Friday, July 29, 2011

Messy Mathematics

This post is really a vehicle to get a comment that I received on my blog more face time. I thought the potency of the words could not be ignored. It was in response to my post entitled Measuring Roots.

To get the full story, first read that particular entry. Basically, I was reliving my encounter with a small boy where he challenged me to answer a square root problem. He let the answer slip prematurely, and quickly rephrased his question. It was very obvious that he had the answer in mind before the question had begun.

Using this idea, I wrote:

"For students, no matter how young, math begins with an answer. You then form a question, jeopardy style, to help disguise the number. In this case, the child's thinking shone through because he leaked his answer prematurely."

If you read on through the post, I refer to this phenomenon as a problem, and I do not agree that answer-first mathematics is a positive thing. (I have since changed the wording in an attempt to make my personal views more apparent.) Shawn Urban (@stefras) posted the following comment on the post; it is a impassioned statement from an educator that we all should read:

"I stopped reading this post as soon as I read the point that, for students, math begins with an answer. (Don't worry, I plan to read the rest of the post. But I needed to respond to this; it is so mind-blowing.)

I learned under Dr. David Pimm of Open University (UK) and the University of Alberta (CAN) during my Diploma in Math Education studies. He argues that math begins with a question; in fact, it does not exist until a question is asked. All the demonstrating and lecturing about math in the world does not involve math until a mathematical question is asked.

This discrepancy is very revealing. It tells us what math is and what math education is. Most students learn to expect math questions and problems to be short, quick, to the point, solvable and structured around "clean" answers (often related in some way to integer components). They anticipate the answers before they anticipate the questions. I am not sure if they even consider the math.

I wonder what they are really learning? Is it math? What to them is math? Is this why so many students are so disconnected with math and why they are proud to have failed it and ashamed to have aced it? After all, from their perspective, if answers they anticipate before math, what have they aced?

I think we have done students a great disservice if they ace math in elementary, secondary and even tertiary school without ever actually learning that math is all about the question, the quest and struggle to tackle it and the discovery of pattern that limits to (an) answer(s). They completely miss the point and the empowering strength of math process and pattern. And in the end they really have nothing to use in their lives beyond the "math" lesson.

So, why do they need to learn this? That question makes so much sense now."

What view of mathematics do you subscribe to? Do you advocate "clean" answers, or embrace the messy nature of mathematics. Something to think about and reflect on. 


Thursday, July 28, 2011

Measuring Roots

I stumbled upon the "root" of this activity late in the school year after I had already taught the unit on radicals and their approximate values to my Grade 9s. I modified its purpose, but the original framework is credited to John Scammell. (@scamdog) I found the concept to be a fairly easy one for the students to grasp once the identity of a root was explored. Students know what a square root is. In fact, I was challenged by a 7 year old boy who I was babysitting just the other day. His older sister--an 8 year old genius--was obviously giving him a crash course in radical mathematics. She had explained to him that square roots can be presented as a problem. He challenged me with this:

"What is the Square Root of 3? I mean, what is the square root of 9?"

This statement is revealing for two reasons. 1) For many students, no matter their age, math begins with an answer. You then form a question, jeopardy style, to help disguise the number. In this case, the child's thinking shone through because he leaked his answer prematurely. 2) There are only certain numbers that can be square roots. This understanding lingers far after they have entered the world of algebra. The child immediately branded the square root of 3 as a false question and quickly changed it. After all, it would be ridiculous to have the square root of three; that is almost as crazy as asking for the square root of 10, or 7. 

The first revelation stems back years into the way mathematics has been taught and viewed. I would hope that every entry on this blog aids in the examination of this problem. This specific entry's focus is on the second problem:

How can teachers of mathematics get students to see radicals as numbers with a set value?

Young students establish a list of numbers that are acceptable square roots. The list includes: 4,9,16,25,36... Ironically, they often leave 1 off this list. I found that linking this established list to the values of all other square roots helps students to see their numerical value. I link them together using the Pythagorean Theorem, the Spiral of Theodorus, and a simple 6 inch ruler. 

I do this activity in pairs. This keeps both parties invested in the activity, and also cuts down on the scissor time. First I distribute each group a piece of card-stock, a 6 inch ruler, and a piece of basic printer paper. I give them basic instructions on how to construct the Spiral of Theodorus. I also take the opportunity to give a brief history lesson and examine the intricacies of the shape. Instructions on constructing the spiral can be found on @scamdog's blog here. I also use the SMART board to show an interactive version of the spiral. My spiral is shown below.

After the spiral is constructed, it is cut out into its individual constituents. Make sure that each triangle has its measurements on the interior or they will be lost when separated. The students use the Pythagorean theorem to label the sides until they find the pattern. When all their triangles are ready, they label one side of their new ruler with the inches marks. 

Right away I get the students to compare the accessible square roots with the ruler. If they put the side lengths of sqrt(4) and sqrt(9) against the ruler they see that they actually measure 2 and 3 inches respectfully. Actually placing various sides against the ruler shows which values the square root falls between. 

After some initial investigations with the ruler and triangles, they begin to mark off the square root's lengths on the other side of the ruler. As they count up, the ruler markings align at spots like 2, 3, 4 etc. I have found that after root(12) the markings become too crowded when 1 inch is used as the base unit. After the markings are done, the ruler may look something like this. 

Now that the relationship between "irregular" roots and the number line is established, the teacher can prompt all kinds of constructions and investigations:

There are no markings before 1. Do any square roots exist here?
How could those roots be constructed?
Where are all the points where both sides synchronize?
How can we use these synchronization points to estimate roots?
Why does the distance between the roots narrow as they get bigger?
How could we use triangles to construct sqrt(20)?
How many new square root lengths can you construct with your current triangles?

This is only the beginning. Students can play with right angled triangles to construct radical lengths. When they begin to take on a concrete measurement, having sqrt(43) for a final answer won't seem as impossible to them. The square roots will take their appropriate spot on the number line--alongside the counting numbers.

Square roots are tricky numbers. They cannot be drawn, but can be constructed. Most can't be rationalized, but can be estimated. Sometimes they don't exist but we use them anyway. The Radical Ruler is a great step toward understanding the true, numerical identity of radicals. 


Friday, July 22, 2011

Jets Logo Not So Golden

The experienced hockey fan will recognize the title of this post as a reference to Bobby Hull, who was a legend with the Chicago Blackhawks before coming to the Winnipeg Jets in 1972. He became one of the biggest draws in the fledgling WHA; his blinding speed and rocket shot gained him the nickname "The Golden Jet" league-wide. The Winnipeg Jets folded in 1995, and only returned weeks ago. Today was the unveiling of the new Jets logo; an event that I refreshed the same page for hours to witness. There had been numerous leaks over the past few weeks, but nothing had been confirmed. When the page loaded at exactly 5:00 PM EST I saw this image:
Take a moment to examine the symbol. What do you notice? Does anything pop out to you? For me, the absence of text was alarming. I am used to the traditional logo with the word "JETS" embossed across the front. After a few minutes of looking, I decided that I didn't like it. Maybe it was me clutching to the traditional logo, or grasping for the memories of Selanne, Hawerchuk, and Hull. Nostalgic feelings aside, something just seemed...unbalanced. 

I voiced two main concerns:

1) The exterior circles seemed far too bulky. 
2) The Jet seemed vertically stretched. 

I decided to test my observations with the Golden Ratio. Maybe it is only fitting that the "golden" analogy fits in this case as well. The Golden Ratio is a naturally occurring ratio that is supposed to be the most appealing to the human eye. Originally born from Fibonacci's famous rabbit problem, the ratio (denoted by the Greek letter phi) can be approximated by dividing consecutive numbers in the Fibonacci Sequence. When the dust settles, it is approximated as 1.618. This ratio exists extensively throughout the plant and animal kingdoms (including humans). 

I decided to test my hypotheses of bulkiness and stretchiness against the mathematics of ancient Italy. I printed off a logo and began measuring lengths of numerous sides. It is important to understand that the size of my print out doesn't matter because we are interested in the ratio. Any enlargement or reduction would produce a similar image with an identical scale factor thus preserving the ratios. The diagram shows the logo with a black dot at each vertex of the design. I took the measurements between each dot tracing the jet. Along with this, I took the diameter of the various circles involved as well as the length and wingspan of the jet. I also measured the horizontal distance across the red leaf. 
I first addressed the issue of the jet's vertical stretch. If you took the length of the jet and compared it to the diameter of the circle (minus the grey strip) you would get a ratio of, you guessed it, 1.61. This is not a surprise because every graphic designer knows of the golden ratio. It seemed as though the jet was in perfect harmony with the circle, but I continued.

The vertical distances all seemed to follow the golden ratio, but the horizontal distances all fell a little short. First off, the wingspan was compared to the inner, white circle's diameter. I thought it would have been a genius move on the behalf of the designers to have the length of the jet correspond with the blue circle and the width correspond with the white, but this was not the case. The ratio between the white diameter and wingspan was 1.83. too large. In order to get closer to phi, the wingspan would need to increase. This strengthened my initial feeling of vertical disproportionally. 

Then it struck me, what if the red leaf next to the wings was there to compensate for the necessary skinniness of the jet (to make it as realistic as possible). When the leaf's width was measured and compared to the white diameter, the ratio was 1.34. So the leaf went too far. In order to comply with the golden ratio, the leaf's width would need to be lessened. I saw this as another "golden" opportunity to make the logo more intricate. 

As the calculations of length continued to lie suspiciously close to phi, and the calculations of width consistently fall short, I began to build a mathematical case for my initial impression of disproportionally. 

The question of bulkiness was addressed with area. I used the measured diameters to get ratios between the areas of the various coloured bands and the middle whitespace designed to fit the jet. The calculations were as follows:

Area(Whole Logo) = 136.85 cm^2
Area(White Inside) = 70.88 cm^2
Area(Blue Strip) = 55.80 cm^2
Area(Grey Strip) = 10.17 cm^2

I figured that the ratio between the coloured strips and the inside was dominated by the strips. This gave the logo the bulky feeling. I took the ratio of their areas:

A(Strips)/A(White) = 70.88/65.97 = 1.07

Miles away from phi. the strips would have to be thinned drastically to reach the ratio. Exactly how much can be figured out algebraically. (I will leave that to the reader if they so please).

Also, the ratio between the total area and the area of the white inside is 1.9. Again, if the strip was narrowed or the white space increased, the ratio would approach a more attractive ratio. According to phi, my impression of bulkiness was grounded. 

Now I understand that there are other factors at play. If the jet was fatter, then it wouldn't look streamlined. Also, the memories of better days of Winnipeg hockey may skew my decision. Even considering those, the fact of the matter is that the logo, devoid of its meaning or significance, seemed unbalanced to me upon first sight. This only opens up a plethora of options for me (or more importantly, my students) to explore. Can we find the perfect sports logo? If it could be created, I would guess that Fibonacci would be making numerous appearances.


Wednesday, July 20, 2011

The "Nearly" in Mathematics

Mathematics is the purest form of science, or at least that is what they tell us in university. This ideology carries over into the school staff; it wasn't long until another member of the staff referred to me as a "math guy". As much as this label is also self-imposed, I still struggle to understand what it means. The labels "english guy", "phys-ed guy", and "science guy" all persist within the building as well, but there is something that about the title of "math guy" that gets me.

My friends and family quickly shunt all calculations to me when needed. When I ask then to justify why I must calculate the tax or tip, they simply reply that I am a "math guy". The public's perception of mathematics is basic arithmetic. I am a "math guy" therefore I am a human computer. In actual fact, I am no such thing; I have never been excellent with those types of calculations. Math has an aura of exactness about it. It is designed to calculate ideal situations with pinpoint accuracy. This image is also at the heart of the Math Wars. Traditionalists believe that the focus belongs on computation--the need for efficient and correct procedures. The reform side values the very diverse process of doing mathematics. This is not the topic today, and so I will digress, but it would be interesting to see someone who believes fluency is the highest aim of math class struggle with the wiggle room probability and statistics afford.

When setting my classroom up last week, I found a package of wooden dice. They were crudely painted and obviously bought in large numbers. Almost by accident, I opened up a can of worms I never wanted to; I wondered if these poorly constructed dice could possibly be fair. It was not long until I had university textbooks, a generous amount of scratch work on scrap paper, and a book by Joseph Mazur out to help me make my decision. The activity only reinforced the beauty of mathematics.

In an ideal world, if I rolled the die 6 times, each number would appear exactly once. If I rolled it 60 times, each would appear 10 times. In essence, I am calculating the expected number of times that each side should show up if a dice is tossed "n" times. Let's say I toss the dice 100 times (n=100) then I would expect each side to turn up 16.67 times. Immediately we see the "nearly" creeping into the math. I can't possibly roll a "3" 16.67 times; each frequency must be a whole number. But in statistics, this is acceptable. The question now becomes:

How many times would a certain number have to appear for me to declare the dice "unfair"?

Certainly, even the most rigid mathematician would not call a dice unfair because a "4" was rolled 18 times in 100, when it should have been rolled 16.67? What about 19, 20, 21, 22? 30? As the process continues, the inexactness of being a "math guy" begins to shine through. Keep in mind, there are very strict laws and formulae that govern the chances of rolling a specific number, but they still only give us chances.

Many of us are familiar with the normal curve shown below. Although rolling a dice is a discrete event (meaning it can be separated into trials) and the curve is based on continuous data, we can still approximate percentages with it.

First we need the mean (average) and standard deviation (distribution from the mean) to calculate values. If 'n' is the number of times you roll the dice, 'p' is the probability of rolling a specific value, and 'q' is the probability of not rolling that value:

Mean = n*p
StDev = Sqrt(n*p*q)

In this case, the mean amount of times we expect to see a value is 16.67, and the standard deviation is 3.73. We use this data to calculate probabilities that a certain number of rolls would occur on a fair dice. I will not go into the calculation of Z-scores and the curve, but any basic statistics book would include that. You can find the table I used to calculate the probabilities here.

It turns out that 95% of the time the frequency will be between 9.21 and 24.1, and 68% of the time the frequency will be between 12.94 and 20.4. The frequency of any number will be greater than 25 only 1.8% if the time, and greater than 20 only 15.39% of the time. These are exact measurements, but the mathematician's job is more challenging than this:

You still need to decide what percentage is acceptable.

Maybe your mark is 20. Surely 85% is certain enough to judge the dice unfair? Maybe you are pickier and will only pronounce a dice unfair when a side appears 25 times. This is only supposed to happen 1.8% of the time with a fair dice. The 'nearly' begins to show.

Now there are other ways to run experiments. You could count the dots on each roll and use that data to make a continuum. In that case the mean number of dots would be 3.5 per roll. How many times would you have to toss the dice, and how far away must the average be to declare the dice unfair with this experiment? It would be a wonderful activity to do with a class of statistics students. It provides them a link between experimental probability and statistics, but, more importantly, teaches them the subjectivity involved in mathematics.

I would introduce them to the normal curve, and then give them each a die. (Some could even be altered to be unfair). I would simply ask to justify if their dice was indeed fair.

It may take a "math guy" to justify the fairness of a dice, but it takes far more than computational fluency to achieve that goal. It just goes to show that sometimes the most innocent wonderings become the richest mathematical experiences. Let's encourage our students to wonder as well.


Saturday, July 16, 2011

Manipulative Revelation

I completed school before manipulatives were in vogue. I am still not sure that they are today (where I teach). I know that my department's manipulatives are locked up in a cupboard. In this Potter-like clandestine state, I didn't even learn of their existence until the end of the year. I was moving classrooms, and found a pile of algebra tiles that the previous teacher had left behind. I didn't discover that I had manipulatives available to me until, ironically, I inquired where I could dispose of this rather large supply of algebra tiles. When I opened the doors of the cupboard, my eyes were bombarded with a vibrant display of primary colours; it is the bright reds, blues, and yellows that initially deter high school students from using these instruments. It creates an aura of immaturity and frivolity. They are coloured in such a way that one may expect students to pack their algebra tiles up neatly and proceed to recess or nap time. Kindergarten students play with blocks; algebra deals with "big-kid' stuff--no use for toys.

Initial impressions aside, the colour of the tiles was not the only thing deterring me from incorporating them into my classroom. It seemed strange to me that someone would try to teach the concept of variability with objects that were horribly static. My "x" block was not "x", it was a certain area just like the other tiles. There is no way to concretely detail to the students the unknown status of such tiles. After all, aren't the tiles designed to give a concrete nature to variability? Simply telling the students this defies the very purpose of the tool.

Secondly, I never understood the connection between the tiles and why they represented the values they did. That isn't completely true, I knew that it was the area of the tile gave it its name. The solution to this problem lies in the way that factoring is introduced to students. It needs to be linked to the factors of numbers, their greatest common factor, multiples, etc. For this geometric tool to be effective, we need to begin to see numbers as geometric bodies. It was twitter that gave me the solution to my 3 hang-ups with algebra tiles; this allowed me to explore some of their interesting uses.

Before we go on, I will introduce the tiles to those who have never seen them. (Chances are they are locked in a cupboard somewhere). There are 3 basic tiles:

1) '1' tile
2) 'x' tile
3) 'x^2' tile
Pictured above, the tiles get their name from their dimensions; it is essentially a measurement of their area. The shorter dimension has a length of 1, the larger has a length of 'x'--an unknown. When you create the 3 possible combinations between the 2, you get:

1*1 - The 1 Piece
1*x - The x Piece
x*x - The x^2 Piece

Each piece has an additive inverse (a negative) which is represented by the same dimensions but a different colour. To get more familiar with this tool, go to:

The site above has forced me to re-think algebra tiles because it addresses their 3 main flaws:

1) You will notice that the blocks are not brightly coloured. They give off a less childish impression. The layout is sophisticated; this goes a long way for a high school student. I understand that this "flaw" is very superficial, but a first impression goes a long way.

2) The elasticity of the tiles allows me to geometrically and concretely show the idea of variability with the tiles. The site allows me to change the dimensions of the tiles while still retaining the same polynomial expression. The site has a built in slider that can increase the length of 'x'. The students can see how the polynomial area changes, based on the value of 'x'. Take the polynomial x^2 + 2x + 1. It is represented below with 2 different values for 'x'. The polynomial name stays the same, but they represent two very different areas.
This feature sets the virtual manipulatives apart from the physical ones. The SMART board allows those students who are tactile to still physically touch and move the tiles, so this element is not lost either. With the power of computers, the tiles have become variable.

3) The idea of area in the tiles has been addressed by the curriculum renewal in our division. Now trinomial factoring is addressed directly after the topics of GCF and LCM and Prime factors. This allows me to introduce numbers as geometric. What does a "perfect square" look like? Does that mean that numbers like 4, 81, and 144 resemble an actual, geometric square? What shape does 8 take? What about 30? Are all other numbers "rectangular" numbers? Can a number be a "perfect square" and a "rectangular" number? Is a square a rectangle? My students are immersed in a geometric concept of numbers. This allows them to bring geometry into the abstract concept of trinomial factoring.

My biggest trouble as a student was the idea of "completing the square". My teacher created a clever acronym that has stuck with me to this day, but I never understood what was going on until last week. That is kind of an embarrassing admission from someone who has a degree in mathematics. I learned to complete the square with a set of rules--meaningless rules. Look how algebra tiles addresses the same topic:

Is this polynomial a "perfect square polynomial"?

x^2 + 4x + 4
A simple re-arrangement of the tiles reveals that this "number" is a "perfect square number". We just don't know which perfect square number it represents, that depends on the value of 'x'? But wait!! Does that mean there are perfect square numbers between the regular ones? Can there be perfect squares with decimals? Try plugging in values for 'x'. Will any 'x' here produce a perfect square? The algebra is becoming an input-output system. Here is the rearrangement:
Now, how could we take: x^2 + 6x + 3 and "complete the square"? Well according to my set of rules we first take the middle number (which for some reason is always the 'x' number) and divide it by 2, then we square it, then we subtract the constant from that number, then we add and subtract the result to complete the square. This works every time? Why am I adding and subtracting the same number? That is a waste of time! Why do we divide it in 2 first? The series of rules can produce correct answers, but does not produce correct understanding. Let's look at the same question geometrically: Complete the square.

well... fill in the gaps.
Here, we see that we physically filled in the pattern until the square was complete. We added 6, so we leave an extra -6 to leave the polynomial unchanged. x^2 + 6x + 3 becomes (x^2 +6x +9) - 6 or (x-3)^2 - 6. We can see the "middle number" as 6, because there are 6 'x' tiles. Why do we divide it is half? Well, we have to split up the tiles between 2 faces of the x^2 tile to make a square. Try other arrangements where you do not divide by 2, what happens? Using this method alongside the rules can shed light on why the rules are what they are. It is also a great test for the teacher to re-learn what they have missed in their schooling.

I have not even introduced the idea of negative tiles; I am still struggling with this myself. Algebra Tiles (especially virtual ones) provide a very powerful learning tool for students (and teachers) when used in the correct way. They can be a gateway to many difficult, and abstract, topics. I am slowly switching camps, and if that means that I have to have recess or nap-time after I am through, both will be welcomed with open arms.


Thursday, July 7, 2011

Odd Factors

I am teaching 5 new classes next year. I am trying not to think of it that way; rather, I am taking it one step at a time. Unfortunately, most of these steps need to be taken during my summer vacation. This isn't the end of the world; I am fairly stationary, and enjoy a mental workout as much as some enjoy time on the beach or in a foreign shopping mall. I began my massive preparation marathon with a unit for Grade 10 Precalculus on factoring. As I dove into the curriculum and textbooks, I found myself actually enjoying the intricacies of the topic...nerdy, I know!

The unit has a very distinct division. Part 1 is a look at numbers and their properties. This examines the relationship between a number, its factors, its multiples, and its relationship to other numbers and their factors and multiples. Part 2 is designed to build skills of factoring trinomials, a far more regimented and abstract topic. A solid base in part 1 will lead to a better understanding of the "rules" for factoring polynomials in Part 2.

Numbers are interesting things. I know a lot about them, and still find times when I am amazed at the way their properties interweave. The curriculum calls for the understanding of prime factors, GCF, and LCM. Topics that can be lectured quite simply. It is a shame that students do not get to play with numbers and discover how factors create relationships between them. About a year ago, I stumbled upon a problem that explores the relationships between factors and their products. It provides an interesting juxtaposition on the typical way of learning about prime factors.

Traditionally, factors are taught with trees--clever, branching diagrams to illustrate the "breaking down" of numbers into their constituents. But we can think of factors in two different ways: a number has a unique prime number factorization (this is known as the Fundamental Theorem of Arithmetic), and every composite number has pairs of factors that multiply to give the original. These factors may or may not be prime. For example:

24 = 2*2*2*3
but also...
= 1 x 24
= 2 x 12
= 3 x 8
= 4 x 6

These are the two concepts of factoring at work. More often than not, the focus in school mathematics is on deconstructing. Take 24 and give me its prime factorization. What are the factors of 24? Our textbook uses the rainbow method where pairs of factors are matched with an arc.
Try using the rainbow method for a prime number? What about a perfect square? What about a perfect cube? This activity is an excellent one to do after the odd factors task. Anyway, after I discuss the two basic types of factors, I switch the focus from decomposing to composing, I give the students this task:

Can you construct me a number with 6 factors? How many solutions are possible?

Immediately the question leaves itself up to interpretation. 6 prime factors? Do they have to be unique? What about 6 composite factors? Is that 3 pairs? What if I mix and match prime and composite factors? My response to all of these queries is, "yes". These questions will help them solve the second half of the problem. When solutions begin to collect on the board, I alter the question a little:

Can you construct me a number with 11 factors? How many solutions are possible?

The same questions emerge, but with a wrinkle. How can you have an odd number of factors in a rainbow? What does this mean? This question takes a while to hammer out. Students can construct a number with 11 prime factors fairly easily, but the composite pairs question is much more difficult. The topic of uniqueness will most likely come up. How many prime factors does 100 have? 2,2,5,5... does that mean 4, or 2? How many pairs of composite factors?

1 and 100
2 and 50
4 and 25
5 and 20
10 and ... 10?

so 9 unique factors in total?

This discussion is very valuable. It leads my students toward the special situation of perfect squares--also a curricular outcome. Odd factors is easy to implement with very little logistical pieces, but the user should be warned: this problem has the habit of opening doorways for divergent discussion. Do not be surprised if you find yourself explaining the existence of an infinite amount of primes. The switch of focus from deconstruction to construction allows students to see the intricacies of numbers and their factors. This deeper understanding should also carry into the topics of GCF, LCM, and Polynomial common factors. Teachers may be surprised with the volume that they learn when they subject themselves to a different perspective on a relatively simple concept like factors.

For the record, I believe that the original problem came from the work of Joan Countryman, but I cannot be sure.


Tuesday, July 5, 2011

The BEDMAS of Broken Keys

It is the end of my first year of teaching, and I am in a reflective mood. The art of "reflection" was one heavily mocked in my professional college. It seemed as though every assignment in the College of Education involved some kind of reflection. Students of other colleges dismissed the idea as elementary. Do something useful, then reflect on it, then reflect on that reflection, etc. The process began to resemble an infinite sequence. It wasn't until the reflections were no longer forced, that I found value in the process.

My year began in chaos. I was hired to teach for a division, but not told where to report until 10 PM the night before staff re-gathered across the city. On 10 hours notice, I went to the school and began my career. They had no classes, students, or space for me. I slowly carved out a niche that included all three. Until this process was complete, I co-taught with 4 different teachers. In this hectic time, I had no time for preparation or archiving. Reflecting on that experience rehashed a very valuable activity I co-taught with a colleague in Mathematics 9.

I was bursting with excitement to try some new things in math class. I was given my first opportunity on the first day of class. We were to start with the teaching of Order of Operations (or BEDMAS, PEMDAS). The acronyms are convenient, but unfortunately suck all meaning out of the process. I decided to take a different approach, got the students into pairs, and handed out a worksheet. It read as follows:

The 6 key on your calculator is broken. Find the answers to the calculations below. Work out which keys to press before trying it on your calculator. Record the calculations that you do.


Most students were excited to see such a simple activity on day 1. For some reason, students feel safe as long as they are permitted to use a calculator. There is a large debate over the use of calculators in math classrooms. One theory insists that the memorization of rote facts is a necessary skill to be numerate. The other recognizes that the virtual cornucopia of available calculators at any given time diminishes the need for rote memorization. It is more important to understand the number system behind the tables, than the tables themselves. Why put our trust in human memory when it proves to be so fallible? If you don't believe this, spend 30 minutes with my dear grandfather; he will remind you of the deficiencies of human memory. The above tangent probably discloses my personal opinion on the debate. I digress...

Back to the task at hand. The questions begin simple. Students break up the "16" in many different ways--mostly additive. This is a great way to start, because correct answers keep curiosity high. The few students who try breaking up "16" into "8x2" quickly discover that their answer differs from the class. This ripple effect begins discussions.

My circulation fuels the fire. What do you mean you got a different answer? Maybe your substitution was incorrect? After checking that indeed 9+7 was equal to 8x2, we had to re-double our efforts. I prod them onto the next calculations, until I hear that student savior from grade 8...

"Do we have to use BEDMAS?"

Jackpot. I now turn the discussion global. What is BEDMAS, and why do we need it? Cue the history lesson. After that is done, I send them back with this knowledge to work out the nuts and bolts of it. You now have a name, now learn why it works. Students use a hierarchy to solve more difficult problems. They are now careful how the represent the numbers. Maybe "9+7" is different than "8x2". The answers are NEVER worked out with working calculators, I instead collect answers from the class and verify with frequency. This helps divergent answers emerge.

I have the students work left-to-right until the topic of BEDMAS comes up. Then they start using operations for their strength. When students begin to emerge as exceptional, I pose further problems:

What if you weren't given a calculator with brackets, could you write your calculations using operations that would work from left-to-right?
What if I also broke your multiplication button? addition button?
What if I banned a Prime number?
Given certain buttons, which numbers can be created?

Playing with operations and their order brings meaning to the memory cue. Certain metaphors may begin to emerge. Brackets are the toughest operations and "beat up" the weaker ones. Addition is the "little brother" of multiplication. Etc. Keep the experimentation alive!

There are some interesting applets that play with broken calculators. These activities do not necessarily enforce BEDMAS, but mingle it with the knowledge of additive and multiplicative factors. One of my favourites is courtesy of

The activity went over very well. I encourage teams of teachers to get together and try exploratory mathematics. The end of the year also reminds me that copious amounts of broken calculators are left for the summer break. The old saying goes that someone's trash is another's treasure. In this case, we can take student waste and turn it into valuable mathematics.