Math Problem Solving in Action

Getting Students to Love Word Problems, Grades K-2

Nicki Newton

• Publisher: Routledge
• Year: 2017
• Print ISBN: 9781138056466
• eBook ISBN: 9781351676793
• Edition: 1
• Topic: Education
• Subtopic: Education General

In this new book from popular math consultant and bestselling author Dr. Nicki Newton, you'll learn how to help students in grades K–2 become more effective and confident problem solvers. Problem solving is a necessary skill for the 21st century but can be overwhelming for both teachers and students alike. Dr. Nicki shows how to make word problems more engaging and fun, starting in the early elementary grades so that students have a strong foundation to build upon.

Topics include:

• Using games, songs, and poems to make complex mathematical concepts more accessible;
• Showing students how to reason about, model, and discuss word problems;
• Implementing problem-solving workshops and workstations to maximize practice time and encourage collaboration;
• Teaching students to recognize and apply elementary mathematical concepts—including addition, subtraction, part-whole relationships, etc.—in real-life situations;
• Incorporating different types of assessment to measure student progress and help them get to the next level.

Each chapter offers examples, charts, and tools that you can use immediately. The book also features a set of action plans so you can move forward with confidence and implement the book's ideas in your own classroom.

Part I

Introduction to Word Problems

1

Play-Doh^®, Puppets and Other Real Stuff Make Problems Make Sense

Play-Doh Matters

Start with Play-Doh! I guarantee that you will immediately have your students’ attention. Next add puppets. You’ll have them at Go! Continue with other real stuff (like stickers, toy cars and marbles) and students will start to love word problems. They will associate them with doing fun stuff. Fun is always a great hook! Although researchers have found that story problems “are notoriously difficult to solve” (Cummins, Kintsh, Reusser and Weimer, 1988), I argue they don’t have to be. The way that we currently teach them is often off-putting to students. We have to shift how we are doing things. Really, more than shift, we need to go back to what we used to do (over 30 years ago).

Students grow to hate problem solving by 3rd grade. But it should be presented as a challenge. Students should look forward to working with word problems because it’s the stuff they do every day. We simply have to mathematize the stuff they do every day. Students should own the problems they solve and pose. They should solve problems about their families, their friends, their daily activities, their school and their lives. If the problems made sense to the students, then students could make sense of the problems.

Wording Matters

Wording matters. There is a famous example of how students understand the meaning of problems but how sometimes the language gets in the way. Hudson (1983) posed this problem to some children: “There are 5 birds and 3 worms. How many more birds are there than worms?” Many of the children could not solve the problem. However, when the problem was reworded, “How many birds won’t get a worm?” many of the students could solve the problem. Riley, Greeno and Heller (1983) said that rewording helps students to understand the problem, and when students understand the problem, they can solve it. Cummins (1991) pointed out that “the data seem to indicate that the knowledge [to solve problems] is there, but is simply is not accessed when problems are worded in certain ways” (p. 267). She argues that students fail to do so because they are missing or “have inadequate mappings of verbal expressions to part-whole structures”. She maintains that “rewording enhances performance.”

So for example, if we say, “There are 8 kids and 5 cupcakes. How many more kids are there than cupcakes?”, students are puzzled. But, if you say, “There are 8 kids and 5 cupcakes. How many kids don’t get a cupcake?”, the students will say 3. If you say, “How many more cupcakes do we need so that everybody gets one?”, the students know the answer. See, this pertains to their everyday lives. They fully understand this scenario and have no trouble reasoning about it. After the understanding is there, the teacher can scaffold the problem with the academic language. There are more kids than cupcakes. There are fewer cupcakes than kids.

Scaffolding Matters

Templates scaffold the process. Pólya (1957) first laid out the process for us with his problem-solving questions with his 4 phases (see Figure 1.1). Templates build on Pólya’s original phases (see Figure 1.2). They help to scaffold the thinking and, with use over time, the students begin to internalize the process. Eventually, the templates are phased out. Have anchor charts up that talk about the process of problem solving. Also, have the students make their own mini anchor charts. Once the template use is phased out, still have the students sketch out the template into their problem-solving notebooks. When students are organized they are much more likely to succeed.

Three Types of Templates

Templates should be used in a gradual release cycle. I do, we do, you do. See Figures 1.2, 1.3 and 1.4 for examples.

Figure 1.1

Figure 1.2 is a fully scaffolded template.

Figure 1.2

Figure 1.3 is a partially scaffolded template. The students get to choose.

Figure 1.3

Figure 1.4 is an unscaffolded template. This template is more of an idea bank with strategies and models that students can choose from.

Figure 1.4

Models Matter

Students need to be able to explain and discuss how they are representing the problem. What models are they using specifically? I define a model as how students are showing their thinking about the problem. This is different than a strategy, which is what they are doing with the numbers. Oftentimes, these words get used interchangeably, but I think it is important to have students explain both what their model is and what their strategy is. Therefore, in the examples in the word problem type chapters, I have shown various models and discussed various strategies.

In terms of models, there are 3 categories: concrete, pictorial and abstract. Concrete models can be cubes, tiles, bears, or any type of real object (marbles, dolls, toy cars, etc.). Pictorial models are drawings or sketches. Students are encouraged to use sketches because these are quick and to the point, whereas drawings can take quite some time (like 10 minutes to draw a marble). So teach the children to do sketches so they can get on with the business of problem solving. The sketch is just a tool to use for thinking. Abstract models include number grids, number lines and tape diagrams.

There are so many types of models, but most state standards name the ones that students must absolutely know at particular grade levels. Most state standards are now putting a heavy emphasis on tape diagrams and number lines (both marked and open). It is important for students to have toolkits with both concrete tools, sketch paper and templates. Templates are also a tool for thinking (number frames, number bonds, part-part whole mats, cube pictures, etc.).

Strategies Matter

Students need to be able to explain how they are thinking about the numbers. Are they counting on, counting up or using known facts? What exactly are they doing with the numbers and can they explain it? Does it make sense? Are they using the facts that they are learning? Because what good does it do for students to learn their doubles facts and then never use them when solving problems? So it is important for teachers and students to talk about the strategies they are using and how we use different strategies for different problems so that we can be efficient problem solvers.

Perseverance Matters

One of the most important things that students need to know about problem solving is that they have to persevere with the problems. They need to get the idea that they have to stick with it and can’t give up. There should be mini-lessons around perseverance. There are so many ways to teach and talk about this now. There are videos, picture books, songs and posters (see www.pinterest.com/drnicki7/growth-mindsetperseverance/). There are several great videos. Sesame Street has done some great videos for the primary grades, including Don’t Give Up featuring Bruno Mars and the Muppets, as well as The Power of Yet featuring Janelle Monáe and the Muppets.

Questioning Matters

Good questions are the building blocks of good problem solving. We shouldn’t ever give answers. We should only scaffold thinking with good questions (see Figure 1.5). When a child says they don’t know, always ask them to look in their toolkit (an actual one with tools appropriate to the grade) or use a template (an actual one that is part of their toolkits). Never ask someone else to help because the minute you do this, you just taught the child who had the question that they don’t have to persevere. You have in essence said, “Don’t stick with it. I’ll send someone in to save you.” You didn’t intentionally say it, but that is the message received. This is how students learn helplessness. Instead, when a child says they are stuck, say, “What could you use to help?” If you need to, suggest a starting point. But never overscaffold. Sometimes teachers will say, “Take out the 10 frames. Now put 4 on the top and 3 on the bottom. Now how many do you have altogether?” Okay—if you do that you are guilty of overscaffolding. Don’t overscaffold. Let your students think. Let them wrestle with the problem. Let them figure out that they can figure it out!

Figure 1.5

Key Points

• Play-Doh Matters
• Wording Matters
• Scaffolding Matters
• Perseverance Matters
• Questioning Matters

Summary

Problem solving is about helping students to see how math is part of our everyday lives. The goal is to foster flexibility, competence and confidence. Since the process is involved, problems should relate to everyday life so that they make sense to the students. Teachers need to word problems in ways that build conceptual understanding. Templates help to scaffold access to all the moving parts of the problem from the beginning to the end. Perseverance should be explicitly taught so that students learn to stick with it when they get stuck. Good questions are powerful tools to scaffold student work around problem solving.

Reflection Questions

1. Do you use real-life stuff when you teach problem solving?
2. Do you carefully choose the wording so that students understand the problem?
3. Do you use templates to scaffold the process?
4. Do you teach perseverance explicitly?
5. What is your big takeaway from this chapter? What one thing in your teaching will you start or expand?

References

Cummins, D. (1991). Children’s interpretations of arithmetic word problems. Cognition and Instruction, 8(3), 261–289.

Cummins, D. D., Kintsh, W., Reusser, K., and Weimer, R. (1988). The role of understanding in solving word problems. Cognitive Psychology, 20(4), 405–438.

Hudson, T. (1983). Correspondences and numerical differences between disjoint sets. Child Development, 54(1), 84–90.

Pólya, G. (1957). How to solve it: A new aspect of mathematical method. Garden City, NY: Doubleday.

Riley, M. S., Greeno, J. G., and Heller, J. I. (1983). Development of children’s problem-solving ability in arithmetic. In H. P. Ginsberg (Ed.), The development of mathematical thinking (pp. 153–196). Orlando, FL: Academic.

2

Problem Solving in Math Workshop

Problem solving should be done every day as a whole class routine (see Figure 2.1). It should also be done in small guided math groups and math workstations. When done in whole groups, the emphasis is on developing the habits of mind and ways of doing that good problem solvers need. The focus here is not on quickly solving a problem but on going through the process of problem solving. It is a practice that is developed over time. Great problem solving is interwoven throughout the math block (see Figure 2.1).

Figure 2.1

Problem Solving in Whole Groups: Rethinking Pro...

Table of contents

• Cover
• Title
• Copyright
• Dedication
• Contents
• Foreword
• Meet the Author
• Acknowledgments
• Introduction: Problem Solving Should Prepare Students for the Future
• Part I Introduction to Word Problems
• Part II A Deeper Dive
• Part III Other Word Problems