**Authors: **Jo Boaler

In her essay “The Having of Wonderful Ideas”

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Eleanor Duckworth, a professor of education at Harvard, makes an extremely important point: the most valuable learning experiences children can have come from their own thoughts and ideas. In Duckworth’s essay she recalls an interview she had with some seven-year-old children in which she asked them to put ten drinking straws, cut into different lengths, in order, from the smallest to the biggest. When Kevin walked into the room, he announced, “I know what I’m going to do,” before Duckworth had explained the task. He then proceeded to order the straws on his own. Duckworth writes that Kevin didn’t mean, “I know what you’re going to ask me to do.” Rather he meant: “I have a wonderful idea

about what to do with these straws. You’ll be surprised by my wonderful idea.” Duckworth describes Kevin working hard to order the straws and then being extremely pleased with himself when he managed it. For Kevin, the experience of ordering straws was much more worthwhile because he was working on his own idea, instead of following an instruction. Research on learning tells us that when children are working on their own ideas, their work is enriched with cognitive complexity and enhanced by greater motivation.

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Duckworth proposes that having wonderful ideas is the “essence of intellectual development” and that the very best teaching that a parent or teacher can do is to provide settings in which children have their most wonderful ideas. All children start their lives motivated to come up with their own ideas—about mathematics and other things—and one of the most important things a parent can do is to nurture this motivation. This may take extra work in a subject such as mathematics, in which children are wrongly led to believe that all of the ideas already have been had and their job is simply to receive them,

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but this makes the task even more important.

Puzzles and Problems

In addition to the provision of interesting settings, another valuable way to encourage mathematical thinking is to give children interesting puzzles to work on. Sarah Flannery has written a fascinating book,

In Code: A Mathematical Journey,

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which describes her mathematical development. It is a very useful resource for parents who would like to encourage the best mathematical start in life. I don’t think that parents need to be the mathematics professor that her father was in order to be successful; they just need the enthusiasm. Sarah Flannery talks about the way her mathematical development was encouraged by working on puzzles in her home. Although she and her

siblings much preferred outdoor sports, her father would give them intriguing puzzles to think about in the evenings and these captured their young minds. Because her father was a mathematics professor and because she was so good at mathematics, people have often assumed that she was given extra math help at home, but Flannery shares something very important with her readers. She says:

Strictly speaking, it is not true to say that I or my brothers don’t get any help with math. We’re not forced to take extra classes, or endure gruelling sessions at the kitchen table, but almost without our knowing we’ve been getting help since we were very young—out-of-the-ordinary help of a subtle and playful kind which I think has made us self-confident in problem solving. Ever since I can remember, my father has given us little problems and puzzles. I have often heard, and still hear, “Dad give us a puzzle.” These puzzles challenged us and encouraged our curiosity, and many of them made math interesting and tangible. More fundamentally they taught us how to reason and think for ourselves. This is how puzzles have been far more beneficial to me than years of learning formulae and “proofs.”

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Sarah Flannery gives examples of the sorts of math problems she worked on as a child that caused her to be so good at mathematics. Here are three of my favorites:

**The Two-Jars Puzzle:**Given a five-liter jar, a three-liter jar, and an unlimited supply of water, how do you measure out four liters exactly?

**The Rabbit Puzzle:**A rabbit falls into a dry well thirty meters deep. Since being at the bottom of a well was not

her original plan, she decides to climb out. When she attempts to do so, she finds that after going up three meters (and this is the sad part), she slips back two. Frustrated, she stops where she is for that day and resumes her efforts the following morning—with the same result. How many days does it take her to get out of the well? Note: This question is assuming that the rabbit jumps up three meters and falls two meters each day.

**The Buddhist Monk Puzzle:**One morning, exactly at sunrise, a Buddhist monk leaves his temple and begins to climb a tall mountain. The narrow path, no more than a foot or two wide, spirals around the mountain to a glittering temple at the summit. The monk ascends the path at varying rates of speed, stopping many times along the way to rest and eat the dried fruit he carries with him. He reaches the temple shortly before sunset. After several days of fasting he begins his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way, finally arriving at the lower temple just before sunset. Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

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Flannery talks about the ways these puzzles encouraged her mathematical mind because they taught her to

think

and

reason,

two of the most important mathematical acts. When children work on puzzles such as these, they are having to make sense of situations, they are using shapes and numbers to solve problems, and they are thinking logically, all of which are critical ways of working in mathematics. Flannery reports that she and her siblings would work on problems that her father had set for them

each night over dinner. I like the sound of this ritual although I also appreciate that it is a difficult one to achieve in a busy household at the end of a tiring day. And puzzles do not have to be set by parents for children to work on. They can be something that children and parents work on together, each week, each month, or more occasionally. Whether a daily ritual or something less frequent, puzzles are incredibly useful in mathematical development, especially if children are encouraged to talk through their thinking and someone is there to encourage their logical reasoning. If children get in the habit of applying logic to problems and to persisting with problems until they solve them, they will learn extremely valuable lessons for learning and for life.

Some recommended books of mathematical puzzles are listed in

appendix C

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Asking Questions

When exploring mathematical ideas with young people it is always good to ask lots of questions. Children often enjoy thinking through questions and it will help them develop mathematical ways of thinking. Good questions are those that give you access to your children’s mathematical thoughts, as these will allow you to support their development. When I am called over to students who are stuck in math classes, I almost always start with “What do you think you should do?” Then, if I can persuade them to offer any ideas, I ask, “Why do you think that?” or “How did you get that?” Often children who have been taught traditionally will think they are doing something wrong at this point and quickly change their answer, but over time my students get used to the idea that I am interested in their thinking, and I will ask the same questions whether they are right or wrong. When students explain their thinking, I am able to help them move forward productively, at the same time helping

them know that math is a subject that makes sense and that they can

reason

their way through mathematics problems.

Pat Kenschaft has written a useful book for parents,

Math Power: How to Help Your Child Love Math, Even If You Don’t,

in which she quotes Swarthmore professor Heinrich Brinkmann. This particular professor was known on the Swarthmore campus for being able to find something right in what every student said. No matter how outrageous a student’s contribution or question, he could respond: “Oh, I see what you are thinking. You’re looking at it as if . . .”

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This is a very important act in mathematics teaching because it is true that unless a child has taken a wild guess, then there will be some sense in what they are thinking—the role of the teacher is to find out what it is that makes sense and build from there. A parent in the home can do the sort of careful inquiring and guiding as they help children with mathematics that it is hard for teachers in a classroom of thirty or more children to do. If children give an answer and just hear that it is wrong, they are likely to be disheartened, but if they hear that their thinking is correct in some ways and they learn about the ways that it may be improved, they will gain confidence, which is critical to math success.

Children should also be encouraged to ask questions of themselves and others. I work with an inspirational teacher, Carlos Cabana, who, when asked for help, prompts students to pose a specific question that he and they can think about. As the students phrase their questions specifically, they are then able to see the mathematics more clearly, and they are often able to answer the questions themselves! Another great teacher I work with, Cathy Humphreys,

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always says that she never asks a question that she knows the answer to. What she means by this is that she always asks students about their mathematical methods and reasons, which she could never know in advance. These are the most valuable questions for any math teacher to ask, as they give the teacher access to students’ developing mathematical ideas. Pat Kenschaft

puts it well: “If you can tap into the real thoughts of the person before you, you can untangle the knots around their mathematical inner light.”

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The ideal way in which to tap into learners’ mathematical thoughts and to discover their mathematical inner light is to provide interesting settings and problems, to gently probe and question, and to encourage their thinking and reasoning.

When you are working with your child on math, be as enthusiastic as possible. This is hard if you have had bad mathematical experiences, but it is very important. Parents, especially mothers of girls, should never, ever say, “I was hopeless at math!” Research tells us that when mothers tell their daughters that they were not good at math in school, their daughter’s achievement immediately goes down that same term.

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It is important not to share your own negative feelings, even if this is difficult. In fact, you may have to fake some enthusiasm and joy around math. I do this with my own children. I genuinely love math, but I do not love a lot of the math homework they bring home, especially when it is pages of repetitive, procedural questions. But whenever my daughters say they have math homework, I always say, “Great! I love that. Let’s look at it together,” or something similar if they want some help or an extra eye. They now often save their math homework for me because they believe I really love it. If you don’t know the math yourself, don’t worry. Ask your children to explain it to you. This is a great experience for children. I often tell my own young children that I don’t understand to give them the opportunity to explain it to me, and they really like to do this. Think of the birth of your own children as the perfect opportunity to start all over again with mathematics, without the people who terrorized you the first time around. I know a number of people who were traumatized by math in school but when they started learning it again as adults, they found it enjoyable and accessible. Parents of young children could make math an adult project, learning with their children or perhaps one step ahead of them each year. I teach online courses, which are available

through www.youcubed.org that give parents a lot of useful information in working with their children.

Mathematical conversations should be relaxed and free from pressure. Fear and pressure impede learning, and children should always feel comfortable when offering their ideas in math. Parents and teachers should never appear irritated or judgmental if children make mistakes. Math, more than any other subject, can cause panic, which stops the mind from working.

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I usually start any work with children by telling them that I love errors because they are really good for learning. I mean this because it is through making errors that children learn the most, as mistakes give them a chance to consider, revise, and learn new things. When I am working with children and they say something that is incorrect, I consider their thinking with them and see this as an important opportunity for learning. When students know that I am not judging them harshly and that I genuinely value errors, they are able to think more productively and learn more.

Number Flexibility

An important mission for all parents and teachers is to steer children away from the mathematical ladder of rules

that I discussed in the previous chapter. Gray and Tall’s research study

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showed that successful students were those who used numbers flexibly, decomposing and recomposing them. This isn’t difficult to do, but it involves children knowing that that is what they

should

be doing. Fortunately, there are specific and enjoyable ways of encouraging this number flexibility that can be used with children of all ages. One of the very best methods I know for encouraging number flexibility is that of

number talks,

the activities I introduced in the previous chapter. The aim of number talks is to get children to think of all the different ways that numbers can be calculated, decomposing and recomposing as they work. For example, you

could ask a child to work out 17 × 5 in their heads without the use of pen and paper. This is a problem that looks difficult but becomes much easier when the numbers are flexibly moved around. So, for example, with 17 × 5 I could work out 15 × 5. I can do this in my head more easily as 10 × 5 is 50 and 5 × 5 is 25, giving me 75. I then need to remember to add 10 as I only worked out 15 × 5 and I need two more 5s. This gives me my answer of 85. Another way of solving the problem is not to work out 17 × 5 but to work out 17 × 10, which is 170, and then halve it. Half of 100 is 50 and half of 70 is 35, so I would get 85. As people work on problems like these, they develop number sense, which is the basis for all higher mathematics. They also develop their mental math abilities. The problem I would set children when working on number talks is to find as many ways as possible of working out the answers. Most children will find this challenging and fun.

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