I made some progress yesterday helping a boy understand the distributive property, and it was mostly due to the use of visual symbols.
I’d like to share the process of my tutoring, for it shows how important it is to break an algebraic procedure into its constituent “baby” steps. I’d also like to share the process because I used a fairly unusual technique — using visual, non-algebraic symbols to take the place of algebraic symbols. As you’ll see, this technique has some interesting advantages.
Students struggle with the distributive property for a mix of reasons, but a key reason is that they’re simply overwhelmed.
It’s easy for us teachers/tutors to forget what it feels like to be an algebra student for the first time. What this lesson taught me is that when students learn this distributive property the first time, they have a great deal to keep track of — not only the pattern of multiplication and addition/subtraction in the distributive property, but also the correct procedure for multiplying the monomials together … all in all, no mean feat.
With all this to deal with, students get intimidated. Before I started helping my tutoree (I’ll call him Kevin), he was making all kinds of mistakes. In fact he grew so confused that for a time he refused to do any problems with the distributive property. That put the ball in my court.
The technique I employed involves using symbols that are visually recognizable but NOT algebraic. My sense of why this works is this: using visual but non-algebraic symbols helps divert the student’s attention from the nettlesome algebraic stuff. Thus diverted, the student can focus more on the overall pattern.
I started by just jotting down a few symbols, such as these: §, √, ¤, ©, ♥, ♦, ♠, ‡, •, etc.
Then I showed Kevin how to work through the distributive property using these visual symbols.
I showed Kevin that he can rewrite:
♥ (§ + ◊)
(♥)· (§) + (♥)· (◊)
In showing this, I talked Kevin through the process, explaining that he takes the single term OUTSIDE the parentheses and MULTIPLIES it by EACH TERM INSIDE parentheses, one at a time, thereby getting two products. And I explained/showed that he connects those two products with a + sign if the original parentheses has a +, and with a – sign if the original parentheses has a – sign.
After the explanation, I asked Kevin to practice the procedure. I asked him to choose any three symbols, and to circle the symbol that he’ll put outside parentheses. Then I asked him to write an equation like the one I created, but with his symbols. I did insist that Kevin include the four sets of parentheses and that he use dots for multiplication symbols. I’ve found that insisting on a consistent format helps students work more carefully.
It took Kevin a short while to get into the flow. Once he did, I asked him to do several more practice problems with nothing more than symbols.
Once I saw that Kevin had this process down, I moved to the next step: having Kevin practice the same process with algebraic symbols (but not doing any real algebraic stuff). I asked Kevin to simply write down how the distributive property would help him re-arrange an expression like this:
3(x + 4)
Based on his prior practice, Kevin successfully wrote this equation:
3(x + 4) = (3) · (x) + (3) · (4)
Note that at this stage I was NOT asking Kevin to multiply the terms together. Doing so would be too much for him to deal with at this time.
[Note: Kevin does have fairly significant math learning challenges, likely more severe than those of the "average" algebra student. Nevertheless, the techniques used in this lesson can still be used successfully with more typical students. As the educator, you will have to decide how slowly or quickly to move along.]
Once I saw that Kevin had gained proficiency, I reviewed the process of multiplying monomials, giving him terms like these to multiply and simplify:
(4x) · (x) = 4x²
(2x) · (5x²) = 10x³
It took a little while for Kevin to remember that he multiplies the coefficients but adds the exponents, and he made a few mistakes in re-learning this. But he eventually did focus and get these two processes untangled.
Once Kevin showed that he remembered how to multiply monomials, I had him put all of the steps together, with the following problem:
2x (3x + 4)
He wrote the steps as follows:
= (2x) · (3x) + (2x) · (4)
= 6x² + 8x
Big “high-five” at this moment! And a shy grin on Kevin’s face.
Then I gave Kevin several more practice problems like this, to get some reinforcement between now and our next session, and he said he would do them.
Going through this process reminds me that we often need to break procedures down into their tiny steps, if we want students to avoid overwhelm and master the procedure.
This lesson also strengthened my sense that moving away from algebraic symbols to purely visual symbols can reliably help students relax and see the big picture. If you have never given this a try, you might want to check it out sometime. If you have and you have any comments on it, feel free to share your thoughts.