Fighting for Understanding versus Doing

A pretty interesting conversation unfolded in Geometry this morning. We are getting ready to explore similarity, so I gave the kiddos a quick assignment on solving proportional equations with one variable. This was meant to be pure review. When we started talking about these problems I, of course, heard talk about cross-multiplying, cross products, and even heard one student exclaim something about the old keep-change-flip idea. I decided to stand firm and talk about why we were able to do what we do with these proportional equations. We started simply with the equation like  frac{x}{5}=frac{3}{7} One of my students was taking a vocal lead in discussing cross products and I asked her what equation to write next. She told me to write 7x=15. I agreed that this was correct and most of my students recognized what she was doing. I then asked them to pause and wrote the following equation frac{x}{35}=frac{6}{35} I asked everyone what they thought the value of x had to be in this situation. They all seemed to agree pretty quickly that x must be 6. So I got them to agree that an equation with one fraction on each side AND the same denominator demanded that the numerators would be equal. They all seemed to think I was making too big a deal out of this. I then asked them if I could do the following to frac{x}{5}=frac{3}{7}. I asked if I could multiply the right hand side byfrac{5}{5} while multiplying the left hand side byfrac{7}{7}. One student protested that I need to do the same thing to each side of the equation. I, of course, agreed with her but I asked her to look more closely at what I was doing. She agreed that I was doing the same thing even though it looked different. Most of my students still seemed to think that I was making too big a deal out of this.  Next came the payoff. I picked the following problem from the homework: frac{x}{4}+5=frac{x}{5}+4 I pointed out that our cross product idea was not really a comfortable fit here. My KCF student quickly suggested that we clear the fractions out of the problem by multiplying by 20. I agreed that this would certainly work but asked if I could try something different. So I wrote the following equation:left ( frac{5}{5}left ( frac{x}{4}+5 right ) right )=left ( frac{4}{4}left ( frac{x}{5}+4 right ) right ) I was immediately met with resistance. I begged for patience and made them a promise. I told them that I would carefully explain why I was doing what I was doing and that if they unanimously decided that they did not like this approach, then I would cease and desist. I pointed out that we were, again, doing the same thing to each side even though it looked different. I made an argument that multiplying by smaller numbers decreased my chances of arithmetic mistakes and I pointed out that this technique made the common denominator for the problem obvious. The equation became frac{5x}{20}+5=frac{4x}{20}+4. I saw some signs of visible relief as they saw that this was now a pretty easy equation to process. Combining like terms gave us frac{x}{20}=-1 and a conclusion that x=-20. I then solved the problem the more standard way by multiplying everything by 20 to begin with. I felt like it was a bit of a triumph when they voted that this new technique did not need to be banned from our vocabulary. I know that this is not revolutionary, but I certainly think that I made some strides here. My students are well-trained in mechanics and they know what works. I want to have serious conversations about the ideas behind why these techniques work.

Back at it again tomorrow!

PS – Thanks to David Wees and Zach Coverstone for valuable assistance in learning some LaTex for this post. I hope it looks right when I hit publish

4 thoughts on “Fighting for Understanding versus Doing”

    1. My first experience with LaTex. I know some things I’ll try next time.

      Thanks for dropping by and commenting! Will I see you in LA this summer at TMC15?!?!

  1. Nice job with the LaTeX.

    By the way, I’ve been having the same issues in my classes, and they’ve been less than convinced. I did something with the KFC method not working (see my blog). I addressed it again yesterday, but I don’t think kids were buying it as well, in spite of about 50% of the kids that participated getting the question wrong. They’re still attached to KFC. 🙁

  2. Your Latex looks good. I recently took an online course (Learning How to Learn — Coursera — out of UCSD by Barbara Oakley) and I think that the issue here is one that they address very well: students don’t spend much (if any) time thinking about their thinking.

    We (as teachers) may be partially guilty of not providing that time to consider or reflect on thinking. I have run into the same problem you discussed above many times. I approach it much like you did, but now I give the students a few minutes to discuss the issue among themselves and I put a couple of problems on the board for them to “try” using the “non-rote” technique on them.

    This doesn’t mean everyone will absorb it, but I want to give them the opportunity to “think” about their thinking.

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