A brief analysis of HPM papers

I completed a review of papers published in the proceedings for the HPM 2000 conference in Taipei and the HPM 2008 conference in Mexico City with the intent of answering three questions:

  1. What kinds of empirical sources are we utilizing in HPM research?
  2. What school levels are we doing research for?
  3. What kinds of teaching with history do we advocate?

I will discuss several ideas from the analysis in this short report.

I analyzed 49 papers from the proceedings of the HPM 2000 conference proceedings and 71 papers from the HPM 2008 proceedings. Without reading all 120 papers in full, I tried to establish what kinds of empirical sources they were based on, the age level of the students for which the research is relevant, and the teaching methods that were either advocated or referred to in the article. Often, no age level is mentioned; in such instances I tried to estimate an age range based on my own experience.

Unsurprisingly, almost all articles were based on empirical sources (primary or secondary) from the field of history of mathematics. The whole field of HPM is meaningless unless the history we teach is founded in historical sources – and indeed, the HPM group has a long tradition of including articles giving historical accounts of the development of concepts that are interesting to mathematics educators.

However, 52 of the articles also included empirical sources outside of the history of mathematics. These are (a) authors’ own experiences with teaching some topic or another (19), (b) planned experiments using history (15), (c) some sort of questionnaire or assessment to obtain qualitative material (11), (d) classroom observations of some sort (8), (e) interviews with teachers or students (8), (f) students’ evaluation of a teaching sequence (6), (g) students’ work samples (4), and (h) students’ written logs (2).

The percentage of articles not based on empirical sources “from the classroom” increased between 2000 and 2008. This could be partly because of an increase in the percentage of articles on the history of mathematics education.

The search for which pupil age groups the papers were relevant – when this information was not explicitly given – was done by estimating. Often it was necessary to estimate with quite a broad age range. I used the grades in the Norwegian school system, where grade 1 corresponds to age 6. In Norway, grade 14 would be the first year of university. The analysis shows that there are many more papers relevant for pupils in grades 8 – 13 (or, pupils aged 13 – 18) than for younger pupils (see Figure 1). There may be good reasons for this, one of which is probably that the researchers in the HPM group have more experience teaching mathematics with history to older students.

Relevance of articles according to pupil age

Fig. 1: Relevance of articles according to pupil age

When it comes to methods of including history of mathematics in teaching, there was a wide variety expressed in the articles, even though the traditional suggestions of giving historical introductions on topics in the curriculum and creating exercises based on the history are the primary ones. (It is worth noting that each of these includes a wealth of different approaches.) The idea of having the students work on original sources is only advocated in eight papers. Other notable suggestions given more than once are for pupils to complete hands-on work (e.g., building things) based on historical methods and using dynamic software along with historical sources to explore the mathematics. But the most striking finding is that most articles leave it up to the reader to figure out how the ideas in the article could be transformed into teaching.

Of course, this very brief analysis does not provide answers to the questions raised in the beginning of this article. It is meant instead as a suggestion and a small contribution to the debate on what kind of research HPM is, and should be, doing.

Bjørn Smestad

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