Tuesday, 20 February 2018

Nine Men in a Trench

Nine Men in a Trench is a math puzzle invented in 1917 by H. E. Dudeney. Historically, it was a dark time. The battles of World War One were never ending, and casualties were horrendous. At the Battle of the Somme, which lasted over four months, the combined casualties for the Allies and the Germans totalled an unimaginable 1,200,000 people.

WWI battles featured trench warfare, which must have given rise to the setting for Dudeney's puzzle.

Photograph from Canadian government archives showing Canadian soldiers at the Somme. Those recesses dug into the sides of the trench are called funk holes, and they are a factor in Dudeney's Nine Men in a Trench puzzle. 

In contrast to all this darkness, we have this wonderful photo of a student from Contentnea-Savannah School in North Carolina. She is displaying her puzzle at a math fair.

Photo posted with permission.

This is the puzzle:

Nine Men in a Trench

Move the red marker from the right end to the left end. You can move markers into the spaces, but cannot double them up. Nor can you jump a marker over another.

(If you look closely at the photo, you can see markers on the puzzle board behind the student.)

The puzzle can be found on our math fair website (link provided below). It is a simplification of Dudeney's puzzle. Here is his original version, complete with funk holes and soldiers:

Nine Men in a Trench

Here are nine men in a trench. Number 1 is the sergeant, who wishes to place himself at the other end of the line––at point 1––all the other men returning to their proper places as at present. There is no room to pass in the trench, and for a  man to attempt to climb over another would be a dangerous exposure. But it is not difficult with those three recesses, each of which will hold a man.

How is it to be done with the fewest possible moves? A man may go any distance that is possible in a move. 

So where's the math??

The Nine Men in a Trench puzzle does not explicitly involve arithmetic. At SNAP, we promote the use of math based puzzles in the classroom, but puzzles like this sometimes prompt the above question.

The truth is that much of mathematics does not involve arithmetic.  Nine Men in a Trench is a sorting problem, and sorting problems are dealt with in both mathematics and computing science courses.

Like all sorting problems, Nine Men deals with a list of objects that have to be put into a specific order, with restrictions on what manoeuvres are allowed when reordering the objects. 

To solve the Nine Men puzzle, students have to figure out procedures to use that will allow them accomplish the sort. Although they are dealing with a 100 year old puzzle, when students work with this puzzle they are truly doing up-to-date mathematics. Beyond memorizing and practicing the usual algorithms of their math courses, when they work on this puzzle they are also beginning to invent algorithms of their own making.

A more challenging adaptation of the puzzle makes it very clear that it is a sorting problem:

Five Men in a Trench

Put the tiles in numerical order from left to right. Stay within the playing board. You can move tiles into the spaces, but cannot double them up. Nor can you jump a tile over another.

We can ask for the minimum numbers of moves required to solve this version (although that should probably not be the point of the puzzle for children). I have been able to do it in twenty-five, but I don't know if that is the smallest possible number. As in Dudeney's puzzle, a move is defined as sliding a single tile any number of positions, following the rules for moving a tile of course.

More about connections between math and puzzles

Teresa Sutherland and I have put together a small booklet about the connections between school mathematics and the puzzles on our math fair website. It includes pointers to both the Alberta Math curriculum and the American Common Core State Standards. Teresa is a Designer/Director/Teacher from Maryland for the STEAM-based Odyssey of the Mind Inspired Spontaneous Challenge Program. 

You can obtain the book from the SNAP math fair website. Or contact me, Teresa, or Sean Graves, respectively at 

(Ted)       tlewis@ualberta.ca

(Teresa)  dsuther@yahoo.com

(Sean)    sgraves@ualberta.ca

The original Dudeney puzzle (and others) can be found in either of the following books:

Friday, 16 February 2018

Who needs algebra anyway?

These five balance and weighing puzzles are intended to be solved using only manipulatives. Some of them are suitable for a puzzle-based math fair. They are fun to solve and require a lot of mathematical thinking.

No algebra is needed, but if you have algebra at your disposal, it is interesting to see how each puzzle can be modelled as a system of linear equations.

There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

How many blue boxes are needed to balance one red box?

So, the solution for puzzle #1 is not difficult: The balance at the top left says that two yellow boxes are the same weight as three blue boxes. So, attending to the balance at the top right, replace the two yellow boxes with three blue boxes to get

By removing three blue boxes from each pan (which  maintains the balance) we see that five blue boxes will balance one red box.

Here's one that is a little more interesting:

There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

How many blue boxes are needed to balance one red box?

And one that is more challenging:

There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

How many blue boxes are needed to balance one red box?

Or this one:

There are three different types of boxes. All boxes with the same colour weigh the same.

 These boxes balance.

How many blue boxes are needed to balance one red box?

Here is one that is about inequalities.

There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

These boxes will not balance.
Which side of the balance will tilt down?

Puzzle #5 is from Sam Loyd, where it was not given as a balance and weighing puzzle, but was set up as a "Tug o' war" problem:
In a tug of war contest, four stout boys could tug just as strong as five plump sisters. Two plump sisters and one stout boy could hold their own against two slim twins. If three plump sisters and the slim twins team up against one plump sister and four stout boys, which side will win?


There are major differences between using manipulatives and using algebra to solve these particular puzzles.

1. There are no negative numbers when using manipulatives. In the following situation, you cannot isolate the red box by removing the yellow box from the left pan because there is no balancing yellow box that can be removed from the other pan.

Whereas with algebra, we readily subtract y from both sides of the equation

r + y = 5b 
                         to get
r = 5b – y.

2. There are no fractions when using manipulatives. In puzzle #4, I did not ask the question
How many blue boxes are needed to balance one yellow box? 
because there is no answering it using manipulatives alone.

It could be deduced that two and one-half blue boxes are needed to balance one yellow box, but you cannot get this via manipulatives unless your manipulatives are sliced into smaller pieces beforehand. (And you have to use at least a minimal amount of algebra to determine the size of the slices.)

If we use algebra to solve the puzzle, we fairly quickly obtain the fact that

 y =  52 b,

even 'though you can argue that this is of dubious meaning in the context of the physical puzzle.

A more complete answer to the question in the post title:

Sometimes you really don't need algebra, but algebra can make thinking a lot easier.

Friday, 12 January 2018

Real world math ??

Here's something that percolates through K-9 math education:

Always relate your projects, exercises, and tests to real-life and the real-worldTeach students that math is necessary in the real-world. Students want to know how math applies to their world. They want to see its practical value.  Doing this will make it appealing to them. 

Well, duh! Looks like an obvious way to make math more appealing. But things "gang aft agley".

Consider this scenario: You are making up a question for elementary students that requires multidigit addition.  So you come up with:

Ageing Relatives

(Also known as "When in doubt, add!")

John’s aunt is 35, Tom’s uncle is 42. What is their combined age?

Without the trigger word "combined" why would you even think of adding the ages?

Young children try to make sense of new experiences. Unfortunately, there is not much opportunity for sense-making with this problem.

The setting (two people of different ages) is real, but the context (the necessity for doing the math) is missing. To solve the Ageing Relatives problem, children are obliged to do math in a setting that does not provide a rationale for actually doing the math. Give them too many problems like this one, and they may start operating according to this principle:

When there are two numbers and it is not clear how to use them, 
the proper thing to do is to add them.

There is evidence that this happens. A while back, this problem swept through the web:

Robert Kaplinsky gave the question to 32 grade eight students, and 24 of his students came up with a numerical answer.  (You can see his discussion here along with reactions from other teachers.) It is not clear whether the students provided numerical answers because they wanted to please the teacher, or whether they acted reflexively and just did some arithmetic with the numbers that happened to be available.

Here's an algebraic variant that my daughter sent me:

Here is a more advanced version of the "math without rationale" syndrome.

Training a Mouse

(A magic function problem)

Alex has been training a mouse to find a reward by carrying out a certain task. He wants to know how long it will take the mouse the find the reward on its fifth try.

The time that it takes the mouse to find the reward is modelled by

T(n) = 0.04n2 – 6n + 30,

where T is the time in seconds and n is the number of times the mouse has previously tried the task.

How long will it take the mouse to find the reward on its fifth try?

A magic function is one with a unfathomable connection to the setting, but which nevertheless must be used to obtain a solution for the problem. Since the derivation of the magic function is unknowable, it carries with it this message:

Any useful parts of mathematics will forever remain beyond our understanding.

In truth, the sole intent of a magic function problem is to have the student do something with the given function –– in this particular example, it is to evaluate the function when n = 4.  In problems like this, the function may or may not be realistic.

I don't know if magic function problems can be fixed without scrapping the function, in which case one should discard the setting and leave the question as a procedural exercise.

Below are a few more vexatious real-world problems. I tried to show how to fix the problem by providing alternate settings that still retain the math task while avoiding the awkwardness of the original setting. Some of the fixes employ a setting that is not real-life. (I believe that most students will accept a pretend-world context as long as it does not attempt to pass itself off as their world.)

The Diner Menu

(Math is everywhere?)

Andrew saw the following sign at a diner. If he bought one of each item and spent $7.50, how much did the drink cost?

Burger3x +  0.05
Drinkx + 0.10

The question is nice and short. But have you ever seen product prices listed like this?

Here's a possible fix that uses a fictitious setting.

(I'll agree with you that the fix is not much better than the original question, but at least it removes that incredibly bogus real-world setting.)

On the forest moon of the planet Endor there lived an Ewok family with 3 children. The parents had given their children some Baubles and Bangles. Here's what each child has:

Infant Ewok1 Bauble
Pre-school Ewok1 Bauble & 2 Bangles
Teen Ewok3 Baubles & 1 Bangle

The children want to exchange their Baubles for Bangles, and the parents have agreed to do this.

Unfortunately, none of the Ewoks could remember how many Bangles a Bauble was worth. All they could remember was that, taken all together, the totality of the children's Baubles and Bangles had the same value as 18 Bangles.

After the exchange, how many Bangles will each Ewok child have?

The following is a different fix, also not real-world, and it's a bit too wordy, but the setting does provide a plausible reason for the way the data is presented.

Mrs Boychuk and her two young daughters were visiting an historical tower that had three observation levels.  Leading between the levels were one or more red staircases sometimes followed by extra wooden steps.

All of the red staircases were identical. Mrs Boychuk asked Freddy to count the number of steps in a red staircase, and asked Frankie to count the number of wooden steps between each level.

Freddy dashed as fast as she could to the top level and enthusiastically reported that there were 90 steps in all. In her exuberance, she didn't count the number of steps in a red staircase.

This is the data they collected:

LevelsRed stairsWooden stairs
Level 0 to Level 11 staircase0
Level 1 to Level 21 staircase10
Level 2 to Level 33 staircases5

Of course, Mrs Boychuk was a bit peeved, since she was going to ask the youngsters to add in columns and find the total number of steps. But then she realized that she had a new problem to ask:

How many steps are there between each observation level?

Cookie Boxes

("The cart before the horse it is!")

Before her coffee break, Ms Kowalchuk had prepared 24 boxes of cookies, and tied each box with a red ribbon. When she came back from her coffee break she noticed that the ribbons were removed from 4 boxes and 7 boxes were missing.

How many boxes with red ribbons were left?

Many of us have had a lot of experience with word problems. When I see one like this, I tend to react reflexively and proceed to solve it without considering the reasonableness of the problem.

How did Ms Kowalchuk know that 7 boxes were missing without first counting how many boxes were left and subtracting that from 24? In other words, the answer to the question had to be known before there was enough information to ask the question.

We can make the math task seem more reasonable by changing things so that the setting itself explains how Ms Kowalchuk noticed that 7 boxes were missing without already knowing the answer she is seeking.

Before her coffee break, Ms Kowalchuk had prepared treats for her community league book club. She put out 24 plates each with a cupcake and two cookies. When she came back, she saw immediately that four plates only had cookies on them, and she counted seven plates that were completely empty.

How many plates were left that still had a cupcake?

Apple Inventory

(Using pliers to hammer a nail?)

A grocery store manager noted that on Tuesday, for every 3 McIntosh apples the store sold, they sold 4 Red Delicious apples. On Tuesday they sold 24 McIntosh apples.

How many Red Delicious apples did they sell?

My friend is an excellent handyman. He vents when he sees someone using the wrong tools. For me, using ratio and proportion to track inventory also calls for a vent. It is a situation where an inappropriate mathematical tool is being used to do a real-world task for which other tools are available and would certainly be used.

As in the previous problem, a change in the setting will fix things and still retain the desired math.

A recipe for making 3 quiches calls for 4 eggs. A pastry shop wants to prepare 24 quiches using that recipe.

How many eggs will they need to make 24 quiches?

The Cargo Truck

(Blind spots.)

Use the following information to answer the question below

A total of 10 packages are arranged in the back of a cargo truck as shown in the diagram below. One large package has the same mass as two medium packages. One medium package has the same mass as three small packages.


How many small packages need to be loaded onto the right side of the truck to balance the load?
A.    8
B.    9
C.   12
D.   13

I had more fun with this problem than any of the others because it illustrates one of the hazards of imposing a real-world setting. The setting has blind spots –– properties that are extraneous to the math task, but which provide alternate and more reasonable approaches to solving the actual real world problem. And when this happens it can turn a good math task into an annoying one.

For the Cargo Truck problem, the real-world setting is a cargo truck loaded with packages of differing masses. The real-world problem is to balance the load. The solution method is imposed by whoever set the problem: add more small boxes to the load.

In real-life, the most obvious way to fix an imbalance would be to reposition the packages already in the truck. (And if that were permitted, then there is an answer where no extra packages are needed.)

Also, a real-life cargo company would know the masses of the individual packages rather than their masses relative to each other. Knowing the individual masses would lead to a different approach to the problem. (Also, this raises the "cart before the horse" objection: it is not clear how the company would obtain the relative masses without first knowing the actual masses of each package.)

The diagram itself could be misleading for some students. It suggests that there is not enough space for the extra small packages that are required as ballast.  (Thirteen are required unless you reposition some packages, but we have already discussed that).

On the other hand, the imposed solution method involves some interesting math, and it worth finding a setting, real-world or not, that works.

Here are two different fixes. This first one retains the notion of balance:

There are three different types of boxes. All green boxes weigh the same,  all blue boxes weigh the same, and all orange boxes weigh the same.

These scales balance. These scales balance.

How many more orange boxes are needed on the right pan to make the scales balance? 

(You cannot move the boxes on the left.)

The second modification drops the demand that the setting must be 100% real-world. This setting uses the concept of money, so it should be familiar to the students, yet different enough to signal that the question is not trying to pass itself off as truly real-world.

A country uses three different coloured coins. Green ones are called buckazoids. Blue ones are called halfzoids. Orange ones are called mini-zoids.

Each buckazoid is worth two halfzoids.
Each halfzoid is worth three minizoids.

Mary and John each have the money shown:

Mary's  money. John's money.

Their mother has promised to give John enough minizoids so that he has the same amount of money as Mary.

How many minizoids should their mother give to John?

Am I against using real-world contexts? No. but, but sometimes they can produce really bizarre questions. Paradoxically, extensive use of real-world settings can cause a dissociation between math and the real world.

For more on this, see this post by Nat Banting and this Globe and Mail column written by Sunil Singh a few years ago.

Problem Sources

The Diner Menu is due to Cathy Yenca,  who posted the question in the sequence of articles by Dan Meyer about pseudocontext in math problems. There are many more examples in those articles.

The Cargo Truck is a question from the sample grade 6 Alberta PAT.  Alberta Education considers this question to be illustrative of the more challenging ones in the PAT.

The other problems are rewordings, paraphrases, or amalgams of questions that I found on worksheets and other resources posted on the web.

Thursday, 14 December 2017

Math fair workshop at BIRS

Every spring for the past 15 years there has been a 2-day math fair workshop at the Banff International Research Station. In 2018, this wonderful event runs from the evening of April 27 until noon April 29.

The workshop is all about how to run a math fair that emphasizes puzzle solving with lots of  interaction between the math fair visitors and the student presenters. The workshop participants will primarily be teachers, including some who have organized highly successful math fairs in their schools. More details about the workshop can be found here. I hope you will consider coming.

Some of the participants at the April 2017  workshop

The workshop sessions are held in the TransCanada Pipelines Pavilion. It's a superb venue, with the all the capabilities and facilities you would expect from a leading research institution. 

TransCanada Pipelines Pavilion. Photo Courtesy of The Banff Centre.


Here are some examples of the sort of puzzles that we will be working with. These happen to involve arithmetic operations, but that is not obligatory –– all that is required is that the puzzles be mathematically based. 

Fill in the digits

(a) (b)


(a) Put the digits 1, 2, 3, and 4 into the squares to make a correct sum. Use all four digits.

(b) Put the digits 1, 3, 6, and 8 into the squares to make a correct sum. Use all four digits.

(c) Put the digits 1 through 7 into the squares to make a correct sum. Use all seven digits.

The following two examples are both based on the same idea and show how puzzles can be adapted to challenge students at different levels.

Crosses and Sums

In the cross above, the numbers from 9 through 12 have been placed in the squares in such a way that horizontal and vertical sums are the same.

( 10 + 8 + 11 =  9 + 8 + 12. ) 

In each of the crosses below, the squares have to be filled with all of the digits from 1 through 5.

(a) (b) (c)

In each of (a), (b), and (c) put the remaining digits from 1, 2, 3, 4, 5 into the empty squares to make the horizontal and vertical sums the same.


In the following figure, the digits from 1 through 7 have been placed in the circles so that the sums along the lines are the same:

1 + 7 + 6 = 2 + 7 + 5 = 4 + 7 + 3.

In each of the following, the circles have to be filled with the digits from 1 through 7. In each case, three of the circles have already been filled.


(a) Place the numbers 1, 2, 3, and 7 in the circles so that the sums along the lines are the same.

(b) Place the numbers 4, 5, 6, and 7 in the circles so that the sums along the lines are the same.

Both the Crosses and Sums and the Spokes puzzles are variations of the Spoke Sums puzzle from our math fair website, which is in turn a simpler version of a much older puzzle Henry Ernest Dudeney and/or Boris A. Kordemsky. I don't know who originated the puzzle –– during that era, puzzlers frequently "borrowed" from each other without giving credit.

For more examples of math fair puzzles, visit our SNAP math fair website.

If you are planning to come to this year's workshop and have some favourite math-based puzzles or games, please bring them and share how you have used them in your teaching.

Friday, 10 November 2017

The no-calculator part of the Alberta PAT

Under pressure from people who want the mathematics curriculum to revert to an earlier age, the Government of Alberta introduced a fifteen minute no-calculator section as part of the grade 6 Provincial Achievement Test. The results for 2016 were not spectacular, unless you think mediocrity is worthy of applause.

Because of my association with the SNAP Mathematics Foundation, which promotes the use of math based puzzles in the K-9 classroom, I am very interested in K-9 math education.  And I am dejected when I read that our students are doing poorly.

I have recently taken a look at the grade 6 Math PAT. It did not leave me feeling warm and fuzzy.  The test by itself cannot account for the less than stellar results, but I do have some quibbles.

Part A questions

The Alberta Grade 6 math PAT has two parts. The questions for Part A, the no-calculator part,
are described as being of low cognitive difficulty. For example: 

It is difficult to believe that a typical grade-sixer would stumble over questions like these. But apparently a lot did: almost 35 % failed to meet the acceptable standard.


Are you still unwilling to jump onto the back-to-basics bandwagon? 

When the results of Part A were released, David Eggen, our education minister, reacted like this:
And there it was. Boom. Big place for room for improvement for basic skills
BUT . . . He also said that he expected to see poor results on Part A of the test.

I like what David Eggen is doing with education, but I didn't expect that last comment.

Permit me to digress a bit.  A few years back, my gas company sent consultants to do a free inspection of my heating system.  You can guess what happened: my system was found to need rehabilitation, and they tried to convince me to buy an expensive high-efficiency furnace.

In case you somehow miss my point, I reacted to Part A as if Alberta Education was trying to sell me a costly high-efficiency basic math curriculum.

What is the test like?

The best way to understand what a test is like is to try it yourself. That means you should actually work out the answers, not just look at them and judge whether they are too hard or too easy for a grade six student. Here's a sample test you can download and try:

Ted's Sample Part A test

It should take less than fifteen minutes. Be sure to read and follow the instructions to the letter.
The time you take to check your answers should be part of the 15 minutes.

(My sample test is a very slight modification of the one provided by Alberta Education.  Their sample test may be found in the link at the end of this post.)

The full PAT has two parts. The questions for Part A, the no-calculator part, have no context. They are arithmetic computations. The questions for Part B are placed in "real-life contexts," which is to say that they are arithmetic computations dressed in real world clothing. (To call them real-life is a stretch. You're not fooling the kids, ABED)

Sad to say, the grade 6 math PAT sample questions quite accurately portray the math world I lived in when I was in grades 6 and 7.  Sixty-five.  Years.  Ago. 

I have met many K-6 teachers through our SNAP workshops, and the math environment that they create in their classrooms is not at all like the one evoked by the Grade 6 math PAT.  (This is a good thing! The effect of my elementary math education was that, afterwards, throughout all of high-school, I habitually brushed off math as a possible source of interesting material. I certainly did not anticipate that math would become a major part of my life. )

To be candid, I'm not an "evaluation" expert. But my intuition suggests that any gulf between the math world of the classroom and the math world of the test will negatively affect the PAT outcomes.

The format of Part A

The heavy promotion of Part A as a no-calculator test suggests that it is a pencil and paper exam. Well it is, sort of. But, like Part B, it is essentially a fill-in-the-bubbles test. It is what you would get if you transferred a computer-based test to paper, with the students shading in the bubbles by pencil instead of mouse clicking the appropriate radio buttons.

Apart from the possibility of grading rough work (and I don't know if that happens), the main difference between Part A and Part B is that Part B is multiple choice. In Part A, students get to enter a numerical answer. 

Students are expected to answer each question in Part A using 
two different formats. 

First, by filling in boxes with written numerals,  and then by shading in 
the appropriate bubbles. The answer template for each question is as shown 
to the right. (In my sample test, I provided only the boxes, because the students were able to complete the bubble portion after the fifteen minutes.)

The numerical answer has to be written in the set of four boxes as follows:
One digit per box, beginning in the left-hand box.
A decimal point, if needed, goes in its own box.
Unused boxes must be left blank.
So the "correct" way to fill in the the boxes for the question

        "What is 7 ÷ 2 ?" 

is as follows:

Hmm. What about the smart-ass student who decides to write the answer in one of the following ways:

(By the way, what is an unused box? If it is unused is it not already blank??)

The bubble portion is to enable machine grading. 

The instructions for completing the bubble portion are:
You may fill in the bubbles below each of your answers as you do the test; however, you may also fill in the bubbles after you have completed both Part A and Part B of the Grade 6 Provincial Achievement Test and your teacher has collected your test booklets.
So the completely finished answer should look like this:

I'm not sure if any of the students wasted part of the 15 minutes filling in the bubbles. The box and bubble answer templates for all fifteen questions are on an answer sheet separate from the examination booklet. 

About learning the basics

Children do need to learn the basics –– but you and I will probably disagree about what the basics are. Students should be fluent with the basics, but you and I might disagree on what fluency means.

The Part A questions were easy, and the grades should be higher. But I'm not sure that fifteen isolated numerical responses to Part A reveal the extent of a student's ability.  And, as I intimated above, I have the uncomfortable suspicion that Part A was intended to prove incompetence rather than competence. 


Alberta Grade 6 Math PAT Bulletin

Saturday, 21 January 2017

Changing percent of students at each PISA math proficiency level

This is a graphical summary of how Canadian students have trended in the triennial math tests offered by the PISA/OECD consortium.

It is not so much about their international standing, but about how their achievement levels have varied over the five rounds from 2003 through 2015.

For each participating region or country, PISA publishes how the students’ grades are distributed across seven different proficiency levels. The levels are the same for all jurisdictions, and they have remained the same for the five rounds since 2003.

The grades defining the boundaries of the proficiency levels are shown in boldface. (The ones in parentheses are what you get when the PISA-assigned grades are converted in a reasonable way to grades out of 100.)  The yellow and blue levels identify low performing and and high performing students. The interval widths for each of Level 1 through Level 5 are the same. PISA does not define the widths for the lowest (less than Level 1) and highest (Level 6) proficiency levels.

The table below summarizes the distributions for Canadian students for each of the five PISA rounds.

In effect, each row of the table is an estimated probability distribution. For example, if you were to randomly choose a Canadian student in 2003, the probability that he or she would have achieved Level 5 is 14.8 percent.

Actually, the reason I wrote this post was to try to figure out how to use slides in blogger -- I think slides give a more vivid visual depiction of the changes than a table does.

The following slideshow displays the table as a sequence of histograms. There is some concern that Canadian students are trending in the wrong direction. Perhaps this graphical representation may help you decide if the trend is cause for alarm.