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Re: Mathematical understanding
Posted:
Nov 8, 2010 12:25 PM
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Phil,
From your words below, it appears you are
attempting to teach understanding before students
practice. Or maybe the word "explain" would be
more appropriate than teaching understanding? I
don't know. Anyway, there is lots to be said
about what you are doing in the lesson from a
brain function perspective, but I am confident no
one is interested. However, let me offer just one
thought about practice. When a person practices
something, the neural circuits that process the
practicing become myelinated. Myelination makes
the flow of information in the circuits easier to
pass along. This process also makes the circuits
more likely to "fire" at the appropriate times.
There is nothing else happening in the brain.
There is no "understanding" circuit that is
created or invoked. The neurons/synapses involved
in practicing the procedure are just more likely
to fire in the near future. I use the words near
future because the practicing may not be
connected to anything else. Long-term memory with
recall requires connections. Maybe more
importantly however, as soon as the practicing
stops, the neural circuits start to loose the
myelin. With no myelination, recall becomes more difficult.
Just one more quote, but this time from the MathAMATYC Educator:
FIRST ABOUT HIGH SCHOOL TEACHING:
"The TIMSS study showed that "the most common
teaching methods used in the U.S. focus almost
entirely on practicing routine procedures, with
virtually no emphasis on understanding of core
mathematics concepts that might help students
forge connections
Given that U.S. students are
taught mathematics as a large number of
apparently-unrelated procedures that must be
memorized, it is not surprising that they forget them
"
AND THEN TEACHING/EDUCATION IN A COMMUNITY COLLEGE:
all the evidence we have shows that although
community college faculty are far more
knowledgeable about mathematics than are their
K-12 counterparts (Lutzer et al., 2007), their
teaching methods may not differ much from those
in K-12 schools (Grubb, 1999).
Stigler, J.W., Givvin, K. B., Thompson, B. J.
(2010). What Community College Developmental
Mathematics Students Understand about
Mathematics, 4-5. MathAMATYC Educator, 1(3).
========================================
And from the Carnegie Foundation:
Forty-two percent (42%) of students fail
beginning algebra in community college and 38%
fail intermediate algebra.
up to 70 percent of
community college students referred to
developmental mathematics do not successfully
complete the sequence of required courses. Some
spend semesters repeating courses, others simply drop out.
http://www.carnegiefoundation.org/problem-solving/developmental-math
Take Care,
Ed
============================
At 10:50 AM 11/8/2010, Phil Mahler wrote:
>[snip]
>When I teach why we add exponents to multiply
>(this is in the context of low level courses,
>whole number exponents), I think the students
>"understand" it when I draw the usual "picture",
>for example 3 x's followed by 4 x's is 7 x's for
>factors. Then they practice. But many will get
>it wrong on the next test. Where did the
>understanding go? It's the practice that helps
>them get it right, not the theory. Hopefully
>after a few weeks of practice, and my
>reiteration of the theory, it does click. Of
>course that still doesn't mean that next
>semester they won't get it wrong. Even knowledge
>of a reason requires refreshing.
>
>OK, by for the day.
>
>Phil
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