An Unconvincing Argument
One of the most common arguments for learning math (or computer programming or chess or <insert your favorite subject here>) is that math teaches you to think. This argument has a long history of failing to convince skeptical students and adults especially where more advanced mathematics such as algebra and calculus is concerned.
The “math teaches you to think” argument has several problems. Almost any intellectual activity including learning many sports teaches you to think. Reading Shakespeare teaches you to think. Playing Dungeons and Dragons teaches you to think. What is so special about math?
Math teaches ways of thinking about quantitative problems that can be very powerful. As I have argued in a previous post Why Should You Learn Mathematics? mathematics is genuinely needed to make informed decisions about pharmaceuticals and medical treatments, finance and real estate, important public policy issues such as global warming, and other specialized but important areas. The need for mathematics skills and knowledge beyond the basic arithmetic level is growing rapidly due to the proliferation of, use, and misuse of statistics and mathematical modeling in recent years.
Book Smarts Versus Street Smarts
However, most math courses and even statistics courses such as AP Statistics teach ways of thinking that do not work well or even work at all for many “real world” problems, social interactions, and human society.
This is not a new problem. One of Aesop’s Fables (circa 620 — 524 BC) is The Astronomer which tells the tale of an astronomer who falls into a well while looking up at the stars. The ancient mathematics of the Greeks, Sumerians, and others had its roots in ancient astronomy and astrology.
Why does mathematical thinking often fail in the “real world?” Most mathematics education other than statistics teaches that there is one right answer which can be found by precise logical and mathematical steps. Two plus two is four and that is it. The Pythagorean Theorem is proven step by step by rigorous logic starting with Euclid’s Postulates and Definitions. There is no ambiguity and no uncertainty and no emotion.
If a student tries to apply this type of rigorous, exact thinking to social interactions, human society, even walking across a field where underbrush has obscured a well as in Aesop’s Fable of the Astronomer, the student will often fail. Indeed, the results can be disastrous as in the fable.
In fact, at the K-12 level and even college, liberal arts such as English literature, history, debate, the law do a much better job than math in teaching students the reality that in many situations there are many possible interpretations. Liberal arts deals with people and even the most advanced mathematics has failed to duplicate the human mind.
In dealing with other people, we can’t read their minds. We have to guess (estimate) what they are thinking to predict what they may do in the future. We are often wrong. Mathematical models of human behavior generally don’t predict human behavior reliably. Your intuition from personal experience, learning history, and other generally non-quantitative sources is often better.
The problem is not restricted to human beings and human society. When navigating in a room or open field, some objects will be obscured by other objects or we won’t happen to be looking at them. Whether we realize it or not, we are making estimates — educated guesses — about physical reality. A bush might be just a bush or it might hide a dangerous well that one can fall into.
The Limits of Standard Statistics Courses
It is true that statistics courses such as AP Statistics and/or more advanced college and post-graduate statistics addresses these problems to some degree: unlike basic arithmetic, algebra, and calculus. The famous Bayes Theorem gives a mathematical framework for estimating the probability that a hypothesis is true given the data/observations/evidence. It allows us to make quantitative comparisons between competing hypotheses: just a bush versus a bush hiding a dangerous well.
However, many students at the K-12 level and even college get no exposure to statistics or very little. How many students understand Bayes Theorem? More importantly, there are significant unknowns in the interpretation and proper application of Bayes Theorem to the real world. How many students or even practicing statisticians properly understand the complex debates over Bayes Theorem, Bayesian versus frequentist versus several other kinds of statistics?
All or nearly all statistics that most students learn is based explicitly or implicitly on the assumption of independent identically distributed random variables. These are cases like flipping a “fair” coin where the probability of the outcome is the same every time and is not influenced by the previous outcomes. Every time someone flips a “fair” coin there is the same fifty percent chance of heads and the same fifty percent chance of tails. The coin flips are independent. It does not matter whether the previous flip was heads or tails. The coin flips are identically distributed. The probability of heads or tails is always the same.
The assumption of independent identically distributed is accurate or very nearly accurate for flipping coins, most “fair” games of chance used as examples in statistics courses, radioactive decay, and some other natural phenomena. It is generally not true for human beings and human society. Human beings learn from experience and change over time. Various physical things in the real world also change over time.
Although statistical thinking is closer to the “real world” than many other commonly taught forms of mathematics, it still in practice deviates substantially from everyday experience.
Teaching Students When to Think Mathematically
Claims that math (or computer programming or chess or <insert your favorite subject here>) teaches thinking should be qualified with what kind of thinking is taught, what are its strengths and weaknesses, and what problems is it good for solving.
(C) 2017 John F. McGowan, Ph.D.
About the author
John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing gesture recognition for touch devices, video compression and speech recognition technologies. He has extensive experience developing software in C, C++, MATLAB, Python, Visual Basic and many other programming languages. He has been a Visiting Scholar at HP Labs developing computer vision algorithms and software for mobile devices. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech).
The image of a Latin proof of the Pythagorean Theorem with diagrams is from Wikimedia Commons and is in the public domain. The original source is a manuscript from 1200 A.D.