tape.

Next, students develop a resource list for the business, research prices, and tally everything up. They calculate a fair price point for the good they’re selling and the sales quantity needed to turn a profit. As a wrap-up, they generate financial statements examining how their profits stack up against the sales figures they had projected.

HELP OTHER TEACHERS OUT!

We’d love this article to be an evolving document of lesson ideas that make math relevant to kids. So, teachers, please tell us about your go-to activities that connect math to kids’ real world experiences.

Teaching Is Problem Solving

Guidance for Parents to Help Their Children Learn Mathematics

by Robert Schoen | October 5, 2020 Blog

In many households, expectations for parents to be involved in helping their children learn mathematics feels higher now than ever.

We think that all parents can benefit from having a few guiding principles for supporting their child’s math learning at home.

Claire Riddell and Laura Steele are two experienced teachers who compiled some recommendations to share with parents. Their recommendations are consistent with a CGI approach to teaching. The recommendations are provided below.

We are always on the lookout for good resources and ideas to support parent involvement in their child’s mathematics learning. We may expand the TiPS website to create a section that focuses directly on parent involvement.

If you know of useful resources to support parent involvement, please let us know so that we can share them with others. You can send them directly to me at [email protected] .

For now, I hope you Claire and Laura’s tips to be useful.

Helping Your Child Learn Math at Home

All children are natural problem solvers. We offer a few suggestions to help you work with your child at home to learn math.

Three Things to Consider

Let your child do the thinking and the talking. One of the most important things you can do is listen. Your role as a listener is crucial to assisting your child’s development. Invite your child to share his or her strategy for solving problems and be patient while they try to explain. As you know, trying to explain something that you are still learning can be difficult and uncomfortable. You can help your child to learn by being a patient listener and providing opportunities for them to try to explain multiple times in different ways.

Everyday items in your household can be math tools. Regardless of the problem’s context, many household items can help your child solve problems. Cheerios, pennies, rocks, dry beans, toys, and so much more can be used by your child to solve problems. Before your child starts working, gather a set of items to have within reach for your child to use while solving problems.

Focus more on the process and less on the answer. Compliment your child’s effort, sense making, and attempts to express their ideas in words and in writing. While we ultimately want children to arrive at a correct answer, the real learning and thinking happens while they are solving the problem. Celebrate the process.

Three Questions to Ask

  • “How did you get [your answer]?” Avoid asking “what is your answer” and instead ask your child to explain his or her thinking. Children sometimes arrive at an incorrect answer, but when given the opportunity to explain their solution, they identify and correct their own mistakes.
  • “How did you know to do that?” Children often think if you ask them a question about how they solve a problem, it indicates their answer is wrong. Asking, “How did you know to do that?” encourages children to share their reasoning and thinking. Get in the habit of asking questions often so your child knows that explaining themselves is just a part of doing math.
  • “Can you explain how you did that?” Children memorize math facts, which is good, but we can learn about the depth of their understanding when we ask children to demonstrate on paper or with math tools what their thinking looks like in a concrete form. Answers like, “I am smart” or “I just knew that” do not show a depth of knowledge. However, when asked to demonstrate their thinking in concrete ways or through verbal explanations, you’ll have a better understanding of their child’s mathematics knowledge.

Three Things to Try (When Your Child Seems Stuck)

  • Focus on what your child does know and understand. Does your child understand the problem? Instead of showing them how to solve it, ask them to explain what they know about the problem. If it is a story problem, ask your child to picture the story in their mind or draw a picture of what is happening.
  • Invite your child to draw a picture of the problem or act it out with objects. By creating a concrete representation, the problem can become less abstract and children are better able to access what the problem is asking and find possible solutions.
  • Don’t be afraid to leave a problem. Ask your child, “Would you like to continue with this problem, or would you like to come back to it later?” Giving your child the permission to choose whether to continue or not is empowering. Being a part of the decision-making process may empower your child and help them to grow as a person. Of course, if it is a mandatory homework assignment, it will need to be revisited in the allotted time (if the choice was to come back later), and the child will need to consider that too.

Download these tips as a PDF

Robert C. Schoen, Ph.D., is an associate professor of mathematics education in the School of Teacher Education at Florida State University. He is also the Associate Director of LSI’s Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR–STEM) and the founder and director of Teaching is Problem Solving . His research involves mathematical cognition, the mathematical education of teachers, the development and validation of educational and psychological measurement instruments, and evaluation of the effectiveness of educational interventions.

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Elementary teachers’ experience of engaging with Teaching Through Problem Solving using Lesson Study

Mairéad hourigan.

Department of STEM Education, Mary Immaculate College, University of Limerick, Limerick, Ireland

Aisling M. Leavy

For many decades, problem solving has been a focus of elementary mathematics education reforms. Despite this, in many education systems, the prevalent approach to mathematics problem solving treats it as an isolated activity instead of an integral part of teaching and learning. In this study, two mathematics teacher educators introduced 19 Irish elementary teachers to an alternative problem solving approach, namely Teaching Through Problem Solving (TTP), using Lesson Study (LS) as the professional development model. The findings suggest that the opportunity to experience TTP first-hand within their schools supported teachers in appreciating the affordances of various TTP practices. In particular, teachers reported changes in their beliefs regarding problem solving practice alongside developing problem posing knowledge. Of particular note was teachers’ contention that engaging with TTP practices through LS facilitated them to appreciate their students’ problem solving potential to the fullest extent. However, the planning implications of the TTP approach presented as a persistent barrier.

Introduction

A fundamental goal of mathematics education is to develop students’ ability to engage in mathematical problem solving. Despite curricular emphasis internationally on problem solving, many teachers are uncertain how to harness students’ problem solving potential (Cheeseman, 2018 ). While many problem solving programmes focus on providing students with step-by-step supports through modelling, heuristics, and other structures (Polya, 1957 ), Goldenberg et al. ( 2001 ) suggest that the most effective approach to developing students’ problem solving ability is by providing them with frequent opportunities over a prolonged period to solve worthwhile open-ended problems that are challenging yet accessible to all. This viewpoint is in close alignment with reform mathematics perspectives that promote conceptual understanding, where students actively construct their knowledge and relate new ideas to prior knowledge, creating a web of connected knowledge (Hiebert, 2003 ; Lester, 2013 ; Takahashi, 2006 ; Watanabe, 2001 ).

There is consensus in the mathematics education community that problem solving should not be taught as an isolated topic focused solely on developing problem solving skills and strategies or presented as an end-of-chapter activity (Takahashi, 2006 , 2016 ; Takahashi et al., 2013 ). Instead, problem solving should be integrated across the curriculum as a fundamental part of mathematics teaching and learning (Cai & Lester, 2010 ; Takahashi, 2016 ).

A ‘Teaching Through Problem Solving’ (TTP) approach, a problem solving style of instruction that originated in elementary education in Japan, meets these criteria treating problem solving as a core practice rather than an ‘add-on’ to mathematics instruction.

Teaching Through Problem Solving (TTP)

Teaching Through Problem Solving (TTP) is considered a powerful means of promoting mathematical understanding as a by-product of solving problems, where the teacher presents students with a specially designed problem that targets certain mathematics content (Stacey, 2018 ; Takahashi et al., 2013 ). The lesson implementation starts with the teacher presenting a problem and ensuring that students understand what is required. Students then solve the problem either individually or in groups, inventing their approaches. At this stage, the teacher does not model or suggest a solution procedure. Instead, they take on the role of facilitator, providing support to students only at the right time (Hiebert, 2003 ; Lester, 2013 ; Takahashi, 2006 ). As students solve the problem, the teacher circulates, observes the range of student strategies, and identifies work that illustrates desired features. However, the problem solving lesson does not end when the students find a solution. The subsequent sharing phase, called Neriage (polishing ideas), is considered by Japanese teachers to be the heart of the lesson rather than its culmination. During Neriage, the teacher purposefully selects students to share their strategies, compares various approaches, and introduces increasingly sophisticated solution methods. Effective questioning is central to this process, alongside careful recording of the multiple solutions on the board. The teacher concludes the lesson by formalising and consolidating the lesson’s main points. This process promotes learning for all students (Hiebert, 2003 ; Stacey, 2018 ; Takahashi, 2016 ; Takahashi et al., 2013 ; Watanabe, 2001 ).

The TTP approach assumes that students develop, extend, and enrich their understandings as they confront problematic situations using existing knowledge. Therefore, TTP fosters the symbiotic relationship between conceptual understanding and problem solving, as conceptual understanding is required to solve challenging problems and make sense of new ideas by connecting them with existing knowledge. Equally, problem solving promotes conceptual understanding through the active construction of knowledge (Hiebert, 2003 ; Lambdin, 2003 ; Takahashi, 2006 ). Consequently, students simultaneously develop more profound understandings of the mathematics content while cultivating problem solving skills (Kapur, 2010 ; Stacey, 2018 ).

Relevant research affirms that teachers acknowledge the merits of this approach (Sullivan et al., 2014 ) and most students report positive experiences (Russo & Minas, 2020 ). The process is considered to make students’ thinking and learning visible (Ingram et al., 2020 ). Engagement in TTP has resulted in teachers becoming more aware of and confident in their students’ problem solving abilities and subsequently expecting more from them (Crespo & Featherstone, 2006 ; Sakshaug & Wohlhuter, 2010 ).

Demands of TTP

Adopting a TTP approach challenges pre-existing beliefs and poses additional knowledge demands for elementary teachers, both content and pedagogical (Takahashi, 2008 ).

Research has consistently reported a relationship between teacher beliefs and the instructional techniques used, with evidence of more rule-based, teacher-directed strategies used by teachers with traditional mathematics beliefs (Stipek et al., 2001 ; Swan, 2006 ; Thompson, 1985 ). These teachers tend to address problem solving separately from concept and skill development and possess a simplistic view of problem solving as translating a problem into abstract mathematical terms to solve it. Consequently, such teachers ‘are very concerned about developing skilfulness in translating (so-called) real-world problems into mathematical representations and vice versa’ (Lester, 2013 , p. 254). Early studies of problem solving practice reported direct instructional techniques where the teacher would model how to solve the problem followed by students practicing similar problems (Chapman, 2015 ; Hiebert, 2003 ; Lester, 2013 ). This naïve conception of problem solving is reflected in many textbook problems that simply require students to apply previously learned routine procedures to solve problems that are merely thinly disguised number operations (Lester, 2013 ; Singer & Voica, 2013 ). Hence, the TTP approach requires a significant shift for teachers who previously considered problem solving as an extra activity conducted after the new mathematics concepts are introduced (Lester, 2013 ; Takahashi et al., 2013 ) or whose personal experience of problem solving was confined to applying routine procedures to word problems (Sakshaug &Wohlhuter, 2010 ).

Alongside beliefs, teachers’ knowledge influences their problem solving practices. Teachers require a deep understanding of the nature of problem solving, in particular viewing problem solving as a process (Chapman, 2015 ). To be able to understand the stages problem solvers go through and appreciate what successful problem solving involves, teachers benefit from experiencing solving problems from the problem solver’s perspective (Chapman, 2015 ; Lester, 2013 ).

It is also essential that teachers understand what constitutes a worthwhile problem when selecting or posing problems (Cai, 2003 ; Chapman, 2015 ; Lester, 2013 ; O’Shea & Leavy, 2013 ). This requires an understanding that problems are ‘mathematical tasks for which the student does not have an obvious way to solve it’ (Chapman, 2015 , p. 22). Teachers need to appreciate the variety of problem characteristics that contribute to the richness of a problem, e.g. problem structures and cognitive demand (Klein & Leiken, 2020 ; O’Shea & Leavy, 2013 ). Such understandings are extensive, and rather than invest heavily in the time taken to construct their mathematics problems, teachers use pre-made textbook problems or make cosmetic changes to make cosmetic changes to these (Koichu et al., 2013 ). In TTP, due consideration must also be given to the problem characteristics that best support students in strengthening existing understandings and experiencing new learning of the target concept, process, or skill (Cai, 2003 ; Takahashi, 2008 ). Specialised content knowledge is also crucial for teachers to accurately predict and interpret various solution strategies and misconceptions/errors, to determine the validity of alternative approaches and the source of errors, to sequence student approaches, and to synthesise approaches and new learning during the TTP lesson (Ball et al., 2008 ; Cai, 2003 ; Leavy & Hourigan, 2018 ).

Teachers should also be knowledgeable regarding appropriate problem solving instruction. It is common for teachers to teach for problem solving (i.e., focusing on developing students’ problem solving skills and strategies). Teachers adopting a TTP approach engage in reform classroom practices that reflect a constructivist-oriented approach to problem solving instruction where the teacher guides students to work collaboratively to construct meaning, deciding when and how to support students without removing their autonomy (Chapman, 2015 ; Hiebert, 2003 ; Lester, 2013 ). Teachers ought to be aware of the various relevant models of problem solving, including Polya’s ( 1957 ) model that supports teaching for problem solving (Understand the problem-Devise a plan-Carry out the plan-Look back) alongside models that support TTP (e.g., Launch-Explore-Summarise) (Lester, 2013 ; Sullivan et al., 2021 ). While knowledge of heuristics and strategies may support teachers’ problem solving practices, there is consensus that teaching heuristics and strategies or teaching about problem solving does not significantly improve students’ problem solving ability. Teachers require a thorough knowledge of their students as problem solvers, for example, being aware of their abilities and factors that hinder their success, including language (Chapman, 2015 ). Knowledge of content and student, alongside content and teaching (Ball et al., 2008 ), is essential during TTP planning when predicting student approaches and errors. Such knowledge is also crucial during TTP implementation when determining the validity of alternative approaches, identifying the source of errors (Explore phase), sequencing student approaches, and synthesising the range of approaches and new learning effectively (Summarise phase) (Cai, 2003 ; Leavy & Hourigan, 2018 ).

Supports for teachers

Given the extensive demands of TTP, adopting this approach is arduous in terms of the planning time required to problem pose, predict approaches, and design questions and resources (Lester, 2013 ; Sullivan et al., 2010 ; Takahashi,  2008 ). Consequently, it is necessary to support teachers who adopt a TTP approach (Hiebert, 2003 ). Professional development must facilitate them to experience the approach themselves as learners and then provide classroom implementation opportunities that incorporate collaborative planning and reflection when trialling the approach (Watanabe, 2001 ). In Japan, a common form of professional development to promote, develop, and refine TTP implementation among teachers and test potential problems for TTP is Japanese Lesson Study (LS) (Stacey, 2018 ; Takahashi et al., 2013 ). Another valuable support is access to a repository of worthwhile problems. In Japan, government-authorised textbooks and teacher manuals provide a sequence of lessons with rich well-tested problems to introduce new concepts. They also detail alternative strategies used by students and highlight the key mathematical aspects of these strategies (Takahashi, 2016 ; Takahashi et al., 2013 ).

Teachers’ reservations about TTP

Despite the acknowledged benefits of TTP for students, some teachers report reluctance to employ TTP, identifying a range of obstacles. These include limited mathematics content knowledge or pedagogical content knowledge (Charalambous, 2008 ; Sakshaug & Wohlhuter, 2010 ) and a lack of access to resources or time to develop or modify appropriate resources (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Other barriers for teachers with limited experience of TTP include giving up control, struggling to support students without directing them, and a tendency to demonstrate how to solve the problem (Cheeseman, 2018 ; Crespo & Featherstone, 2006 ; Klein & Leiken, 2020 ; Takahashi et al., 2013 ). Resistance to TTP is also associated with some teachers’ perception that this approach would lead to student disengagement and hence be unsuitable for lower-performing students (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2010 ).

Problem solving practices in Irish elementary mathematics education

Within the Irish context, problem solving is a central tenet of elementary mathematics curriculum documents (Department of Education and Science (DES), 1999 ) with recommendations that problem solving should be integral to students’ mathematical learning. However, research reveals a mismatch between intended and implemented problem solving practices (Dooley et al., 2014 ; Dunphy et al., 2014 ), where classroom practices reflect a narrow approach limited to problem solving as an ‘add on’, only applied after mathematical procedures had been learned and where problems are predominantly sourced from dedicated sections of textbooks (Department of Education and Skills (DES), 2011 ; Dooley et al., 2014 ; National Council for Curriculum and Assessment (NCCA), 2016 ; O’Shea & Leavy, 2013 ). Regarding the attained curriculum, Irish students have underperformed in mathematical problem solving, relative to other skills, in national and international assessments (NCCA, 2016 ; Shiel et al., 2014 ). Consensus exists that there is scope for improvement of problem solving practices, with ongoing calls for Irish primary teachers to receive support through school-based professional development models alongside creating a repository of quality problems (DES, 2011 ; Dooley et al., 2014 ; NCCA, 2016 ).

Lesson Study (LS) as a professional development model

Reform mathematics practices, such as TTP, challenge many elementary teachers’ beliefs, knowledge, practices, and cultural norms, particularly if they have not experienced the approach themselves as learners. To support teachers in enacting reform approaches, they require opportunities to engage in extended and targeted professional development involving collaborative and practice-centred experiences (Dudley et al., 2019 ; Murata et al., 2012 ; Takahashi et al., 2013 ). Lesson Study (LS) possesses the characteristics of effective professional development as it embeds ‘…teachers’ learning in their everyday work…increasing the likelihood that their learning will be meaningful’ (Fernandez et al., 2003 , p. 171).

In Japan, LS was developed in the 1980s to support teachers to use more student-centred practices. LS is a school-based, collaborative, reflective, iterative, and research-based form of professional development (Dudley et al., 2019 ; Murata et al., 2012 ). In Japan, LS is an integral part of teaching and is typically conducted as part of a school-wide project focused on addressing an identified teaching–learning challenge (Takahashi & McDougal, 2016 ). It involves a group of qualified teachers, generally within a single school, working together as part of a LS group to examine and better understand effective teaching practices. Within the four phases of the LS cycle, the LS group works collaboratively to study and plan a research lesson that addresses a pre-established goal before implementing (teach) and reflecting (observe, analyse and revise) on the impact of the lesson activities on students’ learning.

LS has become an increasingly popular professional development model outside of Japan in the last two decades. In these educational contexts, it is necessary to find a balance between fidelity to LS as originally envisaged and developing a LS approach that fits the cultural context of a country’s education system (Takahashi & McDougal, 2016 ).

Relevant research examining the impact of LS on qualified primary mathematics teachers reports many benefits. Several studies reveal that teachers demonstrated transformed beliefs regarding effective pedagogy and increased self-efficacy in their use due to engaging in LS (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ). Enhancements in participating teachers’ knowledge have also been reported (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ; Murata et al., 2012 ). Other gains recounted include improvements in practice with a greater focus on students (Cajkler et al., 2015 ; Dudley et al., 2019 ; Flanagan, 2021 ).

Context of this study

A cluster of urban schools, coordinated by their local Education Centre, engaged in an initiative to enhance teachers’ mathematics problem solving practices. The co-ordinator of the initiative approached the researchers, both mathematics teacher educators (MTEs), seeking a relevant professional development opportunity. Aware of the challenges of problem solving practice within the Irish context, the MTEs proposed an alternative perspective on problem solving: the Teaching Through Problem Solving (TTP) approach. Given Cai’s ( 2003 ) recommendation that teachers can best learn to teach through problem solving by teaching and reflecting as opposed to taking more courses, the MTEs identified LS as the best fit in terms of a supportive professional development model, as it is collaborative, experiential, and school-based (Dudley et al., 2019 ; Murata et al., 2012 ; Takahashi et al., 2013 ). Consequently, LS would promote teachers to work collaboratively to understand the TTP approach, plan TTP practices for their educational context, observe what it looks like in practice, and assess the impact on their students’ thinking (Takahashi et al., 2013 ). In particular, the MTEs believed that the LS phases and practices would naturally support TTP structures, emphasizing task selection and anticipating students’ solutions. Given Lester’s ( 2013 ) assertion that each problem solving experience a teacher engages in can potentially alter their knowledge for teaching problem solving, the MTEs sought to explore teachers’ perceptions of the impact of engaging with TTP through LS on their beliefs regarding problem solving and their knowledge for teaching problem solving.

Research questions

This paper examines two research questions:

  • Research question 1: What are elementary teachers’ reported problem solving practices prior to engaging in LS?
  • Research question 2: What are elementary teachers’ perceptions of what they learned from engaging with TTP through LS?

Methodology

Participants.

The MTEs worked with 19 elementary teachers (16 female, three male) from eight urban schools. Schools were paired to create four LS groups on the basis of the grade taught by participating class teachers, e.g. Grade 3 teacher from school 1 paired with Grade 4 teacher from school 2. Each LS group generally consisted of 4–5 teachers, with a minimum of two teachers from each school, along with the two MTEs. For most teachers, LS and TTP were new practices being implemented concurrently. However, given the acknowledged overlap between the features of the TTP and LS approaches, for example, the focus on problem posing and predicting student strategies, the researchers were confident that the content and structure were compatible. Also, in Japan, LS is commonly used to promote TTP implementation among teachers (Stacey, 2018 ; Takahashi et al., 2013 ).

All ethical obligations were adhered to throughout the research process, and the study received ethical approval from the researchers’ institutional board. Of the 19 participating LS teachers invited to partake in the research study, 16 provided informed consent to use their data for research purposes.

Over eight weeks, the MTEs worked with teachers, guiding each LS group through the four LS phases involving study, design, implementation, and reflection of a research lesson that focused on TTP while assuming the role of ‘knowledgeable others’ (Dudley et al., 2019 ; Hourigan & Leavy, 2021 ; Takahashi & McDougal, 2016 ). An overview of the timeline and summary of each LS phase is presented in Table ​ Table1 1 .

Lesson Study phases and timeline

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LS phase 1: Study

This initial study phase involved a one-day workshop. The process and benefits of LS as a school-based form of professional development were discussed in the morning session and the afternoon component was spent focusing on the characteristics of TTP. Teachers experienced the TPP approach first-hand by engaging in the various lesson stages. For example, they solved a problem (growing pattern problem) themselves in pairs and shared their strategies. They also predicted children’s approaches to the problem and possible misconceptions and watched the video cases of TTP classroom practice for this problem. Particular focus was placed on the importance of problem selection and prediction of student strategies before the lesson implementation and the Neriage stage of the lesson. Teachers also discussed readings related to LS practices (e.g. Lewis & Tsuchida, 1998 ) and TTP (e.g., Takahashi, 2008 ). At the end of the workshop, members of each LS group were asked to communicate among themselves and the MTEs, before the planning phase, to decide the specific mathematics focus of their LS group’s TTP lesson (Table ​ (Table1 1 ).

LS phase 2: Planning

The planning phase was four weeks in duration and included two 1½ hour face-to-face planning sessions (i.e. planning meetings 1 and 2) between the MTEs and each LS group (Table ​ (Table1). 1 ). Meetings took place in one of the LS group’s schools. At the start of the first planning meeting, time was dedicated to Takahashi’s ( 2008 ) work focusing on the importance of problem selection and prediction of student strategies to plan the Neriage stage of the TTP lesson. The research lesson plan structure was also introduced. Ertle et al.’s ( 2001 ) four column lesson plan template was used. It was considered particularly compatible with the TTP approach, given the explicit attention to expected student response and the teacher’s response to student activity/response.

The planning then moved onto the content focus of each LS group’s TTP research lesson. LS groups selected TTP research lessons focusing on number (group A), growing patterns (group B), money (group C), and 3D shapes (group D). Across the planning phase, teachers invested substantial time extensively discussing the TTP lesson goals in terms of target mathematics content, developing or modifying a problem to address these goals, and exploring considerations for the various lesson stages. Drawing on Takahashi’s ( 2008 ) article, it was re-emphasised that no strategies would be explicitly taught before students engaged with the problem. While one LS group modified an existing problem (group B) (Hourigan & Leavy, 2015 ), the other three LS groups posed an original problem. To promote optimum teacher readiness to lead the Neriage stage, each LS group was encouraged to solve the problem themselves in various ways considering possible student strategies and their level of mathematical complexity, thus identifying the most appropriate sequence of sharing solutions.

LS phase 3: Implementation

The implementation phase involved one teacher in each LS group teaching the research lesson (teach 1) in their school. The remaining group members and MTEs observed and recorded students’ responses. Each LS group and the MTEs met immediately for a post-lesson discussion to evaluate the research lesson. The MTEs presented teachers with a series of focus questions: What were your observations of student learning? Were the goals of the lesson achieved? Did the problem support students in developing the appropriate understandings? Were there any strategies/errors that we had not predicted? How did the Neriage stage work? What aspects of the lesson plan should be reconsidered based on this evidence? Where appropriate, the MTEs drew teachers’ attention to particular lesson aspects they had not noticed. Subsequently, each LS group revised their research lesson in response to the observations, reflections, and discussion. The revised lesson was retaught 7–10 days later by a second group member from the paired LS group school (teach 2) (Table ​ (Table1). 1 ). The post-lesson discussion for teach 2 focused mainly on the impact of changes made after the first implementation on student learning, differences between the two classes, and further changes to the lesson.

LS phase 4: Reflection

While reflection occurred after both lesson implementations, the final reflection involved all teachers from the eight schools coming together for a half-day meeting in the local Education Centre to share their research lessons, experiences, and learning (Table ​ (Table1). 1 ). Each LS group made a presentation, identifying their research lesson’s content focus and sequence of activity. Artefacts (research lesson plan, materials, student work samples, photos) were used to support observations, reflections, and lesson modifications. During this meeting, teachers also reflected privately and in groups on their initial thoughts and experience of both LS and TTP, the benefits of participation, the challenges they faced, and they provided suggestions for future practice.

Data collection

The study was a collective case study (Stake, 1995 ). Each LS group constituted a case; thus, the analysis was structured around four cases. Data collection was closely aligned with and ran concurrent to the LS process. Table ​ Table2 2 details the links between the LS phases and the data collection process.

Overview of data collection methods across the research cycles

* MTE  mathematics teacher educator

The principal data sources (Table ​ (Table2) 2 ) included both MTEs’ fieldnotes (phase (P) 1–4), and reflections (P1–4), alongside email correspondence (P1–4), individual teacher reflections (P1, 2, 4) (see reflection tasks in Table ​ Table3), 3 ), and LS documentation including various drafts of lesson plans (P2–4) and group presentations (P4). Fieldnotes refer to all notes taken by MTEs when working with the LS groups, for example, during the study session, planning meetings, lesson implementations, post-lesson discussions, and the final reflection session.

Teacher reflection focus questions

The researchers were aware of the limitations of self-report data and the potential mismatch between one’s perceptions and reality. Furthermore, data in the form of opinions, attitudes, and beliefs may contain a certain degree of bias. However, this paper intentionally focuses solely on the teachers’ perceived learning in order to represent their ‘lived experience’ of TTP. Despite this, measures were taken to assure the trustworthiness and rigour of this qualitative study. The researchers engaged with the study over a prolonged period and collected data for each case (LS group) at every LS phase (Table ​ (Table2). 2 ). All transcripts reflected verbatim accounts of participants’ opinions and reflections. At regular intervals during the study, research meetings interrogated the researchers’ understandings, comparing participating teachers’ observations and reflections to promote meaning-making (Creswell, 2009 ; Suter, 2012 ).

Data analysis

The MTEs’ role as participant researchers was considered a strength of the research given that they possessed unique insights into the research context. A grounded theory approach was adopted, where the theory emerges from the data analysis process rather than starting with a theory to be confirmed or refuted (Glaser, 1978 ; Strauss & Corbin, 1998 ). Data were examined focusing on evidence of participants’ problem solving practices prior to LS and their perceptions of their learning as a result of engaging with TTP through LS. A systematic process of data analysis was adopted. Initially, raw data were organised into natural units of related data under various codes, e.g. resistance, traditional approach, ignorance, language, planning, fear of student response, relevance, and underestimation. Through successive examinations of the relationship between existing units, codes were amalgamated (Creswell, 2009 ). Progressive drafts resulted in the firming up of several themes. Triangulation was used to establish consistency across multiple data sources. While the first theme, Vast divide between prevalent problem solving practices and TTP , addresses research question 1, it is considered an overarching theme, given the impact of teachers’ established problem solving understandings and practices on their receptiveness to and experience of TTP. The remaining five themes ( Seeing is believing : the value of practice centred experiences ; A gained appreciation of the relevance and value of TTP practices ; Enhanced problem posing understandings ; Awakening to students’ problem solving potential ; and Reservations regarding TTP) represent a generalised model of teachers’ perceived learning due to engaging with TTP through LS, thus addressing research question 2. Although one of the researchers was responsible for the initial coding, both researchers met regularly during the analysis to discuss and interrogate the established codes and to agree on themes. This process served to counteract personal bias (Suter, 2012 ).

As teacher reflections were anonymised, it was not possible to track teachers across LS phases. Consequently, teacher reflection data are labelled as phase and instrument only. For example, ‘P2, teacher reflection’ communicates that the data were collected during LS phase 2 through teacher reflection. However, the remaining data are labelled according to phase, instrument, and source, e.g. ‘P3, fieldnotes: group B’. While phase 4 data reflect teachers’ perceptions after engaging fully with the TTP approach, data from the earlier phases reflect teachers’ evolving perceptions at a particular point in their unfolding TTP experience.

Discussion of findings

The findings draw on the analysis of the data collected across the LS phases and address the research questions. Within the confines of this paper, illustrative quotes are presented to provide insights into each theme. An additional layer of analysis was completed to ensure a balanced representation of teachers’ views in reporting findings. This process confirmed that the findings represent the views of teachers across LS groups, for example, within the first theme presented ( Vast divide ), the eight quotes used came from eight different teacher reflections. Equally, the six fieldnote excerpts selected represent six different teachers’ views across the four LS groups. Furthermore, in the second theme ( Seeing is believing ), the five quotes presented were sourced from five different participating teachers’ reflections and the six fieldnote excerpts included are from six different teachers across the four LS groups. Subsequent examination of the perceptions of those teachers not included in the reporting of findings confirmed that their perspectives were represented within the quotes used. Hence, the researchers are confident that the findings represent the views of teachers across all LS groups. For each theme, sources of evidence that informed the presented conclusions will be outlined.

Vast divide between prevalent problem solving practices and TTP

This overarching theme addresses the research question ‘What were elementary teachers’ reported problem solving practices prior to engaging in LS?’.

At the start of the initiative, within the study session (fieldnotes), all teachers identified mathematics problem solving as a problem of practice. The desire to develop problem solving practices was also apparent in some teachers’ reflections (phase 1 (P1), N  = 8):

I am anxious about it. Problem solving is an area of great difficulty throughout our school (P1, teacher reflection).

During both study and planning phase discussions, across all LS groups, teachers’ reports suggested the almost exclusive use of a teaching for problem solving approach, with no awareness of the Teaching Through Problem Solving (TTP) approach; a finding also evidenced in both teacher reflections (P1, N  = 7) and email correspondence:

Unfamiliar, not what I am used to. I have no experience of this kind of problem solving. This new approach is the reverse way to what I have used for problem solving (P1, teacher reflection) Being introduced to new methods of teaching problem solving and trying different approaches is both exciting and challenging (P1: email correspondence)

Teachers’ descriptions of their problem solving classroom practices in both teacher reflections (P1, N  = 8) and study session discussions (fieldnotes) suggested a naïve conception of problem solving, using heuristics such as the ‘RUDE (read, underline, draw a picture, estimate) strategy’ (P1, fieldnotes) to support students in decoding and solving the problem:

In general, the problem solving approach described by teachers is textbook-led, where concepts are taught context free first and the problems at the end of the chapter are completed afterwards (P1, reflection: MTE2)

This approach was confirmed as widespread across all LS groups within the planning meetings (fieldnotes).

In terms of problem solving instruction, a teacher-directed approach was reported by some teachers within teacher reflections (P1, N  = 5), where the teacher focused on a particular strategy and modelled its use by solving the problem:

I tend to introduce the problem, ensure everyone understands the language and what is being asked. I discuss the various strategies that children could use to solve the problem. Sometimes I demonstrate the approach. Then children practice similar problems … (P1: Teacher reflection)

However, it was evident within the planning meetings, that this traditional approach to problem solving was prevalent among the teachers in all LS groups. During the study session (field notes and teacher reflections (P1, N  = 7)), there was a sense that problem solving was an add-on as opposed to an integral part of mathematics teaching and learning. Again, within the planning meetings, discussions across all four LS groups verified this:

Challenge: Time to focus on problems not just computation (P1: Teacher reflection). From our discussions with the various LS groups’ first planning meeting, text-based teaching seems to be resulting in many teachers teaching concepts context-free initially and then matching the concept with the relevant problems afterwards (P2, reflection: MTE1)

However, while phase 1 teacher reflections suggested that a small number of participating teachers ( N  = 4) possessed broader problem solving understandings, subsequently during the planning meetings, there was ample evidence (field notes) of problem-posing knowledge and the use of constructivist-oriented approaches that would support the TTP approach among some participating teachers in each of the LS groups:

Challenge: Spend more time on meaningful problems and give them opportunities and time to engage in activities, rather than go too soon into tricks, rhymes etc (P1, Teacher reflection). The class are already used to sharing strategies and explaining where they went wrong (P2, fieldnotes, Group B) Teacher: The problem needs to have multiple entry points (P2, fieldnotes: Group C)

While a few teachers reported problem posing practices, in most cases, this consisted of cosmetic adjustments to textbook problems. Overall, despite evidence of some promising practices, the data evidenced predominantly traditional problem solving views and practices among participating teachers, with potential for further broadening of various aspects of their knowledge for teaching problem solving including what constitutes a worthwhile problem, the role of problem posing within problem solving, and problem solving instruction. Within phase 1 teacher reflections, when reporting ‘challenges’ to problem solving practices (Table ​ (Table3), 3 ), a small number of responses ( N  = 3) supported these conclusions:

Differences in teachers’ knowledge (P1: Teacher reflection). Need to challenge current classroom practices (P1: Teacher reflection).

However, from the outset, all participating teachers consistently demonstrated robust knowledge of their students as problem solvers, evidenced in phase 1 teacher reflections ( N  = 10) and planning meeting discussions (P2, fieldnotes). However, in these early phases, teachers generally portrayed a deficit view, focusing almost exclusively on the various challenges impacting their students’ problem solving abilities. While all teachers agreed that the language of problems was inhibiting student engagement, other common barriers reported included student motivation and perseverance:

They often have difficulties accessing the problem – they don’t know what it is asking them (P2, fieldnotes: Group C) Sourcing problems that are relevant to their lives. I need to change every problem to reference soccer so the children are interested (P1: teacher reflection) Our children deal poorly with struggle and are slow to consider alternative strategies (P2, fieldnotes: Group D)

Despite showcasing a strong awareness of their students’ problem solving difficulties, teachers initially demonstrated a lack of appreciation of the benefits accrued from predicting students’ approaches and misconceptions relating to problem solving. While it came to the researchers’ attention during the study phase, its prevalence became apparent during the initial planning meeting, as its necessity and purpose was raised in three of the LS groups:

What are the benefits of predicting the children’s responses? (P1, fieldnotes). I don’t think we can predict- we will have to wait and see (P2, fieldnotes: Group A).

This finding evidences teachers’ relatively limited knowledge for teaching problem solving, given that this practice is fundamental to TTP and constructivist-oriented approaches to problem solving instruction.

Perceived impacts of engaging with TTP through LS

In response to the research question ‘What are elementary teachers’ perceptions of what they learned from engaging with TTP through LS?’, thematic data analysis identified 5 predominant themes, namely, Seeing is believing : the value of practice centred experiences ; A gained appreciation of the relevance and value of TTP practices ; Enhanced problem posing understandings ; Awakening to students’ problem solving potential ; and Reservations regarding TTP.

Seeing is believing: the value of practice centred experiences

Teachers engaged with TTP during the study phase as both learners and teachers when solving the problem. They were also involved in predicting and analysing student responses when viewing the video cases, and engaged in extensive reading, discussion, and planning for their selected TTP problem within the planning phase. Nevertheless, teachers reported reservations about the relevance of TTP for their context within both phase 2 teacher reflections ( N  = 5) as well as within the planning meeting discourse of all LS groups. Teachers’ keen awareness of their students’ problem solving challenges, coupled with the vast divide between the nature of their prior problem solving practices and the TTP approach, resulted in teachers communicating concern regarding students’ possible reaction during the planning phase:

I am worried about the problem. I am concerned that if the problem is too complex the children won’t respond to it (P2, fieldnotes: Group B) The fear that the children will not understand the lesson objective. Will they engage? (P2, Teacher reflection)

Acknowledging their apprehension regarding students’ reactions to TTP, from the outset, all participating teachers communicated a willingness to trial TTP practices:

Exciting to be part of. Eager to see how it will pan out and the learning that will be taken from it (P1, teacher reflection) They should be ‘let off’ (P2, fieldnotes: Group A).

It was only within the implementation phase, when teachers received the opportunity to meaningfully observe the TTP approach in their everyday work context, with their students, that they explicitly demonstrated an appreciation for the value of TTP practices. It was evident from teacher commentary across all LS groups’ post-lesson discussions (fieldnotes) as well as in teacher reflections (P4, N  = 10) that observing first-hand the high levels of student engagement alongside students’ capacity to engage in desirable problem solving strategies and demonstrate sought-after dispositions had affected this change:

Class teacher: They engaged the whole time because it was interesting to them. The problem is core in terms of motivation. It determines their willingness to persevere. Otherwise, it won’t work whether they have the skills or not (P3, fieldnotes: Group C) LS group member: The problem context worked really well. The children were all eager and persevered. It facilitated all to enter at their own level, coming up with ideas and using their prior knowledge to solve the problem. Working in pairs and the concrete materials were very supportive. It’s something I’d never have done before (P3, fieldnotes: Group A)

Although all teachers showcased robust knowledge of their students’ problem solving abilities prior to engaging in TTP, albeit with a tendency to focus on their difficulties and factors that inhibited them, teachers’ contributions during post-lesson discussions (fieldnotes) alongside teacher reflections (P4, N  = 9) indicate that observing TTP in action supported them in developing an appreciation of value of the respective TTP practices, particularly the role of prediction and observation of students’ strategies/misconceptions in making the students’ thinking more visible:

You see the students through the process (P3, fieldnotes: Group C) It’s rare we have time to think, to break the problem down, to watch and understand children’s ways of thinking/solving. It’s really beneficial to get a chance to re-evaluate the teaching methods, to edit the lesson, to re-teach (P4, teacher reflection)

Analysis of the range of data sources across the phases suggests that it was the opportunity to experience TTP in practice in their classrooms that provided the ‘proof of concept’:

I thought it wasn’t realistic but bringing it down to your own classroom it is relevant (P4, teacher reflection).

Hence from the teachers’ perspective, they witnessed the affordances of TTP practices in the implementation phase of the LS process.

A gained appreciation of the relevance and value of TTP practices

While during the early LS phases, teachers’ reporting suggested a view of problem solving as teaching to problem solve, data from both fieldnotes (phases 3 and 4) and teacher reflections (phase 4) demonstrate that all teachers broadened their understanding of problem solving as a result of engaging with TTP:

Interesting to turn lessons on their head and give students the chance to think, plan and come up with possible strategies and solutions (P4, Teacher reflection)

On witnessing the affordances of TTP first-hand in their own classrooms, within both teacher reflections (P4, N  = 12) and LS group presentations, the teachers consistently reported valuing these new practices:

I just thought the whole way of teaching was a good way, an effective way of teaching. Sharing and exploring more than one way of solving is vital (P4, teacher reflection) There is a place for it in the classroom. I will use aspects of it going forward (P4, fieldnotes: Group C)

In fact, teachers’ support for this problem solving approach was apparent in phase 3 during the initial post-lesson discussions. It was particularly notable when a visitor outside of the LS group who observed teach 1 challenged the approach, recommending the explicit teaching of strategies prior to engagement. A LS group member’s reply evidenced the group’s belief that TTP naturally exposes students to the relevant learning: ‘Sharing and questioning will allow students to learn more efficient strategies [other LS group members nodding in agreement]’ (P3, fieldnotes; Group A).

In turn, within phase 4 teacher reflections, teachers consistently acknowledged that engaging with TTP through LS had challenged their understandings about what constitutes effective problem solving instruction ( N  = 12). In both teacher reflections (P4, N  = 14) and all LS group presentations, teachers reported an increased appreciation of the benefits of adopting a constructivist-oriented approach to problem solving instruction. Equally for some, this was accompanied by an acknowledgement of a heightened awareness of the limitations of their previous practice :

Really made me re-think problem solving lesson structures. I tend to spoon-feed them …over-scaffold, a lot of teacher talk. … I need to find a balance… (P4, teacher reflection) Less is more, one problem can be the basis for an entire lesson (P4, teacher reflection)

What was unexpected, was that some teachers (P4, N  = 8) reported that engaging with TTP through LS resulted in them developing an increased appreciation of the value of problem solving and the need for more regular opportunities for students to engage in problem solving:

I’ve come to realise that problem solving is critical and it should be focused on more often. I feel that with regular exposure to problems they’ll come to love being problem solvers (P4, teacher reflection)

Enhanced problem posing understandings

In the early phases of LS, few teachers demonstrated familiarity with problem characteristics (P2 teacher reflection, N  = 5). However, there was growth in teachers’ understandings of what constitutes a worthwhile problem and its role within TTP within all LS groups’ post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 10):

I have a deepened understanding of how to evaluate a problem (P2, teacher reflection) It’s essential to find or create a good problem with multiple strategies and/or solutions as a springboard for a topic. It has to be relevant and interesting for the kids (P4, teacher reflection)

As early as the planning phase, a small group of teachers’ reflections ( N  = 2) suggested an understanding that problem posing is an important aspect of problem solving that merits significant attention:

It was extremely helpful to problem solve the problem (P2, teacher reflection)

However, during subsequent phases, this realisation became more mainstream, evident within all LS groups’ post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 12):

During the first planning meeting, I was surprised and a bit anxious that we would never get to having created a problem. In hindsight, this was time well spent as the problem was crucial (P4, teacher reflection) I learned the problem is key. We don’t spend enough time picking the problem (P4, fieldnotes: Group C).

Alongside this, in all LS groups’ dialogues during the post-lesson discussion and presentations (fieldnotes) and teacher reflections (P4, N  = 15), teachers consistently demonstrated an enhanced awareness of the interdependence between the quality of the problem and students’ problem solving behaviours:

Better perseverance if the problem is of interest to them (P4, teacher reflection) It was an eye-opener to me, relevance is crucial, when the problem context is relevant to them, they are motivated to engage and can solve problems at an appropriate level…They all wanted to present (P3, fieldnotes: Group C)

The findings suggest that engaging with TTP through LS facilitated participating teachers to develop an enhanced understanding of the importance of problem posing and in identifying the features of a good mathematics problem, thus developing their future problem posing capacity. In essence, the opportunity to observe the TTP practices in their classrooms stimulated an enhanced appreciation for the value of meticulous attention to detail in TTP planning.

Awakening to students’ problem solving potential

In the final LS phases, teachers consistently reported that engaging with TTP through LS provided the opportunity to see the students through the process , thus supporting them in examining their students’ capabilities more closely. Across post-lesson discussions and presentations (fieldnotes) and teacher reflections (P4, N  = 14), teachers acknowledged that engagement in core TTP practices, including problem posing, prediction of students’ strategies during planning, and careful observation of approaches during the implementation phase, facilitated them to uncover the true extent of their students’ problem solving abilities, heightening their awareness of students’ proficiency in using a range of approaches:

Class teacher: While they took a while to warm up, I am most happy that they failed, tried again and succeeded. They all participated. Some found a pattern, others used trial and error. Others worked backward- opening the cube in different ways. They said afterward ‘That was the best maths class ever’ (P3, fieldnotes: Group D) I was surprised with what they could do. I have learned the importance of not teaching strategies first. I need to pull back and let the children solve the problems their own way and leave discussing strategies to the end (P4, teacher reflection)

In three LS groups, class teachers acknowledged in the post-lesson discussion (fieldnotes) that engaging with TTP had resulted in them realising their previous underestimation of [some or all] of their students’ problem solving abilities . Teacher reflections (P4, N  = 8) and LS group presentations (fieldnotes) also acknowledged this reality:

I underestimated my kids, which is awful. The children surprised me with the way they approached the problem. In the future I need to focus on what they can do as much as what might hinder them…they are more able than we may think (P4, reflection)

In all LS groups, teachers reported that their heightened appreciation of students’ problem solving capacities promoted them to use a more constructivist-orientated approach in the future:

I learned to trust the students to problem solve, less scaffolding. Children can be let off to explore without so much teacher intervention (P3, fieldnotes: Group D)

Some teachers ( N  = 3) also acknowledged the affective benefits of TTP on students:

I know the students enjoyed sharing their different strategies…it was great for their confidence (P4, teacher reflection)

Interestingly, in contrast with teachers’ initial reservations, their experiential and school-based participation in TTP through LS resulted in a lessening of concern regarding the suitability of TTP practices for their students. Hence, this practice-based model supported teachers in appreciating the full extent of their students’ capacities as problem solvers.

Reservations regarding TTP

When introduced to the concept of TTP in the study session, one teacher quickly addressed the time implications:

It is unrealistic in the everyday classroom environment. Time is the issue. We don’t have 2 hours to prep a problem geared at the various needs (P1, fieldnotes)

Subsequently, across the initiative, during both planning meetings, the reflection session and individual reflections (P4, N  = 14), acknowledgements of the affordances of TTP practices were accompanied by questioning of its sustainability due to the excessive planning commitment involved:

It would be hard to maintain this level of planning in advance of the lesson required to ensure a successful outcome (P4, teacher reflection)

Given the extensive time dedicated to problem posing, solving, prediction, and design of questions as well as selection or creation of materials both during and between planning meetings, there was agreement in the reflection session (fieldnotes) and in teacher reflections (P4, N  = 10) that while TTP practices were valuable, in the absence of suitable support materials for teachers, adjustments were essential to promote implementation:

There is definitely a role for TTP in the classroom, however the level of planning involved would have to be reduced to make it feasible (P4, teacher reflection) The TTP approach is very effective but the level of planning involved is unrealistic with an already overcrowded curriculum. However, elements of it can be used within the classroom (P4, teacher reflection)

A few teachers ( N  = 3) had hesitations beyond the time demands, believing the success of TTP is contingent on ‘a number of criteria…’ (P4, teacher reflection):

A whole-school approach is needed, it should be taught from junior infants (P4, teacher reflection) I still have worries about TTP. We found it difficult to decide a topic initially. It lends itself to certain areas. It worked well for shape and space (P4, teacher reflection)

Conclusions

The reported problem solving practice reflects those portrayed in the literature (NCCA, 2016 ; O’Shea & Leavy, 2013 ) and could be aptly described as ‘pendulum swings between emphases on basic skills and problem solving’ (Lesh & Zawojewski, 2007 in Takahashi et al., 2013 , p. 239). Teachers’ accounts depicted problem solving as an ‘add on’ occurring on an ad hoc basis after concepts were taught (Dooley et al., 2014 ; Takahashi et al., 2013 ), suggesting a simplistic view of problem solving (Singer & Voica, 2013 ; Swan, 2006 ). Hence, in reality there was a vast divide between teachers’ problem solving practices and TTP. Alongside traditional beliefs and problem solving practices (Stipek et al., 2001 ; Swan, 2006 ; Thompson, 1985 ), many teachers demonstrated limited insight regarding what constitutes a worthwhile problem (Klein & Lieken, 2020 ) or the critical role of problem posing in problem solving (Cai, 2003 ; Takahashi, 2008 ; Watson & Ohtani, 2015 ). Teachers’ reports suggested most were not actively problem posing, with reported practices limited to cosmetic changes to the problem context (Koichu et al., 2013 ). Equally, teachers demonstrated a lack of awareness of alternative approaches to teaching for problem solving (Chapman, 2015 ) alongside limited appreciation among most of the affordances of a more child-centred approach to problem solving instruction (Hiebert, 2003 ; Lester, 2013 ; Swan, 2006 ). Conversely, there was evidence that some teachers held relevant problem posing knowledge and utilised practices compatible with the TTP approach.

All teachers displayed relatively strong understandings of their students as problem solvers from the outset; however, they initially focused almost exclusively on factors impacting students’ limited problem solving capacity (Chapman, 2015 ). Teachers’ perceptions of their students’ problem solving abilities alongside the vast divide between teachers’ problem solving practice and the proposed TTP approach resulted in teachers being initially concerned regarding students’ response to TTP. This finding supports studies that reported resistance by teachers to the use of challenging tasks due to fears that students would not be able to manage (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2010 ). Equally, teachers communicated disquiet from the study phase regarding the time investment required to adopt the TTP approach, a finding common in similar studies (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Hence, the transition to TTP was uneasy for most teachers, given the significant shift it represented in terms of moving beyond a teaching to problem solve approach alongside the range of teacher demands (Takahashi et al., 2013 ).

Nevertheless, despite initial reservations, all teachers reported that engagement with TTP through LS affected their problem solving beliefs and understandings. What was particularly notable was that they reported an awakening to students’ problem solving potential . During LS’s implementation and reflection stages, all teachers acknowledged that seeing was believing concerning the benefits of TTP for their students (Kapur, 2010 ; Stacey, 2018 ). In particular, they recognised students’ positive response (Russo & Minas, 2020 ) enacted in high levels of engagement, perseverance in finding a solution, and the utilisation of a range of different strategies. These behaviours were in stark contrast to teachers’ reports in the study phase. Teachers acknowledged that students had more potential to solve problems autonomously than they initially envisaged. This finding supports previous studies where teachers reported that allowing students to engage with challenging tasks independently made students’ thinking more visible (Crespo & Featherstone, 2006 ; Ingram et al., 2020 ; Sakshand & Wohluter, 2010 ). It also reflects Sakshaug and Wohlhuter’s ( 2010 ) findings of teachers’ tendency to underestimate students’ potential to solve problems. Interestingly, at the end of LS, concern regarding the appropriateness of the TTP approach for students was no longer cited by teachers. This finding contrasts with previous studies that report teacher resistance due to fears that students will become disengaged due to the unsuitability of the approach (challenging tasks) for lower-performing students (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ). Hence, engaging with TTP through LS supported teachers in developing an appreciation of their students’ potential as problem solvers.

Teachers reported enhanced problem posing understandings, consisting of newfound awareness of the connections between the quality of the problem, the approach to problem solving instruction, and student response (Chapman, 2015 ; Cai, 2003 ; Sullivan et al., 2015 ; Takahashi, 2008 ). They acknowledged that they had learned the importance of the problem in determining the quality of learning and affecting student engagement, motivation and perseverance, and willingness to share strategies (Cai, 2003 ; Watson & Oktani, 2015 ). These findings reflect previous research reporting that engagement in LS facilitated teachers to enhance their teacher knowledge (Cajkler et al., 2015 ; Dudley et al., 2019 ; Gutierez, 2016 ).

While all teachers acknowledged the benefits of the TTP approach for students (Cai & Lester, 2010 ; Sullivan et al., 2014 ; Takahashi, 2016 ), the majority confirmed their perception of the relevance and value of various TTP practices (Hiebert, 2003 ; Lambdin, 2003 ; Takahashi, 2006 ). They referenced the benefits of giving more attention to the problem, allowing students the opportunity to independently solve, and promoting the sharing of strategies and pledged to incorporate these in their problem solving practices going forward. Many verified that the experience had triggered them to question their previous problem solving beliefs and practices (Chapman, 2015 ; Lester, 2013 ; Takahashi et al., 2013 ). This study supports previous research reporting that LS challenged teachers’ beliefs regarding the characteristics of effective pedagogy (Cajkler et al., 2015 ; Dudley et al., 2019 ; Fernandez, 2005 ; Gutierez, 2016 ). However, teachers communicated reservations regarding TTP , refraining from committing to TTP in its entirety, highlighting that the time commitment required for successful implementation on an ongoing basis was unrealistic. Therefore, teachers’ issues with what they perceived to be the excessive resource implications of TTP practices remained constant across the initiative. This finding supports previous studies that report teachers were resistant to engaging their students with ‘challenging tasks’ provided by researchers due to the time commitment required to plan adequately (Ingram et al., 2020 ; Russo & Hopkins, 2019 ; Sullivan et al., 2015 ).

Unlike previous studies, teachers in this study did not perceive weak mathematics content or pedagogical content knowledge as a barrier to implementing TTP (Charalambous, 2008 ; Sakshaug & Wohlhuter, 2010 ). However, it should be noted that the collaborative nature of LS may have hidden the knowledge demands for an individual teacher working alone when engaging in the ‘Anticipate’ element of TTP particularly in the absence of appropriate supports such as a bank of suitable problems.

The findings suggest that LS played a crucial role in promoting reported changes, serving both as a supportive professional development model (Stacey, 2018 ; Takahashi et al., 2013 ) and as a catalyst, providing teachers with the opportunity to engage in a collaborative, practice-centred experience over an extended period (Dudley et al., 2019 ; Watanabe, 2001 ). The various features of the LS process provided teachers with opportunities to engage with, interrogate, and reflect upon key TTP practices. Reported developments in understandings and beliefs were closely tied to meaningful opportunities to witness first-hand the affordances of the TTP approach in their classrooms with their students (Dudley et al., 2019 ; Fernandez et al., 2003 ; Takahashi et al., 2013 ). We suggest that the use of traditional ‘one-off’ professional development models to introduce TTP, combined with the lack of support during the implementation phase, would most likely result in teachers maintaining their initial views about the unsuitability of TTP practices for their students.

In terms of study limitations, given that all data were collected during the LS phases, the findings do not reflect the impact on teachers’ problem solving classroom practice in the medium to long term. Equally, while acknowledging the limitations of self-report data, there was no sense that the teachers were trying to please the MTEs, as they were forthright when invited to identify issues. Also, all data collected through teacher reflection was anonymous. The relatively small number of participating teachers means that the findings are not generalisable. However, they do add weight to the body of relevant research. This study also contributes to the field as it documents potential challenges associated with implementing TTP for the first time. It also suggests that despite TTP being at odds with their problem solving practice and arduous, the opportunity to experience the impact of the TTP approach with students through LS positively affected teachers’ problem solving understandings and beliefs and their commitment to incorporating TTP practices in their future practice. Hence, this study showcases the potential role of collaborative, school-based professional development in supporting the implementation of upcoming reform proposals (Dooley et al., 2014 ; NCCA, 2016 , 2017 , 2020 ), in challenging existing beliefs and practices and fostering opportunities for teachers to work collaboratively to trial reform teaching practices over an extended period (Cajkler et al., 2015 ; Dudley et al., 2019 ). Equally, this study confirms and extends previous studies that identify time as an immense barrier to TTP. Given teachers’ positivity regarding the impact of the TTP approach, their consistent acknowledgement of the unsustainability of the unreasonable planning demands associated with TTP strengthens previous calls for the development of quality support materials in order to avoid resistance to TTP (Clarke et al., 2014 ; Takahashi, 2016 ).

The researchers are aware that while the reported changes in teachers’ problem solving beliefs and understandings are a necessary first step, for significant and lasting change to occur, classroom practice must change (Sakshaug & Wohlhuter, 2010 ). While it was intended that the MTEs would work alongside interested teachers and schools to engage further in TTP in the school term immediately following this research and initial contact had been made, plans had to be postponed due to the commencement of the COVID 19 pandemic. The MTEs are hopeful that it will be possible to pick up momentum again and move this initiative to its natural next stage. Future research will examine these teachers’ perceptions of TTP after further engagement and evaluate the effects of more regular opportunities to engage in TTP on teachers’ problem solving practices. Another possible focus is teachers’ receptiveness to TTP when quality support materials are available.

In practical terms, in order for teachers to fully embrace TTP practices, thus facilitating their students to avail of the many benefits accrued from engagement, teachers require access to professional development (such as LS) that incorporates collaboration and classroom implementation at a local level. However, quality school-based professional development alone is not enough. In reality, a TTP approach cannot be sustained unless teachers receive access to quality TTP resources alongside formal collaboration time.

Acknowledgements

The authors acknowledge the participating teachers’ time and contribution to this research study.

This work was supported by the Supporting Social Inclusion and Regeneration in Limerick’s Programme Innovation and Development Fund.

Declarations

We have received ethical approval for the research presented in this manuscript from Mary Immaculate College Research Ethical Committee (MIREC).

The manuscript has only been submitted to Mathematics Education Research Journal. All authors have approved the manuscript submission. We also acknowledge that the submitted work is original and the content of the manuscript has not been published or submitted for publication elsewhere.

Informed consent has been received for all data included in this study. Of the 19 participating teachers, 16 provided informed consent.

The authors declare no competing interests.

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Voices | Teaching and Learning

We need more math teachers. here’s how to prepare them for life in the classroom., by cicely woodard     mar 27, 2024.

We Need More Math Teachers. Here’s How to Prepare Them for Life in the Classroom.

Monkey Business Images / Shutterstock

During the day, I teach Algebra I classes to high school freshmen in Springfield, Missouri. One night per week, I teach preservice elementary school teachers who serve as paraprofessionals at K-12 schools in Louisiana, Arkansas, Alabama and California through Reach University. Reach University offers adults employed in schools and other workplaces the opportunity to earn a unique bachelor’s degree that embraces work experience as part of the learning process. After earning this degree and passing certification tests required by some states, candidates are qualified to teach.

The ninth-graders in my classes and the preservice teachers I teach have one thing in common: math has not always come easy for them, and for many in both groups, learning math can be overwhelming.

Over the past 20 years of my education career, I've heard my ninth grade students and preservice teachers say things like, “I have never been good at math” and “I am not a math person.” Confirmingly, research shows that adult learners self-report lower levels of math self-efficacy and higher levels of math anxiety than traditional undergraduates. These findings further exacerbate the challenge of training math-confident educators as our nation works to address unfinished math learning throughout K-12 due to the pandemic.

As a professor of practice whose goal is to prepare preservice teachers to lead their own classrooms, how do I ensure that these preservice teachers know math content well and feel equipped and prepared to teach math to students?

My answer: an immersive, 15-credit-hour semester of math. During the 15 weeks, there is an intentional focus on learning math content through a math reasoning course aligned to content that preservice teachers will see on the Praxis Elementary Education: Mathematics Subtest 5003 , exploring math pedagogy through a math methods course, and practicing math teaching strategies with students on their jobs as paraprofessionals through a math placements course.

A Typical Semester for Future Math Teachers

A standard math semester for preservice teacher candidates includes three key components that improve their learning: building a mathematical mindset, forming a sense of belonging that extends beyond math, and focusing on the connection between learning math content, exploring math pedagogy, and practicing teaching strategies.

The first key component is a focus on a mathematical mindset. Preservice teachers start the semester considering what it means to think like a mathematician and exploring math classroom norms created by Jo Boaler, a Stanford University mathematics education professor. As Boaler asserts, “Everyone can learn math to high levels. Mistakes are valuable. Math is about creativity, making sense, connections, and communicating.” Preservice teachers review these norms at the beginning of every class and determine what resonates with them based on the day's topic.

Second, preservice teachers need to feel a sense of connection and belonging. Adult learners often have low self-confidence when learning math; many have not been students for several years, and they report feeling anxious about taking a math class. Focusing on connection and belonging helps to raise their self-efficacy as it relates to learning math. To start each class, I ask a check-in question that has nothing to do with math:

  • What is bringing you joy right now?
  • If you could travel back in time five years, what would you tell yourself?
  • How would you describe how you are managing your workload right now?

In our virtual math reasoning course, preservice teachers can answer in the chat or share their thoughts verbally. I often get favorable reviews about this part of the class. In a recent survey, one preservice teacher wrote, “I love the beginning check-ins, not all professors care about your well-being.”

Finally, as a math department, we have intentionally created engaging lessons in math methods and math placement courses that are directly connected to what students are learning in the math reasoning course during the same semester. Math teachers need to have a deep understanding of math content and effectively teach mathematics. For that reason, I firmly believe that for preservice teachers to learn math, they must do it.

In our classes, preservice teachers do math individually so that they can develop their own reasoning, think and discuss in small groups to compare strategies and then engage in whole group discussions where their thinking is illuminated. Preservice teachers in our classes appreciate having opportunities to discuss their own ideas and analyze their classmates’ work, bringing them to the conclusion that math problems can be solved in different, creative ways.

Over the last two and a half years since I began teaching at Reach, I've seen this sequence of math courses have a tremendous impact on the preservice teachers in our classes. They spend 15 weeks thinking deeply about understanding math content while considering what it means to be an effective math teacher.

I’ve watched them transform their thinking about what it means to be a mathematician. Instead of declaring that they are not math people, by the end of the semester, they feel more confident in their math skills and have sharpened their ability to teach math to students.

What Preservice Math Teachers Need

My experience as a high school teacher and college professor has led me to three conclusions about preparing preservice teachers to teach mathematics:

  • Belonging matters in math class. Preservice teachers need to feel a sense of belonging in math class, even if they haven’t been successful at it in the past. When students feel connected to each other and the professor, walls are broken down and they are able to engage in the challenging work of learning math. Even as adults, knowing that others care about them helps them feel comfortable enough to learn.
  • Math discourse impacts what and how preservice teachers learn. Talking about math opens up new perspectives. The preservice teachers in my class get to develop their own reasoning, justify their thinking and critique the reasoning of others. Communicating about math helps candidates compare strategies, broaden their thinking and develop their own questions. Discourse also reveals misconceptions; they make mistakes and realize that their mistakes are tools for learning.
  • To deepen understanding and learning, professors must find ways to engage students in thinking. Learning math requires being allowed the time and the space to think critically about connections between concepts. During our math reasoning classes, we use various websites where students in our virtual environment can do the math, discuss their thinking, and ask questions. Desmos , Peardeck and Nearpod all have effective ways to increase engagement beyond lectures in a virtual environment. We must increase opportunities for thinking, not just mimicking in math class.

Math is hard. Teaching math is even harder. Yet, at the end of the semester, the preservice teacher candidates in my classes feel much more empowered to teach math. Our schools desperately need more math teachers, and as we’ve learned by implementing this semester-long math learning course, we can prepare preservice teachers to meet students' needs by ensuring that they leave teacher prep programs believing in themselves. Their ability to teach math gives them ample opportunities to discuss their thinking and be intentional about helping them focus on math content and pedagogy simultaneously.

When I consider the implications of classrooms being led by teachers who are masters of math content and effective practitioners, I cannot help but think of the positive impact on students like the freshmen in my algebra classes. Having teachers who foster a sense of belonging and identity in the math classroom would make a significant and lasting difference in students' lives. So many more students would be proud to declare that they are math people, prepared to think critically and empowered to face challenges wherever life takes them after high school.

Cicely Woodard (she/her) teaches Algebra I at Kickapoo High School and is a math faculty member at Reach University in Springfield, Missouri.

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Teaching mathematics through problem posing: insights from an analysis of teaching cases

  • Original Paper
  • Published: 12 April 2021
  • Volume 53 , pages 961–973, ( 2021 )

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  • Huirong Zhang 1 &
  • Jinfa Cai 2  

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In this study we aimed to understand teaching mathematics through problem posing based on an analysis of 22 teaching cases. Teaching mathematics through problem posing starts with problem-posing tasks. This study provides not only specific examples of problem-posing tasks used in classrooms but also related task variables to consider when developing problem-posing tasks. This study also contributes to our understanding of how teachers can deal with student-posed problems in the classroom. In these 22 teaching cases, there was a typical pattern to how teachers dealt with the students’ posed problems in the classroom according to the instructional goals. For future research, we need to accumulate additional teaching cases and explore possible discourse patterns concerning how teachers handle students’ posed problems, as well as identify the most effective discourse patterns when teaching mathematics through problem posing.

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Zhang, H., Cai, J. Teaching mathematics through problem posing: insights from an analysis of teaching cases. ZDM Mathematics Education 53 , 961–973 (2021). https://doi.org/10.1007/s11858-021-01260-3

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How can Australia solve the math teacher shortage? It can start by training more existing teachers to teach math

by Ian Gordon, Mary P. Coupland and Merrilyn Goos, The Conversation

math

Imagine if you enrolled your child in swimming lessons but instead of a qualified swimming instructor, they were taught freestyle technique by a soccer coach.

Something similar is happening in classrooms around Australia every day. As part of the ongoing teacher shortage, there are significant numbers of teachers teaching " out-of-field ." This means they are teaching subjects they are not qualified to teach.

One of the subjects where out-of-field teaching is particularly common is math.

A 2021 report on Australia's teaching workforce found that 40% of those teaching high school mathematics are out-of-field (English and science were 28% and 29%, respectively).

Another 2021 study of students in Year 8 found they were more likely to be taught by teachers who had specialist training in both math and math education if they went to a school in an affluent area rather than a disadvantaged one (54% compared with 31%).

Our new report looks at how we can fix this situation by training more existing teachers in math education.

Why is this a problem?

Mathematics is one of the key parts of school education. But we are seeing worrying signs students are not receiving the math education they need.

The 2021 study of Year 8 students showed those taught by teachers with a university degree majoring in math had markedly higher results, compared with those taught by out-of-field teachers.

We also know math skills are desperately needed in the broader workforce. The burgeoning worlds of big data and artificial intelligence rely on mathematical and statistical thinking, formulae and algorithms. Math has also been identified as a national skill shortages priority area .

What do we do about this?

There have been repeated efforts to address teacher shortages, including trying to retain existing mathematics teachers, having specialist teachers teaching across multiple schools and higher salaries . There is also a push to train more teachers from scratch, which of course will take many years to implement.

There is one strategy, however, that has not yet been given much attention by policy makers : upgrading current teachers' math and statistics knowledge and their skills in how to teach these subjects.

They already have training and expertise in how to teach and a commitment to the profession. Specific training in math will mean they can move from being out-of-field to "in-field".

How to give teachers this training

A new report commissioned by mathematics and statistics organizations in Australia (including the Australian Mathematical Sciences Institute) looks at what is currently available in Australia to train teachers in math.

It identified 12 different courses to give existing teachers math teaching skills. They varied in terms of location, duration (from six months to 18 months full-time) and aims.

For example, some were only targeted at teachers who want to teach math in the junior and middle years of high school. Some taught university-level math and others taught school-level math. Some had government funding support; others could cost students more than A$37,000.

Overall, we found the current system is confusing for teachers to navigate. There are complex differences between states about what qualifies a teacher to be "in-field" for a subject area.

In the current incentive environment, we found these courses cater to a very small number of teachers. For example, in 2024 in New South Wales this year there are only about 50 government-sponsored places available.

This is not adequate. Pre-COVID, it was estimated we were losing more than 1,000 equivalent full-time math teachers per year to attrition and retirement and new graduates were at best in the low hundreds.

But we don't know exactly how many extra teachers need to be trained in math. One of the key recommendations of the report is for accurate national data of every teacher 's content specializations.

We need a national approach

The report also recommends a national strategy to train more existing teachers to be math teachers. This would replace the current piecemeal approach.

It would involve a standard training regime across Australia with government and school-system incentives for people to take up extra training in math.

There is international evidence to show a major upskilling program like this could work.

In Ireland, where the same problem was identified, the government funds a scheme run by a group of universities. Since 2012, teachers have been able to get a formal qualification (a professional diploma). Between 2009 and 2018 the percentage of out-of-field math teaching in Ireland dropped from 48% to 25%.

To develop a similar scheme here in Australia, we would need coordination between federal and state governments and universities. Based on the Irish experience, it would also require several million dollars in funding.

But with students receiving crucial math lessons every day by teachers who are not trained to teach math, the need is urgent.

The report mentioned in this article was commissioned by the Australian Mathematical Sciences Institute, the Australian Mathematical Society, the Statistical Society of Australia, the Mathematics Education Research Group of Australasia and the Actuaries Institute.

Provided by The Conversation

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COMMENTS

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    As is the case in teaching mathematics through problem solving, engaging in problem posing offers a number of advantages for students (Cai et al., 2015). For example, problem-posing tasks are usually cognitively demanding, because problem posing often requires posers to reflect on the broader aspects of structure and goal.

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  29. PDF MATH 2121 Mathematical Modeling and Simulation Spring 2024 Problem Set 7

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