A Mind for Numbers
How to Excel at Math and Science (Even if you Flunked Algebra)
by Barbara Oakley, Ph.D.
JEREMY P. TARCHER/ PENGUIN
Published by the Penguin Group Penguin Group (USA) LLC
New York, New York 10014
My Rating: 3/5
A Mind for Numbers is a good but not great book on techniques for learning math and science. The book covers a large range of methods including alternating between “diffuse” and “focused” thinking, “chunking,” practicing recall to firmly imprint knowledge in long term memory, the “memory palace” method of memorizing large amounts of information, and methods to avoid procrastination and overcome test anxiety. The book is primarily aimed at high school and college students, but will be useful for anyone trying to learn math and science. Unfortunately, the book has a number of weaknesses and readers should be cautious with some of its claims and complement the book with other sources of advice on how to learn math and science.
Unlike some books which focus narrowly on one specific technique, often touting it as the “one size fits all” solution to learning math, science, or other topics, A Mind for Numbers covers several different methods as well as other common issues such as procrastination and test anxiety — failing due to nervousness even though the student knows the material. The book has many references at the back so that interested readers can investigate each method in more detail and examine the original research behind the methods and suggestions.
I particularly liked the discussion of alternating between “focused” study and thinking and so-called “diffuse” study and thinking. This is something that I do a lot and have found is very effective. Focused study refers to conscious focused study and drilling on a topic. Diffuse study can refer to taking a walk, thinking — more like daydreaming about the big picture. Take a break and come back to the problem again and often the topic or problem will become clear. This is rather counter-intuitive, but nonetheless it works. As A Mind for Numbers points out, many key insights enabling major scientific discoveries and technological inventions were made during a walk or while taking a break after many years of intensive study and struggle. Of course, there is a danger that this method can turn into a pretext for goofing off or procrastinating.
I found the discussion of practicing recalling new knowledge to get it to imprint and remain in long term memory valuable. This was a new insight for me and seems to be true.
The book has a number of significant weaknesses. While these should not stop someone from reading the book, they should be kept in mind.
Lack of Specific Examples
A Mind for Numbers has surprisingly few specific examples of the methods in action. Remarkably for a book on learning math and science, the few specific examples are mostly not examples from math and science. For example, the book has a mnemonic for remembering the number of days in the months (January with 31 days, February with 28/29 days, March with 31 days, etc.) using the knuckles of the two hands. While this is a useful mnemonic, it is not closely connected to math and science. There are almost no specific examples of how to learn algebra, trigonometry, calculus, physics or other sciences. This makes the advice rather abstract in many cases and may disappoint some readers.
Learning Math is Very Sequential
As I discussed in my article How To Learn Math math is very sequential. Learning addition depends on already knowing at least the natural numbers (1,2,3…). Modern addition depends on already knowing the natural numbers and zero, a rather abstract concept that took thousands of years to invent. Learning subtraction depends on already knowing addition. Learning multiplication depends on already knowing addition. Learning division depends on already knowing multiplication which depend on knowing addition which depends on knowing at least the natural numbers. Raising a number to a power depends on already knowing multiplication. Taking a root, the square root of a number for example, depends on already knowing how to raise a number to a power. High school algebra depends on knowing all of these steps. Introductory calculus requires a basic proficiency in algebra. And so on.
This strictly sequential characteristic of mathematics has strong implications for learning mathematics. A Mind for Numbers is aware of this. It is mentioned briefly in passing in a few places, but negligible time is devoted to the subject. The key point is that if you get lost or don’t get something in learning mathematics, it is very important to back up to what you do know, start over, and be sure to master each step in turn. Otherwise, in most cases, you will simply get more lost. It also has the practical implication that students should avoid situations in which the mathematics is coming too rapidly for them to absorb each step, generally with a time cushion for those inevitable distractions and problems that life throws in our way.
The Night of the Living Dead
For some reason, the author refers to zombies and uses zombies as metaphors and illustrations for a number of points in the books, whether because the author is a fan of zombies or perhaps in an attempt to connect with younger readers who are into the current zombie fiction craze. I personally found the zombies a little distracting.
Unicorns and Careers
A Mind for Numbers has many inspirational stories along the lines of “Chauncey Wigginbotham III dropped out of Harvard and founded Colossal Software at age eighteen and became a billionaire by age twenty-seven through hard work, grit and determination” … and a million dollars in seed financing from the Bank of Mom and Dad followed by ten million dollars in Series A funding from the noted venture capital firm of Dad’s Old Buddy and Partners. In the Silicon Valley, these astonishing and extremely rare successes like Colossal Software are often called “unicorns.” The book also has similar inspirational stories about Nobel Laureates like the bongo-playing physicist Richard Feynman. The lesson is usually “you can do it — by following the morally upright precepts of The Book of Proverbs.”
Inspirational “unicorn stories” (minus the seed financing from the Bank of Mom and Dad) are a common element of some textbooks and many self-help books and materials. The relevance of these stories to most people and most students is highly questionable. In some respects it is analogous to pointing to someone who won $100 million in the state lottery as a role model.
In the Silicon Valley, most startups with strong founding teams, good ideas, substantial financing, plenty of hard work, grit, and determination … fail. Venture capital firms often informally claim an eighty to ninety percent failure rate and some informed observers suspect the actual failure rate might be ninety-five percent or even ninety-nine percent. It is common for failing venture-backed startups to merge or be acquired rather than go bankrupt or formally dissolve which makes it difficult to precisely define or identify the failures.
Of the ten to twenty percent of investments that succeed, most return modest returns on the initial investment. By most accounts, only a few percent become big successes that return five or ten times the venture capital firm’s investment. According to the Kaufmann Foundation, a major investor in venture capital funds, the overall returns of venture capital investing over the last decade have been disappointing: “After Fees and Expenses, Most Investors Will Do Better in Public Markets .”
The big successes typically achieve a market capitalization of around one-hundred million dollars and sometimes comparable actual revenues and a few percent actual profits: for example, five million dollars in net income with annual revenues of one-hundred million dollars. It should be pointed out many of the “successes” are successful stock market transactions, some people made over a hundred million dollars selling the stock but the actual business was not successful.
The unicorns — Facebook, Google, my fictional Colossal Software — are a tiny fraction of the small number of already rare and unusual successes. In her article “Welcome To The Unicorn Club: Learning From Billion-Dollar Startups,” venture capitalist Aileen Lee estimated that unicorns comprised “about .07 percent of venture-backed consumer and enterprise software startups”
Similar comments can be made about Nobel Prizes. As I have discussed in other articles, the fraction of Ph.D.’s in physics who become tenured professors is small, let alone the fraction who win a Nobel Prize like Richard Feynman (infinitesimal). How relevant to most math and science students, including truly exceptional students, is Richard Feynman’s career in fact? This does not mean that people should not learn math and science, any more than that people should not learn to read and write because they will never become the next Ernest Hemingway or J.K. Rowling.
Students and others are well-advised to look carefully and critically at the statistics, the actual distribution of outcomes for all students, workers, and “entrepreneurs” rather than extremely rare, exceptional cases. Most people who play state lotteries lose money over their lifetime! A Mind for Numbers would be a stronger book if it included statistics and graphs on career prospects for the different fields that it discusses as well as data on the value of math and science education for those who do not pursue a math or science related career, which often ultimately means software engineering today.
One Size May Not Fit All
The book tends to present the methods as scientifically proven methods that will surely work for everyone. Debatable theories such as the “working memory” theory of short and longer term memory, reasoning, and cognition are often presented as proven facts. Readers should keep in mind that our understanding of learning and cognition is limited and often tends to be derived from rather unrealistic experiments and activities. For example, studies of playing chess play a large and disproportionate role in research into learning, cognition and “expert” performance, notably in the work of Herbert Simon and Anders Ericsson (popularized by Malcolm Gladwell is his book Outliers).
Nonetheless, there are many observations and experiments that raise questions about many prominent theories of cognition and learning. How are some top chess players able to play multiple simultaneous chess games with several opponents? Does this really fit with the theory of working memory? Why do I seem to be able to keep track of many more than the 4-9 slots in my supposed working memory while in a deep state of concentration but can’t keep track of more than a few items under most conditions? Darned if I know, but it doesn’t fit with the simple theory of working memory espoused in the book.
People may differ at a fundamental organic level and different learning methods and techniques may work — or work differently — for different people. Readers and students should keep this in mind while reading the book or trying to put its suggestions into practice.
Overall, I give A Mind for Numbers a rating of three (3) out of a possible five (5), where five represents the best or an exceptionally good book. I would recommend that people read it if they are trying to learn how to learn math and science. Most readers are likely to find some useful information and some methods they do not know, as I did. However, it has some notable weaknesses and readers should both take its claims with a grain of salt as well as look to other complementary resources on learning math and science.
© 2015 John F. McGowan
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). He can be reached at email@example.com.
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