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On Paramount Importance of Mental Math

Summary:  Mental math teaches students to see short, efficient solutions—rather than to blindly follow the brute-force, cookie-cutter, one-size-fit-all, show-all-your-work procedures taught at school.
 
To my youngest students, I lie—by omission—that vertical arithmetic does not exist.  I can usually get away with it for about a year. Until the school shows them the light.  Say, how to add 25 and 8 vertically, with the carry-over 1 carefully written on top of the 2.  By that time, my students are proficient in mental addition and subtraction of 3-digit numbers: carrying, borrowing, and all.  My goal though is by no means to turn them into human calculators.  So then, why bother?

Vertical arithmetic is a convenient method for computing numerical answers.  Especially when the numbers to manipulate are multidigit.  But it is a procedure, requiring—once learned—little thought.  The entire process is delegated to the mechanics of paper—and, essentially, to the spinal cord. It is almost as useless an exercise for one's brain as clicking away on a calculator.  Besides, in the 21st century, being able to follow a procedure is not exactly a marketable skill: where there is a procedure, there is a computer ready to do a much better job than a multitude of humans.
 
Further, in carrying out vertical arithmetic, one rarely deals with more than two digits at a time.  This encourages a math version of the tunnel vision: focusing on a small, narrow portion of the task, at the expense of seeing the big picture.  Finally, with the exception of division, vertical arithmetic is performed right-to-left, which is unnatural.  Indeed, we read numbers left-to-right, we call our answers left-to-right, and, perhaps most importantly, numbers diminish in significance left-to-right.  Consequently, when right-to-left vertical arithmetic is performed, the feeling of whether the answer makes sense has typically been lost along the way.  Combine this with the tunnel vision, and the groundwork for making “silly mistakes” is laid.
 
The situation is quite different with mental arithmetic.  When performed correctly, it is done left-to-right.  Thus, one typically begins with the most significant portion of the answer, and then augments it with less and less significant modifications, as necessary.  The sense of magnitude is maintained at each step—reducing the possibility of “silly mistakes.”  As an added bonus, one can usually begin calling the answer before its less-significant digits have been figured out—creating the impression that the answer has been obtained almost instantly.

Next, mental arithmetic requires mathematical imagination and short-term memory—therefore developing both.  A bad short-term memory is responsible for those “silly mistakes” which one makes due to forgetting—or rather incorrectly remembering—the results of prior steps.  A long-term memory benefits as well, as some numbers begin ringing the bell.  For example, I know off the top of my head that 16 × 3 = 48,  17 × 3 = 51,  252 = 625,  38 / 2 = 19, etc.

Yet, the main benefit of mental arithmetic is found elsewhere.  Nobody is thrilled about doing a lot of work in his mind.  Thus, one is forced to look for easier ways to approach a particular computation.  (Such ways include performing the operations in a more convenient order, using the distributive property and factoring, rounding, reducing, doubling and halving, etc.)  Each computation is approached as a puzzle in its own right, in which not just any, but the most efficient, solution is sought.  In that, the entire problem must be taken in as a whole—as opposed to the vertical arithmetic’s 2-digits-at-a-time tunnel vision.

The importance of this search for an efficient solution is hard to overemphasize.  It seems that during their entire secondary-school tenure, children are taught, and required to follow, brute-force, cookie-cutter, one-size-fit-all, show-all-your-work procedures.  A student who sees a shortcut, a simpler way to solve a problem, is more likely to be penalized than rewarded.  Gradually, even the best students tend to abandon what we loosely term “thinking”—a search for the simplest and most elegant solution—in favor of what the school miscaracterizes as “applying learned skills”: almost mindlessly following prescribed protocols.
 
To repeat, there is little if any virtue in learning how to blindly follow procedures, without asking questions or deviating from what one is told to do.  In the grown-up life, where there is an algorithms to follow, there is a computer.  The same can be said about punching the keys on a calculator: an under-educated cash-register operator at a fast-food restaurant is pretty much as good at punching them keys as one can hope to ever get.
 
It is in overcoming these school-instilled misconceptions of what it means to learn in mathematics, and the misguided study habits that accompany them, that the work is cut out for a math tutor.  This is when mental arithmetic—and, more generally, mental math—come to the rescue. 
 
The math portion of the SAT is a case in point.  I have tutored some [gifted] 7th graders to get the perfect score of 800 on this test.  It would seem then that everybody who is above average should be able to get the 800 say by grade 10.  Yet, achieving the 800 is beyond their wildest dreams for 98% of high-school seniors. Why is that?
 
A typical SAT problem can be solved as follows: one reads it slowly and thoughtfully (say, 10–20 seconds), ponders it briefly (5–10 seconds), sees a simple solution, and computes the answer mentally (5-10 seconds). The total time: 20 to 40 seconds. No calculator. Nothing is written.

By contrast, upon reading the problem, a typical high-school student plunges himself into quickly writing lines upon lines of mathematical expressions, or clicking away on his calculator.  Those 20 to 40 seconds later, he may still have precious little idea of where he is heading with all that work, rather hoping that something good is going to come out of it eventually.

The first thing which I tell my high-school students when I tutor them for a standardized test in mathematics, is to put away their calculators.  It is very uncomfortable for them at first. For they had sworn, and have been intending to stay true to, the school-math version of the Rifleman’s Creed: “This is my calculator.  There are many like it, but this one is mine. My calculator is my best friend.  It is my life.  I must master it as I must master my life. My calculator, without me, is useless. Without my calculator, I am useless.”  After the initial shock has subsided, I begin insisting that they do not write down anything, doing not just the arithmetic part, but the entire solution, mentally.

I have found this approach indispensable even with the most gifted of my students, when preparing them for math competitions such as the AMC.  School habits die hard.  I keep on reminding them that when practicing, the objective is not to arrive at the answer at any cost, but to learn from each problem.  Solving introductory to intermediate problems mentally forces them to abandon brute-force routes in favor of gems of solutions.
 
Q.E.D.
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Andrei K.

Yale Math PhD: SAT prep, gifted-child math and physics

3000+ hours
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