Thursday, 28 January 2016

How did you learn your times tables?

When I was a kid, I liked playing ice hockey. I was actually not very good at it — no Connor McDavid here!  But I did acquire the basic skills. For example, I figured out how to lift the puck. (For you non-hockey players, that means shooting the puck in such a way that it flies off the ice into the air. It’s a essential skill if you want to be able to score goals.)

I practiced that skill a lot. Whatever I did, it worked. I could lift the puck consistently without thinking about it. I haven’t shot a puck for many decades, but whenever I imagine doing so, I swear can feel the memory in my triceps.

Of course, I really did not "figure out" how to lift the puck. I did not know the theory behind the lifting action. And to the extent that the skill was necessary, I didn’t need to understand the theory.

The lesson is this:

When learning something new that you will need for later use, master the mechanics first. You can learn why it works later.

* * * Warning: possible straw man ahead * * * 

It’s a useful lesson.  It helps me understand the approach to mathematics teaching advocated by the back-to-basics people: You can be successful by learning the how without understanding the why. Just learn the essential basic facts and algorithms. Don’t worry about why the puck flies into the air — just practice shooting enough so that you can lift it consistently and effortlessly.

Reasonable advice? Maybe. But, no matter how hard I practiced, I could not always "lift" the multiplication tables. As far as the basic multiplication facts are concerned, I do not have what some people call rote recall — I do not have the ability to rapidly and effortlessly retrieve all of the basic learned facts from memory.

A great chunk of my own elementary math education was founded on the contrary belief, that rote recall is, in fact, achievable by everyone — that all it takes is practice.  Accordingly, my classmates and I were regularly drilled and tested on the multiplication tables. I did not do well, and I argued with my teachers. Ultimately, I was punished for my inability to memorize the 12 x tables.

I don’t believe that my recall difficulties are exceptional. The more blogs I read, the more I suspect that there are many people who, no matter how much they practice, will never possess rote recall of the basic arithmetic facts. In that sense, those people can never know the basic facts.

So, it was with interest that I read that Nikki Morgan, the secretary of state for education in the UK, has decreed that:

"we are introducing a new check to ensure all pupils know their times tables by age 11"

An interesting post by @thatboycanteach asks what it means to "know" the times tables. Like me, he suffers from what might be described as rote recall deficiency. And like me, he survived (and even thrived) by using various work-arounds to compensate.

The UK times-table test will be computerized and time-restricted. It looks like it will be based on pure rote recall. For the flunkies, there will undoubtedly be some sort of penalty. It’s unlikely that they will be physically punished like I was, but even non-corporeal punishment can inflict great stress and harm and, in the end, may prevent them from learning mathematics.

What is of concern to me in Alberta is that, however sincere the back-to-basics people may be, they seem to be basing their reform efforts on the very thing that caused me difficulties, namely, the belief that all students can and must achieve rote recall, that this is the only way to know the basic facts.

That is the conclusion that I draw from reading their petitions and press releases. If I’m wrong, if I am raising a straw man, it is difficult to understand why they also want to banish the teaching of alternate approaches to the basic facts and algorithms that are needed so that people like me can compensate for our deficiencies.

Friday, 1 January 2016

Math fair workshop at Banff

The 14th annual math fair workshop at BIRS will take place over the weekend of May 6/7/8, 2016.
(BIRS = the Banff International Research Station.)

Right off, let me say that I have a pretty bad attitude about school science fairs.  You know — those competitions with poster sessions, baking soda volcanoes, and parent-created displays. The ones that end with an obligatory showcasing of a winner — a bright student who looks like he/she will go on to become the next Neil deGrasse Tyson, and who, for a short while, will be a poster-person for our education system.

OK, that's harsh, but it is still very much the norm to single out a winner.

How about having one that does not overly favour the highly talented? One that even a less confident student would enjoy and not end up feeling like a failure because he or she did not win a medal.

If you’re like me, you do not enjoy being tagged as a loser, and you would likely withdraw from a situation where that is liable to occur. Aviva Dunsiger touched upon this in her blog. Although her post is about phys-ed rather than mathematics, she paints a clear picture of the response to anticipated failure:
Yes, there were always strong athletes, but those that struggled (and I was one of them) wanted nothing to do with phys-ed. With my visual spatial difficulties, games like volleyball, basketball, and baseball were a tremendous struggle. I certainly never got picked for a team, and I couldn’t blame anyone. Why would I want to be physically active if I was only going to meet with failure?

[the emphasis is Aviva's]

Can we have a math fair where students can be mathematically active without the anticipation of failure?  One where students do not need a badge or ribbon to confirm that their efforts have paid off ?

Such math fairs do exist. They’re called SNAP math fairs because they are Student-centred, Non-competitive, All-inclusive, and Problem-based.

The fairs are built around math-based puzzles. The students first solve the puzzles[1] and afterwards prepare the artwork and puzzle pieces that are required to display them.  

Visit such a fair and you will find students manning their puzzles. But, you will not see them exhibiting the solutions. Instead, they will invite you to try the puzzles yourself, and they will give you hints and help when you run into difficulty. The math fair is very interactive. It is much more than a poster-session.

* * *

Here are a couple of puzzles from past math fairs. The first one is for younger students to solve. 

Cats Pigs and Cows

A farmer has nine animal pens arranged in three rows of three. 

Each pen must contain a cat, a pig, or a cow. 

There is already a pig and a cat in two of the pens. 

The farmer wants you to fill the remaining pens so that no row or column contains two of the same animal.

The second puzzle is for older students.[2]

The Sword of Knowledge

The dragon of ignorance has three heads and three tails. 

You can slay it with the sword of knowledge by chopping off all of its heads and all of its tails. 

With one stroke of the sword, you can chop off either one head, two heads, one tail, or two tails.

But the dragon is hard to slay !! 

  • If you chop off one head, a new one grows in its place. 
  • If you chop off one tail, two new tails replace it. 
  • If you chop off two tails, one new head grows. 
  • If you chop off two heads,  nothing grows.

Show how to slay the dragon of ignorance.

* * *

A SNAP math fair is remarkably adaptable to many different circumstances. If you are interested in learning about how you can incorporate a SNAP math fair into your own teaching environment, come to the BIRS workshop. You will meet teachers who have organized math fairs in their own schools. You will also meet a few mathematicians who have taught courses in which a math fair was key ingredient. 

As well, there will be math fair resources available, and the participants will be involved in puzzle-solving sessions.  

The BIRS workshop has room for about 20 participants, and it is oriented towards (but not limited to) K-9 teachers.

For more details about SNAP math fairs, visit the SNAP math fair site. And while you are there, take a look at the Gallery to see how students react.

For more information about the workshop, and who to contact, the link is here.

End notes

[1] The solving part is a crucial element of the math fair.  Ideally, students solve the puzzle by themselves. They are surprisingly persistent.

[2] I imagine the Sword of Knowledge puzzle would work with junior high or high school students. However, I once visited a SNAP math fair where two grade five students had solved it. Their teacher told me that they struggled with the problem but solved it after one of them grabbed a handful of pencils (tails) and erasers (heads).