This article describes a study in which measures of mathematical knowledge for teaching developed in the United States were
adapted to measure mathematical knowledge for teaching in Ireland. When adapting the measures it was not assumed that the
mathematical knowledge used by Irish and U.S. teachers is the same. Instead psychometric and interview-based methods were
used to determine a correspondence between the constructs being measured, and ensure the integrity of item performance in
the Irish context. The study found overlap between the knowledge that is used to teach in both Ireland and the United States,
and that the items tapped into this knowledge. However, specific findings confirm the usefulness of conducting extensive checks
on the validity of items used in cross-national contexts. The process of adaptation is described to provide guidance for others
interested in using the items to measure mathematical knowledge for teaching outside the United States. The process also enabled
the authors to raise questions about the assumptions that lie behind the practice-based construct of mathematical knowledge
The findings presented in this article were derived from a 3- year study aimed at examining issues associated with the use of computers for secondary mathematics learning in Victorian (Australia) schools. Gender and other equity factors were of particular interest. In this article, the focus is on the participating mathematics teachers. Data on their perceived competence levels with technology, and their use of and beliefs about computers for their male and female students’ mathematics learning were gathered. A clear majority of teachers felt comfortable about, and did use, computers for teaching mathematics, and believed that computers helped students’ mathematical learning. Generally, the teachers considered boys to be more confident and capable than girls with computers. The results have implications for pre-service education programs and for the professional development of practicing secondary mathematics teachers.
In this article, I draw on post-structural and feminist epistemologies to analyse interview data from two prospective teachers
on a primary education degree. Specifically I use Foucauldian critical discourse analysis to discuss the competing discourses
of the masculine mathematician and the feminine primary school teacher. The initial purpose of the article is to deconstruct
the themes of control, choice and confidence, which I argue are prevalent within mathematical discourses within our current
neoliberal society. A further aim of the article is to explore the representation of discourse and data within educational
texts, which I do by experimenting with the language used throughout.
This study investigated the effects of a classroom intervention on prospective elementary teachers’ ability to evaluate evidence
of student achievement of mathematical learning goals. The intervention was informed by a framework for teacher education
which aims to provide prospective teachers (PTs) with the skills needed to systematically learn from their own teaching practice.
Prospective teachers (N=160) participated in an intervention aimed at addressing their misconceptions about evidence of student learning. Results
revealed that after the intervention, PTs were less likely to consider teacher behaviors to be evidence of student learning
and more likely to discount student responses that were irrelevant to a specified learning goal. However, PTs were still likely
to take procedural fluency as evidence of conceptual understanding and may have become overly skeptical of student understanding.
Implications of the study suggest new ways of developing prospective teachers’ ability to systematically study and improve
KeywordsProspective teachers–Evaluating teaching–Analyzing teaching–Analyzing student work–Learning to teach–Teacher preparation
In this article we analyze the relations between academic mathematical knowledge and the mathematical knowledge associated
with issues mathematics school teachers face in practice, according to the specialized literature, and restricted to the theme
“number systems”. We present examples that illustrate some areas of conflict between those forms of knowledge. We point out
some implications of our study for teacher education, such as: 1) the importance of making conflicts explicit and of discussing
them with prospective teachers in order to develop a professionally relevant perception of academic mathematics; 2) the relevance
of further research in order to better understand the extent of those conflicts and their effects on the process of integrating,
in a body of professional knowledge, the different kinds of mathematical knowledge presented to prospective teachers.
This report presents an account of one teacher's mathematics teaching and a perspective that underlies his teaching. Nevil
was a fifth grade teacher participating incurrent mathematics education reforms in the United States. Through the account,
we make distinctions about teachers' thinking and practice that can inform teacher education efforts. We constructed an account
by analyzing four sets of classroom observations and interviews. We observed that Nevil decomposed his understandings of the mathematics into smaller components and connections among those components. He created situations that
he believed made those components and connections transparent and attempted to elicit those connections from the students.
This account illustrates a practice that is different both from traditional practice and the type of practice that we would
envision as a goal for teacher development. We contribute two important aspects of mathematics teacher development from traditional
to reform-oriented teaching. In particular, we describe teachers' perspectives – assimilatory structures that constrain and
afford (a) the sense they make of professional development opportunities and (b) their potential learning in teacher education
Teacher learning through professional development is a complex process and is not yet well understood. Some features of professional
development programs are known to be important, such as a focus on learner needs, design of and reflection on classroom artefacts,
and the creation and sustaining of communities of support for teacher professional learning. In this paper, we describe the
workings of such communities in a teacher professional development program, which focused on learner errors in a well-researched
mathematical topic—the equal sign. Drawing on data from program sessions where teachers discussed their lesson designs and
reflections on their teaching with each other, we develop the notions of challenge and solidarity as important in developing
accountability conversations among teachers. We show how our program supported teachers to challenge each other and to build
solidarity with each other and in so doing to develop accountability to each other and the profession, for their practices
and their learning.
KeywordsProfessional development–Professional learning communities–Accountability
Grade 6 teachers (N=106) in one school district were randomly assigned to early or late professional development (PD) groups. The program focused
on reform communication and incorporated principles of effective PD recommended by researchers, although the duration of the
treatment was modest (one full day and four after school sessions over a ten-week period). At the post-test, there were no
statistically significant differences in student achievement. Although it could be argued that the result demonstrates that
PD resources should be redirected to more intensive PD delivered over longer periods, we claimed that the PD was assessed
prematurely. After the completion of the study, the external assessments administered by the province showed a significant
increase in student achievement from one year to the next involving both the early and late treatment groups, an increase
that was not found for the same students in other subjects. The study had high ecological validity: it was delivered by district
curriculum staff to all grade 6 teachers, volunteers and conscripts alike. The cost to the district, less than CAN$14 [9 euros]
per student, was comparable to the modest expenditures typically available for professional development in
Canadian school districts.
This article examines the pedagogical tensions involved in trying to usestudents' ideas as the basis for class discussion while also ensuring thatdiscussion is productive mathematically. The data for this study of theteaching of one middle-school teacher come from observations andvideotapes of instruction across a school year as well as interviews with theparticipating teacher. Specifically, the article describes the teacher'sattempts to support a student-centered process of mathematicaldiscourse and, at the same time, facilitate discussions of significantmathematical content. This tension in teaching was not easily resolved;throughout the school year the teacher shifted his emphasis betweenmaintaining the process and the content of the classroom discourse.Nevertheless, at times, the teacher balanced these competing goals by usinga ``filtering approach'' to classroom discourse. First multiple ideas aresolicited from students to facilitate the process of student-centeredmathematical discourse. Students are encouraged to elaborate theirthinking, and to compare and evaluate their ideas with those that havealready been suggested. Then, to bring the content to the fore, the teacherfilters the ideas, focusing students' attention on a subset of themathematical ideas that have been raised. Finally, the teacher encouragesstudent-centered discourse about these ideas, thus maintaining a balancebetween process and content.
The aim of this study is to characterize the discourse of two problem-solving courses for prospective teachers. The data,
consisting of audio recordings and field notes, were examined from a dialogical approach combined with the theory of contextualization.
I show not only the substantial differences between the two classroom discourses but also elaborate on plausible reasons for
the divergency found. The findings then serve as a basis for a discussion of how to develop a problem-solving course within
the mathematics teacher program.
This article discusses the design of tasks for teacher education. It focuses on tasks that are used in a university course
for pre-service secondary school mathematics teachers. Special attention is given to tasks that use analogical thinking in
their construction or implementation. These tasks are categorized by type of teacher education goal and analyzed with respect
to the use of analogical thinking. Short examples are presented for three of the goal categories, while an elaborated example
is given for the fourth one. The detailed example describes the goals and design of a task sequence following an emergent
pedagogical need. Using the ad hoc constructed task-sequence the teacher educator avoids “telling” while demonstrating an
alternative instructional approach, and seizing the opportunity to bring up additional pedagogical issues.
The goals of this study are to understandelementary school teachers' beliefs andpractices and to unveil factors that influencethe way teachers adapt mathematics reformrhetoric when trying to adopt it. In theresearch, I searched for beliefs beyondmathematics that influence teachers' decisionsand choices for teaching mathematics. Workingwith children from different socioeconomicbackgrounds, teachers interpret reform indifferent ways. Based on their concept of students' needs, teachers select which partsof the reform documents are appropriate fortheir students. While children from uppersocioeconomic backgrounds experience problemsolving, those from lower socioeconomicbackgrounds undergo rote learning. Because notall children have the opportunity to learn thesame quality mathematics, the emerging concernof this study is the issue of equity inmathematics teaching.
Computational estimation has not yet established a place in the Kuwaiti national curriculum. An attempt was made to include
it during the early 1990s, but it was dropped by the Kuwaiti Ministry of Education because of the difficulties teachers had
teaching it. In an effort to provide guidance for reintroducing the concept into the curriculum, this study reports on mathematics
teachers’ understanding of the meaning of computational estimation and their views about its significance in the elementary
and middle school curricula in Kuwait. Data gathered from 59 elementary and middle schools teachers in Kuwait revealed that
more than 60% of teachers equate computational estimation with rounding. While two-thirds of the teachers viewed computational
estimation to be an important skill for daily life; only one-fifth (20%) saw it as important in mathematics education. More
than half of the teachers either disagreed with the idea of teaching computational estimation or only wanted to teach it in
limited situations. Most were concerned about the difficulty of learning computational estimation or feared that teaching
computational estimation would cause problems with students’ development of standard algorithms for determining an exact answer.
These findings reveal the challenge that mathematics educators face in introducing computational estimation into the mathematics
curriculum. In order for computational estimation to be taught in elementary and middle school classrooms, teachers need to
understand the concept and its value in education. Teacher education needs to focus on helping teachers better understand
the concept of computational estimation and appreciate its value for instruction.
Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly
rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or
handheld calculators create new possibilities for teaching and learning algebraic manipulation. This study investigated the
views of Turkish prospective secondary mathematics teachers on the use of advanced calculators with CAS in algebra instruction.
An open-ended questionnaire and group interviews revealed prospective teachers’ views and beliefs about when and why they
prefer three possible uses of CAS—black box, white box, or Symbolic Math Guide (SMG). The results showed that participants
mainly preferred the white box methods and especially SMG to the black box method. They suggested that while the black box
method could be used after students mastered the skills, the general white box method and SMG could be used to teach symbolic
KeywordsProspective teachers-Graphing calculators-Computer Algebra Systems-Algebra instruction-Secondary mathematics education
This article arises from a study whose overall purpose was to investigate the relationship between Colombian mathematics teachers’
conceptions of beginning algebra and their conceptions of their own teaching practices. The teachers’ understandings of their teaching practices were explored with a view to unravelling their
conceptions of change in their teaching. Focusing on the perspectives of teachers afforded opportunities that exposed the
powerful role that the teachers’ conceptions of social/institutional factors of teaching played in their conceptions of their
practices. The degree to which the teachers attributed these (external) factors as crucial reasons for what they do in their
teaching was the basis of a categorisation of their conceptions of the crucial determinants of their teaching practices into
three types. The findings are particularly relevant to our understanding of the stability of mathematics teaching approaches
in the Colombian context but have likely implications for a range of international education contexts. Specific implications
for the development of the research into teachers’ conceptions of mathematics and its teaching, and for teacher education
programmes are presented.
In this study we investigate the arithmetic andalgebra word problem-solving skills andstrategies of future primary and secondaryschool teachers in Flanders (Belgium).Moreover, we describe the evolution of theseskills and strategies from the beginning to theend of their teacher education. The resultsshow that future secondary school mathematicsteachers preferred the use of algebra, evenwhen an arithmetic solution was morestraightforward. The solutions of futureprimary school teachers were more diverse: onesubgroup tended to apply exclusively arithmeticmethods (which led to failures on the mostdifficult word problems), whereas anothersubgroup was more adaptive in its strategychoices. Finally, student teachers evolved intheir problem-solving skills during theirteacher education, but not in their strategypreferences. The research findings indicatethat, in the education of pre-service primaryand secondary school teachers, there is a needfor an explicit treatment of pupils' transitionfrom arithmetical to algebraic thinking.
In this paper, we describe a one-day professional development activity for mathematics teachers that promoted the use of comparison
as an instructional tool to develop students’ flexibility in algebra. Effective use of comparison in mathematics instruction
involves using side-by-side presentation of problems and solution methods and subsequent student discussion of these multiple
solution methods to highlight the similarities and differences among problem-solving techniques. The goals of the professional
development activity were to make teachers aware of how to use comparison effectively in their instruction, as well as to
impact teachers’ own flexibility in algebra by using comparison instructionally during the professional development. Our analysis
of teachers’ experiences in the professional development activity suggests that when teachers were presented with techniques
for effective use of comparison, their own understanding of multiple solution methods was reinforced. In addition, teachers
began to question why they relied exclusively on one familiar method over others that are equally effective and perhaps more
efficient and started to draw new connections between problem-solving methods. Finally, as a result of experiencing instructional
use of comparison, teachers began to see value in teaching for flexibility and reported changing their own teaching practices.
KeywordsComparison–Flexibility–Mathematics teacher professional development–Multiple solution methods–Algebra
In this paper we describe a strand of activities for teachers of mathematics that we used with two cohorts of participants
in a professional development program called Revitalizing Algebra (REAL). We first discuss our goals and describe the participants,
and then we describe the construction and selection of the tasks followed by teacher responses. Finally, we reflect on different
iterations of the tasks, their impact on the teachers’ thinking and practice, and the role of school and department culture
in the process of change.
The purpose of this study was to explore teachers’ growth in understanding of algebra using concept maps. The study was set
in the context of a five-year National Science Foundation funded teacher retention and renewal professional development project.
In this project both beginning and experienced teachers are supported as they increase their understanding about mathematics,
their ability to implement effective mathematics practices in their classrooms, and their knowledge of working with English
Learners. Results indicate that teachers’ algebraic knowledge structures became more complex and connected as a result of
their professional development. In addition, they were able to adapt their knowledge networks to incorporate important aspects
of algebra into them. Concept maps are recommended to other leaders of mathematics professional development as a means of