Sandra Crespo and Nathalie Sinclair poses this very interesting question in an article that has recently been published in Journal of Mathematics Teacher Education. The entire title of the article is: What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems.

Mathematical problems are an integral part of mathematical learning, and although most pupils encounter mathematical problems as they are posed in textbooks, the teachers have an important role in assigning appropriate problems for the students to solve. Prospective teachers have had few opportunities to focus on problem posing in their studies, and their experience with mathematical problems are mostly in connection with the solving of problems that are posed by the teacher or a textbook. The authors of this article “consider the practice of problem posing to be especially important for prospective teachers because a great deal of the work of teaching entails the posing and generation of what the mathematics education community often refers to as “good” questions—questions that aim to support students’ mathematical work”.

The main research questions in the study described in this article are:

1. What is the role of exploration in the problem-posing process? (What happens when prospective teachers pose problems with and without first exploring the situation that could motivate their questions? What kinds of questions do they pose in each of these two kinds of structured problem-posing setting?)
2. How do prospective elementary teachers decide on the quality of the questions they pose? (What rationale do they provide when asked to justify what makes their questions mathematically interesting? What is the effect of making explicit some of the qualities that make mathematics problems interesting and worth solving?)

The questions were investigated in a course that Sandra Crespo taught herself, and the course was offered in the fourth year of a 5-year teacher preparation program. A central theme in the course was a “pedagogy of inquiry” rather than one of presentation, and the students were given the opportunity to investigate different forms of mathematics teaching. There were 22 students in the course, and the researchers used four tasks and two classroom interventions in the study. The data consisted of written work from the students as well as field notes from observations of the students’ work with the given tasks, and from discussions in class.

Here is the abstract:

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’

Kelli Nipper and Paola Sztajn have written an article that was recently published in Journal of Mathematics Teacher Education. The article is entitled: Expanding the instructional triangle: conceptualizing mathematics teacher development.

Abstract As mathematics educators think about teaching that
promotes students’ opportunities to learn, attention must be given to
the conceptualization of the professional development of teachers and
those who teach teachers. In this article, we generalize and expand the
instructional triangle to consider different interactions in a variety
of teacher development contexts. We have done so by addressing issues
of language for models of teachers’ professional development at
different levels and by providing examples of situations in which these
models can be applied. Through the expansion of our understanding and
use of the instructional triangle we can further develop the concept of
mathematics teacher development.

Teachers are professionals with a rich knowledge that is both content specific and general. They shape instruction by the way they interpret and respond to students and materials (p. 2). The notion of “the instructional triangle” is based on the definition of instruction as (they refer to Cohen and Ball, 1999, p. 5 here): the interaction between teachers and students around educational material. These ideas are also shared by other researchers. One of them, Barbara Jaworski, created the teaching triad, consisting of:

• management of student learning
• sensitivity to students
• engagement in challenging mathematics

Nipper and Sztajn describe how they have tried to expand this instructional triangle to teacher education, and as a response to language issues, they suggest to replace the ordinary triangle: teacher – student – mathematics with the more general: organizer – participants – content. For a further elaboration of their analysis and theoretical suggestions, you should dig deeper into the article!

Lyndon C. Martin has written an article called Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie–Kieren Theory. The article is going to be published in The Journal of Mathematical Behavior, and it was available online yesterday at ScienceDirect. Here is the abstract:

The study reported here extends the work of Pirie and Kieren on the nature and growth of mathematical understanding. The research examines in detail a key aspect of their theory, the process of ‘folding back’, and develops a theoretical framework of categories and sub-categories that more fully describe the phenomenon. This paper presents an overview of this ‘framework for folding back’, illustrates it with extracts of video data and elaborates on its key features. The paper also considers the implications of the study for the teaching and learning of mathematics, and for future research in the field.

For another article discussing the Pirie-Kieren theory and related theories, you might want to take a look at this article by Droujkova et al. from PME29.

Göta Eriksson from Stockholm University has written an article in The Journal of Mathematical Behavior. The article is entitled: Arithmetical thinking in children attending special schools for the intellectually disabled, and it was available online yesterday. The entire article is available at the above link, but here is the abstract:

This article focuses on spontaneous and progressive knowledge building in “the arithmetic of the child.” The aim is to investigate variations in the behavior patterns of eight pupils attending a school for the intellectually disabled. The study is based on the epistemology of radical constructivism and the methodology of multiple clinical interviews. Theoretical models elucidate behavior patterns and the corresponding mental structures underlying them. The individual interviews of the pupils were video recorded. The results show that the activated behavior patterns, which are responses to well-adapted contexts presented by the researcher, are compatible with findings in Swedish compulsory schools. Six of the pupils’ mental structures in the study are numerical. A substantial implication for special education is the harmonization of the content in teaching with the children’s own ways of operating, which implies a triadic teaching process.

Erdogan Halat has written an article that has recently been published in International Journal of Mathematical Education in Science and Technology (IJMEST). The article is entitled: “The effects of designing Webquests on the motivation of pre-service elementary school teachers“, and here is the abstract:

The purpose of this study was to examine the effects of webquest-based applications on the pre-service elementary school teachers’ motivation in mathematics. There were a total of 202 pre-service elementary school teachers, 125 in a treatment group and 77 in a control group. The researcher used a Likert-type questionnaire consisting of 34 negative and positive statements. This questionnaire was designed to evaluate a situational measure of the pre-service teachers’ motivation. This questionnaire was used as pre- and post-tests in the study that took place in two semesters. It was administered to the participants by the researcher before and after the instruction during a single class period. The paired-samples t-test, the independent-samples t-test and analysis of covariance with = 0.05 were used to analyse the quantitative data. The study showed that there was a statistically significant difference found in participants’ motivation between treatment and control groups favouring the treatment group. In other words, the participants who designed the webquest-based applications indicated positive attitudes towards mathematics course than the others who did the regular course work.

Research in the didactics of mathematics has shown the importance of the problem of the particular and its relation to the general in teaching and learning mathematics as well as the complexity of factors related to them. In particular, one of the central open questions is the nature and diversity of objects that carry out the role of particular or general and the diversity of paths that lead from the particular to the general. The objective of this article is to show how the notion of semiotic function and mathematics ontology, elaborated by the onto-semiotic approach to mathematics knowledge, enables us to face such a problem.