Sunday, September 18, 2011

Shading Squares

I recently finished up a unit on sequences and series with my grade 11 pre-calculus students. The unit is somewhat of an enigma because it contains relatively simple ideas bogged down in complex notation. This coupled with the overlapping definitions makes for a fortnight of rather rigorous cognitive exercise. 

The unit was supported through group tasks as the topics moved along. Arithmetic sequences and series were linked to linear functions through the toothpick problem. Students were asked to arrange toothpicks into boxes and record how many toothpicks it took to make 'x' number of boxes. Their results were extrapolated and tied to variables from the linear functions notation. From there, I introduced the new terms of "common difference" and "term one" instead of slope and y-intercept. The arithmetic portion usually goes smoother than its geometric cousin for two reasons:

1) The members of the sequence are equally spaced in an arithmetic sequence
2) The arithmetic situation avoids large and ugly fractions

I introduced the concept of geometric sequences with a video clip from "Pay it Forward". The clip, found here, used a familiar situation to show the new relationship between terms. Students commented on how quickly the results grew in this new case. One student asked if all the answers were going to be so huge. Before I could respond, another student retorted that very large numbers are better than very small ones. With an inward smile, I seized my chance to begin the shaded square. 

I handed out a simple sheet to every student. On it were 2 squares--one on each side--and room for a few calculations below. I put three steps on the board, modeled the process for one iteration, and stepped back to let them answer the question. 

1) Cut the unshaded area in half.
2) Shade in one side of your bisection.
3) Repeat.

How much area is shaded with each iteration?
What pattern do you notice between each successive number?

From here, I had to work quickly. Students' pages are only large enough to give 6-7 cuts of the square before they run out of feasible space. I circulate to make sure that the cuts are being made as logically as possible. Most students develop a picture much like this:

The space for calculations is used initially to find simple common denominators, but I found that most students pick up on the pattern quickly. Students begin to list the terms in the familiar sequence notation:

(1/2), (1/4), (1/8), (1/16), (1/32), (1/64), (1/128)...

The pattern then needs to be translated from a "division by 2" into a "multiplication by 1/2". Other problems are presented to work on geometric sequences, and we return to the square a few days later with the topic of series. The question changes slightly:

"What fraction of the square is newly shaded with each iteration?"
"What total fraction of the square is shaded after each iteration?"

We check the sums after a few iterations, but the students soon grow sick of calculating common denominators. The true value of the shaded square problem comes when I introduce infinite geometric series; the concrete nature of the problem brings some sanity to a counter-intuitive situation:

You can add a positive number forever, and never get past a certain point.

I introduce the words "limit" and "convergent" to the dialogue. Will the shading ever stop? Will you ever be able to shade outside the square? Such questions are trivialized to begin with. The shading won't end unless you stop, and of course you can't shade outside the square if you only cut remaining area. The answer, almost logically, converges to a sum of 1. 

The problem opens a window of understanding into infinite possibilities. Change the problem; allow students to carry out different ratios and estimate their sums. The formula is always there to check answers. If you give the formula without explanation, it becomes a crutch--not a tool. Try dividing the remaining area into 4 pieces and shading one. Carry out this method to get an image like this:


My students immediately noticed that the sum must be more than 1/4 and less than 1/2 based on solid and empty quadrants. What will your students notice? What if the cuts with the 1/2 variation were made vertex to vertex creating identical triangles? I had several students do this. This involves more complex algebraic manipulation. 

It was shocking to see my students' nonchalant response to the questions of convergence. The situation provides an excellent concrete base for a difficult topic. It also provides a loose enough environment to foster group discussion and peer collaboration. The idea of infinity is intriguing, but difficult to grasp. The shaded square is an excellent tool to rationalize the fact that we can add forever and never break (or reach) a certain point. 


Sunday, September 11, 2011

All Factors Considered

I have only been teaching for 2 years, but am already beginning to encounter the recursive nature of the profession. I have had several repeat classes in my 4 semesters of teaching, and they require the achievement of the same outcomes. This does not bother me, in general, because I am excited to see the improvement in my teaching. There is one unit, however, that has already frustrated me. Its ability to sabotage creative exploits is unrivaled throughout the mathematics curriculum; I am speaking of the unit on polynomial factoring. 

The topic was taught in isolation of numerical factors until this year. A rearrangement placed the topic--correctly--after the topics of prime factors, greatest common factor, and lowest common multiple. Students spend the first week or so working with familiar numbers and dissecting them into their constituents. As Patrick Honner (@MrHonner) put it:

"Factoring integers into primes is fun, mysterious, compelling. It's like a number's genome"

This much is true. I went through the integer portion with a high degree of creativity and choice for the students. They were presented group tasks which involved the investigation of patterns in factors and multiples. I left the classroom every period feeling fulfilled. Students inquired after class on extensions of problems, and all was right in math-land.

Enter polynomials.

All intuition that the students had with numbers was immediately lost despite my best efforts to retain the connections with various methods. I showed the organization of base-10 blocks and paralleled that to algebra tiles; We factored variable expressions and subbed back in to check solutions. It only took a couple days to realize that the students had lost track of why we factor; they had become completely reliant on empty algorithms. 

Frustrated, I took the issue to twitter. What was an innocent plea for help turned into a blizzard of opinions--far too many to place in a post. The fire fight was started with a single tweet:

"Anyone in #mathchat want to defend teaching of factoring in Grade 10?"

My goal was to gauge the frustration of colleagues. Many immediately assumed that I was stating that the process of factoring was useless mathematically. This was not my intention, but misunderstandings often creep into play when you are limited to 140 characters. I understand the mathematical importance of the process of factoring; I simply wanted to know if other teachers were able to infuse meaning into the process. My sense from the responses is that not many, if any, do.

I was given some very useful links to explore, but the professional learning will have to wait until I must deliver another factoring unit. As I looked back personally, I saw no improvement on this topic. I had begun using some more technology, but it still delivered the same lessons. Although most of the responses were reassuring me that it is sometimes okay to provide empty learning to students, and that some things just need to be memorized and repeated, I was not deterred. If factoring has such a defendable place in mathematics curriculum, then why is it treated as a "means to an end"? (That phrase was use many times in the responses)

I synthesized the feedback and came up with some improvements that I can make next time I teach the topic. After all, teaching is asymptotic--I never quite get to mastery. 

1) Dynamic algebra tiles
These allow the teacher to arrange the various tiles and then set the 'x' value to various integers. If students are introduced to the various multiplicative relationships beforehand, they may create a stronger connection. The National Library of Virtual Manipulatives contains the best set I have seen yet. 

2) Give students a "what"
Often students ask when they are going to use the skill. Unfortunately, they are usually not ready to know the sophisticated methods that use the knowledge. Students have not seen quadratic equations, so listing it as a "why" would not make sense. I want to begin to give students a "what". What is factoring? It exists, essentially, as an inverse to distribution. This framework should fit perfectly into their math autobiography. Opposite operations exist throughout their education--addition/subtraction, multiplication/division, etc. @tweetpmo's analogy of numbers like clothing was very interesting.

"Organize by sorting, folding, and stacking (factoring). Same items just presented in another way. Ball of clothes = unfactored"

This whole experience challenged me to look back on how far I have come on other topics. Some of my historical methods almost warrant an apology. I do not think that accepting excuses is the answer. Sure, maybe humans rely on memorized algorithms in many areas of life, but very rarely are they executed with no understanding of what they are doing or why they are doing it. My factoring unit may never achieve the goal of complete understanding, but I am committed to consistently improving it. 


Monday, September 5, 2011

In The Footsteps of Gauss

I like to introduce each topic with a task or activity. These do not necessarily have to be long, but should activate mathematical thinking. The idea has slowly evolved for me throughout my short career. They are the amalgamation of the ideas of a "motivational set" and discovery learning. I felt that both components are positive things to include in a math class, but both had severe implementation problems.

The motivational set is far too passive. In my college, a picture, story, or conversation could serve as a motivational set. It was essentially a transition tool that was completely void of any mathematics. Every lesson begins with the same routine whether it be a national anthem, attendance, or a short time of homework recap, but each learning experience needs to begin with an active brain. I found that the purpose of the motivational set was important, but needed a stronger method to get brains engaged in the day's learning. 

Discovery learning has become the hottest issue in mathematics education today. (Barring, possibly, the Khan Academy's claims of educational reform). I was introduced to several activities where the mathematical process is reversed and students play with the math in order to create meaning through manipulation. I noticed a gap widening between two types of high school teachers: there were those that wanted to implement inquiry, but didn't have the class time, and those who didn't want to implement, but gave time as their limiting factor. Although one of these groups consists of liars, the problem is the same. There needed to be a time efficient way to implement discovery learning into the classroom. Prompts and tasks keep a very strong curricular tie, and only take a fraction of the class time. Many of them are well known questions.

This past week, I introduced the topic of Arithmetic Series to my grade 11s by asking them to sum the numbers from 1 to 100. This activity was paired with the story of C.F. Gauss. In elementary school, Gauss was asked the same question and found an ingenious pattern to solve it. My students were told that he found a pattern, but were not given a hint on what it was. They were to get into pairs and try to find a shortcut to generalization.

This is where the criticism comes in from traditional teachers. Why, if I was so concerned with lack of class time, would I send students away for 15 minutes to discover a pattern that may be far over their heads? I have many reasons for this, and all of them were apparent in various students as they attempted to follow in the footsteps of Gauss. 

1) It puts the students at the center of the learning
Immediately, the students took the initiative. They formed their groups and began to converse. They began bouncing ideas off each other, while I circulate and sit in on various group discussions. When I visit, it is not to re-direct thought, but to force them to recap their thoughts. These recaps reveal holes in their logic (if they exist), and serve to trace their progress. 

2) It hooks into prior knowledge
Students bring different math experiences into different tasks. Some have seen a similar problem, and begin to take the role of instructor in their group. One student knew that the numbers needed to be lined up in order to create pairs. Although her memories were foggy, she used this starting point to lead her group into valuable learning. Enough of the previous problem was forgotten, that the group still needed to justify their process. Even if they have not seen a problem before, many numeracy skills are involved. The ideas of T-charts and Odd/Even sums were tossed around by several groups.

3) It builds strong peer discourse in the classroom
Students will gravitate to those that are like-minded, so after the groups have the opportunity to solidify their answers, I call the whole class back and recap all solutions. This begins powerful peer discourse. Students will ask for explanation, and, as the teacher, I never provide it. It it the groups' responsibility to defend their work. I simply act as a moderator if the conversation stagnates or becomes exhaustingly cyclical. 

4) The struggle makes the solution more meaningful
The naysayers are correct; not every student will "discover" the math exactly as the book or curriculum intends. Not every student figured out that the numbers from 1-100 line up nicely into 50 pairs of 101. Many tried a different route. When the group work is discussed as a class, and an elegant solution revealed, everyone shares in the success. When the method of pairs was discussed there was an audible sigh and gasp throughout the class. The solution meant more because they were invested in the problem. One girl even said, "wow, that's a cool way to think about it!".

5) Students may stumble upon an alternate learning
Just like some may not get any answer, some students may come up with powerful answers that you may not expect. I love these occurrences more than any other. One student called me to his desk during the task and declared that he had found "the" pattern. I quickly corrected him:

"You have found 'a' pattern"

He went on to show an interesting result:
Sum (0-9) = 45
Sum (10-19 = 145
Sum (20-29) = 245
he concluded that he could answer the problem by summing the first 10, and then adding 100 for each additional group of 10. I had a little bit of time to probe deeper. What if you only wanted to sum every other number? What if we started at 15 and summed to 100? How can you generalize your pattern using sequence notation? This discovery was a highlight of the class for everyone involved. The fact that I was surprised meant the world to the student.

Prompts provide a unique hybrid to discovery learning and a motivational set. Finding or designing an interesting problem allows students to begin a topic in control. They also provide an invaluable reference tool for the teacher. Asking them to remember the Gauss Activity reverts them to the meaning of a formula or idea. There is no coincidence why these activities remain forefront in their memories--students will remember the concepts that they had a hand in forming. 


Friday, September 2, 2011

Struggling with Infinity

My fascination with infinity began early on in life. I went to a small private school in Prince Edward Island for my entire elementary school career, and it was outside on the playground where I first tasted the enigma of infinity and the power it held. 

Across the cul-de-sac parking lot stood the swings, slide, and monkey bars; I still remember the first time I encountered infinity under those bars. You see, we had been learning the base-10 number system that day, and my friend Jason and I somehow got into a counting contest of sorts. We began at very small numbers, and gradually cycled through the digits at varying positions until we countered each other with unusually large--and most likely inaccurate--numbers. Trillion, Quadrillion, Bazillion all made appearances until Jason ended the contest with one word--infinity. 

I countered with infinity plus one, but was quickly informed that infinity plus one was equal to infinity. Despite my protests, his knowledge won out. Every number seemed to be imbibed into infinity; I had lost the contest. Needless to say, I initiated many other counting contests shortly thereafter. It seemed that no middle school student was willing to fight with infinity.

After Buzz Lightyear popularized infinity in the media with his classic catch phrase, it is the most common response I get from students when I ask for their thoughts on infinity. I have inserted tidbits on the subject throughout this semester to get students thinking about a horribly inhuman topic.

Humans are built on limits; we all have boundaries. Infinity can't occur because we are not infinite. Like today--it was my birthday; I turned 24. I told my students that yesterday my age was prime, and next year it will be a perfect square. From this information, they guessed correctly that I was 24. I asked them if there were any more possibilities. 

Three, Eight, Forty-Eight, Eighty...

I then asked them if they thought that there were an infinite amount of numbers that fit the pattern. Human sense had limited them to really 2 answers (24 and 48), but when I moved them into infinity they reluctantly thought there must be an infinite number of possible ages. I introduced them to the concept of a prime twin, and the open problem that has plagued mathematicians for years. 

A prime twin is a set of two primes that differ by 2.
eg. 3,5
The Twin Prime Conjecture states that there are an infinite amount of such pairs.

It seems likely, but has not been proven. We can, however, prove that there are an infinite number of primes. This short dip into the infinite led back into our studies of primes, factors, and GCFs.

The coolest application of infinity (outside of limits) in our curriculum is infinite geometric series. If I have a fixed distance, and step across exactly half of the remaining distance with each subsequent step, I will get to the other side--if I take infinity steps. How can this be? Won't there always be a little space?

Some tinkering with fractions revealed a new angle. Stepping 1/4 of the way with each step would get me 1/3 of the way across. Mathematically, we say the series converges to 1/3. Such an analogy can be represented in the image below. Shading 1/4 of a square, then 1/4 of 1/4... etc. Iterating an infinite number of times yields an area of 1/3 square units.
That reminded me of another problem from my youth. My grade 10 math teacher once asked us to trisect an angle. After many attempts and trials, I gave up. I could get close but could never prove it geometrically. That was before I had the power of infinity. It is easy to bisect an angle, so any angle can be quickly divided into quarters. Shade one of the quarters and then bisect the adjacent quarter twice to create 1/16 of the original angle. Shade that adjacent 1/16 and then bisect accordingly until the 1/16 is now a 1/64. Doing this an infinite amount of times would yield a trisection of the angle because the sum of the infinite series converges to 1/3. After I had this idea, I researched to find many other ingenious ways people have skirted the original problem. 

It seems ridiculous to continue the shading into infinity; can it even be done? These are great questions for students. Rather than letting the mathematical notation take over, allow the students to grapple with the implications of infinity. Students will be defiant that there is no way to actually reach infinity, so telling them that the formula does reach infinity may ruin that adventurous and discerning spirit. Pose problems, change parameters, and allow infinity's enigmatic persona fascinate your students much like my first experience under the monkey bars.