problem solving strategies for primary maths

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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.

Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

Shametria Routt Banks

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2 Responses

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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Teaching Problem Solving in Math

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

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

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

The Problem Solving Strategies

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

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

There’s also this visuals strategy wheel practice.

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

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

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

The Problem Solving Steps

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

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

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

We did this over and over with example problems.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Act  – Act on a strategy and try it out.

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

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

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

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

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Problem Solving

Problem Solving Strategies

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

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

Pólya’s How to Solve It

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

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

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

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

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

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

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

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

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

Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Problem 2 (Payback)

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

Think/Pair/Share

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

This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

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

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

Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

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

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

Problem 3 (Squares on a Chess Board)

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

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

Think / Pair / Share

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

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

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

Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

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

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

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

Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

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

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

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

Problem 4 (Broken Clock)

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

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

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

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

Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

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

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

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

Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

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

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

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

Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Math Problem Solving Strategies That Make Students Say “I Get It!”

Even students who are quick with math facts can get stuck when it comes to problem solving.

As soon as a concept is translated to a word problem, or a simple mathematical sentence contains an unknown, they’re stumped.

That’s because problem solving requires us to  consciously choose the strategies most appropriate for the problem   at hand . And not all students have this metacognitive ability.

But you can teach these strategies for problem solving.  You just need to know what they are.

We’ve compiled them here divided into four categories:

Strategies for understanding a problem

Strategies for solving the problem, strategies for working out, strategies for checking the solution.

Get to know these strategies and then model them explicitly to your students. Next time they dive into a rich problem, they’ll be filling up their working out paper faster than ever!

Before students can solve a problem, they need to know what it’s asking them. This is often the first hurdle with word problems that don’t specify a particular mathematical operation.

Encourage your students to:

Read and reread the question

They say they’ve read it, but have they  really ? Sometimes students will skip ahead as soon as they’ve noticed one familiar piece of information or give up trying to understand it if the problem doesn’t make sense at first glance.

Teach students to interpret a question by using self-monitoring strategies such as:

• Rereading a question more slowly if it doesn’t make sense the first time
• Asking for help
• Highlighting or underlining important pieces of information.

Identify important and extraneous information

John is collecting money for his friend Ari’s birthday. He starts with $5 of his own, then Marcus gives him another $5. How much does he have now?

As adults looking at the above problem, we can instantly look past the names and the birthday scenario to see a simple addition problem. Students, however, can struggle to determine what’s relevant in the information that’s been given to them.

Teach students to sort and sift the information in a problem to find what’s relevant. A good way to do this is to have them swap out pieces of information to see if the solution changes. If changing names, items or scenarios has no impact on the end result, they’ll realize that it doesn’t need to be a point of focus while solving the problem.

Schema approach

This is a math intervention strategy that can make problem solving easier for all students, regardless of ability.

Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:

[Number/Quantity A] with [Number/Quantity B] removed becomes [end result].

This is the underlying procedure or  schema  students are being asked to use. Once they have a list of schema for different mathematical operations (addition, multiplication and so on), they can take turns to apply them to an unfamiliar word problem and see which one fits.

Struggling students often believe math is something you either do automatically or don’t do at all. But that’s not true. Help your students understand that they have a choice of problem-solving strategies to use, and if one doesn’t work, they can try another.

Here are four common strategies students can use for problem solving.

Visualizing

Visualizing an abstract problem often makes it easier to solve. Students could draw a picture or simply draw tally marks on a piece of working out paper.

Encourage visualization by modeling it on the whiteboard and providing graphic organizers that have space for students to draw before they write down the final number.

Guess and check

Show students how to make an educated guess and then plug this answer back into the original problem. If it doesn’t work, they can adjust their initial guess higher or lower accordingly.

Find a pattern

To find patterns, show students how to extract and list all the relevant facts in a problem so they can be easily compared. If they find a pattern, they’ll be able to locate the missing piece of information.

Work backward

Working backward is useful if students are tasked with finding an unknown number in a problem or mathematical sentence. For example, if the problem is 8 + x = 12, students can find x by:

• Starting with 12
• Taking the 8 from the 12
• Being left with 4
• Checking that 4 works when used instead of x

Now students have understood the problem and formulated a strategy, it’s time to put it into practice. But if they just launch in and do it, they might make it harder for themselves. Show them how to work through a problem effectively by:

Documenting working out

Model the process of writing down every step you take to complete a math problem and provide working out paper when students are solving a problem. This will allow students to keep track of their thoughts and pick up errors before they reach a final solution.

Check along the way

Checking work as you go is another crucial self-monitoring strategy for math learners. Model it to them with think aloud questions such as:

• Does that last step look right?
• Does this follow on from the step I took before?
• Have I done any ‘smaller’ sums within the bigger problem that need checking?

Students often make the mistake of thinking that speed is everything in math — so they’ll rush to get an answer down and move on without checking.

But checking is important too. It allows them to pinpoint areas of difficulty as they come up, and it enables them to tackle more complex problems that require multiple checks  before  arriving at a final answer.

Here are some checking strategies you can promote:

Check with a partner

Comparing answers with a peer leads is a more reflective process than just receiving a tick from the teacher. If students have two different answers, encourage them to talk about how they arrived at them and compare working out methods. They’ll figure out exactly where they went wrong, and what they got right.

Reread the problem with your solution

Most of the time, students will be able to tell whether or not their answer is correct by putting it back into the initial problem. If it doesn’t work or it just ‘looks wrong’, it’s time to go back and fix it up.

Fixing mistakes

Show students how to backtrack through their working out to find the exact point where they made a mistake. Emphasize that they can’t do this if they haven’t written down everything in the first place — so a single answer with no working out isn’t as impressive as they might think!

Need more help developing problem solving skills?

Read up on  how to set a problem solving and reasoning activity  or explore Mathseeds and Mathletics, our award winning online math programs. They’ve got over 900 teacher tested problem solving activities between them!

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10 Strategies for Problem Solving in Math

Updated on January 6, 2024

When faced with problem-solving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them. Therefore, this article discusses math strategies and how your students may use them since instructors often have to lead students through this problem-solving maze.

What Are Problem Solving Strategies in Math?

If you want to fix a problem, you need a solid plan. Math strategies for problem solving are ways of tackling math in a way that guarantees better outcomes. These strategies simplify math for kids so that less time is spent figuring out the problem. Both those new to mathematics and those more knowledgeable about the subject may benefit from these methods.

There are several methods to apply problem-solving procedures in math, and each strategy is different. While none of these methods failsafe, they may help your student become a better problem solver, particularly when paired with practice and examples. The more math problems kids tackle, the more math problem solving skills they acquire, and practice is the key.

Strategies for Problem-solving in Math

Even if a student is not a math wiz, a suitable solution to mathematical problems in math may help them discover answers. There is no one best method for helping students solve arithmetic problems, but the following ten approaches have shown to be very effective.

Understand the Problem

Understanding the nature of math problems is a prerequisite to solving them. They need to specify what kind of issue it is ( fraction problem , word problem, quadratic equation, etc.). Searching for keywords in the math problem, revisiting similar questions, or consulting the internet are all great ways to strengthen their grasp of the material. This step keeps the pupil on track.

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Guess and check.

One of the time-intensive strategies for resolving mathematical problems is the guess and check method. In this approach, students keep guessing until they get the answer right.

After assuming how to solve a math issue, students should reintroduce that assumption to check for correctness. While the approach may appear cumbersome, it is typically successful in revealing patterns in a child’s thought process.

Work It Out

Encourage pupils to record their thinking process as they go through a math problem. Since this technique requires an initial comprehension of the topic, it serves as a self-monitoring method for mathematics students. If they immediately start solving the problem, they risk making mistakes.

Students may keep track of their ideas and fix their math problems as they go along using this method. A youngster may still need you to explain their methods of solving the arithmetic questions on the extra page. This confirmation stage etches the steps they took to solve the problem in their minds.

Work Backwards

In mathematics, a fresh perspective is sometimes the key to a successful solution. Young people need to know that the ability to recreate math problems is valuable in many professional fields, including project management and engineering.

Students may better prepare for difficulties in real-world circumstances by using the “Work Backwards” technique. The end product may be used as a start-off point to identify the underlying issue.

In most cases, a visual representation of a math problem may help youngsters understand it better. Some of the most helpful math tactics for kids include having them play out the issue and picture how to solve it.

One way to visualize a workout is to use a blank piece of paper to draw a picture or make tally marks. Students might also use a marker and a whiteboard to draw as they demonstrate the technique before writing it down.

Find a Pattern

Kids who use pattern recognition techniques can better grasp math concepts and retain formulae. The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition.

Students may use this strategy to spot patterns and fill in the blanks. Over time, this strategy will help kids answer math problems quickly.

When faced with a math word problem, it might be helpful to ask, “What are some possible solutions to this issue?” It encourages you to give the problem more thought, develop creative solutions, and prevent you from being stuck in a rut. So, tell the pupils to think about the math problems and not just go with the first solution that comes to mind.

Draw a Picture or Diagram

Drawing a picture of a math problem can help kids understand how to solve it, just like picturing it can help them see it. Shapes or numbers could be used to show the forms to keep things easy. Kids might learn how to use dots or letters to show the parts of a pattern or graph if you teach them.

Charts and graphs can be useful even when math isn’t involved. Kids can draw pictures of the ideas they read about to help them remember them after they’ve learned them. The plan for how to solve the mathematical problem will help kids understand what the problem is and how to solve it.

Trial and Error Method

The trial and error method may be one of the most common problem solving strategies for kids to figure out how to solve problems. But how well this strategy is used will determine how well it works. Students have a hard time figuring out math questions if they don’t have clear formulas or instructions.

They have a better chance of getting the correct answer, though, if they first make a list of possible answers based on rules they already know and then try each one. Don’t be too quick to tell kids they shouldn’t learn by making mistakes.

Review Answers with Peers

It’s fun to work on your math skills with friends by reviewing the answers to math questions together. If different students have different ideas about how to solve the same problem, get them to share their thoughts with the class.

During class time, kids’ ways of working might be compared. Then, students can make their points stronger by fixing these problems.

Check out the Printable Math Worksheets for Your Kids!

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5 Strategies for Successful Problem Solving

• Powerful teaching strategies
• December 26, 2023
• Michaela Epstein

Blog > 5 Strategies for Successful Problem Solving

Problem solving can change the way students see maths – and how they see themselves as maths learners.

But, it's tough to help all students get the most out of a task.

To help, here are  5 Strategies for Problem Solving Success.

These are 5 valuable lessons I've learned from working with teachers across the globe .  You can use these strategies with all your students, no matter their level.

5 Strategies for Problem Solving Success

Your own enthusiasm is quickly picked up by your students. So, choose a problem, puzzle or game that you’re excited and curious about.

How do you know what will spark your curiosity? Do the task yourself!

(That’s why,  in the workshops I run , we spend a lot of time  actually  exploring problems. It’s a chance to step into students' shoes and experience maths from their perspective.)

Often, curriculum content becomes the goal of problem solving. For example, adding fractions, calculating areas or solving quadratic equations.

But, this is a mistake! Here's why-

Low floor, high ceiling tasks give students choices. Choices about what strategies to use, tools to draw on – and even what end-points to get to.

The most valuable goals focus on building confidence and capability in problem solving. For example:

• To make and break conjectures
• To use and evaluate different strategies
• To organise data in meaningful ways
• To explain and justify their conclusions.

The start of a task is what will get your students curious and hungry to get underway.

Consider: What's the least information your students will need?

At  ​our Members' online PL sessions​ , we look at one of four possibilities for launching a problem:

• Present a mystery to explore
• Present an example and non-example
• Run a demonstration game
• Show how to use a tool.

Keep the launch short – under 5 minutes. This is just enough to keep students’ attention AND share essential information.

Let’s face it, problem solving is hard, no matter your age or mathematical skill set.

Students aren’t afraid of hard work – they’re afraid of feeling or looking stupid. And, when those tricky maths moments do come, you can help.

Using questions, tools and other prompts can bring clarity and boost confidence.

(Here's a  free question catalogue  you might find handy to have in your back pocket.)

This careful support will help your students find problem solving far less daunting. Instead, it can become a chance for wonderous mathematical exploration.

Picture this: Your students are elbows deep in a problem, there’s a buzz in the air – oh, and only a minute until the bell.

The  most important  stage of a problem solving task – right at the end – is often the one that gets dropped off.

Why does ‘wrapping up’ matter?

In the last 10 minutes of a problem, students can share conjectures, strategies and solutions. It's also a chance to consider new questions that may open up further exploration.

In wrapping up, important learning will happen. Your students will observe patterns, make connections and clarify conjectures. You might even notice ‘aha’ moments.

Five strategies for problem solving success:

• Choose a task that YOU'RE keen on,
• Set a goal for strengthening problem solving skills,
• Plan a short launch to make the task widely accessible,
• Use questions, tools and prompts to support productive exploration, and
• Wrap up to create space for pivotal learning.

Join the Conversation

Dear Michaela, Greetings !! Thank you for sharing the strategies for problem solving task. These strategies will definitely enhance the skill in the mindset of young learners. In India ,Students of Grade 9 and Grade 10 have to learn and solve lot of theorems of triangle, Quadrilateral, Circle etc. Being an educator I have noticed that most of the students learn the theorems and it’s derivation by heart as a result they lack in understanding the application of these theorems.

I will appreciate if you can share your insights as how to make these topics interesting and easy to grasp.

Once again thanks for sharing such informative ideas.

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Building Problem-Solvers: Engaging Maths Challenges & Playful Games for Primary Students

Table of Contents

Building Problem-Solvers: Incorporating challenging maths puzzles and games into primary education is a powerful strategy for developing problem-solving skills in young learners. We recognise the importance of engaging pupils with interactive tasks that not only stimulate their mathematical abilities but also build their confidence in tackling complex challenges. By designing diverse mathematical exercises that range from simple to complex, we create opportunities for children to develop fluency as mathematical thinkers and problem solvers, a skill set that is essential in today’s fast-paced, ever-evolving world.

It’s crucial to foster a supportive maths culture within the classroom, one that encourages risk-taking and persistence in overcoming obstacles. Education in primary mathematics should not only focus on procedural fluency but also on the understanding of fundamental mathematical concepts . Tools and strategies such as visual representations and heuristic approaches can be invaluable in promoting this kind of deep comprehension. By integrating these techniques with a varied set of maths puzzles and games, we make learning mathematics an engaging and dynamic experience.

Key Takeaways

• Developing problem-solving skills in maths is essential for primary education.
• Diverse and challenging puzzles enhance mathematical thinking.
• A supportive learning environment encourages perseverance and growth in maths.

Understanding the Fundamentals of Maths

In our pursuit of developing proficient problem solvers, it’s essential to start with a strong foundation in mathematics. We focus on building an understanding of numbers and operations, as well as recognising the significance of patterns and relationships. These components are vital in nurturing a comprehensive mathematical mindset.

Number and Operations

At the core of mathematical understanding are numbers and operations . This involves grasping how numbers work and interact with one another through basic operations : addition, subtraction, multiplication, and division. For example:

Addition: It’s combining quantities to increase the total.

Subtraction: It’s determining how much one quantity is greater than another

Multiplication : It simplifies repeated addition.

Division: It’s finding out how many times a number can be subtracted from another in equal parts.

Understanding these operations enables children to solve real-world problems effectively.

Patterns and Relationships

Patterns and relationships form the building blocks for higher-level mathematical thinking. They involve seeing connections and predicting what comes next, based on a sequence or rule.

Patterns: These may be numerical or geometric, allowing us to anticipate the next element in a sequence.

Relationships: Understanding that certain pieces of information are related, and using this to solve problems, is key to mathematical reasoning . An example might be in understanding that if a car covers more distance in an hour at a faster speed, there’s a direct relationship between speed and distance.

In fostering these fundamental skills, we offer our learners not just theory but engaging challenges through mathematical puzzles and games that reinforce these concepts. By participating in such activities, children can apply these foundational maths skills, making connections between theory and practice in a dynamic and enjoyable way.

Teaching Mathematics Effectively

In our quest to develop sharp problem solvers and creative thinkers, we must focus on how mathematics is taught in primary schools. Our approach bridges the gap between traditional teaching and innovative, engaging methods .

Role of Teachers

We, as educators, play a pivotal role in moulding students into proficient problem-solvers. By incorporating a variety of maths puzzles and games into our classrooms, we make abstract concepts tangible and accessible. We believe in adapting our teaching styles to meet individual learning needs, devising strategies that make mathematics more than just a subject – instead, a captivating journey of discovery.

Curriculum Integration

The curriculum should not solely dictate our teaching, but serve as a dynamic framework that integrates real-world problem-solving and logical reasoning through mathematical activities . It is essential we weave challenging exercises with relevant contexts into lessons, thus aligning with our commitment to a broad and balanced education where every child flourishes.

The Importance of Problem-Solving Skills

Problem-solving skills are integral to learning mathematics, as they enable children to approach complex problems strategically and with confidence. Our goal is to harness these skills through puzzles and games that challenge and engage primary school children.

Developing Critical Thinking

We understand that critical thinking is the foundation of effective problem-solving. Our puzzles and games are designed to stimulate students’ thinking and reasoning processes , encouraging them to make connections and derive solutions based on logical deduction.

Cultivating Patience and Persistence

Throughout the problem-solving journey, we foster a sense of patience and persistence in our young learners. Recognising that solutions are not always immediate or straightforward, our resources teach children the value of perseverance and resilience in the face of challenging tasks.

Engaging Pupils with Interactive Maths Games

Interactive maths games are a brilliant way for us to make mathematics appealing and accessible to primary school pupils. By selecting the right games and integrating 21st-century technology, we can transform the learning experience into an interactive adventure that captivates our young learners.

Selecting Appropriate Games

When choosing maths games, we must ensure that they are appropriately challenging and aligned with the curriculum. The games should be designed to encourage pupils to think critically and develop their problem-solving skills. We often look for options that offer multiple levels of difficulty, which enables us to cater to the diverse abilities within a classroom. The aim is to select games that not only educate but also genuinely engage the students, making the learning process both enjoyable and effective.

Integrating ICT in Maths Games

In today’s digital age, integrating Information and Communication Technology (ICT) in maths games is not just innovative; it’s essential. We leverage ICT to provide simulations and virtual environments where pupils can explore mathematical concepts. This interactive technology helps to create immersive experiences that make abstract ideas more concrete. Using ICT, we can give pupils the chance to practise and hone their maths skills through fun, tech-driven games, fostering a more dynamic and interactive learning environment.

Incorporating interactive maths games within the classroom is a testament to our commitment to engaging and educating our pupils in a way that resonates with their experiences and interests. Through careful selection and the use of modern ICT, we are able to provide a learning experience that is both enriching and exciting.

Designing Challenging Maths Puzzles

In our pursuit to build effective problem-solvers, we focus on introducing maths puzzles that are engaging and non-routine, designed to foster natural curiosity in learners.

Creating Non-Routine Tasks

We believe that the essence of problem-solving lies in the ability to tackle non-routine tasks. These are not your everyday textbook problems, but instead, they present a scenario that requires students to apply concepts in ways they might not have anticipated. In constructing these tasks, it’s vital to strike a balance—too easy and they won’t push the envelope; too hard and they might discourage learners. For example, a puzzle involving pattern recognition might ask pupils to identify the underlying rule of a sequence and predict the next set of numbers.

Fostering Curiosity Through Puzzles

Curiosity drives us to explore the unknown, and with maths puzzles, it translates into learners venturing beyond their comfort zones. We craft puzzles that inherently provoke students’ interest and pique their natural inquisitiveness. A well-thought-out puzzle can act as an open inquiry, where the journey to the solution is as valuable as the solution itself. Consider a jigsaw arithmetic puzzle that requires learners to not only solve for missing pieces but also to understand why those pieces fit together as they do.

Incorporating Diverse Mathematical Tasks

When we introduce a variety of mathematical tasks to primary students, we lay the foundation for robust problem-solving skills. By engaging in an assortment of challenges , children can develop a deeper understanding of numbers, spatial awareness, and logical reasoning.

Exploration and Investigation

We believe that learning should be an adventure, where exploration and investigation play pivotal roles. By offering tasks that encourage students to explore, we open up opportunities for hands-on learning and inquisitive thinking. For instance, we might ask them to investigate the number of ways to reach a total of 10 using only red and blue counters. This simple task invites them to explore addition and the concept of combinations.

Measurement and Geometry

Measurement and geometry are two areas where children can apply maths to real-world scenarios. We often ask our students to measure lengths and widths of classroom objects, using rulers and other tools to relate the numbers to physical attributes. Then, we might move on to explore geometric shapes by examining and constructing models , allowing for a tangible understanding of edges, faces, and vertices. This hands-on experience is invaluable for cementing their conceptual knowledge.

Overcoming Obstacles in Problem Solving

Engaging with maths puzzles and games in primary education can sometimes present challenges. It’s essential we understand these stumbling blocks and employ effective strategies to help our children become confident problem solvers.

Identifying Common Mistakes

Often, obstacles in problem solving arise from common misunderstandings or repeated mistakes . We might see children rush through a problem without fully understanding it, or they may become fixed on one approach and not consider alternative methods. Documenting these mistakes offers us a chance to address them directly.

• Rushing : Not taking enough time to understand the problem.
• Fixation on one strategy : Failing to consider different angles.
• Overlooking details : Missing out on crucial information within the problem.

Strategies to Overcome Challenges

After pinpointing the typical mistakes, our strategies should focus on overcoming these barriers and reinforcing effective problem solving habits.

• Encourage thorough reading : Urge pupils to read problems several times.
• Promote multiple approaches : Introduce a variety of methods to tackle a single problem.
• Detail orientation : Teach children to pay attention to all the information given.

Remember, practice makes perfect, and providing children with a mixture of puzzles and games can build their resilience and adaptability in maths.

Creating a Supportive Maths Culture

At the heart of nurturing future problem-solvers is the establishment of a supportive maths culture . This means creating an environment where every child feels valued and capable of mastering mathematical challenges .

Encouraging Open Communication

We recognise the importance of open communication in the classroom. It’s crucial to foster an atmosphere where pupils feel comfortable to express their ideas, ask questions, and share their experiences. In our classroom, we encourage learners to articulate their thought processes and reasoning. This not only clarifies their understanding but also enriches peer learning, as students learn from each other’s perspectives. Such a culture not only enhances their communication skills but also demystifies complex concepts, making maths more approachable .

Building Confidence and Growth Mindset

We strive to infuse our pupils with confidence and a growth mindset . To do this, we emphasise the belief that abilities can be developed through dedication and hard work. This contrasts with a fixed mindset, where children might believe their skills are static and unchangeable. By celebrating effort rather than innate ability , we inspire our students to embrace challenges and learn from mistakes—an approach that is pivotal for fostering resilient problem solvers . Our aim is to show that mathematics is not about being ‘right’ all the time; instead, it’s about the adventure of learning and improving.

Assessing and Enhancing Problem-Solving Capabilities

In our quest to build adept problem-solvers through maths puzzles and games , it’s vital that we effectively assess and enhance primary students’ problem-solving capabilities. This not only involves gauging their current skill levels but also providing constructive feedback to aid their growth.

Conducting Meaningful Assessments

When it comes to assessing problem-solving activities, our approach is to create evaluations that are as engaging as the learning experiences themselves. We believe that a student’s problem-solving skills are best understood by observing them during actual problem-solving activities. Such assessments might involve practical tasks where pupils apply heuristics or use visual representations to tackle mathematical problems, as highlighted by Developing Mathematical Problem-Solving Skills . These tasks are designed to mirror ‘real life’ scenarios, requiring students to think critically and creatively.

Providing Constructive Feedback

Feedback is a powerful tool in our educational arsenal—it informs students about their performance and provides guidance on how they can improve. Our feedback is specific, timely, and always focused on strategies that students can use to enhance their problem-solving abilities. Whether it’s encouraging greater representational fluency or exploring diverse strategies, we ensure that our feedback helps students reflect and grow as problem solvers. By drawing on resources such as the Real Engagement in Active Problem Solving (REAPS) model, we support teachers in offering feedback that fosters creative problem solving in mathematics.

In all of our endeavours at LearningMole, we strive to provide learning experiences that not only educate but also excite. We’re dedicated to fostering environments where every student can become a confident and proficient problem solver.

Resources for Primary Maths Education

We know that the right resources can make all the difference in empowering primary maths education. So, let’s explore some excellent materials that you can access for free, as well as delve into the treasures offered by NRICH to enrich our young learners’ mathematical journey.

Free Teaching Materials

We pride ourselves on providing a variety of free teaching materials designed to make maths engaging and fun . You’ll find interactive tutorials , activity sheets , and articles, all tailored for the curious minds of primary children. These resources are not only educational, but they also allow children to absorb key maths concepts in a way that feels like play.

• Interactive Tutorials : Engage with our step-by-step guides that bring clarity to complex maths problems.
• Articles : Discover insights and tips targeted at enhancing the teaching experience.
• Activity Sheets : Download and print these for hands-on practice that reinforces mathematical understanding.

One platform we admire is LearningMole , which offers a wealth of content to help children discover the joy of maths through various fun and creative resources.

Utilising NRICH Resources

When it comes to using NRICH resources , we’re looking at a treasure trove of maths puzzles and activities that are perfect for primary pupils. NRICH aims to challenge and excite young minds with games that are both intriguing and highly educational.

• Challenging Puzzles : They stimulate strategic thinking and offer varying degrees of difficulty to suit all levels.
• Classroom Activities : Carefully crafted to promote collaborative problem-solving among students.

These resources from NRICH not only complement our teaching but also bring a new dimension to primary maths, making our lessons more dynamic and effective.

Frequently Asked Questions

In this section, we’ll address some of the most common inquiries related to enhancing problem-solving skills through mathematics puzzles and games in primary education.

What are some engaging activities that can enhance problem-solving abilities in primary school children?

Children in primary school can greatly benefit from activities such as building structures using different shapes or collaborative tasks where they devise solutions to “real life” scenarios. These activities encourage them to apply mathematical concepts in practical ways.

Can you suggest some games that improve problem-solving skills for young learners?

Certainly! Games like chess, Sudoku, and even certain board games that require strategic thinking can improve problem-solving skills . These games challenge young minds to think ahead and plan their moves carefully.

How can puzzles be effectively used to develop mathematical problem-solving competencies in children?

Puzzles can be used in teaching by posing them as interesting challenges that are mathematically meaningful . Puzzles such as tangrams, magic squares, and logic puzzles encourage children to use mathematical reasoning and pattern recognition.

In what ways can teachers incorporate problem-solving exercises into their primary classroom curriculum?

Teachers can introduce problem-solving in the classroom by integrating puzzles and games into lesson plans. They might also ask thought-provoking questions that lead to problem-solving discussions or use differentiated instruction to cater to various learning styles.

Which hands-on group activities can help children build teamwork and problem-solving skills concurrently?

Group activities that involve building projects from common objects like blocks or recycled materials allow children to work together. They can also participate in team-based challenges that require collective problem-solving and decision-making.

What types of maths challenges are suitable for primary students to promote critical thinking and reasoning?

Suitable challenges include pattern identification , sequencing tasks, basic arithmetic puzzles, and solving mathematical puzzles . These can be tailored to match the students’ age and proficiency level, ensuring the tasks remain engaging and appropriately challenging.

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20 Primary Maths Strategies Proven To Improve Progress And Understanding At KS2

David Leighton

Working with thousands of pupils every week we’ve developed a clear schema of the primary maths strategies that really make an impact on pupil attainment and progress in maths at KS2.

These primary maths strategies are drawn from the most effective teaching strategies we believe you need when teaching the primary maths curriculum.

Since the arrival of the new curriculum and primary maths framework here at Third Space Learning we have delivered online maths intervention programmes to over 60,000 primary school pupils. Every lesson is taught by professionally trained maths tutors working one to one with the child.

The Primary School Guide to Effective Maths Interventions

Find out how to plan, structure and teach one to one maths lessons to raise attainment in the pupils. Includes a 20 point checklist for improvement.

Origins of our primary maths approach

As part of our quality control and to ensure the best outcomes for every child we realised early on that we needed a clear framework for our tutors to follow – a rubric that identifies the ‘Third Space Learning’ approach.

So, drawing on what we know about quality first teaching and a maths mastery approach, as well as the specific requirements of online tutoring and tutoring in a one to one environment, this is the list of what we believe to be crucial in primary maths, and indeed, what every subject leader or teacher involved in primary maths teaching should be aware of.

We hope that it proves to be valuable in helping you to reflect on your own primary maths practice and that there will be an idea here that you can take to further support your own pupils with primary maths.

1: Build interest and rapport

2. know your pupils and your resources, 3. recap prior learning, 4. share lesson objectives to improve maths progress, 5. teach key maths vocabulary, 6. use afl to adjust pitch and pace.

• 7. Personalise learning and link questions to pupils’ interests

8. Create more opportunities for pupil talk

9. use ‘concrete pictorial abstract’ method, 10. use variation to improve maths progress, 11. frame questions in a real world context, 12. vary approaches to questions, 13. move from scaffolding to independent practice.

• 14. Use deeper questioning strategies to help pupils progress in Maths

15. Use bar-modelling techniques

16. nurture a growth mindset to nurture maths progress, 17. provide a range of strategies and next steps.

• 18. Encourage metacognition to encourage Maths progress

19. Reward, praise and encourage the pupil

20. teach the teacher, a note on resources, why are these strategies important in primary maths.

Excellence in primary maths depends on a combination of factors – the right primary maths resources, good quality teaching backed up by regular CPD, strong subject leadership in the school from the maths coordinator , and a whole school maths curriculum and progression from EYFS Early Years, through to Key Stage 1 and Key Stage 2 that teachers understand and feel ownership of.

One of the joys of primary maths (unlike other subjects like History or Geography) is that each piece of new knowledge builds on prior knowledge.

Provide children with the building blocks of place value , and explain to them the value of learning their number bonds , and by the time you get to addition and subtraction in KS2 , much of the cognitive load of learning a new technique like column subtraction with exchanging will be lifted.

Similarly without a good grasp of times tables , not only will your pupils struggle on their Multiplication Tables Check , but, arguably more importantly, they will find long division , fractions and even pie charts, significantly harder.

With a whole school approach the child’s learning journey in primary maths can progress logically from early numeracy through to the more sophisticated calculations and reasoning they make as they are ready to transition from Year 6 to Year 7 .

Read more: 9 Things You Should Know About The NCETM Mathematics Guidances

The list of primary maths strategies

These are the proven primary maths strategies that Third Space Learning tutors use to make sure every tutoring session is as impactful as possible .

Nothing captures a pupil’s imagination more than their own interests! While learning your pupils’ hobbies and favourite pastimes can be time-consuming, it should happen naturally over the course of the year. You can then use this knowledge to personalise classroom explanations.

Our 1-to-1 maths specialist tutors prepare for each lesson by thinking about how they can effectively engage pupils and relate content to their interests and use images related to these to personalise the session. It’s simple, but it’s also a great way to turn a new concept from ‘mundane’ to ‘memorable.’

With your own KS2 maths class while you cannot personalise for every child, engaging children with fun real life maths activities and is a great start.

This should be the bread and butter of teaching for interventions (and in general). If you’re not sure if a maths resource is pitched at the right level for your pupil(s), then you need to find out. Without appropriate differentiation in the classroom , time spent on a resource could well be time wasted.

Before your intervention, look at previous session data or pupil profiles. Are there any areas you should be aware of, such as EAL, dyscalculia, etc?

Like our tutors, we advise you tally this against the intervention resources and make notes against any potential areas of difficulty, or likely misconceptions. Remember, forewarned is forearmed and avoiding speed bumps when time is limited is crucial to maximising learning.

With a class of 30 it can be more tricky to ensure you’re aware of every misconception. We recommend using diagnostic assessments  wherever possible at the start and end of every topic. See also this blog post on effective mixed ability grouping in maths .

Particularly in maths, where learning needs to build upon previous learning and foundation concepts, recapping prior knowledge through retrieval practice or other methods is essential.

When teaching pupils through our specialist maths intervention , our tutors always create a linking question from past lessons to kick-start learning in a new one.

This can save precious time and ensure you never try to teach a new concept to a class that they’re simply not ready for.

Don’t assume that just showing your class the learning objectives will make them understand them. Our tutors do this by; asking pupils to identify keywords they don’t understand, and asking open ended questions such as ‘what is the difference’ and ‘what do you think of’, and discussing how the lesson objectives link to prior learning.

Discussing lesson objectives is much better for long term memory retention than just stating them.

If a pupil does not understand key maths vocabulary , it is important to get them up to speed before undertaking new learning. To address any misunderstanding, our tutors will often get pupils to describe mathematical terms in their own words before they start a topic.

Plus if pupils are struggling they will simplify keywords and, for example, discuss ‘sharing’ instead of ‘division’.

Where possible, learning should be paced to the needs of the pupil(s). The last thing any teacher wants is to find that they have moved too quickly through a topic and caused cognitive overload in their pupil(s).

New to cognitive load theory in the classroom ? Read Clare Sealy’s post on the topic to find out more about this valuable educational psychology tool, with a focus on how it impacts primary school pupils.

This is is why our tutors use assessment for learning and formative assessment strategies to adjust the pitch and pace of their lessons.

This is as simple as moving quickly through content that the pupil understands, and slowing down to expand or explain a concept that they’re not getting.

The key element is not to move on from content that a pupil is yet to secure, regardless of how much time you spend on it.

7. Personalise learning and link questions to pupils’ interests

Personally relevant problems are way more fun to solve. Posing a question in the context of the pupil’s name, or their favourite animal, is likely to increase their engagement.

To this end, our tutors often use the pupil’s name, favourite animal, or interests to create personally relevant maths problems for them to solve. As familiarity and rapport with your class builds throughout the year, this should become easier and easier to do.

We’ve got plenty of maths activities you can use to personalise learning to pupils’ interest such as this football maths lesson , this Science in Maths activity or a KS2 maths investigation into famous female mathematicians for International Women’s Day.

Finding time for high quality maths talk in a KS2 class is hard. But, wherever possible, you should encourage active learning and promote discussion rather than lecturing.

Short attention spans are an ever-present hurdle and classroom discussion keeps students from zoning out. Plus it enhances metacognitive development.

Remember, however, that direct instruction is still often your most valuable teaching method. Pupil talk should come after you’ve covered the fundamentals of a topic, not before.

The CPA Concrete Pictorial Abstract method is a staple of many maths teachers’ toolkit. Use real objects to build the pupils’ foundations for conceptual understanding, and follow up with pictorial representations that help them freely conceptualise a problem.

This technique is tried and tested and not just for low ability students ; using it enables all pupils to grasp more challenging content quickly and effectively.

There are two stages of this. Conceptual variation , which shows pupils different ideas that underpin a mathematical concept, can be framed using questions such as ‘what’s the same?’ and ‘what’s different?’ between different representations. It helps pupils distinguish the essential and obscure characteristics of a maths concept.

Procedural variation, which is more useful for multi-step operations and comparing successive procedures (e.g. calculating two different sets of numbers), can be framed as questions such as ‘what do you notice?’ and ‘is there a relationship between?’

Connecting topics to maths in the real world makes them more memorable and engaging. This should be done as often as possible and our tutors are expected to connect all maths learning to real life, even in more abstract topics!

For example, if a tutor is teaching rounding up to 1 million, they make look at different populations in towns or cities or ask pupils to round up the total of people in their hometown.

If you’re looking for quick and easy topical maths investigations to use in the classroom, take a look at these resources for Year 5 and Year 6:

• KS2 Topical Maths Resource for Spring Term
• KS2 Topical Maths Resource for Summer Term

There’s a big difference between learning a method and gaining real understanding.

Our tutors always teach multiple ways to approach the same problem. For example when teaching rounding tutors can use a number line, or ask pupils to simply use the relevant columns to inform their rounding.

They also make frequent use of goal-free problems and open-ended questions to ensure pupils learn the underlying concept, rather than simply performing back what they have seen.

This is especially crucial for Year 6 pupils who need the tools to answer SATs questions independently.

Learning should be structured with gradually decreasing support: from scaffolding to independent learning and finally to stretching. In our intervention, tutors always model answers and show pupils the steps to success before asking them to answer a question.

Moreover, when moving towards the plenary of a lesson, our tutors provide multiple opportunities for pupils to work independently and extend their own learning.

Find out more about the 5 stages of deliberate practice here.

• How to Stretch and Challenge More Able Pupils in KS2 Maths
• The 21 Best Maths Challenges At KS2 To Really Stretch Your More Able Primary School Pupils

14. Use deeper questioning strategies to help pupils progress in maths

We all know verbal reasoning plays a large part in improving metacognition, but it is the quality of a discussion is what counts, not the quantity!

Like our tutors, you can encourage quality pupil talk by questioning in the classroom in maths like ‘how did you get to this answer?’ and ‘how would I extend this question?’ This also addresses the three main aims of the national curriculum: fluency, reasoning and problem-solving .

Bar-modelling is taking the primary maths world by storm for a reason. From routine calculations such as 4 + 3, to more complex multi-step SATs problems, bar models are a fantastic learning aid as they show the inner workings of maths problems. Here are 25 word based  problem solving questions with bar models  to get you started.

Research clearly demonstrates that, compared to other subjects, pupils are most likely to believe they won’t succeed at maths. To build that crucial confidence our tutors always praise the effort a pupil puts in, rather than the pupil.

We find phrases like ‘you found a really good way to do that!’, or ‘I can tell you’ve been practising’, really improve motivation and attitude towards maths.

Read more on how to get growth mindset right in primary schools.

Just as linking from previous concepts gives pupils a sense of familiarity, linking to future concepts ensures that pupils don’t get “lost” in learning. This is why our tutors always signpost the learning in any given lesson.

Signposting can be as simple as telling pupils that ‘we’re going to move on to a rounding numbers to a million, it’ll use the same strategies as rounding to ten thousand, but it’s more of a challenge’. This keeps pupils engaged and on track.

It’s particularly important when developing reasoning skills at KS2.

18. Encourage metacognition to encourage maths progress

Metacognition should be an integral part of lessons and pupils should frequently be assessing their own learning using metacognitive strategies and other thinking skills .

One way to do this is to focus the plenary on what was learnt that day and, more importantly, how it was learnt. When a child can reflect on their own understanding, what they found more challenging or easier, then you will be starting to achieve metacognition in the classroom .

Praise is a great confidence-builder, and corrections can be incorporated constructively within praise. For example, ‘it’s okay that you’ve made a mistake but it’s important that you learn from it’. This is especially crucial with pupils who may have less maths confidence than their peers and those struggling with maths anxiety .

To be effective, pupil talk requires proper structure and expectations. One technique we use is to ask a pupil to teach back to the tutor (towards the end of the lesson). It is the ultimate test of understanding.

It’s particularly effective in 1-to-1 settings, where the pupil is under less pressure than in a group. However, when used as a group activity (with more confident pupils doing the teaching) it can be effective in a classroom setting.

If you’re new to this one, why not start by trying it as a primary maths strategy to help your pupils memorise their number facts .

Those are the 20 primary maths strategies that underpin our interventions.

The only other thing to mention is that in our online maths lessons, tutors are provided with a full range of online lesson slides to follow.

For primary teachers with their own classroom activities to prepare, we’ve provided a number of free maths resources and premium teaching resources (including many designed for those following White Rose Maths and other primary maths schemes of work ) as part of our online maths hub .

And don’t forget to  download our full, FREE guide to effective 1-to-1 maths interventions here .

DO YOU HAVE STUDENTS WHO NEED MORE SUPPORT IN MATHS?

Every week Third Space Learning’s specialist online maths tutors support thousands of students across hundreds of schools with weekly online 1 to 1 maths lessons designed to plug gaps and boost progress.

Since 2013 these personalised one to one lessons have helped over 150,000 primary and secondary students become more confident, able mathematicians.

Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

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Problem solving strategies

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What are problem solving strategies?

Strategies are things that Pólya would have us choose in his second stage of problem solving and use in his third stage ( What is Problem Solving? ). In actual fact he called them heuristics . They are a collection of general approaches that might work for a number of problems. 

There are a number of common strategies that students of primary age can use to help them solve problems. We discuss below several that will be of value for problems on this website and in books on problem solving. 

Common Problem Solving Strategies

• Guess (includes guess and check, guess and improve)
• Act It Out (act it out and use equipment)
• Draw (this includes drawing pictures and diagrams)
• Make a List (includes making a table)
• Think (includes using skills you know already)

We have provided a copymaster for these strategies so that you can make posters and display them in your classroom. It consists of a page per strategy with space provided to insert the name of any problem that you come across that uses that particular strategy (Act it out, Draw, Guess, Make a List). This kind of poster provides good revision for students. 

An in-depth look at strategies                 

We now look at each of the following strategies and discuss them in some depth. You will see that each strategy we have in our list includes two or more subcategories.

• Guess and check is one of the simplest strategies. Anyone can guess an answer. If they can also check that the guess fits the conditions of the problem, then they have mastered guess and check. This is a strategy that would certainly work on the Farmyard problem described below but it could take a lot of time and a lot of computation. Because it is so simple, you may have difficulty weaning some students away from guess and check. As problems get more difficult, other strategies become more important and more effective. However, sometimes when students are completely stuck, guessing and checking will provide a useful way to start to explore a problem. Hopefully that exploration will lead to a more efficient strategy and then to a solution.
• Guess and improve is slightly more sophisticated than guess and check. The idea is that you use your first incorrect guess to make an improved next guess. You can see it in action in the Farmyard problem. In relatively straightforward problems like that, it is often fairly easy to see how to improve the last guess. In some problems though, where there are more variables, it may not be clear at first which way to change the guessing.  
• Young students especially, enjoy using Act it Out . Students themselves take the role of things in the problem. In the Farmyard problem, the students might take the role of the animals though it is unlikely that you would have 87 students in your class! But if there are not enough students you might be able to include a teddy or two. This is an effective strategy for demonstration purposes in front of the whole class. On the other hand, it can also be cumbersome when used by groups, especially if a largish number of students is involved.  Sometimes the students acting out the problem may get less out of the exercise than the students watching. This is because the participants are so engrossed in the mechanics of what they are doing that they don’t see the underlying mathematics. 
• Use Equipment is a strategy related to Act it Out. Generally speaking, any object that can be used in some way to represent the situation the students are trying to solve, is equipment. One of the difficulties with using equipment is keeping track of the solution. The students need to be encouraged to keep track of their working as they manipulate the equipment. Some students need to be encouraged and helped to use equipment. Many students seem to prefer to draw. This may be because it gives them a better representation of the problem in hand. Since there are problems where using equipment is a better strategy than drawing, you should encourage students' use of equipment by modelling its use yourself from time to time.  
• It is fairly clear that a picture has to be used in the strategy Draw a Picture . But the picture need not be too elaborate. It should only contain enough detail to help solve the problem. Hence a rough circle with two marks is quite sufficient for chickens and a blob plus four marks will do a pig. All students should be encouraged to use this strategy at some point because it helps them ‘see’ the problem and it can develop into quite a sophisticated strategy later.
• It’s hard to know where Drawing a Picture ends and Drawing a Diagram begins. You might think of a diagram as anything that you can draw which isn’t a picture. But where do you draw the line between a picture and a diagram? As you can see with the chickens and pigs, discussed above, regular picture drawing develops into drawing a diagram. Venn diagrams and tree diagrams are particular types of diagrams that we use so often they have been given names in their own right.  
• There are a number of ways of using Make a Table . These range from tables of numbers to help solve problems like the Farmyard, to the sort of tables with ticks and crosses that are often used in logic problems. Tables can also be an efficient way of finding number patterns.
• When an Organised List is being used, it should be arranged in such a way that there is some natural order implicit in its construction. For example, shopping lists are generally not organised. They usually grow haphazardly as you think of each item. A little thought might make them organised. Putting all the meat together, all the vegetables together, and all the drinks together, could do this for you. Even more organisation could be forced by putting all the meat items in alphabetical order, and so on. Someone we know lists the items on her list in the order that they appear on her route through the supermarket.  
• Being systematic may mean making a table or an organised list but it can also mean keeping your working in some order so that it is easy to follow when you have to go back over it. It means that you should work logically as you go along and make sure you don’t miss any steps in an argument. And it also means following an idea for a while to see where it leads, rather than jumping about all over the place chasing lots of possible ideas.
• It is very important to keep track of your work. We have seen several groups of students acting out a problem and having trouble at the end simply because they had not kept track of what they were doing. So keeping track is particularly important with Act it Out and Using Equipment. But it is important in many other situations too. Students have to know where they have been and where they are going or they will get hopelessly muddled. This begins to be more significant as the problems get more difficult and involve more and more steps.
• In many ways looking for patterns is what mathematics is all about. We want to know how things are connected and how things work and this is made easier if we can find patterns. Patterns make things easier because they tell us how a group of objects acts in the same way. Once we see a pattern we have much more control over what we are doing.
• Using symmetry helps us to reduce the difficulty level of a problem. Playing Noughts and crosses, for instance, you will have realised that there are three and not nine ways to put the first symbol down. This immediately reduces the number of possibilities for the game and makes it easier to analyse. This sort of argument comes up all the time and should be grabbed with glee when you see it.
• Finally working backwards is a standard strategy that only seems to have restricted use. However, it’s a powerful tool when it can be used. In the kind of problems we will be using in this web-site, it will be most often of value when we are looking at games. It frequently turns out to be worth looking at what happens at the end of a game and then work backward to the beginning, in order to see what moves are best.
• Then we come to use known skills .  This isn't usually listed in most lists of problem solving strategies but as we have gone through the problems in this web site, we have found it to be quite common.  The trick here is to see which skills that you know can be applied to the problem in hand. One example of this type is Fertiliser (Measurement, level 4).  In this problem, the problem solver has to know the formula for the area of a rectangle to be able to use the data of the problem.  This strategy is related to the first step of problem solving when the problem solver thinks 'have I seen a problem like this before?'  Being able to relate a word problem to some previously acquired skill is not easy but it is extremely important.

Uses of strategies                                           

Different strategies have different uses. We’ll illustrate this by means of a problem.

The Farmyard Problem : In the farmyard there are some pigs and some chickens. In fact there are 87 animals and 266 legs. How many pigs are there in the farmyard?

Some strategies help you to understand a problem. Let’s kick off with one of those. Guess and check . Let’s guess that there are 80 pigs. If there are they will account for 320 legs. Clearly we’ve over-guessed the number of pigs. So maybe there are only 60 pigs. Now 60 pigs would have 240 legs. That would leave us with 16 legs to be found from the chickens. It takes 8 chickens to produce 16 legs. But 60 pigs plus 8 chickens is only 68 animals so we have landed nearly 20 animals short.

Obviously we haven’t solved the problem yet but we have now come to grips with some of the important aspects of the problem. We know that there are 87 animals and so the number of pigs plus the number of chickens must add up to 87. We also know that we have to use the fact that pigs have four legs and chickens two, and that there have to be 266 legs altogether.

Some strategies are methods of solution in themselves. For instance, take Guess and improve . Supposed we guessed 60 pigs for a total of 240 legs. Now 60 pigs imply 27 chickens, and that gives another 54 legs. Altogether then we’d have 294 legs at this point.

Unfortunately we know that there are only 266 legs. So we’ve guessed too high. As pigs have more legs than hens, we need to reduce the guess of 60 pigs. How about reducing the number of pigs to 50? That means 37 chickens and so 200 + 74 = 274 legs.

We’re still too high. Now 40 pigs and 47 hens gives 160 + 94 = 254 legs. We’ve now got too few legs so we need to guess more pigs.

You should be able to see now how to oscillate backwards and forwards until you hit on the right number of pigs. So guess and improve is a method of solution that you can use on a number of problems.

Some strategies can give you an idea of how you might tackle a problem. Making a table illustrates this point. We’ll put a few values in and see what happens.

From the table we can see that every time we change the number of pigs by one, we change the number of legs by two. This means that in our last guess in the table, we are five pigs away from the right answer. Then there have to be 46 pigs.

Some strategies help us to see general patterns so that we can make conjectures. Some strategies help us to see how to justify conjectures. And some strategies do other jobs. We’ll develop these ideas on the uses of strategies as this web-site grows.

What strategies can be used at what levels?

In the work we have done over the last few years, it seems that students are able to tackle and use more strategies as they continue with problem solving. They are also able to use them to a deeper level. We have observed the following strategies being used in the stated Levels.

Levels 1 and 2

• Draw a picture
• Use equipment
• Guess and check

Levels 3 and 4

• Draw a diagram
• Guess and improve
• Make a table
• Make an organised list

It is important to say here that the research has not been exhaustive. Possibly younger students can effectively use other strategies. However, we feel confident that most students at a given Curriculum Level can use the strategies listed at that Level above. As problem solving becomes more common in primary schools, we would expect some of the more difficult strategies to come into use at lower Levels.

Strategies can develop in at least two ways. First students' ability to use strategies develops with experience and practice. We mentioned that above. Second, strategies themselves can become more abstract and complex. It’s this development that we want to discuss here with a few examples.

Not all students may follow this development precisely. Some students may skip various stages. Further, when a completely novel problem presents itself, students may revert to an earlier stage of a strategy during the solution of the problem.

Draw: Earlier on we talked about drawing a picture and drawing a diagram. Students often start out by giving a very precise representation of the problem in hand. As they see that it is not necessary to add all the detail or colour, their pictures become more symbolic and only the essential features are retained. Hence we get a blob for a pig’s body and four short lines for its legs. Then students seem to realise that relationships between objects can be demonstrated by line drawings. The objects may be reduced to dots or letters. More precise diagrams may be required in geometrical problems but diagrams are useful in a great many problems with no geometrical content.

The simple "draw a picture" eventually develops into a wide variety of drawings that enable students, and adults, to solve a vast array of problems.

Guess: Moving from guess and check to guess and improve, is an obvious development of a simple strategy. Guess and check may work well in some problems but guess and improve is a simple development of guess and check.

But guess and check can develop into a sophisticated procedure that 5-year-old students couldn’t begin to recognise. At a higher level, but still in the primary school, students are able to guess patterns from data they have been given or they produce themselves. If they are to be sure that their guess is correct, then they have to justify the pattern in some way. This is just another way of checking.

All research mathematicians use guess and check. Their guesses are called "conjectures". Their checks are "proofs". A checked guess becomes a "theorem". Problem solving is very close to mathematical research. The way that research mathematicians work is precisely the Pólya four stage method ( What is Problem Solving? ). The only difference between problem solving and research is that in school, someone (the teacher) knows the solution to the problem. In research no one knows the solution, so checking solutions becomes more important.

So you see that a very simple strategy like guess and check can develop to a very deep level.

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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.

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?

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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More From Forbes

Leveraging systems models: a strategic advantage for it professionals.

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Thomas Lim is the Vice-Dean of Centre for Systems Leadership at SIM Academy. He is an AI+Web3 practitioner & author of Think.Coach.Thrive!

IT professionals increasingly need to navigate projects arising from greater business complexity. The integration of systems models with established methodologies, like design thinking, agile and lean, presents a unique opportunity for IT experts to enhance their capacity and expertise. This systemic integration not only enriches their problem-solving tool kits but also ensures the delivery of innovative, sustainable solutions that align with the strategic objectives of their organizations.

The landscape of IT projects has expanded beyond straightforward software development to include complex ecosystems involving artificial intelligence (AI), cloud computing, big data and more. This complexity necessitates a holistic view of project management and solution development, where understanding interdependencies and emergent behaviors becomes crucial. Systems models, with their focus on the interrelations within and among systems, offer a comprehensive framework for analyzing and addressing these complexities.

Enhancing Design Thinking With Systems Thinking

Within the Design Thinking phases, there are opportunities to embed systems thinking models. For example, during the Ideate phase, using causal loops would help clarify the Theory of Success.

Agile methodologies emphasize flexibility, rapid iteration and stakeholder collaboration. Integrating systems dynamics into agile practices allows IT professionals to model and simulate the behaviors of complex systems over time. This predictive capability can inform sprint planning, risk management and the prioritization of features based on their potential impact on the system. By understanding the feedback loops and causal relationships that drive system behavior, IT teams can anticipate challenges and opportunities, leading to more informed decision-making and effective agile implementation.

Diagnosing With The Levels Of Perspective Model

The Levels of Perspective Systems Model categorizes viewpoints from which a system can be analyzed, ranging from concrete events to deep-rooted structures and mental models. This hierarchical framework includes the Events level, Patterns of Behavior level, Structural level and, sometimes, the Mental Models level. Each level offers a unique lens through which IT developers can understand and address complex challenges.

Enhancing Design Thinking

Design thinking emphasizes empathy, collaboration and iterative learning to solve problems creatively. By applying the Levels of Perspective Model, IT developers can deepen their empathetic understanding by recognizing not only the immediate needs (Events level) but also the underlying patterns and behaviors of users (Patterns of Behavior level). This approach allows developers to design solutions that are more closely aligned with users' evolving behaviors and preferences. Additionally, considering the Structural and Mental Models levels enable developers to identify and challenge assumptions about what users need or value, leading to more innovative and user-centric solutions.

Complementing Agile Methodologies

The Levels of Perspective Model can enhance agile practices by encouraging developers to look beyond the immediate backlog of tasks (Events level) and consider the broader patterns of team dynamics and project progress (Patterns of Behavior level). This wider perspective can help in anticipating challenges and opportunities for improvement.

Stacking Systemic Choices Using The Hierarchy Of Choices

The Hierarchy of Choices model offers a structured approach to decision-making that can greatly enhance the effectiveness of IT developers. By delineating decisions across five levels—fundamental, primary, secondary, tertiary and other choices—this model provides a comprehensive framework that aligns closely with design thinking and agile and lean methodologies, enhancing developers' capacity to deliver impactful, user-centered solutions efficiently.

Fundamental Choice: The Purpose Behind Existence

At the core of every IT project is the fundamental question: Why does this project exist? This aligns with the empathetic core of design thinking, which starts with understanding the user's needs and challenges.

Primary Choice: Defining Desired Outcomes

What results does the project aim to achieve? Identifying primary choices involves setting clear, measurable goals, a principle that resonates with the outcome-focused nature of agile methodologies. For developers, defining these outcomes provides a clear direction for the project, facilitating prioritization and helping to maintain focus on delivering value to the end-users.

Secondary Choice: Crafting The Strategy

Deciding on the strategy—how the project will achieve its goals—ties into lean thinking, which emphasizes value streams and waste elimination. For IT developers, understanding the chosen strategy informs the selection of technologies, architectural patterns and development practices that align with project goals. This strategic alignment ensures that resources are optimized and efforts are directed toward activities that directly contribute to the project's objectives.

Tertiary Choice: Selecting Tactics

Tactical choices concern the specific actions taken to implement the strategy. This is where agile practices shine, offering a flexible framework for iterative development, continuous integration and regular feedback loops. Developers, by making informed tactical decisions, can adapt to changes quickly, experiment with solutions and refine their approach based on real-time feedback, all while ensuring alignment with the overarching strategy.

Finally, the model includes a broad category for other choices, encompassing the operational decisions related to the what, where, how and when of project execution.

The Road Ahead

As IT professionals strive to navigate the complexities of modern projects, the integration of systems models with traditional methodologies like design thinking, agile and lean offers a powerful synergy. This systemic integration enables a holistic understanding of complex environments, fosters innovative problem-solving and supports the agile adaptation to change. By embracing these integrated approaches, IT professionals can elevate their expertise, deliver impactful solutions and achieve strategic alignment in their projects, ensuring their role as key drivers of digital transformation.

In conclusion, the relevance of systems models in the IT domain cannot be overstated. As we advance into a future where projects become increasingly complex and interconnected, the ability to integrate and apply these models alongside established IT methodologies will be a defining skill for IT professionals. This systemic integration not only enhances their capacity to address current challenges but also positions them to lead the way in shaping the digital landscapes of tomorrow.

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Top 10 Tips for GRE Quantitative Exam: Learn Maths Tricks and Shortcuts

Excelling in the GRE Quantitative Exam requires more than just a strong grasp of math concepts; it demands effective strategies and techniques. In our article, “Top 10 Tips for GRE Quantitative Exam: Learn Maths Tricks and Shortcuts,” we provide you with essential GRE math tips to enhance your performance. Discover powerful math tricks and quantitative shortcuts that will streamline your problem-solving process and improve your accuracy.

Our expert strategies cover everything from quick calculation techniques to efficient problem-solving methods, ensuring you’re well-prepared for even the toughest questions. By incorporating these GRE quantitative strategies into your study routine, you’ll boost your speed and precision, making a significant impact on your test score. Whether you’re aiming for a perfect score or just looking to improve your current performance, these GRE quantitative exam tips will set you on the path to success.

GRE Verbal Quantitative Section – Format

The GRE Quantitative Exam section is the third and last part of the whole GRE entrance examination. The section consists of 2 sections where you have to answer 27 questions. As the GRE is a computer-adaptive test, for that the difficulty of the second subpart of the GRE Verbal Reasoning section depends on your performance in the first subpart of the section. You will get 21 minutes to complete the first subsection and 27 minutes for the rest i.e. second subsection. Here we have provided the question types of this section as mentioned below.

As we all have learned the structure of the GRE Quantitative Section, we will go through the subsections and related tips with it.

GRE Quantitative Reasoning Section

In this part you need to solve math problems, interpret data, apply mathematical concepts, and manage your time effectively. There are 4 types of questions in this section as mentioned below.

Quantitative Comparison Questions Multiple choice questions where you select one answer choice Multiple choice questions where you select one or more answer choices Numeric entry questions

The goal in the GRE Quantitative Reasoning Section is to accurately solve mathematical problems and interpret data to demonstrate your quantitative skills.

Best Tips for GRE Quantitative Reasoning Section

Here are some major tips to answer the questions of the Quantitative Reasoning Section as mentioned below.

Slow Down on Word Problems

Make sure you fully understand each question and do not rush through tricky ones. Translate words into math carefully. Check for keywords that can change the meaning and double-check your interpretation.

Use the Calculator Wisely

Practice without a calculator if possible. Use it only for complex calculations during the test. Avoid over-reliance; many problems can be solved with mental math or basic calculations.

Spend 2 Minutes Per Question

Aim to solve each question in 1-2 minutes. If you are stuck for over 2 minutes, make an educated guess and move on. Prioritize questions based on your strengths to manage your time effectively.

Note Key Info in Quant Comparison

Write down important numbers and phrases from the questions. Approach these questions strategically. Often, you do not need to solve mathematically—just understand the relationships between quantities.

Backsolve with Answer Choices

Plug numbers from answer choices into the question to check if they work. Start with middle options. This method can quickly show you which answers are feasible without solving the entire problem.

Pick Numbers When Possible

Use simple numbers in place of variables to make problems easier. Ensure the numbers you choose fit the problem’s conditions. This approach simplifies the math and helps you see patterns.

Stick to Your Methods

Be consistent with your strategies and solve problems methodically, regardless of their difficulty. If you find a method that works, use it for similar problems. Adapt your approach based on the problem type.

Memorize Answer Choices

Get familiar with the wording of answer choices to save time during the test. This will help you quickly identify the correct answer or rule out incorrect ones. Practice recognizing patterns in answer choices.

Analyze Tables/Graphs Carefully

Read all labels and understand the data trends before answering questions. Look for key details like scales and units. Make sure you grasp the overall picture before diving into the questions.

Break Down Complex Figures

Divide tough shapes into smaller parts and label them to make geometry problems easier. Look for familiar shapes and properties within the figure. Redrawing and labeling can reveal hidden relationships.

GRE Ideal Score

Your target GRE score should match the average score of admitted students for the program you want.

Score Over 160: Ideal for most programs. Score Over 150: Still good for many universities. Score Below 150: Might make it harder to get into top universities.

Check the average GRE scores for universities you are interested in to see what score will make you competitive.

Also Read: Introduction of Graduate Record Examinations (GRE) Score High on GRE: How to Get Good Score in GRE What is a Good GRE Score in 2024? Tips to Achieve Your Target GRE Score GRE Exam Fee Waiver 2024: How to Get a GRE Fee Waiver? GRE Geometry | Lines and Angles

Tips For GRE Verbal Reasoning Test- FAQs

Is a 149 verbal gre score good.

A score above the 75th percentile in either section will make you a competitive applicant. That means aiming for at least a 157 in Verbal Reasoning and at least a 163 in Quantitative Reasoning.

Is a 159 verbal GRE score good?

A good GRE score is considered to be 158+ on Verbal Reasoning, 159+ on Quantitative Reasoning, and 4.5+ on Analytical Writing.

Is 292 a bad score in GRE?

A GRE score average of 290 to 300 can take you to a quality university in the USA, which would surely help you take off your career ladder. Not just that, there are a variety of courses and MS programs that you can choose from.

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Harris Chooses Walz

A guide to the career, politics and sudden stardom of gov. tim walz of minnesota, now vice president kamala harris’s running mate..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

Hey, it’s Michael. Before we get started, I want to tell you about another show made by “The New York Times” that pairs perfectly with “The Daily.” It’s called “The Headlines.” It’s a show hosted by my colleague, Tracy Mumford, that quickly catches you up on the day’s top stories and features insights from “The Times” reporters who are covering them, all in about 10 minutes or less.

So if you like “The Daily”— and if you’re listening, I have to assume you do — I hope that means you’re going to “The Headlines” as well. You can now find “The Headlines” wherever you get your podcasts. So find it, subscribe to it, and thank you. And now, here’s today’s “Daily.”

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

[MUSIC PLAYING]

Today, the story of how a little known Midwestern governor became Kamala Harris’s choice for a running mate. My colleague Ernesto Londoño walks us through the career, politics, and sudden stardom of Governor Tim Walz of Minnesota.

It’s Wednesday, August 7.

Ernesto, over the past few days, we watched Vice President Harris bring the final three contenders for her running mate to her house in Washington, DC, for a set of in-person interviews. And then we watched as she seemed to narrow her pool of choices down to a final two — the governor of Pennsylvania, Josh Shapiro, and the governor of Minnesota, Tim Walz. And now, of course, we know that she has made her choice. What has she told us about her campaign strategy, the way she views this race, in ultimately choosing Tim Walz?

Michael, I think what the choice tells us is that Kamala Harris was drawn to two qualities that Governor Walz brings to the table. And what’s interesting is they may seem to be in tension. For starters, here’s the ultimate everyday man, somebody who grew up in a small town in Middle America, served in the National Guard, was a high school teacher, a football coach, very plain-spoken, goes to campaign events wearing T-shirts and baseball caps, is a gun owner and very proud about it. He sort of embodies the Midwest.

And she clearly thinks that that is going to bring the kind of moderate, white, working class voters that the campaign needs in swing states to come to them, to make this feel like a balanced ticket and something that will give her enough of the crucial votes to defeat Donald Trump in the fall.

On the other hand, as governor, he passed a slew of pretty progressive legislation in the past couple of years, everything from abortion rights to gun control. So these things are likely to appeal to bread and butter Democrats.

But the question is, when voters have examined these two facets of Tim Walz, may it bring them enough enthusiasm from the base and enough undecided voters that the campaign desperately needs, or at some point, do these two aspects of him start canceling each other out?

Right. In short, you’re saying Harris is betting on a dual appeal from Walz to two essential constituencies, but the risk is that the appeal to one of them is just much, much greater than to the other.

Right. You could definitely see a scenario where voters, once they’ve examined Tim Walz’s story and legacy, may conclude that both of these candidates are quite liberal.

OK, so tell us the story of Tim Walz, a story that I think a lot of us don’t know because we really don’t know Walz all that well, and how he has come to embody these two qualities and that tension that you just described.

Michael, the origin story of Tim Walz’s political career is quite fascinating.

He and his wife were teachers in a small city south of Minneapolis. And in 2004, when George W. Bush was running for re-election, Walz took a group of his students to a political rally in his hometown. They wanted to just see the president make his case. And a strange scuffle happened when they were trying to get in.

Well, one of the kids had a John Kerry sticker on his wallet. And this is where the individual says, well, you’re not going to be allowed to enter. You’ve been deemed a threat.

Apparently, one of the students had a sticker for Bush’s rival, John Kerry, on his wallet. And security officials at the rally didn’t want to let them in.

And I said, oh, it’s OK. They’re with me. And who are you? And I said, I’m Tim Walz. I’m their teacher here, and showed them my ID. And they said, well, you two have been deemed a threat to the president. And I said, well, that’s not true. And it kind of escalated.

And this really ticked off Tim Walz. He was really upset. There was a fight and a confrontation at the rally.

At this point in time, I’m kind of nervous. I’m getting arrested. So I’m like saying, well, I’m Teacher of the Year in Mankato. And they didn’t care about that. And it was kind of a sad epiphany moment, how it felt for people to be looked right through by people. These people didn’t see me. And this is happening.

And ultimately, he sort of walks away from this moment feeling really sick of the Bush administration, the politics of the day. And he turns around and volunteers for the Kerry campaign.

And then the more interested he becomes in politics in this era, he starts looking around his congressional district, and there’s a Republican who’s held the seat for many, many years. This was a largely rural district in southern Minnesota. And there’s no reason to believe that a newcomer to politics, somebody without a donor base, could make a run for this seat and win.

But Walz signs up for this weekend boot camp, where expert campaigners train newcomers who want to run for office. And he gets really enthused by the idea that he can pull it off. So he starts raising money with the support of an army of students who become so thrilled and energized by the prospect that their nerdy and kind geography teacher is making this uphill bid for a congressional race.

So his campaign staff is basically his former students.

That’s right. And he proves to be a formidable candidate. He draws a lot of attention to his experience in the classroom and as a coach.

When I coached football, these stands held about 3,000 people. That’s a lot. It’s also the number of American soldiers who have died fighting in Iraq.

He’s a very strong advocate for pulling out of the war in Iraq.

Serving right now are kids that I taught, coached, and trained to be soldiers. They deserve a plan for Iraq to govern itself, so they can come home.

And one thing that happens in the campaign that is really surprising to people is he comes out as being in favor of same-sex marriage. Now, it’s useful to remember that this is 2006, when the vast majority of Democrats, Democrats running for most elected office, were not ready to come out in favor of same-sex marriage.

And here’s a guy who’s new to politics, who’s trying to unseat a Republican who’s held on to his seat for more than 12 years, taking what appeared to be a reckless position on something. And when he was asked about it at the time, Tim Walz told a supporter, this just happens to be what I believe in. And I’d rather lose a race that I’ve ran being true and consistent to my values than try to run as somebody I’m not.

And of course, he wins.

Yes. To everybody’s surprise, he pulled it off.

So from the get-go, he shows a kind of maverick, “politics be damned” quality, taking stands that he knows may be unpopular among the voters he’s trying to win over. But he’s got some innate political gifts that are all making it work.

Yeah, I think that first campaign showed us that Tim Walz had real political chops. He was a very effective campaigner. And people really liked him. When he was knocking on doors, when he was introducing himself to voters, they saw him as somebody who was very genuine and who was admirable.

So once he gets elected in this conservative leaning district in Minnesota, what does he actually do in Congress?

In Congress, he develops a reputation for being somebody who can work across the aisle. And this is a period where Democrats and Republicans were deeply polarized over the Iraq War. He spends a lot of his time lobbying to expand benefits for veterans, so it’s easier for them to go to college after their service, and also becomes a leading voice in the quest to repeal Don’t Ask, Don’t Tell, the policy that prohibited openly gay servicemen from serving in uniform.

And he remained really popular. He easily won re-election five times. The last time he runs for his seat happens to be 2016, when President Trump wins his district by about 15 points.

And still, voters kept Tim Walz in office.

I think it’s important to note what you just said. Walz is distinguishing himself as a Democrat who can take some pretty progressive positions, as he did in that first campaign on gay rights, as he did with Don’t Ask, Don’t Tell, and keep winning in very Trump-friendly districts of his state.

That’s right. And as he’s serving his sixth term in office, he sets his sights on the governor’s mansion and decides to run for office in 2018. He wins that race easily. And early on, during his time as governor, the eyes of the world are on Minnesota after a police officer kills George Floyd. And what we see is massive looting and protests in Minneapolis.

Right, and remind us how Governor Walz handles that violence, those protests.

Yeah, I think that’s a crucial chapter in Tim Walz’s political career and one that will come under scrutiny in the days ahead.

After George Floyd was killed on a Monday —

People are upset, and they’re tired. And being Black in Minnesota already has a stigma and a mark on your back.

— protests took root in Minneapolis.

Y’all want to sit out here and shoot off your rubber bullets and tear gas.

And they got progressively larger and more violent.

There comes a point where the mayor and the police chief in Minneapolis plead for help. They ask the governor to send in the National Guard. And crucially, that request was not immediately heeded.

This is the third precinct here. There are fires burning to the left of it at the —

And at the height of the crisis, a police precinct building was abandoned.

There’s someone climbing up the wall right now, kicking the window in, trying to climb up the wall.

Because city officials grew concerned that protesters were about to overrun it and may attack the cops inside their own turf.

[EXPLOSIONS]

And the building is set on fire.

Right, a very memorable image. I can recall it happening in real-time.

Yeah, and in the days that followed, I think there were a lot of questions of why the governor didn’t send in troops earlier and whether a more muscular, decisive response could have averted some of the destruction that spread through the city.

And how does Walz end up explaining his decision not to send in the National Guard more quickly?

The governor and his administration have said that they were really, really dealing with an unprecedented challenge. And I think there was a concern that sending in troops into this really, really tense situation could have done more to escalate rather than pacify things on the street.

But in the weeks and months that followed, there were a lot of questions about Governor Walz’s leadership. And there were critics who said, during what may have been the most challenging week of his life, we saw a governor who was indecisive and who waited too long to send in resources that ultimately allowed the city to get to a semblance of order.

Right, and it feels like this is a moment that will almost assuredly be used against him by Donald Trump and JD Vance, the Republican ticket, which has made law and order so central to their message in this campaign.

Yeah, absolutely. And here in Minnesota, that was certainly a liability for him when he ran for re-election in 2022. But voters kept him in office, and he won that race handily. And not only did he win, but Democrats managed to flip the Senate and have full control of the legislature on his watch.

And that sets in motion one of the most productive legislative sessions in Minnesota history, where Tim Walz and his allies in the House and the Senate managed to pass a trove of really progressive legislation, oftentimes on a party vote.

Tell us about some of that legislation.

Well, Minnesota becomes the first state in the wake of the Supreme Court ending the constitutional right to abortion to actually codify this right under state statute. And they did a lot more stuff. They had a huge budget surplus, and they used that, for instance, to fund meals for all school children.

They managed to pass a couple of gun control laws that were very contentious. They gave the right to undocumented immigrants to get driver’s licenses. They legalized recreational marijuana. And finally, the governor takes a pretty bold stance on this issue of gender affirming care for transgender kids and teenagers, and says that Minnesota will be a safe haven for people who want that health care.

So, Ernesto, so how should we think about that blitz of legislation and the largely progressive tone of it, given the way that Walz had campaigned and succeeded up to that moment as somebody with such broad appeal across the political spectrum?

When the governor was asked whether this had been too much too quickly in terms of progressive legislation, his answer was that these were broadly popular policies, that these are issues Democrats had campaigned on. And here, Democrats had a window of opportunity where they were in control of the governor’s mansion and control of the House, the Senate, and that when you have political capital, you spend it.

But when you start listening to Republicans in Minnesota, they say, here’s a guy who campaigned on this mantra of “One Minnesota.” That was his campaign slogan. And he sort of came into office with this promise that he would govern in a bipartisan way, reach across the aisle.

But when they had all the votes they needed to pass their policies, Republicans felt that Walz was not bothering to bring them into the fold and to pass legislation that was going to be palatable to conservatives in the state. So I think people who once regarded him as a moderate now start seeing him as somebody who, when he had the power, acted in ways that were really progressive and liberal.

So at the height of his power, Governor Walz emerges as somebody who, when given a shot at getting done what he really wants to get done with a Democratic legislature, is a pretty progressive leader, even at the risk of being somewhat at odds with his earlier image as more moderate, because in his mind, enough people in the state are behind these policies.

Yeah, and I think he assumed that he had banked enough goodwill and that people across the state liked him enough to tolerate policies they may have disagreed with. And I think it’s safe to say, among the people who cover him here regularly, there was never any real hint that Tim Walz was eyeing a run for higher office. He’s not somebody who has written the kind of political memoir that oftentimes serves as a case of what you would bring to a national ticket or to the White House. And he seems pretty happy with a state job.

So it was a huge surprise when Tim Walz starts going viral through a string of cable news appearances right after President Biden drops out of the race, and the Democrats are scrambling to put Harris at the top of the ticket. And what becomes clear is that Walz is very forcefully auditioning for the role of vice president, and Vice President. Harris starts taking him very seriously.

We’ll be right back.

So, Ernesto, tell us about this cable news audition that Governor Walz undertakes over the past few weeks and how, ultimately, it seemed to help him land this job of being Harris’s running mate.

I think Walz does something really interesting, and that is that he says that Democrats shouldn’t be talking about Trump and Vance as existential threats. He kind of makes the case that Democrats have been in this state of fear and paralysis for too long, and that it’s not serving them well. So the word he latches onto is “weird.”

Well, it’s true. These guys are just weird.

It is. It is.

And they’re running for he-man women hater’s club or something. That’s what they go at. That’s not what people are interested in.

And I think one other thing we see in Walz is somebody who’s putting himself out there as a foil to JD Vance.

That angst that JD Vance talks about in “Hillbilly Elegy,” none of my hillbilly cousins went to Yale, and none of them went on to be venture capitalists or whatever. It’s not —

I think the case he’s making is that Tim Walz is a more authentic embodiment of small town values.

What I know is, is that people like JD Vance know nothing about small town America. My town had 400 people in it, 24 kids in my graduating class. 12 were cousins. And he gets it all wrong. It’s not about hate.

And behind the scenes, people from Tim Walz’s days on Capitol Hill start calling everybody they know in the Harris campaign and the Harris orbit and saying, here’s a guy who has executive experience as governor, but also somebody who has a really impressive record from his time on Capitol Hill and somebody who could be an asset in helping a Harris administration pass tough legislation. So you should take a hard look at this guy.

Which is, of course, exactly what Harris ends up doing. And I want to talk for a moment about how Harris announces Walz as her running mate on Tuesday morning. She did it in an Instagram message. And it felt like the way she did it very much embraced this idea that you raised earlier, Ernesto, that Walz contains these two appeals, one to the Democratic base, one to the white working class.

Harris specifically cites the work that Walz did with Republicans on infrastructure and then cites his work on gun control. She mentions that he was a football coach and the founder of the high school Gay Straight Alliance. She’s straddling these two versions of Walz.

But I want to linger on the idea for a moment of Walz’s vulnerabilities, because once he becomes Harris’s running mate, Harris and Walz are going to lose a fair amount of control over how they present him to the country, because he’s going to become the subject of very fierce attacks from the Republicans in this race. So talk about that for just a moment.

Yeah, I mean, it’s important to keep in mind that Governor Walz has never endured the scrutiny of a presidential race. So the questions he’s going to be asked and the way his record is going to be looked at is going to be different and sharper. I think the Harris campaign is billing him as, first and foremost, a fighter for the middle class. And I think that certainly will have some appeal.

But I think in coming days, there’s going to be a lot of attention drawn to parts of his record that may be unpopular with many voters. For instance, giving undocumented immigrants driver’s licenses, which Governor Walz championed. It’s likely to provide fodder for an attack ad.

The very dramatic footage of Minneapolis burning in 2020 is also something that I think people will be drawn to. And there’s going to be interest in reexamining what the governor did and what he could have done differently to avert the chaos.

And on Tuesday, we saw that the Trump campaign wasted no time in trying to define Tim Walz as soft on crime, permissive on immigration policy. And they also made clear they wanted to relitigate the era of George Floyd’s killing. And specifically, they want to try to tie him to the effort at the time to defund the police, which is a movement that Walz personally never endorsed.

So the Republican attack here will be pretty simple. Walz is liberal. Harris is liberal. So, in their efforts to speak to especially white working class and rural voters in swing states, the Trump campaign is going to say this is not the ticket for that group of voters. This is the ticket of burning police precincts and gun control. And of course, that may not be fair, but that’s very likely going to be the message over the next couple of months.

Right. I think there’s going to be effort to portray him as a radical liberal who has used his small town roots to put on this sort of veneer of being a moderate and a really sort of understanding and being part of the segments of the electorate that I think are critical in this election.

I want to speak for just a moment about the person Harris did not pick when she chose Walz because many Democrats had felt that Walz was a potentially too liberal seeming running mate for a candidate, Kamala Harris, who herself comes from a blue state and is caricatured by the Republicans as liberal herself.

And the person she didn’t choose was Governor Josh Shapiro of Pennsylvania, who was seen as having a huge appeal in that particular key swing state, but also presented risks of his own of alienating parts of the Democratic base with his well-documented support for Israel and his criticism of campus protesters. How should we think about the fact that, ultimately, Harris chose Walz over Shapiro?

Yeah, I think in the final stretch of this campaign to be the vice presidential pick, we started seeing a lot of acrimony in pockets of the Democratic base, drawing attention to the fact that Governor Shapiro could be divisive on Gaza, which has really sort of split the party in recent months.

So I think at the end of the day, they made a calculation that Tim Walz would be more of a unifying figure and would be somebody who would inspire and energize enough pockets of the electorate that they need, particularly in the Midwest, to make him the stronger and more exciting pick and somebody who wouldn’t force them to go back to defending and relitigating the Biden administration’s record on Israel and on the war in Gaza.

Right, and then, on Tuesday night, we got our first glimpse of Harris and Walz together on stage for the first time at a campaign rally. I’m curious, what struck you about their debut together.

Good evening, Philadelphia.

I think everybody was watching the opening scene of this rally to see what the chemistry between these two people was going to be like. And they both seemed giddy. They were literally, at times, bouncing with enthusiasm.

Since the day that I announced my candidacy, I set out to find a partner who can help build this brighter future.

So Pennsylvania, I’m here today because I found such a leader.

Governor Tim Walz of the great state of Minnesota.

They soon got down to business. And that business was how to define Tim Waltz for voters who don’t know him well.

To those who know him best, Tim is more than a governor.

And right off the bat, we saw that Kamala Harris really highlighted a lot of pieces of his pre-political career.

To his former high school football players, he was Coach.

She repeatedly called him Coach Walz, Mr. Walz, evoking his time in the classroom, and even used his military title from his days in the Army.

To his fellow veterans, he is Sergeant Major Walz.

And then when it came time for Tim Walz to introduce himself on this massive stage —

Welcome the next vice president of the United States, Tim Walz.

— he drew a lot of attention to his small town roots.

I was born in West Point, Nebraska. I lived in Butte, a small town of 400.

He said something that he said repeatedly recently in campaign appearances, which is —

In Minnesota, we respect our neighbors and their personal choices that they make. Even if we wouldn’t make the same choice for ourselves, there’s a golden rule — mind your own damn business.

The golden rule of small towns is you mind your own damn business, which is something he said in the context of his argument that Republicans have been limiting, rather than expanding, people’s rights. But he also drew attention to the fact that he’s a gun owner.

By the way, as you heard, I was one of the best shots in Congress. But in Minnesota, we believe in the Second Amendment, but we also believe in common sense gun violence laws.

And then when it came time to draw a sharp contrast with their opponents, Tim Walz said, these guys are phonies.

Donald Trump is not fighting for you or your family. He never sat at that kitchen table like the one I grew up at, wondering how we were going to pay the bills. He sat at his country club in Mar-a-Lago, wondering how he can cut taxes for his rich friends.

He said it’s actually people like me and Kamala Harris who come from humble origins and showed what is possible in America when you hail from a working class background, and you seize opportunities that were available to you.

Thank you, Philadelphia. Thank you, Vice President. God bless America.

So when it comes to this question of Walz’s dual identities and dual appeals, what did we learn on day one of this new Democratic ticket, do you think?

I think the campaign is trying to convey that these two facets of Tim Walz’s life are not mutually exclusive, that they don’t need to be in tension. They don’t cancel each other out. They’re both part of Tim Walz’s story. And I think that’s how they’re going to present him from now until Election Day.

Ernesto, thank you very much. We appreciate it.

It’s my pleasure, Michael.

Here’s what else you need to know today. On Tuesday, Hamas said that Yahya Sinwar, one of the masterminds behind the deadly October 7 attacks on Israel, had consolidated his power over the entire organization. Until now, Sinwar had held the title of Hamas’s leader in Gaza. But with the assassination of Hamas’s top political leader by Israel last week, Hamas said that Sinwar would take on that title as well. Sinwar remains a major target of Israel and is believed to have been hiding in tunnels underneath Gaza since October 7.

And the US Department of Justice has charged a Pakistani man with ties to Iran with trying to hire a hitman to assassinate political figures in the United States. The man recently traveled to the US and was arrested in New York last month. American authorities believe that his potential targets likely included former President Trump.

Today’s episode was produced by Alex Stern, Eric Krupke, and Olivia Natt. It was edited by Lisa Chow and Patricia Willens, contains original music by Pat McCusker and Marion Lozano, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Nick Pittman and Minnesota Public Radio.

That’s it for “The Daily.” I’m Michael Barbaro. See you tomorrow.

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Earlier this summer, few Democrats could have identified Gov. Tim Walz of Minnesota.

But, in a matter of weeks, Mr. Walz has garnered an enthusiastic following in his party, particularly among the liberals who cheer on his progressive policies. On Tuesday, Vice President Kamala Harris named him as her running mate. Ernesto Londoño, who reports for The Times from Minnesota, walks us through Mr. Walz’s career, politics and sudden stardom.

On today’s episode

Ernesto Londoño , a reporter for The Times based in Minnesota, covering news in the Midwest.

Background reading

Who is Tim Walz , Kamala Harris’s running mate?

Mr. Walz has faced criticism for his response to the George Floyd protests.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Michael Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

An earlier version of this episode misstated the subject that Walz’s wife taught. She taught English, not Social Studies.

How we handle corrections

Ernesto Londoño is a Times reporter based in Minnesota, covering news in the Midwest and drug use and counternarcotics policy. More about Ernesto Londoño

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Going Deeper: Achieving Greater Depth in the Primary Classroom

As a mastery approach becomes increasingly common in many schools, we have been working with teachers on a project to uncover effective ways to support their learners working 'at greater depth'. In this article, we will share our key findings and some of our favourite whole-class teaching resources which offer opportunities for teaching 'at greater depth'. (Several of these are linked from the NCETM's resources which exemplify their 'ready to progress criteria'.)

Teaching at 'greater depth'

One of the challenges facing many teachers is engaging all learners throughout the lesson when adopting a whole-class teaching approach. As guidance issued to schools indicates, "It is inevitable that some pupils will grasp concepts more rapidly than others and will need to be stimulated and challenged to ensure continued progression” (Askew et al., 2015, p.6).

Aware of these concerns, the guidance adopted the phrase ‘mastery at greater depth’ and suggested that pupils working at this level should be able to:

• solve problems of greater complexity (i.e. where the approach is not immediately obvious), demonstrating creativity and imagination;

• independently explore and investigate mathematical contexts and structures, communicate results clearly and systematically explain and generalise the mathematics.

(Askew et al. 2015, p.7)

The requirement for learners working 'at greater depth' to investigate more complex problems, work independently and communicate their ideas to others are the very foundations of our Low Threshold High Ceiling (LTHC) approach at NRICH: a low threshold ensures that every learner can get started on a problem and the high ceiling ensures that there is a suitable level of challenege, often unseen, for the learner to consider if and when they are ready to do so.

We teamed up with a group of schools based in Tower Hamlets, who were participating in a joint professional development programme with NRICH and the Tower Hamlets Education Partnership, to invesigate ways to maximise the potential of our curriculum-mapped LTHC resources for teaching 'at greater depth'. In this article we share some of their top tips.

• Be prepared to be surprised

During a visit, we joined a very engaged class who quickly got started on the challenge. Working in pairs, with their teacher taking reguar opportunities to pause the groups and allow time to share their ideas with the wider class, the learners began to organise their dominoes so that they could more easily check which, if any, were missing. Some learners who had completed the initial task were further challenged to consider how many dominoes they might have if the maximum number of dots on each side was increased to seven:

Teacher: We’re doing the seven row. Prove to me that there’s eight in the seven row.

Learner: It will only be like this, seven zero add one, seven one, seven two, seven three, seven four, seven five, seven six, seven seven but you can’t go over that.

Teacher:  So when you get to the seven seven you can’t go any further. So you proved it. There’s no doubles and you can’t go any further than seven seven, can you?

The learning did not stop there, the lesson went much deeper as the learners were encouraged by their teacher to set themselves the challenge of thinking about sets of dominoes which go up to 'double 10', or 'double 20' or even 'double 100'. Working with these much larger domino sets encouraged those working 'at greater depth' to generalise their ideas, and their enthusiasm led them to continue working through their break time! 

Reflecting on the lesson, the teacher noted that one member of that day's 'at greater depth' group had previously struggled to engage with their mathematics:

What’s great about NRICH activities is that it just shows children can shine even if we’ve got them in a box under 'expected'. That’s what I love about it... You know, he completely steps out of the box and he’s shining.

Clearly, one of the many benefits of working with LTHC resources is their potential for challenging all learners and enabling them to reveal their potential. Be prepared to be surprised.

Encourage a growth mindset

No-one enjoys trying to do too much at once, any activity can become frustrating if it is too far beyond our comfort zone. This is true in our classrooms too. In our mathematics lessons, our curriculum challenges teachers to develop fluency, reasoning and problem-solving skills, and trying to find a balance within a particular lesson can be a challenge (this is an aspect we explore more fully in this article by Clare Lee and Sue Johnston-Wilder). By choosing LTHC activities, learners can easily get started on a task, building their confidence and willingness to engage with further challenges later on.

One of the strategies we used to solve the coded 100 chart was to look for patterns. We did this by looking in each column and finding that the symbols end with the same shape. For example, the first column of numbers always end in a rhombus shape. The same thing with the rows. Each row, we noticed while putting the symbols together, begins with the same shape. To solve this problem, we think it's best to work with the shape you first put in the 100 chart, and keep building off of that one. The first piece you out in the 100 chart works best if it fits into a corner of the chart.

In contrast, Nathan adoted a very different approach:

Start by placing any one piece and think which one will fit. Before putting it there, think if any other shape could fit.  Then, if not try another shape that could fit. After the one you choose is put on, repeat until the coded hundred chart is filled.  If you mess up, click on pieces to show the full piece and see if you can change a piece for another piece. Remember the first strategy to check if another one can fit after that one and then repeat again and again until you're done. After I fit the pieces together, I looked down each column. I checked that all the symbols in the ones placed matched. I looked across each row and checked that the symbols in the ones places were in order (1,2,3,4...).

• Celebrate mathematical thinking

For learners to be working 'at greater depth', they need to be communicating their ideas clearly to others. In our project schools, the teachers often encouraged their learners to reflect on their problem-solving by recording their ideas in a class book. These books were proudly shared with visitors on arrival to their classes, including the NRICH team. Although we cannot visit every classroom, we do enjoy reading solutions to our problems. For example, Jordan, Juoiana and Nathan all submitted their ideas about the Coded Hundred Square to the team and their ideas were published on NRICH. Your classes are very welcome to share their ideas about our problem-solving tasks too, simply visit our Live Problems page for our very latest opportunities to communicate mathematically with the team. We publish a selection of the submissions we receive on our website.

There are 10 leaves per twig There are 10 twigs per branch 10 leaves x 10 twigs = 100 leaves per branch There are 10 branches per trunk 100 leaves x 10 branches = 1000 leaves per trunk There are 10 trunks per tree 1000 leaves x 10 trunks = 10 000 leaves on the tree

Cut off one trunk: 10 000 - 1000 = 9000 leaves left Cut off one branch: 9000 - 100 = 8900 leaves left Cut off one twig: 8900 - 10 = 8890 leaves left Pull off one leaf: 8890 - 1 = 8889 leaves left

There are 8889 leaves left on the tree.

Another learner called Rachel seemed to approach the problem in the same way as Kirsty, but she found the total number of leaves which had been pulled off the tree before finding the total number of leaves on the tree initially. 

Submitting solutions often leads to published solutions, and the teachers in our project schools also reported on the usefulness of the solutions accompanying our resources. Some teachers set aside time the following day after introducing an NRICH problem to their classes to enable their learners to compare their solutions with those from other classes:

I really like the solution thing now that I know how to use it and things. Getting the language out of it and stuff and using it after they’ve done it. Maybe tomorrow... we might start by looking at the solution that was there... Everyone will understand it tomorrow and then we can really, you know, look at it and decide whether we like it and pull it apart.

Allow time for you to think mathematically too

Another key finding from our project teachers was the importance of setting aside time to explore each LTHC activity for themselves before the lesson. This approach enabled the teachers to consider the possibilities for extending the learners where needed, to enable teaching 'at greater depth', but it was often seen as a thoroughly enjoyable experience too:

I mean, I consider myself greater depth, okay? ... I do have, you know, quite a high level of maths. So, basically, if I have a greater depth and I’m enjoying it and taking it on and forward progressing, clearly those children can do the same, can’t they?

Our school-based research revealed these four 'tips' for schools to support their learners working 'at greater depth' using NRICH tasks:

• Encourage a growth mindset
• Allow time for you to think mathematically too.

We hope that reading this article will inspire readers to explore the potential of using NRICH tasks to support their own learners working 'at greater depth'.

This project would not have beeen possible without the generous support of the team at the Tower Hamlets Education Partnership and the teachers who welcomed the NRICH team into their classrooms. We were delighted to share our findings at the British Society for Research into Learning Mathematics Day (BSRLM) Conference in November 2020 - you can access a copy of the accompanying research paper which was published in the BSRLM's Proceedings  here .

Askew, A., Bishop, S., Christie, C., Eaton, S., Griffin, P. and Morgan, D. (2015).  Teaching for Mastery: Questions, tasks and activities to support assessment . Oxford University Press.

Lee, C. and Johnson-Wilder, S. (2018). Getting into and staing in the Growth Zone. Retrieved from https://nrich.maths.org/13491