disadvantages of problem solving method in mathematics

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5 Advantages and Disadvantages of Problem-Based Learning [+ Activity Design Steps]

Written by Marcus Guido

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• Teaching Strategies

Advantages of Problem-Based Learning

Disadvantages of problem-based learning, steps to designing problem-based learning activities.

Used since the 1960s, many teachers express concerns about the effectiveness of problem-based learning (PBL) in certain classroom settings.

Whether you introduce the student-centred pedagogy as a one-time activity or mainstay exercise, grouping students together to solve open-ended problems can present pros and cons.

Below are five advantages and disadvantages of problem-based learning to help you determine if it can work in your classroom.

If you decide to introduce an activity, there are also design creation steps and a downloadable guide to keep at your desk for easy reference.

1. Development of Long-Term Knowledge Retention

Students who participate in problem-based learning activities can improve their abilities to retain and recall information, according to a literature review of studies about the pedagogy .

The literature review states “elaboration of knowledge at the time of learning” -- by sharing facts and ideas through discussion and answering questions -- “enhances subsequent retrieval.” This form of elaborating reinforces understanding of subject matter , making it easier to remember.

Small-group discussion can be especially beneficial -- ideally, each student will get chances to participate.

But regardless of group size, problem-based learning promotes long-term knowledge retention by encouraging students to discuss -- and answer questions about -- new concepts as they’re learning them.

2. Use of Diverse Instruction Types

You can use problem-based learning activities to the meet the diverse learning needs and styles of your students, effectively engaging a diverse classroom in the process. In general, grouping students together for problem-based learning will allow them to:

• Address real-life issues that require real-life solutions, appealing to students who struggle to grasp abstract concepts
• Participate in small-group and large-group learning, helping students who don’t excel during solo work grasp new material
• Talk about their ideas and challenge each other in a constructive manner, giving participatory learners an avenue to excel
• Tackle a problem using a range of content you provide -- such as videos, audio recordings, news articles and other applicable material -- allowing the lesson to appeal to distinct learning styles

Since running a problem-based learning scenario will give you a way to use these differentiated instruction approaches , it can be especially worthwhile if your students don’t have similar learning preferences.

3. Continuous Engagement

Providing a problem-based learning challenge can engage students by acting as a break from normal lessons and common exercises.

It’s not hard to see the potential for engagement, as kids collaborate to solve real-world problems that directly affect or heavily interest them.

Although conducted with post-secondary students, a study published by the Association for the Study of Medical Education reported increased student attendance to -- and better attitudes towards -- courses that feature problem-based learning.

These activities may lose some inherent engagement if you repeat them too often, but can certainly inject excitement into class.

4. Development of Transferable Skills

Problem-based learning can help students develop skills they can transfer to real-world scenarios, according to a 2015 book that outlines theories and characteristics of the pedagogy .

The tangible contexts and consequences presented in a problem-based learning activity “allow learning to become more profound and durable.” As you present lessons through these real-life scenarios, students should be able to apply learnings if they eventually face similar issues.

For example, if they work together to address a dispute within the school, they may develop lifelong skills related to negotiation and communicating their thoughts with others.

As long as the problem’s context applies to out-of-class scenarios, students should be able to build skills they can use again.

5. Improvement of Teamwork and Interpersonal Skills

Successful completion of a problem-based learning challenge hinges on interaction and communication, meaning students should also build transferable skills based on teamwork and collaboration . Instead of memorizing facts, they get chances to present their ideas to a group, defending and revising them when needed.

What’s more, this should help them understand a group dynamic. Depending on a given student, this can involve developing listening skills and a sense of responsibility when completing one’s tasks. Such skills and knowledge should serve your students well when they enter higher education levels and, eventually, the working world.

1. Potentially Poorer Performance on Tests

Devoting too much time to problem-based learning can cause issues when students take standardized tests, as they may not have the breadth of knowledge needed to achieve high scores. Whereas problem-based learners develop skills related to collaboration and justifying their reasoning, many tests reward fact-based learning with multiple choice and short answer questions. Despite offering many advantages, you could spot this problem develop if you run problem-based learning activities too regularly.

2. Student Unpreparedness

Problem-based learning exercises can engage many of your kids, but others may feel disengaged as a result of not being ready to handle this type of exercise for a number of reasons. On a class-by-class and activity-by-activity basis, participation may be hindered due to:

• Immaturity  -- Some students may not display enough maturity to effectively work in a group, not fulfilling expectations and distracting other students.
• Unfamiliarity  -- Some kids may struggle to grasp the concept of an open problem, since they can’t rely on you for answers.
• Lack of Prerequisite Knowledge  -- Although the activity should address a relevant and tangible problem, students may require new or abstract information to create an effective solution.

You can partially mitigate these issues by actively monitoring the classroom and distributing helpful resources, such as guiding questions and articles to read. This should keep students focused and help them overcome knowledge gaps. But if you foresee facing these challenges too frequently, you may decide to avoid or seldom introduce problem-based learning exercises.

3. Teacher Unpreparedness

If supervising a problem-based learning activity is a new experience, you may have to prepare to adjust some teaching habits . For example, overtly correcting students who make flawed assumptions or statements can prevent them from thinking through difficult concepts and questions. Similarly, you shouldn’t teach to promote the fast recall of facts. Instead, you should concentrate on:

• Giving hints to help fix improper reasoning
• Questioning student logic and ideas in a constructive manner
• Distributing content for research and to reinforce new concepts
• Asking targeted questions to a group or the class, focusing their attention on a specific aspect of the problem

Depending on your teaching style, it may take time to prepare yourself to successfully run a problem-based learning lesson.

4. Time-Consuming Assessment

If you choose to give marks, assessing a student’s performance throughout a problem-based learning exercise demands constant monitoring and note-taking. You must take factors into account such as:

• Completed tasks
• The quality of those tasks
• The group’s overall work and solution
• Communication among team members
• Anything you outlined on the activity’s rubric

Monitoring these criteria is required for each student, making it time-consuming to give and justify a mark for everyone.

5. Varying Degrees of Relevancy and Applicability

It can be difficult to identify a tangible problem that students can solve with content they’re studying and skills they’re mastering. This introduces two clear issues. First, if it is easy for students to divert from the challenge’s objectives, they may miss pertinent information. Second, you could veer off the problem’s focus and purpose as students run into unanticipated obstacles. Overcoming obstacles has benefits, but may compromise the planning you did. It can also make it hard to get back on track once the activity is complete. Because of the difficulty associated with keeping activities relevant and applicable, you may see problem-based learning as too taxing.

If the advantages outweigh the disadvantages -- or you just want to give problem-based learning a shot -- follow these steps:

1. Identify an Applicable Real-Life Problem

Find a tangible problem that’s relevant to your students, allowing them to easily contextualize it and hopefully apply it to future challenges. To identify an appropriate real-world problem, look at issues related to your:

• Students’ shared interests

You must also ensure that students understand the problem and the information around it. So, not all problems are appropriate for all grade levels.

2. Determine the Overarching Purpose of the Activity

Depending on the problem you choose, determine what you want to accomplish by running the challenge. For example, you may intend to help your students improve skills related to:

• Collaboration
• Problem-solving
• Curriculum-aligned topics
• Processing diverse content

A more precise example, you may prioritize collaboration skills by assigning specific tasks to pairs of students within each team. In doing so, students will continuously develop communication and collaboration abilities by working as a couple and part of a small group. By defining a clear purpose, you’ll also have an easier time following the next step.

3. Create and Distribute Helpful Material

Handouts and other content not only act as a set of resources, but help students stay focused on the activity and its purpose. For example, if you want them to improve a certain math skill , you should make material that highlights the mathematical aspects of the problem. You may decide to provide items such as:

• Data that helps quantify and add context to the problem
• Videos, presentations and other audio-visual material
• A list of preliminary questions to investigate

Providing a range of resources can be especially important for elementary students and struggling students in higher grades, who may not have self-direction skills to work without them.

4. Set Goals and Expectations for Your Students

Along with the aforementioned materials, give students a guide or rubric that details goals and expectations. It will allow you to further highlight the purpose of the problem-based learning exercise, as you can explain what you’re looking for in terms of collaboration, the final product and anything else. It should also help students stay on track by acting as a reference throughout the activity.

5. Participate

Although explicitly correcting students may be discouraged, you can still help them and ask questions to dig into their thought processes. When you see an opportunity, consider if it’s worthwhile to:

• Fill gaps in knowledge
• Provide hints, not answers
• Question a student’s conclusion or logic regarding a certain point, helping them think through tough spots

By participating in these ways, you can provide insight when students need it most, encouraging them to effectively analyze the problem.

6. Have Students Present Ideas and Findings

If you divided them into small groups, requiring students to present their thoughts and results in front the class adds a large-group learning component to the lesson. Encourage other students to ask questions, allowing the presenting group to elaborate and provide evidence for their thoughts. This wraps up the activity and gives your class a final chance to find solutions to the problem.

Wrapping Up

The effectiveness of problem-based learning may differ between classrooms and individual students, depending on how significant specific advantages and disadvantages are to you. Evaluative research consistently shows value in giving students a question and letting them take control of their learning. But the extent of this value can depend on the difficulties you face.It may be wise to try a problem-based learning activity, and go forward based on results.

Create or log into your teacher account on Prodigy -- an adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s used by more than 350,000 teachers and 10 million students. It may be wise to try a problem-based learning activity, and go forward based on results.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

5 Teaching Mathematics Through Problem Solving

Janet Stramel

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

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

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

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

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

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

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

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

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

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

Key features of a good mathematics problem includes:

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

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

But look at the b ack.

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

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

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

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

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

Low Floor High Ceiling Tasks

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

The strengths of using Low Floor High Ceiling Tasks:

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

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

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

Math in 3-Acts

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

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

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

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

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

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

Number Talks

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

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

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

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

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

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

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

Instead, you and your students could say the following:

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

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

Using “Worksheets”

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

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

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

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

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

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

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

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

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

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

when we teach students how to problem solve

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

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

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

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

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The advantages and disadvantages of problem-solving practice when learning basic addition facts

Research output : Chapter in Book/Report/Conference proceeding › Chapter (Book) › Research › peer-review

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• 10.4324/9780429400902-15

T1 - The advantages and disadvantages of problem-solving practice when learning basic addition facts

AU - Hopkins, Sarah

N2 - How children learn to retrieve answers to basic (single-digit) addition problems and how teachers can support children's learning of retrieval has captivated my attention as a researcher and teacher educator for the last 25 years. In this chapter, I describe this research and explain how I got started with the help of Professor Mike Lawson. I then present findings from a series of microgenetic studies to illustrate the different effects problem-solving practice has on children's development of retrieval.

AB - How children learn to retrieve answers to basic (single-digit) addition problems and how teachers can support children's learning of retrieval has captivated my attention as a researcher and teacher educator for the last 25 years. In this chapter, I describe this research and explain how I got started with the help of Professor Mike Lawson. I then present findings from a series of microgenetic studies to illustrate the different effects problem-solving practice has on children's development of retrieval.

U2 - 10.4324/9780429400902-15

DO - 10.4324/9780429400902-15

M3 - Chapter (Book)

SN - 9780367001834

BT - Problem Solving for Teaching and Learning

A2 - Askell-Williams, Helen

A2 - Orrell, Janice

PB - Routledge

CY - Abingdon UK

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Problem-Solving Strategies and Obstacles

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Sean is a fact-checker and researcher with experience in sociology, field research, and data analytics.

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From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

• Discovery of the problem
• Deciding to tackle the issue
• Seeking to understand the problem more fully
• Researching available options or solutions
• Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

• Perceptually recognizing the problem
• Representing the problem in memory
• Considering relevant information that applies to the problem
• Identifying different aspects of the problem
• Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

• Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
• Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
• Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
• Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

• Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
• Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
• Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
• Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

• Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
• Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
• Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
• Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
• Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
• Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. doi:10.3389/fnhum.2018.00261

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1.01: Introduction to Numerical Methods

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Lesson 1: Why Numerical Methods?

Learning objectives.

After successful completion of this lesson, you should be able to: 1) Enumerate the need for numerical methods.

Introduction

Numerical methods are techniques to approximate mathematical processes (examples of mathematical processes are integrals, differential equations, nonlinear equations).

Approximations are needed because

1) we cannot solve the procedure analytically, such as the standard normal cumulative distribution function

\[\Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{x}e^{- t^{2}/2}{dt} \;\;\;\;\;\;\;\;\;\;\;\;(\PageIndex{1.1}) \nonumber\]

2) the analytical method is intractable, such as solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss (Figure \(\PageIndex{1.1}\)).

In the case of Equation (1), an exact solution is not available for \(\Phi(x)\) other than for \(x = 0\) and \(x \rightarrow \infty\) . For other values of \(x\) where an exact solution is not available, one may solve the problem by using approximate techniques such as the left-hand Reimann sum you were introduced to in the Integral Calculus course.

In the truss problem, one can solve \(1000\) simultaneous linear equations for \(1000\) unknowns without using a calculator. One can use fractions, long divisions, and long multiplications to get the exact answer. But just the thought of such a task is laborious. The task may seem less laborious if we are allowed to use a calculator, but it would still fall under the category of an intractable, if not an impossible, problem. So, we need to find a numerical technique and convert it into a computer program that solves a set of \(n\) equations and \(n\) unknowns.

Again, what are numerical methods? They are techniques to solve a mathematical problem approximately. As we go through the course, you will see that numerical methods let us find solutions close to the exact one, and we can quantify the approximate error associated with the answer. After all, what good is an approximation without quantifying how good the approximation is?

Audiovisual Lecture

Title: Why Do We Need Numerical Methods

Summary : This video is an introduction to why we need numerical methods.

Lesson 2: Steps of Solving an Engineering Problem

After successful completion of this lesson, you should be able to:

1) go through the stages (problem description, mathematical modeling, solving and implementation) of solving a particular physical problem.

Numerical methods are used by engineers and scientists to solve problems. However, numerical methods are just one step in solving an engineering problem. There are four steps for solving an engineering problem, as shown in Figure \(\PageIndex{2.1}\).

The first step is to describe the problem. The description would involve writing the background of the problem and the need for its solution. The second step is developing a mathematical model for the problem, and this could include the use of experiments or/and theory. The third step involves solving the mathematical model. The solution may consist of analytical or/and numerical means. The fourth step is implementing the solution to see if the problem is solved.

Let us see through an example of these four steps of solving an engineering problem.

Problem Description

To make the fulcrum (Figure \(\PageIndex{2.2}\)) of a bascule bridge, a long hollow steel shaft called the trunnion is shrunk-fit into a steel hub. The resulting steel trunnion-hub assembly is then shrunk-fit into the girder of the bridge.

The shrink-fitting is done by first immersing the trunnion in a cold medium such as a dry-ice/alcohol mixture. After the trunnion reaches the steady-state temperature, that is, the temperature of the cold medium, the outer diameter of the trunnion contracts. The trunnion is taken out of the medium and slid through the hole of the hub (Figure \(\PageIndex{2.3}\)).

When the trunnion heats up, it expands and creates an interference fit with the hub. In 1995, on one of the bridges in Florida, this assembly procedure did not work as designed. Before the trunnion could be inserted fully into the hub, the trunnion got stuck. Luckily, the trunnion was taken out before it got stuck permanently. Otherwise, a new trunnion and hub would need to be ordered at the cost of \(\$50,000\) . Coupled with construction delays, the total loss could have been more than a hundred thousand dollars.

Why did the trunnion get stuck? Because the trunnion had not contracted enough to slide through the hole. Can you find out why this happened?

Simple Mathematical Model

A hollow trunnion of an outside diameter \(12.363^{\prime\prime}\) is to be fitted in a hub of inner diameter \(12.358^{\prime\prime}\) . The trunnion was put in a dry ice/alcohol mixture (temperature of the fluid - dry-ice/alcohol mixture is \(- 108{^\circ}\text{F}\) ) to contract the trunnion so that it can be slid through the hole of the hub. To slide the trunnion without sticking, a diametrical clearance of at least \(0.01^{\prime\prime}\) is required between the trunnion and the hub. Assuming the room temperature is \(80{^\circ}\text{F}\) , is immersing the trunnion in a dry-ice/alcohol mixture a correct decision?

To calculate the contraction in the diameter of the trunnion, the thermal expansion coefficient at room temperature is used. In that case, the reduction \(\Delta D\) in the outer diameter of the trunnion is

\[\displaystyle\Delta D = D\alpha\Delta T \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.1}) \nonumber\]

\[D = \text{ outer diameter of the trunnion,} \nonumber\]

\[\alpha = \text{ coefficient of thermal expansion coefficient at room temperature, and} \nonumber\]

\[\Delta T = \text{change in temperature.} \nonumber\]

Solution to Simple Mathematical Model

\[D = 12.363^{\prime\prime} \nonumber\]

\[\alpha = 6.47 \times 10^{-6}\ \text{in/in/}^{\circ}\text{F} \text{ at } 80{^\circ}\text{F} \nonumber\]

\[\displaystyle \begin{split} \Delta T&= T_{\text{fluid}} - T_{\text{room}}\\ &= - 108 - 80\\ &= - 188{^\circ}\ \text{F}\end{split} \nonumber\]

\[T_{\text{fluid}}= \text{ temperature of dry-ice/alcohol mixture} \nonumber\]

\[T_{\text{room}}= \text{ room temperature} \nonumber\]

the reduction in the outer diameter of the trunnion from Equation \((\PageIndex{2.1})\) hence is given by

\[\begin{split} \Delta D &= (12.363)\left( 6.47 \times 10^{- 6} \right)\left( - 188 \right)\\ &=- 0.01504^{\prime\prime} \end{split} \nonumber\]

So the trunnion is predicted to reduce in diameter by \(0.01504^{\prime\prime}\) . But is this enough reduction in diameter? As per specifications, the trunnion diameter needs to change by

\[\begin{split} \Delta D &= -\text{trunnion outside diameter} + \text{hub inner diameter} - \text{diametric clearance}\\ &= -12.363 +12.358 - 0.01\\ &= - 0.015^{\prime\prime} \end{split} \nonumber\]

So, according to this calculation, immersing the steel trunnion in dry-ice/alcohol mixture gives the desired contraction of greater than \(0.015^{\prime\prime}\) as the predicted contraction is \(0.01504^{\prime\prime}\) . But, when the steel trunnion was put in the hub, it got stuck. Why did this happen? Was our mathematical model adequate for this problem, or did we create a mathematical error?

Accurate Mathematical Model

As shown in Figure \(\PageIndex{2.4}\) and Table 1, the thermal expansion coefficient of steel decreases with temperature and is not constant over the range of temperature the trunnion goes through. Hence, Equation \((\PageIndex{2.1})\) would overestimate the thermal contraction.

he contraction in the diameter of the trunnion for which the thermal expansion coefficient varies as a function of temperature is given by

\[ \Delta D = D\int_{T_{\text{room}}}^{T_{\text{fluid}}} \alpha dT \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.2}) \nonumber\]

Solution to More Accurate Mathematical Model

So, one needs to curve fit the data to find the coefficient of thermal expansion as a function of temperature. This curve is found by regression where we best fit a function to the data given in Table 1. In this case, we may fit a second-order polynomial

\[\displaystyle\alpha = a_{0} + a_{1} T + a_{2} T^{2}\;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.3}) \nonumber\]

The values of the coefficients in the above Equation \((\PageIndex{2.3})\) will be found by polynomial regression (we will learn how to do this later in the chapter on Nonlinear Regression). At this point, we are just going to give you these values, and they are

\[\begin{bmatrix} a_{0} \\ a_{1} \\ a_{2} \\ \end{bmatrix} = \begin{bmatrix} 6.0150 \times 10^{- 6} \\ 6.1946 \times 10^{- 9} \\ - 1.2278 \times 10^{- 11} \\ \end{bmatrix} \nonumber\]

to give the polynomial regression model (Figure \(\PageIndex{2.5}\)) as

\[\displaystyle \begin{split} \alpha &= a_{0} + a_{1}T + a_{2}T^{2}\\ &= {6.0150} \times {1}{0}^{- 6} + {6.1946} \times {10}^{- 9}T - {1.2278} \times {10}^{- {11}}T^{2} \end{split} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.4}) \nonumber\]

Knowing the values of \(a_{0}\) , \(a_{1}\) , and \(a_{2}\) , we can then find the contraction in the trunnion diameter from Equations \((\PageIndex{2.2})\) and \((\PageIndex{2.3})\) as

\[\begin{split} \displaystyle\Delta D &= D\int_{T_{\text{room}}}^{T_{\text{fluid}}}{(a_{0} + a_{1}T + a_{2}T^{2}}){dT}\\ &= D\left\lbrack a_{0}T + a_{1}\frac{T^{2}}{2} + a_{2}\frac{T^{3}}{3} \right\rbrack\begin{matrix} T_{\text{fluid}} \\ \\ T_{\text{room}} \\ \end{matrix}\\ &= D\lbrack a_{0}(T_{\text{fluid}} - T_{\text{room}}) + a_{1}\frac{({T_{\text{fluid}}}^{2} - {T_{\text{room}}}^{2})}{2}\\ & \ \ \ \ \ + a_{2}\frac{({T_{\text{fluid}}}^{3} - {T_{\text{room}}}^{3})}{3}\rbrack\;\;\;\;\;\;\;\;\;\;\;\;(\PageIndex{2.5}) \end{split} \nonumber\]

Substituting the values of the variables gives

\[\displaystyle \begin{split} \Delta D &= 12.363\begin{bmatrix} 6.0150 \times 10^{- 6} \times ( - 108 - 80) \\ + 6.1946 \times 10^{- 9}\displaystyle \frac{\left( ( - 108)^{2} - (80)^{2} \right)}{2} \\ - 1.2278 \times 10^{- 11}\displaystyle \frac{(( - 108)^{3} - (80)^{3})}{3} \\ \end{bmatrix}\\ &= - 0.013689^{\prime\prime}\end{split} \nonumber\]

What do we find here? The contraction in the trunnion is not enough to meet the required specification of \(0.015^{\prime\prime}\) .

Implementing the Solution

Although we were able to find out why the trunnion got stuck in the hub, we still need to find and implement a solution. What if the trunnion were immersed in a medium that was cooler than the dry-ice/alcohol mixture of \(- 108{^\circ}F\) , say liquid nitrogen, which has a boiling temperature of \(- 321{^\circ}F\) ? Will that be enough for the specified contraction in the trunnion?

As given in Equation \((\PageIndex{2.5})\)

\[\displaystyle \begin{split} \Delta D &= D\int_{T_{\text{room}}}^{T_{\text{fluid}}}{(a_{0} + a_{1}T + a_{2}T^{2}}){dT}\\ &= D\left\lbrack a_{0}T + a_{1}\frac{T^{2}}{2} + a_{2}\frac{T^{3}}{3} \right\rbrack\begin{matrix} T_{\text{fluid}} \\ \\ T_{\text{room}} \\ \end{matrix}\\ &= D\lbrack a_{0}(T_{\text{fluid}} - T_{\text{room}}) + a_{1}\frac{({T_{\text{fluid}}}^{2} - {T_{\text{room}}}^{2})}{2}\\ & \ \ \ \ \ + a_{2}\frac{({T_{\text{fluid}}}^{3} - {T_{\text{room}}}^{3})}{3}\rbrack\;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.5}-repeated) \end{split} \nonumber\]

which gives

\[\displaystyle \begin{split} \Delta D &= 12.363\begin{bmatrix} 6.0150 \times 10^{- 6} \times ( - 321 - 80) \\ + 6.1946 \times 10^{- 9} \displaystyle\frac{\left( ( - 321)^{2} - (80)^{2} \right)}{2} \\ \ - 1.2278 \times 10^{- 11} \displaystyle\frac{(( - 321)^{3} - (80)^{3})}{3} \\ \end{bmatrix}\\ & \\ &= - 0.024420^{\prime\prime} \end{split} \nonumber\]

The magnitude of this contraction is larger than the specified value of \(0.015^{\prime\prime}\).

So here are some questions that you may want to ask yourself later in the course.

1) What if the trunnion were immersed in liquid nitrogen (boiling temperature \(= - 321{^\circ}\text{F}\) )? Will that cause enough contraction in the trunnion?

2) Rather than regressing the thermal expansion coefficient data to a second-order polynomial so that one can find the contraction in the trunnion OD, how would you use the trapezoidal rule of integration for unequal segments?

3) What is the relative difference between the two results?

4) We chose a second-order polynomial for regression. Would a different order polynomial be a better choice for regression? Is there an optimum order of polynomial we could use?

Title: Steps of Solving Engineering Problems

Summary : This video teaches you the steps of solving an engineering problem- define the problem, model the problem, solve, and implementation of the solution.

Lesson 3: Overview of Mathematical Processes Covered in This Course

1) enumerate the seven mathematical processes for which numerical methods are used.

Numerical methods are techniques to approximate mathematical processes. This introductory numerical methods course will develop and apply numerical techniques for the following mathematical processes:

1) Roots of Nonlinear Equations

2) Simultaneous Linear Equations

3) Curve Fitting via Interpolation

4) Differentiation

5) Curve Fitting via Regression

6) Numerical Integration

7) Ordinary Differential Equations.

Some undergraduate courses in numerical methods may include topics of partial differential equations, optimization, and fast Fourier transforms as well.

Roots of a Nonlinear Equation

The ubiquitous formula

\[ x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.1}) \nonumber\]

of finding the roots of a quadratic equation \(\displaystyle ax^{2} + bx + c = 0\) goes back to the ancient world. But in the real world, we get equations that are not just the quadratic ones. They can be polynomial equations of a higher order and transcendental equations.

Take an example of a floating ball shown in Figure \(\PageIndex{3.1}\), where you are asked to find the depth to which the ball will get submerged when floating in the water.

Assume that the ball has a density of \(600\ \text{kg}/\text{m}^{3}\) and has a radius of \(0.055\ \text{m}\) . On applying the Newtons laws of motion and hence equating the weight of the ball to the buoyancy force, one finds that the depth, \(x\) in meters, to which the ball is underwater and is given by

\[\displaystyle 3.993 \times 10^{- 4} - 0.165x^{2} + x^{3} = 0 \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2}) \nonumber\]

Equation \((\PageIndex{3.2})\) is a cubic equation that you will need to solve. The equation will have three roots, and the root that is between \(0\ \text{m}\) (just touching the water surface) and \(0.11\ \text{m}\) (almost submerged) would be the depth to which the ball is submerged. The two other roots would be physically unacceptable. Note that a cubic equation with real coefficients can have a set of one real root and two complex roots or a set of three real roots. You may wonder, why could such an application be important? Let’s suppose you are filling in this tank with water, and you are using this ball as a control so that when the ball goes all the way to the top that the flow of the water stops – say in a fish tank that needs replenishing while the owner is away for a few weeks. So, we do need to figure out how much of the ball is submerged underwater.

A cubic equation can be solved exactly by radicals, but it is a tedious process. The same is true but even more complicated for a general fourth-order polynomial equation as well. However, there is no closed-form solution available for a general polynomial equation of fifth-order or more. So, one has to resort to numerical techniques to solve polynomial and other transcendental nonlinear equations (e.g., finding the nonzero roots of \(\tan x = x\) ).

Simultaneous Linear Equations

Ever since you were exposed to algebra, you have been solving simultaneous linear equations.

Take this problem statement as an example. Suppose the upward velocity of a rocket (Figure \(\PageIndex{3.2}\)) is given at three different times (Table \(\PageIndex{3.1}\)).

The velocity data is approximated by a polynomial as

\[\displaystyle v\left( t \right) = at^{2} + {bt} + c\ {, 5} \leq t \leq {12}.\;\;\;\;\;\;\;\;\;\;\;\;(\PageIndex{3.3}) \nonumber\]

To estimate the velocity at a time that is not given to us, we can set up the equations to find the coefficients \(a,b,c\) of the velocity profile.

The polynomial in Equation (3) is going through three data points \(\left( t_{1},v_{1} \right),\left( t_{2},v_{2} \right),\) and \(\left( t_{3},v_{3} \right)\) where from Table 1.1.3.1

\[\begin{split} t_{1} &= 5,v_{1} = 106.8\\ t_{2} &= 8,v_{2} = 177.2\\ t_{3} &= 12,v_{3} = 600.0 \end{split} \nonumber\]

Requiring that \(v\left( t \right) = at^{2} + bt + c\) passes through the three data points, gives

\[\begin{split} v\left( t_{1} \right) &= v_{1} = at_{1}^{2} + bt_{1} + c\\ v\left( t_{2} \right) &= v_{2} = at_{2}^{2} + bt_{2} + c\\ v\left( t_{3} \right) &= v_{3} = at_{3}^{2} + bt_{3} + c \end{split} \nonumber\]

Substituting the data \(\left( t_{1},\ v_{1} \right),\ \left( t_{2},\ v_{2} \right),\) and \(\left( t_{3},\ v_{3} \right)\) gives

\[\begin{split} a\left( 5^{2} \right) + b\left( 5 \right) + c = 106.8 \\ a\left( 8^{2} \right) + b\left( 8 \right) + c = 177.2 \\ a\left( 12^{2} \right) + b\left( 12 \right) + c = 600.0 \end{split} \nonumber\]

\[\begin{split} 25a + 5b + c = 106.8 \\ 64a + 8b + c = 177.2 \\ 144a + 12b + c = 600.0 \end{split} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.4}) \nonumber\]

Solving a few simultaneous linear equations such as the above set can be done without the knowledge of numerical techniques. However, imagine that instead of three given points, you were given 10 data points. Now the setting up as well as solving the set of 10 simultaneous linear equations without numerical techniques becomes laborious, if not impossible.

Curve Fitting by Interpolation

Interpolation involves that given a function as a set of data points. How does one find the value of the function at points that are not given?. For this, we choose a function, called an interpolant, and make it pass through all the points involved.

You may think that you have already used interpolation in courses such as Thermodynamics and Statistics. After all, it was just taking two points from a table at the back of the textbook or online and finding the value of the function at a point in between by using a straight line.

Take this problem statement as an example. Let’s suppose the upward velocity of a rocket is given at three different times (Table 1.1.3.1).

If one asked you to estimate the velocity at \(7\ \text{s}\) , one might simply use the straight-line formula you are most accustomed to as given below.

Given ( \(t_{1},\ v_{1}\) ) and ( \(t_{2},\ v_{2}\) ), the value of the function \(v\) at \(t\) is given by

\[\displaystyle v = v_{1} + \frac{v_{2} - v_{1}}{t_{2} - t_{1}}(t - t_{1}) \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.5}) \nonumber\]

Although this is possibly enough for courses such as Thermodynamics and Statistics, there are two questions to ask. Is the value calculated accurately, and how accurate is it? To know that, one needs to calculate at least more than one value. In the above example of a rocket velocity vs. time, one can instead use a second-order polynomial interpolant and set up the three equations and three unknowns to find the unknown coefficients, \(a\) , \(b\) , and \(c\) as given in the previous section.

\[v\left( t \right) = at^{2} + {bt} + c{ ,\ 5} \leq t \leq {12}\;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.6}) \nonumber\]

Then, the resulting second-order polynomial can be used to find the velocity at \(t = 7\ \text{s}\) .

The value obtained from the second-order polynomial (Equation \((\PageIndex{3.6})\)) can be considered to be a new measure of the value of \(v( 7)\) , and the first-order polynomial (Equation \((\PageIndex{3.5})\)) result can be used to determine the accuracy of the results.

Numerical Differentiation

You have taken a semester-long course in Differential Calculus, where you found derivatives of continuous functions. So let’s suppose somebody gives you the velocity of a rocket as a continuous and at least once differentiable function of time and wants you to find acceleration. Indeed for this particular problem, you can use your differential calculus knowledge to differentiate the velocity function to get the acceleration and put in the value of time, \(t = 7\ \text{s}\) . What if the velocity vs. time is not given as a continuous and at least once differentiable function? Instead, let’s say the function is given at discrete data points (Table 1.1.3.1). How are you then going to find out what the acceleration at \(t = 7\ \text{s}\) ? Do we draw a straight line from \((5,106.8)\) to \((8,177.2)\) and use the straight-line slope as the estimate of acceleration? How do we know that this is adequate? We could incorporate all three points and find a second-order polynomial as given by Equation \((\PageIndex{3.6})\). This polynomial can now be differentiated to estimate the acceleration at \(t = 7\ \text{s}\) . Now the two values can be used to evaluate the accuracy of the calculated acceleration.

Curve Fitting by Regression

When we talked about curve fitting by interpolation, the chosen interpolant needs to go through all the points considered. What happens when we are given many data points, and we instead want a simplified formula to explain the relationship between two variables. See, for example, in Figure \(\PageIndex{3.4a}\), we are given the coefficient of linear thermal expansion data for cast steel as a function of temperature. Looking at the data, one may proclaim that a straight line could explain the data, and that is drawn in Figure \(\PageIndex{3.4b}\). How we draw this straight line is what is called regression. It would be based on minimizing some form of the residuals between what is observed (given data points) and what is predicted (straight line). It does not mean that every time you have data given to you, you draw a straight line. It is possible that a second-order polynomial or a transcendental function other than the first-order polynomial will be a better representation of this particular data. So these are the questions that we will answer when we discuss regression. We will also discuss the adequacy of linear regression models.

Numerical Integration

You have taken a whole course on integral calculus. Now, why would we need to make numerical approximations of integrals? Just like the standard normal cumulative distribution function

\[\displaystyle\Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{x}e^{- t^{2}/2}{dt} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.7}) \nonumber\]

cannot be solved exactly, or when the integrand values are given at discrete data points, we need to use numerical methods of integration.

In the previous lesson, we looked at the example of contracting the diameter of a trunnion for a bascule bridge fulcrum assembly by dipping it in a mixture of dry ice and alcohol. The contraction is given by

\[\Delta D = D\int_{T_{\text{room}}}^{T_{\text{fluid}}}{\alpha\ dT} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.8}) \nonumber\]

\[D = \text{outer diameter of the trunnion,} \nonumber\]

\[\alpha = \text{coefficient of linear thermal expansion that is varying with temperature} \nonumber\]

\[T_{\text{room}}= \text{room temperature} \nonumber\]

\[T_{\text{fluid}}= \text{temperature of dry-ice alcohol mixture.} \nonumber\]

From Figure \(\PageIndex{3.4a}\), one can note that the coefficient of thermal expansion is only given at discrete temperatures and not as a known continuous function that could be integrated exactly. So we have to resort to numerical methods by approximating the data, say, by a second-order polynomial obtained via regression.

In Figure \(\PageIndex{3.6}\), the thermal expansion coefficient of typical cast steel is approximated by a second-order regression polynomial as given by Equation \((\PageIndex{3.9}\)) (how we get this is a later lesson in regression) as

\[\displaystyle\alpha = - 1.2278 \times 10^{- 11}T^{2} + 6.1946 \times 10^{- 9}T + 6.0150 \times 10^{- 6} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.9}) \nonumber\]

The contraction of the diameter then is given by

\[\displaystyle\Delta D = D\int_{T_{\text{room}}}^{T_{\text{fluid}}}{\left( - 1.2278 \times 10^{- 11}T^{2} + 6.1946 \times 10^{- 9}T + 6.015 \times 10^{- 6} \right){dT}} \;\;\;\;\;\;\;\; (\PageIndex{3.10}) \nonumber\]

and can now be calculated using integral calculus.

Numerical Solution of Ordinary Differential Equations

Taking the same example of the trunnion being dipped in a dry-ice/alcohol mixture, one could ask the question - What would the temperature of the trunnion be after dipping it in the mixture for 30 minutes? The model is given by an ordinary differential equation for the temperature \(\theta\) as a function of time, \({t.}\)

\[\displaystyle -{hA} \left(\theta - \theta_{a} \right) = {mC} \frac{d \theta}{dt}\;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.11}) \nonumber\]

\[h= \text{the convective cooling coefficient,}\ \text{W/m}^{2} \cdot \text{K} \nonumber\]

\[A = \text{surface area},\ \text{m}^2 \nonumber\]

\[\theta_{a} = \text{ambient temperature of dry-ice/alcohol mixture},\ \text{K} \nonumber\]

\[m = \text{mass of the trunnion, kg} \nonumber\]

\[C = \text{specific heat of the trunnion,}\ \text{J/(kg} \cdot \text{K)} \nonumber\]

The differential Equation \((\PageIndex{3.11})\) can be solved exactly by using the classical solution, Laplace transform, or separation of variables techniques. So, where do numerical methods enter into the picture for this problem? For the temperature range of room temperatures to cold media such as dry-ice/alcohol, several of the variables in Equation \((\PageIndex{3.11})\) are not constant but change with the temperature. These include the convection coefficient \(h\) as well as the specific heat \(C\) . Now, this differential equation has turned nonlinear as follows.

\[\displaystyle -h(\theta)A\left( \theta - \theta_{a} \right) = mC(\theta)\frac{d \theta}{dt} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.12}) \nonumber\]

The ordinary differential Equation \((\PageIndex{3.12})\) cannot be solved by exact methods and would need to be solved by a numerical method.

In the above discussion, we have illustrated the need for numerical methods for each of the seven mathematical processes in the course. In the lessons to follow, we will be developing various numerical techniques to approximate the mathematical processes to calculate acceptable accurate values while calculating associated errors.

Title: Overview of Mathematical Processes Covered in This Course

Summary : This lecture shows you four mathematical procedures that need numerical methods - namely, nonlinear equations, differentiation, simultaneous linear equations, and interpolation.

Multiple Choice Test

(1). Solving an engineering problem requires four steps. In order of sequence, the four steps are

(A) formulate, model, solve, implement

(B) formulate, solve, model, implement

(C) formulate, model, implement, solve

(D) model, formulate, implement, solve

(2). One of the roots of the equation \(x^{3} - 3x^{2} + x - 3 = 0\) is

(C) \(\sqrt{3}\)

(3). The solution to the set of equations

\[\begin{split} 25a + b + c &= 25 \\ 64a + 8b + c &= 71\\ 144a + 12b + c &= 155 \end{split} \nonumber\]

most nearly is \(\left( a,b,c \right) =\)

(A) \((1,1,1)\)

(B) \((1,-1,1)\)

(C) \((1,1,-1)\)

(D) does not have a unique solution.

(4). The exact integral of \(\displaystyle \int_{0}^{\frac{\pi}{4}} 2 \cos 2x \ dx\) is most nearly

(A) \(-1.000\)

(B) \(1.000\)

(C) \(0.000\)

(D) \(2.000\)

(5). The value of \(\displaystyle \frac{dy}{dx}\left( 1.0 \right)\) , given \(y = 2\sin\left( 3x \right)\), is most nearly

(A) \(-5.9399\)

(B) \(-1.980\)

(C) \(0.31402\)

(D) \(5.9918\)

(6). The form of the exact solution of the ordinary differential equation \(\displaystyle 2\frac{dy}{dx} + 3y = 5e^{- x},\ y\left( 0 \right) = 5\) is

(A) \(Ae^{- 1.5x} + Be^{x}\)

(B) \(Ae^{- 1.5x} + Be^{- x}\)

(C) \(Ae^{1.5x} + Be^{- x}\)

(D) \(Ae^{- 1.5x} + Bxe^{- x}\)

For complete solution, go to

http://nm.mathforcollege.com/mcquizzes/01aae/quiz_01aae_introduction_answers.pdf

Problem Set

(1). Give one example of an engineering problem where each of the following mathematical procedures is used. If possible, draw from your experience in other classes or from any professional experience you have gathered to date.

a) Differentiation

b) Nonlinear equations

c) Simultaneous linear equations

d) Regression

e) Interpolation

f) Integration

g) Ordinary differential equations

(2). Only using your nonprogrammable calculator, find the root of

\[x^{3} - 0.165x^{2} + 3.993 \times 10^{- 4} = 0 \nonumber\]

by any method. Hint: Find one root by hit and trial, and use long division for factoring the polynomial.

\(0.06237758151,\ 0.1463595047,\ -0.04373708621\)

(3). Solve the following system of simultaneous linear equations by any method

\[\begin{split} 25a + 5b + c &= 106.8\\ 64a + 8b + c &= 177.2\\ 144a + 12b + c &= 279.2 \end{split} \nonumber\]

\(a = 0.2904761905,\ b = 19.69047619,\ c = 1.085714286\)

(4). You are given data for the upward velocity of a rocket as a function of time in the table below. Find the velocity at \(t = 16 \ \text{s}\) .

\(543.0420000\ \text{m/s}\)

(5). Integrate exactly.

\[\int_{0}^{\pi/2}\sin2x \ dx \nonumber\]

\[\frac{dy}{dx}(x = 1.4) \nonumber\]

\[y = e^{x} + \sin(x) \nonumber\]

\(4.225167110\)

(7). Solve the following ordinary differential equation exactly.

\[\frac{dy}{dx} + y = e^{- x}, \ y(0) = 5 \nonumber\]

Also find \(y(0),\ \displaystyle \frac{dy}{dx}\ (0),\ y(2.5),\ \displaystyle\frac{dy}{dx}\ (2.5)\)

\(\displaystyle y(0)=5,\ \frac{dy}{dx}(0)=-4,\ y(2.5)=0.61563,\ \frac{dy}{dx}(2.5)=-0.53355\)

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For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

Inquiry-based approaches to mathematics learning, teaching, and mathematics education research

• Published: 23 March 2021
• Volume 24 , pages 123–126, ( 2021 )

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Three of the four articles in this issue (Andrews-Larson, Johnson, Peterson, & Keller; Wright; Florensa, Bosch, & Gascón) deal explicitly with supporting mathematics teachers to move from teaching approaches characterized as traditional toward approaches based on inquiry. This focus foregrounds concern for pedagogy but in each case teachers’ knowledge, conceptualized in different ways, is also a consideration. For Weiland, Orrill, Nagar, Brown, and Burke, the knowledge upon which middle school teachers can draw in their teaching of proportional reasoning and that constitutes robust understanding of that concept is the central focus. Nevertheless, concern for pedagogy, particularly aspects of pedagogical knowledge, is also apparent in their report. As a collection, the articles offer diverse conceptualizations of mathematical knowledge and raise questions about that knowledge, what it means to do mathematics and the broad social purposes of that activity, the intersection of research and teaching, and the relationship between teachers and researchers. All of the studies reported involved small numbers of participants and detailed analyses of qualitative data and hence questions as to the scalability of the approaches suggested also arise.

Andrews-Larson et al. examined how two groups of undergraduate mathematics instructors engaged in pedagogical reasoning as part of a workshop on inquiry-oriented instruction. Each group engaged with a mathematics task in either abstract algebra or linear algebra firstly in the role of doers of mathematics and then as mathematics instructors as they viewed video of students working on the same task. Andrews-Larson et al. found that the more deeply the mathematicians engaged with the mathematics inherent in the task, the greater was their engagement with the evidence of student mathematical reasoning evident in the video. The group that engaged more deeply with the mathematics and student reasoning were more likely to focus on supporting students in their mathematical work and maintaining the students’ ownership of the ways in which they represented the mathematics. This was in contrast to the group that engaged less deeply with the mathematics and hence with the evidence of student thinking, that focussed more on describing the representational choices that the students made and on evaluating the students’ contributions. Andrews-Larson et al. note subtle differences in the ways in which the facilitators of the two groups oriented the participants to their task and speculate that this may account for the difference in the foci of the activity of the two groups. The authors further venture that an initial focus on mathematics rather than on pedagogical issues may be a useful entry point into examining student reasoning for mathematicians because of the greater likelihood that they will share common understandings and beliefs in this regard compared to in relation to matters related to instruction. In addition, they point to the value of using evidence of students’ partially formulated understandings rather than completed solutions as a prompt for discussion of students’ reasoning.

Andrews-Larson et al. are cognisant of the limitations of their study and of the range of important questions that their findings offer for further research. Although considerable research has been conducted on helping schoolteachers to analyze, respond to, and build upon student thinking, its application in tertiary settings has mainly been to teacher education rather than to undergraduate mathematics teaching (Andrews-Larson et al.). While there is scope to test the applicability of findings from school mathematics to undergraduate mathematics, the reverse should also be considered. To what extent, for example, might the quality of schoolteachers’ engagement with the mathematics inherent in tasks support their capacity to engage with student reasoning about those tasks in productive ways? Certainly, for Weiland et al., middle school teachers’ own robust understanding of proportional reasoning is fundamental to their ability to teach these ideas as it constitutes the knowledge resources upon which they can draw in their teaching.

Like Andrews-Larson et al., Wright frames the need for change to dominant approaches to mathematics teaching in terms of the negative impacts, both cognitive and affective, of such teaching for many students, but goes beyond this to draw attention to the socio-political implications of traditional mathematics teaching practices in which attainment correlates with students’ socioeconomic status, and students deemed less capable experience impoverished curricula and pedagogies. He argues that traditional mathematics education has not recognized its inherently political nature nor sufficiently acknowledged the contextual constraints within which teachers work. Wright reports on the Teaching Mathematics for Social Justice project in which he worked alongside five secondary mathematics teachers using participatory action research to support the teachers to change their practice in line with the aims of the project. He describes how the use of tasks that used mathematics to address questions of justice such as fair trade led to increased student engagement with and enjoyment of mathematics. In addition, he notes shifts in the teachers’ discourse around student ability and attainment that included questioning commonplace practices such as setting (grouping according to prior attainment) and recognizing that, rather than being a meritocracy, education can perpetuate inequity. Teachers began to articulate views consistent with Jorgensen’s ( 2016 ) claim that less advantaged students need to be assisted to develop the skills and attitudes that middle class students bring to their learning and that position them to take full advantage of the opportunities afforded them to learn. Wright acknowledges the role that he played as researcher in bringing an external perspective to the collaboration in a way that is reminiscent of Lerman’s ( 1997 ) notion of a second voice.

Both Wright and Andrews-Larson et al. implicitly acknowledge precursors to changed teaching practices that mathematics educators have been aware of for decades. Jakubowski and Tobin ( 1991 ), for example, recognized that teachers will only do the significant work required to substantially change their classrooms if they are convinced that the effort is warranted, and Nespor ( 1987 ) articulated the additional requirement that change requires the availability of a feasible alternative paradigm. Although these conditions are somewhat obvious, the field is still grappling with how they can be achieved. Whereas Andrews-Larson et al. cite evidence that many mathematicians acknowledge the need to reform undergraduate mathematics teaching and many report using alternatives to traditional lectures alone, Wright presents participatory action research as a means of bringing teachers to a place of problematizing usual practice, thereby motivating change. The provision of tasks that provided an entree into inquiry-oriented instruction (Andrews et al.) or mathematics for social justice (Wright) went some way toward offering an alternative paradigm for the teachers to adopt. Questions remain, however, in both cases, as to the extent to which the teachers concerned were able to extend the new approaches to their teaching more broadly. In the case of Wright’s project, there was encouraging evidence that the tasks used by project teachers were being adopted more widely in their schools but given that the focus of the project was on social justice with mathematics as a powerful tool for investigating issues, concerns about the extent to which such an approach can be applied across the mathematics curriculum arise. It could be that the changes in teachers’ thinking about the capabilities of students deemed low attainers, such as Wright observed, is a key motivator of sustained change (Beswick, 2018 ).

Weiland et al. point to a disparity between the recognized importance of proportional reasoning and the relative lack of research on teachers’ knowledge of proportions. Research that does exist tends to focus on deficits in teachers’ knowledge (e.g. Lobato, Orrill, Druken, & Jacobsen, 2011 ) and the difficulties that it presents for teachers (e.g. Lamon, 2007 ). Weiland et al. drew from the literature to develop a framework for robust understanding of proportional reasoning and from clinical interviews with 32 practising middle school teachers to identify a set of observable knowledge resources that underpin robust understanding of proportional reasoning. Weiland et al.’s conceptualization of robust understanding as an interconnected system of knowledge resources that can be accessed and used in differing combinations in different contexts draws upon ideas Knowledge in Pieces (e.g. diSessa, Sherin, & Levin, 2016 ). They illustrate the potential use of their framework for analyzing teachers’ understanding of proportions and for supporting further development of that understanding. They make the point that has some resonance with Andrews-Larson’s speculation about the potential value of using mathematics as an entry point into analyzing student thinking, that robust understanding of mathematics is a necessary precondition for being able to respond in the moment to students’ ideas. There can be little argument that inquiry-based approaches such as advocated by the Andrews-Larson et al., Wright, and Florensa et al. demand robust understanding, as defined by Weiland et al., of the relevant mathematics.

Florensa et al. address the difficulty that teachers face in making significant changes to their practice in response to curriculum changes or the adoption of unfamiliar pedagogies. They posit Question–Answer (Q–A) maps as a tool to assist teachers to identify and analyze the content they are to teach. Florensa et al.’s concern was also that teachers question taken-for-granted understandings and ways of viewing aspects of content and pedagogy such as the phenomenon of functions being taught in isolation from related mathematical domains. They also challenge the usual research practice of maintaining a separation between researchers’ tools and the tools made available to teachers. From their work with 16 secondary mathematics teachers from four Latin American countries in an online course, Florensa et al. found Q–A maps to be a useful in assisting teachers to problematize existing epistemologies in their schools. In emphasising the questionable and dynamic nature of knowledge, the authors consider Q–A maps to be emancipatory and able to broaden the description of the relevant mathematical knowledge beyond that contained in curricula and other documents. Florensa et al. also point to the adaptation of research tools for teachers as an important way to support teachers in this work.

Juxtaposed with Wright’s article, one wonders how and to what extent if any research such as that reported by Andrews-Larson et al., Weiland et al., and to a lesser extent perhaps Florensa et al. should or might take account of the socio-political environment in which it is conducted and the diverse contexts in which it might be applied. It is also worth remembering that each of these studies, including Wright’s, was conducted in particular countries and schools which will have similarities with and differences from other schools, systems, and cultures. Questions are many and concern, for example, the extent to which the knowledge resources that Weiland et al. observed among US middle school teachers might be unique to that context, the ways in which teachers in countries other than the UK in which Wright worked might respond to the idea of social justice through mathematics education, and how might researchers and teachers in both of these countries learn from the Anthropological Theory of the Didactic (Bosch & Gascón, 2006 ) that underpinned Florensa et al.’s work. This is a thought-provoking collection of articles that highlight how much more we have to learn.

Beswick, K. (2018). Systems perspectives on mathematics teachers’ beliefs: Illustrations from beliefs about students. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd conference of the International Group for the Psychology of Mathematics Education ( Vol. 1 , pp. 3–18). Umeå, Sweden: PME.

Bosch, M., & Gascón, J. (2006). Twenty-five years of the didactic transposition. ICMI Bulletin, 58, 51–65.

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Beswick, K. Inquiry-based approaches to mathematics learning, teaching, and mathematics education research. J Math Teacher Educ 24 , 123–126 (2021). https://doi.org/10.1007/s10857-021-09494-4

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The Algebra Problem: How Middle School Math Became a National Flashpoint

Top students can benefit greatly by being offered the subject early. But many districts offer few Black and Latino eighth graders a chance to study it.

By Troy Closson

From suburbs in the Northeast to major cities on the West Coast, a surprising subject is prompting ballot measures, lawsuits and bitter fights among parents: algebra.

Students have been required for decades to learn to solve for the variable x, and to find the slope of a line. Most complete the course in their first year of high school. But top-achievers are sometimes allowed to enroll earlier, typically in eighth grade.

The dual pathways inspire some of the most fiery debates over equity and academic opportunity in American education.

Do bias and inequality keep Black and Latino children off the fast track? Should middle schools eliminate algebra to level the playing field? What if standout pupils lose the chance to challenge themselves?

The questions are so fraught because algebra functions as a crucial crossroads in the education system. Students who fail it are far less likely to graduate. Those who take it early can take calculus by 12th grade, giving them a potential edge when applying to elite universities and lifting them toward society’s most high-status and lucrative professions.

But racial and economic gaps in math achievement are wide in the United States, and grew wider during the pandemic. In some states, nearly four in five poor children do not meet math standards.

To close those gaps, New York City’s previous mayor, Bill de Blasio, adopted a goal embraced by many districts elsewhere. Every middle school would offer algebra, and principals could opt to enroll all of their eighth graders in the class. San Francisco took an opposite approach: If some children could not reach algebra by middle school, no one would be allowed to take it.

The central mission in both cities was to help disadvantaged students. But solving the algebra dilemma can be more complex than solving the quadratic formula.

New York’s dream of “algebra for all” was never fully realized, and Mayor Eric Adams’s administration changed the goal to improving outcomes for ninth graders taking algebra. In San Francisco, dismantling middle-school algebra did little to end racial inequities among students in advanced math classes. After a huge public outcry, the district decided to reverse course.

“You wouldn’t think that there could be a more boring topic in the world,” said Thurston Domina, a professor at the University of North Carolina. “And yet, it’s this place of incredibly high passions.”

“Things run hot,” he said.

In some cities, disputes over algebra have been so intense that parents have sued school districts, protested outside mayors’ offices and campaigned for the ouster of school board members.

Teaching math in middle school is a challenge for educators in part because that is when the material becomes more complex, with students moving from multiplication tables to equations and abstract concepts. Students who have not mastered the basic skills can quickly become lost, and it can be difficult for them to catch up.

Many school districts have traditionally responded to divergent achievement levels by simply separating children into distinct pathways, placing some in general math classes while offering others algebra as an accelerated option. Such sorting, known as tracking, appeals to parents who want their children to reach advanced math as quickly as possible.

But tracking has cast an uncomfortable spotlight on inequality. Around a quarter of all students in the United States take algebra in middle school. But only about 12 percent of Black and Latino eighth graders do, compared with roughly 24 percent of white pupils, a federal report found .

“That’s why middle school math is this flashpoint,” said Joshua Goodman, an associate professor of education and economics at Boston University. “It’s the first moment where you potentially make it very obvious and explicit that there are knowledge gaps opening up.”

In the decades-long war over math, San Francisco has emerged as a prominent battleground.

California once required that all eighth graders take algebra. But lower-performing middle school students often struggle when forced to enroll in the class, research shows. San Francisco later stopped offering the class in eighth grade. But the ban did little to close achievement gaps in more advanced math classes, recent research has found.

As the pendulum swung, the only constant was anger. Leading Bay Area academics disparaged one another’s research . A group of parents even sued the district last spring. “Denying students the opportunity to skip ahead in math when their intellectual ability clearly allows for it greatly harms their potential for future achievement,” their lawsuit said.

The city is now back to where it began: Middle school algebra — for some, not necessarily for all — will return in August. The experience underscored how every approach carries risks.

“Schools really don’t know what to do,” said Jon R. Star, an educational psychologist at Harvard who has studied algebra education. “And it’s just leading to a lot of tension.”

In Cambridge, Mass., the school district phased out middle school algebra before the pandemic. But some argued that the move had backfired: Families who could afford to simply paid for their children to take accelerated math outside of school.

“It’s the worst of all possible worlds for equity,” Jacob Barandes, a Cambridge parent, said at a school board meeting.

Elsewhere, many students lack options to take the class early: One of Philadelphia’s most prestigious high schools requires students to pass algebra before enrolling, preventing many low-income children from applying because they attend middle schools that do not offer the class.

In New York, Mr. de Blasio sought to tackle the disparities when he announced a plan in 2015 to offer algebra — but not require it — in all of the city’s middle schools. More than 15,000 eighth graders did not have the class at their schools at the time.

Since then, the number of middle schools that offer algebra has risen to about 80 percent from 60 percent. But white and Asian American students still pass state algebra tests at higher rates than their peers.

The city’s current schools chancellor, David Banks, also shifted the system’s algebra focus to high schools, requiring the same ninth-grade curriculum at many schools in a move that has won both support and backlash from educators.

And some New York City families are still worried about middle school. A group of parent leaders in Manhattan recently asked the district to create more accelerated math options before high school, saying that many young students must seek out higher-level instruction outside the public school system.

In a vast district like New York — where some schools are filled with children from well-off families and others mainly educate homeless children — the challenge in math education can be that “incredible diversity,” said Pedro A. Noguera, the dean of the University of Southern California’s Rossier School of Education.

“You have some kids who are ready for algebra in fourth grade, and they should not be denied it,” Mr. Noguera said. “Others are still struggling with arithmetic in high school, and they need support.”

Many schools are unequipped to teach children with disparate math skills in a single classroom. Some educators lack the training they need to help students who have fallen behind, while also challenging those working at grade level or beyond.

Some schools have tried to find ways to tackle the issue on their own. KIPP charter schools in New York have added an additional half-hour of math time to many students’ schedules, to give children more time for practice and support so they can be ready for algebra by eighth grade.

At Middle School 50 in Brooklyn, where all eighth graders take algebra, teachers rewrote lesson plans for sixth- and seventh-grade students to lay the groundwork for the class.

The school’s principal, Ben Honoroff, said he expected that some students would have to retake the class in high school. But after starting a small algebra pilot program a few years ago, he came to believe that exposing children early could benefit everyone — as long as students came into it well prepared.

Looking around at the students who were not enrolling in the class, Mr. Honoroff said, “we asked, ‘Are there other kids that would excel in this?’”

“The answer was 100 percent, yes,” he added. “That was not something that I could live with.”

Troy Closson reports on K-12 schools in New York City for The Times. More about Troy Closson

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Softening the sharp edges in mathematics.

For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start — to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics — a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

“Mathematics should be flexible, conceptual, a place where we play with ideas and make connections," says Professor Jo Boaler. (Photo: Robert Houser Photography)

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced “How to Learn Math,” the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users. 

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer. 

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 — but the most common answer 13-year-olds gave was 19. The second most common was 21. 

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important? 

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mind-set. I think we all need a little more ish in our lives. 

You also argue that mathematics should be taught in more visual ways. What do you mean by that? 

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things — all of that contributes to our understanding of how it works. 

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking. 

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, ‘I’m in geometry class now, and I still remember that  sugar cube, what it looked like and felt like.’ His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing — elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

Click to enlarge: A depiction of various ways to calculate 38 x 5, numerically and visually. (Image: Courtesy of Jo Boaler)

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them. 

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

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