Math Education: An Inconvenient Truth
I've been sitting on this post for a solid two months. I left it out of my baffle-clearing the other day because these thoughts are something I want to keep in text rather than being lost to the ether of ideas.
I am not a teacher, nor a parent of children attempting to learn math in a public school. I did, however, learn math in a public school. In addition (obvious pun), I am a geek who loves numbers and enjoys not needing a calculator for tipping, knowing how much something costs with sales tax, doing my own taxes, etc.
I also annually celebrate π day as well as Almost-Pi day (22/7), which is just about a month away.
I happened upon a youtube video the other day (embedded below) and found its topic irksome1. The idea that math pedagogy this country is trending toward what's "easier" and "fun" for "all students" will never create the next generation of engineers, accountants, computer scientists and other nerdy and non-nerdy professions who use even basic math skills on a daily basis. We need kids to struggle a little to find an answer, not learn shortcuts. The result is certainly important, but being able to readily produce results to other similar problems quickly and understand why you got the consequential result is essential.
Being able to quickly do 2- and 3-digit mathematics is a skill that my current generation arguably lost with the plenitude of calculators being produced. (Curse you Texas Instruments!) Some will disagree, but I think math should actually be a challenge that extends beyond simple addition and subtraction. By the end of 5th grade EVERY student should be able to do long division without hesitation or n-digit multiplication on paper of n+3 lines without fear of "OMG the numbers are so big, I'll never be able to do it!". True, both of these things still rely on simple operations adding/subtracting to a final value, but you have to know why that's so, and know that your method will work reliably every time to attack problems with larger and larger numbers without trepidation.
I also don't think it should be a tough thing to expect a 5th grader to do long division with decimal points, but this is coming from the guy who previously used prime number theory to determine why his php
rand() function call was returning the number 24 somewhere around a quarter of the time it was run. Details.
Anyway, I was appalled. The adage of "I'm going to be an X, why would I ever need to know math" is a fallacious argument at best, and at worst, a total cop-out for the lazy. How about grocery shopping? Living securely on a budget? Buying a house and not getting screwed on your mortgage? Planning for retirement? Investing? Do you not need math for those things?2
Even if you reach for a calculator, you still need to understand what you're calculating! Do you plan to carry a calculator with you everywhere?
From the video, one of the methods is TERC's reasoning method, which seems to suggest that if kids feel their way through a math problem, they'll get smarter. The concept is based on talking out the simpler pieces of a difficult problem which you've seen before and using those simpler problems to build a solution to a harder one. That sounds familiar, but it isn't based on a repetitive set of steps, rather it's based on the pre-existing experiences and knowledge of the student, which will vary from student to student. "Well, I know that 10 x 20 is 200, so 13 x 19 (247) must be about 200."3
Reasoning is actually a really great technique for troubleshooting, attacking a complicated design issue, breaking out of prison with only a paperclip, duct tape and a screwdriver, etc. It is not, however, a smart way to teach arithmetic. You need to understand the fundamentals before you can use them to attack harder problems like, say, designing the next generation of fuel-efficient vehicles, or figuring out how to put a colony on Mars, or Warp Drive. With teaching methods like this, who is going to figure out Warp Drive?!?!
I know that's not really important as the answer is, as we all know, Zephram Cochrane, and he was probably home-schooled anyway.
Even worse, however, is their attitude about algorithmic techniques (e.g. traditional step-wise multiplication and long division):
The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator.
With. A. Calculator.
throws a chair
Seriously? A math book that advocates that students should start relying on a calculator in grade school? Are you kidding me??
throws a second, larger chair
Way to tell kids their way too d–mn stupid to learn how to divide! They even have entire chapters dedicated to calculator use rather than entire chapters dedicated to, say, learning about how to do the problem yourself. Catering to the lowest common denominator4 is the best way to get smart people to stop trying, to disengage the interested few for the sake of the dumbest one.
The narrator's notes on the major problems of today's high school graduates' math skills from her personal experience of going back to college are telling:
- An inability to work alone to solve problems without checking in with other people.5
- A lack of fluency in the symbolic language of math or an ability to think logically.
- Lack of mastery and confidence with basic math skills (trig, algebra and arithmetic)
- Complete dependence on a calculator
All of these are frightening, but to me, someone who considers himself to be a somewhat critical thinker, the scariest is "an ability to think logically". How on earth are we supposed to expect a person to make a difficult decision on their own without a simple understanding of logic? Not even "if a and b then c", but just the basic concept of causality which is utterly crucial to critical thought.
Why should these people vote??? (Ok... that's a different post entirely.)
Every high school math teacher I had would be appalled that dependence on a calculator is now a common ailment of college students. Almost never was I allowed to use, let alone bring, a calculator to math class in high school. The only exception to that was that we used TI-8x calculators for some graphing examples in pre-calculus and calculus for a month or so.6
The TERC method also fails to introduce the simple concept of an algorithm at an early age. There is no concept of "finite steps" in a method requiring you to break a problem down into simpler pieces that you have to reason. Algorithm-based arithmetic is simple, straight-forward, and it always works.
So here’s the crux of what’s bothering me: By teaching someone the basic fundamentals and then the easier/more creative techniques, a few things happen:
- A person appreciates the techniques
- A person might see other better/faster/more productive techniques (read: innovate)
- A person can master a subject and can teach others
- Goto 1
Every single sports coach will tell you the same thing – master the fundamentals and you can be good. If you asked an athlete to go out onto the field and reason their way through the game, what would happen? Imagine tennis. Not so bad. Now imagine football. Rugby.
And, for the sake of further argument – Any student who wants to be a professional athlete and says they don’t need to know math has never seen the contracts those athletes sign. I passed Differential Equations (admittedly by the skin of my teeth) and I don’t understand how a modern baseball player gets paid. Seriously, professional athletes should be forced to have math degrees.
Well, from a school that’s actually going to teach math, anyway. My kids certainly won’t go to a school that teaches them how to “feel their numbers”.
puts both chairs back
The video itself is great.↩
I suppose you can pay someone to advise you, but then how would you know when they are ripping you off?↩
Well, kinda, but what about 13456 x 19344.32? Is that as reason-able? What if you thought, for some reason, that 10 x 20 was 100? What happens to the reliability of your solution then?↩
A concept which you probably won’t learn about from that textbook.↩
Apparently TERC’s method suggests you discuss your thought process with others… what if they ALL think 10 x 20 = 100 ???↩
Those of us who learned TI’s version of BASIC to write neat procedural programs were also given some lenience, but that’s because we already understood the math needed to write said programs. Admittedly, we were writing games…↩