There has been plenty of recent twitter talk about the process of moving the focus of mathematics education away from the “how” and toward the “why”. Traditionally, students have been trained to approach a question–usually given to them by an outside source like a teacher, textbook, or test–with the express intent to show the grader “how” it is answered. Such responses often include the use of algorithms, formulae, or memorized facts we know to be true. (These facts are in no way axiomatic, but constant repetition reduces them to that state. Students have answered them so often, the process loses meaning. Take 2×2 for example.)
The focus on the how encourages a race to the finish. Thomas C. O’Brien calls this phenomenon “Parrot Math”. (Phi Delta Kappan, Feb, 1999) More specifically, it is the process that every student goes through when answering the question “how”–they attempt to repeat or imitate the process that has been shown to them. These carbon-copy answers are created without the knowledge of the mathematics that ensures their success, just as a parrot can possess a large vocabulary without understanding the intricacies of language.
In my class, I attempt to elicit the “why” as much as possible. The solution to “why” is much harder to come by. There are times in class when I challenge students on their mathematical statements–just to keep them honest. A student was working on sketching the graph of a rational function when she called me over. She explained to me that she could not get part c:
c) Find the equation of the Vertical Asymptote(s)
She was having trouble with a “how”. She could describe what a Vertical Asymptote was and how it effected the graph, by had simply forgotten the neat and tidy process to generate them. I explained to her that you set the denominator to zero and solve. Her face instantly renewed with vigor because her question had been solved. I then continued to challenge her with “why”s. I asked why the denominator can’t be equal to zero, but got a very standard, and hollow, response:
“Because you can’t divide by zero.”
Acting as confused as I possibly could, I asked her, “Why can’t you divide by zero?” Her joy instantly drained. It took a 5 minute conversation about piles and sticks before we decided that you could divide zero sticks into 10 piles, but couldn’t divide 10 sticks into zero piles. Division had become a “how”.
Moving on from the distinction, it is plain to see communication as a central element in “why”. This includes student-student and student-teacher communication. The Standards published in 2000 by the NCTM call communication an “essential part of mathematics and mathematical education.” It is these skills that build “meaning and permanence for ideas” in math. (NCTM, 2000) I think most teachers agree that both meaning and permanence are lacking. Just come back from a summer holiday with a group of grade 9s.
The simplest, and most useful, way I open the avenues of communication in my class is through yellow paper. That’s my secret. I grab a couple yellow, lined pads of paper from the office and hand each student a page. We complete the daily task on it, and they are to provide a reflection or explanation when they are through. The students know I read everything handed in on a yellow piece of paper. It took 3-4 practices before the students began to communicate their methods effectively in writing, but my persistence has paid off. I thought it would be interesting to include some student thought from my Mean, Median, and Mode task posted earlier on this blog. (It will be helpful to read that post if you have not already)
The writing created 4 distinct categories of learners, but I would have never made this distinction without the pursuit of “why” and the open communication of the yellow paper. The students were required to fix the set {1,2,3,3,3,4,5,5,6} so the mean, median, and mode are all 3. The mean was the issue.
Some students took out numbers from the high end until they met a target sum. These students knew that the average was dependent on the sum and the number of entries; they altered entries until the mean worked.
“There are 9 numbers in total. Not wanting to change this # that I know will divide by 3. 3 (the # I want to get) timesed by 9 will give me 27. I take the difference of 27 and 32 (32 because thats the original total of these 9 #’s) and I want to get 27 instead of 32 because 27 / 9=3. So I take away 5 (the difference) from 6. 1,2,3,3,3,4,5,5,1 = 27/9 = 3. Mean = 3.”
This explanation above (repeated verbatim) shows excellent mathematical communication. She has shown me the “why” in her process. She has also shown me that she understands how the total sum effects the mean. She uses mathematically rich words like sum, difference, and mean. She has also used less eloquent terms like “timesed” and “take away”. This shows me that she persevered through lapses in thought. She hit a wall in her explanation, and powered through it by sheer determination. This is how I know she can now answer “how” to find the mean, and “why” it is that way.
Other students tried lowering the entries one by one until they got the mean correct. The logic behind this dilution of sorts was they wanted to keep all entries in order so the median was not altered. Genius! I may have seen this process as primitive if they had not communicated their motives. Other students added entries so the large ones took less effect. Others talked of “balancing” the entries. I had never thought of mean as a balance before.
Creating meaning in mathematics is not about throwing the “how” out the window. Mathematicians have worked hundreds of years to establish hows. True meaning comes when students are given the opportunity to both attach a “why” to the “how” and communicate that connection to others.
NatBanting