Thursday, 30 October 2014

Math fairs and puzzles and a weekend workshop at BIRS

(Puzzle number 21 in Boris Kordemsky’s Moscow Puzzles)

Erica doesn't like odd numbers, so the box of chocolates shown above meets with her approval. The problem is that she has to remove six chocolates from the box in such a way that she leaves an even number of chocolates in each row and each column.
Make a 4 by 4 grid, and using pennies or other tokens as chocolates, show how she can do this. There is more than one solution.

I belong to a group of teachers, mathematicians, and puzzle developers who advocate the use of math-based puzzles in the K-12 classroom. To this end, every April for over a decade we have held a weekend math fair workshop at BIRS. (BIRS =  Banff International Research Station). Although BIRS exists to aid research in mathematics and related disciplines, from its inception it has also supported educational initiatives, and our April workshops are an example of its continuing commitment. 

About 20 participants attend the workshops, many of them teachers and mathematicians who share a common interest in enhancing mathematics education. The participants are diverse in background and experience, and the workshops have quite a wide scope. Although the emphasis is on the use of mathematical puzzles and games, the freedom and informality of the workshops allows the discussion to veer off in related directions. For example, when teachers illustrate how they fit puzzles into their teaching, their presentations sometimes spark a discussion on how they have adapted to various other aspects of the math curriculum. 

At every workshop, we promote the use of a non-competitive puzzle-based math fair to elevate the students interest in mathematics. In a math fair of this kind, the students are in charge of booths which present puzzles to passers-by. The students are not there to demonstrate the solutions, but rather to give hints and suggestions to help the visitors solve the puzzle. The whole affair is very interactive. 

A visitor to the math fair might encounter the puzzle at the top of this post, where the students provide a 4 by 4 grid and sixteen removable “chocolates” for the visitors to work with. The students presenting the puzzle would have previously solved it themselves (without the help of parents or guardians) and would have mastered and practiced at least one solution. The students would also be expected to recognize whether or not a visitor has a solution. Depending upon the grade level, students could even be expected to answer a visitor who asks “Is zero an even number?” or “Do I have to worry about the diagonals?” In other words, the students will have become experts on the evensies puzzle. 

If you have tried the evensies puzzle you will know that is not instantaneously solvable. Most people will attack the puzzle using a “trial and error” or a “guess and check” approach—this is the way I first solved it, and, in fact, it is the way most of the participants at one of the recent workshops solved it. However, there are other ways to tackle the puzzle, and asking older students to find an approach that does not depend exclusively on trial and error would be a way to ramp up the mathematics involved. Perhaps one way to invoke a discussion about this would be to ask an entire class to solve the puzzle independently and display the different solutions they have obtained, and then ask them what they they notice. 

Perhaps someone will know a way to prompt the students to ask “What do you mean when you say two solutions are the same?” This is an important question but it may be difficult to answer because understanding “sameness” depends not only the context but also upon the student’s mathematical background. For example, most people agree that the numbers 1/8 and 0.125 are the same but many will balk when told that this is also true for the numbers 1.0 and 0.9999… (where the nines go on forever).  With regard to the evensies puzzle, a grade 6 student would not be expected to understand “sameness” in the same way as a grade 12 student. 

The evensies puzzle involves parity (the properties of even and odd numbers).  Even before solving the puzzle, students may have learned a few facts like “even + even is even” and “odd + even is odd”. The puzzle offers a meaningful context for a deeper investigation. One could ask students if it is possible to remove 5 chocolates and still have an even number in each row and column. Or if it is possible to remove a certain number of chocolates and end up with an odd number in each row and column. Or one could ask students to create a similar puzzle for a 3 by 3, or a 5 by 5, or a 6 by 6 grid. 

Some teachers have come to our math fair workshops specifically because they have either heard about or visited a puzzle-based math fair. Others have come because they have already used puzzles in the classroom, and are willing to share their knowledge. And some have come because they have learned about the workshops from other teachers. Whatever the reasons, feedback from the teachers has always been very positive.  

Some mathematicians have come to the workshop because they teach a math course for Education students who, as part of their course, will be presenting a math fair to local schools. Some mathematicians come because they are interested in K-12 education and will be involved in some way with their school district. If a mathematician is fond of recreational math puzzles, and if he or she is interested in K-12 education, the workshop is a great setting to merge both interests. In addition they will almost certainly encounter some math puzzles and games that they have not seen before, and that’s always interesting. 

The feedback from the mathematicians has also been positive. Some have told me that they were impressed by the quality and enthusiasm of the teachers at the workshops. They say that observing the teachers has made them reflect on their own teaching and assessment methods, and that it has provided ideas on how to help their own students take ownership of their learning. 

For some resources and information about non-competitive puzzle-based math fairs, I would urge you to visit the SNAP math fair website. If you want more information about the math fair workshop, please feel free to contact me at 

Tuesday, 14 October 2014

How to raise our PISA scores

A rather facetious list of recommendations based on what some of the participants have done or are doing.

PISA is the Programme for International Student Assessment. Every three years, the OECD (Organisation for Economic Co-operation and Development) tests 15 year old students around the world in math, reading, and science. A lot of Canadians are upset because Canada ranked 13th out of 65 in the 2012 math test, a slight decline from previous results. The top five ranking countries in the PISA math test were Shanghai, Singapore, Hong Kong, Taiwan, and South Korea.

The PISA math test is a two hour test. There are some multiple choice questions and some long answer ones. I don't know how many there are. A selection of the questions can be found here and here. A quick look reveals that they are predominantly of the "fake world" type that Dan Meyer so dislikes. 

Based on the PISA results, some countries have concluded that their curricula and teaching are deficient. Some, like the U.K and New Zealand, are intending to use the tests as a benchmark for their education systems or at least they are tilting in that direction, and it looks like they are getting ready to retool. (see this BBC report, and this Radio New Zealand report)

To me, using the PISA tests to draw conclusions about either our curriculum or our teachers seems iffy. Judging our entire system by how well a group of teenagers did in a two hour test is like training our athletes to compete in a triathlon and then measuring our success by how well some of them did in a 100-metre sprint. 

However, in case you are really determined to improve our PISA ranking, here are some recommendations that may do the trick without spending millions to revamp everything. But be aware that sprinters do not always turn out to be good triathletes. 

1. Increase the amount of homework that the students have to do. 

Look at the top two finishers: Shanghai students average 13.8 hours of homework per week and Singapore just under 10, while Canadian students only average between 5 and 6 hours (source : Shanghai PISA team). 

2. Set up after-school training clubs. Encourage entrepreneurial teachers to run the clubs and to publish and sell PISA-type practice questions. Encourage parents to hire tutors to help their children.

Parents in the top five jurisdictions spend large sums for extra tutoring and after-school training.  In particular, the top five are know for their notorious private "cram schools". Students attend these schools in the hope of passing exams and achieving either high school or university entrance. 

3. Import instructors from higher ranking jurisdictions to teach our teachers how to teach. 

The UK is  bringing 50 teachers from Shanghai to do just this. Actually the imported teachers will help the Brits reform their math education into a system that is centred around 32 hubs, similar to Shanghai . The cost will be 11 million pounds (CAD 17.75 million), so if we are not careful, importing teachers may lead to expensive restructuring. 

4. Don't let the bottom 20 percent of our students take the PISA tests. We can do this by barring English language learners and students with low socio-economic backgrounds.

It is pretty well established that disadvantaged children do not perform well on tests, so barring them should raise our scores. Shanghai excludes migrant children from even participating in its education system, as was confirmed to me by a teacher who taught in Shanghai. The proportion of excluded students is difficult to determine, but seems to be somewhere between 20 and 50 percent. See the damning report by Tom Loveless.

5. Put pressure on the OECD folks to release the PISA scores for individual students and schools. To further increase competitiveness, institute a set of monetary rewards for schools and provinces whose students perform the best in the PISA tests, and deny those rewards to schools whose students don't do so well. 

I'm a bit late with this recommendation. The OECD has already developed PISA-based tests for schools in the United States, England, and Spain. (Condolences to our American neighbours - more tests, just what you need!) 

Let's be honest about the PISA tests. For most of us, the only thing that counts is where we rank. And this leads to my final recommendation:

6. Let's call the PISA test what it is: a competition, not an assessment. 

I'm not sure that education should be based on competition. 

Monday, 6 October 2014

Don't trust the math prof

Have you read Dale Carnegie's How to Win Friends and Influence People? If you are a math prof, I'll wager that your answer is "No".  Maybe that might explain the unpleasantness that occurred during a math conference in Alberta. (The conference was over a year ago. I was not there, so my comments here are second hand, but my information comes from a reliable source.)

In attendance were both mathematicians and school teachers. During the proceedings, a few mathematicians took to the stage and panned both the math curriculum and the teaching methods being used. In effect they dressed down the teachers and told them how they should be doing their jobs.

I guess this happened more out of arrogance than malice. Yes, I do know that  arrogance does not invalidate the professors opinions, but it makes me think twice about any advice they might offer. 

A lot of math profs agreed with the criticisms raised at the conference. I too have opinions about curriculum and teaching methods, but I would expect a school teacher to be very skeptical about my advice. And there would be a good reason to be skeptical: unlike school teachers, I have not been taught how to teach, and neither have most of my colleagues. This does not mean that our opinions are automatically wrong, but it sure casts a shadow over them. 

Here's the problem. Some math profs have done a lot of teaching and have even become quite good at it. And like most people, they like to dispense the wisdom that they have garnered from their experience. Fair enough, but that experience is limited to university courses whose class members are not typical school students but are, in fact, the cream of the crop.  Although the profs may have developed some very good teaching practices, those practices are geared towards university students and likely won't transfer well to a K-12 classroom. 

(Is there not a little irony here? We mathematicians haven't even been trained how to teach at a university, and yet we are willing to issue directions about how to go about it in a vastly different setting.)

But there must be some value in what math profs say about math education: after all, they are experts at mathematics. This is a slippery argument, and you may have come across something like it before. It appears in different forms:
"I'm the CEO of BigOilCorp. Climate change is bunk."   
"I'm a certified marriage counsellor, and I know what I'm talking about. Children should be spanked for bad behaviour." 
"He says that the Edmonton Oilers stink. He has a degree in sports journalism, so he must be right." 
OK, so you may on board with that last one, but the reasoning is faulty. It's called "argument from authority" and in its undisguised form it goes like this:

  • So-and-so is an authority about topic X.
  • So-and-so makes a statement about topic Y.
  • Therefore the statement must be correct.

Being experts at mathematics in no way confirms that our opinions about how to teach it are valid. Appealing to our mathematical expertise is simply an argument from authority. 

But why are math profs so ready to be critical? I have some thoughts about that.

Some say that their children have not learned the basics in elementary school. It's difficult to comment about this because it is so personal, but it is a concern held by a much larger group of people.

What I am about to say may be educational heresy. I think there will always be a substantial number of children who will have difficulties with math. It was true when I was a student, and it was true when my children were students, and it is true now that my children's children are students. 

I don't think the problem is wholly dependent on either the curriculum or the way it is being taught. From talking to my grandchildren, and from what I have learned from school teachers, (and also from a brief examination of the K-6 curriculum), my own conclusion is that students today are being taught the basics, just in a different way than we were. 

On a less personal level, some professors are concerned that students entering university have not mastered the fundamentals. They perceive that students arriving at university from high school today are not as adept at mathematics as they themselves were in the past. As long as I can remember, math profs, including me, have held that view.  (And in fact you can go back 100 years and read the same complaint.)  

When I first started teaching, our department's concern led to an "advisory exam" that we gave to first year students to check that their background was sufficient. Sometimes it wasn't, and our conclusion then was pretty much the same as what math profs conclude now: there must be a problem with the way math is being taught in school. Sigh. Perfectly logical mathematicians affirming the consequent

There is another thing that bothers some mathematicians. They are worried about Canada's falling rank in international math tests, you know, those PISA tests that have caused so much panic. Some trace the decline back to the introduction of our current elementary math curriculum along with the teaching methods that support it. I don't know if its true that a majority of math profs agree with that viewpoint, but a good many have signed a petition that promotes it, so I assume that plenty actually do believe it. Sigh. Post hoc, ergo propter hoc.

I don't personally think that there is a problem with our PISA rank, but that's a topic for a later discussion. However, in the meantime I would point you to an article by Joanne Jacobs. Take a look at this question:

Did this spark a WTF moment for you like it did for me? Well, what is happening here is that the children are being asked to compute 8 + 5 by splitting the 5 into 2 + 3 as follows:
8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 10 + 3
There's no mystery here: As the teacher's feedback says, take 2 from the 5 and add it to the 8. That's what "making 10" out of 8 + 5 means. It's a method for addition that doesn't rely completely on rote memorization, and it is one of the strategies that some think confuses the children and contributed to our reduced PISA score.  

The comments following the Joanne Jacobs post are worth a look. Although there is the expected outrage, at least one person pointed out that "making 10" is one of the strategies taught to the kids in Singapore. And if you have been following the articles about the PISA math test, you know that Singapore ranked much higher than Canada. I find that somewhat thought-provoking.

That's it. Now, if I could just remember where I put that Dale Carnegie book.