problem solving organizer examples

• 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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15 Free Online Graphic Organizer Types and Examples

Graphic organizer plays a significant role in a presentation or learning materials like books. It is characterized as a convenient means to identify key points and understand concepts in a subject area. In addition, this allows you to effectively create meaningful structures, make relationships of ideas, and have precise information. But depending on the context, the organizers have a variety of uses. Thus, here are some free graphic organizer types and examples to design perfect diagrams or charts according to your needs.

Types of Graphic Organizers to Visualize Ideas

Concept map, hierarchy chart, cause and effect map, timeline chart, venn diagram, sequence chart, idea web graphic chart, analogy organizer, cluster diagram, persuasion map, problem-solving organizer.

You can also use a concept map to brainstorm ideas, discover a number of thoughts, solve problems, and visualize goals. Anyone can utilize this tool including designers, writers, engineers, and students for learning purposes. Similarly, you can branch out many ideas as needed and most likely to end up looking web graphic organizer. In other words, this is suitable for large and complex topics.

Depending on your needs, the hierarchy chart comes with a variety of uses. It is utilized primarily used by Human Resources in creating an organizational chart. The same can be said when visualizing ideas. This graphic organizer lets you understand categories and subcategories by using the top-down structure of the hierarchy chart.

Cause and effect map assists in clearly determining the possible causes of a specific event, problem, or condition and take action to solve the issues. More so, it has a fish skeleton structure in which the bones represent the causes. The fish head to the right should reflect the main effect. Using this diagram, members can also participate in a team discussion when analyzing problems and take corrective actions.

A mind map is another helpful graphic organizer to show relationships between ideas. Instead of sentences, mind map ideally uses keywords per association to summarize information making it easier to memorize and recall. Besides, it allows users to be creative by integrating colors, font style, images along with several kinds of layout structure for a comprehensive mind map.

Another idea organizer that you can use to make a comparison of two options is the T chart. There will be two columns separated by a vertical line in between and a horizontal line at the top allotted for the headings or topics. Under each column are the characteristics, specs, events, situations, etc. This graphic free graphic organizer could help you to make clear decisions by comparing the pros and cons of a given occasion.

This chart is a type of diagram that shows the chronological order or sequence of events. Besides its beneficial impact on studying history, this may also come in handy if you’re trying to display development, progress, or improvement of a business. On some occasions, this graphic organizer is visually presented as a freeform shape but is ideally visualized horizontally or vertically.

Venn diagram helps you visualize comparisons, differences, as well as similarities between two subjects. This has an interesting way of showing differences and similarities. Basically, creates an illustration of two overlapping circles where the similarities are located in the union of two circles or common area. While the unique characteristics or differences are inside its own area or sector. There are several types of Venn diagrams, online graphic organizers, with two or more circles as the topic gets complicated.

Tree chart is a good way to classify information and a practical means in breaking down large projects into more manageable tasks. In addition, this can also be used to create a family tree chart for genealogy and organizing family information. The highest position is the main title and below are the categories. Each category has a list of subtopics to complete the tree graphic organizer.

Sequence chart is a type of diagram that helps you to present the steps of a process in order. You may use it for essay writing, life cycle, or any situation that shows the stage of development. While the timeline chart shows dates and order of events, the sequence chart graphic organizer mainly focuses on details about the set of instructions of a process.

If you are looking for an alternative to T chart and Venn diagram, Idea web graphic chart is nice choice. This is also a type of comparison chart between two topics. There will be two central circles that represents the main ideas. Each topic comes with conjoined branches that forms a single unit to indicate similarities. For the differences, there are another set of branches towards the sides of the circles forming an idea web graphic organizer.

Organizing or conducting biography research can be done with the help of a biography chart. With it, you can easily add the necessary information gathered in books, online resources, and analyze the collected information. You can categorize the data about the character’s accomplishments, experiences, personal information, and a lot more. Also, users may add images to this free graphic organizer to give emphasis as necessary.

Another helpful organizer to help you analyze differences and similarities between two topics. Analogy organizer, in general, compares a new topic and a topic you are familiar with. Then, students or listeners must read and discuss with each other to identify the similarities and differences. This free graphic organizer encourages brainstorming and internalizing knowledge. By then, students will need to write a summary of the new topic.

The next chart you can use to organize your ideas is a cluster diagram. It also helps a group of people in brainstorming and exploring ideas. The central node contains the main topic branching out to show relevant categories and subtopics. This also resembles a web graphic organizer as you continue adding layers to the diagram. So it’s best to use color-coding for each category for a readable and comprehensive organizer.

This organizer is dedicated to creating an essay outline focusing on persuasive writing. As we know, writings like speeches, essays, debates need an outline to lay out the flow of the essay and come up with a coherent outline. And to do that, you need a persuasion map free graphic organizer to determine and write down the main points, facts, and categorize ideas.

Problem-solving organizer is a good strategy to improve your problem-solving skills. With it, you get to record ideas in a systematic way and think more effectively of possible solutions. It is also the best way to make everyone in the team participate by sharing ideas and organize them in structure.

Visual representation in learning plays an important role in improving your study habits and learning. This will help your capability to arrange your thoughts and organize them and give an accurate depiction of given situations. Learning through the use of online graphic organizers will bring out great ideas to have your tasks done in no time.

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Top 5 Math Graphic Organizers

Our kids struggle with math from time to time. The subject gets increasingly more complex the older our kids get. Graphic organizers can help math students slow down to think through a problem in its component parts and, therefore, enhance their problem-solving skills.

How to use math graphic organizers

Graphic organizers help students to logically process the information. Often, by following a step by step process, students find they don’t get lost in the math. A graphic organizer helps students organize the thought process as well as provide a framework to collect, visualize and organize the math information to work through the problem. 

Depending on the type of math problem your child is tackling, there are many different graphic organizers to use. A search on the internet for math graphic organizers is a truly overwhelming experience for parents. Let us try to make your job a little easier. We’ve created five graphic organizers we believe your elementary school children would rely on the most.

Word problem graphic organizer

Word problems are inherently difficult for most students as these problems involve working out what exactly is being asked of them and to apply the correct math equations. This word problem graphic organizer helps students slow down the process and think through the problem carefully. 

Four operations graphic organizer

When solving large number equations, such as 3-digit by 2-digit multiplication, or adding 4-digit by 3-digit numbers, many students have trouble with lining up the numbers properly. Using these grids , students ensure there is one number per box and that they solve each row by their place values (ones, tens, hundreds, thousands, etc.)

Step by step graphic organizer

As students move up the grades, they will find that math problems are solved in many steps. This step by step graphic organizer helps them break down the math problem in those steps and to visualize the process to solve the problem correctly.

Frayer Model graphic organizer

Learning new words or concepts in math is an important part of understanding math. This graphic organizer was developed by Dorothy Frayer and her colleagues at the University of Wisconsin. The model helps students draw on their current knowledge of math terms to build connections to new terms. 

Number lines graphic organizer

Students use number lines to understand number sequences and to visualize counting, comparing, adding, subtracting, multiplying or dividing numbers. Used by younger students for simple whole number counting and math equations, older students continue to use numbers lines for working out equations of negative to positive numbers, as well as for fractions and decimals.

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Problem Solving Map

A graphic organizer that guides the process of problem-solving by identifying the problem, brainstorming possible solutions, and evaluating the pros and cons of each, in order to determine the preferred solution.

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• Problem solving /
• comprehension /
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• story map /

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The graphic organizers are available in Book Creator’s built in App Store . Sign into Book Creator, simply enable the graphic organizer app, then you and your students can add your chosen organizers to your books.

10 graphic organizers are available to all users, with the full set of over 100 organizers available to teachers on a premium or schools and district plan.

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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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8 Graphic Organizers to Improve Learning

A graphic organizer can help students catalog ideas and become stronger writers.

What is a Graphic Organizer?

Why use graphic organizers, different types of graphic organizers.

There are many different forms of graphic organizers, each with its own unique way of organizing and displaying specific bits of information. Read on to learn about the different types of graphic organizers that teachers utilize to help students better organize their thoughts, develop reading strategies, and become stronger writers.

Parents / Julie Bang 

Graphic organizers are visual and graphic displays that organize ideas and demonstrate relationships between different information and concepts.

Graphic organizers are designed to improve learning outcomes for students, help students review information, and are especially helpful to those who struggle with arranging information. It's a great method to keep track of information.

A T-Chart graphic organizer helps students place ideas into two columns and examine two components of an object, concept, or event. For instance, T-charts can be used in any content area to examine the following:

• Pros and cons
• Advantages and disadvantages
• Facts and opinions

Students can select two things to compare (ideas, characters, or events, for example) and write them as headings for the two columns. Then, they can make comparisons or contrasts in both columns.

Concept map

A concept map shows relationships between a main idea and supporting information. Concepts or ideas are represented in circles or boxes and are linked to related ideas with arrows. Most concept maps represent a hierarchical structure, with the most general concepts or ideas presented at the top of the graphic organizer and the more specific sub-ideas presented below.

Another feature of a concept map is the inclusion of cross-links to demonstrate relationships between sub-ideas in different segments of the concept map.

Main idea web

The main idea web starts with a central idea and branches out into related ideas and details (or sub-ideas). Some other names for this type of graphic organizer include:

• Semantic maps

It is used primarily for brainstorming and generating ideas for planning or writing purposes.

Star diagram

Star diagrams are similar to main idea webs; however, this type is typically used to catalog characteristics or traits. For example, star diagrams can help learners create a biography of a historical figure. The subject's name would be in the center of the star, with facts or traits about them located in the star's points.

To make a star diagram, follow these steps:

• Draw a star on a page.
• Use as many points as needed.
• Write the main idea, event, person, or subject in the center of the star.
• Fill in each point in the star with various key traits or facts about the central topic.

Venn diagram

A Venn diagram is used to compare and contrast two or more groups of things by visually displaying their similarities and differences in two or more circles that overlap. Similarities between topics are presented where a circle from one category overlaps with a circle from another category.

Features that do not fit in both categories are placed where the two circles do not overlap. This type of graphic organizer can serve to increase understanding of a relationship between two or more concepts.

Sequence chart

A sequence chart (or flow diagram) presents a series of steps or events in order. Many learners need a visual aid to help clarify a sequence of events in a story or to draw conclusions about different cause-and-effect (or problem-and-solution) relationships between multiple events in a text.

Students can use this type of graphic organizer as a flowchart to organize thoughts as a prewriting activity. It can also be used as part of a classroom activity that makes students responsible for a portion of a classroom assignment (for example, the jigsaw technique).

A cycle diagram is a type of sequential chart used to represent a sequence of stages, tasks, or events in a circular flow. This kind of diagram accentuates the flow and interconnection between things rather than emphasizing the actual stages or steps.

Hexagon organizer

In a hexagon idea organizer, hexagons are arranged on the page with straight edges connected. Facts, ideas, or concepts are put inside hexagons, with related information put in hexagons that are touching each other.

This type of graphic organizer helps learners make sense of how things relate to each other and how ideas unfold and build upon each other. Students can use this information to inform their conclusions and understanding of complex topics.

Problem-solving chart

A problem-solving chart helps students think through issues to find potential solutions. Here is how it works:

• Learners start with a problem, which is written in a box at the top of the page.
• They then fill in a subsequent box with why it's a problem.
• The next boxes contain possible solutions.
• The brainstorming boxes are divided into sections that allow space for possible pros and cons of each idea.
• Lastly, the most promising solution is highlighted at the end.

Problem-solving charts help students brainstorm and learn to evaluate potential solutions for a variety of problems. They also promote critical thinking skills as kids compare the benefits or drawbacks of various options and select the most promising ones.

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Problem Solution Mapping — Everything You Need to Know!

Edraw content team, do you want to make your problem solution mapping.

EdrawMax specializes in diagramming and visualizing. Learn from this article to know everything about what is a problem solution map, problem-solving process, how to make your problem solution map. Just try it free now!

When facing a problem, you have two options. You can either continue untangling the problem in your mind. Or, you can put it all out in a problem-solution map and save time and energy. A problem-solution map helps you find the root cause of the problem. Then, when you know the reasons, you can also find your ultimate solution.

So, let's explore more about problem and solution maps and elements of the problem-solving process in the following guide. We'll also learn about EdrawMax - a free online software to fulfill all your diagramming needs!

So, let's dive into it!

1. What is a Problem Solution Map

A problem solution map represents a logical analysis of available data leading towards a solution. In other words, a problem solution map uses the given data to provide a solution to the problem. A straightforward solution map can have various elements. Typically, it has the following structure:

• a problem statement
• problem definition
• possible causes and effects
• an effective solution

You can change the structure of a solution map according to the problem. However, it should include all the possible laws and principles at all stages to find a practical solution. Solution mapping is crucial and better than thinking about a solution. It's because solution mapping makes you visualize your problem and simplify it.

You consider all the possible reasons behind the problem and milestones to achieve to solve it. As a result, your thoughts become organized and clearer in a visual representation. It also saves your time and effort and increases your productivity.

2. Problem-Solving Process

Practical problem-solving is more than just visualizing and figuring out the solution. It is a process with several steps to guide you to the optimal solution. It includes:

• Problem definition
• Problem analysis
• Possible solutions
• Analysis of the solutions
• Select the best solution
• Execute the solution

Let's explore these!

Problem Definition

Before you start working on your problem solution map, you should define your problem. It might surprise you but defining a problem is more challenging than just writing it on paper. It's because the main problem is often a vague idea merged with unnecessary, distracting conflicts. So, please understand the problem and clear up its confusion.

Problem Analysis

After defining the problem, the next step is to analyze it. When deciding if a solution will truly work, it is vital to recognize where the problem begins, how it fits with the latest advancements, and what the present environment is.

For instance, the following is a problem solution map example. It is a problem tree designed on EdrawMax displaying the causes behind poor health conditions among Gapachi Vileans. You can tell a problem's roots from its consequences (branches) by comparing them to a tree (trunk).

Possible Solutions

In this step, you should focus on solutions. However, it is essential to come up with as many solutions as possible without evaluating them. You should consider it a solution, even if it seems vague. So, think for some time and list all the ideas that come to mind as solutions to the problem.

Analysis of the Solutions

Here, you will go back to all the solutions you wrote in the previous step. Then, you will consider each idea individually and investigate it. When analyzing an argument, assess its positive and negative aspects and whether it addresses the problem entirely. You can also write unique benefits of each idea to differentiate it from others.

For example, consider this Graphic Organizer: Problem Solution diagram by EdrawMax. It simplifies finding a solution as you can write all the possible choices and their pros and cons. Then, you can select the most effective solution as the final one.

Select the Best Solution

After writing the pros and cons of all the ideas side by side, you need to filter the best solutions from the list. You can also improve it by revising the ideas and refining their individual traits. In the end, you might have one or more solutions (the best ones!). If not, consider reevaluating the problem statement or repeating the 3rd and 4rth steps.

Execute the Solution

Once you have come up with the solution, the final step is to execute it. Then, consider what you should do next to follow that particular solution. In other words, you need to plan your solution to solve the problem.

3. Visualize Your Problem Solution Map Process

The best thing about a solution map is that it is open-ended. However, it also means no hard and fast rule or rocket science behind creating a solution map .

Still, creating a problem and solution map from scratch takes a lot of work. The good thing is that you can always take advantage of powerful software like EdrawMax . It is an incredibly powerful online service to tackle all your diagramming needs.

EdrawMax has all the features you need to visualize your problem-solution map. For instance:

• You can insert tables, charts, timelines - anything that can help define and analyze the problem and/or solutions.
• You can add, remove, replace, and do much more with symbols and shapes through Predefine Libraries in EdrawMax.
• You can use the Connector feature and drag it to draw a line (arrows) to a connection point on the shape you want to connect to.
• You can edit color, line weight, dash style, and arrow style using the Line or Line Style feature to customize your solution map.

What else? You can also find hundreds of ready-made solution map templates on the EdrawMax community. These editable templates make everything easy, as you can edit them according to your preference. Creating a solution map from scratch takes less time and effort.

4. Conclusion

Every problem has a solution, but finding that solution can be challenging. The road map to finding the ultimate solution is often chaotic and messy. With a problem and solution map , you can visualize your problem, its root causes, and possible solutions. A problem solution map is a way to analyze the available data and find the perfect solution to solve the problem. With EdrawMax , you can easily create your problem solution map. Or you can also save time by editing an already published template on the EdrawMax community!

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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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Problem and Solution Map can be useful for students to compare different solutions to a problem

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Free Editable Brainstorming Graphic Organizer Examples

1. what is brainstorming graphic organizer.

A brainstorming graphic organizer is a visual representation of ideas and concepts that help create new ideas, solve problems, organize information, and communicate with others. Teachers use these graphic organizers to help students improve their writing skills and organize their ideas. It also helps with idea generation and finding creative solutions for complex problems. Professionals use these organizers to plan their research and create presentations. You can also use it to get out of writer's block and develop fresh ideas.

How to use it?

• Start with picking a graphic organizer template and edit it with a diagramming software such as EdrawMax Online. Or you can grab a blank paper to get started.
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2. The Brainstorming Graphic Organizer Examples

You can use a brainstorming graphic organizer to develop creative ideas for your writing or to brainstorm your thoughts to reach an efficient solution for your problem. Graphic organizers for mapping out ideas and brainstorming concepts come in handy for many things. Based on your subject matter, you can use various formats of brainstorm graphic organizers such as a web chat or a cerebral chart. Our brainstorming graphic organizer examples will help you understand its different formats and uses. Check out our examples and pick a suitable template.

Example 1: Brainstorming Graphic Organizer

A brainstorming graphic organizer is an excellent tool for generating ideas and logically organizing information. Teachers use these organizers to help their students develop out-of-the-box thinking and solve any problem by focusing on the broad picture. The template in this example is a standard graphic organizer to brainstorm ideas. After reading the subject topic, students write whatever comes to their mind and organize these ideas to solve the problem. There are five blank boxes in the template to write their ideas.

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You can use a brainstorming graphic organizer PDF to brainstorm ideas for your reports, make presentations, and write stories and essays. Writers and professionals use this graphic organizer because, in PDF format, you can add a lot of writing space to house multiple ideas. Use it when you don't know how many words you will write in the graphic organizer or whether you will add images. The template in this example follows the standard format, and the topic title goes into the bulb at the center while you write related ideas in the five clouds surrounding it.

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With a brainstorming web graphic organizer, you can list your ideas related to the main topic in the shape of a web. It also helps you organize information about your topic, such as listing attributes, facts and examples on the web to visualize a connection between related themes and find an efficient solution. This graphic organizer comes in a standard format, as you can see in the template. The main idea or title of your topic always goes in the middle. You can add any ideas or details around the main idea in a circle and use lines to connect it with the center. It forms a web image and makes it easier to trace your idea back to the title.

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A brainstorming graphic organizer for writing helps you brainstorm ideas and details about any topic to develop a creative writing framework and escape the writer's block. Teachers use this graphic organizer to help improve their students' reading and writing skills. The template format in this example is simple, and anyone can use it. Students write various ideas for a single topic on the left side of the template. After that, they can add their thoughts and related details about their opinions on the right side to compare which one is better.

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The most common personal narrative brainstorming graphic organizer is a spider map that we can use to plan out writing structure and develop fresh ideas for any subject area. The format of a spider map is similar to a graphic organizer web. The main idea goes at the center, and you can add related themes and various elements or details in a circle. Each idea or element connects directly to the central idea, making it easier to focus on the broad picture while solving problems.

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An essay brainstorming graphic organizer is an excellent tool to outline essays and create a framework for text writing. You can use it when you have problems coming up with fresh ideas, or there are too many ideas in your mind, and you can't decide which one is better. The template in this example is simple and easy to understand. You got the essay topic at the center, where you can also write a little intro to your writing. In the four circles, you will write the primary supporting details. Each circle connects to a specific box where you can write supporting sentences.

Example 7: Student Brainstorming Graphic Organizer

The student brainstorming graphic organizer comes in simple formats compared to professionals' use to make reports and presentations. The purpose of using a graphic organizer is to make brainstorming easy, not to give students a complex blank template that they can't understand. The main idea in the student graphic organizer is at the center, and students can add definitions and attributes in a circle surrounding the main idea. They can also use to organize information logically.

Example 8: Brainstorming Graphic Organizer for Problem Solving

A brainstorming graphic organizer for problem-solving helps you find creative solutions for complex problems. It uses the pros and cons of each idea to determine which one is more useful and related to the solution area of your subject. You can also brainstorm related themes, break down your problem to the lowest level, and find a solution. The problem title goes at the top, and you can list three or more solution choices in the three boxes under the problem title. After that, you can list the pros and cons of each option to find an efficient solution.

Example 9: Graphic Organizer for Brainstorming

A graphic organizer for brainstorming is mainly used to generate creative ideas and for problem-solving. It is also a helpful tool to depict the relationship between various elements related to a similar idea or subject topic. These graphic organizers also help students improve their learning and develop critical thinking. The template in this example is identical to a cerebral chart that uses hierarchies to organize information and represent ideas. The main idea goes to the top, and you can add supporting ideas and concepts to the next level and give details to compare which one is better.

3. Online Brainstorming Tool

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4. Key Takeaways

A brainstorming graphic organizer helps you with concept mapping and generating ideas related to any topic or solving complex problems. Teachers use this tool to help their students improve their narrative writing skills. As there is no specific format for narrative writing, a graphic organizer comes in handy to do that. Professionals also use it to brainstorm their ideas to make reports, product planning and prepare presentations. EdrawMax is the best online brainstorming tool. With EdrawMax, you can create any graphic organizer online to save your time and effort. It also comes with a strong templates community and supports various document formats. Find more graphic organizer templates in the templates community.

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