Tuesday, 30 December 2014

A cheat for the standard multiplication algorithm

Of course, I mean the one that looks like this (pick whatever format you prefer): 

The one on the left is the one that I learned in school. It was one of my great dislikes. I always made errors when I performed it. This post describes a work-around, a “cheat”, if you like, that drastically cut down on my errors. It's a method that I generally hid from my teachers and colleagues so they wouldn't know how weak I was in arithmetic. I have not seen it used elsewhere, but I found it so helpful that I would not be surprised to learn that others have discovered it. 

I have a curious mental glitch. For some reason my brain will occasionally swap digits. For example, when I see 432 on the page, it stays as 432 as long as I look at it, but if I were to avert my gaze in order to record the number elsewhere, I might write 342 and remain totally unaware that I had made an error.

I still make mistakes like this. I'm sure now that this memory quirk is the source of some of my difficulties with arithmetic. However, as a kid, it never crossed my mind that that I might have a twist in my neural circuits. I just accepted that I was not very good in arithmetic.

Here is how my memory quirk could affect me if I were to multiply 68 by 4 following the method that we were taught in elementary school: 
4 times 8 is 32. Write down the 2 and carry the 3. Keep the carry number in memory. Then 4 times 6 is 24, and 24 plus the carry number is 26 so the answer is 262. 
Although my addition skills are no great shakes, in this case I was not adding incorrectly—when I got 26 the addition was perfectly correct: 24 plus 2 is 26. What happened was that in my head I was swapping the memorized "carry number" (the 3) with the number I had just written down (the 2). 

When we were first learning the multiplication algorithm, instead of having to keep the carry number in our heads, we were permitted to write tiny numbers in the appropriate place above the multiplicand like so:

Writing down the carry numbers was not a cure. Even for single digit multipliers, it only helped a little bit. It did not help at all when the multiplier had more than one digit. The jumble of small carry numbers written above the multiplicand actually exacerbated the situation. 

By the time I started high school (grade eight) I had concocted a “cheat”. Instead of separating out the carry numbers either in my mind or on the worksheet, I wrote down the full product of the individual digits. The cheat was not perfect, but it substantially reduced my errors. Using the cheat, here is how to multiply 345 by 6.

The alignment is important. First the product of 6 and 5  is 30, so write 30 diagonally with the 3 in the ten’s column and the 0 in the units column but in the line below. Next, the product of of 6 and 4 is 24, so write 24 diagonally with 2 in the hundred’s column and the 4 in the ten’s column in the line below. This places the 4 directly below the 3, and both 4 and 3 are in the ten's column. Continue in this way, writing the product of 6 and 3 diagonally with the 1 in the thousand’s column and the 8 in the hundred’s column below the 2. Finally, add the columns vertically to get the answer. 

This can be adapted to multi-digit multipliers as shown below, but it starts to get messy.

To avoid the messiness, I usually multiply 345 separately by 4 and 6 on scrap paper as shown below on the right, and then transfer the answers to the proper position in the standard algorithm as shown on the left.  (I still do this when I have to multiply multi-digit numbers without a calculator.)

Why call this a “cheat”? Well, when I was a youngster, doing anything that was outside the rules was considered cheating, and, if nothing else, arithmetic was taught as a set of rules. So in that context, it was cheating.

* * *

You may notice that my cheat has a very strong resemblance to the lattice method of multiplication. Of course, back in my day they did not teach the lattice method, and I didn’t actually know about it until I taught a “math for elementary teachers” course. In such courses I avoided teaching the lattice method because I thought that writing things slantwise in one direction and adding things slantwise in the opposite direction followed by wrapping the answer around two sides of a square would screw up a student's understanding of place value. To be honest, I do not know if the lattice method really does lead to place value confusion. I also confess that I never taught students my cheat method, so, although I believe that the cheat's strong emphasis on the proper location of the digits with regard to place values would be beneficial, I really don't know if  that is the case. 

* * *

Regarding my memory quirk, I don't think it's dyslexia. If it is, then it’s extremely mild. It’s nowhere near as profound as what Toomai describes in his post about the dyslexic mathematician. What little I know about dyslexia is that it also causes troubles with algebra, and I never had difficulty with algebra. Surprisingly, my wonky memory regarding numbers doesn’t apply to letters.

Tuesday, 23 December 2014

Proofs and problem solving, part 3 of 3

First of all there is considerable impatience with the material. Surprisingly, this is often noticeable in the better students. But better students tend to demand instant understanding. Mathematics has always been easy for them. Understanding and intuition have come cheaply. Now as they move into the higher reaches of mathematics, the material is getting difficult. They lack experience. They lack strategies. They don’t know how to fiddle around. 
— Davis and Hersh in "The Mathematical Experience", 1980

Davis and Hersh are talking about students from the late 1970’s, but the passage describes quite accurately the resistance displayed by some of my university students even in this century. 

I particularly like the comment about fiddling around, because that’s what you have to do to create a proof or solve an unfamiliar problem. Throughout my teaching career, whatever the curriculum of the day was, my students seemed to have had very little practice fiddling around. They seemed to have the notion that a solution or proof always follows a pre-ordained script.

I was able to remedy this somewhat by exposing them to mathematical puzzles—ones that were simple, yet which could not be solved by following a memorized procedure. The students pretty well had no choice except to fiddle around. And because the puzzles seemed simple, they were quite willing to try.

I ended the previous post with the following puzzle. It’s simple, but the complete solution is not obvious. It’s one that I frequently used in a university course, and I think it would work for junior high / high school students. Although it is well known, I doubt that many students are familiar with it. 

You are presented with eight coins, numbered from 1 to 8, but otherwise identical in appearance. One of them is fake and weighs slightly more than the others. The real coins are all exactly the same weight.  
You have at your disposal a two-pan balance, and no other weights except the coins themselves. The problem is to find the counterfeit coin using as few weighings as possible. 
Now if you choose two of the coins and by good fortune one of them was the counterfeit, then weighing one against the other would reveal it. However, this really does not count as being a solution. You need something that works all the time and does not depend upon luck.

Here is a possible solution, one that is based on the binary search idea described in my previous post. This is how I did it at first, and so did most of my students. 

Divide the eight coins into two groups of four. Weigh one group of four against the other to find out which group contains the fake. Divide that group in half, and weigh two against two to identify which pair contains the fake. For the third and final weighing, compare the coins in that pair to identify the counterfeit.

A binary search is very efficient in many cases, as it is for this puzzle. But it is a script to be followed, and following a script does not always lead to an optimal solution. We have yet to convince ourselves that three weighings is the best that we can do, that it is impossible find the counterfeit in less than three weighings. There doesn’t seem to be an obvious way to prove this, so here’s where the fiddling around has to occur. And we will see how fiddling around reveals a better solution. 

A tactic that is sometimes useful is to approach a problem from the reverse direction. Instead of trying to find the minimum number of weighings for eight coins, let us ask what is the maximum number of coins we can handle if we are only allowed one weighing. If we have a collection of coins and one is fake and slightly heavier, how many coins can be in the collection?  Can we find the counterfeit with one weighing if we have two coins? three coins? four coins? etc. 

If we have two coins and one of them is fake, putting a coin in each pan will immediately identify the counterfeit. If we have four coins and one is fake, putting two in each pan will allow us to identify which pair contains the fake, but unfortunately it does not reveal which member of the pair is the fake. We would need another weighing. 

So one weighing with four coins (or more) seems out of the question. What if we have three coins? Putting two in one pan and one in the other won’t help. The coins are close in weight, so the pan with two coins would tilt down regardless. So, it is unreasonable to ask about an odd number of coins—we need an even number so that we could put the same number in each pan.  

One of the difficulties that I have when fiddling around is that I am prone to making unwarranted assumptions, and I just made one here. Who says that we have to put all of the coins in the pans? With three coins, why not see what happens if we put one coin in each pan and leave the third coin aside? And that does it! If the balance tilts, we will know immediately what coin is counterfeit. If the balance stays level, then the two coins in the pans must be the same weight and so the coin that we left aside must be the counterfeit. With three coins we can find the counterfeit with one weighing. 

This fiddling around with three coins throws a new light on the entire puzzle. Maybe we don't have to divide the eight coins into two groups of four. What if we divide the eight coins  into three groups, put a group in each pan, and leave the third group aside? 

What if we put three coins in each pan and leave two aside? If the pans don’t balance, this will tell us which group of three contains the fake. If the pans balance, this will tell us that the group of two that we left aside contains the fake. Either way, we will know which group contains the counterfeit, and since the groups contain only two or three coins, with one more weighing we can find the counterfeit. 

So this logical fiddling around leads to a better solution. With eight coins, we can find the counterfeit in two weighings, and the fiddling around has revealed that the counterfeit cannot be found with less than two weighings. 

* * *

In its original form, the counterfeit coin puzzle did not use numbered coins. The reason I specified that they be numbered is that there is another solution, and it’s easiest to explain if you number the coins. This alternate solution came about because I was still fiddling around and changed the question a bit. What if we could not use the information obtained from the first weighing to determine what to do for the second weighing?  In fact, a solution with two weighings is also possible in this case.

Let us imagine that the coins are arranged in an array as follows:. 

The rows and columns will tell us what coins we should weigh. In both weighings we will weigh three against three and leave two aside. 

Using the rows: weigh 1, 2, and 3 against 4, 5, and 6, and leave 7 and 8 aside. That will tell us which row of the array contains the fake. Call it the hot row

Using the columns: weigh 1, 4, and 7 against 2, 5, and 8, leaving 3 and 6 aside. That will tell us which column is the hot column

The counterfeit coin is located at the intersection of the hot row and the hot column, just like finding a point in the Cartesian plane given its x and y coordinates. And we don't have to know the outcome of the first weighing before doing the second weighing. 

* * *

You can search the internet and find this and many other counterfeit coin puzzles. You will no doubt encounter the problem discussed here. Some posts ask why use eight coins when the same solution works for nine. I think that using nine coins gives too strong a hint that the coins should be split into three groups of three, so I prefer the eight coin version.

* * *

It would be wonderful if there was an algorithm that always tells us how to fiddle around. 
Unfortunately, there is as yet no such algorithm. The closest I have seen to this is the collection of strategies described by George Polya in his books about problem solving. What one should do when fiddling around depends upon what the problem is. The best way to learn is to practice on lots of problems.

Wednesday, 17 December 2014

Proofs and problem solving, part 2

This is the second of three posts about the connection between problem solving and proofs. 

A couple of questions tempted me to write these posts:  What is a proof? And why is it difficult to learn how to prove things? 

Michael Pershan asked 37 math teachers why they think kids find proof so hard in geometry. One of their suggestions was that kids have not yet developed logical thinking or deductive reasoning. I am not entirely comfortable with that sentiment, and, like Michael Pershan, I think that that kids already have some logical reasoning abilities. I don’t think that you can sharpen their abilities by making them study and-or / if -then statements or by drawing truth tables. And figuring out how to prove something is not completely a deductive process. 

As to what a proof actually is, most definitions are as vague as you can get, and boil down to some variation of saying that “a proof is just a convincing argument”. Keith Devlin has written several evolving posts about it, and most recently (here) put it this way:
Proofs are stories that convince suitably qualified others that a certain statement is true.
I like Devlin’s discussion of what a proof is, but I worry about the “suitably qualified others” part of the definition. Not because I think it is inaccurate, but because I think that a student who is first learning how to prove things will likely take it to mean “It’s not a proof unless my teacher says it is.”

Some of us may equate “proof” with a notion that I would call “formal proof”, the idea that a proof must follow a specific template or format—which puts yet another hurdle in front of a student trying to learn how to prove things. 

I think you will have great difficulty proving something if you are continually fretting about obtaining some sort of external approval. Before even thinking about convincing someone else, you should at least be satisfied in your own mind that what you are doing is valid. Some sort of “self feedback” if such an oxymoron makes sense. 

Before teaching someone how to prove things, and especially before making students practice templated proofs, it may be beneficial to put them in a position where this self feedback occurs naturally. One way to do this is to provide interesting problems, ones that are within reach but which do not have an immediately obvious solution.  

In the previous post, I included the following puzzle, and I will use it as an example to show how problem solving and proof sometimes go hand in hand.

There are three boxes that contain red and white balls. One box contains 10 red balls, one contains 10 white balls, and the third contains 5 red and 5 white balls. The boxes have been labelled “red”, “white”, and “mixed”, but each label is on a wrong box.  
All of the boxes have lids so you cannot see what is in them, but you are allowed to reach inside the boxes without peeking and take one or more balls out and look at them. Taking out as few balls as possible, figure out what the correct labels should be. 

When I first saw this puzzle, my initial reaction was to start asking “What if” questions:
What if I take a ball from the box that is mislabelled “red”?  Either the box contains only white balls or else it contains a mixture. If I take one ball and it’s red, then I know the box has a mixture. But if it’s a white ball, then I’m in trouble—I could have selected a white ball if the box contains a mixture. So selecting one ball is not enough. If I take even five balls from the box they could all be white, and I still won’t know whether it contains the white balls or a mixture.
What if I also take some balls from the box mislabelled “white”. If I get red balls from the  from the “white” box, I’m in the same pickle.
This is not too promising. Maybe I should take some balls from all three boxes?
What I described above is based partially on my memory of my own experience and also on what I imagine might happen to others. However, even if you had these thoughts, you no doubt moved on. Your next step might have been:  
What if I take a ball from the box mislabelled “mixed”?  That box contains either all red balls or all white balls, so one ball is enough to tell me what its label should be.
At this point you would probably feel that you are on track to a solution. You are also starting to work out why the solution works even though you haven’t yet got a complete solution. 

After realizing that you only need to look at one ball to determine what is in the "mixed" box, you have to imagine what your next step would be. There are several courses of action. My own instinct was to ask myself if I needed to look at ball from another box. Here’s how I remember thinking about it:
What if I now draw a ball from the box mislabelled “red”? I’ve just been through that.  That box contains either all white balls or a mixture of red and white balls. 
Wait a minute. By this time I will know the colour of the ball that I drew from the box mislabelled “mixed”. Suppose it was white. Then this is the only box that contains all white balls, so the box mislabelled “red” can’t contain only white balls. I already know that it can’t contain only red balls, so it would have to contain a mixture of red and white balls. 
And then this leaves only one possibility for the last box, the one mislabelled “white”—it must contain only red balls.
At this point you have a solution: you can figure out the correct labels if you take just one ball from the box labelled "mixed".  At the same time you have a proof that your solution works even though the proof and the solution are a bit disorganized and not quite complete. 

I do not know if all students can hold all this reasoning in their minds while they are solving the puzzle. It would be beneficial to use physical models to play with—actual boxes and balls with labels that could be switched around. Many years ago I saw a grade 5 student demonstrate a solution and explain her reasoning using such props. Her proof convinced me. 

Next post’s puzzle 

The puzzle that is the subject of the next post is interesting to me because of some prior knowledge that I had, and before stating it I want to set you up with the same background. 

When I was in grade seven or eight, we used to play a "guess a number" game. I goes like this (this is not the puzzle for the next post, but just a set up):

I am thinking of a number between 1 and 100. You can ask me questions that have a “yes or no” answer. Try to guess it in as few questions as possible. (Of course by a “number” we mean a whole number, not a fraction.)

We had a variety of approaches and questions.  One of our strategies was to first determine whether it was even or odd, thereby cutting the possibilities in half. Let us suppose the number was 21. The questions might go like this:
Q1: Is the number divisible by 2? Ans: No.  
Q2: Is the number divisible by 3? Ans: Yes.  
Q3: Is the number divisible by 9? Ans: No.
And we could continue in this way trying to find the divisors of the number:
Q4: is the number divisible by 5? Ans: No.
Q5: Is the number divisible by 7? Ans: Yes. (so it's odd and a multiple of 21)
Q6: Is the number 21? Yes.
This strategy works well for some numbers, but the process can be rather lengthy for others. (Try it for a prime number like 59.) 

Eventually, we learned another strategy: keep halving the interval that contains the number.
Q1: Is it bigger than or equal to 50?  Ans: Yes. (so it's from 50 to 100)  
Q2: Is it bigger than or equal to 75? Ans: No. (so it's from 50 to 75) 
Q3: Is it bigger than or equal to 62? Ans: No. (so it's from 50 to 61) 
Q4: Is it bigger than or equal to 56? Ans: No. (So it's from 50 to 55) 
Q5: Is it bigger than or equal to 53? Ans: No. (So it’s, 50, 51 or 52) 
Q6: Is it bigger than or equal to 51? Ans: Yes. 
Q7: Is it 52? No. 
Q8: Is it 51? Yes.
The process being used is called a binary search, and any computing science student will be familiar with it. You can always find a number between 1 and 4 in three guesses, between 1 and 8 in four guesses, between 1 and 16 in five guesses, and so on. 

This sets you up for the following problem which is the subject of the next post. I’ve known this puzzle for years. I’m not sure where it originated, but I think it was with Martin Gardner.

You are presented with eight coins, numbered from 1 to 8, but otherwise identical in appearance. One of them is fake and weighs slightly more than the others. The real coins are all exactly the same weight. 
You have at your disposal a two-pan balance, and no other weights except the coins themselves. The problem is to find the counterfeit coin using as few weighings as possible.
Now if you choose two of the coins and by good fortune one of them was the counterfeit, then weighing one against the other would reveal it. However, this really does not count as being a solution. You need something that works all the time and does not depend upon luck.

Come back in a week or so and see if you got the same solution (and surprise) that I did. 

Wednesday, 10 December 2014

Proofs and problem solving, part 1

The human visual system alters and reshapes our perception of reality. Highlights appear where there are none, shadows are created that do not exist, and identical colours somehow end up looking dramatically different. We are so accustomed to this that it passes without notice. 

The effects can be very strong. I remember being astonished in the mid 1970’s when I first learned about the highlight and shading effects in the Cornsweet illusion and Mach band effect. There are also amazing difficulties in accurately perceiving colours. See for example the examples here (scroll down a screen or so to the "blue and green spirals"). I urge you to visit these illusions before reading the rest of this page. 

There was a time, well before the invention of photography, when artists strived to portray things in a realistic fashion. The perceptual distortions of light and colour mentioned above must have been a challenge: if an artist were to paint what he “saw”, he would include those false visual effects, and a person viewing the painting would be looking at something that differed significantly in light, shade, and colour from the actual scene. As a result, the viewer's perception would likely differ quite a bit from the painter's perception. Sort of like the increased blurriness that you get when you save a jpeg copy of a jpeg, or the degradation that occurs when you make a Xerox copy of a Xerox copy. Passing an already filtered version back through the same filter may have a detrimental effect.

Yet, despite these obstacles, some artists were able to achieve remarkable realism. It makes me think that they were fully aware of the artifacts that their vision introduced, and that they found ways to expunge the distorted light, shade, and colour from their paintings. Exactly how they did this is not clear.

One solution is offered in the film Tim’s Vermeer. It recounts Tim Jenison’s attempt to describe how, in the 17th century, Johannes Vermeer could have used relatively simple optical devices to help him create the light and colour in the painting The Music Lesson. Although not explicitly stated as such, the film is all about how Vermeer could have circumvented the problems inherent in his own visual system and thereby realistically duplicate in his painting the physical highlights, shadings, and colours that were present in the scenes that he observed.  

Viewing the film through my filter as a mathematician, I think of it as somewhat of a sustained metaphor about the process of problem solving and its relationship to proof. In a university lecture, the presentation of a solution frequently follows a path like this: Here’s a problem, here’s my solution, and here’s proof that my solution works. But this is what happens after the solution is obtained. What most likely happened before the solution was completed is that the solving and the proving were thoroughly interweaved.

In the film we see Tim Jenison’s solution to the problem of  accurately reproducing physical light, shade, and colour, along with his proof that his solution works. His proof that Vermeer could have used optical devices is to use those devices himself to recreate Vermeer's Music Lesson.  As the film proceeds, Jenison sticks to his basic premise, but continually adjusts and adapts the process, much like what happens when one is solving a mathematical problem. 

By the time I have solved a problem, my first thoughts about how to solve it have been mostly forgotten. It may be a personal peculiarity, but I find it very difficult to recapture my state of mind as it was when I first considered the problem. When I present my solution, I can polish and burnish it, and can point to logical insights that, once you notice them, inexorably lead to a solution. However, I usually garner these insights as I work my way through the solution; they are not the initial thoughts that led me to the solution.  In other words, the manner in which I present my solution is not always a true reflection of how I solved it. 

This is not a happy state of affairs when teaching. If you show your ultimate solution to a class—no matter how polished and clear it is—if your students do not experience some of  the missteps, stumbles, and side avenues that you took along the way, are you really doing anything different than reading from a textbook? 

George Polya has written books about problem solving, and he gives suggestions on how to get past the initial stages [see the wikipedia article about How to Solve It]. I am not going to dwell too much on that part of problem solving. But what I do want to examine is what happens after you are pretty sure that you are going in the correct direction, how you can continue in that direction, and how during that process you are actually developing an argument that your solution is valid. In other words, I will try to show how proof and problem solving are intimately intertwined just as they are in the the Tim’s Vermeer film.

This will be the content of my next two posts. and I will use two well known simple puzzles to try and illustrate the process. 

Here is the first puzzle:
There are three boxes that contain red and white balls. One box contains 10 red balls, one contains 10 white balls, and the third contains 5 red and 5 white balls. The boxes have been labelled “red”, “white”, and “mixed”, but each label is on a wrong box.   
All of the boxes have lids so you cannot see what is in them, but you are allowed to reach inside the boxes without peeking and take one or more balls out and look at them. Taking out as few balls as possible, figure out what the correct labels should be.
So go ahead and solve it, and convince yourself that you have a solution. Come back in a few days and see if you followed a path similar to mine. 

If you are interested in the Tim's Vermeer film, there is more information about it here and here.  The film is controversial, especially among art historians who view it as an attack upon the artistic integrity of Vermeer. Moreover, in The Music Lesson, Vermeer did not depict everything accurately, but deliberately added and removed light and shade to enhance the painting (see here). This, however, does not disprove that Vermeer used physical devices to help him understand light, shade and colour. And, unlike some of the art historians, I do not view the film as belittling artists: to me it demonstrates the intelligence and ingenuity of visual artists like Vermeer.

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 tjelewis@gmail.com. 

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.

Thursday, 25 September 2014

Just teach, dammit!

Twenty years ago a student had some advice for me:
It's the professor's job to know the theory. It's the student's job to know the facts. You should just tell us the facts and show us how to use them.
That's what the student told me, and this is what I heard:
I don't want to know why things work, I just want you to show me how to do it!
"YESSS!" says the student.

"AArgh!" say I.

I really enjoyed teaching math at university. The students were mostly receptive, most of them worked pretty hard, and I got along with them very well. However, there were always a few that fell outside this norm, and for those few there were typically two things that annoyed them.

The first thing that really yanked their chain was having to learn a proof. The process caused them great agony, and adding to their stress was the fact that a proof often had no immediate use beyond the theorem it was attached to. The other thing that irritated them, not quite so mightily,  but still quite a bit, had to do with what they said they wanted, that is, with what they called the how-to-do-it part of math. They would resist learning a new way of solving a problem when they had an old way at hand, and this was true even when the new way was more efficient. Learning proofs and learning alternate approaches, those two things really rankled them. 

Now, I was actually a pretty good teacher, and I've had some success. As I said, I enjoyed teaching very much, and I'm sure the students knew that and responded to it. However, it was difficult holding my exasperation in check when I met students who did not want to know why something was true or who were unwilling to try different approaches. They just didn't get it. Worse, it seemed like they were not even remotely interested it getting it. 

Interpreting the students' behaviour in a charitable light, I would guess that they were asking for help, but I have to say that it bothered me a lot when they reacted in such an anti-intellectual way. I wonder if they picked up that attitude somewhere, or if their reaction was an inborn one. Does such a response originate in the parenting, or in the school system, or is it really an innate human characteristic? Psychologists tell us that youngsters have an insatiable curiosity, and watching my children and grandchildren grow up tells me the same thing, so it is hard for me to accept that reacting so negatively is part of the human condition. 

So where did the negativity come from? The complete answer is probably quite complicated, and it is outside my expertise. The only way I can understand it is by extrapolating from my own personal experiences. 

Unlike many of my colleagues, becoming a mathematician was not a smooth ride for me. In high school and university I was very good at math but even during those times I did not always like it. There are still parts of math that cause me difficulties (namely arithmetic), and when I think about my past I am quite surprised that I did become a mathematician. 

At the very start, in elementary school, I had some difficulty with arithmetic. Lots of difficulty. I really disliked it and I avoided it whenever I could.  There is no question that I had a bad attitude. 

Some of my troubles arose because my memory for numbers is not always trustworthy. Although I was not aware of this until I was an adult, it certainly must have been a contributing factor in my younger years. However, I somehow became at least marginally competent in arithmetic, and after much reflection I don't believe that an unreliable memory was the main cause of my difficulties. I think my difficulties and my resentment were rooted in the math curriculum and the way we were expected to learn it. 

We learned math in the good old fashioned way,  that is  1) by memorizing the addition and multiplication tables, 2) by practicing such things as adding in columns and performing long division, and  3) by memorizing some procedures to solve some specific problems. 

Some of my classmates thrived under this regime, but a good number of us did not. For me, arithmetic never fully took hold. I never became skilled at it. I found it boring, I found it confusing. The good old fashioned way did not work for me, and it also did not work for a lot of my classmates (so it couldn't have been just my wonky memory that caused my troubles). 

With the good old fashioned way there was an official and unalterable route to the answer. We were taught "This is how you do long multiplication. This is how you do long division. This is how you add a column of numbers." That, together with the practice, the drills, and all that memorization delivered a very strong hidden message:  "You don't need to understand why it works, you just need to learn how to do it." 

Some people, many of them my colleagues, call this "learning the fundamentals". Well, if that's the case, I personally never learned the fundamentals. In retrospect, I eventually did acquire the basics, but not because of the good old fashioned way. I learned them because I was lucky enough to find some ways to work around the procedures that I couldn't master. It would have been a lot easier for me if I had been taught those work-arounds without having to devise them myself.

Things are different in elementary school today. Here's a thumbs up to the teachers and the Education profs who are trying to make kids math life so much more interesting and less punitive. 

Some of you know where I am heading with this. I am distressed with that Alberta back-to-basics petition. Read it, and see if you don't think that they want to reinstitute the "good old fashioned way". Our newly appointed Education minister and the petitioners talk about understanding the basics, but I'm not sure about their commitment to the understanding part.  Despite their disclaimers, I think that, deep down, many of the people who signed the petition want to banish understanding from the classroom by getting rid of anything that offers alternatives and exploration. 

I can't remember much about elementary school math except for the memorization and the drills. It wasn't until I attended high school that I was encouraged to tinker and to try different approaches, and it was there that some remarkable teachers began changing my attitude. Here are three episodes that I remember about learning math in high school. I know these won't seem very novel to today's teachers, but read on anyway. (But be prepared for a little math.)

Episode 1. Mr. Troughton's quiz 

In our first year of high school, in our first day of class, in our first ever Algebra course, our teacher, Mr. Troughton,  gave us a short math quiz. He read the questions aloud, and we wrote down our answers and handed them in. Here was the first question. Although very well known, it was new to us. 
A bottle and a cork cost $1.10. The bottle cost a dollar more than the cork. How much did the cork cost?
I confess that I was sometimes a bit of a smart-ass, and, with a satisfied smirk, I wrote down my answer:  "The cork cost ten cents!!"

Next day, when Mr Troughton handed back the answer sheets he asked "If the bottle cost $1.10, and the cork cost ten cents, how much did the bottle cost?" 

"A dollar," we said.

And then he asked  "So how much more did the bottle cost?" (was he looking directly at me?) 

Ha! I was supposed to be one of the smart ones.

That was first time ever in a math class I was faced with a problem that I had not been taught how to solve. This was a brand new experience, and was a bit of a shock. In mathematics, isn't the teacher supposed to show us how to do the problems? Are we not supposed to always get the method for working out the solution? It took a few more years to realize that the answers will forever be "No."  

Episode 2. Mr. Stirling's geometry challenge

The next year there was Mr. Stirling, who taught us grade 9 Geometry. This was our first exposure to the subject, and early in the course he showed us how to bisect an angle with a compass and a straight edge. Then he challenged us to divide an angle into quarters, and after we figured that out he mentioned, rather off-handedly, that nobody had been able to trisect an angle. He promised that we would become famous if we could do it.

I know external rewards are frowned upon, but the truth is that the prospect of fame was a very tempting lure, and we bit. We spent the next few days exploring the trisection problem. It consumed our lunch hours, and Mr. Stirling let us continue working on it during class.  In the process, we became skillful with the geometric instruments, and we learned what geometric constructions were legal and what types were not. We learned quite a bit of geometry by playing around with it on our own.

[ starting the mathy part . . . .

One bright boy claimed he had a solution. Using a protractor, he measured and drew a trisecting line. We objected. "You can't uses a protractor, you can only use a compass and straight edge," we said. 

"Never mind" he said, "Watch: if you bisect the angle, you get two half-angles, and one of them contains the trisecting line." His argument continued like this:

"Bisect that half-angle and you get two quarter-angles. One of the quarter-angles contains the trisecting line. Now bisect that quarter angle, and you've got a couple of  one-eighth angles and one of them contains the trisecting line. If you keep on like this, you get a one-sixteenth angle, then a one-thirty-secondth angle and you keep getting closer and closer to the trisecting line, so eventually we should get the trisection."

Mr. Stirling gently pointed out that this was not a legal solution because you must be able to finish in a finite number of steps. Nevertheless, the student had discovered something very interesting, namely that 1/2 - 1/4 + 1/8 - 1/16 + 1/32, etc, would get us as close to 1/3 as we desired. Quite a feat on his part, I think, and there is a lot of mathematics going on here.

 . . . . ending the mathy part ]

We never did find the answer to the trisection problem, and I learned many years later that the problem was not merely unsolved, but that the construction has actually been proven to be impossible. I don't know if Mr. Stirling knew that (but I sure hope he didn't).

Episode 3. Mister Watson's Intermediate Algebra course

Mr. Watson walked into the room with the textbook in his hand and sat down at his desk.  We all knew who he was, so he didn't bother to introduce himself. I can't remember what he said about the course but I do remember that there was an uncomfortable silence. We looked at him, but he just sat there. Minutes passed. More minutes passed. Tension grew. Finally, someone put up a hand and asked "Sir, when are you going to start teaching?" 

He smiled and said something like: "Don't you have a textbook? Open it at the beginning, read chapter one, and try solving the problems at the end.  If you have trouble, I'm at the front of the room. Bring your work here and I'll help you."

So we had to learn on our own, and we had to read the textbook by ourselves. Occasionally (maybe a bit more frequently that I remember) there would be a passage in the text or a problem that was difficult, and if Mr. Watson noticed that a lot of us were stuck at that point, he would discuss it with the whole class. 

This continued throughout the course. Sometimes he would teach for the entire period, and sometimes never at all. But what I remember most is working on the problems by myself with hardly any help or instruction from him. 

We finished the course material early, many weeks before end of the school year. This lengthy span of non-course days, however, was not entirely goof-off time. Mister Watson filled the time with a mix of different things. Mostly he lectured about different topics, but what I particularly remember is that he brought math puzzles to the room.

One puzzle stood out, and I'd like to leave it for you. There's a good chance you are familiar with it, but, as was the case with the grade 8 quiz question, it was completely new to us. Here's the puzzle (and no, I'm not going to tell you the solution):
In the following sum, each letter stands for a digit, and different letters represent different digits. The leading digit is never zero, so neither S nor M are zero. Find what each letter stands for.

These three episodes did not turn me into a mathematician, but they did help me realize that I was very good at math and that parts of it could be very interesting. And they did help quash that budding anti-intellectual attitude that I had picked up in elementary school.

Here endeth the lesson.