I was going to file this under “Articles I Never Finished Reading, Dept” but I felt that “Shut the Hell Up” was more apt, considering this quote:
“Often what’s lacking for U.S. students, and Massachusetts students as well, is a conceptual understanding,” Chester said. “They may learn the mechanics, but without that conceptual understanding it’s not as clear to those students how that math gets applied.” Mitchell agrees. It’s the skills to apply math that are lacking. She says students need to be taught how to think with math and not just memorize.”
Given that this has been the complaint for the past 28 years or so, don’t you think the excuse that “we’re just not teaching reform math right” sounds a little lame?
On the issue of “understanding” vs “procedures”, a math teacher I communicate with in New Zealand has this to say:
“I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.
“Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.
“Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding. Most kids really don’t want to understand very much. Every now and then I have a student who refuses to learn a new skill until they “understand” it — and that causes problems, largely because they learn so unnecessarily slowly as a result, which I find difficult as a teacher.”
I recognize that this quote will cause cognitive dissonance. That’s OK. Â Productive struggle is good for you.
traditional math 
They don’t understand that they are just parroting back what they learned by rote in ed school. They need to get out and mix with those who really understand math and how to get there. They need to talk with the AP Calculus track high school teachers (many from industry) to see what they think is wrong with K-8. They need to talk to the parents of the best students.
They even get the “understanding” part wrong. There are many levels of understanding and individual mastery of skills proves that the student has a base level of understanding. Base level. First. Success in math is never rote or zombie. There could be more levels of understanding built on top, but there are NO EXAMPLES of success using their top-down model of “conceptual” understanding. None. Conceptual is not deep. It’s superficial. Mastery proves a much deeper level of understanding. If they showed some examples of success, I would be their biggest fan. However, I’m not a fan of ignorance. I distinctly remember wanting more for my son in math when he was in pre-school. Then I found out our school used Mathland. They got it completely wrong.
So how do you go from understanding half and quarter hours on the clock to simple fractions to mixed fractions to rational expressions with complex factors? Motivating new units with concepts is nice (and done by most textbooks), but deep understanding comes from doing many individual homework problem variations and tests. Individual homework enforces mastery, and that’s how you build success and understanding in math.
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My exams don’t have a mark for understanding either. They don’t pay me enough to be a mind reader.
I *DO*, however, often use questions for which I have a belief that a student whose understanding is deficient will perform poorly. This is the only proxy for understanding that I feel has any value. The task tested is either mechanical or otherwise knowledge & skill based. And understanding will be intrinsic in that task. This has always been basic way understanding contributes to one’s score, and there is nothing new to be learned about this in the 21st century. All rubrics for “understanding” I’ve seen are either re-hashes of age-old standard common-sense approaches … or total nonsense
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None of my students could have passed my math classes with just rote knowledge. That doesn’t mean that they understood everything, but that could be added on top. The biggest problems they had were gaps in knowledge and skills. That would eventually catch up with them, not the understanding.
The opposite is not possible – conceptual understanding with few skills. Successfully doing homework problems and tests proves all of the understanding one needs to be going on with. Conceptual understanding (whatever the hell that is), is nowhere. They can’t place all of their bets on what happens in class while ignoring the need of pushing, testing, and individual mastery of homework sets. The biggest tutoring problems I see are students who don’t value doing and understanding homework sets completely. They can parrot back the concepts, but I tell them they have to redo all of the homework problems on their own. Inductive transference does not get the job done, no matter how much time you are allowed.
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