"The mathematical ideas we want to address are ones that can be engaged with directly and profitably, whether your mathematical training stops at pre-algebra or extends much further."

- How Not to Be Wrong, page 16

"A mathematician is always asking ‘what assumptions are you making? Are they justified?’ This can be annoying. But it can also be very productive.`"- How Not to Be Wrong, page 7

Through all the examples, studies, theories, quotes, and lessons graciously shared by Ellenberg, the most fundamental point being driven home is around being a better critical thinker. The book is a continuum of examples that bake mathematical theory into everyday assumptions like headlines, studies, myths, gambling, religion (yup he goes there!) and everything in between. Within every story the protagonist is the principle, which for the most part are easily understood, who wins every time because they went just that much further to think about it a new way, ask a better question, or clarify what it is we are really getting at here. In the age of information overload, big data, and the reliance on computers and one of my closest friends, Excel, it is easy to forget what it is we are saying or understanding and whether it actually makes sense. The Insights below are some of my personal favourites that demonstrate how digging into what assumptions are we making—and if they are justified—can provide you an outcome you may otherwise not have considered.

"Human beings are quick to perceive patterns where they don’t exist and tend to overestimate their strength where they do."- How Not to Be Wrong, page 129

At the time of this summary the Toronto Raptors are tied in the second round of the NBA playoffs against the Miami Heat. Our nemesis seems to be Dwayne Wade, who most fans would agree in this series appears to not be able to miss a shot. Naturally my interest was piqued when the author dedicated a few pages to dispelling this myth. There are a number of mathematical principles at play here. One being in statistical studies, the phenomenon of an expected size can be underpowered. Think looking at the planets with binoculars instead of a telescope. This little theory reminds us that if we do some statistical sampling around what a hot hand might look like with some real data, there’s a chance that even if this “hot hand” is happening our method may not even allow us to detect it. There are multiple studies referenced that show the progression and refinement of disproving the hot hand myth. Not to leave you hanging, but essentially a 2009 study asserts that players who made a basket are more likely to take a more difficult shot next. Your basketballer friends will argue that they’ve seen this myth happen though! Wade can’t miss! He can, in fact he only made 13 of 24 field goal attempts in the last game. Stretch that over his 11 years in Miami and statistically speaking it will be challenging to find “hot hands”. So what are we taking away from this? The example here is a great reminder about how easily our assumptions can mislead us. You don’t even need to know the statistical theories mentioned, but have you considered the assumptions you are making? Are they justified?

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"It’s the epidemiological equivalent of saying there are -4 grams of water left in the bucket. Zero credit."- How Not to Be Wrong, page 61

This last insight was a humbling read for me. The author builds into the concept of linear regression (think any simple line graph trending in a certain direction). He uses an example of a study out of the US that claims that by 2048 all Americans will be obese. Some of us have probably seen this or similar headlines through many different media outlets. Now let’s consider our assumptions and if they are justified. Continue this trend and by 2060 a cool 109% of Americans will be obese. Wait, what? At some point your proportion needs to actually start to bend before 100%. Why? Because when you really think about it (mathematician or not)–is it actually a good assumption to say that 100% of the United States will be obese? The paper further dives deeper into the population statistics by sex and gender and informs us that 100% of black men will be obese in 2095. Double take. How does that happen when 100% of the population is already obese by 2048? The paper doesn’t acknowledge the mathematical contradiction and we are now left feeling like we’ve been let down by our statistical best friend (and you have). I am not suggesting (nor is the author) that linear regression is bad (phew), but let’s use this as a friendly reminder about the stories we plan to tell with it. What are you assuming? Is it justified?

*How Not to be Wrong* is a great tool for anyone who wants to become a better critical thinker, or perhaps has a staff member that could improve in this area. I love math and numbers and the legitimacy that it brings to a story. The author reminds us as good critical thinkers we need to be employing this method of reasoning to some degree or another. Perhaps even more importantly, consider how we are building this type of thinking within young people or our teams’ today–are we teaching them to consider their assumptions and if they are justified?

Jordan Ellenberg is a professor of mathematics at the University of Wisconsin-Madison, with a Ph.D. in mathematics from Harvard and an MFA in creative writing from Johns Hopkins. His areas of research specialization are number theory and algebraic geometry. He has written articles on mathematical topics in the New York Times, the Washington Post, Wired, the Wall Street Journal, the Boston Globe, and the Believer, and is a regular columnist for Slate.