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I'm moving something I tried to say in a comment. I'm starting to wonder whether there is too much extraneous information in today's pre-calc math such that students are missing the forest for the trees? I was shocked by the story about the boy in 8th grade algebra who couldn't figure a 10% tip.
To me one of the cool things about math is that once you understand a concept, you can often summarize the whole concept in a very few words. Would it help if the students wrote their own summaries?
I tried writing a summary of per cents that covers everything you need to know to figure a tip. It's only a page long. Here's the link.
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You know --- I think that is an excellent question, and it's one of the questions that makes me understand how constructivists ended up where they ended up. (I think.)
The fact is that there is something useful about "putting things into words" for many, many students.....(though I worried that it may not be useful for the "math brains").
And the research on kids-teaching-kids seems, to me, quite strong in terms of showing that when you teach a concept you "understand" it better. Teaching a concept is going to mean putting that concept into words to some degree.
With the constructivist-instructivist debate, we have three intertwined issues:
* "what works"? (summaries? cumulative practice? etc.)
* "what do we want kids to know and be able to do"? (fluency in the multiplication facts or not??)
* freedom, self-determination, & choice
I think the very existence of these three "strands" (STRANDS!) has made the debate more polarized than it absolutely needs to be. Or maybe I mean to say that it has polarized the debate in unproductive, as opposed to productive, ways.
I'm thinking of the "baby with the bathwater" phenomenon.
On the constructivist side, traditional math seems not to have worked for a large segment of the public (I question whether that's true absolutely, but there's no doubt in my mind they've got a case....)
So you get rid of long division.
On the instructivist side, "constructing meaning" is manifestly not working for a lot of our kids so we rapidly reach the point at which we don't want to hear ANOTHER WORD about kids writing essays in math class.
At least, I rapidly reach that point!
Horrors! I wasn't thinking of having the teachers involved this.
Fantastic comment, Catherine!
And Susan, great explanation! Four things I really like about it: (1) It's written as though you were speaking; (2) You take care to explain things (like the slash vs. the fraction bar) that may seem trivial but can derail a student's interest; (3) You use repetition beautifully; and (4) That summary at the end is very smart!
Just about every basal text--no matter where it finds itself on the ideological spectrum--includes (a) some kind of instruction and (b) some kind of practice.
And if you ask just about any textbook editor or teacher for that matter whether or not actual students (at any grade level) actually read any of the instruction in a basal textbook in order to learn, he or she will probably tell you NO.
The instruction is written so that a typical teacher, administrator, or evaluator will flip through it and say, "I like that you have/do this."
Thus,
Some models/diagrams = builds students' conceptual understanding.
The phrases "Step 1, Step 2, etc." anywhere in the book = includes step-by-step instruction.
A phrase such as "Reaching All Learners" = provides for differentiated instruction.
A hands-on activity = helps students with different learning styles, or modalities.
Highlighted vocabulary = helps students build academic vocabulary/good for ELLs.
And I suspect that verbosity = clearly explains concepts.
Hardly a soul sits down to a blank instruction page in a basal text and thinks to himself, "How can I write this so that students will really learn it."
But that's what you did. So, are you looking for a job? : )
Mr. Person, thank you so much for the nice words.
No, I'm not looking for a job. I'm a retiree with a full-time "addiction" to writing open source braille translation software, most especially involving math.
Trying to understand why parents of blind kids were saying that their kids have trouble with math because, "After all, math is highly visual" was what led me to KTM. My younger son graduated from high school 20 years ago and I hadn't realized pre-KTM the extent to which math education had changed since then.
I wrote more about this a while back here .
Wow, that’s a wonderful lesson summary. I absolutely agree about the value of streamlining math lessons so that extraneous information does not confuse the essence. Perhaps in their extreme desire to “engage” students, educators are including too much fluff that just ends up becoming a distraction to learning.
I was wondering if you had a teaching background.
(2) You take care to explain things (like the slash vs. the fraction bar) that may seem trivial but can derail a student's interest;
oh gosh; that jumped out at me, too
it is SOOOO important (it's also one of the core elements of professional writing, according to people who study expertise)
It's not easy to do --- AND it's not easy to know just how much of it to do, and when
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