Subject: Innovation: A Golden Teaching Moment

Posted by George Richardson on 12/1/2010
In Reply To:Innovation: A Golden Teaching Moment Posted by Bob Gorman on 12/1/2010

Message:

On Dec 1, 2010, at 6:57 AM, Niall Palfreyman wrote:

>> How do you get a teacher interested in SD when they have a Honors
>> Algebra 2 class that does not know how to simplify a radical, but
>> they know how to punch the numbers in a calculator? In fact, that is
>> becoming a huge problem, the kids use the calculator and really do
>> not understand what they are doing, garbage-in, garbage-out. I am a
>> firm believer of technology, but wisdom takes place when we balance
>> technology with understanding basic mathematical rules etc.. I do not have the answer.

In what seems like a previous life in the late 1960s, I was pursuing a PhD in mathematics education. At that time there was a robust research result about Bloom's Taxonomy of Educational Objectives in mathematics that seems very relevant here.

SMSG had found that the Bloom sequence (Knowledge, comprehension, application, analysis, synthesis, and evaluation) was *not* sequential at the lower levels when it came to arithmetic. As I recall it, research in the National Longitudinal Study of Mathematics Abilities (NLSMA) appeared to show that

- Knowledge of arithmetic (ability to compute) was *not* required for Comprehension (understanding) of why or how the computation worked, and
- Comprehension in arithmetic (understanding of why and how it works) was *not* required for arithmetic knowledge (ability to compute).

The two were independent! They could be taught in either order, and having either one first didn't seem to help getting the other. One had to teach both and not expect one would help the other.

I was (and probably still am) deeply devoted to "guided discovery" teaching and learning, so I initially found the result to be a crushing blow to my world view, that understanding would help performance. I think I still believe that understanding helps performance, but I'm forced to acknowledge that in arithmetic at least, and maybe algebra and some computational aspects of calculus, it may not.

That may be what's happening here with kids in Honors Algebra who can use a calculator (compute) but don't understand some aspects of algebra (comprehend). The two are probably independent. Both require teaching and learning, and one can't rely on one to help get the other. At least, that's what my old memory of that old research result suggests. I wonder if that old result is still regarded as true after 40 years?

...George

Follow Ups:

Innovation: A Golden Teaching Moment - Alex Leus 12/10/2010 
Innovation: A Golden Teaching Moment - Sharon Villines 12/1/2010