How Not to Be Wrong

I recently read the book titled How Not to Be Wrong, The Power of Mathematical Thinking by Jordan Ellenberg. This book is composed of many examples of different types of mathematics, how that specific topic was used in a context, and basically how they were either wrong or right and how they may fix this issue. Ellenberg really puts the misuse of mathematics into perspective as you go through all of the examples provided in the book. Besides putting the misuses into perspective, I also learned a lot from this book about mathematical ideas and history that I had not previously known before. For example, I did not know that the man, who discovered the square root of 2, named Hippasus, was thrown into the sea to his death for this discovery. The book progresses through a series of realistic, real-life situations that involve mathematics, making it easier to comprehend the complexity of the situations. But I really liked how this book showed how these situations went wrong in the world of mathematics, how they went right, or how mathematics can be applied to other areas of our lives like science, politics, conspiracy theories, and the probability of completely random events.

One chapter that I really enjoyed reading, chapter 15, titled Galton’s Ellipse, really looks into regression and the patterns that you are able to see when you plot data on to a graph. Galton created our idea of a scatterplot by comparing different factors of heredity. Galton compared a father’s height and son’s height from many different families and decided that the best way to arrange the data would be in a graph, therefore creating the scatterplot. This scatterplot made the relation between the to factors easier to see, explained in the book on page 313, “The coxcomb and scatterplot play to our cognitive strengths: our brains are sort of bad at looking at columns of numbers, but absolutely ace at locating patterns and information in a two-dimensional field of vision.” After reading this, I could not agree more. We can be right in mathematics by visualizing things, the book even states that we are more likely to see a pattern when we are seeing something visually rather than seeing a bunch of numbers.

What I took away from this book was not how to be right, even though this is the title that the author had picked. Even in the last pages of the book, Ellenberg states that there is no way to always be right in mathematics, but it’s more about how we learn this mathematics and how we teach it to our future students. “We have to teach a mathematics that values precise answers but also intelligent approximation, that demands the ability to deploy existing algorithms fluently but also the horse sense to work things out on the fly, that mixes rigidity with a sense of play” (Ellenburg pg. 58). There is an entire section dedicated to how you should and should not teach algebra or calculus. Mathematicians and teachers are afraid of creating students who are purely computation based, but they are also afraid of creating students who are only conceptual thinkers, not able to do simple computations. This book is able to give suggestions of where to meet in the middle. Between the different views of how we should or should not teach math, I found it interesting that they were suggesting that we throw out all of the algorithms and procedural connections to math in order to have students stop relying on them. I agree with Ellenburg, that these are necessary but students should also not be dependent on them.

Overall, I thought that this was a really great book. I learned an awesome amount of history from the book and I also learned how to be right and how to be wrong. You have to be very careful when you’re completing computations and different environments bring in different factors that you need to be aware of. I think that all of the ideas really brought light to my eyes about what direction I should head into with mathematics and how to become a better teacher. I think that I would recommend this book because I think it’s filled with a lot of great real-life examples. It is difficult to follow sometimes because it jumps around a lot from topic to topic and it also tends to talk in circles, but I think that he makes great points at the end of each chapter. This book would have been better if it had an overhanging theme, tying every single topic back to this theme since it was easy to get lost in the computations and the long explanations. With a title like How Not to Be Wrong: The Power of Mathematical Thinking, you would think that it would be more about how not to be wrong, instead of these lengthy examples about history and abstract concepts. But still, I would recommend this book to anyone who is interested in mathematics.

Some quotes that I really enjoyed:

“Pythagoras was said to have the ability to talk to cattle and one of the few ancient Greeks to wear pants.” (pg. 34)

“Rule of mathematical life: if the universe hands you a hard problem, try to solve and easier one instead, and hope the simpler problem is close enough to the original problem that the universe doesn’t object.” (pg.35)

“In the mathematical context, the good choices are the ones that settle unnecessary perplexities without creating new ones.” (pg. 47)

“An important rule of mathematical hygiene: when you’re field-testing a mathematical method, try computing the same thing several different ways. If you get several different answers, something’s wrong with your method.” (pg. 64)

“Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.” (pg. 85)

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