Essay #1

7/2/00

 

Who constructs mathematics and for what?

 

 

I am taking a course on “conceptual learning”. But if anybody would ask me what we learn there, I would say it is a full-fledged cognitive psychology course that I have no problem with it. People who are from social sciences would know at least few words about “constructivism”. May be more than they would like to! For the people, whose backgrounds are totally different, I want to explain what it is from the literature.

 

Constructivism seems to take its origins from the work of Piaget. In some books his work is identified as “Psychological Constructivism” (Phillips and Soltis, 1998) as well as cognitive psychology. He was arguing that every child goes through the same developmental stages in consecutive manner. And the process takes two big processes as assimilating the information and adapting yourself to it. Vygostky (1978, cited in Brunning et.al., 1999) emphasized the social environment for learning. His ideas are named as “Dialectical Constructivism”. He was referring to the necessary developmental zone for each children (zone of proximal development) to learn the things by himself/herself or with the help of an adult. Von Glassersfeld (“Radical Constructivism” is on the other hand argues that teachers can not assume that their understanding would resemble to the children’s. Main idea behind this belief is that the meanings that we attach to the events all around us, and the explanations that we come up for the world, can not come to a perfect agreement never, because there is not a sole understanding that can explain all the events. All explanations are socially constructed and there is no one true understanding of the world around us.

 

Duffy and Jonassen (1992) compare the objectivist and constructivist approaches to the instructional design. They state that in the objectivist tradition, the goal is to find or determine the complete and correct understanding as well as understanding entities, attributes, and relations that exist. And this has some implications for the student learning as the aim of the educator should be to find and identify those “need to learn” baseline for a learner. This view has dominated our educational systems for nearly all of the last century. Constructivists argue that there are many ways to structure the world, and there are many meanings attached to it, hence there should not be a correct meaning or explanation that we are striving for.

 

After this brief history, I need to say that I am no longer a mathematician so I should not talk about mathematics solely and I will not. But mathematics education is something I am in and I think we have the opportunity to see mathematics from a different perspective that real mathematicians lack. This may be named as how to learn and what to learn in mathematics. These two questions are intriguing me for the last few years.

 

In our course that I have started with, we had a discussion about “radical constructivism” and its premises. Our professor mentioned that the radical constructivists defend that even mathematics that we have know is a form of explanation and we should not be sure that it would keep its rightness for the rest of the time. Their example was thinking about an explanation other than 1+1=2 for 1+1. It was like that, it has been and it will be !

 

I have a problem with that as they do, since even in mathematics sometimes we change our understanding for 1+1=0 (mod 2) as in clock arithmetic. Hence, even mathematics what we have built up to now, is context specific, and bounded with some rules. And if those rules change, mathematics seem to change! This is really a paradox for a lot of objectivist scientists who believe what they learn from their childhood so fullheartedly. I remember once in our evaluation class, our professor was mentioning that is “objectivism” really objectivist? Or can it be? I believe that mathematics is a collective art, that has been built in centuries. But does this explanation assures its existence as an unchanging entity or an absolute truth? I believe, not.

 

I should mention that I am not a full radical constructivist, but I am not a full objectivist neither! I believe that there is an absolute truth for the universe for each moment that we live by. But, there is also an impossibility to explain it in fully objectivist terms. We can only hope that we are coming close to that absolute truth day by day, but we should also be a little bit skeptical about each explanation that we come by, especially each day following. Since, neither the world nor the universe is once what Galileo, Newton, Leibniz,etc, were trying to explain or were experiencing. We as humans have changed it, are changing it, and will change it!. Need to think about the rules, explanations, formulas, meanings that we have attached to what we see, otherwise we may be named as those “silly” scientists who once were thinking that the world is flat, and on the shoulders of some big elephants!

 

Finally, this essay is a try to get the attention of scientist educators or scientists, not a try to attack them. I think their value as educators is so important for the next generation scientists to emerge without hopeless aimes or so-called statements like "life made me this way, I did not choose it" behavior.

 

References

 

Brunning, R.H., Schraw, G.J., & Ronning, R.R. (1999). Cognitive Psychology and Instruction. Columbus, OH: Prentice Hall.

 

Duffy, T.M. & Jonassen, D. H. (1992). Constructivism and the technology of instruction: A conversation. Hillsdale, NJ: Lawrance Erlbaum Associates, Publishers.

 

Phillips, D. C. & Soltis, J. F. (1998). Perspectives on learning. NewYork, NY: Teachers College, Columbia University.

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Ozlem Cezikturk

 

P.S. Any kind of feedback is welcome, pls e-mail me your comments: oc7815@csc.albany.edu