The Decline and Rise of Geometry in 20th Century North America

This week I read "The Decline and Rise of Geometry in 20th Century North America" by Walter Whitely, a very interesting read from someone on 'the inside' of research in mathematics and in math education.  He begins by discussing the death of geometry in the mid 20th century.  This is most notably chronicled by the decline of its consideration as an important field.  This had a cascade effect: less research was done on the topic, so less graduate curriculum was taught on the topic, so fewer future math educators were able to teach the topic, which eventually relegated geometry to be a supplemental high school mathematics course.  Whitely recounts geometry's fall from grace in wonderful detail, culminating the decline with a discussion of the nature of mathematics, that the public would believe that "mathematics is essentially about logical intelligence," and that "popular culture sees mathematics as detached from spatial intelligence" (p. 3).  As for the rise of geometry in recent years, Whitely credits much to do with the resurgence with computers. As a resource for learning, computers provide dynamic geometry programs which can be used for teaching, learning, and research.  They also provide an unparalleled visual aspect that is so important for developing understanding.              

Personally, geometry is one of my favorite topics within math.   Finding unseen relationships that can be linked to a visual representation is so satisfying.   Geometry gives a way to very easily provide the context for a problem that students can immediately understand, at least in a visual sense.  This understanding can lead to powerful problem posing, and intuitive problem solving. 

The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof.

A very interesting article this week by Jonathan Borwen called "The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof."

The first paragraph of this article had me hooked.  The premise:  computers are becoming so powerful that proving by "quasi-inductive methods" is becoming a viable way for proof.  Where does this leave the mathematician?  Borwen explores the "relationship between proof (deduction) and experiment (induction)."  He carries on to show a large number of examples where hand-deduction, and computer-induction proved successful methods in helping to illuminate a problem.  

          His discussion of large and small numbers I found to be very engaging.  Statistics relies on the ‘law of large numbers,’ where correlations, connections, and relations are forged between quantities of data that humans could not find (potentially).  As mathematics ventures into more abstract domains, it too ventures into large-number-territory, however, not with the quantity of data, but rather simply because often you are dealing with a very large or hard-to-find number (like the trillionth digit of pi).  In both of these cases, computers offer a reliable way to access these kinds of numbers.  Small numbers, however, is where mathematicians can shine.  Small numbers, Borwen argues, are important numbers of which a computer cannot grasp the significance of.  He cites wonderfully concise proofs of √2’s irrationality.

          Concluding with one thought of “there are different versions of proof or rigor” was a rewarding way to finish the article, as Borwen points out that historical proofs have since been reneged and renewed on the basis that the standards of rigor change over time.  Computer-assisted proofs may offer a current ‘standard of rigor,’ but if it will stay the test of time is to be determined.

Mathematics in the Streets and in the Schools

This week I read a very interesting article by Carraher, Carraher, and Schliemann titled "Mathematics in the Streets and in Schools."  This article explains how street vendors in Brazil do mental arithmetic to calculate the prices of goods, and to calculate change.  The catch is, is that these street vendors are children, who's average age is around 11.  The researchers set out to determine the children's abilities to do similar calculations that they would on the streets but in a formal setting.   The researchers found that, across the board, these children were able to do calculations better in the informal setting of street vending than they were when the same or similar questions were posed to them in formal settings.  

These are very interesting results, and they give rise to several interesting discussions surrounding the topic.  In North America we tend to heavily favor first learning procedures in abstract settings before applying them.  One motive for this is that there are fewer factors to take into consideration when solving a problem; they are streamlined for a particular task.  This research, however, shows that this may not be the best method. Having never been taught in this particular way, these Brazilian children are now able to solve problems in a variety of methods depending on the context. 

In the end, I am not surprised with the results from this study.  Children practicing to do math in a specific context will inevitably do better than their peers who practice in a different context.  The flexibility with which the Brazilian children answer problems, however, is something that I think that we should strive to teach to our pupils.