After reading the first column, I predict
that the article will set a framework for publishing in the “mathematics
education” field(s) as we know it (them) today, compiled from three responses
by Pearla Nesher, Alan Bell, and Caleb Gattegno, written for a call for
manuscripts from David Wheeler.
It is likely that the rest of the article
will involve responses concerning the epistemological nature of qualitative
data collection in a field that depends on a “research programme”. Other
concerns likely involve the nature of data collection, number of scholars
researching, and establishing a reputable and beneficial field.
After having read the entire article, the three
main persons’ responses are broken down as follows:
1)
Pearla Nesher raises concerns
that the research programme proposed is in response to the negative reactions
toward the “New Math” movement, explaining that the research programme should
be a revision aimed at improving mathematical understanding as a language
system as opposed to using research for responding publicly to a failed programme.
The use of other disciplines will be important in developing a competency in
mathematics, rather than simply understanding cognitive processes
2)
Alan Bell notes that
mathematics education research should be used to achieve a better understanding
of conceptual tasks long-term, after tricks and memory aids have been
forgotten.
3)
Caleb Gattegno explains the
need for understanding the structure of numbers in classical mathematics, in
that children must be able to consciously create mental structures so as to be
able to produce the current existing (topological, algebraic, order)
structures. Algebra, according to Gattegno, was a pre-existing body formalized
by humans through the senses into a mathematical structure.
Nesher, Bell and Gattegno’s views toward
mathematics education and mathematical knowledge seemed largely positivist,
which I was surprised to find. It makes sense, given New Math hangover of the
1960s, which placed a high value on science and technology development. In
their views, mathematics is pre-existing and must be broken down in structure
and explored scientifically. For example, Nesher writes that mathematics
education must be researched in a “rigorous” manner, which often refers to the
use of scientific method (perhaps this is only my interpretation?). Alan Bell’s
consideration of mathematics education similarly casts doubt on a whole-class
teaching method comparison, and says that “results” (rather than results,
without quotation marks) were obtained through classroom observations.
My
understanding of the current field is that there is a great deal of concern
over validity of results. While obtaining results and considering validity is
always warranted, Bell’s one statement discredits the value of qualitative data
as being inferior to the quantitative. In a similar fashion, Gattegno explains the
connection between algebra and order, that these things are “out there”, to be
learned; it seems, according to Gattegno, that children learn mathematics from
scratch, rather than having a pre-existing knowledge of mathematics – knowing we
were all once children, this implies humans are external to mathematics until
it is bridged for our access.
The three authors' thoughts in your article make me think about our roles as math educators. Do we really have these goals in math education? In Higginson's article, math educator's altimate objective is to provide "an intellectually rich and emotionally satisfying experience for the learner" by providing "novel insights, generating "new questions" and clarifying "old problems". Had we overly emphasized too much on the math element in the math education?
ReplyDeleteAll in all, math learning is a life-long process. I personally believe that math educators should focus more on students' willingness to learn math rather than the math skills they learn from school.
I think that there is an interesting parallel between Wheeler's article and Kilpatrick's. Kilpatrick cites Scriven to suggest that there exists a lack of effectiveness or pay-off from existing research on mathematics education. Along this line, Wheeler's article requests for contributors to indicate tangible deliverables. It seems like both authors were seeking for something practical or realistic to come out of research programs or researchers. Wheeler tempers these expectations by acknowledging that successful research programs require enough committed people. I think that not only do research programs require enough committed people, but research programs also require sufficient resources over an extended period of time to sustain short and long term research studies.
ReplyDeleteThe issue on whether children learn math from scratch or they have a pre-existing knowledge is an interesting point. Children learn mathematics every day informally through interaction with their environments. Their everyday math knowledge serves as a basis for learning and understanding more formal ideas that they are introduced to in school. It is difficult to say whether children are born with a blank tablet in their minds and learn math from scratch. Or do they already have a pre-existing math knowledge from birth?
ReplyDeleteOne important aspect of education is to take students' affect into account. As educators, we need to help them develop a love of learning math using instructional methods that best suit their learning styles and maintain their interest in the subject. Hence, interest may lead to a positive attitude, not math phobia.
Interesting how important it was for these authors at that time to give a 'science-y' tone to mathematics education!
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