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Showing posts with label integrated math. Show all posts
Showing posts with label integrated math. Show all posts
Thursday, July 29, 2010
Monday, April 12, 2010
The "traditional" math sequence
Well first there is algebra in 9th grade. But then take the entire sequence geometry --> algebra I --> algebra II --> precalc --> calc I --> calc II etc. (Maybe if you are advanced enough you get to do Calc III or linear algebra in the last term of high school or maybe the opening semester of college.) Why necessarily in that order? Is it necessary to have fully mastered all the theorems of geometry before moving on to algebra I?
Furthermore, there is a perspective lost in the entire sequence, because generally, a lot of fields from number theory to graph theory are not even considered until college, except as shallow "enrichment" activities. Are kids capable of doing hardcore number theory in eighth grade? Well take the International Math Olympiad (for young students) or your average state math competition. You can get straight As on your calculus exams and totally get creamed at these competitions by a bunch of ninth graders from a school with a more dedicated math team.
Formal linear algebra is mind-expanding, but it occurs to me it doesn't require college-level understanding to master its basic concepts (like transformations). Arguably, you /really/ understand what differentiation and integration is all about when you learn to map from one polynomial space to another. The most important thing about solving math is abstraction, and a rather powerful abstraction is the ability to rotate entire shapes by a simple entity called a matrix; ever had that annoying geometry problem where you had to tediously map all the points of some crazy polygon after it was rotated an arbitrary angle? Well now, what a perfect time to bring up a concept that would systematically map every point to its new point after the rotation...
Here is the place to use a calculator. Take a polygon with 20 points (or some inconvenient number). Students would then have to design a matrix that carried out said transformation. Perhaps an arbitrary transformation whose formula wouldn't generally be memorised (like with standard rotations). The calculator is the "reward": simply input all the points, the appropriate transformation and then report the results of the magic. Here, the calculator ensures that students actually understand how to use the abstraction being learnt (rather than say manually applying an algorithm to each point). On the other hand, it is not being used as a crutch for bad understanding.
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There has been much said about "holistic" educations. Many ostensibly "holistic" curricula are not holistic curricula at all. One of my disenchanted teachers put it -- you can't just toss a bunch of department heads together into biweekly meetings and call the result "holistic". (Now mind you, this teacher was from a top Singaporean school and he had been "released" from duty by being too vocal about certain teaching practices.) There is so much reteaching of math concepts from applied subject to applied subject. And oh, all the forgetting! Imagine what could be done if all the inefficiency was removed...
See, I don't actually disagree with the actual idea of integrated math. Integrated math should be a vehicle for exploring important life concepts (like, ooh, equilibrium! optimisation under constraints! recursion! etc.) taught tightly with useful, complex, mind-expanding applications. You could pack a lot of learning into a year. It shouldn't be about using matrices as some sort of organisational table for keeping scores of random soccer matches.
Furthermore, there is a perspective lost in the entire sequence, because generally, a lot of fields from number theory to graph theory are not even considered until college, except as shallow "enrichment" activities. Are kids capable of doing hardcore number theory in eighth grade? Well take the International Math Olympiad (for young students) or your average state math competition. You can get straight As on your calculus exams and totally get creamed at these competitions by a bunch of ninth graders from a school with a more dedicated math team.
Formal linear algebra is mind-expanding, but it occurs to me it doesn't require college-level understanding to master its basic concepts (like transformations). Arguably, you /really/ understand what differentiation and integration is all about when you learn to map from one polynomial space to another. The most important thing about solving math is abstraction, and a rather powerful abstraction is the ability to rotate entire shapes by a simple entity called a matrix; ever had that annoying geometry problem where you had to tediously map all the points of some crazy polygon after it was rotated an arbitrary angle? Well now, what a perfect time to bring up a concept that would systematically map every point to its new point after the rotation...
Here is the place to use a calculator. Take a polygon with 20 points (or some inconvenient number). Students would then have to design a matrix that carried out said transformation. Perhaps an arbitrary transformation whose formula wouldn't generally be memorised (like with standard rotations). The calculator is the "reward": simply input all the points, the appropriate transformation and then report the results of the magic. Here, the calculator ensures that students actually understand how to use the abstraction being learnt (rather than say manually applying an algorithm to each point). On the other hand, it is not being used as a crutch for bad understanding.
---
There has been much said about "holistic" educations. Many ostensibly "holistic" curricula are not holistic curricula at all. One of my disenchanted teachers put it -- you can't just toss a bunch of department heads together into biweekly meetings and call the result "holistic". (Now mind you, this teacher was from a top Singaporean school and he had been "released" from duty by being too vocal about certain teaching practices.) There is so much reteaching of math concepts from applied subject to applied subject. And oh, all the forgetting! Imagine what could be done if all the inefficiency was removed...
See, I don't actually disagree with the actual idea of integrated math. Integrated math should be a vehicle for exploring important life concepts (like, ooh, equilibrium! optimisation under constraints! recursion! etc.) taught tightly with useful, complex, mind-expanding applications. You could pack a lot of learning into a year. It shouldn't be about using matrices as some sort of organisational table for keeping scores of random soccer matches.
Monday, April 14, 2008
Sunday, February 18, 2007
integrated math, yay or nay?
Linda Moran is looking into it. She cites this post by Matt Abe:
Strands!
whole stuff taught wholly
This is more constructivist Romanticism.
The Romantics believed in wholes (I gather).
Not in parts.
Nature is a whole....therefore reading is a whole & math is a whole etc.
That appears to be the jist (speaking as a person who knows nothing about intellectual history, of course).
Interdisciplinary this, that, and the other is the same idea.
The middle school model, planned for our middle school, was going to be all interdisciplinary teaching all the time.
That's constructivism: teach everything all the time in one big blob.
No component skills!
No starting small and building up!
No baby steps!
This is why we have, in our middle school, a Regents Earth science course that can be handled only by students scoring in the top 10% of the country on the CTBS.
The kids have to swallow the subject whole.
Which probably means they're coming into the course having mastered lots of background knowledge to begin with.
School never looks at that.
We look at "maturity."
"Mature" students can "conceptualize."
"Immature" students can't "conceptualize."
I wonder if this attitude has anything to do with our middle school being worried about not making AYP?
What is integrated ("reform") math? It is a way of teaching math in which traditionally separate subject areas, such as algebra and geometry, are integrated into one course of study; and it integrates math with non-math subjects and real-world experiences.
Strands!
whole stuff taught wholly
This is more constructivist Romanticism.
The Romantics believed in wholes (I gather).
Not in parts.
Nature is a whole....therefore reading is a whole & math is a whole etc.
That appears to be the jist (speaking as a person who knows nothing about intellectual history, of course).
Interdisciplinary this, that, and the other is the same idea.
The middle school model, planned for our middle school, was going to be all interdisciplinary teaching all the time.
That's constructivism: teach everything all the time in one big blob.
No component skills!
No starting small and building up!
No baby steps!
This is why we have, in our middle school, a Regents Earth science course that can be handled only by students scoring in the top 10% of the country on the CTBS.
The kids have to swallow the subject whole.
Which probably means they're coming into the course having mastered lots of background knowledge to begin with.
School never looks at that.
We look at "maturity."
"Mature" students can "conceptualize."
"Immature" students can't "conceptualize."
I wonder if this attitude has anything to do with our middle school being worried about not making AYP?
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