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.

Problem Posing in Mathematics Education

In Brown and Walter's article "Problem Posing in Mathematics Education" they discuss the recurrent themes that should be a part of a student’s mathematical experiences if “learning mathematics is to be viewed as an act of liberation.”  They call these themes sensitivities, and in this article in particular, shy away from the mathematical content, and instead focus on the pedagogical issues.  Each of these sensitivities revolves around the idea that mathematics is, at its essence, a “problem-solving activity.” Filled with examples and anecdotes from their team-teaching experiences, this article was an engaging look at how problem-posing can motivate and involve students, giving them a voice in an, otherwise externally-imposed, curriculum. 

            In their discussion of the efforts that go into creating an atmosphere of problem-posers, they describe three layers: context, posing, and solving.  Typically, a teacher would provide a problem given a particular context, and the students would be asked to solve it. In problem posing, a context, or scenario, is provided for which a problem is to be posed.  This gives the students creative reign over what kind of question they might ask of the scenario.  Brown and Walter hope that these individual efforts, creative endeavors, and opportunities to explore problems of their own choosing will result in students who take more responsibility for their learning. 

            I once taught an extracurricular problem-solving course in mathematics.  The engagement and creativity that students exhibited was truly impressive.  Looking back on my choice of problems, I think that it was their open-ended nature that was a major contributor to the high levels of engagement.  Giving students the choice of how they might attempt the problem was reflected in their range of answers and opened up the door for inspired learning and creative discourse: students defended their own methods and learnt from their classmates.  Open-ended problems could be a good gateway to developing the independence and confidence needed for students to pose their own problems.  

"On the Dual Nature of Mathematical Conceptions"

In the article “On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin” Anna Sfard discusses the dichotomies present in mathemactical concepts, contrasting them with such words as structural and operational, abstract and algorithmic, and declarative and procedural. Sfard focuses on the structural/operational dichotomy and attempts to find the bounds of these two modes, to find where they begin to overlap.  She goes on to rescind  her use, and academia’s use, of the word dichotomy claiming that, in fact, since these two terms are “inseparable, facets of the same thing” it should be called a duality instead as they complement each other.

A very interesting example of this notion in action revolves around the concept of number.  There exists a one-to-one structural relationship between the number of objects in a set and the words ‘one,’ ‘two,’ ‘three,’ but when a child is asked to determine how many in a set, they will repeat the process of counting, showing the operational side of the concept. So number represents both operation, (of measuring how many) and it represents a structure (of representing an amount).

In what order should these two different faces of a concept be learnt?  Or should it be strived to be simultaneous?  Near the end of the article Sfard makes me stop as she mentions that one of the ‘vexing issues’ facing math education is the apparent trade-off between profiency and comprehension, between “promoting the smooth performance of mathematical procedure and developing understanding of how and why that procedure works and what it means.”  What do you think?  Is this an intrinsic issue in learning mathematics, or is it extrinsic of learning math, introduced by ways of how we approach math education?

My personal pedagogic ideology

In Herbal-Eisenmann's paper "Examining the Voice of  Mathematics Textbook" she discusses how a teachers's classroom discourse can limit student learning experiences, and how the language used in textbooks "manifests particular ideological views in mathematics education."  Reading how Herbal-Eisenmann dissected the language of textbooks and how that positions the locus of authority was fascinating and gave rise to a wealth of questions in my own practice.   In particular interest were the use of imperatives in giving exact directing instructions to the student: “make a graph,”  “draw a straight line,” “explain your reasoning.”  Also of interest was the use of pronouns and the absence of the author’s voice, almost as if the textbook were an omniciescent power, a decree from an unknown.

I have been teaching high school mathematics for the last five years in the setting of a small classroom (with 5-10 students) or one-on-one.  Working in a highly supported school environment I realize, after reading this article, that I never assign my students to read through a textbook.  If they have troubles with course material then I will direct them to the wealth of resources we have on hand, namely my colleagues who's job it is to support students with exactly this issue.  This article makes me question my own discourse with my students, and how this is similar or different to what Herbal-Eisenmann describes in her paper.  Being permitted a wealth of time to work with students, I often take a slower, more exploratory approach to learning.  My use of pronouns would differ, obviously, from a textbook, as I am a very real voice in the students learning, not a non-corporeal textbook, and I try to focus on asking ‘how’ questions instead of directing students with imperatives. This personal pedagogic ideology, however, as Herbal-Eisenmann suggests, may certainly be limiting students’ learning in some aspects.  What do you think?

Building from the roof downwards, or the floor upwards?

The introduction to Bill Higginson's "On the Foundations of Mathematics Education" begins with reference to Gulliver's travels, remarking on the far-reaching societies that Gulliver encounters on his voyages. A quoted passage discusses how a math educator attempted to teach his pupils by giving to them a wafer to eat, upon which the theories they were to learn were written. From this passage, I imagine that Higginson will begin to discuss how math education is often conducted en masse, in a one-size-fits-all style, where the teacher is the keeper of the knowledge to be transferred to the pupil. The teacher will offer up their wisdom for the students to nourish themselves on, with hopes that they will be able to digest the offering. Higginson, I imagine, will discuss how math education treats the students who find these offering “too nauseous” to be able to digest, and will offer alternative ways of approaching math education instead of this top-down approach, where the teacher is the purveyor of wisdom.

On further reading, Higginson reveals his thesis statement: “We will not begin to make significant progress in dealing with [the difficulties of learning mathematics] until we more fully acknowledge the foundations of our discipline.” Higginson’s goal in this article is to broach a model of mathematics education called the MAPS-tetrahedral model, where MAPS stands for mathematics, philosophy, psychology, and sociology. He hopes that developing this model as the foundation for math education will help educators better understand their subject, and approach its teaching. Higginson concludes the article in a fascinating, humbling way, claiming that even if this model is shown to be “incomplete, logically flawed, or of very limited use” its ulterior motive, regarding the value it might contribute to help stimulate discussion between educators concerning the foundations of math education, will remain.