This chapter reminded me of a conversation I had with one of my college professors. I was feeling terrible because I was not able to grasp the math concepts he was teaching me. He then said to me that he didn't expect me to understand, he added that most students start understanding the math material of their previous class when they move on to higher classes (i.e. you don't start understanding Algebra I until you take Algebra II).
I used to think that understanding the material meant you can get the right answer on a test. I realized that I used to memorize how to work a problem out without understanding it's use. It wasn't until I started teaching math that I started understanding it more in depth.
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Hello! This blog will contain my thoughts, opinions, and ideas as I continue my career in teaching. Any comments or suggestions are welcome.
2 comments:
Chrisel-
I wonder what your professor meant by that really... It seems a bit odd to me. Personally, I think that understanding in math is EXTRA important and frequently not paid attention to... There seems to be a very heavy emphasis in mathematics instruction on procedural fluency. However, there is a lot of research to suggest that emphasizing procedural fluency (memorize steps) without conceptual understanding (or understanding in the UbD model) may get a student through there current level of mathematics, but that it will severely limit their ability to succeed at more advanced levels. This might be the exact opposite of what your professor was saying.
Here is a passage from National Research Council’s “Adding It Up," which surveys the research on math instruction:
"Our analyses of the mathematics to be learned, our reading of the research in cognitive psychology and mathematics education, our experience as learners and teachers of mathematics, and our judgment as to the mathematical knowledge, understanding, and skill people need today have led us to adopt a composite, comprehensive view of successful mathematics learning. Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to capture what we think it means for anyone to learn mathematics successfully. Mathematical proficiency, as we see it, has five strands:
• conceptual understanding—comprehension of mathematical concepts, operations, and relations
• procedural fluency—skill in carrying out procedures
flexibly, accurately, efficiently, and appropriately
• strategic competence—ability to formulate, represent, and solve mathematical problems
• adaptive reasoning—capacity for logical thought,
reflection, explanation, and justification
• productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
The most important observation we make about these five strands is that they are interwoven and interdependent. This observation has implications for how students acquire mathematical proficiency, how teachers develop that proficiency in their students, and how teachers are educated to achieve that goal."
What do you think?
When I was in high school, I'm sure my math teachers explained why the math I was learning was important, but it wasn't until I got to higher levels of math that I really started appreciating the "basic skills" I learned in high school. I was able to apply what I learned in high school to these advance classes and that's when everything started making sense.
For example, in one class you learn a basic skill let's say solving for x. The teacher shows the students multiple ways of solving and gives some examples of how it relates to the real world. At the end of the chapter the students are tested on solving for x (students should know how to isolate variables). A few classes later (let's say Geometry/Precalculus) students are asked to find the tangent of an angle. In order to find the solution the student must "transfer" their previous knowledge (isolating variables) to this problem. By this time it is assumed that the student has a better understanding/appreciation of isolating variables.
It's also interesting how you mentioned the whole memorizing thing. I used to do this, and it definitely slowed me down further along the road. This was a good thing though because that's when I really needed to prove that I understood the material.
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