Why play with math? Because play is the best way to learn.
From the introduction of the book:
Math, more than any other subject, has to be approached by each student at their own pace, and in their own way. There may be one right answer, but there are more ways to think about the path from question to answer than you’d expect.
What is math?
Most people think it’s adding, subtracting, multiplying, and dividing; knowing your times tables; knowing how to divide fractions; knowing how to follow the rules to find the answer. Math is so much more than that! Math is seeing patterns, solving puzzles, using logic, finding ways to connect disparate ideas, and so much more. People who do math play with infinity, shapes, map coloring, tiling, and probability; they analyze how things change over time, or how one particular change will affect a whole system. Math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways. The stories in this book will be full of examples that show math from these angles.
And yet, the stories our culture tells about what math is may make you think “this isn’t math” when you see those examples. This sort of ‘cultural story’ can be thought of as myth – a story that holds a lot of cultural weight, and that most folks believe. Even when a story like this is not quite true literally, it often holds some deep truth – which is where it gets its power. But the myths about math lead us in some troublesome directions; addressing some of our culture’s math myths now may help you enjoy the rest of this book more.
Some Math Myths
Myth #1: Math is mostly arithmetic, and it’s all about memorizing and following the rules.
Myth #2: Some people have a ‘math mind’ and some don’t.
Myth #3: Math is dry and lifeless, requiring cold logic – intuition and creativity have no place.
Myth #4: Math is hard.
Myth #5: It’s bad to count on your fingers.
Myth #6: It’s always important to get the answer exactly right, and you must always know how you got the answer.
Myth #7: There is one right way to do math problems.
Myth #8: Mathematicians do problems quickly, in their heads, by working intensely, alone, until the problem is solved.
Myth #9: I don’t need to know math – I’ve got my calculator and the Internet.
Myth #10: Men are better at math than women.
Many people will say “I don’t have a math mind”, as if that meant there were no way they could learn math. Math does come more easily to some people than to others, just like singing, and dancing, and drawing do. People are drawn in certain directions, and certain skills come more easily to one person than to another. But almost everyone can sing, dance, draw, swim, ride a bike, and read. How we get good at those things is by practicing. The people who are good at math got that way because they enjoyed playing with the ideas. That got them practicing lots more than the people who don’t enjoy math.
I am a very slow learner when it comes to music. But because I love it, I practice. I play the penny whistle pretty well (at a basic level). With lots of practice, I can also sing well enough to entertain my son or share a song with a friend. If you’re a slow learner with math you can still find pleasure in it and share its delights with friends. The practice it takes to get better comes most easily through problems you find intriguing and delightful; I hope the stories you read in Playing With Math inspire you to want to play, and the puzzles sprinkled through the book give you something good to play with.
You may even believe that only those with a math mind could possibly enjoy doing math. But I’m convinced that’s a myth, too. I think almost every young child delights in thinking about big numbers. But if you were convinced by sufficiently unpleasant experiences in school to believe that math was simply horrid , it would be hard later to remember that delight, because the memory would clash with your belief. Our brains use story to organize what they keep, and a memory that clashes with the organizing story won’t have a place to fit in.
If we taught kids how to ride bikes with ‘objectives’ for each age, and graded their progress, we might have lots of bike phobic kids on our hands. Considering the fear most elementary teachers have of math, it’s easy to see how hard it would be for kids to make it through school with their love of number intact.
Math seems dry and lifeless to many people because of the way it’s taught – you may think there’s no creativity in it. But according to math educator Marilyn Burns: “The secret key to mathematics is pattern”. Finding patterns is a deeply creative and intuitive thinking process. Just like the artist analyzing the problem of balance in the painting she’s beginning to imagine, a mathematician analyzes a puzzle from lots of different perspectives, looking for a way to make the pieces fit together more simply. Just like a poem can have many layers of meaning, a mathematical relationship can be analyzed at many levels of abstraction.
How can kids be creative when they’re learning math? Suppose a group of kids are given a problem that’s outside their comfort zone, maybe a situation that calls for adding big numbers. If the kids are given freedom to come up with their own methods, and then discuss their thinking process, the teacher can help shine a light on place value simply by noticing some of the choices kids make. There will be lots of different choices made. If presented well in a group where kids know how to listen to one another, this can be exciting, hard work. Like any creative endeavor, learning math can be hard work. But if it’s approached right, it’s also deeply rewarding.
Maria Droujkova, of Natural Math, says that to memorize something you must love it first. I have a terrible memory, and chose math over science so I wouldn’t have to memorize long lists of bones, muscles, organs, chemicals, etc. How did I memorize my times tables? I don’t remember that far back, but it was never an issue. I loved the patterns of numbers, and only had trouble with 7×8, which I’ve since learned is the hardest multiplication fact for most people to learn. How can we help a child who’s struggling with those tables to find more love for them? Waldorf schools have students draw and color in a beautiful star in a circle, in which the patterns of the multiples of a number become more visible. A similar technique is used in ‘Vedic math’. (You can read more about Vedic math in Tiffani Bearup’s chapter about how the math haters came around.)
Before the multiplication facts come the addition facts, another memory task that can be kept pleasant (even exciting) through a focus on pattern. Many kids enjoy doubling. My son, who’s not the huge math fan I am, really enjoys it. Here’s a conversation we had when he was seven:
R: Mom, what’s 13 and 13?
R: How about 26 and 26?
S: Well, 25 is like a quarter, and 2 quarters are 50 cents, so 26 and 26 is…
R: 52. 52 and 52 is 104. 104 and 104 is … 208. 208 and 208 is … 316…
S: What’s 200 and 200?
R: 400. Oh, so 208 and 208 is 416. What’s 416 and 416?
I loved it. But no way would he attempt to add two different 3-digit numbers yet.
The best way to learn math is to make connections. If you play with it enough, there’s very little to memorize. (That may not seem possible to you now. I’ll bring this up again at the end of the book; see what you think then.)
Schools spend a lot of time working with young children to get these facts memorized, but many children aren’t ready for that task yet. They’ll count on their fingers, and may be reprimanded for it. What happens when a person becomes embarrassed about counting on their fingers? If they still want to think, they’ll hide it. That’s the better option. The worse option that way too many students choose? They start guessing. When math becomes too incomprehensible, or not living up to someone else’s expectations becomes too painful, many students give up on math, and then they just guess.
We count on our fingers as part of a thinking process – it’s a way to keep track of how many times we’ve done something that repeats. When I’m teaching (college), I usually think out loud so the students can follow my thinking. Talking about powers, I often count on my fingers: 2 (first finger up) times 2 (second finger up) is 4, times 2 again (third finger up for three factors of 2, or ) is 8, (fourth finger up for ) 16, (fifth finger up) 32. I know my students haven’t memorized powers of 2, so the quickest way to figure them is to multiply repeatedly by 2 and keep track on our fingers.
Of course, students who add well but haven’t memorized their times tables can do the same. Fives are the easiest: 7 times 5 is… 5 (first finger up), 10 (second finger up), 15, 20, 25, 30, 35, and we stop when our 7th finger pops up.
So counting on our fingers is useful any time we’re trying to figure something that needs a repeated step. Perhaps the thing I want to figure can be memorized. But if I haven’t memorized it yet myself, the most efficient way to figure it will likely involve fingers. Learning math is a process. Young children can add 2 blocks and 3 blocks by touching each one and counting them together. At some point, they learn that they can imagine the blocks and count them by touching a different finger for each one. And those fingers are one of the first steps they take toward math’s power to generalize.
Like counting on your fingers, you can use a calculator or an abacus as a tool to help you figure out things you couldn’t figure completely on your own. In Japan, children can take special classes to learn how to use an abacus efficiently – later lessons involve imagining the abacus, so the tool moves into your mind. The students progress to moving the imaginary beads with their fingers. Calculators, unfortunately, are not so transparent, and can slip from being a useful exploration tool to becoming a crutch for the basic building blocks that were never internalized.
Although it’s useful, perhaps vital, to have those addition and multiplication facts internalized before approaching algebra, two much bigger barriers to student success are the beliefs many students have that they can’t learn math, and that it’s not important.
Just one answer, just one way?
In basketball, you only get points when your ball goes in the basket, but during practice a ball that wobbles on the rim and falls out can still be pretty exciting. It should be the same way with math – learning can be exciting, even when you haven’t arrived at a right answer. (It’s only later, when you’re programming the Mars lander, or calculating taxes, that right answers are vital.)
Musicians make mistakes, even in performance, and learn how to flow past their mistakes so gracefully only the most astute listeners notice. It’s a similar case in almost any profession. We need to be willing to make mistakes in order to learn. So celebrate the hard work you’re doing learning something new, and give yourself a break when you make mistakes. Notice what you did right.
A similar issue is whether there is one right way to solve a math problem. There is probably never just one way to solve a problem, although there is often a most efficient way. As you learn math, it’s important to make connections. Finding more than one way to do a problem is a great way to make connections, giving you deeper insight into what’s really going on.
Whenever a group of students comes together to think about math, it becomes possible for them to discuss the different approaches they’re taking. These conversations, and the work of explaining and trying to understand others’ explanations, deepen our understanding of math in ways that are not easily available when we work alone.
Should we ever care about the most efficient way to do something? It’s important to let students use the method that makes the most sense to them, but once they have a solid grasp on a subject, considering efficiency is a great math problem in its own right. One of the most basic concerns of computer science is how many computer steps a calculation will take.
Do we need math?
What we really need, we learn when we need it. However, if you want to be able to analyze the accuracy of what you read – you’ll need a feel for the math behind those charts, percentages, and ridiculously big numbers. If you’re considering buying a house, you’ll want to know whether you can really afford it, along with the best way to set up your mortgage (since banks prey on people who don’t understand the intricacies of balloon payment mortgages, and people lose their homes, along with their life savings when they get too far behind on payments). There are lots of financial decisions that may be only arithmetic-level calculations, albeit pretty complicated, but require some careful analysis that math-phobia can get in the way of.
More than what you need, though, this book is about how glorious and delightful mathematical thinking can be – for anyone. Like dancing, swimming, or knowing the stories behind art and music, math is part of the richness of being human.
A real math problem is something you don’t know how to solve when you start – that takes time. Only the exercises we do to practice so we can internalize facts and procedures are quick. Real problems require lots of work, sleeping on it, going away from the problem to return later, discussion, doodling, and eventually writing out the steps in an organized way so you can find the holes.
Andrew Wiles spent x years working on solving Fermat’s Last Theorem. When he presented his solution to other mathematicians at an important conference after years of work, he suddenly realized there was a hole in his reasoning. After a few more years, he managed to get it right.
Most, or possibly all, of the differences we see between men and women in math can be traced to socialization. In Iceland, gender roles in math get reversed – it’s girls who perform better on the international math tests. There are surprising complexities to the gender question, and so a whole chapter is devoted to it .
 John Kellermeier, at Tacoma Community College, calls those unhappy experiences math abuse.
 He goes to a ‘freeschool’, and hasn’t had any pressure to learn facts or standard algorithms (procedures) yet. My goal is to help him maintain his delight in playing with ideas – I trust that the rest will come.