What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (23 page)

One of the great advantages of number talk problems is that they can be posed at all sorts of levels of difficulty, and there is an endless selection of possible problems, so they can be fun for children and adults of all ages. Here are some starters of different levels of difficulty:

Some good prompts to use while you are working with children are:

• How did you think about the problem?

• What was the first step?

• What did you do next?

• Why did you do it that way?

• Can you think of a different way to do the problem?

• How do the two ways relate?

• What could you change about the problem to make it easier or simpler?

Number talks are an excellent way to teach children, of any age, to decompose and recompose numbers, which is extremely valuable in their mathematical development. But there are other great problems that require them to think creatively and use numbers flexibly. Here is a small selection:

Painted Cubes

A 3 × 3 × 3 cube

is painted red on the outside. If it is broken up into 1 × 1 × 1–unit cubes, how many of these small cubes have three sides painted? Two sides painted? One side painted? No sides painted? What about if you start with a larger original cube?

Partitions

You could use Cuisinaire rods to help with this problem.

The number 3 can be broken up into whole numbers in four different ways:

Or maybe you think that 1 + 2 and 2 + 1 are the same, so there are really only three ways to break up the number.

Decide which you like better and investigate partitions for different numbers using your rules.

In this chapter I have outlined some ways of encouraging mathematical thinking in the home or classroom through mathe-
matical settings, puzzles, and strategies. In
appendix C
, I give many different resources and Web sites that can help you in your work with your own children. In the next chapter, I will summarize some critical advice for teachers, parents, and anyone who works with math learners and wants to orient them toward the most productive math learning path possible.

T
hese are exciting times, characterized by new evidence of the vast potential of all children to learn to high levels, widespread access to knowledge, incredible new technologies, and a growing movement of positivity around learning. There is increasing public awareness that the traditional mathematics education system that has been maintained for decades is badly broken, and people from across the spectrum are devoting their time and imagination to bringing about change. In the last few years, I have been fortunate to communicate this to leaders of countries as well as to teachers, district leaders, parents, CEOs, and the general public through mainstream TV shows. I have also been working with a filmmaker who has devoted her new film to the need for mathematics change. You can see the exciting and powerful film at www.youcubed.org. In my experience, I have found that every person to whom I have shown the research evidence and demonstrated the nature of
true mathematics engagement (usually through video) has recognized the need for change and, in many cases, joined forces to bring it about.

Researchers of mathematics education have stood collectively and firmly for many decades to promote change. There is very little variation in the tens of thousands of studies conducted, with almost all pointing to the need for active mathematics engagement.
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In addition to researchers, the people who lead professional development and school districts and who have had access to research knowledge have been trying to bring about the changes that I set out in this book despite the efforts of the antichange movement. But the last seven years have shown me that those who oppose change are a small, politically motivated minority, many of whom oppose an ideal that is foundational to the United States—that of equality.

We are in a time when equality is a national priority; a time with widespread awareness of the damage done by mindless math drills; and a time when those who are experts in education are being recognized and given a platform. When Cathy Williams and I formed YouCubed, we described the math change that we are promoting as a revolution. I believe that we are in the midst of an incredible revolution, one that is led by research showing that all students can reach high levels in mathematics and the nature of the teaching and parenting that brings it about.

I recently published some advice on YouCubed that I am going to share here because I believe it to be important for both teachers and parents. Teachers and parents have the opportunity to shape their children’s mathematical futures. It may not seem like this is the case at times, especially when children are going through bad experiences at school or teachers are being pressured by faulty policies such as pacing guides and
prescriptive curriculum. But I know, both from my extensive work with teachers and from my experience as a mother of two children, that you have the opportunity to make a huge difference in children’s mathematical lives.

One of the most important contributions you can make is to dispel the idea that only some children can be successful at math or that math is a “gift” that some children have and some do not. This idea permeates American society (as well as others), but it has been completely disproved by the science of the brain and learning. The idea that some children can do well in math and some can’t is a damaging myth that is harmful to children’s mathematical development. All students can achieve at the highest levels of math in school if given the right opportunities and support.

Suggested Strategies

1. Never praise children by telling them they are smart. This may seem like it is encouraging but it is a fixed-ability message that is damaging.
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When children are told they are smart they often feel good, but later when they fail in some situation—and everyone does—they think, “Hmm, I am not so smart.” Instead of praising the child, always praise
what
the child has done: for example, “It is wonderful that you have learned how to add numbers,” not “Wow you can add numbers, you are so smart.”

When children know that learning and hard work make them achieve at the highest levels, their achievement increases significantly.
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This idea can be hard to get across to children because mainstream media such as television programs often communicate the opposite message—that some children are smart and some are not. They communicate lots of other damaging ideas too—that math is hard, inaccessible and only for
nerds. It is critical to reject these ideas as often and as loudly as you can. Instead, keep telling children that math is very exciting, and it is important to work hard because it is hard work that leads to high achievement.

2. Never share stories of math failure or even dislike. As I shared in
chapter 8
, research has shown that as soon as a mother says to her daughter, “I wasn’t good at math at school,” her daughter’s achievement goes down.
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Even if you have to put on your best acting skills, always seem happy—even thrilled—when you see math! Don’t worry if you cannot do your children’s homework. Ask them to explain it to you. This can be one of the most encouraging experiences a parent can give their children. I often tell my own children that I don’t know how to do the work they are doing, even if I do, because when they explain it to me, they are learning it at a much deeper level.
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3. Always praise mistakes and say that you are really pleased that your child is making them. Recent research has shown that our brains grow when we make mistakes.
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Scientists have found that when people make a mistake in math, synapses spark, and there is activity in the brain that is absent when people get work correct. What this means is that we want people to make mistakes! In fact, making mistakes in math is one of the most helpful things to do. But many children (and adults) hate to make mistakes, they think it means they are not a “math person.” It is important both to celebrate mistakes and to tell children their brain is growing when they make them.
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When my eldest daughter was young she had some terrible school experiences when teachers decided she was not smart and stopped giving her the same questions they gave to other children. As a result she developed a fixed mind-set about
mathematics and was very anxious about math for some years. Now, after some careful work, she is a great math advocate and has a growth mind-set about math.

When she was ten I remember her working on two math problems, one of which she got right and one wrong. When she got the one wrong she immediately reacted badly, putting her head into her hands saying, “I can’t do math.” I said to her, “Do you know what just happened? When you got the one question right, nothing happened in your brain, but when you got the other question wrong, your brain grew.” I give this message to my children every time they are confused, are struggling, or make a mistake. Now they tell others in their school how valuable mistakes are and how they make your brain grow.

4. Encourage children to work on problems that are challenging. We know that it is really important for students to take risks, engage in “productive struggle,” and make mistakes. Sometimes my daughter asks for help with her homework when it looks difficult. I try to encourage her to have a go first without my help, saying, “I don’t want to take away the opportunity for you to struggle and for your brain to grow!” Keep telling your children or students that struggle is really important. This is a delicate balance as you don’t want to leave students struggling to the point that they feel despondent, but always try to encourage as much struggle as you think they can cope with at that time.

Girls, in particular, have often learned to avoid difficult work because they have more fixed mind-sets than boys—usually because they have been praised for being smart, which makes them worried about losing that label.
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Avoidance of harder work is damaging for children and it is one of the reasons that fewer girls pursue math and science.
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In one of Carol Dweck’s studies, participants were given
math problems that they all solved correctly. Half of the participants were praised for being “smart” and half for “working hard.” When offered a choice of a follow-up problem that was easy or hard, 90 percent of the participants praised for being smart chose the easy problem, whereas most of those who were praised for working hard chose the harder problem.
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This tells us that the praise we give children has an immediate effect on them. It also gives us some important clues into gender inequities in mathematics participation rates.

5. When you help students, do not lead them through work step-by-step. Doing this takes away important learning opportunities for them. We often help children by solving the hard part of a problem, such as working out what the problem is asking, and then get children to do something easier, such as a calculation. For example, consider this scenario:

Question

Carlos started with 12 candies. He gave some to Janice, and then he had 8 candies. How many did he give to Janice?

Child:
I don’t know how to do this.

Parent/Teacher:
Well, Carlos started with 12, and now he has 8, so what is 12 take away 8?

Child:
4.

Parent/Teacher:
That’s right!

In this scenario, the student may feel good, but the parent or teacher has done the hardest part of the problem, which is to make sense of the situation. It would be more helpful to ask the child to draw the problem out, including, if they wish, pictures of the candies. Or ask the child to restate the
problem in her own words or actually manipulate objects that represent the candies. Try not to lower the cognitive demand of a problem when helping. Try not to do the hard thinking for the child, leaving her with a calculation. This, ultimately, is unhelpful for her mathematics learning.

6. Encourage drawing whenever you can. The whole of mathematics could be taught visually, which would help millions of children, but few classrooms encourage drawing, and some students believe it to be babyish. Yet mathematicians draw all the time. They do this because sketching a problem helps them really
see
the important mathematical ideas. Both drawing and restating problems help children understand what questions are asking and how the mathematics fits within them.

7. Encourage students to make sense of mathematics at all times.
Connecting Mathematical Ideas
is a book I have coauthored with Cathy Humphreys, an expert teacher who focuses on sense making at all times. In the book, we share six video cases that show Cathy’s teaching as well as her lesson plans and our shared reflections on the lessons.
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Children should never think that math is a set of rules that they need to follow—although they often have good reason to think this! As they work on math keep asking them: “Does that make sense to you?” Why?” Or “Why not?” Discourage guessing. If children seem to be guessing, say, “Is that a guess? Because this is something we can make sense of and do not need to guess about.” Mathematics is a conceptual subject, and students should be thinking conceptually at all times.
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Here are some questions you can ask children to help them think conceptually:

• What is the question asking you?

• How could you draw this situation?

• How did you get that answer? (Ask this whether the answer is right or wrong.)

• Can you share your method with me?

• Can you try a different way of solving this?

• What does addition/probability/ratio/etc. mean?

• In what other situation could we use this?

• Would this method work with different numbers?

• What is important about this work?

8. Encourage students to think flexibly about numbers. Research has shown that the biggest difference between elementary students who are successful and those who are not is not that the higher achievers know more but that they think flexibly with numbers.
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It is critical that children develop number sense, which means that they think flexibly with numbers and can change and regroup them. For example, a student with number sense faced with a problem like 41 minus 17 would not use an algorithm such as:

Nor would they count up from 17 or down from 41 They would change the numbers to, for example, 40 minus 16, which is a much easier question.

Often when students struggle with math early on they are given more practice with methods, facts, or skills. This is not
what they need. They need a more conceptual understanding of math, and they need to develop number sense.

Many students in the United States fail algebra. The reason for this is not that algebra is really difficult but that the students lack number sense, which is the most important foundational base students can have for all higher mathematics.
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