Things Go on Forever
Ellenberg’s metaphors bring humor to his explanations of complex mathematical concepts. Human bias is toward thinking linearly—that is, assuming a current state or growth at the same rate would continue. For example, as Ellenberg wonders in his book: Can we predict the location of a ballistic missile? Knowing its direction and speed, yes. It is very easy to determine its location. However, missiles don’t move in one direction forever. Engineers design them to turn and land back on earth, preferably some distance from their launch point. This behavior introduces new variables such as gravity and wind resistance. This is one of the best explanations of calculus I’ve read—accounting for factors that determine the value of variables under certain conditions. It’s also a practical application of math.
In a similar vein as his denial of the example of the missile that flies in a straight line forever, Ellenberg critiques a national health study that claimed, by 2048, 100% of Americans would be obese. This is a fairly significant claim that would be sure to get headlines. But again, when we look into the numbers and the regression analysis that the study provided, we see significant problems. For example, the document uses a crude, linear analysis of the data to arrive at its conclusion, which indicates that, by 2060, around 109% of Americans would be obese. As the author notes, the paper includes further errors, indicating that black American males are becoming obese at half the rate of the overall population, so it would not be until 2095 that 100% of black American males would have become obese. But none of this is plausible. The root of the problem is that the report’s authors failed to consider the curve. Rates would slow over time, so the underlying statistical analysis falls apart fairly quickly.
You don’t need to do any complex math to realize that there are some logical contradictions in this example. We should probably be skeptical of these sorts of claims.
The problem of not knowing where we are in a series of events encourages people to make bad decisions every day. We have a tendency to assume that current states of being will continue. People buy into the stock market with enthusiasm when the market is up, invest in companies that are outperforming the market, bet on basketball players who have a hot-hand, bet on numbers that haven’t won recently on roulette wheels, and create algorithms that trick them into believing there are coded messages in the Torah—otherwise known as “The Bible Code.” People try to find patterns in their environment to make sense of things or to find an advantage. But we deceive ourselves by assuming that unrelated events have some bearing on each other.
Ellenberg uses the example of coin flipping to describe the problem of small samples. While we can assume that flipping a fair quarter in the air would result in a 50% chance of the quarter’s landing heads up, this does not mean that two coin flips would result in an equal distribution of heads and tails. Likewise, seeing a coin flip land five times in a row does not mean the sixth flip would land tails up. However, an appropriate sample size—perhaps 100 flips—would be likely to follow our assumption of an equal distribution of heads and tails.
Understanding the rationale and applicability for sample sizes is critical for UX professionals who conduct surveys or analyze large data sets.
Perhaps one of the most challenging aspects of math education relates to word problems. Performing simple—or even complex—mathematical calculations might be relatively simple for most people, but solving word problems is where the idea of mathematical thinking really shines. To be able to successfully solve a word problem, people must identify the relevant factors that support a decision. It is in solving word problems that we truly put our cognitive ability to the test.
While word problems can be challenging, the truth is that virtually all applications of math in the real world are math problems that are just waiting to be articulated.
Although you might assume that this is a book about math, it does not include a particularly heavy amount of arithmetic. Instead, Ellenberg presents mathematical tactics for testing our assumptions. This book encourages divergent thought, while providing tools that promote confidence. It also provides several thought-provoking examples, with analysis that encourages readers to think critically about the logic of our decisions, as well as the information that organizations provide to us.
Although this is not a typical UX book, Ellenberg does discuss how to understand A/B testing in a rational way. Nevertheless, when we, as UX professionals, need to make sound decisions and assess the quality of our assumptions, the tools of mathematical thinking can be invaluable.