In class, we've started talking about "algebrafying" problems which is something I really enjoy, and lots of others hate. It's the process of taking a problem and expressing it in variables to find exact answers by solving for the variables. Coincidentally I had an experience with a student who I interviewed in which I thought he would use algebra to solve a problem, but he was more comfortable guessing and checking. You can see his work in the picture below.

In both of these problems he tried to solve them by guessing dimensions and then checking whether they matched the picture (like 4 would visually be a smaller side than 5). Unfortunately he erased the dimensions that were wrong, so all that's left is the answer that he thought was correct. But I know from sitting with him as he worked through the problems his method was to guess dimensions, like he guessed 10x18 for the bottom picture, and then to check whether it was correct. So for the bottom one he determined 10x18 was wrong because he couldn't divide 18 (along the vertical side) into four equal integers, which represented the long side for the small rectangles.

I think this is interesting because I have learned that the most efficient way to solve a problem like this is to use algebra. So I would label the long side of each small rectangle as x and each shorter side as y and have equations like (x+y)(4x)=180 and 5y=4x, then solve for x and y to get x=5, y=4. This eliminates the possibility of having multiple guesses and having only my eyes to approximate the distances drawn. So while the student was looking at the lengths to see if his answers could make sense, his guesses would've been completely thrown off if the distances weren't drawn to scale. It could have dramatically changed his answer, whereas my answer using algebra would've remained the same and I would still be confident in my answer once I checked whether my math was correct when solving for the variables.

I also think this was eye-opening because he's in Algebra 1 and he has done problems out of the book for months now that use variables and solving systems and creating systems based on story problems. However, he was unable to apply that knowledge to these interview problems. So has he really learned how to use variables? I would lean towards no, he has not mastered the concept. He can do problems that solve for variables, but he cannot apply that to new problems which have no suggestion for how to solve them. However, to be fair it could be that he preferred to guess and check just because that's more comfortable for him and perhaps since I did not tell him how to solve it he didn't even bother with thinking about variables. Either way I think this is a great set of problems to check students understanding of variables or to even introduce the topic of variables. I knew that it wasn't fun for him to try all those different dimensions so it could be a great way to force the topic of algebra so that he could solve this problem once using variables and be done.

Interesting observation. If the student understands variables, but chooses not to use them, what does it mean? After the student had their solution, it might have been interesting to have them watch you solve using algebra, and ask questions about what you're doing or why it works. I'm guessing there's some separation of problem types. Problems with numbers, use numbers; problems with variable, use variables. This is a clear case where mathematical practices would help.

ReplyDeleteHow would you encourage the student to use more algebra by varying the context/problem or assignment?

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Very interesting observation! I wonder if many students do that: guessing-and-checking. it was interesting how you said he has been doing problems all year out of the book and now he cant apply what he learned to a different problem. I wonder how much a student really retains when do problems out of the book.

ReplyDeleteAfter reading this very interesting observation, I can't help but think: is there anything beneficial for the student that comes out of guessing and checking? I think that there is but that is just my opinion. By being able to solve a problem in multiple ways, students knowledge and understanding enhance immensely. However, it does surprise me that the student didn't think to use variables if he had been doing problems out of the book. Does that mean that he is just able to do the specific problems that have been emphasized in class and on the homework he is just guessing and checking? It would be interesting to see past work and compare it to this observation. Good post though.

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