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6 Tips for Teaching Math Problem-Solving Skills

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

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

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

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

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

1. Link problem-solving to reading

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

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

2. Avoid boxing students into choosing a specific operation

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

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

3. Revisit ‘representation’

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

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

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

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

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

4. Give time to process

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

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

5. Ask questions that let Students do the thinking

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

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

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

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

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

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

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

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

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

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

Instead, you and your students could say the following:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Problem-Solving Strategies

October 16, 2019

There are many different ways to solve a math problem, and equipping students with problem-solving strategies is just as important as teaching computation and algorithms. Problem-solving strategies help students visualize the problem or present the given information in a way that can lead them to the solution. Solving word problems using strategies works great as a number talks activity and helps to revise many skills.

Problem-solving strategies

1. create a diagram/picture, 2. guess and check., 3. make a table or a list., 4. logical reasoning., 5. find a pattern, 6. work backward, 1. create a diagram/draw a picture.

Creating a diagram helps students visualize the problem and reach the solution. A diagram can be a picture with labels, or a representation of the problem with objects that can be manipulated. Role-playing and acting out the problem like a story can help get to the solution.

Alice spent 3/4 of her babysitting money on comic books. She is left with $6. How much money did she make from babysitting?

2. Guess and check

Teach students the same strategy research mathematicians use.

With this strategy, students solve problems by making a reasonable guess depending on the information given. Then they check to see if the answer is correct and they improve it accordingly.  By repeating this process, a student can arrive at a correct answer that has been checked. It is recommended that the students keep a record of their guesses by making a chart, a table or a list. This is a flexible strategy that works for many types of problems. When students are stuck, guessing and checking helps them start and explore the problem. However, there is a trap. Exactly because it is such a simple strategy to use, some students find it difficult to consider other strategies. As problems get more complicated, other strategies become more important and more effective.

Find two numbers that have sum 11 and product 24.

Try/guess  5 and 6  the product is 30 too high

  adjust  to 4 and 7 with product 28 still high

  adjust  again 3 and 8 product 24

3. Make a table or a list

Carefully organize the information on a table or list according to the problem information. It might be a table of numbers, a table with ticks and crosses to solve a logic problem or a list of possible answers. Seeing the given information sorted out on a table or a list will help find patterns and lead to the correct solution.

To make sure you are listing all the information correctly read the problem carefully.

Find the common factors of 24, 30 and 18

Logical reasoning is the process of using logical, systemic steps to arrive at a conclusion based on given facts and mathematic principles. Read and understand the problem. Then find the information that helps you start solving the problem. Continue with each piece of information and write possible answers.

Thomas, Helen, Bill, and Mary have cats that are black, brown, white, or gray. The cats’ names are Buddy, Lucky, Fifi, and Moo. Buddy is brown. Thoma’s cat, Lucky, is not gray. Helen’s cat is white but is not named Moo. The gray cat belongs to Bill. Which cat belongs to each student, and what is its color?

A table or list is useful in solving logic problems.

Thomas	Lucky	Not gray, the cat is black
Helen	Not Moo, not Buddy, not Lucky so Fifi	White
Bill	Moo	Gray
Mary	Buddy	Brown

Since Lucky is not gray it can be black or brown. However, Buddy is brown so Lucky has to be black.

Buddy is brown so it cannot be Helen’s cat. Helen’s cat cannot be Moo, Buddy or Lucky, so it is Fifi.

Therefore, Moo is Bill’s cat and Buddy is Mary’s cat.

5. Find a pattern.

Finding a pattern is a strategy in which students look for patterns in the given information in order to solve the problem. When the problem consists of data like numbers or events that are repeated then it can be solved using the “find a pattern” problem-solving strategy. Data can be organized in a table or a list to reveal the pattern and help discover the “rule” of the pattern.

 The “rule” can then be used to find the answer to the question and complete the table/list.

Shannon’s Pizzeria made 5 pizzas on Sunday, 10 pizzas on Monday, 20 pizzas on Tuesday, and 40 pizzas on Wednesday. If this pattern continues, how many pizzas will the pizzeria make on Saturday?

Sunday	5
Monday	10
Tuesday	20
Wednesday	40
Thursday
Friday
Saturday

6. Working backward

Problems that can be solved with this strategy are the ones that  list a series of events or a sequence of steps .

In this strategy, the students must start with the solution and work back to the beginning. Each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backward they will need to subtract. And if they multiply working forwards, they must divide when working backward.

Mom bought a box of candy. Mary took 5 of them, Nick took 4 of them and 31 were given out on Halloween night. The next morning they found 8 pieces of candy in the box. How many candy pieces were in the box when mom bought it.

For this problem, we know that the final number of candy was 8, so if we work backward to “put back” the candy that was taken from the box we can reach the number of candy pieces that were in the box, to begin with.

The candy was taken away so we will normally subtract them. However, to get back to the original number of candy we need to work backward and do the opposite, which is to add them.

8 candy pieces were left + the 31 given out + plus the ones Mary took + the ones Nick took

8+31+5+4= 48   Answer: The box came with 48 pieces of candy.

Selecting the best strategy for a problem comes with practice and often problems will require the use of more than one strategies.

Print and digital activities

I have created a collection of print and digital activity cards and worksheets with word problems (print and google slides) to solve using the strategies above. The collection includes 70 problems (5 challenge ones) and their solution s and explanations.

sample below

How to use the activity cards

Allow the students to use manipulatives to solve the problems. (counters, shapes, lego blocks, Cuisenaire blocks, base 10 blocks, clocks) They can use manipulatives to create a picture and visualize the problem. They can use counters for the guess and check strategy. Discuss which strategy/strategies are better for solving each problem. Discuss the different ways. Use the activities as warm-ups, number talks, initiate discussions, group work, challenge, escape rooms, and more.

Ask your students to write their own problems using the problems in this resource, and more, as examples. Start with a simple type. Students learn a lot when trying to compose a problem. They can share the problem with their partner or the whole class. Make a collection of problems to share with another class.

For the google slides the students can use text boxes to explain their thinking with words, add shapes and lines to create diagrams, and add (insert) tables and diagrams.

Many of the problems can be solved faster by using algebraic expressions. However, since I created this resource for grades 4 and up I chose to show simple conceptual ways of solving the problems using the strategies above. You can suggest different ways of solving the problems based on the grade level.

Find the free and premium versions of the resource below. The premium version includes 70 problems (challenge problems included) and their solutions

There are 2 versions of the resource

70 google slides with explanations + 70 printable task cards

70 google slides with explanations + 11 worksheets

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Math Strategies: Problem Solving by Making a List

As I’ve mentioned many times, one of the main goals in mathematics education is to raise up confident problem solvers . And while there are many ways to go about solving math problems, and we as adults may often see strategies as common sense, these are things that need to be taught. Giving kids as many tools as possible will set them up for success so that you can “let them loose” and see their creative minds work and explore. To continue my series on teaching kids to problem solve, today I’m going to discuss problem solving by making a list .

–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

Making a Meaningful List: 

This was always a hard approach for me personally because it doesn’t seem like math, and can often be time consuming. I mean, who really wants to sit and list out all the possible solutions to a math problem? BUT, it is a very useful strategy, and as we’ll see, learning to be organized and systematic is the key ( and will also save tons of time )!

So when is it useful to make a list? Basically, any time you have a problem that has more than one solution , or you’re trying to solve a combination problem , it’s helpful to make a list.

But not just any list of possibilities. That will feel useless and frustrating if you’re just trying to pull out possibilities from anywhere. And more than that, it’s very likely possibilities will be skipped or repeated, making the final solution wrong.

On top of that, it will probably be more time consuming to make a list if you don’t have a systematic approach to it, which is probably why I was never a fan as a kid. No one wants to just sit and stare at the paper hoping solutions will pop into their brain.

Organizing the information in a logical way keeps you on track and ensures that all the possible solutions will be found.

There are different ways to organize information, but the idea is to exhaust all the possibilities with one part of your list before moving on.

For example, say you’re trying to figure out all the different combinations of ice cream toppings at your local ice cream shop. They have 3 different flavors (chocolate, vanilla and strawberry), but also have 4 different toppings (nuts, whipped cream, chocolate candies and gummy bears).

If you just start listing different possibilities without any kind of structure, you’re bound to get lost in your list and miss something. So instead, list all the possibilities for chocolate ice cream before moving on to vanilla.

Chocolate: just chocolate (no toppings), chocolate with nuts, chocolate with whipped cream, chocolate with candies and chocolate with gummy bears.

Now we see that there are 5 possibilities if you get chocolate ice cream, and so we can move on to vanilla, and then strawberry.

The key is to start with the first flavor and list every possible topping in order . Then move on to the next flavor and go through the toppings in the same order .

Then nothing gets skipped, forgotten or repeated . After completing the list, we see that there are 15 possible combinations.

Some students may even notice that there will be 5 possibilities for each flavor , and thus multiply 3×5 without completing the list. (That’s another great strategy: look for patterns ).

Even if a pattern is not discovered, however, completing the list in an organized, systematic way will ensure all possibilities are covered and the total (15) found.

Another way to organize the list is to make a tree diagram . Here’s another example problem:

Sarah is on vacation and brought 3 pairs of pants (blue, black, and white) and 3 shirts (pink, yellow and green). How many different outfit combinations can she make?

Using a tree diagram is a great way to keep the information organized, especially if you have kids who struggle with keeping track of their list:

Then it is very easy for students to see that there are 9 different outfit combinations .

Was this helpful? Is it a strategy that you share with your kids?

See the rest of the posts in this series and prepare your kids to be great problem solvers:

• Problem Solve by Solving an Easier Problem
• Problem Solve by Drawing a Picture
• Problem Solve by Working Backwards
• Problem Solve by Finding a Pattern
• Problem Solve with Guess & Check

I’m really liking this “problem solving” series! As someone who’s not so great as problem solving, these tips are going to come in handy when helping my daughter! Thanks for sharing at the Thoughtful Spot!

I was never very good at math, and unfortunately, my daughter isn’t great at it either. I came across this post on Hop (on the Hip Homeschool Moms site), and I love it! I’m going to read the other articles in the series too. I would love to help my daughter enjoy and understand math, and I’m hopeful that your posts will help me do that! Thanks so much for sharing your post with us on the Hop!

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Maths with David

Problem solving. be systematic.

Many formal structures can help us to systematically record information, such as lists, tables, charts, Venn diagrams, 2-way diagrams and tree diagrams.

Systematic recording makes it easy to review what has already been done, to easily spot missing or duplicate values, and to identify patterns (which might then be extended).

Worked Examples

First we will read both examples and have a quick think about them and then we will look at how working systematically can help us with each one:

South Korea Example

Below is the South Korean flag:

The trigrams around the outside of the central circle each are made up of three lines. The top left one has solid lines. The one on the top right has broken lines at the bottom an the top and a solid line in the middle.

Using only solid or broken lines, what fraction of all the possible trigrams appear on the South Korean flag.

Postman Example

The postman notice that Suzie had a very large number of cards and packages one day, so guessed that it must be her birthday. He asked her how old she was and she told him that yesterday her age was a square number and today it was a prime number. Would he be able to identify how old Suzie was?

Worked Solutions to Examples

First we must ensure we understand the problem. Lines can be either solid or broken and their are three lines.

To be systematic, we could start with three solid lines and then gradually introduce broken lines. If there were one broken line, there are three positions that it could be in.

If there were two broken lines, there are three positions that the remaining solid line could be in.

And if there were three broken lines, there is only one such arrangement. So all of the possible trigrams are:

As there are 8 in total and the flag contain 4, so the answer to the problem is that 1/2 of the possible trigrams appear on the flag.

We are interested in consecutive numbers where the first number is square and the second number is prime.

Because it is an age and (almost) all ages are from 1 to 121 we only need to check 11 square numbers to have checked all the square numbers that can be ages. For each we simply need to look at the following number to check if it is prime.

We look at the square numbers rather than the prime numbers, because there are fewer and it is easier to identify them. For instance square number 1 is followed by 2, which is prime, and square number 4 is followed by 5 which is prime. Continuing like this we can compile a table of our results:

From the table we can see that possible ages of Suzie are 2, 5, 17, 37 and 101. Hopefully he should be able to identify which of these specific ages was her actual one, based on her appearance.

18 questions in increasing order of difficulty

Solutions to Question Set

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Math Heuristics: Make A Systematic List, Guess And Check, And Restate The Problem In Another Way

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The skills children pick up in math are indispensable; they can be applied to other academic subjects and to solve real-world problems in their daily lives and future work.

The Singapore Math curriculum focuses on problem-solving. Through problem-solving, children develop thinking skills such as creative thinking and critical thinking.

When children analyse math problems, they notice patterns, learn to generalise, form new ideas and activate their creative thinking. Children become critical thinkers when they are able to select the best strategy out of multiple methods to solve word problems.

Singapore Math Heuristics

In part one of our Math Heuristics series , we gave an overview of the 12 problem-solving methods or heuristics taught in the Singapore primary math education syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to use them to solve various math word problems.

In part two of the Math Heuristic series , we zoomed in on the heursitics – Act It Out, Draw A Diagram and Look For Patterns, and also showed how to apply the Polya’s 4-step problem-solving process in sample word problems.

In the third part of this series, we will focus on the next 3 heuristics – Make a systematic list , guess and check , and restate the problem in another way .

Sample word problems are solved using these 3 heuristics and Poly’s 4-step process in the step-by-step worked solutions provided by the curriculum team at S.A.M.

Heuristic: Make a systematic list

Word Problem (Primary 2):

Jimmy uses the number cards given below to form as many 3-digit odd numbers as he can. List all the numbers that Jimmy can form.

1. Understand: What to find: All the 3-digit odd numbers that Jimmy can form from the 4 number cards. What is known: Odd numbers end with 5 or 7.

2. Choose: Make a systematic list

Odd numbers that end with 5: 245 275 425 475 725 745

Odd numbers that end with 7: 247 257 427 457 527 547

Jimmy can form 12 3-digit odd numbers.

4. Check: Did I form 3-digit numbers? Yes Did I form odd numbers? Yes Did I form all possible numbers? Yes

Try solving the following word problem using Polya’s 4-step process.

A shop sells apples in bags of 3. It sells lemons in bags of 4. Paul buys some bags of apples and lemons. He buys the same number of each fruit. He buys more than 20 and fewer than 30 pieces of each fruit. How many apples does Paul buy?

Answer: Paul buys 24 apples. See the solution in part one of our Singapore Math Heuristics series .

Heuristic: Guess and check

Word Problem (Primary 3):

David sold a total of 15 $4 coupons and $5 coupons for a funfair. He received $65 for the sale of the coupons. How many $4 coupons and how many $5 coupons did he sell?

1. Understand: What to find: The number of $4 coupons and the number of $5 coupons David sold? What is known: He sold 15 coupons. He received $65.

2. Choose: Guess and check

$4 + $5 = $9. (7 × 9) = 63 is close to 65. I can start the first guess with 7 $4 coupons.

David sold 10 $4 coupons and 5 $5 coupons.

4. Check: What is the total number of coupons? 10 + 5 = 15 What is the total value of coupons? $40 + $25 = $65

Word Problem (Primary 5):

In a quiz, 5 marks were awarded for each correct answer and 3 marks were deducted for each wrong answer. Darren answered 14 questions and scored 30 marks. How many questions did he answer correctly?

1. Understand: What to find: The number of questions Darren answered correctly. What is known: Add 5 marks for each correct answer. Minus 3 marks for each wrong answer. He answered 14 questions. He scored 30 marks.

I can start the first guess with the same number of correct answers and wrong answers.

4. Check: What is the total number of questions? 9 + 5 = 14 What is the total marks scored? 45 – 15 = 30

Vijay is presented with the equations below. Insert one pair of brackets in each equation to make it true. 4 × 11 + 18 ÷ 3 + 6 = 46

Answer: The equation is 4 × 11 + 18 ÷ (3 + 6) = 46. See the solution in part one of our Singapore Math Heuristics series .

Heuristic: Restate the problem in another way

Sally has some beds and sofas. All of them are equal in length. The total length is 14 metres. Each bed is 2 metres long. Sally has 1 fewer bed than sofas. What is the total length of the sofas?

1. Understand: What to find: Total length of the sofas. What is known: Each bed is 2 metres long. Each sofa is 2 metres long. The total length of beds and sofas is 14 metres. Sally has 1 more sofa than bed.

2. Choose: Restate the problem in another way

If we add 1 more bed, Sally will have the same number of beds and sofas. New total length = 14 m + 2 m = 16 m Sally has the same number of beds and sofas Total length of sofas = 16 m ÷ 2 = 8 m

The total length of the sofas is 8 metres.

4. Check: How many sofas are there? 8 m ÷ 2 m = 4 How many beds are there? 4 – 1 = 3

There are some identical pens and erasers. 2 pens and 3 erasers are 45 centimetres long altogether. 6 erasers and 2 pens are 60 centimetres long altogether. What is the length of 3 erasers?

Answer: The length of 3 erasers is 15 cm. See the solution in part one of our Singapore Math Heuristics series .

These are just a few examples to show you how heuristics are used to solve basic and intermediate word problems in lower primary levels and complex word problems in upper primary levels.

Look out for parts four and five of this series for the other 6 remaining Math Heuristics and word problems with step-by-step worked solutions.

This is the third part to S.A.M Math Heuristics series for expert tips on math heuristics.

Read the rest of the “S.A.M Math Heuristics” five-part series below :

Part 1: What Are Heuristics? Part 2: Math Heuristics: Act It Out, Draw A Diagram, Look For Patterns Part 4: Math Heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards Part 5: Math Heuristics: Draw a Table, Make Suppositions and Use Before-After Concept

Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world’s largest Singapore Math enrichment program for children aged four to 12. The award-winning S.A.M program is based on the global top-ranking Singapore Math curriculum with a focus on developing problem-solving and thinking skills.

The curriculum is complemented by S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

This post is brought to you by Seriously Addictive Mathematics .

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Frustration in mathematical problem-solving: A systematic review of research

• Kaitlin Riegel 1 ,  , 

Department of Mathematics, University of Auckland, Auckland, New Zealand

* Correspondence: [email protected]

Academic Editor: Christopher Tisdell

Emotions are an integral part of problem-solving, but must emotions traditionally conceptualised as "negative" have negative consequences in learning? Frustration is one of the most prominent emotions reported during mathematical problem-solving across all levels of learning. Despite research aiming to mitigate frustration, it can play a positive role during mathematical problem solving. A systematic review method was used to explore how frustration usually appears in students during mathematical problem-solving and the typical patterns of emotions, behaviours, and cognitive processes that are associated with its occurrence. The findings are mixed, which informs the need for further research in this area. Additionally, there are theories and qualitative findings about the potential positive role of frustration that have not been followed up with empirical investigations, which illuminate how our findings about negative emotions may be limited by the questions we ask as researchers. With the support of research, I consider how educators may directly or indirectly address rethinking the role and consequences of frustration during problem-solving with their students.

• frustration ,
• problem-solving ,
• affective pathways ,
• affective domain .

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Table 1.   Literature search and processing of records

Table 2.   Studies selected for the systematic review and the role of frustration

Bjuland [ ]	QL	Student teachers	positive
Carlson & Bloom [ ]	QL	Mathematicians (N = 12)	inconclusive
Chen et al. [ ]	MM	Case study of a 9-year-old boy	negative
DeBellis & Goldin [ ]	QL	High school students (N = 8)	positive
DeBellis & Goldin [ ]	T/QL	9-10 year olds (N = 19)	both
Di Leo & Muis [ ]	MM	Grade 5 students (N = 57)	negative
Di Leo et al. [ ]	MM	Study 1: Grade 5-6 students (N = 138); Study 2: Grade 5 students (N = 79)	both
Galán & Beal [ ]	QN	Undergraduate students (N = 16)	negative
Goldin [ ]	T	n/a	both
Goldin [ ]	T	n/a	both
Goldin [ ]	T	n/a	both
Goldin et al. [ ]	T	n/a	both
Gómez et al. [ ]	QN	Grade 9 students (N = 452)	negative
McCleod [ ]	T	n/a	negative
Muis et al. [ ]	MM	Grade 5 students (N = 79)	negative
Munzar et al. [ ]	MM	Study 1: Grade 3-6 students (N = 136); Study 2: Grade 5 students (N = 80)	negative
O'Dell [ ]	QL	Grade 4-5 students (N = 10)	positive
Presmeg & Balderas-Cañas [ ]	QL	Graduate students (N = 4)	both
Voica et al. [ ]	MM	Pre-service teachers (N = 114)	inconclusive
Weber [ ]	QL	Case study of an undergraduate student	negative
QL = Qualitative, QN = Quantitative, T = Theoretical, MM = Mixed Methods

Table 3.   Summary of the findings on the role of frustration in mathematical problem-solving by study participants

	Positive	Negative	Both	Inconclusive	Total
Primary	1	4	2	-	7
Secondary	1	1	-	-	2
Tertiary	-	2	1	-	3
Student-teachers	1	-	-	1	2
Mathematicians	-	-	-	1	1
Total	3	7	3	2	15
* The exclusively theoretical papers were not applicable so were not included (N = 15)

Table 4.   Summary of the role of frustration in mathematical problem-solving by study methods

	Positive	Negative	Both	Omitted	Total
Qualitative	3	1	1	1	6
Quantitative	-	2	-	-	2
Mixed-Methods	-	4	1	1	6
Theoretical	-	1	5	-	6
Total	3	8	7	2	20
* DeBellis & Goldin [ ] was included as a theoretical study as this is where the discussion of the role of frustration is dominant.

[1]	Hannula, M., Emotions in problem solving, in , S.J. Cho Ed. 2015, pp. 269-288, Springer. .
[2]	, et al., Affect in mathematics education: An introduction, , (2006), 113-122.  doi:
[3]	McCleod, D.B., The role of affect in mathematical problem solving, in , D.B. Mcleod and V.M. Adams Ed. 1989, pp. 20-36, Springer. .
[4]	and  , Dynamics of affective states during complex learning, , (2012), 145-157.  doi:
[5]	, Affective pathways and representation in mathematical problem solving, , (2000), 209-219.  doi:
[6]	Pekrun, R. and Stephens, E.J., Achievement emotions in higher education, in , J.C. Smart Ed. 2010, 25: 257-306, Springer. .
[7]	, Emotion research in education: Theoretical and methodological perspectives on the integration of affect, motivation, and cognition, , (2006), 307-314.  doi:
[8]	,   and  , Stimulating student aesthetic response to mathematical problems by means of manipulating the extent of surprise, , (2017), 42-57.  doi:
[9]	and  , Surprise and the aesthetic experience of university students: A design experiment, , (2016), 127-151.  doi:
[10]	and  , Affect and meta-affect in mathematical problem solving: A representational perspective, , (2006), 131-147.  doi:
[11]	, et al., Curiosity...Confusion? Frustration! The role and sequencing of emotions during mathematics problem solving, , (2019), 121-137.  doi:
[12]	, et al., The role of epistemic emotions in mathematics problem solving, , (2015), 172-185.  doi:
[13]	and  , Confused, now what? A Cognitive-Emotional Strategy Training (CEST) intervention for elementary students during mathematics problem solving, , (2020), 101879.  doi:
[14]	, et al., Elementary students' cognitive and affective responses to impasses during mathematics problem solving, , (2021), 104-124.  doi:
[15]	and  , The roles of aesthetic in mathematical inquiry, , (2004), 261-284.  doi:
[16]	Galán, F.C. and Beal, C.R., EEG estimates of engagement and cognitive workload predict math problem solving outcomes, in , 2012, pp. 51-62, Springer. .
[17]	, et al., Achievement emotions in mathematics: Design and evidence of validity of a self-report scale, , (2020), 233-247.  doi:
[18]	Chen, L., ., Riding an emotional roller-coaster: A multimodal study of young child's math problem solving activities, in , T. Barnes, M. Chi and M. Feng Ed. 2016, pp. 38-45.
[19]	, et al., Beliefs and engagement structures: Behind the affective dimension of mathematical learning, , (2011), 547-560.  doi:
[20]	, Problem solving heuristics, affect, and discrete mathematics, , (2004), 56-60.  doi:
[21]	, Affective issues in mathematical problem solving: Some theoretical considerations, , (1988), 134-141.  doi:
[22]	, The role of affect in learning Real Analysis: A case study, , (2008), 71-85.  doi:
[23]	, Representational systems, learning, and problem solving in mathematics, , (1998), 137-165.
[24]	, Student teachers' reflections on their learning process through collaborative problem solving in geometry, , (2004), 199-225.  doi:
[25]	,   and  , How are motivation and self-efficacy interacting in problem-solving and problem-posing?, , (2020), 487-517.  doi:
[26]	DeBellis, V.A. and Goldin, G. A., Interactions between cognition and affect in eight high school students' individual problem solving, in , R.G. Underhill Ed. 1991, pp. 29-35. Virginia Polytechnic University, Division of Curriculum and Instruction.
[27]	and  , The cyclic nature of problem solving: An emergent multidimensional problem-solving framework, , (2005), 45-75.  doi:
[28]	and  , Visualization and affect in nonroutine problem solving, , (2001), 289-313.  doi:
[29]	O'Dell, J.R., The interplay of frustration and joy: Elementary students' productive struggle when engaged in unsolved problems, in , T.E. Hodges, G.J. Roy and A.M. Tyminski Ed. 2018, pp. 938-945. University of South Carolina & Clemson University.
[30]	, Productive struggle in middle school mathematics classrooms, , (2015), 375-400.  doi:
[31]	,  , Routledge, London, 1960.
[32]	Vygotsky, L.S., , Ed. by R.W. Rieber and A.S. Carton.
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[34]	,   and  , Rethinking stress: The role of mindsets in determining the stress response, , (2013), 716-733.  doi:
[35]	, et al., The role of stress mindset in shaping cognitive, emotional, and physiological responses to challenging and threatening stress, , (2017), 379-395.  doi:

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Evaluating Large Vision-and-Language Models on Children's Mathematical Olympiads

• Cherian, Anoop
• Peng, Kuan-Chuan
• Lohit, Suhas
• Matthiesen, Joanna
• Smith, Kevin
• Tenenbaum, Joshua B.

Recent years have seen a significant progress in the general-purpose problem solving abilities of large vision and language models (LVLMs), such as ChatGPT, Gemini, etc.; some of these breakthroughs even seem to enable AI models to outperform human abilities in varied tasks that demand higher-order cognitive skills. Are the current large AI models indeed capable of generalized problem solving as humans do? A systematic analysis of AI capabilities for joint vision and text reasoning, however, is missing in the current scientific literature. In this paper, we make an effort towards filling this gap, by evaluating state-of-the-art LVLMs on their mathematical and algorithmic reasoning abilities using visuo-linguistic problems from children's Olympiads. Specifically, we consider problems from the Mathematical Kangaroo (MK) Olympiad, which is a popular international competition targeted at children from grades 1-12, that tests children's deeper mathematical abilities using puzzles that are appropriately gauged to their age and skills. Using the puzzles from MK, we created a dataset, dubbed SMART-840, consisting of 840 problems from years 2020-2024. With our dataset, we analyze LVLMs power on mathematical reasoning; their responses on our puzzles offer a direct way to compare against that of children. Our results show that modern LVLMs do demonstrate increasingly powerful reasoning skills in solving problems for higher grades, but lack the foundations to correctly answer problems designed for younger children. Further analysis shows that there is no significant correlation between the reasoning capabilities of AI models and that of young children, and their capabilities appear to be based on a different type of reasoning than the cumulative knowledge that underlies children's mathematics and logic skills.

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• Computer Science - Computer Vision and Pattern Recognition

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Working Systematically - Secondary Teachers

Working Systematically is part of our Developing Mathematical Thinking  collection.

In  Developing Mathematical Thinking - Working Sytematically  we highlight the benefits of working systematically in a variety of contexts. Mathematicians often talk about the importance of working systematically. This means that rather than working in a haphazard and random way, there is a methodical, organised and logical approach. The problems below will challenge students to work systematically, and will help them appreciate the benefits of working in this way.

Year 7 Working Systematically

Year 8 Working Systematically

Year 9 Working Systematically

Year 10 Working Systematically

Year 11 Working Systematically

Extension Working Systematically