Math education: you’re doing it wrong
May 13, 2011 34 Comments
Recent discussion about the problems with our educational system reminded me about the story of why my friend J almost didn’t make it into his high school honors math class. Now, to clarify, J is easily one of the smartest people I know. But he is also a smartass, and thirteen-year-old J was certainly no different.
On the entrance exam for his honors math class, several of the problems asked you to fill in the next number in the sequence, such as: 2, 4, 8, 16, _?_. Obviously, whoever wrote the exam wanted you to complete that sequence with “32,” because the pattern they’re thinking of is powers of 2. For n = 1, 2, 3, 4, 5, the formula 2n = 2, 4, 8, 16, 32. But J didn’t write “32.” He wrote “π.”
When his teacher marked that problem wrong (as well as all of the other sequence questions, which J had answered in similar fashion), J explained that there are literally an infinite number of numbers that could complete that sequence, because there are an infinite number of curves which go through the points (1, 2), (2, 4), (3, 8), and (4, 16). Sure, he said, one of those curves is the obvious one which also goes through (5, 32), but you can also derive a curve which goes through (5, π). He showed her an example:
As you can see if you try plugging in the numbers 1, 2, 3, 4, and 5, to the equation above, you get the sequence 2, 4, 8, 16, π. Here are the two curves plotted on a graph, both the “correct” curve and J’s smartass curve (hat tip to the mathematician at www.askamathematician.com for graphing this for me in Mathematica):
Anyway, after thirteen-year-old J explained the math behind his unconventional, but admittedly accurate, answer to the original problem, his teacher replied, “Oh come on, you knew what it was asking for!” and refused to give him any credit. I can’t think of a better illustration of the triumph of the stick-to-the-book method of teaching over kids’ innate creativity… or of the triumph of math education over actual math skills.
How absurd. If the teacher wants to say “Oh come on, you knew what it was asking for!” it’s admitting that J understands already! Not only does he understand her question and the “right” answer, he gave another answer to show how the question was poorly asked.
He should have gotten extra credit, not zero.
*re-aggravates my longstanding frustration with poor education systems*
Very similar to the smartass method of estimating the height of a tower using a barometer: drop it from the top, count how many seconds it takes to hit the ground, and solve d = 16t^2. Again, “Hey, Teach — this problem has such an obvious answer that it’s boring! Please let me have some fun with it!” As for the specific answer to the question “How can we fill in the blank: 2,4,8,16, ___ ?” — uh, wouldn’t Occam’s Razor impell you toward ’32’?
Or does this example point out the limitations of Occam’s Razor?
If climate scientists take temperature or CO2 data that shows an exponential increase, and they fit a curve that takes a sharp downturn, they better have a good reason for it.
@Max: Actually, if climatologists fit data to an exponential curve that goes to infinity, they will also need a good reason such as “if we ignore…” or “in an idealized world…” Exponential growth in the physical world always either tapers off or “hits a wall” (showing a sharp downturn) due to resource depletion, negative feedbacks, and so on. A model that doesn’t include such a mechanism is bound to be unreliable for long-term predictions. Just one more issue that should receive more attention from educators in several different subjects. How many students finish Econ 101 still believing 3% growth is something that can potentially continue forever?
I was one of those annoying students in HS physics classes, ha. I was told that a big part of what students are supposed to read in high school is actually how to follow the rules around them, for future success in society. 🙁
IMO by J’s standards, his response ought to be something to the effect of “not enough information”, then no? By his own admission, π would only be one possible answer, not really *the* answer. Furthermore, if he sincerely was interested in finding the right answer, he also presumably could have asked the teacher for clarification instead of being confrontational. Just sayin’
PS I do agree that the US educational system is much too heavy on emphasizing blind obedience to authority and not respecting the opinions of students.
What’s truly alarming is that this was the entrance exam for an “honors math class,” since it’s obvious that the teachers couldn’t appreciate the caliber of the 13-year old mind they were dealing with. It would be kind of like blackballing a student from enrolling in an “Ancient Studies Honors” class because he answered all the questions on the entrance test in Greek or Latin.
That is not the problem with today’s math curriculum..I can assure you
I’m reminded of something in the children’s book A Swiftly Tilting Planet, by Madeleine L’Engle. Speaking of one of the characters, a brilliant child named Charles Wallace Murray, the observation was made that he would have to live in a world full of people whose minds didn’t work at all in most of the ways his did, and with whom he would have to learn to get along.
In J’s case, the teacher was obviously in the wrong; and I think most would feel a sense of indignation on J’s behalf. Ideally, a kid of J’s caliber would be praised by his teachers for his creativity and given every encouragement to develop it, even while being taught the at times galling lesson that other people’s often seemingly arbitrary rules really do sometimes have to be respected if one is going to get along in society.
Explain how the teacher was “obviously wrong”.
If J were really smart, he’d first give the obvious answer of 32, and then explain why any number would work.
If really smart, he’d have just answered 32 and moved on. I know this to my cost having myself being a smartarse at that age and never having quite grown out of it. There is still a point of contention between me and my (presumably long dead) chemisty teacher about elephants and electrical conductivity. Perhaps I’ll let it go one day.
But if J really wanted to be a smartarse he should have explained his answer in the exam paper rather than afterwards. It’s hard to see how he can get credit after the paper was already marked. A good teacher with a sense of humour and a genuine desire to teach would recognise that J understood the problem and would hopefully reward it as long as it was done on the actual exam paper. Special pleading after the fact can hardly count toward performance in the exam. The ironic grin on your face as you write down the smartarse answer without explanation doesn’t count for much in the marking scheme.
I have a vaguely similar anecdote, where I was the smartarse. It shows how old I am, I’m afraid. It was a LISP exam. We had to write a program in LISP and run some test data through it to prove it worked in the space of 2 hours. The first hour was to work out how to do it on paper and write down our workings and the second was to write and test the code. It was really, really easy. It took me about five minutes to write down the method. I’m not boasting or anything, it really was stupidly easy. So with 55 minutes to go, I wrote down three other solutions and wrote a brief essay about which I was going to implement and why. Then in the implementation phase I wrote all four programs and tested them and plotted the results in a table and suggested this was evidence that my assumption was likely correct.
I was pretty pleased with myself, as we smartarses usually are. And I got the highest mark on that exam. But I only got 2% more than someone who just answered the question mechanically and obviously and as expected. And someone else who clearly cheated by typing in every single LISP program we’d written in tutorials (presumably from a crib sheet), one of which was similar to the exam question, also, somehow, passed the exam.
My conclusion is that you shouldn’t bother to be smart in exams. Do enough to get the highest mark you can and save any cleverness for when you need it. You’re going to need it to stop every boss you’ll ever have doing stupider things than they’d do if you weren’t there. If that isn’t a noble use for superior intelligence, I don’t know what is.
These sequence questions are more appropriate for an IQ test, which tests pattern recognition skills more than math skills.
Answering pi for every sequence doesn’t demonstrate pattern recognition skills. J should’ve at least given the formula on the exam, not wait until he fails the exam.
No mention of Wittgenstein, Kripkenstein, and the rule-following problem? Really?
Individual teachers can choose to “bend” the system and actually think – like we teach our kids to do. I LOVE those “smartass” kids! They are terrific role models for other kids and they make my day! Besides, it is fun to have those really bright thinkers and watch them go off into their areas of interest then share it with us all. No, they don’t “fit in” to every teacher’s class, but some of us are truly grateful for those wonderful smartasses!
If a genius kid completely blows off homework and exams, and instead goes off and does some brilliant project that no one asked for, should he get an A?
He should drop out of school and start a company.
Putting aside that he’s thirteen years old, who would invest in a company started by a smart-ass high school dropout? Will the smart-ass ignore his customers to do what he feels like? When he gets vague requirements from users, will he work with them to understand what they really want, or will he show them how stupid they are by producing something that technically meets their requirements but isn’t what they want?
He’d probably be better off pursuing an academic career, which is what I’m guessing he did.
(You’re being a bit hasty to assume that a 13-year-old’s smartassery isn’t just a phase, no?)
If he’s a “genius” and the project is “brilliant” then it ought to have a market somewhere. And if he chooses to seriously dedicate himself to success in the market (i.e. to understand his failures as learning experiences), it’ll wring the smartass out of him.
You’re right that another option is academia, but it depends on the kid’s personal ambitions.
I’m not sure this is an indictment of our educational system as much as it is a story about a teacher and a student, with the commenters taking one side or the other.. A similar situation could have happened in an open-ended college program where each student was free to do their own thing and Pass/Fail would be determined solely on the basis of the completion of student-written goals. Or so it seems. Even here, the Pass/Fail would be influenced by the achievements of others in the program. Why? Because the program has an interest in survival, and could be damaged by individual human creativity.
J was working towards his own ends, not the class goals. On the opposite end of the intellectual spectrum, I remember at that age a pop quiz on some Easter story that we were supposed have read, and no idea where this story came from let alone read it. Most of the questions were of the form “What was the …….”, and I answered to each one, “An egg”. I did not mean to be a smartass, just felt I could not hand in a blank paper. I like J, HAD to create.
But it seems that all educational systems can handle just so much creativity or they will fail, bankrupted by the replacement cost of all of Barry’s broken barometers.
But answering “egg” is not a creative act, especially if you put the same thing for every answer. If you’d come up with reasons why the answer should be egg for each one, then I’d grant that you’d been creative. However, unless it was a creativity exam, you should still have got zero marks.
This is not because educational systems ‘can’t handle creativity’ it’s because somebody has to mark your test. That person has to justify the marks given. That person can’t give marks because she happens to know that the student is actually intelligent and knowledgable because that’s her personal, subjective opinion and not fair on the other students.
Every single person on the planet knows that standard tests are only any good at testing how good you are at the standard tests. But this doesn’t make them worthless. Some people’s abilities will be harder than others to measure this way and some people indeed slip through the gaps. It’s a real shame. But the answer is NOT to allow teachers to assign marks based on what they think someone deserves rather than what questions they actually got right.
This has NOTHING to do with creativity. Creativity should indeed be encouraged. However, I’d suggest that a student who decides to express creativity in an exam – when everyone knows what the exam is about and its purpose and how it will be marked – has made a poor choice.
My own calculus professor in college hated those “complete the pattern” problems for precisely this reason. “No sequence can be completely determined by a finite set of its terms,” he used to say.
The real problem statement is to find the simplest rule, or the formula with the fewest parameters that fits the sequence. Think of it as data compression. In the above case, 2^n is more compact than 2, 4, 8, 16, whereas the smartass formula is less compact.
Finding a compact representation demonstrates familiarity with a variety of functions and concepts: polynomials, exponentials, sinusoids, prime numbers, etc.
Well put. But there is a smartass solution which is more ‘compact’ … depending again on how you define ‘compact’. I believe a continuous solution with the lowest total power – sum of y^2 from -inf to +inf – is the one below which gives a result of zero as the next item… (but then, this approach gives you zero as the next element for all sequences which are assumed to occur at integer values of x).
http://www.wolframalpha.com/input/?i=sinc%5Bpi*x%5D%2B2+sinc%5Bpi*(x-1)%5D+%2B+4+sinc%5Bpi*+(x-2)%5D+%2B+8+sinc%5Bpi*(x-3)%5D+%2B+16*sinc%5Bpi*(x-4)%5D
The square brackets in the url got mangled by the blog (but it’s a nice picture). This is what was meant:
http://bit.ly/mqtRkF
On the general subject of killing mathematical creativity with teaching, I would really recommend A Mathematician’s Lament by Paul Lockhart – a brilliant little essay describing much of whats wrong with K-12 math education today.
Enh, I’m actually not 100% sure I agree with you here. I think it was clever, but on the other hand, recognizing obvious patterns is a useful skill in math. I also think there’s a difference between being a smartass on an individual project, and on a test, where administrability matters.
I think your larger point that clearly J didn’t deserve to be kept out of higher level math seems obvious enough, but if smart-alecking your way through one question is enough to keep you in on-level classes, that displays some pretty big problems with the whole testing system in general. (Granted, I say this as someone who was always near the top of her classes in elementary and middle school but who always choked on standardized tests – which I think informs my skepticism about things like NCLB.)
I hope your friend becomes a high school math teacher, math needs out-of-the-box thinking. In the real world, things are not always so straightforward, there’s no ‘back of the book” answer – you have to sit down and actually THINK. The US education system needs to wake up if this economy is going to ever recover. We can’t afford to raise dumb kids anymore!
People are making much ado about nothing. Here is the clue in the statement “my friend J almost didn’t make it into his high school honors math class.” Notice the word “almost”. One can assume that young J was able to accurately determine which of the other questions were answered correctly and would also be able to establish how many questions he could answer incorrectly for the sake of humor. A smartass must keep up appearances you know.
When I first went to take a driver’s license exam that was given on a computer terminal me and friend who was taking the test at the same time staged a race to see who could complete the test fastest. We were not worried about missing one or two questions as it was a ridiculously easy test.
I once confused an EE professor by solving simultaneous equations representing a network on a test using Gaussian elimination instead of using determinants as the syllabus taught. It did not get marked wrong but he came and asked me what I had done as he was unfamiliar with the method.
When I was in 3rd grade, I overheard my teacher telling another girl that you can’t divide 1 by 2. “Yes, you can,” says I. “it’s 1/2.” All I did was think of one candy bar, 2 people, and it was obvious. But the teacher said, “No, you can’t. We haven’t gotten there yet.”
I had a similar experience, a girl who could count by 5 (and I could too) was told sit on a chair and be quiet by my grade 2 teacher.
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Honestly I gotta say I need to look up stuff smartasses did more often this conversation throughout the comments has been something to read and something I’ve been missing to read about. You guys have no idea how boggling these comments have been to try and read thorough.