Friday, 29 May 2015

Why I won’t go to that meeting

The meeting I'm referring to is 
Albert Math and Education Forum: Restoring Excellence and Evidence-Based Teaching in the K-12 Curriculum
which is being held today in Edmonton Alberta.

I imagine I have a decade or two left. My time is somewhat precious to me. I’m an evolutionist and I consider it a gross waste of my time to attend a forum organized by young earth creationists. Because, education-wise, that’s what you are, you back-to-basics people. You’re young earth creationists. Or flat earthers. 

In the last few decades, there have been tremendous advances made in understanding how children learn mathematics. The research is still at the beginning, but it is getting results. And if we mathematicians don’t acknowledge it, the Faculties of Education will soon take over teaching math to preservice teachers, just like Psychology and Economics departments now teach Elementary Statistics and Introductory Game Theory.

Don’t take this rant to mean that I’m against direct instruction — I’m not. But I’m all for teaching being informed by research on how kids learn. It’s inevitable that it is going to transform how we teach mathematics. 

I recommend that you read this post by Keith Devlin about how teaching math has changed.

I also wrote a bit about the harm being caused by forums like these some time ago. 

But if you’re a Young Earther, I suspect you will just dismiss these as irrelevant.

Wednesday, 27 May 2015

Hoist by my own petard

The expression in the title, which nowadays means “blown up by one’s own devices”, first occurred in Hamlet. It is interesting to note that Shakespeare dropped the “d” and used the word “petar” which apparently means “fart”. 

When solving problems, my students were sometimes blinkered by unacknowledged assumptions that really had little to do with the mathematics at hand. To highlight the situation, I used to distribute a sheet of puzzles that could be difficult to solve if you did not recognize your hidden assumption. Most of the puzzles were cribbed from Martin Gardner’s books. Here is one of my favourites:

I propped up four cards against the blackboard as follows.

The puzzle here is to determine what cards that you have to turn over to make sure that the following statement is true:
    If a card has a blue back then it has a 2 on the other side.
Note that we are not asking whether the statement is true. We are being asked which of the four cards must be turned over to enable us to determine if the statement is true. Here is how we analyzed it in class.

Blue card: We all agreed that you do have to turn over the blue card

Red card: We eventually agreed that you do not have to turn over the red card. Several students initially said that you had to turn it over because the statement would be false if there is a 2 on the other side. This is a familiar logical error whereby a statement and its converse are considered to be the same. The statement at issue does not say “If one side has a 2 on it then the card has a blue back.” It says “If a card has a blue back then the other side has a 2 on it.” It’s the same as the difference between the two statements “If your car is out of gas then it won't start,” and “If your car won't start then it's out of gas.”  So we agreed that we do not have to turn over the red card — it doesn’t matter what number is on the other side. 

The 3 card: All agreed that you do have to turn over the 3 card. You have to check that the other side is not blue, for if it were, you would have a card with a blue back that did not have a 2 on other side and the statement would be false. 

The 2 card: We agreed that you do not have to turn over the 2 card. It doesn’t matter what is on the other side. If the other side is blue, it’s fine; if it’s anything else it doesn’t matter. 

So there we have it. We only need to turn over the blue card and the 3 card to determine whether the statement is true.  

Here’s what happened next: I turned over the blue card and it had a 2 on the other side. I turned over the 3 card and the other side was red. At this point we agreed that we had enough information to conclude that the statement had to be true.

Well, of course, this was a flagrant set-up. When I turned over the remaining cards it revealed that the statement was false. How could this be? Is our logic not impeccable? Keep reading.

There’s actually nothing wrong with our logic. There is however the assumption that the four cards were standard playing cards. They were not. The red card was a two-backed card — red on the visible side and blue on the other. So it was a card with a blue back that did not have a 2 on the other side. So we do have to turn over the red card.

This a routine that I used for many years, and you can imagine my delight when I encountered Shecky Riemann’s recent post that links to the Wason selection task in Wikipedia. The Wason selection task goes like this:
You are shown a set of four cards placed on a table, each of which has a number on one side and a colored patch on the other side. The visible faces of the cards show 3, 8, red and brown. Which card(s) must you turn over in order to test the truth of the proposition that if a card shows an even number on one face, then its opposite face is red?
In the Wason puzzle, the roles of the numbers and colours are reversed from what I did in class. The Wason task is about what colour is opposite the number rather than what number is opposite the colour. The statement we want to check in the Wason task is this:
If a card shows an even number on one face, then its opposite face is red.
The answer given in the Wikipedia article is that to check the veracity of this statement you only have to turn over the 8 card and the brown card. 

Except for the reversal of colours and numbers, the Wason task appears to be identical to  the Martin Gardner puzzle. I gleefully dashed off an email to SheckyR and pointed out that in the Wason task you would also need to turn over the 3 card, because there is a possibility that there is an even number on the other side. 

SheckyR quickly replied that this is not possible because the statement of the Wason task very clearly states that 
You are shown a set of four cards placed on a table, each of which has a number on one side and a colored patch on the other side
He has a point. It is pretty hard to interpret this as allowing numbers on both sides of a card.  
The Gardner puzzle, on the other hand, does not preclude this from being the case, and in fact it is deliberately set up to tempt you to assume that it is the case. My familiarity with the Gardner puzzle and my careless reading of the Wason puzzle caused me to bring unwarranted assumptions to the Wason puzzle and resulted in pretty hefty brain fart.


Martin Gardner, Combinatorial Card Problems, in Time Travel and other mathematical bewilderments. 

The playing card version described above is a bit different than the one in Gardner's book. There, the puzzle involves five playing cards, and Gardner attributes it to Tom Ransom, a Canadian amateur magician. 

Shecky Riemann’s blogs are here and here. If you like things mathematical in the Martin Gardner vein, you should visit his blogs.

Tuesday, 12 May 2015

That student again!

The student asked questions. Lots of questions. Even the simplest topics caused her problems. She was not “getting it”. The questions continued for the entire term. I was quite frustrated. There was a curriculum to be covered, and this was slowing me down. The lectures dragged and lost continuity. Other students began to roll their eyes.

I was relieved when the course ended, and, since she found it difficult, I think she was relieved as well. There was a followup course that I was slated to teach in the second term, but it was not required for her degree. I imagined that she would be happy that she didn’t need to take it.

You know where this is going, don’t you? She took the followup course. 

When I walked in to the classroom on the first day, there she was again. I got through the first lecture, handed out the first assignment, and began to worry about the pending onslaught of questions. 

I arrived early for the next lecture. She was there, discussing something with two other students, apparently seeking help. But as I watched, it became clear that she was not seeking help—she was doing the helping. 

This continued for the rest of the course. She became the go-to person for those who were having difficulty. 

My initial judgment of this student’s abilities was way off the mark. And for me it raised doubts about how I assessed all of my students.  

I don’t like the high stakes nature of final exams and I think they often provide an inaccurate measure of a student’s learning. But, for this particular student, the exams and tests were more reliable than my in-class observations. In this case, classroom activity was not a valid indicator of the student’s progress. 

When there is a discrepancy between how a student performs in class and how a student performs on the tests and exams, what assessment do you trust? Quite a dilemma.