## Wednesday, February 11, 2015

### How has teaching changed in the last five decades?

I've been trying to get my colleagues to change the way they teach university courses. So far, I haven't had much success.

Part of the problem is the culture of the university (University of Toronto, Toronto, Canada). Here's a description of the problem. It was sent to me by a former public school teacher (thanks, Helen) but it captures the essence of what's happening in higher education—especially the last decade.1
1. Teaching Maths In 1950s

A logger sells a truckload of lumber for \$100. His cost of production is 4/5 of the price. What is his profit?

2. Teaching Maths In 1970s

A logger sells a truckload of lumber for \$100. His cost of production is 4/5 of the price, or \$80. What is his profit?

3. Teaching Maths In 1980s

A logger sells a truckload of lumber for \$100. His cost of production is \$80. Did he make a profit ? Yes or No

4. Teaching Maths In 1990s

A logger sells a truckload of lumber for \$100. His cost of production is \$80 and his profit is \$20 Your assignment: Underline the number 20.

5. Teaching Maths In 2000s

A logger cuts down a beautiful forest because he is selfish and inconsiderate and cares nothing for the habitat of animals or the preservation of our woodlands.

He does this so he can make a profit of \$20. What do you think of this way of making a living? Topic for class participation after answering the question: How did the birds and squirrels feel as the logger cut down their homes? (There are no wrong answers, and if you feel like crying, it's ok).
I don't think the last part is quite accurate. In a real modern classroom we would refer to the logger as "she" or at least "he/she."

1. Normally I don't reproduce these internet clips but this one is so relevant.

1. The funniest part of that joke is that it existed back in the late 60s. The same questions for the decades, but of course starting earlier and ending with the 60s. I don't think mathematics curricula have changed that much over time and this kind of joke is mostly a conservative gesture saying that the methods used these days are worse than those used in days past. Because Cross-multiplication is so much better than understanding how to solve a linear equation by algebra is...

1. I would suspect that if students don't know how to do cross multiplication, linear algebra won't come easy.

2. I don't think anybody is teaching linear algebra at that point. But Cross-multiplication used to be taught by rote. Now you start with the notion that if you have an equality you can perform the same operation on both sides and retain the equality and then use this to solve the problem. It sets up the use of this method for other problems, where the old route of rote learning didn't.
In a similar way multiplication tables used to be taught by rote learning. These days time is spent on having students find the patterns in multiplication. Things like "even multiples of 5 all end in 0, while odd multiples of 5 end in 5". My mother was an elementary school teacher when this was introduced and got endless complaints by parents who felt that not doing it by rote was watering things down. The response always has been: There are studies on this that show that the better students do not show any change in their ability to do multiplication when taught this way, but they do significantly better when they later have to understand things like associativity. And the worse students even do better on the multiplication table tests. The reason for this is that the good students find these patterns even when not encouraged and they will use them. But since they aren't encouraged they do not necessarily understand that they are a legit part of maths. The worse students when not encouraged to look for them do not find the patterns and it turns out that rote learning can not make up for this. So the performance of the worst students goes up when they find the patterns.
What people begrudge students is that they have an easier time grasping things. And they confuse the ease of learning with watering down the subject. But the whole point of working on didactics is to make knowledge more accessible, not by dumbing down the subject, but by removing obstacles that are purely the result of how the subject is presented.

3. This is a very interesting comment. Though I have to say I have witnessed the opposite: rote memorization taking the place of finding the patterns.

My students had such watered down math that they can't do quite simple stuff, like simplifying an equation before starting putting numbers in their calculators. Not enough practice. So, neither the patterns, nor the rote. :/

2. Your right wing is visible through a crack in the closet door.

I spent a few months as an intern teaching math to mildly retarded eighth kids. I found they could solve math problems at near grad level as long as they weren't "word problems."

I think much of the uproar over "common core" or "new math" has to do with the difficulty of teaching reasoning, as opposed to teaching computation.

Here's a problem some people find difficult:

The price of a cheeseburger is \$2.20, which is the price of a plain hamburger plus the price of the added cheese. The price of a plain hamburger is two dollars more than the price of the added cheese. What is the price a plain hamburger?

1. Am I walking into the trap if I say the plain hamburger is \$2.10? But this is exactly the sort of word problem that I think is unhelpful because it is so artificial -- nobody in real life would need to calculate a hamburger cost based on the cheeseburger cost (the other way, yes, if it is just the cost of adding cheese). In my opinion, math should either be taught just as pure calculation (to stress that math has no basis in the physical world but is abstract) or if word problems used, these should reflect actual applications of math -- knowing how much 25% off is, and so on.

3. I don't think the last part is quite accurate. In a real modern classroom we would refer to the logger as "she" or at least "he/she."

Oh, no. A she-logger would never be that selfish! At the very least, she would hug the trees before cutting them down, and comfort the squirrels!

4. I wanted to be... a lumberjack!

Leaping from tree to tree, as they float down the mighty rivers of British Columbia. The Giant Redwood. The Larch. The Fir! The mighty Scots Pine! The lofty flowering Cherry! The plucky little Apsen! The limping Roo tree of Nigeria. The towering Wattle of Aldershot! The Maidenhead Weeping Water Plant! The naughty Leicestershire Flashing Oak! The flatulent Elm of West Ruislip! The Quercus Maximus Bamber Gascoigni! The Epigillus! The Barter Hughius Greenus!

1. I wanted to be... a lumberjack!

Or lumberjill.

2. It's every man's right to have babies if he wants them.

5. We had the exact. same. joke, only with a different profession, in my home country twenty years ago. Somehow I still learned about calculus, imaginary numbers and stochastics...

6. There is another pressing issue. Do we teach Maths or Math?

7. I do deplore the decline in math intensity in schools. If that makes me right wing. So be it.