Two interesting articles on teachers’ knowledge

In the recent issue of Journal for Research in Mathematics Education, two interesting articles about teachers’ mathematical knowledge for teaching are published. One of these articles, “The nature and predictors of elementary teachers’ mathematical knowledge for teaching“, was written by Heather C. Hill. Here is the abstract of her article:

This article explores elementary school teachers’ mathematical knowledge for teaching and the relationship between such knowledge and teacher characteristics. There were few substantively significant relationships between mathematical knowledge for teaching and teacher characteristics, including leadership activities and self-reported college-level mathematics preparation. Implications for current policies aimed at improving teacher quality are addressed.

The other article was written by Courtney A. Bell, Suzanne Wilson, Traci Higgins and D. Betsy McCoach, and this article is entitled “Measuring the effects of professional development on teacher knowledge: the case of developing mathematical ideas“. The abstract of their article can be found below:

This study examines the impact of a nationally disseminated professional development program, Developing Mathematical Ideas (DMI), on teachers’ specialized knowledge for teaching mathematics and illustrates how such research could be conducted. This study adds to our understanding of the ways in which professional development program features, facilitators, and issues of scale interact in the development of teachers’ mathematical knowledge for teaching. Study limitations and challenges are discussed.

Preservice teachers’ conceptions of multidigit wholenumbers

Eva Thanheiser (Portland State University) has written an interesting article that was published online in Educational Studies in Mathematics this week. The article is entitled Investigating further preservice teachers’ conceptions of multidigit whole numbers: refining a framework, and in the article, Thanheiser digs into the domain of (preservice) teachers’ content knowledge of mathematics. Here is the abstract of Thanheiser’s article:

This study was designed to investigate preservice elementary school teachers’ (PSTs’) responses to written standard place-value-operation tasks (addition and subtraction). Previous research established that PSTs can often perform but not explain algorithms and provided a four-category framework for PSTs’ conceptions, two correct and two incorrect. Previous findings are replicated for PSTs toward the end of their college careers, and two conceptions are further analyzed to yield three categories of incorrect views of regrouped digits: (a) consistently as 1 value (all as 1 or all as 10), (b) consistently within but not across contexts (i.e., all as 10 in addition but all as 1 in subtraction), and (c) inconsistently (depending on the task).

How to develop mathematics for teaching and understanding

Susanne Prediger has written an article about How to develop mathematics-for-teaching and for understanding: the case of meanings of the equal sign. The article was published online in Journal of Mathematics Teacher Education on Thursday last week. Point of departure in her article is the very important question about what mathematical (content) knowledge prospective teachers need. A main claim which is raised already in the introduction is: “Listen to your students!” In the theoretical background, Prediger takes Shulman’s classic conceptualization of three main categories of content knowledge in teaching as point of departure:

  1. Subject-matter knowledge
  2. Pedagogical-content knowledge
  3. Curricular knowledge

She continues to build heavily on the work done by Hyman Bass and Deborah Ball (e.g. Ball & Bass, 2004), and she goes on to place her own study in relation to the work of Bass and Ball:

Whereas Bass and Ball (2004) concentrate on the first part of their program, namely, identifying important competences, this article deals with both parts, the analytical study of identifying, and the developmental study of constructing a sequence for teacher education, exemplified by a sequence in the course entitled school algebra and its teaching and learning for second-year, prospective middle-school teachers.

Here is the abstract of Prediger’s article:

What kind of mathematical knowledge do prospective teachers need for teaching and for understanding student thinking? And how can its construction be enhanced? This article contributes to the ongoing discussion on mathematics-for-teaching by investigating the case of understanding students’ perspectives on equations and equalities and on meanings of the equal sign. It is shown that diagnostic competence comprises didactically sensitive mathematical knowledge, especially about different meanings of mathematical objects. The theoretical claims are substantiated by a report on a teacher education course, which draws on the analysis of student thinking as an opportunity to construct didactically sensitive mathematical knowledge for teaching for pre-service middle-school mathematics teachers.

Bass, H., & Ball, D. L. (2004). A practice-based theory of mathematical knowledge for teaching: The case of mathematical reasoning. In W. Jianpan & X. Binyan (Eds.), Trends and challenges in mathematics education (pp. 107–123). Shanghai: East China Normal University Press.

Knowledge and beliefs

Much of my own research the last years has been related to knowledge and beliefs concerning mathematics, teaching and learning of mathematics. In the most recent issue of Instructional Science, Angela Boldrin and Lucia Mason have written an article that caught my attention: Distinguishing between knowledge and beliefs: students’ epistemic criteria for differentiating. Here is the abstract of this highly interesting article:

“I believe that he/she is telling the truth”, “I know about the solar system”: what epistemic criteria do students use to distinguish between knowledge and beliefs? If knowing and believing are conceptually distinguishable, do students of different grade levels use the same criteria to differentiate the two constructs? How do students understand the relationship between the two constructs? This study involved 219 students (116 girls and 103 boys); 114 were in 8th grade and 105 in 13th grade. Students had to (a) choose which of 5 graphic representations outlined better the relationship between the two constructs and to justify their choice; (b) rate a list of factual/validated, non-factual/non-validated and ambiguous statements as either knowledge or belief, and indicate for each statement their degree of truthfulness, acceptance and on which sources their views were based. Qualitative and quantitative analysis were performed. The data showed how students distinguish knowledge from belief conceptually and justify their understanding of the relationship between the two constructs. Although most students assigned a higher epistemic status to knowledge, school grade significantly differentiated the epistemic criteria used to distinguish the two constructs. The study indicates the educational importance of considering the notions of knowledge and belief that students bring into the learning situation.

Proof constructions and evaluations

Andreas J. Stylianides and Gabriel J. Stylianides have written an article called Proof construction and evaluations. The article was published online in Educational Studies in Mathematics on Friday. Here is a copy of their article abstract:

In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation” activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid. Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof, the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account of our findings, and we discuss implications for research and instruction.

Preservice teachers’ subject matter knowledge of mathematics

Ramakrishnan Menon has written an article entitled Preservice teachers’ subject matter knowledge of mathematics. The article has been published in International Journal for Mathematics Teaching and Learning. Here is the abstract of the article:

Sixty four preservice teachers taking a mathematics methods class for middle schools were given 3 math problems: multiply a three digit number by a two digit number; divide a whole number by a fraction; and compare the volume of two cylinders made in different ways from the same rectangular sheet. They were to a) solve them, explaining their solution, b) classify them as easy, of medium difficulty, or difficult, explaining the rationale for their classification, and c) explain how they would teach/help children to solve them. Responses were classified under three categories of subject matter knowledge, namely traditional, pedagogical, and reflective. Implications of these categories to effective math teaching are then discussed.

Activating mathematical competencies

César Sáenz from the Autonomous University of Madrid, Spain, has written an article called The role of contextual, conceptual and procedural knowledge in activating mathematical competencies (PISA). This article describes and analyzes the difficulties that Spanish student teachers had when attempting to solve the released items from PISA 2003. The student teachers (n=140) were first-year students, and they had not taken any mathematics courses in their teacher training at the time of the study. They didn’t have any experience with the PISA tests, and they had no more than secondary-level mathematics studies before they started their teacher education. The test they took was made from a collection of 39 released items from PISA 2003.

The article was published in Educational Studies in Mathematics on Sunday. Here is the article abstract:

This paper analyses the difficulties which Spanish student teachers have in solving the PISA 2003 released items. It studies the role played by the type and organisation of mathematical knowledge in the activation of competencies identified by PISA with particular attention to the function of contextual knowledge. The results of the research lead us to conclude that the assessment of the participant’s mathematical competencies must include an assessment of the extent to which they have school mathematical knowledge (contextual, conceptual and procedural) that can be productively applied to problem situations. In this way, the school knowledge variable becomes a variable associated with the PISA competence variable.

ZDM, No 5, 2008

For some reason, ZDM has published two December issues this year. I have already covered one of them, which is actually No 6, but I have not covered No 5 (both are December issues). ZDM, No 5 has a focus on Empirical Research on Mathematics Teachers and their Education, and it is a very interesting issue (for me at least), with 14 articles:

So, if you (like me) you are interested in research related to mathematics teachers and/or mathematics teacher education, this would certainly be an issue to take a closer look at!

A large part of the articles in this issue are related to the international comparative study: “Mathematics Teaching in the 21st Century (MT21)”. This study, according to the editorial, is the first study that has a focus on “how teachers are trained and how they perform at the end of their education”.

Diagnostic competentces of future teachers

Björn Schwarz, Björn Wissmach and Gabriele Kaiser have written an article entitled “Last curves not quite correct”: diagnostic competences of future teachers with regard to modelling and graphical representations. The article was published online in ZDM last week. Here is the abstract of their article:

The article describes the results of a national enrichment to the six-country study Mathematics Teaching in the 21st century (MT21)—an international comparative study about the efficiency of teacher education. The enrichment focuses on the diagnostic competence of future mathematics teachers as sub-component of teachers’ professional competence for which the evaluation of students’ solutions of a modelling task about the course of a racetrack is demanded. In connection with two sub-facets of the diagnostic competence, namely the competence to recognise students’ misconceptions and the competence of criteria-guided assessment of students’ solutions, typical answer patterns are distinguished as well as the frequency of their occurrence with regard to future teachers’ phase of teacher education and the level of school teaching they are going to teach in.

Future teachers’ professional knowledge on argumentation and proof

Björn Schwarz, Issic K.C. Leung, Nils Buchholtz, Gabriele Kaiser, Gloria Stillman, Jill Brown and Colleen Vale have written an article about Future teachers’ professional knowledge on argumentation and proof: a case study from universities in three countries, which was also published online in ZDM last week. It appears that a forthcoming issue of ZDM will have a strong focus on teacher education and teachers’ mathematical content knowledge!

Here is the abstract of the article:

In this paper, qualitative results of a case study about the professional knowledge in the area of argumentation and proof of future teachers from universities in three countries are described. Based on results of open questionnaires, data about the competencies these future teachers have in the areas of mathematical knowledge and knowledge of mathematics pedagogy are presented. The study shows that the majority of the future teachers at the participating universities situated in Germany, Hong Kong and Australia, were not able to execute formal proofs, requiring only lower secondary mathematical content, in an adequate and mathematically correct way. In contrast, in all samples there was evidence of at least average competencies of pedagogical content reflection about formal and pre-formal proving in mathematics teaching. However, it appears that possessing a mathematical background as mandated for teaching and having a high affinity with proving in mathematics teaching at the lower secondary level are not a sufficient preparation for teaching proof.