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Math: Memorizing procedures vs understanding concepts. What are the signs?

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My question: What are the signs that a kid at the precalculus(ish) level is learning procedures without true understanding? What types of errors would you expect to see?


Some background--my 16yo son entered a small private school last January. Prior to that he was homeschooled for 8.5 years. In the fall semester prior to school entry, he completed the first semester of precalculus with Derek Owens and got an A. He has dyslexia and ADHD, but he has always been very good at math, working 1-3 years ahead over the course of his homeschooling career and always scoring at the 99th+ percentile on standardized tests. He just about always got As on his math tests when I was teaching him and I believe that my grading procedures were fairly typical.


In January, he started "10th grade math" at this new school. It is an integrated math program that culminates in IB SL/HL Math in the 11th and 12th grades. He got Cs on the first several tests last year, and finally, I stepped in and started preteaching and reteaching each lesson, which helped immensely. All along he has been complaining that the teacher never teaches concepts, just procedures.


Fast forward to this year. They have had three tests so far. He failed (<50%) the first and third and got an 80 on the second one. He does all the homework and is attentive in class.


His pattern of mistakes seems to indicate that he is not learning the math conceptually, but at this level, my math is spotty and so it's difficult for me to judge what's going on. His dyslexia and ADHD make him *very bad* at memorizing procedures without the concepts to make them make sense.


I hope this makes sense. I'd appreciate any thoughts you may have.



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My question: What are the signs that a kid at the precalculus(ish) level is learning procedures without true understanding? What types of errors would you expect to see?


You know a student is memorizing instead of understanding if, when asked to explain his solutions, he is unable to thoroughly explain the reasoning behind each step.

If you have him narrate and he says things like "we need to use this formula", "that's how the teacher told us ", "that's how they did it in the example problem", he is not understanding what he is doing. This is actually what I would have my student do if I was trying to judge whether he understands the concepts or not.

I would expect such a student to perform reasonably well on problems that follow the same style and wording as examples he has practiced, but to be stumped if the problem was worded completely differently and he would have to think to recognize that it is, in fact, a similar problem.


For a student with a thorough conceptual understanding, the mistakes I would expect to see would be more of a careless nature: sign and arithmetic mistakes, but nothing general about setting up the problem and following the correct procedure.

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If it is doing precalculus by procedure, usually I would see friends trip up on questions from other precalculus textbooks, and also on physics questions. If I don't remember wrongly there are plenty of precalculus type questions in the mechanics section of physics.


What my lecturer said was that it is possible to give all the formulas needed for precalculus in the exams and people would still use the wrong one or substitute wrongly. So it is applying the knowledge to different types/styles of problems that my lecturer test on.


I did my precalculus with Singapore Additional Mathematics. Now I am reading Larsons Precalculus for fun.

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