Math for Life
Crucial Ideas You Didn't Learn in School
By Jeffrey Bennett, Joan Marsh, Lynn Golbetz
Big Kid Science Copyright © 2014 Jeffrey Bennett
All rights reserved.
(Don't Be) "Bad at Math"
Nothing in life is to be feared. It is only to be understood.
— Marie Curie
Equations are just the boring part of mathematics.
— Stephen Hawking
Let's start with a multiple-choice question.
Question: Imagine that you're at a party, and you've just struck up a conversation with a dynamic, successful businesswoman. Which of the following are you most likely to hear her say during the course of your conversation?
a. "I really don't know how to read very well."
b. "I can't write a grammatically correct sentence."
c. "I'm awful at dealing with people."
d. "I've never been able to think logically."
e. "I'm bad at math."
We all know that the answer is E, because we've heard it so many times. Not just from businesswomen and businessmen, but from actors and athletes, construction workers and sales clerks, and sometimes even teachers and CEOs. Somehow, we have come to live in a society in which many otherwise successful people not only have a problem with mathematics but are unafraid to admit it. In fact, it's sometimes stated almost as a point of pride, with little hint of embarrassment.
It doesn't take a lot of thought to realize that this creates major problems. Mathematics underlies nearly everything in modern society, from the daily financial decisions that all of us must make to the way in which we understand and approach global issues of the economy, politics, and science. We cannot possibly hope to act wisely if we don't have the ability to think critically about mathematical ideas.
This fact takes us immediately to one of the main themes of this book. Look again at our opening multiple-choice question. It would be difficult to imagine the successful businesswoman admitting to any of choices A through D, even if they were true, because all would be considered marks of ignorance and shame. I hope to convince you that choice E should be equally unacceptable. Through numerous examples, I will show you ways in which being "bad at math" is exacting a high toll on individuals, on our nation, and on our world. Along the way, I'll try to offer insights into how we can learn to make better decisions about mathematically based issues. I hope the book will thereby be of use to everyone, but it's especially directed at those of you who might currently think of yourselves as "bad at math." With luck, by the time you finish reading, you'll have a very different perspective both on the importance of mathematics and on your own ability to understand it.
Of course, I can't turn you into a mathematician in a couple hundred pages, and a quick scan of the book should relieve you of any fear that I'm expecting you to repeat the kinds of equation solving that you may remember from past math classes. Instead, this book contains a type of math that you actually need for life in the modern world, but which you probably were never taught before.
Best of all, this is a type of mathematics that anyone can learn. You don't have to be a whiz at calculations, or know how to solve calculus equations. You don't need to remember the quadratic formula, or most of the other facts that you were expected to memorize in high school algebra. All you need to do is open your mind to new ways of thinking that will enable you to reason as clearly with numbers and ideas of mathematics as you do without them.
The Math Recession
For our first example, let's consider the recent Great Recession, which left millions of people unemployed, stripped millions of others of much of their life savings, and pushed the global financial system so close to collapse that governments came in with hundreds of billions of dollars in bailout funds. The clear trigger for the recession was the popping of the real estate bubble, which ignited a mortgage crisis. But what created the bubble that popped? I believe a large part of the answer can be traced to poor mathematical thinking.
Take a look at Figure 1, which shows one way of looking at home prices during the past few decades. The bump starting in 2001 represents the housing price bubble. Let's use some quantitative reasoning to see why it should have been obvious that the bubble was not sustainable.
Here's how to think about it. As its title indicates, the graph shows the ratio of the average (median) home price to the average income. For example, if the average household income were $50,000 per year, then a ratio of 3.0 would mean that the average home price was three times the average income, or $150,000. The graph shows that the average ratio for the three decades prior to the start of the bubble was actually about 3.2, which means someone with an income of $50,000 typically purchased a house costing about $160,000 (which you find by multiplying 3.2 by $50,000).
Now look at what happened during the housing bubble. After increasing modestly in the 1990s, the ratio began shooting upward in 2001, reaching a peak of about 4.7 in 2005. This was nearly a 50% increase from the historical average of 3.2, which means that relative to income, the average home was about 50% more expensive in 2005 than it was before the bubble. In other words, a family that previously would have bought a house costing $160,000 was instead buying one that cost nearly $240,000.
With homes so much more expensive relative to income, families had to spend a higher percentage of their income on them. In general, a family can spend a higher percentage of its income on housing only if some combination of the following three things happens: (1) its income increases; (2) it cuts expenses in other areas; or (3) it borrows more money. Other statistics showed clearly that average income was not rising significantly, and that while homeowners gained some benefit from relatively low mortgage interest rates, overall consumer spending actually increased. We are therefore left with the third possibility: that the housing bubble was fueled primarily by borrowing. With little prospect that incomes would rise dramatically in the future, it was inevitable that this borrowing would be unaffordable and that loan defaults and foreclosures would follow. The only way to restore equilibrium to the system was for home prices to fall dramatically.
Lest you think that this is a case of hindsight being 20/20, keep in mind that these kinds of data were available throughout the growth of the bubble. Anyone willing to think about it should therefore have known that the bubble would inevitably pop, and, indeed, you can find many articles from the time that pointed out this obvious fact. So how did everyone else manage to miss it?
Although it's tempting to blame the problem on a failure of "the system," it was ultimately the result of millions of individual decisions, most of which involved a real estate agent arguing that prices could only go up, a mortgage broker offering an unaffordable loan, and a customer buying into the real estate hype while ignoring the fact that the mortgage payments would become outsized relative to his or her income. In short, many of us ignored the mathematical reality staring us in the face.
That is why I think of the Great Recession as a "math recession": It was caused by the fact that too many of us were unwilling or unable to think mathematically. Perhaps I'm overly idealistic, but I believe that with better math education — and especially with more emphasis on quantitative reasoning — many more people would have questioned the bubble before it got out of hand. We can't change the past, but I hope this lesson will convince you that we all need to get over being "bad at math."
Fear and Loathing of Mathematics
If we as a society (or you as an individual) are going to overcome the problems caused by being "bad at math," a first step is understanding why this form of ignorance has become socially acceptable. This social acceptance is not as natural as it might seem, and in fact is relatively rare outside the United States. Research has shown that infants have innate mathematical capabilities, and it's difficult to find kindergartners who don't get a thrill out of seeing how high they can count; both facts suggest that most of us are born with an affinity for mathematics. Even many adults who proclaim they are "bad at math" must once have been quite good at it. After all, the successful businesswoman of our multiple-choice question probably could not have gotten where she is without decent grades.
My own attempt to understand the origins of the social acceptance of "bad at math" began with surveys of students who took a course in quantitative reasoning that I developed and taught at the University of Colorado. This course was designed specifically for students who did not plan to take any other mathematics courses in college, and the only reason they took this one was because they needed it to fulfill a graduation requirement. In other words, it was filled with students who had already decided that math wasn't for them. When asked why, the students divided themselves roughly into two groups, which I call math phobics and math loathers. The math phobics generally did poorly in their high school mathematics classes and therefore came to fear the subject. The math loathers actually did pretty well in high school math but still ended up hating it.
Probing further, I asked students to try to recall where their fear or loathing of mathematics may have originated. Interestingly, the most common responses traced these attitudes to one or a few particular experiences in elementary or secondary school. Many of the students said they had liked mathematics until one adult, often a teacher but sometimes a parent or a family friend, did something that turned them off, such as telling the student that he or she was no good at math, or laughing at the student for an incorrect solution. Dismayingly, women were far more likely to report such experiences than men. Apparently, it is still quite common for girls as young as elementary age to be told that, just because they are girls, they can't be any good at math.
Who would say such things to young children, thereby afflicting them with a lifelong fear or loathing of mathematics? Certainly, there are cases where the offending adult is a math teacher with some sort of superiority complex. But more commonly, it appears that the adults who turn kids off from mathematics are those who are themselves afflicted with the "bad at math" syndrome. Like an infectious disease, "bad at math" can be transmitted from one person to another, and from one generation to the next. Its social acceptance has come about only because the disease is so common.
Caricatures of Math
My students taught me another interesting lesson: While they professed fear and loathing of mathematics, they didn't really know what math is all about. Most of their fears were directed at a caricature of mathematics, though admittedly one that is often reinforced in schools.
The students saw mathematics as little more than a bunch of numbers and equations, with no room for creativity. Moreover, they assumed that mathematics had virtually no relevance to their lives, since they didn't plan to be scientists or engineers. It's worth a moment to consider the flaws in these caricatures.
Numbers and equations are certainly important to mathematics, but they are no more the essence of mathematics than paints and paintbrushes are the essence of art. You can see its true essence by looking to the origin of the word mathematics itself, which derives from a Greek term meaning "inclined to learn." In other words, mathematics is simply a way of learning about the world around us. It so happens that numbers and equations are very useful to this effort, but we should be careful not to confuse the tools with the outcomes.
Once we see that mathematics is a way of learning about the world, it should be immediately clear that it is a highly creative effort, and that while equations may offer exact solutions, the same may not be true of the mathematical essence. Consider this example: Suppose you deposit $100 into a bank account that offers a simple annual interest rate of 3%. How much will you have at the end of one year?
Because 3% of $100 is $3, the "obvious" answer is that you'll have $103 at the end of a year. This is probably also the answer that would have gotten full credit in your past math classes. But, of course, it's only true if a whole range of unstated assumptions holds. For example, you have to assume that the bank doesn't fail and doesn't change its interest rate, and that you don't find yourself in need of the money for early withdrawal. In the real world, these assumptions are the parts that require far more thought and study — more real mathematics — than the simple percentage calculation.
As to my students' assumption that mathematics had no relevance to their lives, our hhousing bubble example should already show that this is far from the truth. Today, mathematics is crucial to almost everything we do. We are regularly faced with financial choices that can make anyone's head spin; just consider the multitude of cell phone plans you have to select from, the many options you have for education and retirement savings, and the implications of how you deal with medical insurance for both your bank account and your health. Looking beyond finance, we are confronted almost daily with decisions that we can make thoughtfully only if we understand basic principles of statistics, which is another important part of mathematics. For example, your personal decision on whether to use a cell phone while driving should surely be informed by the statistical research into its dangers, and hardly a day goes by without someone telling you why you need this or that to make you healthier or happier — claims that you ought to be able to evaluate based on the quality of the statistical evidence backing them up.
The issues go even deeper when we look at the choices we face as voting citizens. We're constantly bombarded by competing claims about the impacts of proposed tax policies or government programs; how can you vote intelligently if you don't understand the nature of the economic models used to make those claims, or if you don't really understand the true meaning of billions and trillions of dollars? And take the issue of global warming: On one side, you're told that it is an issue upon which our very survival may depend, and on the other side that it is an elaborate hoax. Given that global warming is studied by researchers almost entirely through statistical data and mathematical models, how can you decide whom to believe if you don't have some understanding of those mathematical ideas yourself?
Getting Good at Math
If you have suffered in the past from fear or loathing of mathematics, then I may be making you nervous. Although you may now accept that mathematics is important to your life, a book about math can still seem scary. But it shouldn't. A simple analogy should help.
Just as you don't have to be the Beatles to understand their music, you don't have to be a mathematician to understand the way mathematics affects our lives. That is why you won't see a lot of equations in this book: The equations in mathematics are like the notes in music. If you want to be a songwriter, you'll need to learn the notes, and if you want to be a mathematician (or a scientist or engineer or economist), you'll need to learn the equations. But for the kinds of mathematics that we all encounter every day — the "math for life" that we'll discuss in this book — all you need are those things that we talked about before: an open mind and a willingness to learn to think in new ways.
In fact, I'll go so far as to make you the same promise that I've made to my students in the past. If you read the whole book, and think carefully as you do so, I promise that you'll find not only that you can understand the mathematics contained here, but that you'll find the topics both useful and fun.
I have just one favor to ask in return: Help in the cause of battling an infectious disease that has been crippling our society by promising that you'll never again take pride in being "bad at math," and that you'll do what you can to help others realize that being bad at math should be considered no less a flaw than being bad at reading, writing, or thinking.
Crucial Ideas You Didn't Learn in School
Before we delve into all the fun parts, there's one more bit of background we should discuss: why you haven't learned all this stuff previously.
Consider again the housing bubble example. It is clearly mathematical; its analysis requires a variety of different mathematical concepts, including ratios, percentages, mortgages (which use what mathematicians call exponential functions), statistics, and graphing. Its practical nature is also clear, since it affected people's lives all over the world. But now ask yourself: Where in the standard mathematics curriculum do we teach students how to deal with such issues? (Continues...)
Excerpted from Math for Life by Jeffrey Bennett, Joan Marsh, Lynn Golbetz. Copyright © 2014 Jeffrey Bennett. Excerpted by permission of Big Kid Science.
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