examples of teaching through problem solving

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving 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.

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

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 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

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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Center for Teaching

Teaching problem solving.

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Tips and Techniques

Expert vs. novice problem solvers, communicate.

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

Encourage Independence

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

Be sensitive

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

Encourage Thoroughness and Patience

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

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

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

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6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

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Teaching Through Problem-solving

• TTP in Action

What is Teaching Through Problem-Solving?

In Teaching Through Problem-solving (TTP), students learn new mathematics by solving problems. Students grapple with a novel problem, present and discuss solution strategies, and together build the next concept or procedure in the mathematics curriculum.

Teaching Through Problem-solving is widespread in Japan, where students solve problems before a solution method or procedure is taught. In contrast, U.S. students spend most of their time doing exercises– completing problems for which a solution method has already been taught.

Why Teaching Through Problem-Solving?

As students build their mathematical knowledge, they also:

• Learn to reason mathematically, using prior knowledge to build new ideas
• See the power of their explanations and carefully written work to spark insights for themselves and their classmates
• Expect mathematics to make sense
• Enjoy solving unfamiliar problems
• Experience mathematical discoveries that naturally deepen their perseverance

Phases of a TTP Lesson

Teaching Through Problem-solving flows through four phases as students 1. Grasp the problem, 2. Try to solve the problem independently, 3. Present and discuss their work (selected strategies), and 4. Summarize and reflect.

Click on the arrows below to find out what students and teachers do during each phase and to see video examples.

• 1. Grasp the Problem
• 2. Try to Solve
• 3. Present & Discuss
• 4. Summarize & Reflect
• New Learning

WHAT STUDENTS DO

• Understand the problem and develop interest in solving it.
• Consider what they know that might help them solve the problem.

WHAT TEACHERS DO

• Show several student journal reflections from the prior lesson.
• Pose a problem that students do not yet know how to solve.
• Interest students in the problem and in thinking about their own related knowledge.
• Independently try to solve the problem.
• Do not simply following the teacher’s solution example.
• Allow classmates to provide input after some independent thinking time.
• Circulate, using seating chart to note each student’s solution approach.
• Identify work to be presented and discussed at board.
• Ask individual questions to spark more thinking if some students finish quickly or don’t get started.
• Present and explain solution ideas at the board, are questioned by classmates and teacher. (2-3 students per lesson)
• Actively make sense of the presented work and draw out key mathematical points. (All students)
• Strategically select and sequence student presentations of work at the board, to build the new mathematics. (Incorrect approaches may be included.)
• Monitor student discussion: Are all students noticing the important mathematical ideas?
• Add teacher moves (questions, turn-and-talk, votes) as needed to build important mathematics.
• Consider what they learned and share their thoughts with class, to help formulate class summary of learning. Copy summary into journal.
• Write journal reflection on their own learning from the lesson.
• Write on the board a brief summary of what the class learned during the lesson, using student ideas and words where possible.
• Ask students to write in their journals about what they learned during the lesson.

How Do Teachers Support Problem-solving?

Although students do much of the talking and questioning in a TTP lesson, teachers play a crucial role. The widely-known 5 Practices for Orchestrating Mathematical Discussions were based in part on TTP . Teachers study the curriculum, anticipate student thinking, and select and sequence the student presentations that allow the class to build the new mathematics. Classroom routines for presentation and discussion of student work, board organization, and reflective mathematics journals work together to allow students to do the mathematical heavy lifting. To learn more about journals, board work, and discussion in TTP, as well as see other TTP resources and examples of TTP in action, click on the respective tabs near the top of this page.

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Teaching problem solving: Let students get ‘stuck’ and ‘unstuck’

Subscribe to the center for universal education bulletin, kate mills and km kate mills literacy interventionist - red bank primary school helyn kim helyn kim former brookings expert @helyn_kim.

October 31, 2017

This is the second in a six-part  blog series  on  teaching 21st century skills , including  problem solving ,  metacognition , critical thinking , and collaboration , in classrooms.

In the real world, students encounter problems that are complex, not well defined, and lack a clear solution and approach. They need to be able to identify and apply different strategies to solve these problems. However, problem solving skills do not necessarily develop naturally; they need to be explicitly taught in a way that can be transferred across multiple settings and contexts.

Here’s what Kate Mills, who taught 4 th grade for 10 years at Knollwood School in New Jersey and is now a Literacy Interventionist at Red Bank Primary School, has to say about creating a classroom culture of problem solvers:

Helping my students grow to be people who will be successful outside of the classroom is equally as important as teaching the curriculum. From the first day of school, I intentionally choose language and activities that help to create a classroom culture of problem solvers. I want to produce students who are able to think about achieving a particular goal and manage their mental processes . This is known as metacognition , and research shows that metacognitive skills help students become better problem solvers.

I begin by “normalizing trouble” in the classroom. Peter H. Johnston teaches the importance of normalizing struggle , of naming it, acknowledging it, and calling it what it is: a sign that we’re growing. The goal is for the students to accept challenge and failure as a chance to grow and do better.

I look for every chance to share problems and highlight how the students— not the teachers— worked through those problems. There is, of course, coaching along the way. For example, a science class that is arguing over whose turn it is to build a vehicle will most likely need a teacher to help them find a way to the balance the work in an equitable way. Afterwards, I make it a point to turn it back to the class and say, “Do you see how you …” By naming what it is they did to solve the problem , students can be more independent and productive as they apply and adapt their thinking when engaging in future complex tasks.

After a few weeks, most of the class understands that the teachers aren’t there to solve problems for the students, but to support them in solving the problems themselves. With that important part of our classroom culture established, we can move to focusing on the strategies that students might need.

Here’s one way I do this in the classroom:

I show the broken escalator video to the class. Since my students are fourth graders, they think it’s hilarious and immediately start exclaiming, “Just get off! Walk!”

When the video is over, I say, “Many of us, probably all of us, are like the man in the video yelling for help when we get stuck. When we get stuck, we stop and immediately say ‘Help!’ instead of embracing the challenge and trying new ways to work through it.” I often introduce this lesson during math class, but it can apply to any area of our lives, and I can refer to the experience and conversation we had during any part of our day.

Research shows that just because students know the strategies does not mean they will engage in the appropriate strategies. Therefore, I try to provide opportunities where students can explicitly practice learning how, when, and why to use which strategies effectively  so that they can become self-directed learners.

For example, I give students a math problem that will make many of them feel “stuck”. I will say, “Your job is to get yourselves stuck—or to allow yourselves to get stuck on this problem—and then work through it, being mindful of how you’re getting yourselves unstuck.” As students work, I check-in to help them name their process: “How did you get yourself unstuck?” or “What was your first step? What are you doing now? What might you try next?” As students talk about their process, I’ll add to a list of strategies that students are using and, if they are struggling, help students name a specific process. For instance, if a student says he wrote the information from the math problem down and points to a chart, I will say: “Oh that’s interesting. You pulled the important information from the problem out and organized it into a chart.” In this way, I am giving him the language to match what he did, so that he now has a strategy he could use in other times of struggle.

The charts grow with us over time and are something that we refer to when students are stuck or struggling. They become a resource for students and a way for them to talk about their process when they are reflecting on and monitoring what did or did not work.

For me, as a teacher, it is important that I create a classroom environment in which students are problem solvers. This helps tie struggles to strategies so that the students will not only see value in working harder but in working smarter by trying new and different strategies and revising their process. In doing so, they will more successful the next time around.

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Teaching Problem-Solving Skills

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

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

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

Principles for teaching problem solving

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

Woods’ problem-solving model

Define the problem.

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

Think about it

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

Plan a solution

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

Carry out the plan

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

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

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

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

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

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Teaching Problem Solving in Math

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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.

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.

There’s also this visuals strategy wheel practice.

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.

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!

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.

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.

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.

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.

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.

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.

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.

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!

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.

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Teaching problem solving

Strategies for teaching problem solving apply across disciplines and instructional contexts. First, introduce the problem and explain how people in your discipline generally make sense of the given information. Then, explain how to apply these approaches to solve the problem.

Introducing the problem

Explaining how people in your discipline understand and interpret these types of problems can help students develop the skills they need to understand the problem (and find a solution). After introducing how you would go about solving a problem, you could then ask students to:

• frame the problem in their own words
• define key terms and concepts
• determine statements that accurately represent the givens of a problem
• identify analogous problems
• determine what information is needed to solve the problem

Working on solutions

In the solution phase, one develops and then implements a coherent plan for solving the problem. As you help students with this phase, you might ask them to:

• identify the general model or procedure they have in mind for solving the problem
• set sub-goals for solving the problem
• identify necessary operations and steps
• draw conclusions
• carry out necessary operations

You can help students tackle a problem effectively by asking them to:

• systematically explain each step and its rationale
• explain how they would approach solving the problem
• help you solve the problem by posing questions at key points in the process
• work together in small groups (3 to 5 students) to solve the problem and then have the solution presented to the rest of the class (either by you or by a student in the group)

In all cases, the more you get the students to articulate their own understandings of the problem and potential solutions, the more you can help them develop their expertise in approaching problems in your discipline.

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Problem-solving skills are necessary in all areas of life, and classroom problem solving activities can be a great way to get students prepped and ready to solve real problems in real life scenarios. Whether in school, work or in their social relationships, the ability to critically analyze a problem, map out all its elements and then prepare a workable solution is one of the most valuable skills one can acquire in life.

Educating your students about problem solving skills from an early age in school can be facilitated through classroom problem solving activities. Such endeavors encourage cognitive as well as social development, and can equip students with the tools they’ll need to address and solve problems throughout the rest of their lives. Here are five classroom problem solving activities your students are sure to benefit from as well as enjoy doing:

1. Brainstorm bonanza

Having your students create lists related to whatever you are currently studying can be a great way to help them to enrich their understanding of a topic while learning to problem-solve. For example, if you are studying a historical, current or fictional event that did not turn out favorably, have your students brainstorm ways that the protagonist or participants could have created a different, more positive outcome. They can brainstorm on paper individually or on a chalkboard or white board in front of the class.

2. Problem-solving as a group

Have your students create and decorate a medium-sized box with a slot in the top. Label the box “The Problem-Solving Box.” Invite students to anonymously write down and submit any problem or issue they might be having at school or at home, ones that they can’t seem to figure out on their own. Once or twice a week, have a student draw one of the items from the box and read it aloud. Then have the class as a group figure out the ideal way the student can address the issue and hopefully solve it.

3. Clue me in

This fun detective game encourages problem-solving, critical thinking and cognitive development. Collect a number of items that are associated with a specific profession, social trend, place, public figure, historical event, animal, etc. Assemble actual items (or pictures of items) that are commonly associated with the target answer. Place them all in a bag (five-10 clues should be sufficient.) Then have a student reach into the bag and one by one pull out clues. Choose a minimum number of clues they must draw out before making their first guess (two- three). After this, the student must venture a guess after each clue pulled until they guess correctly. See how quickly the student is able to solve the riddle.

4. Survivor scenarios

Create a pretend scenario for students that requires them to think creatively to make it through. An example might be getting stranded on an island, knowing that help will not arrive for three days. The group has a limited amount of food and water and must create shelter from items around the island. Encourage working together as a group and hearing out every child that has an idea about how to make it through the three days as safely and comfortably as possible.

5. Moral dilemma

Create a number of possible moral dilemmas your students might encounter in life, write them down, and place each item folded up in a bowl or bag. Some of the items might include things like, “I saw a good friend of mine shoplifting. What should I do?” or “The cashier gave me an extra $1.50 in change after I bought candy at the store. What should I do?” Have each student draw an item from the bag one by one, read it aloud, then tell the class their answer on the spot as to how they would handle the situation.

Classroom problem solving activities need not be dull and routine. Ideally, the problem solving activities you give your students will engage their senses and be genuinely fun to do. The activities and lessons learned will leave an impression on each child, increasing the likelihood that they will take the lesson forward into their everyday lives.

You may also like to read

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• Activities for Teaching Tolerance in the Classroom
• 5 Problem-Solving Activities for Elementary Classrooms
• 10 Ways to Motivate Students Outside the Classroom
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• How to Engage Gifted and Talented Students in the Classroom

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Mathematics for Teaching

This site is NOT about making mathematics easy because it isn't. It is about making it make sense because it does.

Teaching through Problem Solving

Problem solving is not only the reason for teaching and learning mathematics. It is also the means for learning it. In the words of Hiebert et al:

Students should be allowed to make the subject problematic. … Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students. (Hiebert, et al, 1996, p. 12)

For years now, UP NISMED in-service training programs for teachers have organized mathematics lessons for teachers using the strategy we call Teaching through Problem Solving (TtPS). This teaching strategy had also been tried by teachers in their classes and the results far outweighed the disadvantages anticipated by the teachers.

Teaching through problem solving provides context for reviewing previously learned concepts and linking it to the new concepts to be learned. It provides context for students to experience working with the new concepts before they are formally defined and manipulated procedurally, thus making definitions and procedures meaningful to them.

What are the characteristics of a TtPS?

• main learning activity is problem solving
• concepts are learned in the context of solving a problem
• students think about math ideas without having the ideas pre-explained
• students solve problems without the teacher showing a solution to a similar problem first

What is the typical lesson sequence organized around TtPS?

• An which can be solved in many ways is posed to the class.
• Students initially work on the problem on their own then join a group to share their solutions and find other ways of solving the problem. (Role of teacher is to encourage pupils to try many possible solutions with minimum hints)
• Students studies/evaluates solutions. (Teacher ask learners questions like “Which solutions do you like most? Why?”)
• Teacher asks questions to help students make connections among concepts
• Teacher/students extend the problem.

What are the theoretical underpinnings of TtPS strategy?

• Constructivism
• Vygotsky’s Zone of Proximal Development ( ZPD )

Click here for sample lesson using Teaching through Problem Solving to teach the tangent ratio/function .

The best resource for improving one’s problem solving skills is still these books by George Polya.

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14 thoughts on “ Teaching through Problem Solving ”

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A fun addition to this, I have found, is to get the class to solve a mastermind game as a group. Cracking the code involves a reasonable amount of logical thinking and playing it as a group encourages people to learn from each other.

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Phillips Exeter Academy has their whole math curriculum designed around a problem-based system. I have adopted/adapted this for my calculus and geometry classes.

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Module 1: Problem Solving Strategies

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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 solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s 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

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

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!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (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 thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

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.

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.

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

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)

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)

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.

“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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• learning math
• making sense of math
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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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1.5: Teaching Mathematics Through Problem Solving

• Last updated
• Save as PDF
• Page ID 156541

• Janet Stramel
• Fort Hays State University

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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.

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

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 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

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 ”.

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

• Original Article
• Published: 13 May 2022
• Volume 35 , pages 901–927, ( 2023 )

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• Mairéad Hourigan   ORCID: orcid.org/0000-0002-6895-1895 1 &
• Aisling M. Leavy   ORCID: orcid.org/0000-0002-1816-0091 1  

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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.

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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 1 .

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 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 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 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 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 2 details the links between the LS phases and the data collection process.

The principal data sources (Table 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 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.

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 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 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.

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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.

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Hourigan, M., Leavy, A.M. Elementary teachers’ experience of engaging with Teaching Through Problem Solving using Lesson Study. Math Ed Res J 35 , 901–927 (2023). https://doi.org/10.1007/s13394-022-00418-w

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Top 10 Challenges to Teaching Math and Science Using Real Problems

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Nine in ten educators believe that using a problem-solving approach to teaching math and science can be motivating for students, according to an EdWeek Research Center survey.

But that doesn’t mean it’s easy.

Teachers perceive lack of time as a big hurdle. In fact, a third of educators—35 percent—worry that teaching math or science through real-world problems—rather than focusing on procedures—eats up too many precious instructional minutes.

Other challenges: About another third of educators said they weren’t given sufficient professional development in how to teach using a real-world problem-solving approach. Nearly a third say reading and writing take priority over STEM, leaving little bandwidth for this kind of instruction. About a quarter say that it’s tough to find instructional materials that embrace a problem-solving perspective.

Nearly one in five cited teachers’ lack of confidence in their own problem solving, the belief that this approach isn’t compatible with standardized tests, low parent support, and the belief that student behavior is so poor that this approach would not be feasible.

The nationally representative survey included 1,183 district leaders, school leaders, and teachers, and was conducted from March 27 to April 14. (Note: The chart below lists 11 challenges because the last two on the list—dealing with teacher preparation and student behavior—received the exact percentage of responses.)

Trying to incorporate a problem-solving approach to tackling math can require rethinking long-held beliefs about how students learn, said Elham Kazemi, a professor in the teacher education program at the University of Washington.

Most teachers were taught math using a procedural perspective when they were in school. While Kazemi believes that approach has merit, she advocates for exposing students to both types of instruction.

Many educators have “grown up around a particular model of thinking of teaching and learning as the teacher in the front of the room, imparting knowledge, showing kids how to do things,” Kazemi said.

To be sure, some teachers have figured out how to incorporate some real-world problem solving alongside more traditional methods. But it can be tough for their colleagues to learn from them because “teachers don’t have a lot of time to collaborate with one another and see each other teach,” Kazemi said.

What’s more, there are limited instructional materials emphasizing problem solving, Kazemi said.

Though that’s changing, many of the resources available have “reinforced the idea that the teacher demonstrates solutions for kids,” Kazemi said.

Molly Daley, a regional math coordinator for Education Service District 112, which serves about 30 districts near Vancouver, Wash., has heard teachers raise concerns that teaching math from a problem-solving perspective takes too long—particularly given the pressure to get through all the material students will need to perform well on state tests.

Daley believes, however, that being taught to think about math in a deeper way will help students tackle math questions on state assessments that may look different from what they’ve seen before.

“It’s myth that it’s possible to cover everything that will be on the test,” as it will appear, she said. “There’s actually no way to make sure that kids have seen every single possible thing the way it will show up. That’s kind of a losing proposition.”

But rushing through the material in a purely procedural way may actually be counterproductive, she said.

Teachers don’t want kids to “sit down at the test and say, ‘I haven’t seen this and therefore I can’t do it,’” Daley said. “I think a lot of times teachers can unintentionally foster that because they’re so urgently trying to cover everything. That’s where the kind of mindless [teaching] approaches come in.”

Teachers may think to themselves: “’OK, I’m gonna make this as simple as possible, make sure everyone knows how to follow the steps and then when they see it, they can follow it,” Daley said.

But that strategy might “take away their students’ confidence that they can figure out what to do when they don’t know what to do, which is really what you want them to be thinking when they go to approach a test,” Daley said.

Data analysis for this article was provided by the EdWeek Research Center. Learn more about the center’s work.

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