Article: The notion of historical "parallelism" revisited: historical evolution and students' conception of the order relation on the number line

Yannis Thomaidis and Constantinos Tzanakis: The notion of historical "parallelism" revisited: historical evolution and students' conception of the order relation on the number line, Educational studies in mathematics (2007) 66: 165-183.

This article combines historical study with classroom study, to examine the "relation between historical evolution of mathematical concepts and the process of their teaching and learning". They wanted to look at both whether there is a "parallelism" here and what such a parallel "between a creative mathematician of the past and a student learning mathematics in a modern classromm" might consist of.

The mathematical topic considered is the order relation on the number line - that is (for instance); is -4 greater than or smaller than -2? In section 2 of the article, the authors trace the development of mathematicians' thought on this. For instance, Newton wrote that "the greatest negative [root]" was the one "most remote [from zero]", that is -4 > -2. Also, Bolzano used the notation e<±1 for which we would write -1< e< 1.

The study of students posed three questions, of which the first one was:
"What are the solutions of the inequality x² > 9 when x ∈ ℜ?"

Lots of interesting answers were given to the questions. For instance, it did turn out that some students had the same basic ideas as Newton and Bolzano mentioned above.

The authors argue that there are parallelisms that could be exploited to

"forsee possible persistent difficulties of the students; and to make teachers more tolerant towards their students' errors, by increasing their awareness that these errors and difficulties do not simply mean that "the student has not studied enough" but may have deeper epistemological roots which should be explored and understood thoroughly."

but also

"become more tolerant towards non-conventional, but essentially correct, views of their students, even though they may be wrong or insufficient by modern standards of logical rigor and clarity; [...] encourage these and other students to express their possibly idiosyncratic views on specific mathematical problems and in this way to implicitly guide them towards conceiving mathematics as an exciting human activity".

Reason enough to include history of mathematics in teacher education, in my opinion...

I will derive

Article: Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking

Issue number 2 of Educational studies in mathematics 2007 was a special issue on history of mathematics. For some reason, I haven't read it in full before, but now is as good a time as any. In the following days I will blog about the articles in this special issue.

Luis Radford and Luis Puig: Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking, Educational studies in mathematics (2007) 66: 145-164.

There are a few main concepts in this article, and by pointing to these, I will give an idea of what the article is about:
The Embedment Principle: "our cognitive mechanisms (e.g. perceiving, abstracting, symbolizing) are related, in a crucial manner, to a historical conceptual dimension ineluctably embedded in our social practices and in the signs and artifacts that mediate them."

Zones of proximal development of the culture: "new mathematical ideas are answers worked out in the historically situated zones of proximal development of their cultures".

The authors argue that "phylogenesis cannot recapitulate ontogenesis". "To learn algebra is not to construct the objects of knowledge (for they have already been constructed) but to make sense of them."

The article goes on to look at how this can be explored in an 8th grade class' work on fractional equations. The authors look at a word problem, in which students successfully writes the algebraic equation. In this form, the students recognize every part of the equation as a representation of a corresponding part of the word problem. However, when trying to solve the equation, the correspondence with the word problem is lost, and the operations take over as the "main personages".

This is an interesting point which I will try to keep in mind next time I teach equations.

The point about learners not "constructing" the objects of knowledge ("for they have already been constructed") is not entirely convincing to me, however. Why not "construct anew"? I agree that when students learn algebra, they do not on their own construct the objects of knowledge from nothing, but the word "construct" still makes sense to me. Or maybe I would prefer to think of it as a combination - partly (re)constructing and partly making sense?

(Obviously, I need to think more about this...)

Teacher pay

Freakonomics writes about an idea for teacher pay: "What if teachers were paid based on the future income their students make."

It's interesting to see this idea. I wrote about the same idea in a Dagbladet article in 1999".

And I continued: "This would have the interesting consequence that a teacher that encourages his pupils to become stock brokers, can get a better salary than a teacher that gets his pupils to become nurses. And that is good - the society does award stock brokers better than nurses because stock brokers are more productive, doesn't it? But if the teacher himself becomes a role model for his pupils, so that they also become teachers, he would probably not get a raise - again exactly what society wants, isn't it?"

I do hope the irony shines through...

Ken ken

This summer, I tried Ken Ken for the first time. This is a puzzle related to Sudoku, but with even more connection to mathematics. Depending on the difficulty level, addition, subtraction, multiplication and division can be involved.

I think this could be a welcome diversion when there is time to kill – if a few pupils get hooked on Ken Ken, their mathematics skill may improve considerably. Have a look at KenKen.com.

A Mathematician Reads the Newspaper

I came across “A Mathematician Reads the Newspaper” by John Allen Paulos in a bookstore the other day, and had to pick it up, partly because of the blurbs on the back, for instance: “A wise and thoughtful book, which skewers much of what everyone knows to be true.” (Los Angeles Times)

I do understand that I’m not in the target group for the book. For me, quite a lot of what the book says is things that I already knew (which may not be surprising, as I have worked in mathematics education for 12 years, following my mathematics studies). The book is still interesting to read, however, as there are many interesting examples included. A bit more worrisome is that Paulos obviously struggled to fill a whole book, and at times the link between the newspaper and the mathematics he wants to discuss is a bit strained (as in the incidence matrices on page 189-).

However, many of the topics he covers are important and well worth revisiting: voting, chaos, coincidences, Ponzi schemes, statistical tests and meaningless precision, to name just a few.

(To exemplify the two last ones from the list: He is wary of journalists writing about polls without mentioning the margin of errors and of recipes stating the number of calories as 761, for instance.)

To conclude: it was an interesting book, but also a bit disappointing, given the hype on the cover.

InSITE: Didactics of ICT

Said Hadjerrouit of University of Agder had a talk on Didactics of ICT in Secondary Education. Hadjerrouit first described a lack of research on the didactics of ICT, and that he believed that there would be much to learn from the didactics of mathematics. I agree. In mathematics, there has been much research done on separate areas of mathematics (algebra, numbers, geometry, probability etc), and this has meant that the field of mathematics education has never become a purely theoretical subject, but has remained close to the actual teaching tasks. It seems to me that ICT does not have such a clear division into subfields, and that therefore, didactics of ICT may risk becoming too general and not too useful.

Hadjerrouit has taken part in small-scale projects with teachers, students and pupils in secondary school to learn ICT (not just use it), but from the presentation I got no clear sense of what the outcome of these was. Surely, the paper could be consulted for more on this.

InSITE: Ground Rules in Team Projects

Janice Whatley’s talk titled Ground Rules in Team Projects was based on her PhD. She has created a computer system to automate parts of the process of deciding on “ground rules” when students are to start group work. The students answer a poll on which ground rules they find useful for their group, and based on the answers of the students in the group, a set of rules is suggested. The students concluded that it was useful to have a discussion on group rules early on, but it is a bit unclear for me whether the use of the computer system was a critical factor.