How not to teach mathematics.

[catsidhe]

1. Set a problem for homework.
   
2. Provide a suggested method for finding a solution of, essentially, "randomly shuffle these numbers until it kinda looks right".
   
3. ...
   
4. Profit! End up with children who are frustrated and angered by the sight of numbers, and have little to no idea that there are ways in which this sort of problem can be approached, let alone relatively simple and rigorous ways to prove them correct, let alone that the solution raises all sorts of other questions, which can themselves be answered...

The actual problem was "Take the nine numbers 2 to 10, and arrange them in three groups of three so that each group adds to the same number."

The suggested approach was to "write the numbers on pieces of paper, and arrange them into the right groups."

No, seriously, the suggested approach was to randomly shuffle the numbers until they (magically) come out in the right order. Personally, I'm wondering if there is a worse possible approach to the problem.

When I sat down with Miss A to approach this, my first question was: so, what is the number they have to add up to?

What you're looking for is

x = a + b + c
  = d + e + f
  = g + h + i

So the first thing to notice is that

a + b + c + d + e + f + g + h + i = 3x

The sum of 2..10 is 54, so the answer to each group of three must be 54/3 = 18.

So then we need an algorithm to fill in the blanks. Start with the biggest number, so

18 = 10 + b + c

b≠9, because that is already too big. And while 10+8 = 18, that only works if c=0, which isn't an available value. Neither is 1, so b≠7. So by elimination, we have a=10, b=6, c=2. Then do the same with the remaining numbers (d=9, e=5, f=4), and the remaining three must be g, h and i. Luckily, when you check, they are.

There was a secondary part to do the same thing with the set B = [3..11]. And yes, we showed that the algorithm still works. Only now the sum to each group is 21.

Hang on, 21 = 18+3, and we're dealing with groups of three... that can't be a coincidence, can it? It turns out, if you compare the ordered sets, then you see that each number B_x is just A_x+1. So if each number has 1 added, then each group must have 3 added to the total for it to work out.

And if we've just solved this problem for the set N₂ = [2..10], and for N₂₊₁, then we've demonstrated that the solution will work for N_x, where x is any positive integer. So for the set [1..9], the sum to each group should be 15... and when you check, it is.

But wait... what we've got can be drawn in a grid

10	6	2	=18
9	5	4	=18
8	7	3	=18

If we re-arrange the numbers within each row, then we get

10	6	2	=18
5	4	9	=18
3	8	7	=18
= 18	= 18	= 18

And if you do a bit of matrix manipulation, then you get a Magic Square, where the rows, columns and diagonals all add up to the same magic number.

And we've proved that this pattern is a Magic Square whether you pick your nine numbers starting from 2, 3, 1, 512, 100473, or whatever. I wonder if it works for other progressions? Say, N⁵₅ = [5, 10, 15, ..., 45]? (It does, but proof is an exercise for the reader.) Or for negative integers? What would we have to do to the algorithm to make it work? What about magic squares of order 4, 5, 19? What about...?


Just look at all this number theory we got from a question where the suggested approach was to "fiddle randomly and hope you trip over the right answer."

I'm sure there's some sort of pedagogical approach which calls for the systematic frustration of children, and the comprehensive murder of any potential joy of mathematics, but for the life of me I can't think what it is.

Comments

[etfb]

This is the homeschooler's mantra: Children start out with a natural love for learning. It can take years of schooling to fully eliminate it.

Not saying "get your kids out of school and homeschool them", because their boredom and misery has to be weighed against the whole "sorry, kids, the bank owns our house now" issue, but just know that you're not alone in ranting at the stupidity of the "education" system.

[armourer]

I've been thinking about your post and I feel a bit dissatisfied with your assertion. On the one hand your maths is (mostly(ha ha)) fine, but on the other hand I wouldn't be so quick to dismiss the method suggested by the teacher. It may be that it is only one of a number to be introduced. Solving a problem in different ways is very useful as it can allow a problem to be solved with an "efficient" method. Having practice at using different techniques is, of course, pretty powerful too.

The suggested solution changes the problem from a numeric to a visual one, where the position of the paper is an analogue of the arithmetic steps that you describe.

The exercise of shuffling bits of paper can lead refinement and insight into some everyday activities. e.g. pairing socks or putting away cutlery. i.e. general organisational problems.

Famously, David Bowie would write songs lyrics and then cut each line separately and rearrange them to break the orthodoxy of his writing.

Dealing with the physicality of the problem is equally as useful as the brainwork. It may reveal techniques that are more practical in the real world than in the ideal.

Cheers
JB

[sjl]

So, just how old is Miss A again?

I'd have no qualms about teaching this sort of technique to a nine or ten year old kid who's reasonably quick on the uptake. Younger than that, and I'd be cautious.

But yeah. I hear you. Sigh.

"Weep for the future, Na'toth. Weep for us all."

[catsidhe]

Miss A is nine. (Although I did have to ask her... she looked at me quizzically and said "I'm nine... did the fever give you amnesia, Daddy?")

I've been trying to come up with a cogent response to Armourer above, in between bouts of fever. I think at least part of why I'm so upset is that I can see in this a part of what was so fundamentally screwed up in my own mathematics education (primary, secondary and tertiary): the assumption that if you just shut up and do the set exercises, then you will magically learn the underlying concepts by osmosis.

Which might work fine for some people, but almost never has for me: I needed to know why it worked, and once I had that, the rest was simply practice. But without that, then it was merely drills, and I was too trapped in trying to remember the immediate equation to be able to step back and figure out what was going on.

The only single time I remember having a but why? question actually answered, and in such a way as to make my other questions of detail redundant was when a teacher was trying to persuade us that 10⁰ = 1. I argued vehemently that this made no sense, that if 10³ = 10×10×10, and 10² = 10×10, and 10¹ = 10, then 10⁰ should equal "", that is, the null string: nothing. Such was my understanding of the process.

He thought about it for a moment, and came at it from a different angle.

12,345
= 1 × 10000 + 2 × 1000 + 3 × 100 + 4 × 10 + 5 × 1
= 1 × 10⁴ + 2 × 10³ + 3 × 10² + 4 × 10¹ + 5 × 10⁰
QED.

and then it made sense, and thence came the understanding that just as ±0 is a null operation, so is ×1, and much that had been a matter of remembering the equation (don't question, just memorise the equation) became obvious, and I didn't need to memorise the equation, because I could re-derive it when needed.

[sjl]

Indeed.

I was a very big fan for understanding the principles underlying the rote exercises, so I didn't need to remember the rules - just the basic principles, which I could then use to derive the rules I needed in an exam situation. Which worked brilliantly for me, right up until I ran slap-bang into electromagnetic theory. (That was painful. Scored 38% for that subject.)

You pay peanuts, you get monkeys, and that's just what we have in our school system right now. There are those teachers that struggle against it, but unfortunately, they're too few and too far between. Remind me to rant about a friend of mine, who trained as a maths and science teacher, who's likely to drop out of the system because she's too long out of uni for grad positions, but too inexperienced for anything else...