math problem solving organizer

Download: Graphic organizers to help kids with math

By Jenn Osen-Foss, MAT

The Frayer model PDF - 46.9 KB

Step-by-step PDF - 52.5 KB

Line it up PDF - 62.2 KB

Graph it out PDF - 59.5 KB

Break it down PDF - 59.5 KB

If your child has trouble with math , a graphic organizer can help. Graphic organizers help kids break down math problems into steps. They’re great tools for figuring out what’s being asked in a word problem or which operation to use. Kids can also use them to learn new math concepts.

There are many types of graphic organizers. The five here could work for students of all ages. You and your child can adjust them depending on your child’s needs and the math problem or topic.

Each graphic organizer comes with instructions and one or two samples of what it looks like filled in. You can print them out and have your child fill them in by hand. (If your child struggles with writing, you can write in the problem and let your child do the rest.)

Key takeaways

Use a homework contract to help your child stay on top of assignments.

Try graphic organizers to help your child with writing .

Print lunchbox notes to give your child confidence in the middle of the school day.

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• Post published: September 25, 2020
• Post category: Blog / Math

Do your students struggle to show and explain their thinking when solving word problems? This problem solving graphic organizer will give your students a structured way to do just that.

So many students equate being successful in math with being able to solve a problem the fastest. The kids who put their hands up first are the best at math. That’s what they think but it’s not true. We need to let our students know that it’s okay to slow down; that we want them to think deeply about mathematical concepts and struggle through.

So, how do you get them to slow down? You make them use this tool. But honestly, they aren’t going to want to do it.

Step 1: Use a problem solving graphic organizer

This is the graphic organizer is what we use. Try it with your class and watch the magic happen . You can get a free version of the problem solving organizer  here . But first, make sure you  read this post  all about how to introduce problem solving to your students.  Here’s the breakdown:

Box 1: Understand – students restate the problem in their own words and list what they know from the problem and what they need to know.

Box 2: Strategize – students make a plan and list strategies that they think they might use to solve the problem before they actually start solving it.

Box 3: Solve – students carry out their plan here, making adjustments as necessary.

Box 4: Reflect – students explain what they did and why and also reflect on why their solution makes sense.

Step 2: Model how to use it

Here’s my biggest takeaway from the past few years:

Until I know what my students know I need to assume that they don’t know what I assume they should know.

Does that make sense?

I don’t think you should always model math concepts first but in this case I wanted to explicitly teach the problem solving process and I wanted them to do it my way.

The strategies that they use to solve the problem – that’s all them – but my students needed this scaffold until they internalized the process.

1. Explain to your students that mathematicians use an organized process to record their thinking when solving problems and that today they are going to learn how to use a problem solving graphic organizer.

This will help them to organize their thinking in a way that will make it easier for others to understand and will help them to slow down and think deeply about the problem that they are required to solve.

2. Post a problem on the board. Here’s the one I used:

3. Use the graphic organizer and model how you would solve this problem. Your students may want to jump in at this point and help solve it, which is great, but encourage them to just listen and absorb. They will get a chance to practice soon!

4. Have students think-pair-share what they noticed you doing and why they thought you did it.

Step 3: Have students work in groups using the graphic organizer

1. Post another problem on the board. If you are looking for some open-ended problems for groups you can check out my product  here .

2. Split students up into groups of four and give them the first box of the problem solving graphic organizer. I would enlarge this onto 11×14 paper so that group members can see and to facilitate sharing later on.

I purposely only give one box at a time, otherwise the students want to race ahead to the solving part. Have students complete Box 1 and then meet with another group to share.

3. Put up a blank version of the problem solving graphic organizer and fill in Box 1 together. Then pass out Box 2 and repeat the process until all four boxes have been completed.

Step 4: Have students use the problem solving graphic organizer independently

Now it is Independent Practice time. Post three problems on the board. An example is included with the problem solving graphic organizer freebie.

Explain that the problems are of varying levels of difficulty and they can choose to complete any one of the three.

This lesson may take more than one class period but it is so worth it to take the time to teach your students how to use a problem solving graphic organizer properly. You are setting them up for success and helping to change their mindset about math. You are doing big things!

Looking for more great tips? Sign up for the Middle School Mashup Email Club below to get some more ideas in your inbox every week!

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Teach Word Problems with a Graphic Organizer [Free Download]

Scottie Altland · November 11, 2016 · 1 Comment

Word Problem Challenges

Many students love to read stories, play with puzzles, and creatively approach problems in their everyday life, so why do they have such an aversion to math word problems?

Word problems often pose a challenge because they require reading and comprehending the text of the problem, identifying the question that needs to be answered, and finally creating and solving a numerical equation.

If your student answers a word problem correctly, you assume he has a grasp of the concept. However, if it’s incorrect, you are left with many questions:

• Does he realize the answer doesn’t make sense? • Did he not understand the context? • Did he simply compute all the numbers in some way just to finish? • Did he truly not know where to begin?

Most importantly: How can I help?

If you are lucky, you can identify the mathematical misconception and work from there. Oftentimes, however, the student’s answer isn’t even reasonable, and it can be difficult to pinpoint the exact area of his confusion. Then what can you do? If you find it challenging to teach your child how to solve word problems, try one or more of the suggestions below to help your student gain confidence in his problem solving skills.

Word Problem Strategies

Be more relaxed & playful.

If your student has anxiety about word problems, try approaching them in a more relaxed and playful way by presenting problems that have more than one right answer, like a math puzzle or game. This can relieve the pressure of having to find “the right answer” and place the emphasis on the problem solving skills.

For example, consider this word problem: Shawn had $156. He went shopping for new hockey skates and now has $12 left. Find the cost of Shawn’s new hockey skates.

You can revise the problem to something like: The answer to a subtraction problem is 12. Tell what the equation might be.

The second question gives your student a springboard to work from, since it already indicates the operation. It also allows her to begin at her level of numerical comfort. In addition, because there are many answers to this type of problem, you can discuss with her how she went about solving it and gain some insight to her conceptual understanding and the strategies she used.

As an extension, you can ask her to write a word problem to match the math problem she created. These techniques can help her build confidence in her problem solving abilities and provide insight to you as to how she approaches solving a problem. Gradually she can work up to solving more problems like the first one presented above.

Extract the Numbers

Another way to increase comfort with word problems is to extract the numbers – in other words, create “numberless” problems. This is a great way to help your student notice the relationships in a problem and to observe how the language can help him understand those relationships.

For example, think about a problem such as: There are 125 girls participating in a choral competition. Twenty-nine more boys than girls are participating. Find the number of boys in the choral competition. Extract the numbers so it reads: Some girls are participating in a choral competition. More boys than girls are participating.

Then discuss with your child how he could find the number of boys participating in the choral competition.

By doing this you are helping him to focus on the situation of the problem, which will lead him to the necessary computation. He can insert his own numbers, or you can slowly introduce the numbers back into the problem for him to work with when he is ready to test a strategy. This method can also help your student determine when a problem contains extra information that is not necessary for solving the problem.

Graphic Organizer Example

Representing the information in the math problem with manipulatives, a drawing, or a diagram can also help make sense of the situation.

For example: Sara has 15 grapes. How can she divide the fruit between herself and two friends evenly?

Your student can show the 15 grapes with cubes or other small objects to represent the grapes and move them into three groups to find the answer. Discuss with your child how the picture, diagram, chart, expression, or equation relates to the situation in the problem. Ask her to explain why she chose it or why she thinks it is a good mathematical expression to use for the problem.

A graphic organizer can help develop the habit of visualizing a problem in smaller chunks. This can help shape your student’s thinking strategies to begin to think about where to start the problem solving process. Graphic organizers can also aid in brainstorming different thoughts and ideas as well as analyzing and synthesizing the mathematical concepts and procedures he selects to solve a problem.

Free Graphic Organizer Download

Click on the sample graphic organizer to see how it can be used to solve a word problem.

A blank version is also available for you to use with your student.

Word problems help your student to see math applied in the real world, and they encourage and give a reason to learn the underlying concepts and operations.

You can help your student make sense of these problems by helping and encouraging them with the problem solving processes. As your student moves forward in his mathematical learning, he will need to apply problem solving processes to more complex situations. By practicing on a regular basis, your child can strengthen and develop confidence in his problem solving skills.

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Math Graphic Organizers

Explain math concepts effectively, simplify complex mathematical problems and describe concepts graphically in an easily consumable manner..

• Customizable templates to organize math concepts and solve problems
• In-app video conferencing to facilitate teaching in a remote environment
• JPEG, PNG, & PDF exports to print and distribute physical copies

Present Math Concepts Visually

Multiple math graphic organizer templates to help you start explaining math concepts quickly.

Smart formatting and preset color themes to highlight, color-code, and differentiate concepts on the math graphic organizer.

Import or drag and drop images and icons to the canvas or use built-in Google image search to browse the internet for graphics to further customize your graphic organizers.

Built-in tools to instantly arrange visuals and information on the canvas and create interactive presentations and graphic organizers for teaching.

Over 50 types of diagrams to visualize, analyze, classify, and organize information; Euler diagrams, Venn diagrams, KWL charts, concept maps, storyboards, and more.

Centralize All Learning Resources

Insert documents and assets with in-app previews to provide more context to the graphic organizers and create resourceful instructional material.

Embed your graphic organizers in any site , or in Google docs, sheets, or slides, and Powerpoint presentations.

Export your math graphic organizers as SVGs, PDFs, and PNGs to publish, present, print, and share.

Infinite canvas that scales up to 1000s of items in a single view to create a central repository of resources, learning material, instructions, and assignments.

Collaborate with Students Remotely

Interactive online whiteboard with freehand drawing to conduct collaborative brainstorming sessions and facilitate classroom discussions.

Video conferencing baked into the platform to feel like you are in the same classroom while engaging with students.

Real-time cursors for any number of participants . Collaborate with students on a shared canvas during lessons.

Comment with context . Full comment threads and discussions for async collaboration to streamline giving students feedback or obtaining feedback from them.

Multiple access and role levels to streamline sharing, reviewing, and editing graphic organizers with students, parents, or colleagues.

What Is a Math Graphic Organizer?

Math graphic organizers are used to describe math concepts graphically to students. It helps with simplifying and solving complex math problems.

How to Create a Math Graphic Organizer?

• Select the math problem you want to identify and a relevant graphic organizer that you can use to solve it. For example, you can use a Venn diagram template.
• If you want to get a quick start, use Creately math graphic organizer templates, or you can start from scratch.
• Invite your students or colleagues to collaborate as you wish. You can send them either a Review diagram or Edit diagram link. Any changes they make will be applied to the diagram in real-time.
• Personalize your math graphic organizers with Creately color schemes. You can also import images or find them with the built in image search option.
• If you wish to take print-outs, export your diagram as a PDF or an image. You can also add them to your PowerPoint presentations and Word docs.
• The Creately Viewer is a handy way to embed your math graphic organizers in a website or your blog. This will display an always updated diagram and give quick access to students.

Math Graphic Organizers Gallery

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Free Word Problems for Math Intervention Activities

Is it a challenge for your students to break down word problems? Do they struggle with identifying the operation, solving the problem, justifying their answer, or all of the above? This can definitely be a challenge for your students and you when this happens (and it seems to happen every year). This post will share some strategies for teaching word problems and a set of free word problems that are perfect for math intervention activities and remediation.

Strategies for Word Problem Math Intervention

Here are some of the strategies and activities that I use to support my students with word problems:

• Introduce new math problems with a context in the form of a relatable word problem
• Expose my students to a variety of word problems regularly and consistently > The free set of word problems will help with this.
• Model my thinking when reading and solving word problems
• Provide my students with a step-by-step way to work through understanding and solving a word problem > T he free set of word problems will help with this.
• Get hands-on…use hands-on sorts and moveable digital activities to have my students interact with word problems > The free set of word problems will help with this.
• Teach situations and contexts and not key words when it comes to determining the operation needed. Click here to read a post about using situations and not relying on key words to solve word problems.
• Allow for opportunities for students to share and discuss word problem solving strategies

Now, let’s take a look at the free set of word problems that will give you some free math intervention activities to help your students master word problems.

About the Free Word Problems

This set of free word problems includes two different math intervention activities that focus on word problems.

• The first set of word problems help students conquer word problems by helping them break down the problem and organize their plan for solving.
• The second set of word problems is a sort where students sort word problems according to the operation needed to solve them.

The free word problems focus on basic multiplication and division math facts.

This will allow you to focus on teaching your students the strategy of breaking down a word problem (with the help of a graphic organizer) without being bogged down by difficult math concepts.

Math Intervention Word Problems – Graphic Organizer Activity

The first set of word problems includes 10 multiplication and division word problems with a graphic organizer. The organizer helps students break down the steps of solving the word problem.

The graphic organizer for each problem asks the students to:

• Retell the problem
• Select the operation and reason for selecting that operation
• Solve the problem
• Defend why the answer makes sense
• Prove the answer

I have also included the word problems in a copy friendly version with 8 word problems on a page and blank templates of the graphic organizer. You can use laminated copies of the blank template and have students write on them with dry erase markers. This makes it super easy to use during stations and guided math groups.

Note: It can be time-consuming to complete all parts of the graphic organizer each time if used regularly. Feel free to have students only complete specific sections as needed. You could even start with just having them retell the problem, select the operation, and then solve the problem. Then you can move to the other sections later on. Alternatively, you can skip the retelling and focus on the defending and proving sections.

Math Intervention Word Problems – Sort Activity

Word problems are a difficult math skill to make hands-on but anytime I can make something hands-on, the better my students do.

The second set of word problems included in this free resource are for a sort activity where the students sort six word problems by operation (multiplication or division).

You can use this to focus just on understanding the problems and determining the operation or you can take it a step further and have the students solve the problems after sorting.

Digital Versions

If you want to assign this as a digital math activity, both sets of word problems (the 10 graphic organizer word problems and the word problem sorts) are included in digital format on Google Slides.

For the graphic organizer word problems, students will complete their responses in the provided text boxes on each slide and then drag and drop the operation needed to solve the problem.

For the word problem sorts, students will drag and drop the problems to columns labeled “multiplication” or “division” to sort by operation.

Get the FREE Word Problems Here!

In the download, you will find a PDF containing the printable version. Digital access links are on page #4 of the PDF.  To get this free Word Problem Intervention resource, simply enter your email address in the box below.  You’ll be subscribed to a special 4-day email sequence where you’ll receive the free intervention word problems, word problems organizer mats, tips for teaching word problems, and more!

For a detailed step-by-step guide to assigning Google Slides, click here.

Want more math word problem resources for 4-5 grades?

If these word problem activities are helpful to your students and you want more to use throughout the year for a variety of math skills and operations, including decimals and fractions, check out the full grade level bundles below!

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4th Grade Word Problem Intervention Activities

5th Grade Word Problem Intervention Activities

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November 14, 2022 at 7:23 pm

Do you have an editable version of the problem solving template? I love it but want to make some changes.

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Word problem solving strategies for students in grades k–4 [free templates], by: jeff todd.

Math problem solving strategies should begin as early as Kindergarten or Grade One! As nonfiction reading has seen a stronger emphasis in recent years, we can think of word problems as part of the genre of nonfiction. The downloads for today’s post include several templates or graphic organizers to help students make the connection between word problems and equations that represent those problems.

As a math teacher, I have heard many times that “we are all teachers of reading,” and this post will show how the two areas relate, both math and reading, as students create representations to help them move from words to equation and vice versa. Plus, grab my downloadable templates for multiple representations below! By using these templates to design lessons, you are able to address many of the Standards for Mathematical Practice that undergird math word problem solving strategies.

• SMP 1: Make sense of problems and persevere in solving them
• SMP 2: Reason abstractly and quantitatively
• SMP 4: Model with mathematics
• SMP 7: Look for and make use of structure

Manipulative and visual representation of math word problems are closely related. These representations are the math problem solving strategies that students can employ. I hope you’ll bear with me as I give a bit of the history of what I have learned about teaching students using manipulatives and representations. In the 1960s, Jerome Bruner coined the terms enactive, iconic, and symbolic to describe how students progress from using manipulatives, to making drawings based on the manipulatives, to using numbers and symbols alone. Today, we might call these steps concrete, representational (semi-concrete), and abstract. Singapore Math uses the terms concrete, pictorial, and abstract. These three sets of terms all refer to the same core strategy of using manipulatives with mastery to show a mathematical idea, then having students represent that idea using paper and pencil (re-presenting it), and, finally, using only numbers and symbols to represent it.

I would encourage you to have students first work with manipulatives such as teddy bear counters, little cubes, or even beans. These help to show the relationships between the situation students are reading about in a word problem. It is best to have them use the template to represent their idea using a ten frame, number bond, array or area model, and tape diagram (semi-concrete, pictorial or iconic representations). Finally, they will understand the meaning of the equation (abstract or symbolic representation) when they write it.

Math Word Problem Solving Strategies for Students

If you are looking for math word problem strategies in Kindergarten through Grade 4, you will find the downloadable templates below very helpful. By using the templates, you can give students strategies to read word problems and create representations to solve them, or even give them a representation and have them create word problems. Use these downloadable templates to give students math problem solving strategies involving addition, subtraction , multiplication, and division. Print them and use them today in your class!

Kindergarten and Grade 1—Adding

In the earliest grades, students are only expected to add. A typical word problem might be, “Chris has three oranges and two apples. How many pieces of fruit does Chris have all together?” Students can model the problem using cubes of different colors. The downloadable template has a spot for the question, then students can make a drawing based on their manipulatives. The key semi-abstract representations for these students are ten frames and number bonds. Particularly with number bonds, students are having to think about parts and totals. Finally, students write an addition sentence.

There are two templates available for adding . The first has one ten frame targeted for Kindergarten where students only add within ten. The second has two ten frames targeting first grade, where students add within twenty. Advanced students could be pushed to represent their addition sentences using a number line, but that is not included with this download.

Grades 1 and 2—Adding & Subtracting

As students progress through Grade 1 and into Grade 2, they are learning about the relationship between addition and subtraction. Conceptually, this is different from early work with just adding. Strategies for solving addition word problems with two addends can be formulaic. The two numbers in the word problem must be added, but when students encounter word problems with a missing part, they must have strategies and representations to think about parts and wholes.

On the template for adding and subtracting , you will find number bonds and a tape diagram. Each template has a frame with two number bonds, one with the “whole” x-ed out, the other with one of the “parts” x-ed out. Students need to read the problem and decide whether it is a missing-part or a missing-whole type problem. This is where we need to tie in the reading with the math. Similarly, students should complete the tape diagram using the part and whole ideas, but this time using a “?” or a letter as a variable to represent the unknown.

Finally, students should write at least one addition or subtraction sentence to represent the problem using a “?” or variable for the unknown. Then, they can write the number sentence showing the “solution” in place of the question mark or variable. Advanced students could be pushed to represent their number sentence using a number line, but that is not included with this download.

Grade 3 and 4—Multiplying & Dividing

Building on work in second grade, Grade 3 and Grade 4 students need to adopt strategies for solving word problems that involve multiplication and division. These problems require different representations than the strategies for math word problems involving addition and subtraction.

The downloadable template for Grades 3 and 4 include a space for an array model, an area model and a tape diagram. Just to be clear, students can represent multiplication and division word problems using any of these three representations:

You can see that this series of abstract representations of multiplication and division move from the more concrete (semi-abstract) versions where you can count dots or squares, to more abstract versions where students move away from counting to finding solutions. This also aids students in the beginning use of variables to represent unknowns, as they can label missing parts of the area or array models with a letter.

In the last box of the downloadable template, students are asked to write the equation using a variable or question mark for the unknown and then to “solve” it. By solving, I mean not using algebraic steps (i.e., divide both sides by three), but instead just to write “x = 7” in the case of the example immediately above. Students could use any form of reasoning, including going back to using physical counters and dividing them into equal groups.

How The Standards For Math Practice Relate To Using The Templates

I want to share some thoughts about how these downloadable templates can be used to develop students’ strategies to solve math word problems and tap into the Standards for Mathematical Practice (SMPs).

SMP 1: Make sense of problems and persevere in solving them.

When students are asked to make a diagram, they must be clear about what parts and wholes are. Giving them representations such as number bonds, area and array models, or tape diagrams helps them make sense of the problems and the relationships of the elements that they find when reading the word problem.

SMP 2: Reason abstractly and quantitatively.

When students create a representation such as in Download 4, (3 boxes of “x” equal to 21), this is an abstract representation. It doesn’t say anything about what the problem is about. When students read a word problem involving quantities (three toys that cost $21 total) and make the tape diagram they move from quantities to abstractions. Another way to use these templates is to complete the tape diagram (or array or area model) and ask students to fill in the other boxes. In other words, students will create their own word problems from the tape diagram. They start with the abstract representation and come up with a quantitative idea (this problem could be 21 apples and three people or 21 chocolates and three boxes, etc).

SMP 4: Model with mathematics.

These kinds of models, such as area models and tape diagrams, if introduced early, will help students when they use area models in upper grades to model more complex problems.

SMP 7: Look for and make use of structure.

Reading word problems and then making representations using the templates will help students look for keywords and how they relate to the structure of parts and wholes, rows and columns, factors, totals, and divisors. Seeing the common underlying structures using number bonds, ten frames, area and array models, and tape diagrams helps to reinforce common underlying structures that appear in various word problems.

Download and use my free templates to help students make connections between word problems and equations that represent those problems. When you do this, you will engage students in the use of the Standards for Mathematical Practice outlined above, giving them ways to picture word problems in their heads and create representations that show the relationships of the quantities involved.

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Math Graphic Organizers

What are Math Graphic Organizers? Does your child face a problem in dealing with mathematics? If your kid is having trouble in mathematics, let's give you a piece of good news. There is a graphic math organizer that can help your kid. Mathematical Graphic organizers help kids break down math problems into easy steps. They are the go-to problem-solving tool for parents and teachers. Since it's a way to visualize your problem, it strategizes the problem to solve it quickly. They serve as a great tool for figuring out the problem that is required to be answered in the mathematical operation. These organizers can also be used by Kids to learn about new mathematical concepts. It is particularly great for young learners as they are not only able to arrive at the solution of the problem quickly but also find out how they arrived at it. The organizers are adjustable, and kids can adjust them as depending on their needs or requirements. There are different types of graphic organizers, and some of them work for all types of ages of kids. Each of these graphic organizers come with instructions. You can print these out and also have your child fill it in by hand. Graphic organizers can really help students grasp complex concepts, relative to their skill level, at a faster pace. We have seen these specific organizers to be the most helpful for students.

Elementary Level

• Long Division - A nice way to approach this topic. Can be be applied to most three and four digit values.
• Prime Factorization - A way to break it all down for you. Start with the number you are working with and go from there.
• Common Factor Forest - It's a thick forest out there. Factors are broke up like little branches of a tree.
• Bubble Organizer - Great for fact families. I also find this helpful for plotting out word problems.
• Create & Solve - Create math problems (addition, subtractions, multiplication, and division), and switch papers with a partner. Solve the math problems.
• Solve a Problem 3 Ways - Show us three different ways to solve that problem.
• Addition Chart - Get your add on here. These are great to practice with a partner. Take turns placing sums and addends of your choice.
• Balancing Equations - You can even team up with a partner and us this as a review.
• Division Chart - Fill in one column and go from there. You can work with friend on this and place numbers in any of the boxes and then complete it.
• Box-Style Organizer - A way to style things differently. This can be used for any number of operations.
• Multiplication Chart - Another fun partner project for you. You get time to work out your times tables here.
• Geometry Terms - Go after all these terms. You will see them after as you advance with geometry.
• Subtraction Chart - A way to style your subtraction problems. A better title may have been, "What's the Difference?"
• Number Lines - Four different point values and thicknesses. They all fit different types of occasions.
• Coin Place Value Chart - This can be used as a counting or tally sheet, your choice.
• Dollar Place Value Chart - We move on to working with paper. Hopefully you will get a chance to count out to the last place.
• Finding the Greatest Common Factor - A nice way to line it up for yourself.
• Place Value Chart - From the ones place to the millions. A great way to classify values that you may come across.
• Understanding Basic Math Symbols - What does each symbol mean and use it in a sentence. A math sentence that is...
• Identifying Shapes - We name shapes based on their stated qualities.
• Venn Diagram 1 - A double Venn. Two of them on a page with a title area. Great for comparing concepts.
• Venn Diagram 2 - A big triple Venn diagram. This helps you understand crossover between three concepts.
• Ways To Make a Number - Use what ever operations you would like here.
• Word Problem Breakdown - What is the word problem about and asking you to do?
• Inequalities - This give you room for four constants or variables.

Middle School

• Adding & Subtracting Negative Integers - A different way to think about it.
• Converting Fractions to Percents - A way to show your work.
• Dividing Mixed Numbers - A paced approach with extra steps to simplify if they are needed.
• Multiplying Mixed Numbers - There is a great deal of value conversion on this organizer.
• Order of Operations (PEMDAS) - The steps in order for you to work through.
• Simplifying Fractions - This allows for four progressions. A great way to present reducing or equivalent values.
• Double Bar Graph - You can display multiple values here. It provides with a legend to work off of.
• Single Bar Graph - This is great for comparing two different values.
• Improper Fractions & Mixed Numbers - The conversion process. This all starts with a top heavy fraction that needs to be grounded.
• Single-Variable Equations - This shows you how to line them up.
• Line Graph with 3 Lines - This can work for bar, line, and heck even pie charts.
• Solving Inequality Equations - Placement is key here. Just focus on the comparison symbol first.
• Making Pie Charts - Three, four, and even five slice varieties.
• Square Roots - For perfect or natural roots.
• Math with Decimals - All operations need apply. There is space for two sums and differences.
• Using a Protractor - Find all these angles please. You can also use it to model a system.
• Multiplying & Dividing Negative Integers - Going in both directions. This makes a great practice sheet with a partner.

High School Level

• Circumference & Perimeter - The basics are covered here. It is a good way to review the concept of Pi.
• Possibility Diagrams - Great for stats. problems. This is also called a fishbone model.
• Congruency Bubble Diagram - This is helpful for triangles. This will help you stage your thought flow.
• Pythagorean Theorem - An in depth analysis for you to work off of.
• Finding Area - A quick cheat sheet for you. We have it aligned to the different geometric shapes.
• Solving a System of Equations - A way to map out your solutions.
• Finding Slope - The old run over rise guys. It also guides you through the process of determining points.
• Using FOIL - First, Outside, Inside, Last. This worksheet walks you through the entire thing.
• Mapping Diagrams - A ton of different application here. use your imagination.
• Solving Circle and Cylinder Volumes - The equations you will need to work with those shapes.

How Math Graphic Organizer Can Really Make a Difference for Students

We tend to think of graphic organizers to just help us in language arts and social studies class, but the truth is they can help our comprehension and understanding of topics in many aspects of life, including math class. The big way these organizers help is to break math into smaller steps that make the steps to solving the problem more obvious and clearer for students. As students begin to progress towards algebra, they will often come across the dreaded math word problem. If you use this approach to help you organize yourself, life becomes much easier. Math graphic organizers promote critical thinking in areas that address concepts and relationships between terms in an expression or equation. It will also help identify anything that may be missing or simply absent. We have seen in recent years where test makers have been using graphic organizers to assess if students have mastered concepts. This should tell teachers and tutors to start making sure that students have many quality experiences with math in this form. The main reason being that until they get some practice with it, students will often struggle to pull a visual concept out of a mathematical expression or system. As always, practice makes perfect.

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The Frayer Model for Math

ThoughtCo. / Deb Russell

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The Frayer Model is a graphic organizer that was traditionally used for language concepts, specifically to enhance the development of vocabulary. However, graphic organizers are great tools to support thinking through problems in math . When given a specific problem, we need to use the following process to guide our thinking which is usually a four-step process:

• What is being asked? Do I understand the question?
• What strategies might I use?
• How will I solve the problem?
• What is my answer? How do I know? Did I fully answer the question?

Learning to Use the Frayer Model in Math

These 4 steps are then applied to the Frayer model template ( print the PDF ) to guide the problem-solving process and develop an effective way of thinking. When the graphic organizer is used consistently and frequently, over time, there will be a definite improvement in the process of solving problems in math. Students who were afraid to take risks will develop confidence in approaching the solving of math problems.

Let's take a very basic problem to show what the thinking process would be for using the Frayer Model.

Sample Problem and Solution

A clown was carrying a bunch of balloons. The wind came along and blew away 7 of them and now he only has 9 balloons left. How many balloons did the clown begin with?

Using the Frayer Model to Solve the Problem:

• Understand :  I need to find out how many balloons the clown had before the wind blew them away.
• Plan:  I could draw a picture of how many balloons he has and how many balloons the wind blew away.
• Solve:  The drawing would show all of the balloons, the child may also come up with the number sentence as well.
• Check : Re-read the question and put the answer in written format.

Although this problem is a basic problem, the unknown is at the beginning of the problem which often stumps young learners. As learners become comfortable with using a graphic organizer like a  4 block method  or the Frayer Model which is modified for math, the ultimate result is improved problem-solving skills. The Frayer Model also follows the steps to solving problems in math.

• Basic Subtraction Fact Worksheets to 20
• Worksheets for Elementary Math: Doubles Addition
• Timestable Facts to 10
• Factor Tree Worksheets
• Learning Long Division
• Free Printable 3-Digit Subtraction Worksheets
• 3-Digit Addition Worksheets
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• 2 Times Tables Fact Worksheets
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Printable — problem solving organizer for math.

Print this problem solving mat and use as is, or make a copy, and change what you need to meet the needs of your own students. This organizer also includes a quick self-reflection on the student's understanding.

You can find this printable along with 5 more covering several other subjects.  All free!  All for you to download and enjoy with your class today. Just click below to see the whole bundle of printables we’ve made available to you thanks to TeacherSherpa.

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SimpleK12 is excited to enter into a partnership with TeacherSherpa to bring you a collection of some really interesting, fun, cool, and very useful printables.  From time to time, we will provide a printable that you are free to download and use however you would like.  We’ll provide downloadable printables that cover everything from Social Emotional Learning (SEL) to Behavior Management.  Teacher Sherpa is a company that provides unlimited access to thousands of teacher-created materials and an all-in-one design tool preloaded with royalty free clipart, fonts, and additonal components so you can create your own materials.

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The 3 Reads Protocol for Solving Word Problems

Raise your hand if you wish your students were more confident and successful in solving word problems. Right, that’s what I thought. And the answer probably doesn’t change much based on your grade level. Face it, word problems are just plain hard!

What DOESN’T WORK

Over the years, very well-intentioned teachers have developed strategies designed to help students solve word problems . Two such strategies that are still quite prevalent are “problem-solving” models and the use of keywords. The idea is that if you follow these steps and look for these keywords, you will be able to solve any word problem. Unfortunately, it’s just not that simple, and despite their widespread use, these strategies are not very effective.

If you look at the CUBES problem-solving model, reading the problem is not even one of the steps! And if you’re thinking,  Well, of course students know to read the problem! you might want to watch this model in action. I have more often than not seen students just literally start circling numbers (and not even the labels that go with the numbers…) without ever having read the problem. And keywords are not reliable either. Some word problems have no keywords, and keywords in multi-step problems end up confusing students because of the mixed messages they send.

So can we just agree that something else is needed and put these “strategies” to rest? Students fail at solving word problems for one reason—they don’t understand what the problem is asking them to do. It’s a comprehension problem, so students need reading comprehension skills.

The 3 Reads Protocol

Let me first say that if you search the Internet for 3 Reads Protocol , you’ll find that there are slightly differing versions. What I’m about to describe is the version that I find to be particularly effective. Regardless of the version, we are reading the problem three different times and each reading has a different focus.

The 3 Reads Protocol is a guided learning experience. Students are presented with the problem in stages, and with each read the teacher asks probing questions. Looking at an example is probably the easiest way to understand the protocol, so let’s dive in.

To begin the 3 Reads Protocol, the teacher presents the students with a problem, and the class reads the problem together. Probably the easiest way to do this is with a PowerPoint or Google Slides file. Notice that with the first read, there are no numbers and no question. We just want the students to understand what the story is about and make a mental picture. Without numbers, students have to focus on the meaning of the words! After reading the problem together, the teacher asks what the story is about and calls on students for responses. Don’t be surprised if the responses are very general at first ( girls, flowers,  etc.). Ask for additional details, if necessary. Ideally, for this problem, you’d like the students to offer the names of the girls and the types of flowers.

For the second read, the problem is again presented to the students, but this time it includes the numbers. Read the problem again whole class. The questions you will ask now are all related to the numbers in the story. Our goal is for the students to understand that it’s not just 10, it’s 10 daisies . Students might also offer relationships—e.g., Natassja picked more daisies than Ayriale.

Finally, with the third read, students are asked to generate questions that could be answered using the information in the problem. Even though the problem looks just like it did for the second read, don’t skip the reading part! Some problems won’t lend themselves to very many different questions. I like to use this problem as an example because many different questions can be generated. Why? Because there are lots of different numbers in the problem. Here’s a sampling of questions that could be asked. I’m sure you can think of many others.

• How many flowers did Ayriale pick?
• How many flowers did Ayriale and Natassja pick?
• Which girl picked more flowers? How many more?
• How many daisies did the girls pick?

That’s the protocol in a nutshell! Once a question or questions have been generated, you can have students go on to solve the problem.

Frequently asked questions

1. When students are solving word problems independently, do I ask them to ignore the numbers and the question?

No! That would be pretty much impossible for them to do. By routinely solving problems using the 3 Reads Protocol with either the whole class or in small groups, you are helping students develop good reading habits that will transfer to their independent work. When they are working independently, the idea is that they will automatically think about the context, identify what the numbers mean within that context, and better understand what the question is asking them to find.

2. Where do I find problems for the 3 Reads Protocol?

I’m sure you can find some that have already been prepared, but it’s super easy to make your own! Just set up a PowerPoint or Google Slides file and format it however you like. Maybe you want a colorful border or a particular font. Use problems that you already have from your resources—textbooks, supplemental books, etc. You’ll need two slides for each problem. On the first slide, type the problem from your resource, leaving out the numbers and the question. On the second slide, add in the numbers. Use a nice big font so students can easily read the problem when projected on your interactive whiteboard. That’s all there is to it!

If you have other questions, add them in the comments below, and I’ll add them to the FAQ. I’d also love to hear how the 3 Reads Protocol is working out in your classroom!

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16 Comments

fantastic thoughts! I LOVE this approach. I’ve used a resource from Lucky Little Learners that is called Numberless Word Problems. This 3 reads protocol will work perfectly with that resource!

Yes! It does include elements of numberless word problems, which I was first introduced to by Brian Bushart .

I appreciate your thoughts on problem solving however the use of CUBES has been totally effective for the past three years with my students who never attempt word problems. You stated that not circling the “labels”, I suppose you mean units connected to the problem. I use this strategy daily and we circle both numbers and units. I’ve made it my own. Let me also reiterate that this strategy works exceptionally well with students who have low reading abilities too. I spend 5 weeks on CUBES and we slowly work our way into the Three Read Protocol which also helps the “non-reader”

Thank you for sharing your experiences! Since you transition to Three Reads, I’m curious why you wouldn’t start with it. I’d also be interested to know what grade level you teach.

I teach both 7th and 8th grade, and many times they have not had any experience with either. CUBEs is embedded with 3-read. I read the question. Students read the questions, we use cubes to understand quantities and the operations ( C and B) used to bring these quantities together. Then we underline the question to provide clarity and to determine what type of answer we are seeking. In our data, we found that our students do not attempt word problems on any formative assessments so we had to start with decreasing student fear of problem-solving and reading. When I used CUBES, I found that many students could answer the question however they had no idea what to do (operations) with the numbers due to a lack of experience,(key operational words) not knowledge ( they always wanted to add). Immediately, students would attempt word problems because CUBES allowed them to pull out what they needed, producing increased math confidence. As we continued with problem-solving, the 3 read protocol was introduced when solving multi-step word problems, student growth was tremendous!

Excellent post!

Thank you! This strategy has so much potential for teaching students to approach word problems in a much more meaningful way!

Hello! This strategy looks great and reaffirms how I’ve been teaching my son to read the problems through as many times as he needs to understand. Now I have the right questions to ask with each reading! Thank you! My question, are there still some strategies to go along with this protocol that you have found to be helpful to use in conjunction i.e. drawing a picture, making a chart, etc. Thanks so much!

I’m so glad this provides you with the missing link! Drawing a picture or diagram is another powerful strategy that goes really well with 3 Reads because it helps students visualize what’s going on in the problem.

Thank you for your post. The Three Read strategy is a gentle reminder of the power of cross-curricular reading strategies; coupling this annotation and graphic organizers will help students develop ways to process and solve simple to more complex multi-step word problems.

I am a math interventionist at an elementary school. We see students K-5 and realize how important it is for us to have a common process/language when it comes to word problems. Looking forward to learning more about this and helping guide our school, thank you!

I absolutely agree about using common language and strategies campus-wide! It’s so much more effective than students having to adjust to their teacher’s language/strategies every new year.

Thank you! I am going to try Three Reads Protocol for the first time next week with my 4th grade students. I am excited. I have had students read word problems all the way through before trying to solve the problem, however, I love the 3 Reads approach and the questions that go with each read. Thank you for breaking it down.

It’s a very powerful strategy! Good luck.

Many of the problems already include the question or questions. When first teaching the strategy, I can take the questions away and then we can compare to see if one of our questions matched the questions posed, but how to do facilitate the strategy when the question is there and you can’t take it away (thinking of assessments).

The idea is that when done regularly, students will develop comprehension skills that will transfer to solving “normal” word problems. They’ll know, for example, that reading it through the first time and ignoring the numbers helps them focus on what’s happening in the problem. You certainly want to help them make that connection, though. You might even show them a normal problem and explicitly model how to use 3 read strategies when the numbers and question are present.

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Students Use Graphic Organizers to Improve Mathematical Problem-Solving Communications

By: Alan Zollman

“Let me give you a math story problem.” This sentence often strikes fear in many middle grades students as well as some teachers. As international comparisons, national commissions, and state assessment results confirm, students have difficulty solving mathematical applications problems (Lester, 2007; U. S. Department of Education Institute of Educational Science, 2007; TIMMS, 2003; McREL, 2002; National Research Council, 2002; Illinois State Board of Education, 1997).

Improving students’ problem-solving abilities is a major, if not  the  major, goal of middle grades mathematics (National Council of Teachers of Mathematics, 2000; 1995; 1989). To address this goal, the author, who is a university mathematics educator, and nine inner-city middle school teachers developed a math/science action research project. This article describes our unique approach to mathematical problem solving derived from research on reading and writing pedagogy, specifically, research indicating that students who use graphic organizers to organize their ideas improve their comprehension and communication skills (Goeden, 2002 ; National Reading Panel, 2000).

Many teachers and students use graphic organizers to enhance the writing process in all subject areas, including mathematics. Graphic organizers help students organize and then clarify their thoughts, infer solutions to problems, and communicate their thinking strategies.

We designed a classroom action research project to study a problem-solving instructional approach in which students used graphic organizers. Our goal was to improve student achievement in three areas of our state’s math assessment in open-response problems: mathematics knowledge, strategic knowledge, and mathematical explanation. In this article, we discuss graphic organizers and their potential benefits for both students and teachers, we describe the specific graphic organizer adaptations we created for mathematical problem solving, and we discuss some of our research results of using the  four corners and a diamond  graphic organizer.

Benefits of using graphic organizers in mathematics learning

A graphic organizer is an instructional tool students can use to organize and structure information and concepts and to promote thinking about relationships between concepts. Furthermore, the spatial arrangement of a graphic organizer allows the student, and the teacher, to identify missing information or absent connections in one’s strategic thinking (Ellis, 2004).

Middle grades teachers already use many different types of graphic organizers in the writing process. All share the common trait of depicting the process of thinking into a pictorial or graphic format. This helps students reduce and organize information, concepts, and relationships. When a student completes a graphic organizer, he or she does not have to process as much specific, semantic information to understand the information or problem (Ellis, 2004). Graphic organizers allow, and often require, the student to sort information and classify it as essential or non-essential; structure information and concepts; identify relationships between concepts; and organize communication about an issue or problem.

Consider the middle grades math problem in Example 1 from a recent state assessment.

How many vertices (corners) are there in 1, 2, 3, 4, 5, 6 … n squares when they are arranged in the following way?

What did you first think when reading the problem? Did you first think of the meaning of the term “vertices” or that this is a mathematical pattern problem? Did you first think of counting the corners or that this looks like an arrangement of tables? Did you first think to discuss in your solution why you are not just adding four with every square? Did you first try to think of the singular form of the word vertices?

Initial thinking is not a linear activity, especially in mathematical problem solving. Yet, the result of problem solving—the written solution—often looks like a linear, step-by-step procedure. Good problem solvers brainstorm different thoughts and ideas when first presented with a problem, and these may or may not be useful. Problem solvers can use a graphic organizer to record random information but not process it. A student can later reflect upon usefulness of the information and ideas. If the information and ideas help the student make relationships between concepts, then they are essential. A graphic organizer allows a student to quickly organize, analyze, and synthesize one’s knowledge, concepts, relationships, strategy, and communication. It also gives every student a starting point for the problem-solving process.

Adapting a graphic organizer for mathematical problem solving

Figure 1:   Four Corners and a diamond mathematics graphic organizer

Figure 1 depicts the four corners and a diamond graphic organizer. This graphic organizer was modified from the four squares writing graphic organizer described by Gould and Gould (1999). The four square writing method is a formulaic writing approach, originally designed to teach essay writing to children in a five paragraph, step-by-step approach. The graphic organizer portion of the method specifically assists students with prewriting and organizing. We saw beneficial problem-solving aspects in the graphic organizer portion of this writing method for mathematics.

Our four corners and a diamond graphic organizer has five areas:

• What do you need to find?
• What do you already know?
• Brainstorm possible ways to solve this problem.
• Try your ways here.
• What things do you need to include in your response? What mathematics did you learn by working this problem?

Actually, the form in Figure 1 does not have to be given to the students each time. Figure 2 shows how students, using a blank piece of paper, make the four corners and a diamond graphic organizer template. The student folds the paper into fourths, first folding the paper horizontally (“hot dog style”), then vertically (“hamburger style”), and finally the inner corner is folded up. When the paper is unfolded, the creases form the four corners and the “diamond” rhombus in the middle. The teachers reported that students later (e.g., during state testing) often folded or drew the five areas on their paper to begin problem solving.

Figure 2:   Four Corners and a diamond folding template

So how does the use of the four corners and a diamond graphic organizer differ from the traditional Polya’s four-step mathematical problem-solving hierarchy? In terms of objectives, it does not. Obviously, the four corners and a diamond graphic organizer is designed to help students understand the problem, devise a plan, carry out the plan, and look back (Polya, 1944). However, by having the non-linear layout of the graphic organizer, the student is not expected to do these “steps” in a hierarchical, procedural order that some students misapply. It is the implementation process, how students form their response, that is the important aspect of the four corners and a diamond graphic organizer (Zollman, 2006a).

The pictorial orientation allows students to record their ideas in whatever order they occur. If students first think of the unit for their final answer, then this is recorded in the fifth, bottom-right area. This idea (the unit), then, is not needed in the short-term memory because a reminder is recorded. If students first think of a possible procedure for their answer, this is recorded in the third, upper-right area. The four corners and a diamond graphic organizer allows, and even encourages, students to use their problem-solving strategies in a non-hierarchical order. A student can work in one area of the organizer and later work a different area. It also shows that completing a problem-solving response has several different, but related, aspects.

Students do not begin writing a response until some information or ideas are in all five areas. The four corners and a diamond graphic organizer especially encourages students to begin working on a problem before they have an identified solution method. As in the four square writing method, the students then organize and edit their thoughts by writing their solution in the traditional linear response, using connecting phrases and adding details and relationships. The steps for the open response write-up are as follows: (1) state the problem; (2) list the given information; (3) explain methods for solving the problem; (4) identify mathematical work procedures; and (5) specify the final answer and conclusions.

The graphic portion of the organizer allows all students to fill in parts of the solution process. It encourages all students to persevere—to “muck around” working on a problem. Further, teachers quickly can identify where students are confused when solving a problem by simply examining the graphic organizer.

The teacher should model proper use of the four corners and a diamond graphic organizer and have students work in groups when introducing this tool. Working in groups allows students to see that many problems can be worked in more than one way and that different people start in different places when solving a problem. In their small-group discussions, students identify relationships between the areas in the graphic organizer and among the various solutions.

Four corners and a diamond provides students with a logical framework for writing about problem-solving tasks.

Graphic organizers can benefit students when they take standardized state mathematics assessments, specifically open-response problem-solving items. Most states use a scoring rubric for these types of items. In Illinois, for example, the scoring rubric has three categories: mathematical knowledge, strategic knowledge, and explanation (Illinois State Board of Education, 2005). Responses are scored on a four-point scale for each category, with scores ranging from zero for “no attempt” to four for “complete.” Typically, low-ability students do not attempt to show any work in one or more response categories, while average-ability students often have disorganized responses. Higher-ability students sometimes skip steps in their explanations. The four corners and a diamond graphic organizer helps each type of student produce a more complete response in each of the three categories and, thus, receive a higher score.

Impact of graphic organizers

Nine middle school teachers decided to use the open-response mathematics questions as the focus of their action research on the effects of using graphic organizers. Teachers administered pre- and post-tests with their students to see if using the four corners and a diamond graphic organizer impacted their performance.

All teachers reported dramatic improvements in students’ mathematics scores on open-response items after implementing the four corners and a diamond graphic organizer. The percentage of students (N=186) who scored at the “meets” or “exceeds” levels on each of the open-response item categories on the pre-test was 4% for math knowledge, 19% for strategic knowledge, and 8% for explanation. After instructing students to use the graphic organizer in mathematical problem solving, the percentage of students scoring “meets” or “exceeds” on the post-test improved to 75% for math knowledge, 68% for strategic knowledge, and 68% for explanation (Zollman, 2006a; 2006b).

Each teacher self-collected and self-scored these data using the state’s scoring rubric. Overall scores increased from a 27% average on the pre-test to a 70% average on the post-test. Data collected, analyzed, and triangulated from three sources—the teachers, the action research pre- and post-test data, and the students’ work—suggests that the use of the graphic organizer in mathematical problem solving may significantly help students coordinate their mathematical ideas, methods, thinking, and writing. The graphic organizer helped students coordinate various parts of mathematical problem solving: (a) What is the question? (b) What information is known? (c) What strategies might be used? (d) Which operations, procedures, or algorithms of the strategy need to be shown? (e) What explanations and reflections are needed to communicate the method(s) of solution? (Zollman, 2006a; 2006b).

Graphic organizers help students communicate their thinking when they solve problems.

The teachers found the use of graphic organizers in mathematical problem solving to be very efficient and effective for all levels of students. The teachers saw that their lower-ability students, who normally would not have attempted problems, had now written partial solutions. The organizer appeared to help average-ability students organize thinking strategies and help high-ability students improve their problem-solving communication skills (Zollman, 2006b). Students now had an efficient and familiar method for writing and communicating their thinking in a logical argument.

Samples of students’ work

The samples of student work in Figures 3 and 4 are from an open-response squares and vertices problem before and after the use of graphic organizers in the classroom.

Figure 3:   Samples 1 & 2

Sample 1 shows the work of a student who was presented the problem before becoming familiar with the four corners and a diamond graphic organizer. Sample 2 shows the same student’s work later in the semester, after learning how to problem solve using the graphic organizer. The student’s strategy on the pre-test was to count the individual vertices in the picture, then add these numbers. This work shows a misunderstanding of the problem, limited strategy, and no explanation. On the post-test, this same student’s work shows a complete understanding of the problem presented (10 squares) and a complete explanation of a correct strategy that will transfer to other problems, however, it lacks a concluding algebraic formula to demonstrate mathematical knowledge. While it is not a perfect response, understanding, organization, development, and reflection are all strongly represented on the graphic organizer.

Figure 4:   Samples 3, 4 & 5

The second student’s pre-test (Sample 3) shows the common incorrect strategy of just counting the total vertices in the picture. It appears that the student then attempted to “add” the individual pictures in the student’s own drawing to again count the vertices. However, without any explanation, the teacher cannot know what strategy, if any, the student was attempting. Again, this work shows a misunderstanding of the problem, limited strategy, and no explanation. This student’s post-test (Sample 4) illustrates excellent understanding, organization, development and reflection of the problem presented (10 squares). The graphic organizer shows the student’s complete, correct strategy, solution, and explanation of the problem. For mathematical knowledge, the formula is well explained in words, not as an algebraic expression. This would be acceptable on state assessments, as the problem did not specifically ask for an algebraic expression.

Sample 5 is the post-test work of a higher-ability student. This student’s work demonstrates a full understanding of the problem, a correct solution, and a complete explanation. The drawings also suggest that the student feels a sense of ownership of and satisfaction with the solution and probably finished the problem with plenty of time to spare.

We hoped the students in our action research study would improve their problem solving with an instructional intervention from pre-test to post-test; however, no single instructional method directly affects learning. Rather, instruction is one of many factors that may influence learning. Others include the curriculum, the student, the class, and the teacher. Nevertheless, the teachers who conducted the action research described in this article believed the graphic organizer was associated with many of the positive outcomes in their students’problem-solving ability (Zollman, 2006b).

The crucial factor in the effectiveness of any instructional method is how it is implemented. If four corners and a diamond graphic organizer is used as a linear, systematic procedure to teach problem solving, it will succeed sporadically. In fact, any direct teaching about problem solving is likely to have intermittent success. Giving students a chart of Polya’s (1944) four steps in problem solving or a graphic organizer sheet may help students learn the steps of problem solving. However, students may remain uncertain about where to start a problem, confused by essential versus non-essential information, or unaware how to communicate important steps and reflections in their solutions. We found that graphic organizers aid students in all three of these areas.

Allowing students to first use their own thinking–and then reflect, revise, and re-organize their knowledge, strategies, and communication–helps them improve their problem-solving abilities. Initially, teaching about problem solving as a hierarchy of procedural steps is neither efficient nor effective. Our results confirm other studies that found teaching  via  problem solving is the key instructional process (Lester, 2007).

As our work suggests, effective reading and writing strategies like graphic organizers may have crossover effects in mathematics for students of all ability levels. We found that four corners and a diamond, when properly used, was an extremely useful instructional method in the middle grades mathematics classroom. Our instructional approach helped students construct content knowledge and strategic knowledge and, we contend, it also improved their mathematical communication skills. In addition, four corners and a diamond allowed teachers to quickly identify the weaknesses and strengths of students’ problem solving abilities. As teachers seek to expand and improve students’mathematical knowledge to help them solve problems, they may find that good teaching in reading and writing is good teaching in math.

Extensions The author shows how graphic organizers that are typically used to help students organize their thoughts while writing in ELA can also be used to help them think through problem-solving tasks in mathematics.

How can teachers use graphic organizers to actively engage students in thinking and problem-solving activities in all areas of the curriculum?

Author note I would like to thank the students and teachers of the East Aurora (IL) middle schools, and especially teacher Karen Lopez, for their assistance.

Acknowledgement This work was supported in part by the Illinois Mathematics and Science Partnerships Program/ISBE/US Department of Education, funded by NCLB, Title II, Part B, US DOE.

Ellis, E. (2004).  What’s the big deal about graphic organizers?  Retrieved July 7, 2003, from  http://www.Graphic Organizers.com

Goeden, J. (2002).  Using comprehension frames (graphic organizers) to impact students’ reading comprehension . Unpublished thesis. Black Hills State University.

Gould, J., & Gould, E. (1999).  Four square writing method for grades 1–3 . Carthage, IL: Teaching and Learning Company.

Illinois State Board of Education. (2005).  Mathematics scoring rubric.  Springfield, IL: Author.

Illinois State Board of Education. (1997).  Illinois learning standards . Springfield, IL: Author.

Lester, F. (Ed.). (2007).  Second handbook of research on mathematics teaching and learning . Greenwich, CT: Information Age.

Mid-continent Research for Education and Learning (McREL). (2002).  ED thoughts: What we know about mathematics teaching and learning . Aurora, CO: Author.

U.S. Department of Education Institute of Educational Sciences. (2007).  The nation ‘s  report card . Retrieved January 28, 2008, from  http://nces.ed.gov/nationsreportcard

National Council of Teachers of Mathematics. (1989).  Curriculum and evaluation standards for school mathematics . Reston, VA: Author.

National Council of Teachers of Mathematics. (1995).  Assessment standards for school mathematics . Reston, VA: Author.

National Council of Teachers of Mathematics. (2000).  Principles and standards for school mathematics . Reston, VA: Author.

National Reading Panel. (2000).  Teaching children to read: An evidence-based assessment of the scientific research literature on reading and its implications for reading instruction . Washington DC: U.S. Department of Health and Human Services.

National Research Council. (2002).  Learning and understanding: Improving advanced study of mathematics and science in U.S. high schools . Washington DC: National Academy Press.

Polya, G. (1944).  How to solve it . Garden City, NY: Doubleday.

Trends in International Mathematics and Science Study (TIMSS). (2003).  Trends in international mathematics and science study . Washington, DC: International Association for the Evaluation of Educational Achievement.

Zollman, A. (2006a, April). Four-corners is better than Four-squares: Assessment connection for solving mathematics story problems. Presented at the Annual Conference of the National Council of Teachers of Mathematics, St. Louis, MO.

Zollman, A. (2006b, October). Write is right: Improving students’ problem solving using graphic organizers. Presented at the 105th Annual Convention of the School Science and Mathematics Association, Missoula, MT.

Alan Zollman  is an associate professor of mathematics education in the department of mathematical sciences at Northern Illinois University, DeKalb. E-mail:  [email protected]

Published in  Middle School Journal , November 2009.

Copyright © 2009, Association for Middle Level Education

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Opening up students’ mathematical thinking using open middle math problems

When paired with number talks , open middle problems can help teachers embed rich conversation throughout math class.

Defining the problem

So, what exactly are open middle math problems? In an open middle problem, students are given starting information and either an answer or a definition of the answer. By “definition of the answer,” we mean that students are given a description of the type of answer toward which they are driving. For example, many open middle math problems ask students to find the least or greatest solution or a number that is closest to a given number. How they get from start to end is up to them. Rather than following a solution path that was stepped out for them by the teacher, the path is completely under their control.

The best way to understand an open middle math problem may be to compare one to a typical textbook problem for the same topic. For a lesson on three-digit subtraction, a standard textbook might ask students to find 539 – 286 either as a straight computation item or embedded within a word problem. By contrast, an open middle math problem for the same topic asks students to use the digits 1–9, at most one time each, to make two three-digit numbers with a difference as close to 329 as possible.

Although understanding of place value, regrouping, the relative magnitude of numbers, and the relationship between addition and subtraction underlie both types of problems, with the more traditional problem, a student can answer correctly by following a set of learned steps without truly understanding the conceptual underpinnings. However, students must actively engage with these concepts to solve the open middle math problem. The open structure forces students to consider these concepts and constantly make decisions and adjust their solution path. Here are some of the questions students might consider when solving the open middle math problem:

• What strategies can be applied to solve? Should I subtract? Should I add up? Can estimation help?
• Should I focus on a certain place value for each number first, or should I create one number completely?
• Is it possible to make two numbers where regrouping will not be required?
• Is it possible to get exactly 329?

Open middle problems can also be word problems . While some open middle math problems are structured this way, others can easily be turned into word problems. For example, the three-digit subtraction problem above can be rewritten as a word problem such as this: Jared and Suni are playing a video game. Each of their scores were three-digit numbers. Jared’s score was 329 less than Suni’s. Use the digits 1–9, at most one time each, to make two three-digit numbers that could be Jared and Suni’s scores.

It’s the journey, not the destination

Because of the way they are designed, open middle math problems emphasize process over product. While students are still expected to arrive at a defensible answer, open middle problems push students to utilize a variety of mathematical skills and understandings to wrestle with the problem. The goal is not to be the quickest to follow a predetermined solution path but, rather, to allow for a more iterative approach where students continually test and revise strategies. Dan Meyer terms this “patient problem solving,” where students bring all their mathematical knowledge and intuition to bear on solving non-routine problems. The active engagement that this engenders supports rich mathematical thinking and discussion amongst students.

One of the co-developers of open middle problems, Nanette Johnson, describes their value this way: “This structure requires students to prove to themselves and others that they have truly found the best possible answer. Rather than putting down their pencils and saying, ‘I’m done,’ students continue to think, argue, and work. They develop the habit of making multiple attempts to solve the problem, each time wondering if they can come up with an even better solution than their last.”

The fact that students can arrive at different answers naturally prompts discussion between students of how they approached the problem, why they think their answer best meets the criteria, and whether other answers could get closer to the goal. As Johnson states, this serves to foster increased engagement and persistence. It also provides a vehicle for the rich mathematical discourse we were seeking in our districts.

Low floors, high ceilings, and big concepts

By design, open middle math problems allow easy entry for all students, but they also provide the flexibility to allow students to stretch themselves. For this reason, open middle problems have been described as easy in terms of getting an answer, but not so easy in terms of getting the ideal or “best” answer. Take, for example, that problem that challenges students to use the digits 1–9, at most one time each, to create two mixed numbers with the least possible difference. Students are given a template like the one on the Open Middle website in which to write their numbers.

Students who are still developing their understanding of fraction operations might take a simple approach of having the digits in the second mixed number be one less than the digits in the first mixed number: 9 5/7 – 8 4/6. This results in a difference of 1 1/21. Another student might opt to use the greatest digits available, 8 and 9, to make the denominators of the fractions, knowing that the greater number of parts a fraction is divided into, the smaller it is. Perhaps they create the expression 2 7/9 – 1 6/8 , which results in a difference of 1 1/36. At this point, the teacher might ask these two students which of their answers is the least, sparking a conversation about the relative magnitude of the numbers, benchmark fractions, and estimation. The teacher could then push students by challenging them to see if it is possible to get an answer that is less than one.

A student further along in their understanding of fractions might recognize that the need to regroup could reduce the size of the final number. Perhaps they create the problem 3 1/9 – 2 7/8 with the result of 17/72. Student interactions or well-placed teacher questions, such as, “Is it possible to get an answer of 0?” might prompt some students to realize using improper fractions can further reduce the magnitude of the answer, indeed all the way down to zero.

The nature of the problem allows all students entry without limiting students with more advanced knowledge of the involved concepts. Rich conversations and strategic teacher questions help all students expand and grow their understanding of concepts beyond the idea of fraction subtraction, while providing teachers insights into each students’ level of understanding of these concepts. In the iterative process of testing ideas, discussing different approaches, and trying new solution paths, students explore far deeper concepts of the meaning of fractions, number magnitude, and equivalent forms.

Impact of using open middle math problems

While there is no research that specifically examines open middle problems, there are studies that show the efficacy of components of the approach. In a research paper on equitable teaching approaches to math , Jo Boaler and Megan Staples call out that several studies have demonstrated that “conceptually oriented mathematics materials, taught well and with consistency, have shown higher and more equitable results for participating students than procedure-oriented curricula taught using a demonstration and practice approach.” Open middle math problems support such a conceptual orientation and, by design, break the “demonstration and practice” model.

Having students discuss and compare different solution approaches is a critical component of open math problems. Although students are not working in groups, they are also not working in isolation. Part of the iterative process of using open middle math problems includes time to share and discuss the relative merits of various solution approaches and use these to iterate new approaches. Actively comparing solution methods has been shown to increase procedural understanding , particularly for lower achieving students. Another study found that students who learned by comparing alternative solution approaches demonstrated greater conceptual knowledge and more flexible problem-solving than students who reflected on different approaches one at a time.

Open middle math problems are well aligned to NCTM’s process standards . They provide rich opportunities to practice:

• Building new mathematical knowledge through problem-solving
• Monitoring and reflecting on the problem-solving process
• Making and exploring mathematical conjectures
• Developing, evaluating, and communicating mathematical reasoning and analyzing the reasoning of others

Getting started

When working on incorporating using open middle math problems in your classroom, we encourage you to determine how to integrate them into existing math lesson plans so they aren’t one more thing you feel you have to do. Consider, for example, how open middle math tasks build off number talks but take you and your students to the next level.

Both open middle math problems and number talks encourage rich dialogue between students. While number talks are conversation-base only, open middle math problems involve both conversation and written responses. Number talks are brief 10–15-minute warmups, while open middle math problems give students more time to explore their thinking and the thinking of others. They also provide teachers with a written record of this thinking and how it evolves over the course of iterative problem-solving.

You might also try working on open middle problems yourself, which can help you understand the power of the approach.

Adding doesn’t necessarily mean more

We also encourage you to focus on potential roadblocks to making open middle math problems a part of your classroom. In our experience, time can be a big concern. If that’s worrying you as well, we suggest you focus on the relative merit of having students deeply engage in a single rich mathematical problem in place of completing 10 rote problems. This can help you move away from a “mile-wide, inch deep” approach to math.

Remember, too, that these problems do not need to be completed in a day. Take the time to let students work through possible solutions, share, discuss, and revise over the course of one or more days.

Finally, time you previously spent looking for better resources could be repurposed into unpacking open middle math problems to prepare you for facilitating a rich classroom discussion. Here are some questions to get you started:

• What big mathematical ideas do you hope students will bring into discussions?
• What models or mathematical terms might students use in their explanations?
• What are examples of clear justification or correct reasoning you hope to see?
• What are some possible solution approaches and what do each of them reveal about students’ understanding of the underlying concepts?

Give it a try!

Adding open middle tasks to your math routines can truly deepen both math conversations and student engagement in your classroom. Ready to try them yourself? Use these links to get more information about open middle math problems and supporting mathematical writing and discourse.

Open middle resources

• Open Middle This site, created by Open Middle co-creators Nanette Johnson and Robert Kaplinsky, contains a wealth of open middle tasks organized by grade and domain as well as worksheets in English, French, and Spanish and printable number tiles.
• Open Middle Exercises GeoGebra has converted many of the paper versions of open middle exercises into interactive versions.
• Open Middle Math: Problems that Unlock Student Thinking, Grades 6–12 You can preview Robert Kaplinsky’s book about open middle math here.

Classroom discussion and mathematical writing resources

• “4 ways to engage students with writing in math class” This article delves into a different way to support and encourage student writing about math, including how to write high-quality explanations for how they solved a problem.
• “Eliciting, supporting, and guiding the math: Three key functions of the teacher’s role in facilitating meaningful mathematical discourse” This piece defines what makes math discourse meaningful, why it’s important, and what the teacher’s role is in facilitating such discourse.
• “Mathematical discourse in the classroom: A prime time for discussion” This article provides strategies for facilitating meaningful mathematical discussion with the goal of helping students develop a deeper understanding of mathematical concepts.
• “Using the 5 practices in mathematics teaching” This article provides an overview of how teachers can successfully orchestrate classroom discussions by anticipating possible student solutions, monitoring their work, selecting and sequencing how students share their work, and asking questions that help students make connections between mathematical ideas.

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