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4th grade classroom, students have demonstrated mastery of basic multiplication facts and some conceptual understanding of multiplication as arrays and repeated addition.On the board, the teacher writes a column of 15’s (there are ten in all)151515…15——She instructs students to find the sum mentally but not to raise their hands with any answers at that moment. After a minute, she directs students to discuss their ideas in small groups of 3-4. She tells them to compare their results, not just the sum but HOW they did this. After 2 minutes, she leads a discussion, inviting students from each group to share their ideas. She comments and adds other ways, insuring that students see at least 4 methods. She asks, “Which method is easiest for you? Why?” etc… This does not take more than 15 minutes in all.”
The way it really works:
4th grade classroom, students have demonstrated mastery of basic multiplication facts and some conceptual understanding of multiplication as arrays and repeated addition. On the board, the teacher writes a column of 15’s (there are ten in all) 151515…15——She instructs students to find the sum mentally but not to raise their hands with any answers at that moment. Most of the students breathe a sigh of relief, because they have no idea of the answer. After a minute, she directs students to discuss their ideas in small groups of 3-4. She tells them to compare their results, not just the sum but HOW they did this. The kids form groups and immediately start goofing around. Luckily most of the groups end up with at least one kid who can figure out the problem because their parents tutor them at home. After 2 minutes, she leads a discussion, inviting students from each group to share their ideas. No one answers at first, but she calls on a kid anyway. He mumbles something about doing it in his head. She comments and adds other ways, insuring that students see at least 4 methods. The students furiously copy what she is doing on the board, but not quite sure what she is talking about. She asks, “Which method is easiest for you? Why?” etc… This does not take more than 15 minutes in all. Most of the kids didn’t quite understand the lesson, but that's OK… they will spiral through the curriculum and hit it up several more times over the next year. Eventually by 8th grade 20% of the kids will master the concept. The majority of them will still be adding up the numbers the long way.
Meanwhile over in Sir Zigs class across town. Zig has demonstrated to his students that that if you have the same list of numbers, you can count up how many numbers there are and its the same as a multiplication problem. The kids quickly calculate 10 x 15. Upon Zigs signal, in unison they shout out the answer. Zig gives them several other similar problems, to test their mastery. After 5 minutes, he is sure that they understand. All the kids seem to get it, and he compliments them on how smart they are. It seems like they can spit out the answer almost as fast as he can write problems. This has not taken more 15 minutes. Now that 100% of the kids have mastered this concept, he can move on to something else. He makes a mental note to revisit this concept tomorrow morning, to reinforce it. Eventually, 90+% of his kids will go on to pass the state proficiency exam.
6 comments:
I want to know what the 3 other methods are.
Even if you figure out something like grouping 15s together in groups of twos to get thirty, then what do you do with five 30s?
“Which method is easiest for you? Why?”
Meaning that the kids don't have to be able to count up the 15s and multiply?
This is not pedagogy. It's stupidity.
The NCTM/Constructivist mystique is built upon the works of the famous 19th century mathematician Charles Dodgson (Lewis Carroll) drawing from one of his more famous works "Alice in Wonderland". In particular, they evoke the Queen, who at Alice's trial, pronounced the trial's one and only rule:
"Sentence first; verdict afterward!"
This is how math is taught. Concepts and complex/creative problem solving techniques first; basic skills afterward, though with this extra added twist: "As necessary".
"As necessary".
For "balance", this means at home.
You left out the part about how it's okay if the students still don't understand what's going on, what they're supposed to be doing, or how to solve math problems even after the math lessons spiral for years - because the kids are going to be assessed in cooperative groups, too.
As long as one kid in each group is lucky enough to have stumbled upon how to solve the math problems, or has been tutored in math at home, everyone will test as proficient.
Yay!
This spiraling business is spiraling out of control.
Teach to mastery and then move on. Throw in previously learned material periodically so the pupils don't forget what they have learned.
You would think this is obvious. But you can't get this advice to be absorbed by the numbskulls in edland. Educationist fog serves like and impenetrable shield. The military may want to take notice.
The approach isn't even very mathematical.
The special thing about maths is its proofs.
If, for some reason, it is important for students to work out which method is most efficient, telling them to add the 15s while keeping track of the number of operations required, and then comparing this to the number of operations required to multiply 15 by 10 would be more interesting mathematically than just discussing it. Then they'd have an exact basis for saying which method is easiest.
I have no idea if 4th graders could actually do that sort of work of course. I only ran across that sort of thing in software engineering courses at uni.
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