what is teaching mathematics 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?

what is teaching mathematics through problem solving

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

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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.

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

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.

Photo of elementary school teacher with students

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.

The Lesson Study Group

at Mills College

Teaching Through Problem-solving

An elementary-age male student points while his female teacher stands beside him and observes

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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1.5: Teaching Mathematics Through Problem Solving

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Janet Stramel
Fort Hays State University

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There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

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.

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

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

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.

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

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”

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?

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 ”.

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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Original research article, mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

1 Department of Education, Uppsala University, Uppsala, Sweden
2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

Introduction

The research on instruction in mathematical problem-solving has progressed considerably during recent decades. Yet, there is still a need to advance our knowledge on how teachers can support their students in carrying out this complex activity ( Lester and Cai, 2016 ). Results from the Program for International Student Assessment (PISA) show that only 53% of students from the participating countries could solve problems requiring more than direct inference and using representations from different information sources ( OECD, 2019 ). In addition, OECD (2019) reported a large variation in achievement with regard to students’ diverse backgrounds. Thus, there is a need for instructional approaches to promote students’ problem-solving in mathematics, especially in heterogeneous classrooms in which students with diverse backgrounds and needs are educated together. Small group instructional approaches have been suggested as important to promote learning of low-achieving students and students with special needs ( Kunsch et al., 2007 ). One such approach is cooperative learning (CL), which involves structured collaboration in heterogeneous groups, guided by five principles to enhance group cohesion ( Johnson et al., 1993 ; Johnson et al., 2009 ; Gillies, 2016 ). While CL has been well-researched in whole classroom approaches ( Capar and Tarim, 2015 ), few studies of the approach exist with regard to students with special educational needs (SEN; McMaster and Fuchs, 2002 ). This study contributes to previous research by studying the effects of the CL approach on students’ mathematical problem-solving in heterogeneous classrooms, in which students with special needs are educated alongside with their peers.

Group collaboration through the CL approach is structured in accordance with five principles of collaboration: positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing ( Johnson et al., 1993 ). First, the group tasks need to be structured so that all group members feel dependent on each other in the completion of the task, thus promoting positive interdependence. Second, for individual accountability, the teacher needs to assure that each group member feels responsible for his or her share of work, by providing opportunities for individual reports or evaluations. Third, the students need explicit instruction in social skills that are necessary for collaboration. Fourth, the tasks and seat arrangements should be designed to promote interaction among group members. Fifth, time needs to be allocated to group processing, through which group members can evaluate their collaborative work to plan future actions. Using these principles for cooperation leads to gains in mathematics, according to Capar and Tarim (2015) , who conducted a meta-analysis on studies of cooperative learning and mathematics, and found an increase of .59 on students’ mathematics achievement scores in general. However, the number of reviewed studies was limited, and researchers suggested a need for more research. In the current study, we focused on the effect of CL approach in a specific area of mathematics: problem-solving.

Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017 ). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016 ). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007 ; Davydov, 2008 ).

Problem-solving processes have been studied from different perspectives ( Lesh and Zawojewski, 2007 ). Problem-solving heuristics Pólya, (1948) has largely influenced our perceptions of problem-solving, including four principles: understanding the problem, devising a plan, carrying out the plan, and looking back and reflecting upon the suggested solution. Schoenfield, (2016) suggested the use of specific problem-solving strategies for different types of problems, which take into consideration metacognitive processes and students’ beliefs about problem-solving. Further, models and modelling perspectives on mathematics ( Lesh and Doerr, 2003 ; Lesh and Zawojewski, 2007 ) emphasize the importance of engaging students in model-eliciting activities in which problem situations are interpreted mathematically, as students make connections between problem information and knowledge of mathematical operations, patterns, and rules ( Mousoulides et al., 2010 ; Stohlmann and Albarracín, 2016 ).

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study focused on supporting students in problem-solving, through instruction in problem-solving principles ( Pólya, 1948 ), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality.

Students’ problem-solving may be enhanced through participation in small group discussions. In a small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

However, group collaboration is not only associated with positive experiences. In fact, studies show that some students may not be given equal opportunities to voice their opinions, due to academic status differences ( Langer-Osuna, 2016 ). Indeed, problem-solvers struggling with complex tasks may experience negative emotions, leading to uncertainty of not knowing the definite answer, which places demands on peer support ( Jordan and McDaniel, 2014 ; Hannula, 2015 ). Thus, especially in heterogeneous groups, students may need additional support to promote group interaction. Therefore, in this study, we used a cooperative learning approach, which, in contrast to collaborative learning approaches, puts greater focus on supporting group cohesion through instruction in social skills and time for reflection on group work ( Davidson and Major, 2014 ).

Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991 ), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992 ; Cohen, 1994 ). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013 ) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012 ). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships.

The Present Study

In previous research, the CL approach has shown to be a promising approach in teaching and learning mathematics ( Capar and Tarim, 2015 ), but fewer studies have been conducted in whole-class approaches in general and students with SEN in particular ( McMaster and Fuchs, 2002 ). This study aims to contribute to previous research by investigating the effect of CL intervention on students’ mathematical problem-solving in grade 5. With regard to the complexity of mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach in this study was combined with problem-solving principles pertaining to three underlying models of problem-solving—multiplication/division, geometry, and proportionality. Furthermore, considering the importance of peer support in problem-solving in small groups ( Mulryan, 1992 ; Cohen, 1994 ; Hwang and Hu, 2013 ), the study investigated how peer acceptance and friendships were associated with the effect of the CL approach on students’ problem-solving abilities. The study aimed to find answers to the following research questions:

a) What is the effect of CL approach on students’ problem-solving in mathematics?

b) Are social acceptance and friendship associated with the effect of CL on students’ problem-solving in mathematics?

Participants

The participants were 958 students in grade 5 and their teachers. According to power analyses prior to the start of the study, 1,020 students and 51 classes were required, with an expected effect size of 0.30 and power of 80%, provided that there are 20 students per class and intraclass correlation is 0.10. An invitation to participate in the project was sent to teachers in five municipalities via e-mail. Furthermore, the information was posted on the website of Uppsala university and distributed via Facebook interest groups. As shown in Figure 1 , teachers of 1,165 students agreed to participate in the study, but informed consent was obtained only for 958 students (463 in the intervention and 495 in the control group). Further attrition occurred at pre- and post-measurement, resulting in 581 students’ tests as a basis for analyses (269 in the intervention and 312 in the control group). Fewer students (n = 493) were finally included in the analyses of the association of students’ social acceptance and friendships and the effect of CL on students’ mathematical problem-solving (219 in the intervention and 274 in the control group). The reasons for attrition included teacher drop out due to sick leave or personal circumstances (two teachers in the control group and five teachers in the intervention group). Furthermore, some students were sick on the day of data collection and some teachers did not send the test results to the researchers.

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

As seen in Table 1 , classes in both intervention and control groups included 27 students on average. For 75% of the classes, there were 33–36% of students with SEN. In Sweden, no formal medical diagnosis is required for the identification of students with SEN. It is teachers and school welfare teams who decide students’ need for extra adaptations or special support ( Swedish National Educational Agency, 2014 ). The information on individual students’ type of SEN could not be obtained due to regulations on the protection of information about individuals ( SFS 2009 ). Therefore, the information on the number of students with SEN on class level was obtained through teacher reports.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

Intervention

The intervention using the CL approach lasted for 15 weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018 ). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7 weeks. Then, 2 days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a ). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948 ) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8 weeks.

Solving a problem is a matter of goal-oriented reasoning, starting from the understanding of the problem to devising its solution by using known mathematical models. This presupposes that the current problem is chosen from a known context ( Stillman et al., 2008 ; Zawojewski, 2010 ). This differs from the problem-solving of the textbooks, which is based on an aim to train already known formulas and procedures ( Hamilton, 2007 ). Moreover, it is important that students learn modelling according to their current abilities and conditions ( Russel, 1991 ).

In order to create similar conditions in the experiment group and the control group, the teachers were supposed to use the same educational material ( Karlsson and Kilborn, 2018a ; Karlsson and Kilborn, 2018b ), written in light of the specified view of problem-solving. The educational material is divided into three areas—multiplication/division, geometry, and proportionality—and begins with a short teachers’ guide, where a view of problem solving is presented, which is based on the work of Polya (1948) and Lester and Cai (2016) . The tasks are constructed in such a way that conceptual knowledge was in focus, not formulas and procedural knowledge.

Implementation of the Intervention

To ensure the implementation of the intervention, the researchers visited each teachers’ classroom twice during the two phases of the intervention period, as described above. During each visit, the researchers observed the lesson, using a checklist comprising the five principles of the CL approach. After the lesson, the researchers gave written and oral feedback to each teacher. As seen in Table 1 , in 18 of the 23 classes, the teachers implemented the intervention in accordance with the principles of CL. In addition, the teachers were asked to report on the use of the CL approach in their teaching and the use of problem-solving activities embedding CL during the intervention period. As shown in Table 1 , teachers in only 11 of 23 classes reported using the CL approach and problem-solving activities embedded in the CL approach at least once a week.

Control Group

The teachers in the control group received 2 days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948 ). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1 , only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving.

Tests of Mathematical Problem-Solving

Tests of mathematical problem-solving were administered before and after the intervention, which lasted for 15 weeks. The tests were focused on the models of multiplication/division, geometry, and proportionality. The three models were chosen based on the syllabus of the subject of mathematics in grades 4 to 6 in the Swedish National Curriculum ( Swedish National Educational Agency, 2018 ). In addition, the intention was to create a variation of types of problems to solve. For each of these three models, there were two tests, a pre-test and a post-test. Each test contained three tasks with increasing difficulty ( Supplementary Appendix SA ).

The tests of multiplication and division (Ma1) were chosen from different contexts and began with a one-step problem, while the following two tasks were multi-step problems. Concerning multiplication, many students in grade 5 still understand multiplication as repeated addition, causing significant problems, as this conception is not applicable to multiplication beyond natural numbers ( Verschaffel et al., 2007 ). This might be a hindrance in developing multiplicative reasoning ( Barmby et al., 2009 ). The multi-step problems in this study were constructed to support the students in multiplicative reasoning.

Concerning the geometry tests (Ma2), it was important to consider a paradigm shift concerning geometry in education that occurred in the mid-20th century, when strict Euclidean geometry gave way to other aspects of geometry like symmetry, transformation, and patterns. van Hiele (1986) prepared a new taxonomy for geometry in five steps, from a visual to a logical level. Therefore, in the tests there was a focus on properties of quadrangles and triangles, and how to determine areas by reorganising figures into new patterns. This means that structure was more important than formulas.

The construction of tests of proportionality (M3) was more complicated. Firstly, tasks on proportionality can be found in many different contexts, such as prescriptions, scales, speeds, discounts, interest, etc. Secondly, the mathematical model is complex and requires good knowledge of rational numbers and ratios ( Lesh et al., 1988 ). It also requires a developed view of multiplication, useful in operations with real numbers, not only as repeated addition, an operation limited to natural numbers ( Lybeck, 1981 ; Degrande et al., 2016 ). A linear structure of multiplication as repeated addition leads to limitations in terms of generalization and development of the concept of multiplication. This became evident in a study carried out in a Swedish context ( Karlsson and Kilborn, 2018c ). Proportionality can be expressed as a/b = c/d or as a/b = k. The latter can also be expressed as a = b∙k, where k is a constant that determines the relationship between a and b. Common examples of k are speed (km/h), scale, and interest (%). An important pre-knowledge in order to deal with proportions is to master fractions as equivalence classes like 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 6/18 = 7/21 = 8/24 … ( Karlsson and Kilborn, 2020 ). It was important to take all these aspects into account when constructing and assessing the solutions of the tasks.

The tests were graded by an experienced teacher of mathematics (4 th author) and two students in their final year of teacher training. Prior to grading, acceptable levels of inter-rater reliability were achieved by independent rating of students’ solutions and discussions in which differences between the graders were resolved. Each student response was to be assigned one point when it contained a correct answer and two points when the student provided argumentation for the correct answer and elaborated on explanation of his or her solution. The assessment was thus based on quality aspects with a focus on conceptual knowledge. As each subtest contained three questions, it generated three student solutions. So, scores for each subtest ranged from 0 to 6 points and for the total scores from 0 to 18 points. To ascertain that pre- and post-tests were equivalent in degree of difficulty, the tests were administered to an additional sample of 169 students in grade 5. Test for each model was conducted separately, as students participated in pre- and post-test for each model during the same lesson. The order of tests was switched for half of the students in order to avoid the effect of the order in which the pre- and post-tests were presented. Correlation between students’ performance on pre- and post-test was .39 ( p < 0.000) for tests of multiplication/division; .48 ( p < 0.000) for tests of geometry; and .56 ( p < 0.000) for tests of proportionality. Thus, the degree of difficulty may have differed between pre- and post-test.

Measures of Peer Acceptance and Friendships

To investigate students’ peer acceptance and friendships, peer nominations rated pre- and post-intervention were used. Students were asked to nominate peers who they preferred to work in groups with and who they preferred to be friends with. Negative peer nominations were avoided due to ethical considerations raised by teachers and parents ( Child and Nind, 2013 ). Unlimited nominations were used, as these are considered to have high ecological validity ( Cillessen and Marks, 2017 ). Peer nominations were used as a measure of social acceptance, and reciprocated nominations were used as a measure of friendship. The number of nominations for each student were aggregated and divided by the number of nominators to create a proportion of nominations for each student ( Velásquez et al., 2013 ).

Statistical Analyses

Multilevel regression analyses were conducted in R, lme4 package Bates et al. (2015) to account for nestedness in the data. Students’ classroom belonging was considered as a level 2 variable. First, we used a model in which students’ results on tests of problem-solving were studied as a function of time (pre- and post) and group belonging (intervention and control group). Second, the same model was applied to subgroups of students who performed above and below median at pre-test, to explore whether the CL intervention had a differential effect on student performance. In this second model, the results for subgroups of students could not be obtained for geometry tests for subgroup below median and for tests of proportionality for subgroup above median. A possible reason for this must have been the skewed distribution of the students in these subgroups. Therefore, another model was applied that investigated students’ performances in math at both pre- and post-test as a function of group belonging. Third, the students’ scores on social acceptance and friendships were added as an interaction term to the first model. In our previous study, students’ social acceptance changed as a result of the same CL intervention ( Klang et al., 2020 ).

The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012 ). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise.

What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

As seen in the regression coefficients in Table 2 , the CL intervention had a significant effect on students’ mathematical problem-solving total scores and students’ scores in problem solving in geometry (Ma2). Judging by mean values, students in the intervention group appeared to have low scores on problem-solving in geometry but reached the levels of problem-solving of the control group by the end of the intervention. The intervention did not have a significant effect on students’ performance in problem-solving related to models of multiplication/division and proportionality.

TABLE 2 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.

The question is, however, whether CL intervention affected students with different pre-test scores differently. Table 2 includes the regression coefficients for subgroups of students who performed below and above median at pre-test. As seen in the table, the CL approach did not have a significant effect on students’ problem-solving, when the sample was divided into these subgroups. A small negative effect was found for intervention group in comparison to control group, but confidence intervals (CI) for the effect indicate that it was not significant.

Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

As seen in Table 3 , students’ peer acceptance and friendship at pre-test were significantly associated with the effect of the CL approach on students’ mathematical problem-solving scores. Changes in students’ peer acceptance and friendships were not significantly associated with the effect of the CL approach on students’ mathematical problem-solving. Consequently, it can be concluded that being nominated by one’s peers and having friends at the start of the intervention may be an important factor when participation in group work, structured in accordance with the CL approach, leads to gains in mathematical problem-solving.

TABLE 3 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving, including scores of social acceptance and friendship in the model.

In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015 ), and for students with SEN in particular ( McMaster and Fuchs, 2002 ), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019 ). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007 ). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012 ; Hwang and Hu, 2013 ), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not.

The results of the study confirm previous research on the effect of the CL approach on students’ mathematical achievement ( Capar and Tarim, 2015 ). The specific contribution of the study is that it was conducted in classrooms, 75% of which were composed of 33–36% of students with SEN. Thus, while a previous review revealed inconclusive findings on the effects of CL on student achievement ( McMaster and Fuchs, 2002 ), the current study adds to the evidence of the effect of the CL approach in heterogeneous classrooms, in which students with special needs are educated alongside with their peers. In a small group setting, the students have opportunities to discuss their ideas of solutions to the problem at hand, providing explanations and clarifications, thus enhancing their understanding of problem-solving ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ).

In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999 ; Gravemeijer, 1999 ).

Furthermore, for successful instruction in mathematical problem-solving, scaffolding for content needs to be combined with scaffolding for dialogue ( Kazak et al., 2015 ). From a dialogical perspective ( Wegerif, 2011 ), students may need scaffolding in new ways of thinking, involving questioning their understandings and providing arguments for their solutions, in order to create dialogic spaces in which different solutions are voiced and negotiated. In this study, small group instruction through CL approach aimed to support discussions in small groups, but the study relies solely on quantitative measures of students’ mathematical performance. Video-recordings of students’ discussions may have yielded important insights into the dialogic relationships that arose in group discussions.

Despite the positive findings of the CL approach on students’ problem-solving, it is important to note that the intervention did not have an effect on students’ problem-solving pertaining to models of multiplication/division and proportionality. Although CL is assumed to be a promising instructional approach, the number of studies on its effect on students’ mathematical achievement is still limited ( Capar and Tarim, 2015 ). Thus, further research is needed on how CL intervention can be designed to promote students’ problem-solving in other areas of mathematics.

The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012 ). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994 ; Mulryan, 1992 ; Langer Osuna, 2016 ) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013 ), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015 ; Jordan and McDaniel, 2014 ), and peer support may be essential in such situations.

Limitations

The conclusions from the study must be interpreted with caution, due to a number of limitations. First, due to the regulation of protection of individuals ( SFS 2009 ), the researchers could not get information on type of SEN for individual students, which limited the possibilities of the study for investigating the effects of the CL approach for these students. Second, not all teachers in the intervention group implemented the CL approach embedded in problem-solving activities and not all teachers in the control group reported using educational materials on problem-solving. The insufficient levels of implementation pose a significant challenge to the internal validity of the study. Third, the additional investigation to explore the equivalence in difficulty between pre- and post-test, including 169 students, revealed weak to moderate correlation in students’ performance scores, which may indicate challenges to the internal validity of the study.

Implications

The results of the study have some implications for practice. Based on the results of the significant effect of the CL intervention on students’ problem-solving, the CL approach appears to be a promising instructional approach in promoting students’ problem-solving. However, as the results of the CL approach were not significant for all subtests of problem-solving, and due to insufficient levels of implementation, it is not possible to conclude on the importance of the CL intervention for students’ problem-solving. Furthermore, it appears to be important to create opportunities for peer contacts and friendships when the CL approach is used in mathematical problem-solving activities.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics Statement

The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

NiK was responsible for the project, and participated in data collection and data analyses. NaK and WK were responsible for intervention with special focus on the educational materials and tests in mathematical problem-solving. PE participated in the planning of the study and the data analyses, including coordinating analyses of students’ tests. MK participated in the designing and planning the study as well as data collection and data analyses.

The project was funded by the Swedish Research Council under Grant 2016-04,679.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.710296/full#supplementary-material

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Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms, hierarchical linear regression analysis

Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

Reviewed by:

Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nina Klang, [email protected]

Young Students Gravitate to Math. How Teachers Can Build on That Curiosity

Zachary Champagne’s 3rd and 4th graders figure out early on that this math class will be different when their teacher tells them: “I don’t care about the answer.”

The goal is to shift his elementary students’ thinking from some numerical endgame toward the problem-solving process itself. In his more than two decades as a classroom teacher and math researcher, Champagne has found this strategy can be a balm for math anxiety, spur students’ creativity, and pique their curiosity about a subject many find boring and irrelevant.

Telling students the answer doesn’t matter—or throwing it out early on, then working backwards, another of Champagne’s go-to strategies—"can reframe the way we think about mathematics,” said Champagne, who teaches at The Discovery School, a private school in Jacksonville, Fla.

Photo illustration of chemistry teacher working with young student.

“If we’re thinking about math where the solving is actually the interesting, important part, it frees kids from the stigma of ‘I’m not good at this because I always get things wrong,’” said Champagne, who spent more than a decade working in Florida’s Duval County public schools and served as a math researcher at Florida State University.

This problem-solving or open-ended approach, which emphasizes flexible thinking and real-world situations, is a powerful strategy for engaging the youngest learners in math . Kindergarten through 5th grade is an important time for building students’ skills, confidence, and interest in math—the critical building blocks for middle- and high-school-level math and science, experts say.

Though the approach has been around for decades, districts are striving to incorporate more real-world problem-solving into math class in recent years. California, for instance, recently adopted a controversial framework that puts a heavy emphasis on the approach . And there’s new urgency to get students motivated in math as federal data show students’ math achievement plummeting.

The vast majority of educators—92 percent—say students are more motivated to learn math and science if teachers employ a problem-solving approach, according to a survey of 1,183 district and school leaders and teachers, conducted by the EdWeek Research Center in April. Despite the fact that this approach is highly popular among educators, many have not been trained in how to use it, the same survey found.

Does using real-world problems motivate students?

The Canadian province of Quebec has been using a problem-based approach for decades—and it helps students connect with math and understand how to use it in the real world, said Krista Muis, a professor at McGill University in Montreal, who has studied student perceptions of the teaching strategy.

“When you look at the motivational profiles of students who are just given traditional word problems, or more standard types of math problems, or math content, their motivation is really low when it comes to the value of what they’re learning,” Muis said. “The main question they ask is, ‘why should I care? How is this relevant to me? How am I ever going to use this?’”

But when students tackle common real problems—a favorite of Muis’ asks elementary schoolers to map out the trick-or-treating route that nets the most Halloween candy—they get excited.

“They see the value in it,” Muis said. “And they’re fun problems. They can do them in groups together collaboratively, they can do them individually.”

Quebec students’ higher motivation in math may explain why the province outperforms the rest of Canada—and the United States— on the Trends in International Mathematics and Science Study or TIMMS , Muis said.

In 2019, the most recent year the test was given, Quebec’s 4th graders didn’t perform statistically differently from their U.S. counterparts in math. But 8th graders from the province scored significantly better than their U.S. peers . One reason may be the increased motivation to learn math that Muis believes stems from exposing students to a problem-solving approach early on.

To be sure, a problem-based or open-ended approach to teaching math is often pitted against more traditional, procedural methods—think of the math worksheets full of equations without context.

But many experts and educators see value in exposing students to both strategies.

“I think, really, these things can mutually support one another. And both are necessary,” said Julia Aguirre, a professor and the faculty director of teacher certification programs at the University of Washington Tacoma. “I think we can all agree that a math class that’s only about worksheets would not be a very fulfilling or interesting math class.”

Promote young students’ natural curiosity and creativity

The approach is most effective when teachers apply it to students’ existing interests.

That’s especially important for elementary school students, who start school with a natural curiosity that often dissipates by the time they get to high school, said Molly Daley, a regional math coordinator for Education Service District 112, which serves about 30 districts near Vancouver, Wash.

Thinking about “math is a universal human behavior, and people of all ages engage in math for their own purposes,” Daley said.

Students are using math when they play games and make crafts, she said, or even just look at the landscape.

For instance, a preschool teacher might take a picture of the classroom shoe rack and ask students questions like: How many shoes are there? What patterns do you notice? What shapes do you see?

“If we can honor the math that kids are doing beyond the classroom, then we’re more likely to create a mathematical connection and really allow every person to see how math is not just useful but enjoyable,” Daley said.

In Champagne’s mixed age classroom of 3rd and 4th graders in Florida—which he co-teaches with another educator—students turn to math early in the day, the time when younger students tend to be most able to focus on the subject, in Champagne’s view.

Champagne typically kicks off with a 10- or 15-minute math routine as a warm-up. That might be a “number talk” in which Champagne will put an equation on the board, say 29 plus 15, and then students will solve the problem in their heads.

They’ll spend the next few minutes comparing strategies for finding a solution. One student might have added 30 plus 15 and subtracted one, while another might have added 9 and 5, then 30.

The exercise is aimed at promoting flexibility and the idea that there are multiple ways to solve a problem, Champagne said. It lets students know: “I don’t have to revert to just one strategy. I can think about it in different ways,” he said. The idea is to gives students a chance to use their creative thinking skills in math class.

Students still learn the basics, but lessons are structured so that students can see how seemingly simple problems play out in different, real-world contexts. For instance, if students are learning about dividing with remainders, they may consider how four people can share 31 balloons. In that case, each person gets seven balloons, with three left over.

But what if it were 31 dollars instead of balloons? How does that change the answer? Or what if 31 people needed to get somewhere in four cars? How could they divide up?

Problems can also get more complex—and interdisciplinary—as students advance in elementary school.

Teachers need more training in the problem-solving approach

Tackling big problems with no clear answer is another way to engage elementary school students in math.

Last school year, Aguirre worked with Janaki Nagarajan’s 3rd graders outside Seattle on a project involving a real-life problem with salmon the students had raised and planned to release.

Inexplicably, the fish began dying. So Nagarajan’s students used mathematical modeling to estimate how quickly they were losing salmon, answering questions like: Will we have enough salmon for each student on release day? What can we do if we don’t? Students worked on the problem in groups, and then presented their answers. The class voted on the solution they thought would work best.

The project was “really engaging,” said Nagarajan. She believes that students will be motivated to learn math if they “feel the skills have some purpose outside the classroom.”

But she thinks that many teachers don’t know how common procedures learned in math class could be applied in the real world, so they struggle to make those connections for their students.

Nagarajan began teaching in Renton, a different, Seattle-area school district this school year, largely because it provides more support for teachers to use the real-world problem-solving approach in elementary school math.

Though the approach was encouraged in her previous school, Nagarajan said her new district uses a curriculum that embraces problem-solving and provides coaches who can help her implement the strategy.

Professional development in the problem-solving approach remains uneven. About one in five educators said they “completely agree” that their districts have offered deep and sustained professional development into how to teach math and science from a problem-solving perspective, while just over 40 percent said they disagree—at least somewhat—that they’ve been offered that level of support.

That professional development can be particularly important for elementary school teachers who typically “aren’t math specialists, right? They are generalists,” said Muis of McGill University. “Often, teachers who are not comfortable with mathematics don’t necessarily understand it fully themselves. And so when you bring in complexity that scares them. And then you see teachers kind of stepping back going, ‘I can’t really teach this, I don’t really know what I’m doing.’”

And the approach requires teachers to respond to what students see or notice, which can be stressful for some, Daley said.

“We can get too hyper focused on ‘this is my goal’” in a particular lesson, she said. That can look like: “We’re learning about fractions, but the student made a comment about multiplication. I gotta ignore that.’”

Teachers need to learn not to be afraid if students go off script, Daley said. A problem-solving approach is about “creating more space for students’ ideas and students’ thinking versus just letting your own dominate.”

Making that shift isn’t easy. But if teachers are successful, they positively shape their students’ relationship with math, potentially for years, Daley said.

“Especially with younger learners, when we’re following their lead, that’s how we’re going to tap into their connection and their motivation to engage with math and build up their sense of themselves as a mathematician,” she said.

Coverage of problem solving and student motivation is supported in part by a grant from The Lemelson Foundation, at www.lemelson.org . Education Week retains sole editorial control over the content of this coverage.

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Opinion | Teaching and Learning

When teaching students math, concepts matter more than process, by nicola hodkowski jun 5, 2024.

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As a mathematics education researcher, I study how math instruction impacts students' learning, from following standard math procedures to understanding mathematical concepts. Focusing on the latter, conceptual understanding often involves understanding the “why” of a mathematical concept ; it’s the reasoning behind the math rather than the how or the steps it takes to get to an answer.

So often, in mathematics classrooms, students are shown steps and procedures for solving math problems and then required to demonstrate their rote memorization of these steps independently.

As a result, students' agency, knowledge and ability to transfer the concepts of mathematics suffer. Specifically, students experience diminished confidence in tackling mathematical problems and a decreased ability to apply mathematical reasoning in real-world situations. In addition, students may struggle with more advanced mathematical concepts and problem-solving tasks as they progress in their education.

While procedural fluency is important, conceptual understanding provides a framework for students to build mental relationships between math concepts. It allows students to connect new ideas to what they already know , creating increasing connections toward more advanced mathematics.

If we want mathematics achievement to improve, we need instruction to begin focusing on concepts instead of procedures.

Why Concept Matters More Than Procedure

Conceptual understanding builds on existing understanding to advance knowledge and focuses on the student’s ability to justify and explain. Procedural fluency, on the other hand, is about following steps to arrive at an answer and accuracy.

When considering how students will learn more advanced mathematics concepts, it is important to consider how they will engage with the problems presented to them in class and how those problems will contribute either to their greater conceptual understanding or greater procedural fluency. For example, consider these two math questions and ask yourself: What knowledge is needed to solve each problem?

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What Is DeltaMath?

By Med Kharbach, PhD | Last Update: March 23, 2024

$DeltaMath$

DeltaMath is a teacher-created math platform used by millions of students and teachers from all around the globe. Founded in 2009 by Zach Korzyk, a visionary math teacher, DeltaMath has since garnered a global user base, captivating millions of students and teachers with its comprehensive, practice-oriented approach to mathematics.

DeltaMath offers vast repository of over 1,800 types of math problems, meticulously aligned with the Common Core standards for grades 6 through 12. The platform provides randomized problems, ensuring that students have endless opportunities to hone their skills and receive immediate, insightful feedback on their performance. Such features not only facilitate a deep, nuanced understanding of mathematical concepts but also foster a culture of self-directed learning and continuous improvement.

Delta Math Features

DeltaMath offers a wide array of features tailored to enhance the teaching and learning of mathematics. Here are the key features that make DeltaMath stand out:

Over 1,800 Problem Types: Offers a vast collection of math problems, covering grades 6 through 12, aligned with Common Core standards, ensuring a wide range of topics and difficulties.
Randomized Problems: Provides students with unlimited practice opportunities through randomized math problems, enhancing learning outcomes by preventing memorization.
Auto-Grading System: Features an integrated auto-grading system that gives students instant feedback on their performance, allowing for immediate correction and understanding.
Custom Graph Creation: Enables students to create and automatically grade graphs, offering instant feedback along with the correct answers for improved learning.
Teacher Customization: Teachers can create assignments and tailor specific problem types to students. The Plus version further allows for the creation of self-graded and custom problems for individual or group assignments.
Differentiated Instruction: Facilitates differentiated teaching and learning by allowing teachers to use site tracking data to analyze students’ performance and tailor assignments for remedial work or additional practice.
Interactive Problem Features: Incorporates interactive features and visual cues in math problems to guide students toward the correct answers, making complex concepts more understandable.
Visual and Illustrative Components: A growing number of math problems include visuals and illustrative components to aid in explaining the procedural steps and facilitating the comprehension of abstract concepts.
Instructional Videos: DeltaMath Plus provides instructional videos for every problem type, and teachers can also assign videos from YouTube, enriching the learning experience with multimedia resources.
Customizable Assignments: Enables easy creation and sharing of assignments with students, including setting specific due dates, selecting skills, and monitoring student progress with detailed timestamps and completion grades.
Aggregate Data and Export Options: Offers teachers the ability to view and export aggregate student completion grades and detailed performance data for comprehensive analysis and feedback.
Cheat Prevention Tools: Includes tools designed to prevent cheating, ensuring that students engage with the material authentically and responsibly.
Flexible Subscription Options: Available in both a free version for basic use and a Plus version for individual teachers, offering additional features for a more customized teaching experience. School/District licenses are also available, providing wider access and further capabilities.

DeltaMath Pricing

The platform offers a DeltaMath Teacher Free version, which includes unlimited assignments, unlimited student practice, over 1,800 premade problem types aligned to Common Core, autograding, detailed student data with timestamps, evidence of student progress, cheat prevention tools, and the ability to copy and share assignments—all at no cost.

For educators seeking enhanced features, the DeltaMath Plus (Individual Teacher) subscription is available at $95 per teacher. This premium option adds instructional videos for every problem type, the ability to assign videos from YouTube, create tests and test corrections, select specific problem sub-types, assign specific problems to individuals or groups, and create custom problems. For schools or districts needing wider access, DeltaMath Plus (School/District License) offers a tailored solution, with pricing available upon request.

Related: Best Math Apps for iPad

Closing thoughts

For educators, DeltaMath serves as a versatile and powerful instructional tool. Its ability to create customized assignments, coupled with the Plus version’s enhanced features, allows for targeted teaching strategies that can accommodate diverse learning needs. Teachers can leverage this platform to track student performance, tailor their instruction based on real-time data, and provide differentiated support through ad hoc assignments. Moreover, the inclusion of step-by-step problem-solving guides and interactive, visually rich problems demystifies complex concepts, making math more accessible and engaging for students.

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Meet Med Kharbach, PhD

Dr. Med Kharbach is an influential voice in the global educational technology landscape, with an extensive background in educational studies and a decade-long experience as a K-12 teacher. Holding a Ph.D. from Mount Saint Vincent University in Halifax, Canada, he brings a unique perspective to the educational world by integrating his profound academic knowledge with his hands-on teaching experience. Dr. Kharbach's academic pursuits encompass curriculum studies, discourse analysis, language learning/teaching, language and identity, emerging literacies, educational technology, and research methodologies. His work has been presented at numerous national and international conferences and published in various esteemed academic journals.

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The Algebra Problem: How Middle School Math Became a National Flashpoint

Top students can benefit greatly by being offered the subject early. But many districts offer few Black and Latino eighth graders a chance to study it.

The arms of a student are seen leaning on a desk. One hand holds a pencil and works on algebra equations.

By Troy Closson

From suburbs in the Northeast to major cities on the West Coast, a surprising subject is prompting ballot measures, lawsuits and bitter fights among parents: algebra.

Students have been required for decades to learn to solve for the variable x, and to find the slope of a line. Most complete the course in their first year of high school. But top-achievers are sometimes allowed to enroll earlier, typically in eighth grade.

The dual pathways inspire some of the most fiery debates over equity and academic opportunity in American education.

Do bias and inequality keep Black and Latino children off the fast track? Should middle schools eliminate algebra to level the playing field? What if standout pupils lose the chance to challenge themselves?

The questions are so fraught because algebra functions as a crucial crossroads in the education system. Students who fail it are far less likely to graduate. Those who take it early can take calculus by 12th grade, giving them a potential edge when applying to elite universities and lifting them toward society’s most high-status and lucrative professions.

But racial and economic gaps in math achievement are wide in the United States, and grew wider during the pandemic. In some states, nearly four in five poor children do not meet math standards.

To close those gaps, New York City’s previous mayor, Bill de Blasio, adopted a goal embraced by many districts elsewhere. Every middle school would offer algebra, and principals could opt to enroll all of their eighth graders in the class. San Francisco took an opposite approach: If some children could not reach algebra by middle school, no one would be allowed to take it.

The central mission in both cities was to help disadvantaged students. But solving the algebra dilemma can be more complex than solving the quadratic formula.

New York’s dream of “algebra for all” was never fully realized, and Mayor Eric Adams’s administration changed the goal to improving outcomes for ninth graders taking algebra. In San Francisco, dismantling middle-school algebra did little to end racial inequities among students in advanced math classes. After a huge public outcry, the district decided to reverse course.

“You wouldn’t think that there could be a more boring topic in the world,” said Thurston Domina, a professor at the University of North Carolina. “And yet, it’s this place of incredibly high passions.”

“Things run hot,” he said.

In some cities, disputes over algebra have been so intense that parents have sued school districts, protested outside mayors’ offices and campaigned for the ouster of school board members.

Teaching math in middle school is a challenge for educators in part because that is when the material becomes more complex, with students moving from multiplication tables to equations and abstract concepts. Students who have not mastered the basic skills can quickly become lost, and it can be difficult for them to catch up.

Many school districts have traditionally responded to divergent achievement levels by simply separating children into distinct pathways, placing some in general math classes while offering others algebra as an accelerated option. Such sorting, known as tracking, appeals to parents who want their children to reach advanced math as quickly as possible.

But tracking has cast an uncomfortable spotlight on inequality. Around a quarter of all students in the United States take algebra in middle school. But only about 12 percent of Black and Latino eighth graders do, compared with roughly 24 percent of white pupils, a federal report found .

“That’s why middle school math is this flashpoint,” said Joshua Goodman, an associate professor of education and economics at Boston University. “It’s the first moment where you potentially make it very obvious and explicit that there are knowledge gaps opening up.”

In the decades-long war over math, San Francisco has emerged as a prominent battleground.

California once required that all eighth graders take algebra. But lower-performing middle school students often struggle when forced to enroll in the class, research shows. San Francisco later stopped offering the class in eighth grade. But the ban did little to close achievement gaps in more advanced math classes, recent research has found.

As the pendulum swung, the only constant was anger. Leading Bay Area academics disparaged one another’s research . A group of parents even sued the district last spring. “Denying students the opportunity to skip ahead in math when their intellectual ability clearly allows for it greatly harms their potential for future achievement,” their lawsuit said.

The city is now back to where it began: Middle school algebra — for some, not necessarily for all — will return in August. The experience underscored how every approach carries risks.

“Schools really don’t know what to do,” said Jon R. Star, an educational psychologist at Harvard who has studied algebra education. “And it’s just leading to a lot of tension.”

In Cambridge, Mass., the school district phased out middle school algebra before the pandemic. But some argued that the move had backfired: Families who could afford to simply paid for their children to take accelerated math outside of school.

“It’s the worst of all possible worlds for equity,” Jacob Barandes, a Cambridge parent, said at a school board meeting.

Elsewhere, many students lack options to take the class early: One of Philadelphia’s most prestigious high schools requires students to pass algebra before enrolling, preventing many low-income children from applying because they attend middle schools that do not offer the class.

In New York, Mr. de Blasio sought to tackle the disparities when he announced a plan in 2015 to offer algebra — but not require it — in all of the city’s middle schools. More than 15,000 eighth graders did not have the class at their schools at the time.

Since then, the number of middle schools that offer algebra has risen to about 80 percent from 60 percent. But white and Asian American students still pass state algebra tests at higher rates than their peers.

The city’s current schools chancellor, David Banks, also shifted the system’s algebra focus to high schools, requiring the same ninth-grade curriculum at many schools in a move that has won both support and backlash from educators.

And some New York City families are still worried about middle school. A group of parent leaders in Manhattan recently asked the district to create more accelerated math options before high school, saying that many young students must seek out higher-level instruction outside the public school system.

In a vast district like New York — where some schools are filled with children from well-off families and others mainly educate homeless children — the challenge in math education can be that “incredible diversity,” said Pedro A. Noguera, the dean of the University of Southern California’s Rossier School of Education.

“You have some kids who are ready for algebra in fourth grade, and they should not be denied it,” Mr. Noguera said. “Others are still struggling with arithmetic in high school, and they need support.”

Many schools are unequipped to teach children with disparate math skills in a single classroom. Some educators lack the training they need to help students who have fallen behind, while also challenging those working at grade level or beyond.

Some schools have tried to find ways to tackle the issue on their own. KIPP charter schools in New York have added an additional half-hour of math time to many students’ schedules, to give children more time for practice and support so they can be ready for algebra by eighth grade.

At Middle School 50 in Brooklyn, where all eighth graders take algebra, teachers rewrote lesson plans for sixth- and seventh-grade students to lay the groundwork for the class.

The school’s principal, Ben Honoroff, said he expected that some students would have to retake the class in high school. But after starting a small algebra pilot program a few years ago, he came to believe that exposing children early could benefit everyone — as long as students came into it well prepared.

Looking around at the students who were not enrolling in the class, Mr. Honoroff said, “we asked, ‘Are there other kids that would excel in this?’”

“The answer was 100 percent, yes,” he added. “That was not something that I could live with.”

Troy Closson reports on K-12 schools in New York City for The Times. More about Troy Closson

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

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

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Huirong Zhang 1 &
Jinfa Cai 2

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

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

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Explore how Microsoft's partnership with Khan Academy is enhancing the future of education with AI innovation and tools for teachers >

Published Nov 3, 2022

Empowering student learning with Math Assistant

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Math teachers, what if there was a way to help students who say “I’m not a math person” or “I’m just not good at math”? When a student lacks math confidence, every challenge along the way simply confirms their self-doubt. These challenges only intensify as students move through their K-12 school experience and their needs and the course content continue to evolve. The good news is that Math Assistant , found in Microsoft OneNote, helps students find their math confidence and gain important math skills through inclusive, individualized features.

Explore the scenarios below to learn how Math Assistant can help transform K-12 students’ math experience, accelerate their learning, and help them enjoy math too.

Capture notes using Ink to Math

Ellie is a 5th grade student who is currently learning about multiplying fractions. Ellie has consistently enjoyed learning the steps for multiplication and understood the concept of fractions almost immediately. However, combining the two ideas into multiplying fractions has been challenging, even when using a calculator.

As a learner who constantly takes notes to remember her teacher’s instructions, Ellie can quickly draw equations on her Windows 11SE device’s touchscreen and convert the image into a mathematical format using Ink to Math . Ellie, like most of her classmates, is still developing her handwriting skills and she can select which characters she is writing to ensure that each problem comes out exactly right.

Math Assistant features that impact Ellie’s skills and confidence:

Ink to Math helps Ellie capture class notes in real-time. Later, she can convert her hand-written equations into math.
Ink to Math lets Ellie select the correct character from a list.

Instant practice with Generate Quiz

Sadie is a 7th grader whose class just started its algebra unit. She uses OneNote Class Notebook to keep up with resources and example problems that she can access at home or in class. Tomorrow is the big quiz and Sadie is still nervous about how to solve for x in linear inequalities, so she opens OneNote to practice.

Sadie starts by using her Math Assistant’s Ink to Math feature to create a sample problem. As this is her first practice problem, Sadie starts by using Math Solver for a step-by-step explanation of how to find x.

Now that Sadie is ready for some independent practice, she clicks the Generate Quiz button to create a custom set of no-stakes practice problems based on her example equation. She can even determine how many questions the practice set should have.

Using tools like Math Assistant’s Generate Quiz, Sadie creates personalized, on-demand assessments that help her accelerate her learning when she needs it regardless of if she is in class or at home. The next day in class, Sadie confidently completes her algebra quiz using the skills she developed using Math Assistant.

Math Assistant features that impact Sadie’s skills and confidence:

Ink to Math allows Sadie to naturally write equations on her touchscreen device and then convert it into math for additional AI features.
Math Solver offers step-by-step explanations of solving algebraic equations.
Generate Quiz creates on-demand practice challenging topics.

Graphing and analysis in OneNote Notebook

Guillermo is an 11th grade Algebra II student who is learning about graphing equations. His teacher uses Microsoft Teams for Education and a OneNote Class Notebook to share resources, post assignments, and offer virtual office hours for students who need extra help or want to check in outside of class.

Earlier this week, Guillermo’s teacher shared a linear equation. Now Guillermo is trying to remember the steps to solving the equation and how to graph his solution. Guillermo uses OneNote Math Assistant’s graphing feature to graph both sides in 2D and insert the graph into his OneNote.

Later, Guillermo explores other features of his graph like the y-intercept and x-y values at different points along the graph line.

As the year continues, Guillermo feels self-assured knowing that he can always return to his linear equation notes to review past concepts. He increasingly trusts his math skills without anxiousness knowing that he can use other graphing features in Math Assistant when his class covers polynomial arrays, systems of equations, and inequalities.

Math Assistant features that impact Guillermo’s skills and confidence:

OneNote Class Notebook organizes and structures Guillermo’s notes.
Math Assistant graphs equations and inserts them directly into Guillermo’s notebook.
Math Assistant analyzes the graph for features like y-intercept and x-y values.

Inclusion and accessibility with Immersive Reader for Math

Mei is a 6th grade student who primarily speaks Mandarin at home. While skilled in math, she feels unsure of herself due to the challenge of accessing domain-specific vocabulary in English. Mei’s teacher recently attended a virtual training led by Microsoft Innovative Educators and was introduced to Immersive Reader for Math , a built-in feature of Math Assistant.

Inclusive-by-design, Immersive Reader for Math allows students like Mei to access math content through equations read aloud, a feature that many browser-based screen readers lack. Beyond simply reading the equations aloud, Mei can select her preferred language to access the content using familiar vocabulary and phrases. By removing language barriers, Mei is confident that her math skills will be on full display.

Math Assistant features that impact Mei’s skills and confidence:

Immersive Reader for Math makes math text and equations inclusive and accessible with read aloud and other features.
Math Assistant allows Mei to explore a step-by-step explanation of the equation to practice the concept.
Immersive Reader for Math translates the step-by-step process into Chinese. Mei can customize the voice, speed, and other features to her preferences.

Meeting students’ unique math needs

Just like a student’s personality or interests, their learning needs are unique. Math Assistant in OneNote offers inclusive-by-design instructional tools that empower learners to gain the important confidence and skills that they need to thrive.

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What IS Problem-Solving?
Tips To Teach Kids Math Problem Solving Skills
Teaching Mathematics Through Problem Solving
Teaching Mathematics Through Problem-Solving

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Supporting mathematics and statistics students: is it all about the teaching?
Pedagogical content knowledge in Mathematics
Problem Solving and Reasoning: Polya's Steps and Problem Solving Strategies
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Exploring Mathematics through Problem Solving

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Teaching Mathematics Through Problem Solving
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 ...
NAIS
In "teaching through problem solving," on the other hand, the goal is for students to learn precisely that mathematical idea that the curriculum calls for them to learn next. A "teaching through problem solving" lesson would begin with the teacher setting up the context and introducing the problem. Students then work on the problem for ...
Mathematics Through Problem Solving
What Is A 'Problem-Solving Approach'? As the emphasis has shifted from teaching problem solving to teaching via problem solving (Lester, Masingila, Mau, Lambdin, dos Santon and Raymond, 1994), many writers have attempted to clarify what is meant by a problem-solving approach to teaching mathematics.The focus is on teaching mathematical topics through problem-solving contexts and enquiry ...
Teaching Mathematics through Problem Solving- An Upside-Down Approach
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 ...
Learning to Teach Mathematics Through Problem Solving
Teaching and learning mathematics through problem solving supports learners' development of deep and conceptual understandings (Inoue et al., 2019 ), and is regarded as an effective way of catering for diversity (Hunter et al., 2018 ). While the importance and challenge of mathematical problem solving in school classrooms is not questioned ...
Teaching Mathematics Through Problem-Solving
Teaching Mathematics Through Problem-Solving gives educators the tools to restructure their lesson and curriculum design to make creative and adaptive problem-solving the main way students learn new procedures. Takahashi showcases TTP lessons for elementary and secondary classrooms, showing how teachers can create their own TTP lessons and ...
Problem Solving
Problem solving plays an important role in mathematics and should have a prominent role in the mathematics education of K-12 students. However, knowing how to incorporate problem solving meaningfully into the mathematics curriculum is not necessarily obvious to mathematics teachers. (The term "problem solving" refers to mathematical tasks that ...
Problem Solving in Mathematics Education
It has infused mathematics curricula around the world with calls for the teaching of problem solving as well as the teaching of mathematics through problem solving. And as such, it has been of interest to mathematics education researchers for as long as our field has existed. More relevant, mathematical problem solving has played a part in ...
PDF Teaching Math Through Problem Solving
1999). However, teaching mathematics through problem solving is a relatively new idea in the history of problem solving in the mathematics curriculum (Lester, 1994). In fact, because teaching mathematics through problem solving is a rather new conception, it has not been the subject of much research.
Teaching Mathematics Through Problem-Solving
This engaging book offers an in-depth introduction to teaching mathematics through problem-solving, providing lessons and techniques that can be used in classrooms for both primary and lower secondary grades. Based on the innovative and successful Japanese approaches of Teaching Through Problem-solving (TTP) and Collaborative Lesson Research (CLR), renowned mathematics education scholar ...
6 Tips for Teaching Math Problem-Solving Skills
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.
What research tells us about teaching mathematics through problem solving
1999). However, teaching mathematics through problem solving is a relatively new idea in the. history of problem solving in the mathem atics cu rriculum (Lester, 1994). In fact, because teaching ...
Problem solving in the mathematics curriculum: From domain‐general
'Teaching mathematics through problem solving', while an excellent approach to teaching mathematical content, does not seem to guarantee that all students will develop the necessary toolbox of specific tactics that will enable them to tackle a wide range of unseen problems confidently and reliably, and risks leaving content knowledge inert.
Teaching Mathematics Through Problem-Solving
Teaching Mathematics Through Problem-Solving gives educators the tools to restructure their lesson and curriculum design to make creative and adaptive problem-solving the main way students learn new procedures. Takahashi showcases TTP lessons for elementary and secondary classrooms, showing how teachers can create their own TTP lessons and ...
Overview
Download. 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.
(PDF) Teaching Mathematics Through Problem-Solving: A Pedagogical
Teaching Mathematics Through Problem-Solving gives educators the tools to restructure their lesson and curriculum design to make creative and adaptive problem-solving the main way students learn ...
1.5: Teaching Mathematics Through Problem Solving
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.
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.
Elementary teachers' experience of engaging with Teaching Through
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 ...
Frontiers
Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students' mathematical problem-solving in heterogeneous classrooms in ...
Teaching mathematics through problem posing: Elements of the task
5. Discussion. Although problem posing is an ancient activity key to the evolution of science, problem posing as a task to be carried out by students as a means of instruction in mathematics classes is a more recent activity, and problem-posing research still has many questions that remain unanswered.
Young Students Gravitate to Math. How Teachers Can Build on That Curiosity
Kindergarten through 5th grade is an important time for building students' skills, confidence, and interest in math—the critical building blocks for middle- and high-school-level math and ...
PDF Teaching Through Problem Solving
The Problem of Teaching. (Teaching as Problem Solving) Can/should tell. Conventions [order of operation, etc.] Symbolism and representations [tables, graphs, etc.] Present and re‐present at times of need. Can/should present alternative methods to resolve.
When Teaching Students Math, Concepts Matter More Than Process
As a teacher, I know it can be difficult to know how to respond in the moment of teaching to make sure the students' learning continues to improve toward the goal. Additionally, it demands a high level of flexibility and creativity from educators as students can contribute many unanticipated responses as they work through solving math problems.
What Is DeltaMath?
DeltaMath offers a wide array of features tailored to enhance the teaching and learning of mathematics. Here are the key features that make DeltaMath stand out: Over 1,800 Problem Types: Offers a vast collection of math problems, covering grades 6 through 12, aligned with Common Core standards, ensuring a wide range of topics and difficulties.
The Algebra Problem: How Middle School Math Became a National
Teaching math in middle school is a challenge for educators in part because that is when the material becomes more complex, with students moving from multiplication tables to equations and ...
Teaching mathematics through problem posing: insights from ...
Teaching mathematics through problem posing is similar to teaching mathematics through problem solving (Schroeder & Lester, 1989), but, in teaching through problem posing, learning takes place during the process of students' posing of mathematical problems and classroom discussion of posed problems. With respect to understanding how problem ...
Empowering student learning with Math Assistant
The good news is that Math Assistant, found in Microsoft OneNote, helps students find their math confidence and gain important math skills through inclusive, individualized features. Explore the scenarios below to learn how Math Assistant can help transform K-12 students' math experience, accelerate their learning, and help them enjoy math too.
Memorial of St Boniface, Bishop & Martyr
Memorial of St Boniface, Bishop & Martyr | June 5, 2024 We welcome our Holy Family 8th Grade students as they receive their diplomas after the Mass

5 Teaching Mathematics Through Problem Solving

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Low Floor High Ceiling Tasks

Math in 3-Acts

Number Talks

Saying “This is Easy”

Using “Worksheets”

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Teaching Mathematics through Problem Solving- An Upside-Down Approach

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Published by Caty Romero - Math Specialist

6 Tips for Teaching Math Problem-Solving Skills

1. Link problem-solving to reading

2. Avoid boxing students into choosing a specific operation

3. Revisit ‘representation’

4. Give time to process

5. Ask questions that let Students do the thinking

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

The Lesson Study Group

Teaching Through Problem-solving

What is Teaching Through Problem-Solving?

Why Teaching Through Problem-Solving?

Phases of a TTP Lesson

WHAT STUDENTS DO

WHAT TEACHERS DO

How Do Teachers Support Problem-solving?

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1.5: Teaching Mathematics Through Problem Solving

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Low Floor High Ceiling Tasks

Math in 3-Acts

Number Talks

Saying “This is Easy”

Using “Worksheets”

Teaching through Problem Solving

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Introduction

The Present Study

Participants

Intervention

Implementation of the Intervention

Control Group

Tests of Mathematical Problem-Solving

Measures of Peer Acceptance and Friendships

Statistical Analyses

What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

Limitations

Implications

Data Availability Statement

Ethics Statement

Author Contributions

Conflict of Interest

Publisher’s Note

Acknowledgments

Supplementary Material

Young Students Gravitate to Math. How Teachers Can Build on That Curiosity

Does using real-world problems motivate students?

Promote young students’ natural curiosity and creativity

Teachers need more training in the problem-solving approach

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