RF03: INTERNATIONAL PERSPECTIVES ON THE NATURE OF
MATHEMATICAL KNOWLEDGE FOR SECONDARY
TEACHING: PROGRESS AND DILEMMAS
Coordinators: Helen M. Doerr, USA and Terry Wood, USA
This research forum addresses the question: what is the nature of the mathematical
knowledge that is needed for secondary teaching? Six international contributors
respond by making two claims (one related to an area where progress in research
has been made and the other related to dilemmas facing researchers): preparing
teachers, teaching practice, and research designs and methodologies. This structure
provides a way of focusing the discussion among forum participants and a means to
develop international points of view on the nature of the mathematical knowledge
that is needed for secondary teaching.
GENERAL FRAMEWORK
Over the past two decades, international perspectives on research about the teaching
of mathematics have received considerable and increasing attention at PME and by
the research community in mathematics education (Ellerton, 1998; Jaworski, Wood
& Dawson, 1999). Yet, progress towards changes in teaching practices remains slow
and large gaps exist between the highest achieving schools and countries and the
lowest achieving schools and countries. Substantial progress has been made in many
areas of research related to students’ learning along with the emergence of curricular
materials and standards documents that reflect findings of this research (e.g., the
early numeracy projects in the United Kingdom, New Zealand and Australia).
Nevertheless, translating research on mathematical learning into forms that are useful
for teaching practice continues to be a difficult problem that varies substantially
across schools and countries and progress has been elusive. Difficulties in preparing
new teachers are compounded by the disconnection that pre-service teachers can
experience between their teacher preparation programs and their experiences in
practice. Furthermore, the complexity that characterizes teaching and learning seems
to have yielded a multiplicity of research designs and methodologies with insufficient
coherence across these research designs to support the development of a shared
knowledge base for teaching.
KEY QUESTIONS AND THEMES
There is substantial agreement among mathematics educators that the quality of
teachers’ subject matter knowledge is necessary but not sufficient for effective
teaching. Subject matter knowledge is just one category among many that attempt to
capture the complexity of the nature of the mathematical knowledge base that is
needed for teaching (Hiebert, Gallimore & Stigler, 2002; Shulman, 1986). Hence, the
central focus of this research forum is the nature of the mathematical knowledge that
is needed for teaching in secondary schools.
Proceedings of the 28th Conference of the International
Group for the Psychology of Mathematics Education, 2004 Vol I pp 167–196
, A recent National Research Council report in the United States (NRC, 2003)
described mathematical proficiency for students as the simultaneous and integrated
acquisition of five strands: (1) conceptual understanding, (2) procedural fluency, (3)
strategic competence, (4) adaptive reasoning, and (5) productive disposition. These
proficiencies provide one possible framework for considering the mathematical
knowledge that is needed by secondary teachers. However, in addition to teachers
having this kind of mathematical proficiency, they must also understand (1) how such
mathematical proficiencies are developed in curricular materials, (2) the ways in
which students’ thinking might reveal students' mathematical proficiencies, and (3)
how students from diverse cultural and ethnic backgrounds develop these
mathematical proficiencies.
Another possible framework comes from the KOM project (Niss, 2003) which
provides eight competencies for students, such as ‘think mathematically and make
use of different representations and translate between them,’ that describe the main
components for mastering mathematics derived from the work of mathematicians. In
addition to having these competencies, teachers must also have competencies in
curriculum, teaching, student learning and assessment. We will use these
proficiencies and competencies as background for considering the nature of teachers’
mathematical knowledge, and then we will describe the challenges and difficulties in
designing and implementing research in this area.
This research forum will focus on the following central question: What is the nature
of the mathematical knowledge that is needed for secondary teaching?
Our goal in this forum is to stimulate discussion on this question through a reporting
of research findings that identify areas in which significant progress has been made
and where difficulties and persistent obstacles to progress continue to exist. To
initiate the discussion, contributors from six different countries that represent
international differences in contexts and perspectives report their findings. Each
contributor addresses the question from three views: (a) preparing teachers; (b)
supporting teachers in practice; and (c) research design and methodologies. Each
contributor makes two key claims related to each of the three views of the above
question. The first claim reflects an area where substantial research progress has been
made in the contributor’s country with respect to the nature of the mathematical
knowledge that is needed for teaching secondary mathematics. These claims reflect
findings that are of significance to the field and are based on a substantial body of
research. The second claim reflects a significant dilemma in research or an area
where progress has remained elusive. This structure provides both a broad view of
the field (as it is seen internationally) and a way of focusing the discussion among
forum participants. The contributors’ claims are presented in the next section.
Following those contributions, we provide a tentative synthesis of the claims and
pose some cross-cutting questions that will provide a beginning point for work of the
participants in this forum.
1–168 PME28 – 2004
, PREPARING TEACHERS—PROGRESS AND DILEMMAS
Australia (Kaye Stacey)
Claim 1 Progress: The corpus of research on students’ conceptions, thinking and
learning in mathematical content areas provides foundation knowledge for a greatly
improved teacher education.
Claim 2 Dilemma: This corpus of knowledge needs to undergo substantial didactic
transposition before it is maximally useful.
Claim 1 is about creating the scientific basis of a discipline of mathematics didactics
(pedagogy) for teacher education. The established sciences and humanities have an
accumulated set of well-tested research results, which have been codified and
simplified to create learnable disciplines. In mathematics education, we are now
reaching a point where we too have a sufficiently strong scientific foundation to
undertake this task. We know enough about students’ thinking patterns, conceptions
and their development to begin the “didactic transposition” from raw research results
to learnable and organised material which could form the basis of a new teacher
education. I expect these outcomes to be very much more effective than teacher
education based around general theories of mathematical development (as was tried,
for example, with Piagetian research in times past).
What is the evidence for Claim 1? The extent of the research knowledge is evident
from the accumulated proceedings of PME, the handbooks of reviews of research and
so forth. The need for this material to undergo a didactic transposition is evident in
the lack of textbooks on student’s thinking and learning for secondary mathematics
teacher education (indeed no textbook is widely used in Australia) and the
consequent practice of referring teacher education students directly to research
reports rather than to scholarly accounts written for them.
My claim also requires evidence that this new content of teacher education would
“make a difference.” Two large scale elementary teacher development projects
provide some confirmation. Count Me In Too (Bobis, 1999) is a New South Wales
government professional development initiative where mathematics education
researchers turned international research on children’s early number development
into support material for professional development. Teachers learned about how
children’s knowledge progressed, assessed children’s learning carefully and selected
teaching materials to move them along the framework. The Early Numeracy
Research Project in Victoria had a similar mission and adopted a similar approach,
although differing in detail. Both projects, although focused on elementary schooling,
demonstrate improved outcomes for students across large numbers of schools, some
of them sustained. A difficulty with using a program evaluation as evidence for my
claim is that improved learning outcomes are a result of the whole program, rather
than one component, such as improved teacher knowledge.
PME28 – 1–
MATHEMATICAL KNOWLEDGE FOR SECONDARY
TEACHING: PROGRESS AND DILEMMAS
Coordinators: Helen M. Doerr, USA and Terry Wood, USA
This research forum addresses the question: what is the nature of the mathematical
knowledge that is needed for secondary teaching? Six international contributors
respond by making two claims (one related to an area where progress in research
has been made and the other related to dilemmas facing researchers): preparing
teachers, teaching practice, and research designs and methodologies. This structure
provides a way of focusing the discussion among forum participants and a means to
develop international points of view on the nature of the mathematical knowledge
that is needed for secondary teaching.
GENERAL FRAMEWORK
Over the past two decades, international perspectives on research about the teaching
of mathematics have received considerable and increasing attention at PME and by
the research community in mathematics education (Ellerton, 1998; Jaworski, Wood
& Dawson, 1999). Yet, progress towards changes in teaching practices remains slow
and large gaps exist between the highest achieving schools and countries and the
lowest achieving schools and countries. Substantial progress has been made in many
areas of research related to students’ learning along with the emergence of curricular
materials and standards documents that reflect findings of this research (e.g., the
early numeracy projects in the United Kingdom, New Zealand and Australia).
Nevertheless, translating research on mathematical learning into forms that are useful
for teaching practice continues to be a difficult problem that varies substantially
across schools and countries and progress has been elusive. Difficulties in preparing
new teachers are compounded by the disconnection that pre-service teachers can
experience between their teacher preparation programs and their experiences in
practice. Furthermore, the complexity that characterizes teaching and learning seems
to have yielded a multiplicity of research designs and methodologies with insufficient
coherence across these research designs to support the development of a shared
knowledge base for teaching.
KEY QUESTIONS AND THEMES
There is substantial agreement among mathematics educators that the quality of
teachers’ subject matter knowledge is necessary but not sufficient for effective
teaching. Subject matter knowledge is just one category among many that attempt to
capture the complexity of the nature of the mathematical knowledge base that is
needed for teaching (Hiebert, Gallimore & Stigler, 2002; Shulman, 1986). Hence, the
central focus of this research forum is the nature of the mathematical knowledge that
is needed for teaching in secondary schools.
Proceedings of the 28th Conference of the International
Group for the Psychology of Mathematics Education, 2004 Vol I pp 167–196
, A recent National Research Council report in the United States (NRC, 2003)
described mathematical proficiency for students as the simultaneous and integrated
acquisition of five strands: (1) conceptual understanding, (2) procedural fluency, (3)
strategic competence, (4) adaptive reasoning, and (5) productive disposition. These
proficiencies provide one possible framework for considering the mathematical
knowledge that is needed by secondary teachers. However, in addition to teachers
having this kind of mathematical proficiency, they must also understand (1) how such
mathematical proficiencies are developed in curricular materials, (2) the ways in
which students’ thinking might reveal students' mathematical proficiencies, and (3)
how students from diverse cultural and ethnic backgrounds develop these
mathematical proficiencies.
Another possible framework comes from the KOM project (Niss, 2003) which
provides eight competencies for students, such as ‘think mathematically and make
use of different representations and translate between them,’ that describe the main
components for mastering mathematics derived from the work of mathematicians. In
addition to having these competencies, teachers must also have competencies in
curriculum, teaching, student learning and assessment. We will use these
proficiencies and competencies as background for considering the nature of teachers’
mathematical knowledge, and then we will describe the challenges and difficulties in
designing and implementing research in this area.
This research forum will focus on the following central question: What is the nature
of the mathematical knowledge that is needed for secondary teaching?
Our goal in this forum is to stimulate discussion on this question through a reporting
of research findings that identify areas in which significant progress has been made
and where difficulties and persistent obstacles to progress continue to exist. To
initiate the discussion, contributors from six different countries that represent
international differences in contexts and perspectives report their findings. Each
contributor addresses the question from three views: (a) preparing teachers; (b)
supporting teachers in practice; and (c) research design and methodologies. Each
contributor makes two key claims related to each of the three views of the above
question. The first claim reflects an area where substantial research progress has been
made in the contributor’s country with respect to the nature of the mathematical
knowledge that is needed for teaching secondary mathematics. These claims reflect
findings that are of significance to the field and are based on a substantial body of
research. The second claim reflects a significant dilemma in research or an area
where progress has remained elusive. This structure provides both a broad view of
the field (as it is seen internationally) and a way of focusing the discussion among
forum participants. The contributors’ claims are presented in the next section.
Following those contributions, we provide a tentative synthesis of the claims and
pose some cross-cutting questions that will provide a beginning point for work of the
participants in this forum.
1–168 PME28 – 2004
, PREPARING TEACHERS—PROGRESS AND DILEMMAS
Australia (Kaye Stacey)
Claim 1 Progress: The corpus of research on students’ conceptions, thinking and
learning in mathematical content areas provides foundation knowledge for a greatly
improved teacher education.
Claim 2 Dilemma: This corpus of knowledge needs to undergo substantial didactic
transposition before it is maximally useful.
Claim 1 is about creating the scientific basis of a discipline of mathematics didactics
(pedagogy) for teacher education. The established sciences and humanities have an
accumulated set of well-tested research results, which have been codified and
simplified to create learnable disciplines. In mathematics education, we are now
reaching a point where we too have a sufficiently strong scientific foundation to
undertake this task. We know enough about students’ thinking patterns, conceptions
and their development to begin the “didactic transposition” from raw research results
to learnable and organised material which could form the basis of a new teacher
education. I expect these outcomes to be very much more effective than teacher
education based around general theories of mathematical development (as was tried,
for example, with Piagetian research in times past).
What is the evidence for Claim 1? The extent of the research knowledge is evident
from the accumulated proceedings of PME, the handbooks of reviews of research and
so forth. The need for this material to undergo a didactic transposition is evident in
the lack of textbooks on student’s thinking and learning for secondary mathematics
teacher education (indeed no textbook is widely used in Australia) and the
consequent practice of referring teacher education students directly to research
reports rather than to scholarly accounts written for them.
My claim also requires evidence that this new content of teacher education would
“make a difference.” Two large scale elementary teacher development projects
provide some confirmation. Count Me In Too (Bobis, 1999) is a New South Wales
government professional development initiative where mathematics education
researchers turned international research on children’s early number development
into support material for professional development. Teachers learned about how
children’s knowledge progressed, assessed children’s learning carefully and selected
teaching materials to move them along the framework. The Early Numeracy
Research Project in Victoria had a similar mission and adopted a similar approach,
although differing in detail. Both projects, although focused on elementary schooling,
demonstrate improved outcomes for students across large numbers of schools, some
of them sustained. A difficulty with using a program evaluation as evidence for my
claim is that improved learning outcomes are a result of the whole program, rather
than one component, such as improved teacher knowledge.
PME28 – 1–
