On Teaching Math Correctly

I think I’ll make a great math teacher, not because I can teach well, but because the math education in this country is so fucked up I can’t possibly do worse.

For those of you already teaching pre-college math, here are ten good places to start:

1)  Ban calculators:  This, I think, is an essential starting point.  I experienced the shock of my life in seventh grade when my math class was assigned calculators.  Not a single math problem in that grade level through high school is appropriate for the use of calculators.

Allowing students of that age to use calculators is so preposterous on so many levels, but let’s begin with the most obvious:  it makes kids dumb.  In college, I graded calculus homeworks.  As those who took calculus know, one of the popular word problem questions requires you to find an area of so and so.  Students come up with a negative answer and moves on to the next question.  It is mindlessness to the umpteenth degree.  And yet, students repeatedly do it because they get into the habit of thinking, “If I punch a number into a calculator and it spits out a number, it must be the answer.”  In American math education starting at seventh grade, no thinking is required.

It also undermines teaching.  What’s the point of teaching kids graphing methods and graphing principles when you allow them to use a graphing calculator to graph everything for them?   The importance of x-intercept, y-intercept, horizontal shift and vertical shift are all lost because there is no need to understand these concepts for a graphing calculator to graph.  I haven’t met a single American (except some gifted students majoring in math) who can do as simple a task as graphing a quadratic equation.  Utterly shocking.

The practicality of math is also lost.  Calculators handle fractions very poorly because they handle everything in the decimals through the binary system.  Because of this, Americans are incapable of working in the world of fractions, and thus producing an “exact” answer.  Everything is in decimals.  1/3 is always (incorrectly) answered as .333.  I tried to cure this while I was grading, but simply gave up because it was such a futile fight.  Moreover, Americans’ ability to do do mental math is greatly reduced because of their reliance on decimals.  I had to do 1.25 x 1.25 in my head the other day and realized how difficult that is.  Fractions easily give you the answer.  5/4 x 5/4 = 25/16 = 1 9/16 = 1.625 or, if this is too hard,  little greater than 1 1/2 or 1.5.

The importance of banning calculators to encourage mental math was hammered home in one of the education law classes I was taking.  This class, composed of principals and superintendents (I was the only law student), required each student to give a presentation on some educatio topic of his/her choice.  A principal chose the Massachusetts high school standard exam as his topic and discussed why calculators should be permitted on the exam.  As an example, he pointed to the multiple choice question which went as follows:  “Which of the following is closest to the square root of 1641?,” with answer choices 30, 40, 50 and 60.  His point was that no one uses square roots and this question is best suited for a calculator.  And I say, that attitude is precisely what’s wrong with the American educational system.  The point of the question is not to calculate the square root of 1641; it’s to estimate it using your knowledge of the concept of square roots.  This question is no different from asking the student to estimate a 15% tip from a restaurant bill.  The principal’s suggestion, while with good intentions (he had a grander policy reason why he was endorsing his view, but that’s not important for this discussion), is simply disturbing.

2)  Emphasize the importance of memorization:  I know many teachers are opposed to this, but this is actually common sense.  To be sure, it’s important to know concepts and principles.  But the fact is, you have to know 5 x 4 = 20 without adding 5 apples 4 times.  Similarly, you have to memorize the quadratic formula because even if you knew how to derive it (as you should), you shouldn’t be deriving it every time you need it.  You know what happens when you don’t memorize?  You end up like me, drawing a triangle ever time to figure out what sine of 30 degrees is.  Sine may be opposite over hypotenuse, but in higher levels of math, that’s really not that relevant.  Failure to memorize the basics hampers your ability in the future as it did me.

3)  Do not teach Pi as 22/7.  Of all the things that are simply wrong in American text books, this one tops them all.  22/7 is a rational number; pi is an irrational number.  They are fundamentally conceptually distinct numbers.  In fact, they have absolutely no relation to each other except that when you convert them to decimals, in both, the first three numbers are a 3, a 1, and a 4.  Pi is the first of many irrational numbers that kids learn.  It’s little wonder that they grasp no concepts of numbers when they were taught to estimate the first irrational number they saw by a rational number.

4)  Get rid of that = sign with a question mark over it.  To this day, I haven’t the slightest idea what the sign is supposed to mean.  I think it’s supposed to be a problem asking the student to calculate whether what’s on the left is equal to what’s on the right.  If it is, it’s a very unmathematical way of asking it.  Either two things are equal or they’re not–with consequences flowing from either.  There shouldn’t be a sign to indicate neither, as in, “we don’t know what the mathematical significance of this statement is.”  Absolute travesty is what that symbol is.

5)  Use a pencil.  This may be a Japanese thing, but I think it’s a minor point that’s rather important.  People make mistakes all the time in math, especially in early stages of algebra.  Students need to get in the habit of going back, finding what you did wrong, erasing it, and getting it right.  It is simply a terrible habit to cross out what’s wrong and write what’s correct in illegible manner.  Which leads me to the next point:

6)  Write cleanly and legibly:  I wasn’t particularly brilliant in math, but I got as far as I did because I had strong fundamentals and good habits.  If students write cleanly, the student (or the teacher) can go back through the answer and find out what he/she did wrong.  Students can also eliminate careless mistakes like mistaking a plus for a minus, which is an error made far too often.  Arithmetic is actually the easy part of math because the student has already translated the actual problem into numbers, but too many kids are atrocious with arithmetic.  The deficiency is caused by lack of practice, but also from sloppiness.

7)  Force students to write down, not sideways.  This is a pet peeve of mine.  Far too often, I see the following when kids solve equations:  5x+5=10=5x=5=x=1.  Good god.  This isn’t even about concepts and principles.  It’s kids not learning how to do math.  I was taught that in solving equations, = sign should line up top to bottom.  It is an excellent advice that majority of kids are never taught.

8)  Answers to word problems should contain units.  When the question asks “What is the area?”, the answer cannot be “2.”  That’s simply wrong.  Is it 2 square feet?  2 square miles?  2 square meters?  Does the kid even know?  I deeply suspect not, because he’s never been taught to care.  It’s terrible teaching not only because the answer is incomplete, but because it fails to teach an important principle.  You can add 2 and 3.  You cannot add 2 apples and 3 oranges.  (But you can add 2 fruits and 3 fruits).  On the other hand, you can multiple 2 apples and 3 oranges, but the unit of the answer depends on the question:  “How many apples are there if there are two apples for each of the three oranges?” or “How many oranges are there is there are two oranges for each of the two apples?”

9)  “Guess and check” is not a proper method to solve a problem.  I actually saw this as an option in a “box of tools” to use in a textbook when I was in second grade.  It encourages poor habits like mindlessly picking a number to see if it’s the answer.  The only time I can ever think of where “guess and check” was a proper method was in middle school when I was taught to factor polynomials.  Because there are always several combination of factors to choose from, the only way to see if it works was to actually pick one and try.  It’s worth noting that even in the realm of factoring, Americans teach it poorly, but that’s a whole another issue.

10)  Correctly teach the concept of percentages.  This is a substantive issue, but because American deficit in their knowledge of percentages is so staggering, I had to add it to the list.  If something doubles (i.e. multiply by 2), there is an increase of 100%.  Thus, if something quadruples, there is an increase of 300%.  Hence, if something increases by 800%, the original must be multiplied by 9, not 8.

This is not even the worst mistake.  If Apple’s market share of the U.S. personal computer market is 2% in one year and 5% the next year, by what percent did the market share increase?  The answer is not 3%; it’s 150% (or 2.5 times).  This concept is so fleeting that American newspapers consistently make this error.  Think about polls.  If McCain received 47% support in prior poll and 45% in the new poll, newspapers describe it as McCain’s support failing by 2%.  No! No! No!  His support fell by 2 POINTS, as Japanese news correctly reports.  This mistake drives me crazy to no end because the fact that it’s so prevalent proves most Americans don’t even realize there’s a problem.

Following these 10 points may not solve all the problems, but it’ll sure solve the most egregious ones.


9 Responses to “On Teaching Math Correctly”

  1. 1 Chris Schroeck December 3, 2008 at 12:52 pm

    My high school was really great at teaching math, and my teachers emphasized a lot of these same points. And yes, I can graph a quadratic equation by hand. At least, I could, when I used to do math, and I could probably still do it if I tried.

    The one place that I will quibble a bit is use of calculators. I don’t think it should be commonplace, but when I was taking calculus in high school, my teacher would let us use calculators to graph and do some other advanced things after we had first learned to do it by hand. That helps when the problem isn’t “graph this equation”, but instead is “graph this equation so that you can do something much harder”, and you are already good enough at graphing that it’s essentially busy work.

  2. 2 joesas December 3, 2008 at 4:21 pm

    Chris, you would truly be the first one I’ve met who is able to graph a quadratic equation (although, in fairness, I suspect Tom and maybe Tristan can too, but I wouldn’t know because I never inquired).

    I see your point about the calculator, but I think the problem is that’s a difficult line to draw. I can see how you can achieve higher level of math with a help of a graphing calculator, but I think it’s far more likely that calculators will be abused rather than effectively used. Perhaps Honors Calculus students like yourself can be excused from the ban?

  3. 3 Joseph Lee December 4, 2008 at 3:45 pm

    You have a lot of good points. I would like to point out, however, that as a student all but nine of your ten points were part of my math education, and I still failed at math. I have a card in my wallet that tells me how much I should tip because I can’t figure it out on my own (that’s not entirely true, but pretty close). Luckily, since it is me, you can just blame that on sheer stupidity, the sad but incurable disease, so your argument still stands. My mind is also blown by your bit about percentages at the end (that would be one out of the ten that wasn’t part of my education).

  4. 4 joesas December 4, 2008 at 4:48 pm

    I merely said these 10 points are necessary conditions for learning math, not sufficient. Clearly, you prove the “insufficient” aspect of my 10 points. LOL.

    Joseph, if you were in a math class that I taught, you would love math and be good at it. And you’ll understand the concept of percentages. (You seriously don’t follow what I wrote?)

  5. 5 Joseph Lee December 5, 2008 at 1:34 pm

    Oh no, I follow completely. I just never thought about it. The whole “it dropped 2%” never stuck out to me as completely and totally wrong… which it is. I just find it hard to wrap my mind around how the media could make such a large mistake like that without someone pointing it out sooner. You should write a letter to all major newspapers and news stations. Or make a youtube video about it. Maybe you can get your 15 seconds of fame. And you’d have to be God incarnate to be able to get me to like math.

  6. 6 joesas December 7, 2008 at 2:23 pm

    Haha. Me and a youtube video! Maybe I can turn it into a multi-million dollar sponsorship deal from Scholastic books or something. LOL

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