Math tricks do not teach students how to complete questions, the tricks are actually cheating the students of the real learning that could occur. In one of my classes, students needed to square (x - 2) to complete the question. When a group of students progressed to this step they were struggling on how to advance. One student, Timmy, wrote (a - b) squared is equal to "a squared, minus 2 times ab, plus b squared". As I was inhaling to ask him "why", one of his group members beat me to the question. Timmy was questioned, by a peer, "Why does that trick work?". Immediately, I saw confusion across Timmy's face. After looking at me for an answer and I informed Timmy, "For you to be able to use a trick in my class, you must be able to explain the reasoning behind it".
"Well Mr. Martin, FOIL, mean First, Outside, Inside, Last, and that will work for any multiplication we need."
As if right on cue, Timmy's peer asked, "Always?". For the second time, Timmy's face was showing confusion, when he replied "I think so, oh wait, not for a bracket with three things". Now I did not want to stop the thinking process and correct Timmy with the word "terms", as I saw Timmy was trying to logically work out this trick he had pulled from his brain.
I could provide more and more anecdotes of the same events. When students learn formulas, shortcuts, or tricks, and cannot explain why, we are essentially leading them down a path into an abyss. I have been guilty of this many times in the past, but I am now forcing students to construct their own shortcuts, tricks, and formulas based on inductive reasoning that they complete themselves. I am then often asked by students, "Does this shortcut work?", for which I will usually respond with "You tell me."
As educators, we need to stop giving tricks and allow the students to complete the process the "long way". This idea will allow for students to start creating a stronger foundation of learning, and which will then allow for higher level thinking to occur. Students need to discover these math concepts first and then allowed to create their own shortcuts, however when they create the tricks they are no longer magical to them.
Don't know where this comes from, but this is a quote I often share with my learners:
ReplyDelete"People who know what to do and people who know how to do it will always be working for those who know why it is being done."
Your post is interesting because we are trying to apply the same idea to developing teaching approaches. No short cuts - just progress. Check out my colleagues blog at http://mathhombre.blogspot.com/2011/01/planning-for-engagement.html
Great point, Dave - looking for the consistent cognitive frameworks inherent in the mathematical concepts can often serve to reveal the developmental problems of teaching 'tricks' in math.
ReplyDeleteGOOD tricks are simply distillations of the methods that can be used interchangeably with the 'expected' algorithms in appropriate contexts... they highlight, not conceal, student understandings.
In your FOIL example, a student that understood the concept of 'everything times everything else' would be able to transparently apply the 'trick' across a wide range of situations without losing sight of the ball.
Math is, after all, an ongoing game of 'I wonder if...?'
Have a good one!