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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE : 8 Common Core math examples

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies of problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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Math Problem Solving Strategies

In these lessons, we will learn some math problem solving strategies for example, Verbal Model (or Logical Reasoning), Algebraic Model, Block Model (or Singapore Math), Guess & Check Model and Find a Pattern Model.

Related Pages Solving Word Problems Using Block Models Heuristic Approach to Problem-Solving Algebra Lessons

Problem Solving Strategies

The strategies used in solving word problems:

• What do you know?
• What do you need to know?
• Draw a diagram/picture

Solution Strategies Label Variables Verbal Model or Logical Reasoning Algebraic Model - Translate Verbal Model to Algebraic Model Solve and Check.

Solving Word Problems

Step 1: Identify (What is being asked?) Step 2: Strategize Step 3: Write the equation(s) Step 4: Answer the question Step 5: Check

Problem Solving Strategy: Guess And Check

Using the guess and check problem solving strategy to help solve math word problems.

Example: Jamie spent $40 for an outfit. She paid for the items using $10, $5 and $1 bills. If she gave the clerk 10 bills in all, how many of each bill did she use?

Problem Solving : Make A Table And Look For A Pattern

• Identify - What is the question?
• Plan - What strategy will I use to solve the problem?
• Solve - Carry out your plan.
• Verify - Does my answer make sense?

Example: Marcus ran a lemonade stand for 5 days. On the first day, he made $5. Every day after that he made $2 more than the previous day. How much money did Marcus made in all after 5 days?

Find A Pattern Model (Intermediate)

In this lesson, we will look at some intermediate examples of Find a Pattern method of problem-solving strategy.

Example: The figure shows a series of rectangles where each rectangle is bounded by 10 dots. a) How many dots are required for 7 rectangles? b) If the figure has 73 dots, how many rectangles would there be?

a) The number of dots required for 7 rectangles is 52.

b) If the figure has 73 dots, there would be 10 rectangles.

Example: Each triangle in the figure below has 3 dots. Study the pattern and find the number of dots for 7 layers of triangles.

The number of dots for 7 layers of triangles is 36.

Example: The table below shows numbers placed into groups I, II, III, IV, V and VI. In which groups would the following numbers belong? a) 25 b) 46 c) 269

Solution: The pattern is: The remainder when the number is divided by 6 determines the group. a) 25 ÷ 6 = 4 remainder 1 (Group I) b) 46 ÷ 6 = 7 remainder 4 (Group IV) c) 269 ÷ 6 = 44 remainder 5 (Group V)

Example: The following figures were formed using matchsticks.

a) Based on the above series of figures, complete the table below.

b) How many triangles are there if the figure in the series has 9 squares?

c) How many matchsticks would be used in the figure in the series with 11 squares?

b) The pattern is +2 for each additional square.   18 + 2 = 20   If the figure in the series has 9 squares, there would be 20 triangles.

c) The pattern is + 7 for each additional square   61 + (3 x 7) = 82   If the figure in the series has 11 squares, there would be 82 matchsticks.

Example: Seven ex-schoolmates had a gathering. Each one of them shook hands with all others once. How many handshakes were there?

Total = 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes.

The following video shows more examples of using problem solving strategies and models. Question 1: Approximate your average speed given some information Question 2: The table shows the number of seats in each of the first four rows in an auditorium. The remaining ten rows follow the same pattern. Find the number of seats in the last row. Question 3: You are hanging three pictures in the wall of your home that is 16 feet wide. The width of your pictures are 2, 3 and 4 feet. You want space between your pictures to be the same and the space to the left and right to be 6 inches more than between the pictures. How would you place the pictures?

The following are some other examples of problem solving strategies.

Explore it/Act it/Try it (EAT) Method (Basic) Explore it/Act it/Try it (EAT) Method (Intermediate) Explore it/Act it/Try it (EAT) Method (Advanced)

Finding A Pattern (Basic) Finding A Pattern (Intermediate) Finding A Pattern (Advanced)

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Problem Analysis in Math: Using the 5-Step Problem-Solving Approach

This blog will explore the ways in which problem analysis of student mathematics difficulties can be applied within a problem-solving approach. We will review the core principles of a multi-tier system of support (MTSS) framework, identify the steps within a problem-solving approach, and explore the ways in which problem analysis helps to inform intervention development within the context of mathematics instruction.

Core Principles of an MTSS Framework

A variety of definitions of an MTSS framework exist within the field of education; however, several common principles are apparent and have helped to shape much of the work within this area. The National Association of State Directors of Special Education (Batsche et al., 2006) define eight core principles that capture some of the most important aspects and core beliefs of an MTSS framework:

• We can effectively teach all children.
• Intervene early.
• Use a multi-tier model of service delivery.
• Use a problem-solving model to make decisions within a multi-tier model.
• Use scientific, research-based validated intervention and instruction to the extent available.
• Monitor student progress to inform instruction.
• Use data to make decisions.
• Use assessment for screening, diagnostics, and progress monitoring.

The fourth core principle refers to utilizing a problem-solving model to make decisions. More specifically, educators and administrators should use a clearly defined problem-solving process that guides their team in identifying the problem, analyzing the size and effect of the problem, developing a plan for intervention to address the problem, implementing the plan, and examining the effectiveness of the intervention plan.

5-Step Problem-Solving Approach

The problem-solving approach utilized by the FastBridge Learning ® system includes the following five steps:

Problem identification

• Problem analysis
• Plan development
• Plan implementation
• Plan evaluation

Following this 5-step problem-solving approach helps to guide school teams of educators and administrators in engaging in data-based decision making: a core principle of the MTSS framework.

The first step in this problem-solving approach is problem identification. Christ and Arañas (2014) define a problem as a discrepancy between observed and expected performance. Regularly scheduled universal screening plays an important role in problem identification. The FastBridge Learning ® system offers a variety of screening measures for mathematics, which are summarized in the table below. While the results of regularly scheduled universal screening help to inform whether or not a discrepancy between observed and expected performance exists, problem analysis aims to identify the size and effects of the problem.

Problem Analysis

After a problem has been identified, the problem must be defined through the method of problem analysis. Only after a problem has been sufficiently analyzed and defined, can the significance of a problem be understood (Brown-Chidsey & Bickford, 2016). Problem analysis can occur at both the individual level and the group level.

Problem Analysis at the Individual Leve l

At the individual level, a discrepancy between observed and expected performance may appear within an individual’s universal screening results. FastBridge Learning ® Individual Skills Reports can provide detailed information to support problem analysis. The Individual Skills Report provides a snapshot of a given student’s risk, relative to benchmark goals, and also provides detailed results on an item-by-item basis. This item-by-item analysis provides insight into the skills the student demonstrates and which skills may require additional instructional support.

If the student’s performance is relatively close to the benchmark goal, the problem is likely to be understood as a minor problem. Item-by-item analysis can help a teacher determine if targeted reteaching of specific content may help the student to reach the goal, or if more intensive intervention may be necessary. An example of a FastBridge Learning ® Individual Skills Report for the earlyMath subtest Numeral Identification (Kindergarten) may be seen below.

The above report suggests that the sample student is at “some risk” for difficulty with numeral identification. It also reveals that the student was able to identify the given numerals with 91% accuracy. Additionally, the item-by-item analysis indicates that the student struggled to identify the following numerals: 12, 14, and 19. In this case, problem analysis may suggest that this is a minor problem, which may be remedied through targeted reteaching of the numerals 12, 14, and 19. In contrast, if during problem analysis, an Individual Skills Report for Numeral Identification indicated very low accuracy and a substantial number of misidentified numerals, the results would suggest a more significant problem. Significant problems warrant planning for more intensive intervention.

Problem analysis at the group level

At the group level, problem analysis seeks to determine the size and effects of a problem at the class, grade, or school-wide level. FastBridge Learning ® Group Screening Reports and Group Skills Reports help to provide insight about groups of students at risk for learning difficulties. An example of a Group Screening Report for earlyMath may be seen below.

The Group Screening Report pictured above indicates that approximately 77% of Ms. Horst’s kindergarten class scored at or above the benchmark goal on the earlyMath Composite during the fall benchmark period, while 6% and 18% were identified as being at “some risk” and “high risk” respectively.

All students should demonstrate growth in math achievement throughout the school year. If there are students who met the fall benchmark for mathematics, but not the winter and/or spring benchmarks, problem analysis must occur. In Ms. Horst’s class, we can see by the spring benchmark period, only 12% of her kindergarten class scored at or above the benchmark goal on the earlyMath composite, while 18% and 71% were identified as being at “some risk” and “high risk” respectively.

Problem analysis at the group level may aim to answer questions such as the following:

• What type of mathematics instruction was provided for these students?
• Did other kindergarten classrooms within the school experience a similar pattern of mathematics difficulty?
• With which specific mathematics skills (e.g., decomposing, number sequencing, numeral identification) did the group demonstrate success and difficulty?
• How close to the end-of-year mathematics goals are the students now?

In order to complete a thorough problem analysis, supplemental data about student performance and instructional practices may need to be collected.

For instance, classroom observations and/or a teacher interview would provide insight into the first question. Group Screening Reports for the other kindergarten classrooms in the school could help identify if this is a schoolwide pattern of difficulty or a classroom-specific pattern of difficulty. An analysis of the Group Skills Report for Ms. Horst’s class would provide insight into students’ strengths and weaknesses across the earlyMath Composite’s subtests. Additional, targeted, follow-up assessment using selected FastBridge Learning ® earlyMath subtests could help to gather information about the final question.

The goal of problem analysis is to determine the significance of a problem and to develop a hypothesis about why a problem has occurred. The information gathered during the problem analysis stage is then used to inform the third step of the 5-step problem-solving approach: plan development.

Batsche, G., Elliott, J., Graden, J. L., Grimes, J., Kovaleski, J. F., Prasse, D., Schrag, J., & Tilly, W.D. (2006). Response to intervention: Policy considerations and implementation. Alexandria, VA: National Association of State Directors of Special Education, Inc.

Brown-Chidsey, R. and Bickford, R. (2016). Practical handbook of multi-tiered systems of support: Building academic and behavioral success in schools. New York, NY: The Guilford Press.

Christ, T.J., & Arañas, Y.A. (2014). Best practices in problem analysis. In A. Thomas & J. Grimes (Eds.), Best Practices in School Psychology VI . Bethesda, MD: National Association of School Psychologists.

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What Is a Bar Model? How to Use This Maths Problem-Solving Method in Your Classroom

Written by B

Teaching mathematical problem-solving to your students is a crucial element of mathematical understanding. They must be able to decipher word problems and recognise the correct operation to use in order to solve the problem correctly (rather than freaking out as soon as they see a mathematical problem in words). But what is the best strategy for embedding this knowledge into your students? Enter the Bar Model Method , also known in education as a tape diagram, strip diagram, or tape model.

The model has gained traction in classrooms around the world, so you may be thinking it’s time to introduce it in your classroom. But what is the bar model all about, and what are the advantages of using this method with primary students? The maths teachers on the Teach Starter team are here with a quick primer to help you decide if this is the right path for your maths class!

Explore the latest maths teaching resources from Teach Starter!

What Is a Bar Model?

The Bar Model is a mathematical diagram that is used to represent and solve problems involving quantities and their relationships to one another.

It was developed in Singapore in the 1980s when data showed Singapore’s primary school students were lagging behind their peers in math. An analysis of testing data at the time showed less than half of Singapore’s students in years 2-4 could solve word problems that were presented without keywords such as ‘altogether’ or ‘left.’ Something had to be done, and that something was the introduction of the bar model, which has been widely credited with rocketing the kids of Singapore to the top of maths scores for kids all around the globe.

At its core, the bar model is an explicit teaching and learning strategy for problem-solving. The actual bar model consists of a set of bars or rectangles that represent the quantities in the problem, and the operations are represented by the lengths and arrangements of the bars. Among its strengths is the fact that it can be applied to all operations, including multiplication and division . They’re also useful when it comes to teaching students more advanced maths concepts, such as ratios and proportionality.

The Bar Model combines the concrete (drawings) and the abstract (algorithms or equations) to help the student solve the problem.

How to Use a Bar Model in Math

Whether you’re calling it a bar model, a strip diagram, or a tape diagram, the concept is the same – you have rectangular bars (or strips) that are laid out horizontally to represent quantities and the relationships between them.

• The bars themselves — horizontal rectangles—  represent the problem.
• The length of the bar(s) represents the quantity.
• The locations of the bars show the relationship between the quantities.

Visualising this relationship helps students decide which operation to use to solve the problem. The student then labels the known quantities with numbers and labels the unknown quantities with question marks.

The three basic structures are:

• Part-Part-Whole
• Equal Parts

Our printable  Bar Model Poster Pack is a perfect way to introduce your students to the basics of the Bar Model!

Many teachers (particularly in the primary years) will recognise elements of the Bar Model as being similar to the Part-Part-Whole method we’ve been teaching forever.

The Bar Model could be looked at as an extension of this concept. It can be used by students right through primary school (and beyond) not only to solve addition and subtraction problems but to tackle multiplication and division word-based problems as well.

Teaching with the Bar Model

The Bar Model can easily be incorporated into your primary maths instruction, from simple addition to more complex multiplication and division and so on. Here are just a few ideas from our teacher team:

• Use part–whole bar models to show word problems with a missing number element to teach addition or subtraction.
• Incorporate into your  CUBES strategy  — The E in CUBES stands for Evaluate and Draw. As a Bar Model is a drawing, this is the perfect place to use it.
• Use the bar model method to help students see how a bar must be cut into equal parts when multiplying and dividing.
• Help students visualise  fractions with the bar model — Use the rectangles to help students see how the fractions relate to whole numbers by showing the relationship between the numerator and denominator.

Explore our complete collection of curriculum-aligned resources for teaching about operations !

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One of them was literally The Singapore Bar Model and I can’t remember the other. I’ve moved schools in between time and left the books at my previous school.?

Thanks for your comment Karen. We hope this blog was helpful.

I'm so pleased to see this on Teach Starter. I have been 'fiddling' around with this strategy for a couple of years now. We have recently bought some books to help us finally do this a bit better.

Hi, Karen! Thanks for your positive comment. It's a really interesting strategy and it's just starting to pick up popularity. I'd love to hear which books you've bought!

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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.

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Explicitly Model Mathematics Concepts/Skills & Problem Solving Strategies

What is the purpose of explicitly modeling mathematics concepts/skills and problem solving strategies.

The purpose of explicitly modeling mathematics concepts/skills and problem solving strategies is twofold. First, explicit modeling of a target mathematics concept/skill provides students a clear and accessible format for initially acquiring an understanding of the mathematics concept/skill. Explicit modeling by you provides students with a clear, accurate, multi-sensory model of the skill or concept. Students must first be able to access the attributes of a concept/skill before they can be expected to understand it and be able to use it in meaningful ways. Explicit teacher modeling does just that. Second, by explicitly modeling effective strategies for approaching particular problem solving situations, you provide students a process for becoming independent learners and problem solvers. While peers can sometimes be effective models for students, students with special needs require a well qualified teacher to provide such modeling, at least in the initial phases of instruction.

What is Explicit Modeling?

Explicit modeling involves well-prepared teachers employing a variety of instructional techniques to illuminate the key attributes of any given mathematics concept/skill. In a sense, you serve as a "bridge of learning" for your student, an accessible bridge between the student and the particular mathematics concept/skill they are learning:

The level of teacher support you provide your students depends on how much of a learning bridge they need. In particular, students with learning problems need a well-established learning bridge (teacher model). They learn most effectively when their teacher provides clear and multi-sensory models of a mathematics concept/skill during math instruction.

What are some important considerations when implementing Explicit Modeling?

The teacher purposefully sets the stage for understanding by identifying what students will learn (visually and auditorily), providing opportunities for students to link what they already know (e.g. prerequisite concepts/skills they have already mastered, prior real-life experiences they have had, areas of interest based on your students' age, culture, ethnicity, etc.), and discussing with students how what they are going to learn has relevance/meaning for their immediate lives.

• Teacher breaks math concept/skill into learnable parts/steps. Think about the concept/skill and break it down into 3-4 features or parts.
• Teacher clearly describes features of the math concept or steps in performing math skill using visual examples.
• Teacher describes/models using multi-sensory techniques. Use as many "input" pathways as possible for any given concept/skill including auditory, visual, tactile, and kinesthetic means. For example, when modeling how to compare values of different fractions to determine "greater than," you might verbalize each step of the process for comparing fractions while pointing to each step written on chart paper (auditory and visual), represent each fraction using fraction circle pieces, running your finger around the perimeter of each piece, laying one fraction piece over the other one and running your finger along the space not covered up by the fraction of lesser value/area; "thinking aloud" by saying your thoughts aloud as you examine each fraction piece (visual, kinesthetic, auditory), verbalizing your answer and why you determined why one fraction was greater than the other, and having students run their fingers along the same fraction pieces and uncovered space (auditory, visual, tactile, kinesthetic).
• Teacher provides both examples and non-examples of the mathematics concept/skill. For example, in the above example, you might compare two different fractions using same process but place the fraction of greater value/area on top of the fraction of lesser value/area. Then prompt student thinking of why this is not an example of "greater than."
• Explicitly cue students to essential attributes of the mathematics concept/skill you model. For example, when associating the written fraction to the fraction pieces and their respective values, color code the numerator and denominator in ways that represent the meaning of the fraction pieces they use. Cue students to the color-coding and what each color represents. Then demonstrate how each written fraction relates to the "whole' circle:

2/4 = 2 of four equal pieces

• Teacher engages students in learning through demonstrating enthusiasm, through maintaining a lively pace, through periodically questioning students, and through checking for student understanding. Explicit modeling is not meant to be a passive learning experience for students. On the contrary, it is critical to involve students as you model.
• After modeling several examples and non-examples, begin to have your students demonstrate a few steps of the process.
• As students demonstrate greater understanding, ask them to complete more and more of the process.
• When students demonstrate complete understanding, have various students "teach" you by modeling the entire process.
• Play a game where you and your students try to "catch" each other making a mistake or leaving out a step in the process.

How do I implement Explicit Modeling?

• Select the appropriate level of understanding to model the concept/skill or problem solving strategy (concrete, representational, abstract).
• Ensure that your students have the prerequisite skills to perform the skill or use the problem solving strategy.
• Break down the concept/skill or problem solving strategy into logical and learnable parts (Ask yourself, "What do I do and what do I think as I perform the skill?"). The strategies you can link to from this site are already broken down into steps.
• Provide a meaningful context for the concept/skill or problem solving strategy (e.g. word or story problem suited to the age and interests of your students. Invite parents/family members of your students or members of the community who work in an area that can be meaningfully applied to the concept/skill or strategy and ask them to show how they use the concept/skill/strategy in their work.
• Provide visual, auditory, kinesthetic (movement), and tactile means for illustrating important aspects of the concept/skill (e.g. visually display word problem and equation, orally cue students by varying vocal intonations, point, circle, highlight computation signs or important information in story problems).
• "Think aloud" as you illustrate each feature or step of the concept/skill/strategy (e.g. say aloud what you are thinking as you problem-solve so students can better "visualize" the metacognitive aspects of understanding or doing the concept/skill/strategy).
• Link each step of the problem solving process (e.g. restate what you did in the previous step, what you are going to do in the next step, and why the next step is important to the previous step).
• Periodically check student understanding with questions, remodeling steps when there is confusion.
• Maintain a lively pace while being conscious of student information processing difficulties (e.g. need additional time to process questions).
• Model a concept/skill at least three times.

How does Explicitly Modeling Mathematics Concepts/Skills and Problem Solving Strategies help students who have learning problems?

• Teacher as model makes the concept/skill clear and learnable.
• High level of teacher support and direction enables student to make meaningful cognitive connections.
• Provides students who have attention problems, processing problems, memory retrieval problems, and metacognitive difficulties an accessible "learning map" to the concept/skill/strategy.
• Links between parts/steps are directly made, making confusion and misunderstanding less likely.
• Multi-sensory cueing provides students multiple modes to process and thereby learn information.
• Teaching students effective problem solving strategies provides them a means for solving problems independently and assists them to develop their metacognitive awareness.

What Mathematics Problem Solving Strategies can I teach my students?

Mathematics problem solving strategies that have research support or that have been field tested with students can be accessed by clicking on the link below. These strategies are organized according to mathematics concept/skill area. Each strategy is described and an example of how each strategy can be used is also provided. 

What are additional resources I can use to help me implement Explicitly Modeling Mathematics Concepts/Skills and Problem Solving Strategies?

MathVIDS is an interactive CD-ROM/website for teachers who are teaching math to students who are having difficulty learning mathematics. The development of MathVIDS was sponsored through funding by the Virginia Department of Education.

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Unit 12: Modeling

About this unit.

Let's dive even deeper into the world of modeling. We'll take our knowledge about all the different function types we were exposed to so far, and use it to model and analyze various phenomena, from heart rates to business profits.

Modeling with function combination

• Modeling with function combination (Opens a modal)
• Model with function combination Get 3 of 4 questions to level up!

Interpreting features of functions

• Periodicity of algebraic models (Opens a modal)
• End behavior of algebraic models (Opens a modal)
• Symmetry of algebraic models (Opens a modal)
• Periodicity of algebraic models Get 3 of 4 questions to level up!
• End behavior of algebraic models Get 3 of 4 questions to level up!

Manipulating formulas

• Manipulating formulas: perimeter (Opens a modal)
• Manipulating formulas: area (Opens a modal)
• Manipulating formulas: temperature (Opens a modal)
• Manipulate formulas Get 3 of 4 questions to level up!

Modeling with two variables

• Graph labels and scales (Opens a modal)
• Rational equation word problem (Opens a modal)
• Quadratic inequality word problem (Opens a modal)
• Exponential equation word problem (Opens a modal)
• Graph labels and scales Get 3 of 4 questions to level up!
• Equations & inequalities word problems Get 3 of 4 questions to level up!

Modeling with multiple variables

• Modeling with multiple variables: Pancakes (Opens a modal)
• Modeling with multiple variables: Roller coaster (Opens a modal)
• Modeling with multiple variables: Taco stand (Opens a modal)
• Modeling with multiple variables: Ice cream (Opens a modal)
• Interpreting expressions with multiple variables: Resistors (Opens a modal)
• Interpreting expressions with multiple variables: Cylinder (Opens a modal)
• Modeling: FAQ (Opens a modal)
• Modeling with multiple variables Get 3 of 4 questions to level up!
• Interpreting expressions with multiple variables Get 3 of 4 questions to level up!

• Our Mission

Using Mathematical Modeling to Get Real With Students

Unlike canned word problems, mathematical modeling plunges students into the messy complexities of real-world problem solving.  

How do you bring math to life for kids? Illustrating the boundless possibilities of mathematics can be difficult if students are only asked to examine hypothetical situations like divvying up a dessert equally or determining how many apples are left after sharing with friends, writes third- and fourth- grade teacher Matthew Kandel for Mathematics Teacher: Learning and Teaching PK-12 .

In the early years of instruction, it’s not uncommon for students to think they’re learning math for the sole purpose of being able to solve word problems or help fictional characters troubleshoot issues in their imaginary lives, Kandel says. “A word problem is a one-dimensional world,” he writes. “Everything is distilled down to the quantities of interest. To solve a word problem, students can pick out the numbers and decide on an operation.” 

But through the use of mathematical modeling, students are plucked out of the hypothetical realm and plunged into the complexities of reality—presented with opportunities to help solve real-world problems with many variables by generating questions, making assumptions, learning and applying new skills, and ultimately arriving at an answer.

In Kandel’s classroom, this work begins with breaking students into small groups, providing them with an unsharpened pencil and a simple, guiding question: “How many times can a pencil be sharpened before it is too small to use?”

Setting the Stage for Inquiry 

The process of tackling the pencil question is not unlike the scientific method. After defining a question to investigate, students begin to wonder and hypothesize—what information do we need to know?—in order to identify a course of action. This step is unique to mathematical modeling: Whereas a word problem is formulaic, leading students down a pre-existing path toward a solution, a modeling task is “free-range,” empowering students to use their individual perspectives to guide them as they progress through their investigation, Kandel says. 

Modeling problems also have a number of variables, and students themselves have the agency to determine what to ignore and what to focus their attention on. 

After inter-group discussions, students in Kandel’s classroom came to the conclusion that they’d need answers to a host of other questions to proceed with answering their initial inquiry: 

• How much does the pencil sharpener remove? 
• What is the length of a brand new, unsharpened pencil? 
• Does the pencil sharpener remove the same amount of pencil each time it is used?

Introducing New Skills in Context

Once students have determined the first mathematical question they’d like to tackle (does the pencil sharpener remove the same amount of pencil each time it is used?), they are met with a roadblock. How were they to measure the pencil if the length did not fall conveniently on an inch or half inch? Kandel took the opportunity to introduce a new target skill which the class could begin using immediately: measuring to the nearest quarter inch. 

“One group of students was not satisfied with the precision of measuring to the nearest quarter inch and asked to learn how to measure to the nearest eighth of an inch,” Kandel explains. “The attention and motivation exhibited by students is unrivaled by the traditional class in which the skill comes first, the problem second.” 

Students reached a consensus and settled on taking six measurements total: the initial length of the new, unsharpened pencil as well as the lengths of the pencil after each of five sharpenings. To ensure all students can practice their newly acquired skill, Kandel tells the class that “all group members must share responsibility, taking turns measuring and checking the measurements of others.” 

Next, each group created a simple chart to record their measurements, then plotted their data as a line graph—though exploring other data visualization techniques or engaging students in alternative followup activities would work as well.

“We paused for a quick lesson on the number line and the introduction of a new term—mixed numbers,” Kandel explains. “Armed with this new information, students had no trouble marking their y-axis in half- or quarter-inch increments.” 

Sparking Mathematical Discussions

Mathematical modeling presents a multitude of opportunities for class-wide or small-group discussions, some which evolve into debates in which students state their hypotheses, then subsequently continue working to confirm or refute them. 

Kandel’s students, for example, had a wide range of opinions when it came to answering the question of how small of a pencil would be deemed unusable. Eventually, the class agreed that once a pencil reached 1 ¼ inch, it could no longer be sharpened—though some students said they would be able to still write with it. 

“This discussion helped us better understand what it means to make an assumption and how our assumptions affected our mathematical outcomes,” Kandel writes. Students then indicated the minimum size with a horizontal line across their respective graphs. 

Many students independently recognized the final step of extending their line while looking at their graphs. With each of the six points representing their measurements, the points descended downward toward the newly added horizontal “line of inoperability.” 

With mathematical modeling, Kandel says, there are no right answers, only models that are “more or less closely aligned with real-world observations.” Each group of students may come to a different conclusion, which can lead to a larger class discussion about accuracy. To prove their group had the most accurate conclusion, students needed to compare and contrast their methods as well as defend their final result. 

Developing Your Own Mathematical Models

The pencil problem is a great starting point for introducing mathematical modeling and free-range problem solving to your students, but you can customize based on what you have available and the particular needs of each group of students.

Depending on the type of pencil sharpener you have, for example, students can determine what constitutes a “fair test” and set the terms of their own inquiry. 

Additionally, Kandel suggests putting scaffolds in place to allow students who are struggling with certain elements to participate: Simplified rulers can be provided for students who need accommodations; charts can be provided for students who struggle with data collection; graphs with prelabeled x- and y-axes can be prepared in advance.

.css-1sk4066:hover{background:#d1ecfa;} 7 Real-World Math Strategies

Students can also explore completely different free-range problem solving and real world applications for math . At North Agincourt Jr. Public School in Scarborough, Canada, kids in grades 1-6 learn to conduct water audits. By adding, subtracting, finding averages, and measuring liquids—like the flow rate of all the water foundations, toilets, and urinals—students measure the amount of water used in their school or home in a single day. 

Or you can ask older students to bring in common household items—anything from a measuring cup to a recipe card—and identify three ways the item relates to math. At Woodrow Petty Elementary School in Taft, Texas, fifth-grade students display their chosen objects on the class’s “real-world math wall.” Even acting out restaurant scenarios can provide students with an opportunity to reinforce critical mathematical skills like addition and subtraction, while bolstering an understanding of decimals and percentages. At Suzhou Singapore International School in China, third- to fifth- graders role play with menus, ordering fictional meals and learning how to split the check when the bill arrives. 

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Module 1: Problem Solving Strategies

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

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

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

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

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

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

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

Powerful online learning at your pace

What IS Problem-Solving?

Ask teachers about problem-solving strategies, and you’re opening a can of worms! Opinions about the “best” way to teach problem-solving are all over the board. And teachers will usually argue for their process quite passionately.

When I first started teaching math over 25 years ago, it was very common to teach “keywords” to help students determine the operation to use when solving a word problem. For example, if you see the word “total” in the problem, you always add. Rather than help students become better problem solvers, the use of keywords actually resulted in students who don’t even feel the need to read and understand the problem–just look for the keywords, pick out the numbers, and do the operation indicated by the keyword.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

Another common strategy for teaching problem-solving is the use of acrostics that students can easily remember to perform the “steps” in problem-solving. CUBES is an example. Just as with keywords, however, students often follow the steps with little understanding. As an example, a common step is to underline or highlight the question. But if you ask students why they are underlining or highlighting the question, they often can’t tell you. The question is , in fact, super important, but they’ve not been told why. They’ve been told to underline the question, so they do.

The problem with both keywords and the rote-step strategies is that both methods try to turn something that is inherently messy into an algorithm! It’s way past time that we leave both methods behind.

First, we need to broaden the definition of problem-solving. Somewhere along the line, problem-solving became synonymous with “word problems.” In reality, it’s so much more. Every one of us solves dozens or hundreds of problems every single day, and most of us haven’t solved a word problem in years. Problem-solving is often described as  figuring out what to do when you don’t  know what to do.  My power went out unexpectedly this morning, and I have work to do. That’s a problem that I had to solve. I had to think about what the problem was, what my options were, and formulate a plan to solve the problem. No keywords. No acrostics. I’m using my phone as a hotspot and hoping my laptop battery doesn’t run out. Problem solved. For now.

If you want to get back to what problem-solving really is, you should consult the work of George Polya. His book, How to Solve It , which was first published in 1945, outlined four principles for problem-solving. The four principles are: understand the problem, devise a plan, carry out the plan, and look back. This document from UC Berkeley’s Mathematics department is a great 4-page overview of Polya’s process. You can probably see that the keyword and rote-steps strategies were likely based on Polya’s method, but it really got out of hand. We need to help students think , not just follow steps.

I created both primary and intermediate posters based on Polya’s principles. Grab your copies for free here !

I would LOVE to hear your comments about problem-solving!

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Do you tutor teachers?

I do professional development for district and schools, and I have online courses.

You make a great point when you mentioned that teaching students to look for “keywords” is not teaching students to become better problem solvers. I was once guilty of using the CUBES strategy, but have since learned to provide students with opportunity to grapple with solving a problem and not providing them with specified steps to follow.

I think we’ve ALL been there! We learn and we do better. 🙂

Love this article and believe that we can do so much better as math teachers than just teaching key words! Do you have an editable version of this document? We are wanting to use something similar for our school, but would like to tweak it just a bit. Thank you!

I’m sorry, but because of the clip art and fonts I use, I am not able to provide an editable version.

Hi Donna! I am working on my dissertation that focuses on problem-solving. May I use your intermediate poster as a figure, giving credit to you in my citation with your permission, for my section on Polya’s Traditional Problem-Solving Steps? You laid out the process so succinctly with examples that my research could greatly benefit from this image. Thank you in advance!

Absolutely! Good luck with your dissertation!

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Family of LLMs for mathematical reasoning.

mathllm/MathCoder

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This repo is for " MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning "

• [2023.05.20] 🤗 MathCodeInstruct Dataset-Plus is available now! 🔥🔥🔥
• [2023.04.29] 🤗 MathCodeInstruct Dataset is available now! 🔥🔥🔥
• [2023.02.27] 🚀 MathGenie achieves an accuracy of 87.7% on GSM8K and 55.7% on MATH. 🎉 Congratulations!
• [2023.02.27] The inference and evaluation code for MathCoders is available now.
• [2023.01.16] 🌟 Our MathCoder and CSV has been accepted at ICLR 2024 ! 🎉 Cheers!
• [2023.10.05] Our work was featured by Aran Komatsuzaki . Thanks!
• [2023.10.05] Our 7B models are available at Huggingface now.
• [2023.10.05] Our paper is now accessible at https://arxiv.org/abs/2310.03731 .

Datasets and Models

Our models are available at Hugging Face now.

🤗 MathCodeInstruct Dataset

Training Data

The models are trained on the MathCodeInstruct Dataset.

Introduction

The recently released GPT-4 Code Interpreter has demonstrated remarkable proficiency in solving challenging math problems, primarily attributed to its ability to seamlessly reason with natural language, generate code, execute code, and continue reasoning based on the execution output. In this paper, we present a method to fine-tune open-source language models, enabling them to use code for modeling and deriving math equations and, consequently, enhancing their mathematical reasoning abilities.

We propose a method of generating novel and high-quality datasets with math problems and their code-based solutions, referred to as MathCodeInstruct. Each solution interleaves natural language , code , and execution results .

We also introduce a customized supervised fine-tuning and inference approach. This approach yields the MathCoder models, a family of models capable of generating code-based solutions for solving challenging math problems.

Impressively, the MathCoder models achieve state-of-the-art scores among open-source LLMs on the MATH (45.2%) and GSM8K (83.9%) datasets, substantially outperforming other open-source alternatives. Notably, the MathCoder model not only surpasses ChatGPT-3.5 and PaLM-2 on GSM8K and MATH but also outperforms GPT-4 on the competition-level MATH dataset. The proposed dataset and models will be released upon acceptance.

Model deployment

We use the Text Generation Inference (TGI) to deploy our MathCoders for response generation. TGI is a toolkit for deploying and serving Large Language Models (LLMs). TGI enables high-performance text generation for the most popular open-source LLMs, including Llama, Falcon, StarCoder, BLOOM, GPT-NeoX, and T5. Your can follow the guide here . After successfully installing TGI, you can easily deploy the models using deploy.sh .

We provide a script for inference. Just replace the ip and port in the following command correctly with the API forwarded by TGI like:

We also open-source all of the model outputs from our MathCoders under the outs/ folder.

To evaluate the predicted answer, run the following command:

Please cite the paper if you use our data, model or code. Please also kindly cite the original dataset papers.

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In this article, we explore problem-solving strategies and models that can be applied to solve problems. Then, we practice applying these models in some example exercises.

Problem-solving strategies and model descriptions

Oftentimes in mathematics, there is more than one way to solve a problem. Using problem-solving strategies can help you approach problems in a structured and logical manner to improve your efficiency.

Problem-solving strategies are models based on previous experience that provide a recommended approach for solving problems or analyzing potential solutions.

Problem-solving strategies involve steps like understanding, planning, and organizing, for example. While problem-solving strategies cannot guarantee an easier solution to a problem, they do provide techniques and tools that act as a guide for success.

Types of problem-solving models and strategies

Many models and strategies are developed based on the nature of the problem at hand. In this article, we discuss two well-known models that are designed to address various types of problems, including:

Polya's f our-step problem-solving model

• IDEAL problem-solving model

Let's look at these two models in detail.

A mathematician named George Polya developed a model called the Polya f our-step problem-solving model to approach and solve various kinds of problems. This method has the following steps:

Understand the problem

Devise a plan, carry out the plan.

John Bransford and Barry Stein also proposed a five-step model named IDEAL to resolve a problem with a sound and methodical approach. The IDEAL model is based on the following steps:

• Identify The Problem
• Define An Outcome
• Explore Possible Strategies
• Anticipate Outcomes & Act
• Look And Learn

Using either of these two models to help you identify and approach problems methodically can help make it easier to solve them.

• Polya's four-step problem-solving model

Polya's f our-step problem-solving model can be used to solve day-to-day problems as well as mathematical and other academic problems. As seen briefly, the steps of this problem-solving model include: understanding the problem, creating and carrying out a plan, and looking back. Let's look at these steps in more detail to understand how they are used.

This is a critical initial step. Simply put, if you don't fully understand the problem, you won't be able to identify a solution. You can understand a problem better by reviewing all of the inputs and available information, including its conditions and circumstances. Reading and understanding the problem helps you to organize the information as well as assign the relevant variables.

The following techniques can be applied during this problem-solving step:

Read the problem out loud to process it better.

List or summarize the important information to find out what is given and what is still missing.

Sketch a detailed diagram as a visual aid, depending on the problem.

Visualize a scenario about the problem to put it into context.

Use keyword analysis to identify the necessary operations (i.e., pay attention to important words and phrases such as "how many," "times," or "total").

Now that you have taken the time to properly understand the problem, you can devise a plan on how to proceed further to solve it. During this second step, you identify what strategy to follow to arrive at a solution. When considering a strategy to use, it's important to consider exactly what it is that you want to know.

Some problem-solving strategies include:

Identify the pattern from the given information and use it.

Use the guess-and-check method.

Work backward by using potential answers.

Apply a specific formula for the problem.

Eliminate the possibilities that don't work out.

Solve a simpler version of the problem first.

Form an equation and solve it.

During this third step, you solve the problem by applying your chosen strategy. For example, if you planned to solve the problem by drawing a graph, then during this step, you draw the graph using the information gathered in the previous steps. Here, you test your problem-solving skills and find if the solution works or not.

Below are some points to keep in mind when solving the problem:

Be systematic in your approach when implementing a strategy.

Check the work and see whether the solution works in all relevant cases.

Be flexible and change the strategy if necessary.

Keep solving and don't give up.

At this fourth step, you check your solution. This can be done by solving the problem in another way or simply by confirming that your solution makes sense. This step helps you decide if any improvements are needed for your solution. You may choose to check after solving an individual problem or after solving an entire set. Checking the problem carefully also helps you to reflect on the process and improve your methods for future problem solving.

The IDEAL problem-solving model was developed by Bransford and Stein as a guide for understanding and solving problems. This method is used in both education and industry. The IDEAL problem-solving model consists of five steps: identifying the problem, describing the outcome, exploring the possible strategies, anticipating the outcome, and looking back to learn. Let us explore these steps in detail by considering them one by one.

I dentify the problem - In this first step, you identify and understand the problem. To do this, you evaluate which information is provided and available, and you identify the unknown variables and missing information.

D escribe the outcome - In this second step, you define the result you are seeking. This matters because a problem might have multiple potential results, so you need to clarify which outcomes in particular you are aiming for. Defining an outcome clarifies the path that must be taken to solving the problem.

E xplore possible strategies - Now that you have considered the desired outcome, you are ready to brainstorm and explore different strategies and techniques to solve your particular problem.

A nticipate outcomes and act - From the previous step, you already have explored different strategies and techniques. During this step, you review and evaluate them in order to choose the best one to act on. Your selection should consider the benefits and drawbacks of the strategy and whether it can ultimately lead to the desired outcome. After making your selection, you act on it and apply the technique to the given problem.

L ook and learn - The final step to solving problems with this method is to consider whether the applied technique worked and if the needed results were obtained. Also, an additional step is learning from the current problem and its methods to make problem solving more efficient in the future.

Examples of problem-solving models and strategies

Here are some solved examples of the problem-solving models and strategies discussed above.

Find the number when two times the sum of \(3\) and that number is thrice that number plus \(4\). Solve this problem with Polya's f our-step problem-solving model .

Solution: We will follow the steps of Polya's f our-step problem-solving model as mentioned above to find the number.

Step 1 : Understand the problem.

By reading and understanding the question, we denote the unknown number as \(x\).

Step 2 : Devise a plan.

We see that two times \(x\) is added to \(3\) to make it equal to thrice the \(x\) plus \(4\). So, we can determine that forming an equation to solve the mathematical problem is a reasonable plan. Therefore, we form an equation by going step by step:

First we add \(x\) with \(3\) and multiply it with \(2\).

\begin{equation}\tag{1}\Rightarrow 2(x+3)\end{equation}

Then, we form the second part of the equation for thrice the \(x\) plus \(4\).

\begin{equation}\tag{2}\Rightarrow 3x+4\end{equation}

Hence, equating both sides \((1)\) and \((2)\) we get,

\[2(x+3)=3x+4\]

Step 3 : Carry out the plan.

Now, we algebraically solve the equation above.

\begin{align}2(x+3) &=3x+4 \\2x+6 &= 3x+4 \\3x-2x &= 6-4 \\x &=2\end{align}

Step 4 : Look back.

By inputting the value of 2 in our equation, we see that two times \(2+3\) is \(10\) and three times \(2\) plus \(4\) is also 10. Hence, the left side and right side are equal. So, our solution is satisfied.

Hence, the number is \(2\).

A string is \(48 cm\) long. It is cut into two pieces such that one piece is three times that of the other piece. What is the length of each piece?

Solution : Let us work on this problem using the IDEAL problem-solving method.

Step 1 : Identify the problem.

We are given a length of a string, and we know that it is cut into two parts, whereby one part is three times longer than the other. As the length of the longer piece of string is dependent on the shorter string, we assume only one variable, say \(x\).

Step 2 : Describe the outcome.

From the problem, we understand that we need to find the length of each piece of string. And we need the results such that the total length of both the pieces should be \(48 cm\).

Step 3 : Explore possible strategies.

There are multiple ways to solve this problem. One way to solve it is by using the trial-and-error method. Also, as one length is dependent on another, the other way is to form an equation to solve for the unknown variable algebraically.

Step 4 : Anticipate outcomes and act.

From the above step, we have two methods by which we can solve the given problem. Let's find out which method is more efficient and solve the problem by applying it.

For the trial-and-error method, we need to assume value(s) one at a time for the variable and then solve for it individually until we get the total of 48.

That is, suppose we consider \(x=1\).

Then, by the condition, the second piece is three times the first piece.

\[\Rightarrow 3x=3(1)=3\]

Then the length of both pieces should be:

\[\Rightarrow 1+3=4\neq 48\]

Hence, our assumption is wrong. So, we need to consider another value. For this method, we continue this process until we find the total of \(48\). We can see that proceeding this way is time-consuming. So, let us apply the other method instead.

In this method, we form an equation and solve it to obtain the unknown variable's value. We know that one piece is three times the other piece. Therefore, let the length of one piece be \(x\). Then the length of the other piece is \(3x\).

Now, as the string is \(48 cm\) long, it should be considered as a sum of both of its pieces.

\begin{align}&\Rightarrow x+3x=48 \\&\Rightarrow 4x=48 \\&\Rightarrow x=\frac{48}{4} \\&\Rightarrow x=12 \\\end{align}

So, the length of one piece is \(12cm\). The length of the other piece is \(3x=3(12)=36cm\).

Step 5: Look and learn

Let's take a look to see if our answers are correct. The unknown variable value we obtained is \(12\). Using it to find the other piece we get a value of \(36\). Now, adding both of them, we get:

\[\Rightarrow 12+36=48\].

Here, we got the correct total length. Hence, our calculations and applied method are right.

Problem-solving strategies and models - Key takeaways

• Problem-solving strategies are models developed based on previous experience that provide a recommended approach for analyzing potential solutions for problems.
• Two common models include Polya's Four-Step Problem-Solving Model and the IDEAL problem-solving model.
• Polya's Four-Step Problem-Solving Model has the following steps: 1) Understand the problem, 2) Devise a plan, 3) Carry out the plan, and 4) Looking back.
• The IDEAL model is based on the following steps: 1) Identify The Problem, 2) Define An Outcome, 3) Explore Possible Strategies, 4) Anticipate Outcomes and Act, 5) Look And Learn.

Flashcards in Problem-solving Models and Strategies 15

What are the steps to solve a problem efficiently?

1. Understand the problem

Name the two problem-solving models.

State two problem-solving strategies when devising a plan.

Apply the specific formula for the problem.

Step 01: What do you know? 

• Mrs. Grave gives 1 penny on Day 1, 2 pennies on Day 2, and 4 pennies on Day 3.
• Each day the amount of money will double. 
• Paul does his tasks for 5 days.

Step 02: What do you want to know?

You curious to figure out how much money will Paul have in total after 5 days of doing his tasks. We want to solve the problem by formulating a simpler one.  

Step 01: What does David know? 

• The number of players starting the tournament:8 
• Only winners can advance to the next round

Step 02: What does David want to know?

David wants to compare the number of players in the second round to the number that starts the tournament. To solve the problem, David can use a diagram. 

Step 01: What do you know?

• Slices of tomatoes and cucumber were used.
• The total number of slices used is 60.
• The ratio of cucumbers to tomatoes is 4:6
• The ratio simplifies to 2:3

Step 02: What do you want to know? 

We need to know the number of both cucumber and tomato slices. We want to solve the problem by doing a table.  

24 cucumber slices and 36 tomato slices is one solution

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Frequently Asked Questions about Problem-solving Models and Strategies

What are problem-solving models?

Problem-solving models are models developed based on previous experience that provide a recommended approach for solving problems or analyzing potential solutions.

What are types of problem-solving?

The most basic types of problem-solving are Polya's four-step problem-solving model and the IDEAL problem-solving model.

What are the strategies to problem-solve efficiently?

The strategies to solve a problem efficiently are to understand it, determine the correct method, solve it and verify and learn from it.

What are the lists of problem-solving models in algebra?

In algebra, any problem can be solved using Polya's four-step problem-solving model and IDEAL problem-solving model.

What are the 5 problem-solving strategies?

The 5 problem-solving strategies are 1. Identify The Problem, 2. Define An Outcome, 3. Explore Possible Strategies, 4. Anticipate Outcomes & Act, 5. Look And Learn.

Test your knowledge with multiple choice flashcards

Step 01: What do you know? Mrs. Grave gives 1 penny on Day 1, 2 pennies on Day 2, and 4 pennies on Day 3.Each day the amount of money will double. Paul does his tasks for 5 days.Step 02: What do you want to know?You curious to figure out how much money will Paul have in total after 5 days of doing his tasks. We want to solve the problem by formulating a simpler one.  

Step 01: What does David know? The number of players starting the tournament:8 Only winners can advance to the next roundStep 02: What does David want to know?David wants to compare the number of players in the second round to the number that starts the tournament. To solve the problem, David can use a diagram. 

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