On Fri, 17 Jun 2011 13:33:04 -0700, Rick Froman wrote: >Mike Palij quoted an article from the New York Times by Benedict >Carey that said: > >"For years school curriculums have emphasized top-down instruction, >especially for topics like math and science. Learn the rules first - the >theorems, the order of operations, Newton's laws - then make a run >at the problem list at the end of the chapter. Yet recent research has >found that true experts have something at least as valuable as a mastery >of the rules: gut instinct, an instantaneous grasp of the type of problem >they're up against." > >Although I don't know where they go from there (and I know Mike >isn't quoting this in the context of agreeing with it), this is almost a >humorous >description of the confusion of cause and effect.
When I was in grade school I remember getting new yellow paperback books for math. These books used the "New Math" approach and they tried to provide basic rules, often in abstract form, to use on different problems. Looking back, I felt that the presentation was too abstract and, as Holyoak and Gick among others would show, it was hard to see the math operating in problems that had the same deep structure but different surface structures (the novice-expert research shows that experts do use deep structure relationships while novices rely upon surface features as a guide to problem solving). But this realization is not an old one. The Wikipedia entry (yadda-yadda) on "New Math" reviews some of the history and why it was ultimately rejected; see: http://en.wikipedia.org/wiki/New_Math Here's the Straight Dope's take on the New Math: http://www.straightdope.com/columns/read/1529/what-exactly-was-the-new-math >Just because experts may have developed a quick heuristic for learning >something (probably by starting with the traditional method of learning >the rules first, along with some examples to illustrate the rules), that >doesn't mean that the best way to teach it or learn it is to teach this >instantaneous grasp of the problem (or intuition). Maybe the so-called >intuition is actually the result of learning the other information and then >just learning to use the information most efficiently. When research on the nature of expertise was hot back in the 1980s and 1990s, (e.g., see Chi, Glaser, & Farr 1988 http://tinyurl.com/chi-expertise ) a key difference between novices and experts was the ability to classify problems according to general principles and/or the deep structure of the problem (the research with novices/students and experts in physics has shown that experts ignored the surface structure and define the problem in terms of the laws and principles involved -- one way to think about this is that the pattern recognition abilities of novices and experts are fundamentally different). One implication of this is if you want to use the heuristics that experts use, then you may have to have the knowledge that experts have that serves as the basis for the use of the heuristic. >To apply this to inferential statistics, I do feel sometimes that, certainly >compared to beginning students, I have an intuitive grasp on the type of >statistical analysis that will be appropriate for a given hypothesis. I don't >know that there is a shortcut to that result which I know in my case started >with a very traditional statistical education (doing HW exercises on paper >each >night in a five day a week class at my community college). I can't guarantee, >however, that I developed any kind of intuitive sense about it until I began >teaching Statistics in my second job out of grad school. I have developed a >decision tree that I think is more of an algorithm than a heuristic and I >don't >think anyone feels they are intuitively solving the problem when they are >using >the decision tree developed on the basis of the rules. Remember that people can use algorithmic processing to solve a problem in a well-define domain (e.g., math, logic, etc.) but heuristics will provide a quicker solution even if one doesn't know enough to solve the problem or even understand the problem. If you learn the relevant algorithms, then you should be able to apply them to the relevant problems you come across (but one does have to distinguish between problem where the algorithms are relevant from those when they are not). Achieving this probably requires a fair amount of rote work as well as problem solving that involves the use of insight to come up with a solution. >There is another part of the article (near the beginning) that I can imagine >can be useful for developing what I would call a conceptual understanding. > >"For about a month now, Wynn, 17, has been practicing at home using an unusual >online program that prompts him to match graphs to equations, dozens upon >dozens of them, and fast, often before he has time to work out the correct >answer. An equation appears on the screen, and below it three graphs (or vice >versa, a graph with three equations). He clicks on one and the screen flashes >to tell him whether he's right or wrong and jumps to the next problem." > >This sounds remarkably similar to one of the ways I teach the meaning of >scatterplots and correlation coefficients. There is a game at: > > http://www.stat.uiuc.edu/courses/stat100/java/GCApplet/GCAppletFrame.html > >that generates scatterplots and allows you to guess the corresponding >correlation coefficient. It produces them randomly so sometimes you will be >asked to make almost ridiculous distinctions, (like to visually distinguish >the >scatterplot of an r of .83 from a scatterplot representing an r of .84). It >can >be fun and does allow students to eventually come to an understanding of >scatterplots and correlation coefficients. Although this is good for its >purpose, I think it is also important to help students understand how a >correlation coefficient is calculated and how you can predict one variable by >knowing the score of another variable. I agree. Being able to go through a large number of examples/instances and being able to see what changes and why is what I would say is part of the dedicated practice that one needs to engage in in order to be able to learn (a) the general rules (what a linear relationship is like, especially when the relationship is only partially linear) and (b) examples of what different degrees of linear relationship look like. -Mike Palij New York University [email protected] --- You are currently subscribed to tips as: [email protected]. To unsubscribe click here: http://fsulist.frostburg.edu/u?id=13090.68da6e6e5325aa33287ff385b70df5d5&n=T&l=tips&o=11042 or send a blank email to leave-11042-13090.68da6e6e5325aa33287ff385b70df...@fsulist.frostburg.edu
