**Authors: **Jo Boaler

Consider for a moment this mathematics problem:

A woman is on a diet and goes into a shop to buy some turkey slices. She is given 3 slices that together weigh

1

⁄

3

of a pound, but her diet says that she is allowed to eat only ¼ of a pound. How much of the 3 slices she bought can she eat while staying true to her diet?

This is an interesting problem and I urge readers to try it before moving on. It was posed by Ruth Parker, a wonderful teacher of teachers who has spent many years working with parents to help them understand the benefits of inquiry approaches. In one of her public sessions with children and parents, she posed this problem and asked people to solve it. Her purpose in doing so was to see what kind of solutions people offered and how these compared to their school experiences. Many of the adults who had experienced passive approaches were unable to solve the problem because they could not apply a rule they had learned. Some of them tried 1/4 × 1/3, as they knew that something should be multiplied, but they recognized that their answer of 1/12 was probably incorrect. Some tried 1/4 × 3, but their answer of 3/4 of a pound also did not make sense. To use a rule they needed to set up the following equation:

3 slices =

1

⁄

3

x slices = ¼

Once Ruth told them this, the people who had remembered rules and methods were able to do the rest—to cross multiply and say that

1

⁄

3

x = ¾

so x =

9

⁄

4

slices

But as she pointed out, the most important part of the mathe-matics that is needed is to be able to set up the equation. This is something that children get very little experience in—they either use the same equation over and over again in a math lesson and so do not focus on how to set one up, or they are given equations that are already set up for them and they rehearse how to solve them, over and over again.

But look at some of the wonderful solutions offered by young children who had not yet been subjected to rule-bound passive approaches at school:

One fourth grader said,

If 3 slices is

1

⁄

3

of a pound, then 9 slices is a pound. I can eat ¼ of a pound, and ¼ of 9 slices is

9

⁄

4

slices.

Another solved the problem visually by representing a pound:

and then a quarter of a pound:

These elegant solutions involve the sorts of methods that are suppressed by passive, rule-bound mathematics approaches that teach only one way to solve problems and discourage all

others. We can only speculate as to whether these same young people would be able to think of solutions such as these after future years of passive mathematics approaches. The fact that many students learn to suppress their thoughts, ideas, and problem-solving abilities in math classes is one of the most serious problems in American math education.

Learning without Talking

Another major problem with passive approaches to mathematics is that students work in silence. This may, to some, seem to be the optimum learning condition, but in fact this is far from the truth. I have visited hundreds of classrooms in which students sit in rows at individual desks, silently watching the teacher work on math and silently copying the methods. But this approach is flawed for a number of reasons. One problem is that students often need to talk through methods to know whether they really understand them. Methods can

seem

to make sense when people hear them, but explaining them to someone else is the best way to know whether they are really understood.

When two famous mathematicians from very different circumstances reflected on the conditions that allowed them to succeed in mathematics, I was struck by the similarity in their statements. Sarah Flannery is a young Irish woman who won the European Young Scientist of the Year award for the development of a “breathtaking” mathematical algorithm. In her autobiography she writes about the different conditions that promoted her learning, including the “simple math puzzles” that she worked on as a child, which I shall talk more about in

chapter 8

. Flannery writes: “The first thing I realized about learning mathematics was that there is a hell of a difference between, on the one hand,

listening

to math being talked about by somebody else and thinking that you are understanding, and, on the other, thinking about math and understanding it yourself and

talking

about it to someone else.”

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Reuben Hersh, an American mathematician, wrote the book

What Is Mathematics, Really?

In it he also talks about the source of his mathematical understanding, saying that “Mathematics is learned by computing, by solving problems, and by

conversing,

more than by reading and

listening.

”

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Both of these successful mathematicians highlight the role of talking over listening, yet listening is the signature of the passive mathematics approaches that are the norm for American students. The first faulty learning condition that Flannery describes (“listening to math being talked about by someone else”) is the quintessence of passive math approaches. The second condition that enabled her to understand (“thinking about math” and “talking about it to someone else”) is what students should be doing in classrooms and homes and is essential to the active approach I will set out in

chapter 3

. When students listen to someone laying out mathematical facts (a passive act that does not necessarily involve intellectual engagement), they usually think that it makes sense, but such thoughts are very different from understanding, as the following true story illustrates.

Soon after Emily’s school converted back to traditional classes, I visited an algebra lesson there. The teacher was an “old-school” math teacher who had been hired to teach the traditional approach. He strode up and down at the front of the room, filling the board with mathematical methods that he explained to students. He joked occasionally and peppered his sentences with phrases such as “this is easy” and “just do this quickly.” The students liked him because of his jokes, cheery outlook, and clear explanations, and they would watch and listen carefully and then practice the methods in their books. The students had all been involved in the very public debate at the school between those

supporting a traditional math approach and those supporting the newer curriculum materials. One day when I was visiting the algebra class that the students knew as “traditional math,” I stopped and knelt by the side of one boy and asked him how he was getting on. He replied enthusiastically, “Great. I love traditional math. The teacher tells it to you and you get it.” I was about to go to another desk when the teacher came around handing tests back. The boy’s face fell as he saw a large F circled in red on the front of his test. He stared at the F, looked through his test, and turned back to me, saying, “Of course, that’s what I hate about traditional math—you think you’ve got it when you haven’t!” This afterthought, given with a wry smile, was amusing, but it also communicated something very important about the limitations of the approach that he was experiencing. Students do think that they “get it” when methods are shown to them on the board and they repeat them lots of times, but there is a huge difference between seeing something that appears to make sense and understanding it well enough to use it a few weeks or days later or in different situations. To know whether students are understanding methods as opposed to just thinking that everything makes sense, they need to be solving complex problems—not just repeating procedures with different numbers—and they need to be talking through and explaining different methods.

Another problem with the silent approach is that it gives students the wrong idea about mathematics. One of the most important parts of being mathematical is an action called

reasoning.

This involves explaining

why

something makes sense and how the different parts of a mathematical solution lead from one to another. Students who learn to reason and to

justify

their solutions are also learning that mathematics is about making sense. Reasoning is central to the discipline of mathematics. Scientists work to prove or disprove theories by finding new cases, but mathematicians prove their work through reasoning. Mathematicians make logically connected statements and reason their way to a proof. When parents ask me why their children should “waste time” in class “explaining their work” when they “know the answers,” I tell them that they need to explain their work because this is the most mathematical of acts. If students are not reasoning, then they are not thinking and working mathematically. Whenever students offer a solution to a math problem, they should know why the solution is appropriate, and they should draw from mathematical rules and principles when they justify the solution rather than just saying that a textbook or a teacher told them it was right. Reasoning and justifying are both critical acts, and it is very difficult to engage in them without talking. If students are to learn that being mathematical involves making sense of their work and being able to explain it to someone else, justifying the different moves, then they need to talk to one another and to their teacher.

Another reason that talking is so critical in mathematics classrooms is that when students discuss mathematics, they come to know that the subject is more than a collection of rules and methods set out in books—they realize that mathematics is a subject that they can have their own ideas about, a subject that can invoke different perspectives and methods, and a subject that is connected through organizing concepts and themes. This is important for all learners but perhaps none more so than adolescents. If young people are asked to work in silence and they are not asked to offer their own ideas and perspectives, they often feel disempowered and disenfranchised, ultimately choosing to leave mathematics even when they have performed well.

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When students are asked to give their ideas on mathematical problems, they feel that they are using their intellect and that they have responsibility for the direction of their work, which is extremely important for young people.

YanLev

Mathematical discussions are also an excellent resource for student understanding. When students explain and justify their work to each other, they get to hear each other’s explanations, and there are times when students are much more able to understand an explanation from another student than from a teacher. The students who are talking are able to gain deeper understanding through explaining their work, and the ones listening are given greater access to understanding. One of the reasons for this is that when we verbalize mathematical thoughts, we need to reconstruct them in our minds, and then when others react to them, we reconstruct them again. This act of reconstruction deepens understanding.

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When we work on mathematics in solitude, there is only one opportunity to understand the mathematics. Of course, discussions need to be organized well. I will explain in later chapters how successful discussions are managed.

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