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 unrivalled 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.
NatBanting