Successful adaptive behavior presupposes having at your disposal a veritable array of tools for adaption and the mental knowhow to selectively apply or modify those tools. In math, these tools include a good conceptual understanding of the mathematical concepts being tested, as well as the ability to play around, combine, re-fashion these concepts in new/novel problems.
Renaming numbers or “borrowing” can in a sense be an adaptive practice. A student of mine added 2 6/11 and 3 7/11 by “borrowing” 4/11 from 6/11, added 4/11 to 7/11 to make a whole, and gave an answer of 6 2/11. He did not go thru the usual route of 5 13/11 = 6 2/11. When asked how he learned this approach, he pointed to the fraction tower that I extensively use with my students, and explained that it’s easier to regroup to make whole than to add improper fractions. This student has a better conceptual understanding of fractions than some Grade 12 students I know.
While routine expertise is prized by many especially come exam time, it is of limited value in some tests (GRE, GMAT or SAT) when an increasing number of questions tests the level of your adaptive expertise. Many questions force you to view the problem from a different perspective, perhaps less analytically and more holistically, perhaps less algorithmically and more relationally (among variables). That requires a great deal of flexibility to switch between viewpoints, and creativity to re-invent your take of the problem compared to similar ones encountered in the past.
Adaptive thinking can be a habit, an evolving process to greater adaptive expertise. It is not necessarily linked to the years of math schooling, as I’ve seen veteran math teachers and postgraduate math graduates unable to discern patterns and relationships in non-routine problems. It behooves us to develop adaptive habits in our students.
Tuesday, November 16, 2010
Tuesday, November 9, 2010
Best Experience in 2-week practicum
The best part is observing other non-Math classes and the insights they afforded me to be a better math teacher. I was thrilled to observe a Music 9 class when the fledgling musicians took up their individual instruments and played as a band a classical music, a upbeat dance and a moving sentimental piece. After that class, I was bent to discover the mathematical dimension of music, or even play some classic background music at appropriate stages of class time.
Then I observed a Foods 10 class, where measurement, proportions and even shapes (e.g. why are English pears roundish compared to Asian pears?) of ingredients rule. Then off to a PE 10 basketball class which had a jogging warm-up, then were led to the different mechanics of proper shooting. I intently watched how parabolic the students shots were, as well as the recommended leg and arm angles lectured by the instructor.
When asked what they felt were the most serious learning obstacle in learning math, a number of math teachers mentioned the language component. Upon learning this, the principal arranged for me to sit in an ESL class of mostly Chinese and Japanese students. Noting how serious the language gaps of some students are, I will try to re-phrase concepts when I'm actually teaching, or write key words and explanations in the board.
These invaluable insights will help me integrate other school subjects to math, and also find ways to link outside students' interests (like computer games, sports) to mymath teaching.
Then I observed a Foods 10 class, where measurement, proportions and even shapes (e.g. why are English pears roundish compared to Asian pears?) of ingredients rule. Then off to a PE 10 basketball class which had a jogging warm-up, then were led to the different mechanics of proper shooting. I intently watched how parabolic the students shots were, as well as the recommended leg and arm angles lectured by the instructor.
When asked what they felt were the most serious learning obstacle in learning math, a number of math teachers mentioned the language component. Upon learning this, the principal arranged for me to sit in an ESL class of mostly Chinese and Japanese students. Noting how serious the language gaps of some students are, I will try to re-phrase concepts when I'm actually teaching, or write key words and explanations in the board.
These invaluable insights will help me integrate other school subjects to math, and also find ways to link outside students' interests (like computer games, sports) to mymath teaching.
Word problems
Word problem from Mathpower 11 (1999), p. 221, #98.
#98. Scaffold Construction: two Scottish construction workers broke the world record for scaffold construction by building a 2-storey scaffold in 25 min 53 sec. The volume occupied by the scaffold was 100 m3. The length of the scaffold was 4 times the height. The width was 4 meters less than the height. Find the dimensions of the scaffold.
Comments:
1. The concept of a scaffold may not be accessible to most students. “Scaffold” will most likely not be in the working vocabulary of many Grade 11 students. From the problem, scaffold has obviously something to do with construction. It can be surmised that this may be a 3-dimensional solid because volume is given.
2. Transforming a verbose problem into a more visually pleasing and compact yet concise pictorial representation is one of the most powerful problem solving tools. Since a good drawing captures the essence of the problem, it strips the unessentials, and greatly clarifies in the resolution of the problem. Not to be able to draw a geometric –based problem (and rely only in algebraic manipulation) diminishes the holistic understanding of the problem. A student who does not have any idea of a scaffold will not be able to draw one.
3. This is a case of dressing up only to undress the problem. Upon seeing the words: volume, length, height, width and dimensions (in that order), the auto response will be to simply plug in numbers and variables into the volume formula. Chances are, students will get the right answer. This reinforces their previous experience that you do not really need to understand the scenario presented in the problem; you simply need to be creative to mix-match the numbers given.
#98. Scaffold Construction: two Scottish construction workers broke the world record for scaffold construction by building a 2-storey scaffold in 25 min 53 sec. The volume occupied by the scaffold was 100 m3. The length of the scaffold was 4 times the height. The width was 4 meters less than the height. Find the dimensions of the scaffold.
Comments:
1. The concept of a scaffold may not be accessible to most students. “Scaffold” will most likely not be in the working vocabulary of many Grade 11 students. From the problem, scaffold has obviously something to do with construction. It can be surmised that this may be a 3-dimensional solid because volume is given.
2. Transforming a verbose problem into a more visually pleasing and compact yet concise pictorial representation is one of the most powerful problem solving tools. Since a good drawing captures the essence of the problem, it strips the unessentials, and greatly clarifies in the resolution of the problem. Not to be able to draw a geometric –based problem (and rely only in algebraic manipulation) diminishes the holistic understanding of the problem. A student who does not have any idea of a scaffold will not be able to draw one.
3. This is a case of dressing up only to undress the problem. Upon seeing the words: volume, length, height, width and dimensions (in that order), the auto response will be to simply plug in numbers and variables into the volume formula. Chances are, students will get the right answer. This reinforces their previous experience that you do not really need to understand the scenario presented in the problem; you simply need to be creative to mix-match the numbers given.
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