Oct 14, 2009
My friend and former mentor, John Golden, has a great math ed blog where you can find interesting problems, comics, and thoughts from Grand Valley's math department.
Mar 17, 2009
Dec 08, 2008
What is Mathematics?
I believe that a primary source of the problem of mathematics' standing as a despised (perhaps the despised) school subject is misaligned definitions between laypeople and mathematicians of what mathematics is. Said another way, the public has a deficienct conception of mathematics as being roughly synonomous with calculation/computation or at least having a lot to do with numbers.
This morning I woke up to NPR radio and heard them interviewing a "math prodigy" talking about the benefits and burdens of being such. This prodigy is essentially a 17-year-old human calculator. If you are a mathematician, you will be interested to know that he is not especially proficient in formulating useful and non-redundant definitions of mathematical entities, grasping the logical structure of axiomatic systems, or generating insightful and elegant proofs (or if he is they didn't talk about it on the radio program). So where is the math in his math prodigy? Or is he really a computing sensation?
The young man said that it all started when he was 3 years old with his father who taught him to add and subtract. This got me thinking -- yes, it can start with addition and subtraction, but the question is: where does it go from there? I think that for many people, math goes from addition and subtraction to much bigger and faster addition and subtraction as well as multiplication and division. But for me, math goes from addition and subtraction through logical progression of ideas and concepts that continually shed light on the structure of time, quantity, space, and shape. Addition and subtraction are a fine starting place as these operations give us an abstract handle on the phenomenon of joining and separating, for instance. But a true mathematical prodigy would not be happy merely becoming very good at performing such operations --- a mathematical prodigy would ever seek to reveal, to himself and perhaps to others, more and more of the structure of the world in which we live (that's the "math part"), and s/he would reach these revelations remarkably quickly and efficiently (that's the "prodigy" part).
Oct 15, 2008
Research and Practice Discussion
I had another interesting confluence of course ideas this month as I read the latest edition of Educational Researcher. Bulterman-Bos (2008) wrote an article presented her ideas about how a clinical approach can be used to blend the disparate roles of researcher and practitioner, thus making education research more relevant. She works from Labaree's (2003) distinctions between the world of research (analytic, intellectual, universal, theoretical) and the world of teaching (normative, personal, particular, experiential). (In proseminar we've been thinking about the link between research-practice, and last week in my research design/method class we focused on clinical interviewing.)
After this article, the issue included responses from several other education researchers. Labaree himself argued that the solution to the relevance problem is not to merge the two perspectives, because they each have value in their independent state. Rather, the barrier between the two camps needs to be reduced and communication/collaboration should be increased. Labaree also argues that attempts within research to be immediately relevant can often lead to stale results, whereas basic research often "ages well." So relevance is not only impossible to define, but is also not necessarily predicted by the intention of relevance.
I did not read the responses from Lagemann and Noffke, but I did read the one from Dylan Wiliam. He referred to Pasteur's Quadrant and also brought up Aristotle's intellectual virtues of episteme, techne, and phronesis -- with phronesis (practical wisdom, the ability to weigh contingent and variable conditions particular to the situation together with rationality and general principles) being identified as a way in which to conceptualize Bulterman-Bos's call. I especially connected with the point that Wiliam made when he pointed out that episteme and phronesis are not equivalent to quantitative and qualitative approaches, respectively. Phronesis rejects the either/or in favor of a both/and approach. "Phronetic social science is problem-driven and not methodology-driven," so it is appropriate to use whatever research methodology can best investigate particular research questions. This is often a mixture of the methods, which agrees with my budding personality as a "mixed methods man."
Sep 25, 2008
Sfard and Watson
This week I have experienced a sort of confluence of readings that was quite interesting. In CEP 911 (Intellectual History of Educational Psychology) we have recently read some of Watson and Thorndike's work which was the foundation of behaviorism. I took many things away from the readings of these historic pieces, two of which were the idea that educational research should be based on observable phenomena (i.e., behaviors) and the definition of thinking as "talking to oneself". Of course, behaviorism has been demonized by many educational researchers and practitioners nowadays because they are too restrictive and their vision of learning is oversimplified. (Whether or not this demonization is fair is not for me to say, though the case could be made that what has been demonized is not the original theory but a strawman.)
Just last night, I was reading some recent work by Anna Sfard on her commognitive framework (from the Journal of the Learning Sciences). Interestingly, she said that research should focus on phenomena that is investigable (observable?) and she defined thinking as communicating with oneself. Now, I am certainly not saying that Sfard is a behaviorist, but I do find this return to a somewhat familiar core that she is trying to usher in quite intriguing from a historical perspective.
Sep 19, 2008
I am currently a research assistant on two math ed studies -- both heavily qualitative, and I am a big fan of the work we're doing. However, I am at a point in my career (read: very early on) where I should not just take methodological philosophies for granted and should think reflectively in a way so that I may come to my own professional position. (Not to spoil the ending, but I expect that I will be a "mixed methods" man who believes that both quantitative and qualitative methods are appropriate, depending on the situation and the questions being asked).
Here are a few brief thoughts I had while reading a chapter from Creswell's book on Research Design.
"I can understand the value of researching in the 'natural settings' where the things are really going on, but I am a bit more hesitant about the researcher being so subjectively integral in the data collection, analysis, interpretation, etc. This seems to open qualitative research up to the same charge that was leveled by the behaviorists against introspective psychology; namely, that there are as many analyses as there are individual researchers. But, my question is, is it even possible to conduct quantitative (or objective) research in natural settings?
Regarding the issue of particularity (the aim of qualitative research) and generality (the aim of quantitative research), I was wondering if it is somewhat disingenuous to conduct exclusively particular research. What I'm thinking is that human beings have a strong, almost automatic, tendency to generalize things we see -- this is how we make sense of the world. So even if something is presented as 'particular', the tendency will be for people to generalize it anyway (maybe even the researchers themselves, as was seen at the colloquium on video clubs last Wednesday). Wouldn't it, then, be more fitting to have generality as the aim so that we don't all have to fight our urges? (I probably already know the answer to this, and it's the same answer as always -- balance -- but it's interesting to think about, nonetheless.)"