Justifications first, shortcuts second.

“When you divide by a fraction, you multiply by the reciprocal.”  This is a mathematical truth, but when I asked my students to explain why I was amazed by all the tricks.  Answers ranged from “you kiss and flip” to “you can’t divide fractions.”  The latter bothered my mathematical core greatly.

After hearing these “math tricks”, I decided to write about my interpretation on why you multiply by the reciprocal and how I explained it to my students.

I started with a simple question, “8 pies were baked and cut into thirds, how many pieces are there?”  My students answered 24 quickly.  We then had a brief discussion on why when you divide a number greater than 1 by a number between 0 and 1, the answer increases. 

We then looked at the thought process of the above question. “8 divided by 1/3, is actually the equivalent of multiplying 8 by 3”.  I still, however, was not happy about the explanation.  Lucy then shouted out “each pie will create 3 pieces, and since there are 8 pies we can multiply 3 by 8”.  I smiled.

Now the tough question, “If you made 8 pies, and everyone you invited over ate 2/3 of a pie, how many people could you invite such that all the pies were eaten?”  Students realized that the work was 8 divided by 2/3, but were having troubles explaining how to do this in words.  After 5 minutes of struggling, the light bulbs started to come on. 

Paraphrased here is the explanation:

First we need to figure out how many thirds are in 8 pies.  We multiply 8 by 3 (using the same logic from above) and the product is 24. However this time, each person needs 2 pieces, therefore we now divide 24 by 2, and the answer is 12. 

In the end, 8 divided by 2/3 is the same as 8 times 3, divided by 2. 

Did my students know this already? Yes, but none of them could explain why. 

Through my own experiences, I have realized the mathematical shortcuts, tricks and magic need to stop, while rationalizations, explanations, and justifications need to begin.

Math concepts first, tricks second.

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.