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[U]nderstanding and mastery are tightly linked. Curricula like Everyday Math try their best to unlink them. They think that understanding concepts with just a little bit of practice is all that you need. This conveniently fits their dislike of drill and kill and how they believe that there is no one way to solve a problem.
Mastery is not just about speed. It's directly related to understanding.
My friend K. came up with an apt definition just the other day.
She said that understanding without mastery is a case of "That makes sense."
It's grasping the logic.
Being able to grasp the logic in a number of different realms is an important goal of a liberal education; it's my liberal education that allows me to read articles about the economic crisis and follow the argument reasonably well.
But following the logic of an article about the economy is completely different from understanding economics with any degree of sophistication.
It was for this reason that, a few years ago, I stopped reading books explaining "math for non-mathematicians" and started working my way through math textbooks.
9 comments:
You've explained it well. Understanding is not mastery, not at all. This is why a kid can sit through an algebra lecture and understand everything the teacher is demonstrating. Each step is fine, especially if the teacher is working the problem.
But when the student must work a problem on their own, with no one prompting them to "get all the x's on one side" or "what can you multiply both sides by to eliminate the divisor?" -- that's the point where the student can't do it. They have practiced to automaticity the skills of properly balancing an equation. That's the point at which understanding can't be remediated easily. A graphing calculator will mask the problem a little longer, but eventually, the lack of mastery will form an impenetrable brick wall.
Well, I don't mean to quibble about words, but if you don't have mastery, you don't REALLY understand.
you may be able to say "that makes sense", but not enough to question the argument, or find a flaw, or go to the next level.
I'd use the word "familiarity". If you can do the "that makes sense", you've got familiarity. You've got basic ratiocination, maybe. But if you ACTUALLY UNDERSTAND an algebra lecture, you darn well know how to solve for X, and you know what roots are.
Most engineers use a heuristic like "if you understood a lecture on a given topic, you should be able to ask 3 questions about it immediately". That's why when you hear a talk, and no one asks any questions, it's indicative that it was a bad talk: because no one understood well enough to ask a question.
This is the problem I have with the idea of "understanding first". It's too vague. My point is that when reform math people use understanding, they really mean conceptual understanding.
When they talk about understanding place value, they are at a simplistic level for third or fourth grade. That's OK, perhaps, but there are other levels, such as an algebraic view or linear space view, or a different base view (binary, octal, hexadecimal, etc.).
Allison says it in few words:
"Well, I don't mean to quibble about words, but if you don't have mastery, you don't REALLY understand."
I was trying to point out that all levels of understanding require mastery. I didn't want to suggest that understanding just means "that makes sense". I would call that conceptual understanding (maybe). Reform math is stuck at the conceptual understanding level; pictures of pie and manipulables.
As LynnG says, what happens when students work problems on their own. Even a conceptual understanding lets you down. If you look at any reasonably good algebra textbook, the whole point of the homework sets is to make sure students go from a vague, conceptual understanding to something more solid. The homework sets tackle the same concept from all directions using mastery. This is nothing new. This is what real math classes do when you get past the fluff of K-6.
I don't want to reduce the importance of understanding. I want to define it in terms of mastery. Reform math advocates want to isolate (unlink) procedural knowledge from understanding, but it can't be done.
They explain this effect by showing how students can do some calculations, but don't really understand what they are doing. The premise is that there is some sort of understanding that will fix this. Students will be able to figure things out. That supports their position that there is no one way to solve problems. You just think mathematically and apply your understandings.
That's not what math is all about. Math is about applying specific tools and skills. You have to understand your tools and why your skills work, but that is achieved with mastery. If a student tries to apply skills without a full understanding, then it will look like they just have procedural understanding. However, the solution is not some "other kind" of conceptual understanding. the solution is more practice; work, not talk.
Reform advocates are bound and determined to make math into something that it isn't; a sort of magical thinking process that can solve anything.
Look for the pattern.
I see a pattern when I look at reform math. Ignorance.
I understand music.
I just can't sing, play an instrument, identify notes, or read sheet music.
By constructivist standards, I should be at Carnegie hall any day.
"I understand music."
It's interesting how they treat music and sports. They require skill, but no deep understanding, like academics. That seems to be their position. That's why they see no problem with skills on the ball field, but get weird about drill and kill in academics. You don't want to screw up the magical something-or-other that's going on.
Then again, drill and kill does apply to sports and music. Coaches try anything they can think of to make it easier and more fun, but they don't pretend that it isn't necessary or that there is some way to avoid the hard work, unless you want to sit in the stands.
The drill and kill position in academics requires spoon-feeding, some magical understanding that makes mastery easy and fun, or the belief that mastery is just not that important. Rather than confront the issues of drill and kill directly, they pretend that mastery is not that important.
Even at the under-10 level, many sports require significant individual effort outside of team practices or the kid doesn't make the top-level team. Any 10-12 year old boy playing Division I elite soccer in a strong soccer area has spent uncounted hours on his own, practicing juggling, dribbling, trapping, shooting and "moves"; the practice never ends because if he doesn't, he falls behind his teammates who are working and he gets cut from the team. If he isn't willing - and parents can't push successfully for very long- he leaves for some other activity. That's the difference from school; school isn't voluntary. Both processes require concentration, dedication and dealing with frustration to succeed (mastery). (Lest I be accused of sexism, I am less familiar with girls' soccer, but it used to be the norm that girls didn't spend time learning to juggle the way boys did - I don't know why not).
I started taking martial arts classes over the summer. Before doing anything fun, we spent the first couple weeks entirely on footwork. Two weeks teaching educated adults how to walk. It was not only boring as heck, it was sometimes painful and exhausting.
Nevertheless, everyone stuck it out because it was well understood that we had to master these basics - how to advance, retreat, and sidestep without falling on your face - before we could get to the fun stuff. Moreover, we kept practicing footwork drills even as we moved to the more advanced techniques - because it wasn't enough to simply know how to move, it needs to be automatic.
Even after it's automatic, you continue to practice the fundamentals constantly because it's really easy to get lazy. When you get good at throwing punches, you tend to think only about the punch, and not about returning to your guard position afterwards. You learn to guard against a takedown, and you forget to protect your head.
The importance of fundamentals became obvious when my work schedule eventually prevented me from attending class regularly - and my performance dropped precipitously. The things which had once been automatic now required my conscious attention. Instead of learning new skills - or applying them in new combinations - I had to spend twice as much time re-learning the old ones that I "understood", but clearly hadn't mastered.
But, of course, all of this is totally different from academic subjects.
This is why a kid can sit through an algebra lecture and understand everything the teacher is demonstrating. Each step is fine, especially if the teacher is working the problem.
This has happened to me SO many times.
"Familiarity" is a good word for this phenomenon, too.
I don't want to reduce the importance of understanding. I want to define it in terms of mastery.
Absolutely.
A related term is "appreciation" as in art appreciation, etc.
btw, I'm not contemptuous of superficial understanding; I'm glad I have it in the number of realms I do.
But it is in no way the same animal as real understanding - as "mastery/understanding."
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