Table of Contents (click a chapter to jump down to that excerpt)
PREFACE
CHAPTER 1 (DON’T BE) “BAD AT MATH”
CHAPTER 2 THINKING WITH NUMBERS
CHAPTER 3 STATISTICAL THINKING
CHAPTER 4 MANAGING YOUR MONEY
CHAPTER 5 UNDERSTANDING TAXES
CHAPTER 6 THE U.S. DEFICIT AND DEBT
CHAPTER 7 ENERGY MATH
CHAPTER 8 THE MATH OF POLITICAL POLARIZATION
CHAPTER 9 THE MATHEMATICS OF GROWTH
EPILOGUE GETTING “GOOD AT MATH”
PREFACE
So what exactly will you learn about “math for life” in this short book? Perhaps the best way for me to explain it is to list my three major goals in writing this book:
- On a personal level, I hope this book will prove practical in helping you make decisions that will improve your health, your happiness, and your financial future. To this end, I’ll discuss some general principles of quantitative reasoning that you may not have learned previously, while also covering specific examples that will include how to evaluate claims of health benefits that you may hear in the news (or in advertisements) and how to make financial decisions that will keep you in control of your own life.
- On a societal level, I hope to draw attention to what I believe are oft-neglected mathematical truths that underlie many of the most important problems of our time. For example, I believe that far too few of us (and far too few politicians) understand the true magnitude of our current national budget predicament, the true challenge of meeting our future energy needs, or what it means to live in a world whose population may increase by another 3 billion people during the next few decades. I hope to show you how a little bit of quantitative reasoning can illuminate these and other issues, thereby making it more likely that we’ll find ways to bridge the political differences that have up until now stood in the way of real solutions.
- On the level of educational policy, I hope that this book will have an impact on the way we think about mathematics education. As I’ll argue throughout the book, I believe that we can and must do a much better job both in teaching our children traditional mathematics — meaning the kind of mathematics that is necessary for modern, high-tech careers — and in teaching the mathematics of quantitative reasoning that we all need as citizens in today’s society. I’ll discuss both the problems that exist in our current educational system and the ways in which I believe we can solve them.
CHAPTER 1 (DON’T BE) “BAD AT MATH”
[From the beginning of the chapter]
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?
Answer choices:
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.
CHAPTER 2 THINKING WITH NUMBERS
[From the middle of the chapter, in which we are discussing ways of understanding large numbers]
A billion here, a billion there. Now let’s move into the realm of the “real money” alluded to in the famous aphorism that opens this chapter. The same math that shows that $100 million could hire 500 scientists means that $1 billion could hire 5,000 of them. Going a step further, the $23 billion that Goldman Sachs initially set aside for its bonus pool in one recent year would allow the hiring of more than 100,000 scientists. Even if you change the assumption from $200,000 to $2 million per scientist, thereby allowing plenty of money for building construction, staff expenses, and higher salaries, you could still hire more than 10,000 scientists. In other words, if the $23 billion were sustainable year after year, “Goldman Scientific” could become the largest single research institution in the world, with an annual operating budget roughly ten times that of major research institutions such as MIT or the University of Texas at Austin. Since I’m a fan of human space exploration, I’ll also point out that $23 billion is about 25% larger than NASA’s budget (roughly $18 billion in 2013), which means it is somewhat more than a presidential commission said would have been needed to keep NASA’s cancelled “return to the Moon” program on track. So it seems to me that Goldman missed an opportunity to be on the forefront of future business opportunities in space, opportunities likely to offer far more long-term benefit for shareholders than lavishing large paychecks on wizards of finance.
Government money. Even Goldman pales in comparison to the sums that we regularly hear about with government programs. The biggest sum that’s regularly in the news is the federal debt, for which you might want to calculate your share. If you divide the roughly $17 trillion debt (late 2013) by the roughly 315 million people in the United States, you’ll find that each person’s share of the debt is more than $50,000, which means that an average family of four owes more than $200,000 to future generations — significantly more than it owes for its home. And at the risk of really depressing you, I’ll remind you that the debt is not only a burden on the future, but also a burden today because the government must pay interest on it. In 2012, for example, the interest totaled $360 billion — which is more than the total spent by the federal government on education, transportation, and scientific research combined. Worse, the only reason the interest payment was so “low” was because of record low interest rates. If interest rates rise back up to something more like their average for recent years, the annual interest payments on the current debt could easily double or triple, and that’s before we even consider the fact that the debt is still rising. Perhaps, as some politicians argue, we’ve had no choice but to borrow (and continue to borrow) so much money. But when you consider what else we might do with the money going to interest alone, it sure makes you think that there ought to be a better way.
CHAPTER 3 STATISTICAL THINKING
[From the middle of the chapter, in the section dealing with correlation and causality]
The issue of cell phones and driving has followed a similar trajectory. The correlation between cell phone use and accidents was also thought to be coincidental at first, since talking on a cell phone doesn’t seem to be so different from talking to a passenger, and the latter did not appear to be correlated with accident rates. As evidence for the correlation accumulated, the idea of coincidence became harder and harder to accept, so researchers began to look for other explanations, and one seemed obvious on the surface: most cell phone users were holding the cell phones in their hands, which seemed likely to mean less control while driving. This led to calls for laws requiring hands-free devices in cars. However, while those devices probably can’t hurt, further studies showed that they didn’t make the correlation go away; that is, it was the act of talking on the cell phone that was correlated with accident rates, not the act of holding the phone.
At this point, researchers began to question the assumption that talking on a cell phone is just like talking to a passenger. Brain scans conducted during simulated driving sessions soon showed that talking on a cell phone activates different areas of the brain than talking with someone sitting next to you. This offered a potential explanation for why talking on a cell phone might have different effects than talking to a passenger. Follow-up studies provided additional evidence for the idea that cell phone use is a source of distraction that can cause accidents with or without hands-free devices. Most shockingly, some studies have found that the distraction caused by talking on a cell phone while driving — even with a hands-free device — can make you as dangerous as a drunk driver. Texting or using other computing devices while driving is even worse, because it causes the same type of distraction but also forces your eyes off the road. The National Safety Council now estimates that approximately 1.6 million car crashes each year, more than a quarter of the total, are caused by some type of distraction.
To sum up this discussion, you should focus on two main points. First, it’s not easy to establish causality. Finding a correlation is only the first step in a long process, and we can be confident in causality only after careful and in-depth study. Second, where we can establish causality, knowing the causal relationship gives us a strong basis for making decisions. For example, although we may not yet have political agreement on what to do about cell phones and driving, your personal decision should be easy: Hang up the phone. Knowing that cell phone use, texting, and other distractions make you as dangerous as a drunk driver means there’s just no justification for them. If there’s a call you just can’t miss or a message you just must send, or you just have to put a new destination into your GPS, then find a place to pull off the road.
CHAPTER 4 MANAGING YOUR MONEY
[From the very end of the chapter]
When I talk to people about the material that we’ve covered in this chapter, one of the most common reactions that I get is, “But doesn’t everyone already know all this stuff?” After all, we’ve been discussing ideas that arise in nearly every major financial decision that any of us ever make. My response is always simple: Just look around at the economic mess we’ve made as a nation, and it’s pretty clear that a lot of people don’t really understand basic finance.
Of course, this begs the question of why not, and to answer that one, ask yourself (or someone else for whom this chapter was all review) where you first learned about it. Unless you were an economics or business major, the answer is unlikely to be “in school.” More likely, you learned it from reading the newspaper, or in the course of your own personal financial experiences; sometimes, you may have learned it the hard way.
The sad fact is that our schools have been just as negligent in teaching basic financial literacy as they have been at teaching how to think with numbers, or statistics, or any of the other topics we’ll cover in this book. Indeed, I’ve spoken to friends with PhD’s in mathematics or science who didn’t know the meaning of the consumer price index, or the difference between stocks and bonds, or what lenders mean by “points” on a home mortgage.
So in closing, I want to return to an idea I’ve mentioned before: We can’t expect people to know things that they’ve never been taught. Given the importance of financial literacy, it’s critical that we start teaching it in school. I hope you’ll join in the growing movement to get high schools and colleges to incorporate financial literacy into their curricula. Better yet, don’t do it in isolation from other “math for life” topics; instead, introduce it as part of a general course in quantitative reasoning, and then hammer it home by making sure that it is integrated throughout the school curriculum.
CHAPTER 5 UNDERSTANDING TAXES
[From the beginning of the chapter]
Question: Based on statistical averages, rank the following individuals according to the percentage of income they pay in taxes to the federal government, from the lowest percentage to the highest.
Answer choices:
A. A fast-food worker earning $9 per hour
B. A teacher earning $60,000 per year
C. A self-employed businesswoman earning $110,000 per year
D. A midlevel executive earning $220,000 per year
E. A billionaire investor
…If you put A in the first spot, you’re off to a good start. The $9-per-hour fast-food worker almost certainly pays the lowest percentage of income in federal taxes. In fact, depending on the specific circumstances, he or she might pay nothing at all, and could even be receiving money from the federal government through the earned income tax credit.
The rest of the positions are more difficult. You might guess that the percentage would go up in income order, but taxes are much more complicated than that. We’ll spend much of this chapter discussing the details, but for now let me just give you the basic answer, which is A-E-B-D-C under tax law as it stood from 2001 through 2012, and A-B-E-D-C after tax rate changes that took effect in 2013. (Depending on when you read this, it is possible the answer has changed again, but we’ll discuss based on 2013 law.) If you’re like most people, your next thought is probably along the lines of “You’ve got to be kidding.”
But I’m not. The surprisingly low tax rate of the billionaire investor (E) comes about because investment income, called capital gains, is taxed at a much lower rate than earned income from a job. For most capital gains, the maximum tax rate was a flat 15% before 2013, and 20% starting in 2013.
Now consider the teacher (B). Again, the details will vary greatly with personal circumstances, but the teacher’s relatively low income will probably yield an overall income tax rate between about 10% and 15%. Wait, you say: that’s lower than the billionaire’s tax rate, not higher. But note the key words “income tax.” In addition to income tax, earned income (but not capital gains) is subject to FICA taxes, which pay for Social Security and Medicare. (FICA stands for the Federal Insurance Contributions Act.) In contrast, capital gains were exempt from FICA before 2013, and are now subject to a maximum 3.8% tax for Medicare. When you add in the FICA tax, the teacher ended up with a higher overall tax rate than the billionaire prior to 2013, while the billionaire has a slightly higher overall tax rate under the new law.
We’re left with the self-employed businesswoman (C) and the executive who earns twice as much (D). Surely, you might think, the higher-earning executive must pay more. But no … Again, FICA creates a surprising result. The executive will almost certainly pay a higher rate of income tax, though not by more than about 5 to 10 percentage points. While this might sound like a significant difference (it could be the difference between 15% and 25%), it’s easily overwhelmed by the difference in FICA payments. Although most people don’t realize it, the 7.65% that gets pulled out of your paycheck for FICA must be matched by your employer. That means the self-employed businesswoman must pay both her personal share and the employer share, for a total rate of 15.3%. Moreover, the Social Security portion of the FICA payment, which is most of it, goes away above a certain income level. For 2013, that level was $113,700. Therefore, the $220,000 executive pays the 7.65% FICA rate on less than half of his income (and only the Medicare portion of 1.45% on the rest), for an effective rate of less than 5% on his full income. In contrast, the businesswoman’s income is just short of the cutoff, which means she pays the 15.3% self-employed FICA tax on her entire income.
Of course, all this is based on statistical averages; like dietary supplements, taxes come with the caveat that “individual results may vary.” Your personal situation matters. Married people pay at a different rate than singles; having more children at home usually reduces your tax bill (though it increases your other bills). Most critically, a long list of special tax breaks, including numerous tax credits and deductions, means that two people with exactly the same income can have vastly different tax bills. In recent years, this has been further exacerbated by the alternative minimum tax, or AMT, which was originally designed to prevent the rich from using so many loopholes that they ended up with no tax bill, but which now ensnares millions of taxpayers of more modest means.
You’ve probably now guessed why I’ve included an entire chapter on taxes in this book, but just in case, let me be clear about my two main reasons. First, if you’re like most people, taxes are either your largest or second-largest (after housing) personal expenditure, which means you can’t possibly do the kind of budgeting we discussed in the last chapter unless you understand what you are paying in taxes. Second, I challenge anyone of any political persuasion to consider the answer to our ranking task (not to mention the AMT and other complications) and claim that our current system makes sense and meets reasonable standards of fairness. Our tax system needs changes, and while coming to agreement on the changes is going to be very difficult, understanding what we’re dealing with is a clear first step.
CHAPTER 6 THE U.S. DEFICIT AND DEBT
[From the middle of the chapter, in the section discussing Social Security]
With all this talk of government giving IOUs to itself, you may be starting to think that something’s fishy with Social Security. You’d be right, but you haven’t seen the half of it yet. Social Security is usually presented to the public as a retirement plan, in which you pay in through your FICA taxes and later collect based on what you put in. The Social Security Administration even provides handy little reports of how much you’ve put in and the payments you’ll therefore be entitled to when you retire. This tends to make current retirees think that they’re just getting back their own money each month, and it also explains why many people think FICA taxes are somehow different from other income taxes. But while there’s some linkage between past payments and future benefits, the linkage is quite weak. Today, most retirees collect far more in benefits than they ever paid in, even when you account for inflation, in part because longer lifetimes mean many more years spent in retirement. Moreover, Social Security is not just for retirees; it’s also a disability program, and the difference between past payments and future benefits is even greater for people in this category.
The truth is that while Social Security acts somewhat like a national retirement and disability plan — it currently provides monthly payments to more than 50 million retired and disabled workers — it is not at all like an individual retirement plan, in which you deposit money into an account while you work and withdraw it later when you retire. This difference between Social Security and personal retirement plans isn’t necessarily a bad thing, and it might actually work quite well if the Social Security trust fund held real money. But as we’ve discussed, it’s actually filled with IOUs.
We can use an analogy to see just how bad this situation is. Imagine that, at a young age, you decide to set up a retirement savings plan that will allow you to retire comfortably at age 65. Based on your best guesses about future interest rates, you calculate that you can achieve your retirement goal by making monthly deposits of $250 into your retirement plan. So you start the plan by making your first $250 deposit.
However, the very next day, you decide you want a new TV and find yourself $250 short of what you need. You therefore decide to “borrow” back the $250 you just deposited into your retirement plan. Because you don’t want to fall behind on your retirement savings, you write yourself an IOU promising to put the $250 back. Moreover, recognizing that you would have earned interest on the $250 if you’d left it in there, at the end of the month you write yourself an additional IOU to replace this lost interest.
Month after month and year after year, you continue in the same way, always diligently depositing your $250, but then withdrawing it so you can spend it on something else, and replacing it with IOUs for the withdrawn money and the lost interest. When you finally reach age 65, your retirement plan will contain IOUs that say you owe yourself enough money to retire on — but you’ll obviously find it very difficult to live off them.
No rational person would ever treat a retirement plan in the way we’ve just described, yet this is essentially the way the Social Security trust fund works: The government has been diligently depositing the excess money collected through FICA into the Social Security trust fund, then immediately withdrawing it for other purposes while replacing it with Treasury bills that are nothing more than IOUs. It was easy to ignore this insanity as long as the IOUs kept piling up without anyone trying to collect on them. But that has now changed. High unemployment has lowered FICA tax collections, so that in 2010, for the first time, Social Security payments exceeded Social Security tax collections. The difference was small enough to barely register in the federal deficit, but this tolerable situation won’t last. Without changes, Social Security’s payments are expected to exceed its tax collections by ever-larger amounts in the coming decades.
To see the problem vividly, consider the year 2036, which is approximately when the government’s “intermediate” projections (meaning those that are neither especially optimistic nor especially pessimistic) say the Social Security trust fund will finally run dry. That year, projected Social Security payments will be about $600 billion more than collections from Social Security taxes, which means the government will be redeeming its last $600 billion in IOUs from the Social Security trust fund. But since the government owes this money to itself, it will have to find some other source for this $600 billion. Generally speaking, the government could find this money through some combination of the following three options: (1) it could cut spending on discretionary programs; (2) it could borrow the money from the public by selling more debt (in the form of Treasury bills, notes, and bonds); or (3) it could raise other taxes.
Unfortunately, none of the options are viable. The $600 billion is more than the total amount of all non-defense discretionary spending, so the government would have to eliminate all these programs and substantially cut the military to save this much money. Borrowing an additional $600 billion might be viable if it were a one-time event, but these IOU redemptions will be an ongoing trend; I don’t believe there’s any chance that U.S. Treasuries would still be considered sound investments if we kept borrowing so much year after year. An increase in other (non-FICA) taxes won’t work either; if you go through the analysis, you’ll find that the government would have to raise individual income tax revenues so dramatically to collect an extra $600 billion that the tax increases might well wreck the economy. We might try to get $200 billion from each of the three sources, but I’m afraid that wouldn’t work either, because the cumulative effects over many years would be too devastating to both the government budget and the economy.
That’s not even the worst of it. Medicare is expected to face a similar crisis, and many economists believe that rapidly rising health costs will make that crisis much more severe than the one for Social Security. Moreover, I’m skeptical that the trust fund will last even to 2036 (which is still long before today’s young workers will retire). Why am I skeptical? Because of life expectancies. When predicting future Social Security payments, the government has to make assumptions about future life expectancies; after all, if people live longer, then more people will still be alive collecting benefits in the future. To take an example, the current projections assume that American women will not reach a life expectancy of eighty-two until about 2030 — but that’s already the life expectancy of women in France. In fact, during the twentieth century, U.S. life expectancies rose an average of about three years per decade. If that trend continues, life expectancy for women will reach eighty-six by about 2030, which throws the current Social Security (and Medicare) projections almost completely out the window.
The bottom line is that the Social Security trust fund is a myth. People can (and do) argue about whether it is fair or right that Social Security money has been used for other purposes, but this won’t change the reality that the trust fund contains no money. As a result, the system is headed for dramatic failure, and we either act to fix it now, or suffer the consequences later.
CHAPTER 7 ENERGY MATH
[From the beginning of the chapter]
Question: Suppose that we had the ability to generate power through nuclear fusion (which is different from the fission used in current nuclear power plants), using hydrogen extracted from ordinary water as the fuel. If you had a portable fusion power plant and hooked it up to the faucet of your kitchen sink, how much power could you generate from the hydrogen in the water flowing through it?
Answer choices:
A. Enough to provide for all the electricity, heat, and air conditioning you use in your house
B. Enough to provide for the energy needs of everyone on your block
C. Enough to provide for the energy needs of approximately 500 homes
D. Enough to provide for the energy needs of approximately 5,000 homes
E. Enough to provide for all the energy needs of the entire United States
…let’s think about how we could actually figure it out. There are only two steps.
First, we need to know how much water flows through your kitchen faucet. The easiest way to learn this is to turn it on, place a pitcher under it, and see how much water you can collect in a fixed amount of time. You’ll find that a typical kitchen faucet pours out about three quarts of water per minute, or just over one and a half ounces per second. The second step is to calculate the amount of energy that can be generated by fusing the hydrogen in this water. This step takes a little more work, since it requires data about the amount of hydrogen in each ounce of water and the amount of energy released by fusing that hydrogen. You can find the necessary data easily on the Web, but I’ll just tell you the answer: If you could fuse all the hydrogen in the water flowing from your kitchen faucet, you would generate about three terawatts of continuous power.
What are terawatts? A terawatt is a trillion watts, which means that continuous power of one terawatt is enough power to keep 10 billion 100-watt light bulbs, or the equivalent thereof, turned on for as long as the power is turned on. So three terawatts is enough power to light the equivalent of 30 billion 100-watt light bulbs, which turns out to be roughly equal to the total amount of continuous power used by the entire United States. …
Think about what we’ve just found out. If we had the technological capability for fusion power, and if you were willing to allow us to borrow your kitchen sink, then we could stop drilling for oil, we could stop digging for coal, we could dismantle all the dams on our rivers, we could take down all the wind turbines, and we could even turn off all the currently operating nuclear power plants. All we’d need is the power coming from fusion at your kitchen sink. We’d use that power to generate electricity, which would in turn power all our cars, homes, and industry. (Fusion turns out to be somewhat easier if, instead of using ordinary hydrogen, we use an isotope of hydrogen called deuterium. About 1 in 50,000 hydrogen atoms is deuterium, so to use deuterium we’d need a water flow equivalent to about 50,000 kitchen sinks — which is still only about the flow rate of a small creek. Fusion is even easier with helium-3, but we’d have to get that from the Moon — which may not be as ridiculous a suggestion as it might at first sound.)
The potential of nuclear fusion is truly mind-boggling, an example of what is sometimes called a game-changing or disruptive technology — one that would fundamentally change the way we approach energy, both economically and politically. So why aren’t we building nuclear fusion power plants? Because we don’t yet know how. Whether we can figure it out soon enough to make a difference to our current energy problems is a topic of great debate, and one that we’ll return to later in this chapter. But I hope I’ve at least opened your mind a bit to what may be possible if we work hard enough. As Mike Bowlin says in the quote that opens this chapter, a key to solving our problems lies in finding ways to “embrace the future.”
CHAPTER 8 THE MATH OF POLITICAL POLARIZATION
[From the middle of the chapter, in the section on redistricting]
The practice of drawing district boundaries for political advantage is so common that it has its own name — gerrymandering. The term originated in 1812, when Massachusetts Governor Elbridge Gerry created a district that critics ridiculed as having the shape of a salamander. A political cartoon of the time used the governor’s last name to make the word “gerrymander” as a play on “salamander,” and the name has stuck ever since.
Let’s do a simple example to show how it works. Figure 16 shows a hypothetical state with only 16 voters (represented by little houses), half Democrats and half Republicans. The left side shows how they are distributed geographically. On first glance, it appears that the geographical distribution is even, since Democrats and Republicans live in alternating houses by row. But look what happens if you draw the districts as shown on the , in the section on redistricting. Because District 1 is 100% Republican, the Democrats end up with a clear majority in Districts 2 and 4, and a tie in District 3. If we now assume that each little house in the figure represents the general preferences of a couple hundred thousand voters, then it takes only a bit more planning for the Democrats to virtually ensure that they’ll win three of the four districts, despite the 50:50 split in overall voter preferences.
You’re probably wondering: Is all this really legal? That is, just because one party controls the legislature at the time of a census, does it really have the right to divide up districts to maintain that control? The answer is both no and yes. The Supreme Court has ruled that gerrymandering for partisan political purposes violates the Constitution, which makes it illegal in principle. In practice, however, politicians can usually come up with an alternate rationale that accomplishes the same partisan effect, but to which the courts have been much less willing to object. For example, if the politicians actually said that their aim in Figure 16 was to pack Republicans into one district, the courts would not allow it. But if they said they drew the district to reflect a well-traveled business corridor, or to give a voice to a group of voters who share similar concerns about schools, then they could probably get away with it.
It’s worth noting that party power is not the only objective for which gerrymandering can be used, especially when compromise is required. For example, suppose the Republicans and Democrats share power in a particular state (such as by one controlling the legislature and the other the governorship). The governor would veto a formula that heavily favors the opposing party, so they’ll have to compromise — and a likely compromise will be one that gives powerful incumbents easy-to-win districts. Similarly, suppose that some powerful legislators have a grudge against some particular congressman of the other party. They might decide to redraw his district boundaries so that he no longer lives within it, making him ineligible for reelection to his current seat. Another common strategy is to draw district boundaries that force two popular incumbents from the opposing party into the same district, ensuring that they cannot both be reelected.
Complicating things even further, redistricting applies not only to congressional districts, but also to districts drawn for state legislatures. The number of possible ways of drawing district boundaries for both state and national offices is enormous. Today, the parties typically employ mathematicians or statisticians, along with computer programmers, to create sophisticated computer models with which they can test the effects of millions of different possible boundaries; they then choose the model that seems most likely to achieve their political goals without going so far as to risk losing a court challenge. If they have the power, they can put these boundaries in place; if they don’t, they can make deals to accomplish as many of their goals as possible. We have indeed reached a point where, through mathematics, representatives essentially choose their own voters.
CHAPTER 9 THE MATHEMATICS OF GROWTH
[From the middle of the chapter, in the section on world population, building on an earlier example of the growth of a population of bacteria]
…To make the arithmetic a little easier, let’s call the doubling time about 60 years and start from the world population of close to 7 billion at the end of 2010. If the annual population growth rate held steady at 1.1%, then the human population would double to 14 billion by 2070, double again to 28 billion by 2130, and reach 56 billion by 2190.
It’s probably already obvious that our numbers cannot really go so high, but let’s keep going anyway, just to see where it leads. The total land surface area of Earth is about 150 million square kilometers. If you extend human population growth with a growth rate of 1.1%, you’ll find that, in a mere 900 years, our entire planet would be standing room only. Even if we could somehow colonize the rest of the planets and moons in our solar system, we’d have standing room only on all of them, too, just a couple hundred years after that. And like the bacteria in the bottle, we’d reach the impossible limit of filling the universe within a few thousand years.
People can and do argue about the relative merits of population growth versus population stability, or about how many people Earth can actually support. But these arguments are a luxury of our time. After many thousands of years of human civilization, our numbers have reached the point where they cannot continue their exponential growth much longer. We cannot really stand elbow to elbow on every bit of land on Earth, so a roughly 1% growth rate cannot possibly continue for another 900 years. It’s almost inconceivable to imagine the planet supporting a population of 56 billion, so this type of growth for another 180 years is also out of the question. In fact, very few experts believe that our planet could sustain a population of 14 billion people, which means that our current growth cannot continue even for another 60 years.
The lesson should be clear. We are like the bacteria in the bottle a minute or two before midnight, because one way or another, the exponential growth of world population will stop within the next few decades. Fortunately, we differ from the bacteria in one crucial way: While there is nothing that either we or the bacteria can do to change the fact that exponential growth will stop, we have the ability to choose how it stops. There are two basic ways to slow or stop the growth of a population: (1) decrease the birth rate, or (2) increase the death rate. Because no one wants to see an increase in the death rate, our only real choice is to reduce the birth rate.
The good news is that most people are already choosing to reduce the birth rate. Before the twentieth century, the average woman gave birth to more than 6 children during her lifetime. By 1950, that average had fallen to about 5 children. It has fallen more rapidly since then, and today the global average is about 2.5 children. It needs to fall just a little more (to about 2.0 to 2.1 children, depending on assumptions for death rates) for world population to stop growing and stabilize; in some nations, including Japan and several nations in Europe, the birth rate has already fallen below this threshold, leading to declining populations. If the downward trend in birth rates continues to follow its pattern of recent decades, human population will stabilize at about 9 to 10 billion people by around 2050.
Before we become too complacent, however, we should keep two important facts in mind. First, even if the population does indeed stabilize at 9 to 10 billion by 2050, that still means an increase of 2 to 3 billion people over the next 40 years. That’s an average of 50 to 75 million more people each year. At 75 million, which is about the current annual increase (it would begin to drop as the population leveled out), we find that every four years the world is adding a population nearly equivalent to that of the entire United States. Think about the infrastructure, food, education, energy, and jobs that will be needed for all these people, and you can see that we face a daunting global challenge.
This brings us to the second important fact, which is that this daunting scenario is probably the best-case scenario, and it will happen only if people all around the world make a conscious decision to make it happen. After all, having fewer children is a choice that requires action such as the use of birth control; without such action, the birth rate would go back to its natural high average of 6 or more children per woman. So the next time you get involved in a debate about family planning, birth control, teen pregnancies, single motherhood, or any other population-related topic, please remind everyone of the following indisputable facts about world population growth:
The exponential growth of world population will come to a stop, and it will come to a stop soon. It will do so through either a decrease in the birth rate or an increase in the death rate. This statement is not a threat, a warning, or a prophecy of doom. It is simply a law of nature, because exponential growth always stops. As human beings, we can choose to bring it to a gradual halt through conscious effort to reduce the birth rate to the point of population stability. But if we do not make this conscious choice, then the death rate will have to go up dramatically, and this is a fact over which we have no more control than we do over hurricanes, tornadoes, earthquakes, or the explosions of distant stars. In the end, then, we need only answer the simple question of whether we can be smarter than the bacteria in the bottle.
EPILOGUE GETTING “GOOD AT MATH”
[From the middle of the chapter, in the section on how to improve mathematics in college]
The issues change a bit once students enter college, because they generally are adults and therefore are responsible for their own choices. By and large, this means that college students split into two groups: those who have made the decision to pursue a major or career that will require calculus or more advanced math, and those who have not.
The first group is often known to educators by the acronym STEM, which stands for science, technology, engineering, and mathematics majors; personally, I prefer to think of them as the students on the calculus track. This group is easy for colleges to work with, at least in principle, because college mathematics departments have been built around the calculus track for decades. The major problems that colleges face in working with these students are that we’d like to have more of them to begin with (since careers in fields requiring advanced mathematics will be in high demand) and that too many current students drop off this track. As at lower levels, I believe that a little rethinking of the way we teach calculus and other advanced math classes could go a long way, particularly if we move to a more context-driven approach and make better use of technologies that can help students visualize mathematical ideas.
The more difficult challenge for colleges comes in deciding what to do with the students who have already made the choice to leave the calculus track. Most colleges require all students to complete at least one college course in mathematics, which is a very good thing given the importance of mathematics to our lives. Unfortunately, I believe that most colleges are still wasting this opportunity by teaching students something other than the “math for life” that they’ll really need. The important point is that, for the vast majority of these students, their single required college math course will be the last math course they ever take in their lives. We therefore owe it to these students — and to the nation and world — to make the best possible use of the time the students will spend in this last math course. With that in mind, I’ll offer a few specific suggestions.
Make college algebra an oxymoron. Nationwide, the majority of students who are not on the calculus track currently fulfill their college mathematics requirement by taking a course in “college algebra.” This is pointless, for at least two reasons. First, most college students already have taken at least two years’ worth of algebra in middle or high school; if it hasn’t already sunk in, it’s difficult to believe that one last semester of it will make a huge difference. (I once heard an algebra textbook author answer a question about the difference between “high school algebra” and “college algebra” approximately as follows: “The difference is simple. In college algebra, we teach students the same things that we taught them in high school algebra, only this time we teach it to them LOUDER.”) Second, students who don’t plan to take more advanced math will never again use most of what we teach in algebra. Let’s recognize “college algebra” for what it really is: high school algebra that is taught in college. As such, it should be considered a remedial course for those who need it because they hope to move on to more advanced math courses. As a remedial course, it should not count toward any graduation requirement.
Focus on quantitative reasoning. If you accept my rationale for no longer allowing algebra to fulfill the college math requirement, then the question becomes what to replace it with. To me, the answer is clear: quantitative reasoning. As you’ve seen throughout this book, most of the mathematical skills needed for quantitative reasoning are fairly basic, but the level of conceptual thinking can be quite advanced. This means that quantitative reasoning can be taught at a clearly collegiate level, and there is plenty to cover in a semester-long or even a yearlong course; this entire book contains only about 5% to 10% as much material as a quantitative reasoning course typically covers. Moreover, because quantitative reasoning is so important to modern life, I believe it is a great disservice to make the requirement anything else. For example, some colleges have recently introduced course requirements in financial literacy, while others offer courses in statistical literacy; both types of course are clearly useful, but neither covers the breadth of topics that we’ve covered in this book, which means they are not by themselves enough. (Note, however, that such courses can be great options for one semester of a two-semester quantitative reasoning requirement.) Still other colleges offer courses giving students a brief introduction to some of the esoteric branches of mathematics that mathematicians study. These courses can be immensely interesting, but I don’t think they are covering the material that students need for their everyday lives; for that reason, I’d make such courses electives, to come after a quantitative reasoning requirement is fulfilled.
You can’t learn if you don’t study. The single biggest problem in college mathematics education is the same problem that is harming all college education: the downward spiral in how much students are studying. Surveys show that the average number of hours that college students study outside class has fallen from about 25 hours each week in the 1960s to about 14 hours today. Unless you believe that students of today study much more efficiently than students of the past — and given the distractions that students now face from their electronic devices, it’s far more likely that the opposite is true — then this dramatic reduction in study time can only mean that college students today are learning much less than their counterparts of the past. While it’s easy to see the pressures on college faculty that have led to these reduced expectations of students, it’s equally easy to see how detrimental this fact is both to students and to society at large. The solution, of course, is for colleges to institute policies to ensure that all courses require students to put in a traditional level of collegiate effort, which means two to three hours of study outside class for each hour in class. This solution admittedly will be difficult to implement in practice, but if we don’t implement it, then college will increasingly become a waste of time and money for everyone involved.