Mr. Jabez Wilson laughed heavily. “Well, I never!” said he. “I thought at first that you had done something clever, but I see that there was nothing in it after all.”
“I begin to think, Watson,” said Holmes, “that I make a mistake in explaining. ‘Omne ignotom pro magnifico,’ you know, and my poor little reputation, such as it is, will suffer shipwreck if I am so candid….”
The Red-Headed League, Arthur Conan Doyle
I recently was asked by a manager, an English major, who struggled with the quantitative part of the GMAT (Graduate Management Admissions Test) how to learn math. Many people struggle with mathematics in our increasingly mathematical world, ranging from balancing their checkbook to managing a budget at work to understanding the abstruse mathematical models that are invoked more and more often in public policy debates such as global warming. This article is an expanded version of my answer.
The most important rule to master mathematics is: if you get lost (and most people including “experts” get lost a lot), back up to what you know and start over. Don’t try to keep going; in most cases, you will only get more lost. If necessary, when you back up and start over, take smaller steps, find and use simpler, more concrete, and more specific learning materials and examples, and practice more at each step. Repeat this process, backing up, simplifying and practicing more until you find yourself making progress. In a nutshell, this is the secret of mastering mathematics for most people.
Mathematics is not Like English
Many mathematically oriented people have weak verbal skills. The verbal SAT scores for students at engineering and science schools such as MIT, Caltech, and Carnegie-Mellon are generally much weaker than their impressive quantitative/math scores. Conversely, many people with strong verbal skills are weak at mathematics. I am somewhat unusual in that I scored in the top 99th percentile on the verbal sections of both the undergraduate SAT and graduate GRE exams. I can compare and contrast learning Mathematics and learning English (and other humanities) better than most.
Mathematics differs from English and many other humanities. In mathematics, each step depends critically on each preceding step. Learning addition depends on knowing the numbers, being able to count. Multiplication is meaningless without a mastery of addition: three times four means “add three fours together (4 + 4 + 4)” or “add four threes together (3 + 3 + 3 + 3)”. Division is defined in terms of multiplication: twelve divided by three is the number which when multiplied by three gives twelve (the answer is four). This critical dependence of each step on a preceding step or steps is found in most mathematics from basic arithmetic to algebra and calculus to proving theorems in advanced pure mathematics to performing complex calculations by hand or using a computer.
In English and many other humanities, missing a step — not knowing the definition of a new word, skipping a few sentences or even pages in a rush, etc. — is frequently not a show-stopper. You can keep going. The meaning of the unknown word or the skipped passages will often become clear from context. It is important to get the big picture — the gist of a passage, article, or book — but specific details frequently can be missed or poorly understood without fatal consequences. You can still get an A in school or perform well at work. Of course, it is better to read and understand every word and detail, but it is usually not essential.
In mathematics, when you encounter a unknown term or symbol, it is critical to master its meaning and practical use before continuing. Otherwise, in the vast majority of cases, you will become lost and get more and more lost as you proceed. If a single step in a calculation, derivation of a formula, or proof of a theorem doesn’t make sense, you need to stop, back up if necessary, and master it before proceeding. Otherwise, you will usually get lost. This is a fundamental, qualitative difference between Mathematics and English (and many other humanities).
Don’t Compare Yourself to Prodigies
The popular image of mathematics and mathematicians is that mathematics is akin to magic and mathematicians are anti-social weirdos born with a magical power that enables them to solve differential equations in the cradle — no practice or hard work required. The movie Good Will Hunting (1997) features Matt Damon as a self-taught mathematical genius from a tough, poor Irish neighborhood in Boston and janitor at MIT who solves world-class math problems left on classroom blackboards while cleaning. The hit situation comedy The Big Bang Theory features Jim Parsons as Sheldon Cooper, a crazed theoretical physicist with the supposed symptoms of Asperger’s Syndrome who apparently published breakthrough research as a teenager. The 1985 movie Real Genius, set at a fictional university very loosely based on Caltech, features Gabriel Jarret as Mitch Taylor, a fifteen year old self-taught child prodigy with a terrible relationship with his unsupportive parents who is shown performing breakthrough research for the CIA as a (15 year old) freshman at “Pacific Tech.” Many more examples may be cited in movies, television, and popular culture.
Two points. First, these popular, mostly fictional portrayals of math and science prodigies are greatly exaggerated compared to actual prodigies, as impressive and intimidating as the real prodigies can sometimes be. Fictional prodigies, like Matt Damon’s Will Hunting, are frequently depicted as arising as if by magic or divine intervention in highly unlikely families and circumstances. In contrast, the most common background for math or science prodigies seems to be an academic family — Dad or Mom or both parents are professors — or a similar math and science rich family environment. Many prodigies that I met at Caltech or other venues have academic or other knowledge-rich family backgrounds. Not a single janitor from MIT 🙂 .
Fictional prodigies are also frequently portrayed making major scientific or technological breakthroughs as teenagers. This is exceptionally rare in the real world. It is true there have been a fair number of major scientific and technological breakthroughs by people in their twenties, but teenagers are quite rare. Even Philo Farnsworth, who is often credited with devising electronic television at fourteen, did not have a working prototype of the electronic television set until he was twenty.
Most real math prodigies, like most or all chess prodigies, appear to achieve their remarkable performance through extensive study and practice, even if they have some inborn knack for mathematics. Curiously, many real prodigies do not achieve the accomplishments that one might expect later in life.
Second, genuine prodigies are very rare. Despite the portrayal in Real Genius most undergraduates at Caltech in the 1980’s were not real-world prodigies, let alone exaggerated fictional prodigies like Mitch Taylor and Chris Knight (played by Val Kilmer). Historically, especially prior to the transformation of math and science during and immediately after World War II which made it more difficult to pursue a career in math or science without very high quantitative scores on standardized tests and exams, many breakthroughs in mathematics and highly mathematical sciences were made by non-prodigies. The mathematician Hermann Grassmann was described as “slow” by some of his teachers. Minkowski famously referred to Einstein as “that lazy dog.” Grassmann and Einstein are both examples of “late bloomers” in math and physics.
In learning math, don’t compare yourself to prodigies, especially fictional prodigies. Most people who are proficient in math weren’t prodigies.
How to Learn Math
Again, to learn math, if you get lost, which is common and natural, back up to what you know, make sure you really know it, practice what you know some more and then work forward again. You may need to repeat this many times.
Sometimes a step may be difficult. Try to break the difficult step down into simpler steps if possible. Learn each simpler step in sequence, one at a time. Mathematics textbooks and other learning materials sometimes skip over key steps, presenting two or more steps as a single step, assuming this is obvious to the student (it often is not) or will be explained further in the classroom (it often is not). Consequently, be on alert that a single confusing step may hide several steps. If a single step is confusing, try to find a teacher, another student, or learning materials that can explain the step more clearly and in more specific detail.
Mathematics is an abstract subject and suffers from excessive abstraction in learning materials and teaching. A notorious example of this is the “New Math” teaching experiment of the 1960’s.
Some of you who have small children may have perhaps been put in the embarrassing position of being unable to do your child’s arithmetic homework because of the current revolution in mathematics teaching known as the New Math. So as a public service here tonight I thought I would offer a brief lesson in the New Math. Tonight we’re going to cover subtraction. This is the first room I’ve worked for a while that didn’t have a blackboard so we will have to make due with more primitive visual aids, as they say in the “ed biz.” Consider the following subtraction problem, which I will put up here: 342 – 173.
Now remember how we used to do that. three from two is nine; carry the one, and if you’re under 35 or went to a private school you say seven from three is six, but if you’re over 35 and went to a public school you say eight from four is six; carry the one so we have 169, but in the new approach, as you know, the important thing is to understand what you’re doing rather than to get the right answer. Here’s how they do it now….
Tom Lehrer, Introduction to New Math (The Song)
The rule (for most people) in math is: if a step proves too abstract, look for more concrete, specific learning materials and examples. If “balls in urns” (a notorious cliche in probability and statistics) is too abstract for you, look for explanations and examples with “cookies in jars” or something else more concrete and relevant to you. Something you can easily visualize or even get from the kitchen and use to act out the problem.
The simpler and the more concrete and specific you can make each step in learning math, the easier it will be for most people. Practice, practice, practice until you have mastered the step. It usually takes a minimum of three worked examples or other repetitions to remember something. Often many more repetitions, followed by continued occasional use, is needed for full mastery. Then, and only then, move on to the next step in the sequence.
Start with the simple, the concrete, and the specific. In time the abstract and the more complex will come. Don’t start with the abstract or the complex. If something is too abstract or complex for you, make it concrete and simplify if possible. Search the library, the used book store, the Web, anywhere you can, for simpler, more concrete learning materials and examples that work for you. Practice, practice, practice. Today, many tutorials, videos of lectures, and other materials (of widely varying quality) are available for free on the Web.
The Perils of Drinking from a Firehose
The critical dependency of each step on mastering the preceding step in learning math has some strong consequences for education. When I applied to Caltech many moons ago, the promotional materials from the university included a phrase comparing learning at Caltech to “drinking from a firehose.” This is the sort of rhetoric that appeals to young people, especially young men. Of course, no one in their right mind would try to drink from a firehose. This did not occur to me at the time.
In the 1980’s, perhaps to the present day, Caltech had a stunning drop out rate of about a third of its top rated, highly intelligent students.
It soon became apparent that much of the teaching by big-name researchers was rather mediocre. It did not compare well to the math and science teaching I had experienced previously. At the time, I lacked an adequate understanding of how math and science topics are successfully taught and learned to explain what the professors were doing wrong. It should be noted that success as a researcher or scholar appears to be unrelated to the ability and skill to actually teach in one’s field 🙂 .
What was the problem? In general, the professors were rushing through the material, especially many foundational topics and concepts that they considered basic and obvious — sometimes even skipping them entirely. They often gave extremely advanced, complex, sometimes “trick” problems as introductory examples, homework, and exam problems. The problems may have been intellectually fascinating to a researcher with years of experience but quite inappropriate for an undergraduate class in mathematics or physics.
I have a still vivid memory of a second year mathematics course professor droning on about “linear functions” and “linear operators” until one frustrated student finally spoke up and asked: “What is linear?” Now, the professor did give a pretty good answer as to what linear meant in mathematics, but the point is that the idea was taken so much for granted by the big-name mathematics faculty that they had not even bothered to teach it in their introductory classes. 🙂
Looking back, most of the undergraduates at Caltech were coming from schools with excellent math and science teaching which followed many of the rules outlined in this article. There was enough use of simple examples and repetition built into the classes to ensure that a motivated student would absorb and master the material. In fact, in many cases, the very bright students who went on to Caltech probably felt they could go faster, hence the appeal of “drinking from a firehose.”
The lesson for anyone learning math is to make sure whatever course or curriculum that you take goes slow enough, taking the time to present each step in a simple, clear way, that you can fully absorb the material — learn and master each step before advancing to the next step. It should not be “drinking from a firehose.” Rather, you should feel you could go a bit faster. Not ten times faster, but there should be a cushion, more time and repetition than absolutely necessary, in case you have difficulties learning a particular step, get sick, break up with your girlfriend/boyfriend, or something else happens. Real life is full of unexpected setbacks.
Each step in learning mathematics depends critically on learning and mastering a preceding step or steps. The most important rule to master mathematics is: if you get lost (and most people including “experts” get lost a lot), back up to what you know and start over. Don’t try to keep going; in most cases, you will only get more lost. If necessary, when you back up and start over, take smaller steps, find and use simpler, more concrete, and more specific learning materials and examples, and practice more at each step. Repeat this process, backing up, simplifying and practicing more until you find yourself making progress. Don’t try to “drink from a firehose.” Be patient. Take your time and learn and master each step in sequence. In a nutshell, this is the secret of mastering mathematics for most people.
© 2014 John F. McGowan
About the Author
John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. 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 and a Visiting Scholar at HP Labs working on computer vision applications for mobile devices. 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).
Dear John, Just wanted to thank you for taking the time to write these wonderful articles. They are full of great information and are lucidly written for the layman as opposed to most math blogs. Kudos to you.
Correction: CalTech has been corrected to Caltech in the article. Caltech is the capitalization used by the California Institute of Technology in its official publications.
SAT and GRE Scores
I received a number of comments on my use of SAT and GRE scores in the article.
SAT scores were rescaled by the College Board in 1994. There was a small shift in math SAT scores and a large shift in the Verbal SAT scores. One of the effects of this rescaling was to squash together verbal SAT scores at top schools, giving Caltech and top humanities schools similar median verbal SAT scores. When men were men and giants walked the Earth, verbal SAT scores were 1-200 points lower at Caltech than competing top humanities schools. This difference is harder to see today due to the rescaling of the verbal SAT scores. There are those sceptics who suggest the College Board rescaled the verbal SAT scores because it showed a long term steady decline since the 1940’s which was unpopular with politically powerful interest groups.
I used SAT scores as objective evidence in support of my general observation that people with strong quantitative skills tend to be significantly weaker in verbal skills. This is not always true and I have good reason including my verbal SAT and GRE scores to believe this is not the case for me.
I agree with comments that I received that SAT and GRE scores have significant weaknesses and are not necessarily an accurate measure of either quantitative, verbal, or other cognitive skills.
Thanks for this article; I know mathematics comes easily for me, and it is mostly because of my father and brother discussing and working with me as a child. It is very interesting that you point out that prodigies are most likely in a family where that certain brain function is nurtured most.
Same here, my mom knew good math and she teached me and now i believe that if you are in touch with math when you are still child, then they will come out in natural way when you mature. 😛
nice inspiring article by the way
Excellent article. It attracted me to read it for the whole morning. Thanks for sharing it with us. I may gather more confidence in Maths learning=)
Thank you. A well-written, well-thought-out article, and very true. Math, simply taught and without students fearing the ever-present possibility of trick questions, is attainable, but not by the standard texts/methods. These seem to be written by people who feel it necessary to make things tricky or complicated in order to prove they are smart enough to write a math text.
I graduated from high school without ever really grasping such simple basics as fraction to decimal conversion (both ways), and other general fraction manipulation.
After high school I took some placement tests at some temp agencies and realized that if I was to get the better jobs I needed to learn enough to pass the tests. So the next test I took, I wrote down the problems from a test, went home and figured them out and how to do these basics – in a day! I went back, re-took the test and aced it.
Doesn’t say much for my math teachers, because I repeatedly begged for help in one class, and finally gave, stopped trying, just showed up every class, handed back in blank tests and deliberately failed, hoping someone would finally notice and give me the kind of help I needed. Nope. The teacher was more concerned with following the text than teaching understanding and competency. A shame, or else I could have pursued engineering or architecture.
thanks for posting, helpful.
Being a math major, I realized how difficult it is for me to formulate a well-structured sentence without getting stuck. I find myself contemplating what words fit and whether or not everything makes sense. My lack of writing skills was the main reason I became a math major, and also because my love of math. Then I realize how odd it is that people are not good at math. That things that make perfect sense to me, are completely insane to someone who is not good at math. Math is like formulating a sentence. You have to have proof and follow a certain system. However, these two things are polar opposites of each other. I believe that math can be easy to follow if you are willing to put in the time and effort to understand it. Trust me, it is easier said than done.
Dear John, Thank you for an article so well written. So informative and insightful. Three cheers to you
This is a great tool for elementary school/high school teachers who aren’t too adept in the ways of math. I’m currently a student teacher and I have yet to sit in on a math class that lets the children know that it’s okay to step back from too hard a question and go back to what they know to start over. Knowing that both children and adults alike have this issue is a great resource for those who are struggling in the math department. It give them the chance to see that, even those at university level and beyond sometimes struggle. It also gives them a “work-around” their issue of not getting a problem – Stop. Start from what you do know. And then try again when you’re ready.
I loved math until I got to grade 9. I had a horrible experience in math 9 due to a teacher who rushed us through the basics. Then all of a sudden instead of looking forward to my math homework, I dreaded it. Math 10 came along and, guess what, I was terrible! I didn’t understand anything and the teacher didn’t understand why I hated doing the homework. There was never the time or knowledge to step back to what I knew and re-learn what I needed to.
I wish someone had showed me this article when I was going through this – just to let me know that it isn’t just me, that all I needed was a little more time to develop and learn what I needed to. Hopefully the children I teach will be able to have some solace in this article.
Hi John! I was always inclined to learn languages, but one day I decided that I am interested in mathematics. And I started learning it using my school books and other studying materials. Now I even think about getting a math degree.
Especially useful for me is “MATHEMATICS IS NOT LIKE ENGLISH” paragraph 🙂 Thanks for this great article.