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Making Progress on Mathematical Knowledge for Teaching

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Making Progress on Mathematical Knowledge for Teaching

Abstract

Although the field lacks a theoretically grounded, well-defined, and shared conception of mathematical knowledge required for teaching, there appears to be broad agreement that a specialized body of knowledge is vital to improvement. Further, such a construct serves as the foundation for different kinds of studies with different agendas. This article reviews what is known and needs to be known to advance research on mathematical knowledge for teaching. It argues for three priorities: (i) finding common ground for engaging in complementary studies that together advance the field; (ii) innovating and reflecting on method; and (iii) addressing the relationship of such knowledge to mathematical fluency in teaching and to issues of equity and diversity in teaching. It concludes by situating the articles in this special issue within this emerging picture.
e Mathematics Enthusiast
Volume 13
Number 1 Numbers 1 & 2 Article 3
2-2016
Making Progress on Mathematical Knowledge for
Teaching
Mark Hoover
Reidar Mosvold
Deborah L. Ball
Yvonne Lai
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Recommended Citation
Hoover, Mark; Mosvold, Reidar; Ball, Deborah L.; and Lai, Yvonne (2016) "Making Progress on Mathematical Knowledge for
Teaching," e Mathematics Enthusiast: Vol. 13: No. 1, Article 3.
Available at: h=p://scholarworks.umt.edu/tme/vol13/iss1/3
TME, vol. 13, no. 1&2, p.
The Mathematics Enthusiast, ISSN 1551-3440, vol. 13, no. 1&2, pp. 334
2016© The Author(s) & Dept. of Mathematical Sciences-The University of Montana
3
Making Progress on Mathematical Knowledge for Teaching
Mark Hoover
University of Michigan, USA
Reidar Mosvold
University of Stavanger, Norway
Deborah Loewenberg Ball
University of Michigan, USA
Yvonne Lai
University of Nebraska-Lincoln, USA
Abstract: Although the field lacks a theoretically grounded, well-defined, and shared
conception of mathematical knowledge required for teaching, there appears to be broad
agreement that a specialized body of knowledge is vital to improvement. Further, such a
construct serves as the foundation for different kinds of studies with different agendas.
This article reviews what is known and needs to be known to advance research on
mathematical knowledge for teaching. It argues for three priorities: (i) finding common
ground for engaging in complementary studies that together advance the field; (ii)
innovating and reflecting on method; and (iii) addressing the relationship of such
knowledge to mathematical fluency in teaching and to issues of equity and diversity in
teaching. It concludes by situating the articles in this special issue within this emerging
picture.
Keywords: mathematical knowledge for teaching, MKT, specialized knowledge,
pedagogical content knowledge, PCK, mathematics teacher education, method,
mathematical fluency, equity, diversity.
Introduction
A century ago, a central focus of teacher education in the United States was on
developing a thorough understanding of subject matter, but the mid-twentieth century
witnessed a steady shift to an emphasis on pedagogy generalized to be largely
independent of subject matter. By the 1980s, an absence of content focus was so
prevalent that Shulman (1986) referred to this as a “missing paradigm” in teacher
education. A similar tendency can be seen in other countries. For example, a few decades
ago, it was possible to become qualified for teaching mathematics in grades 1–9 in
Norway with no more mathematics than a short course in didactics. A widespread
Hoover, Mosvold, Ball, & Lai
assumption seemed to be that prospective teachers already knew the content they needed,
from their experiences as students, and they only required directions in how to teach this
content. Shulman’s call for increased attention to subject matter reoriented research and
practice. However, the connection between the formal education of mathematics teachers
and the content understanding important for their work is not straightforward. Teachers’
formal mathematics education is not highly correlated with their students’ achievement
(Begle, 1979) or with the depth of understanding they seem to have of the mathematical
issues that arise in teaching (Ma, 1999).
One of Shulman’s (1986) most important contributions was the suggestion that
the work of teaching requires professional knowledge that is distinctive for the teaching
profession. He proposed different categories of professional knowledge for teaching. One
of these categories was distinctive content knowledge, which Shulman described as
including a deep knowledge of the structures of the subject (e.g., Schwab, 1978), beyond
procedural and factual knowledge. Another category of knowledge was what Shulman
termed “pedagogical content knowledge,” which is aspects of the content most germane
to its teaching (1986, p. 9). The idea about an amalgam of subject matter knowledge and
pedagogical knowledge has continued to appeal to researchers working in different
subject areas, and Shulman’s foundational publications are among the most cited
references in the field of education. (Google Scholar identifies over 13000 publications
that cite his 1986 article.)
In the last two decades, researchers and mathematics educators have increasingly
emphasized the significance of mathematical knowledge that is teaching-specific. Such
knowledge is seen as different from the mathematics typically taught in most collegiate
mathematics courses and from the mathematics needed by professionals other than
teachers. Although it includes knowing the mathematics taught to students, the kind of
understanding of the material needed by teachers is different than that needed by the
students. Even though the literature suggests a general consensus that mathematics
teaching requires special kinds of mathematical knowledge, agreement is lacking about
definitions, language, and basic concepts. Many scholars draw on Shulman’s notion of
pedagogical content knowledge (or PCK) and view this knowledge as being either a kind
of “combined” knowledge or a kind of “transformed” knowledge. Grounded in
Shulman’s proposals, the phrases “for teaching” and “practice-based” have been
emphasized to indicate the relationship of the knowledge to specific work of teaching
(e.g., Ball, Thames, & Phelps, 2008). For this article, we adopt these phrases but maintain
an ecumenical view of a more extended literature.
With growing interest in ideas about specialized professional content knowledge,
the early 2000s saw a spate of large-scale efforts to develop measures of such knowledge
and the use of such a construct as the basis for a wide range of research studies, such as
evaluating professional development (e.g., Bell, Wilson, Higgins, & McCoach, 2010),
examining the impact of structural differences on the mathematical education of teachers
(e.g., Kleickmann et al., 2013), arguing for policies and programs (e.g., Hill, 2011), and
investigating the role of professional content knowledge on mathematics teaching
practice (e.g., Speer & Wagner, 2009). Instruments for measuring such knowledge
represent a crucial tool for making meaningful progress in a field. They operationalize
emerging thinking, invite scrutiny, and support the investigation of underlying models.
TME, vol. 13, no. 1&2 p.
5
This special issue on developing measures and measuring development of mathematical
knowledge for teaching continues this focus on instruments, along with a concomitant
regard for broader purposes and potential ways to advance the field. In an effort to situate
this special issue, this introductory article provides some selected highlights from the
field — focusing on what is being studied, how, and to what ends. To accomplish this, we
draw on both a detailed review of articles sampled from 2006 to 2013 and our wider
reading of the literature. We then nominate some key areas for making progress on
research and development of the specialized mathematical knowledge teachers need and
we use this framing to characterize the agendas and contributions of the collection of
articles assembled in this special issue. The article consists of three major sections.
1. Lessons from Empirical Research
2. Next Steps for the Development of Mathematical Knowledge for Teaching
3. Articles that Develop Measures and Measure Development
The first describes a review we conducted and discusses three broad arenas of work
suggested by this review. The second discusses three proposals for future research. The
last briefly situates the articles in this issue within the lessons and directions discussed.
Lessons from Empirical Research
In our reading of empirical literature concerned with the distinctive mathematical
knowledge requirements for teaching, several broad strands of research stand out. We
begin by describing a formal review we conducted of empirical research that began
appearing in about 2006, in the wake of a number of conceptual proposals (beyond PCK),
and that began using these proposals as a conceptual basis for empirical study. This
review informs our overall reading of the field. Combining this review with our wider
reading in the field, we then identify and discuss three major arenas of work.
Reviewing the literature. In the course of other research we were conducting, we
reviewed international empirical literature published in peer-reviewed journals in English
between 2006 and 2013.1 Wanting to survey the topic across theoretical perspectives, we
developed and tested inclusive search terms:
Mathematics
o math* (the asterisk is a placeholder for derived terms)
Content knowledge
o know* AND (content OR special* OR pedagog* OR didact* OR math*
OR teach* OR professional OR disciplin* OR domain) OR “math for
teaching” OR “mathematics for teaching” OR “math-for-teaching” OR
“mathematics-for-teaching
1 This more formal review, which we use to inform our wider reading of the literature, was funded
by the National Science Foundation under grant DRL-1008317 and conducted in collaboration with Arne
Jakobsen, Yeon Kim, Minsung Kwon, Lindsey Mann, and Rohen Shah, who we wish to thank for their
assistance with searching, conceptualizing codes, coding, and analysis. The opinions reported here are
those of the authors and do not necessarily reflect the views of the National Science Foundation or our
colleagues.
Hoover, Mosvold, Ball, & Lai
Teaching
o teaching OR pedagog* OR didact* OR instruction*
These search terms initially yielded over 3000 articles from the following six databases:
PsycInfo
Eric
Francis
ZentralBlatt
Web of Science
Dissertation Abstracts
Broadened search terms, additional databases, and inclusion of earlier publication years
yielded none to negligible additional articles.
Based on a reading of abstracts, 349 articles were identified as potential empirical
articles (as characterized by the American Educational Research Association, 2006) in
which some concept of distinctive mathematics needed for teaching was used as a
conceptual tool to formulate research questions or structure analysis. Our goal was not to
reach high standards of reliability, but rather to use a systematic process to collect a
corpus of relevant studies representing the literature from this period. In coding the
articles, we sought to be descriptive rather than evaluative and iteratively worked
between an inductive examination of a sample of articles and initial conceptualizations of
empirical research combined with a basic model of educational change. After reading full
articles, 190 of the 349 remained in the final set. A set of core codes were developed for
the following categories:
1. Genre of the study
2. Research problem used to motivate the study
3. Variables used
4. Whether or not and how causality was addressed
5. Findings
Additional codes included sample size, instruments used for measuring
mathematical knowledge for teaching, school level or setting, professional experience of
the teachers, geographic region, and mathematical area addressed. Each article was read
and coded by two team members, with a decision as to whether it satisfied our inclusion
criteria, and if so, codes were reconciled. (For a more detailed description of the methods
used, see Kim, Mosvold, and Hoover (2015).)
In table 1, we present some patterns that emerged from some of the additional,
descriptive codes.
TME, vol. 13, no. 1&2 p.
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Table 1. Selected descriptive codes for sample size, instrument used, level of schooling,
and geographic context.
Categories and codes
Number
of papers
Sample size
Small scale (<10)
60
Medium 1 (1029)
51
Medium 2 (3070)
34
Large scale (>70)
43
None
2
Instrument
COACTIV
4
CVA
3
DTAMS
3
LMT (including adaptations)
31
TEDS-M
2
Non-standardized
56
None
91
Level of teachers
Primary (K8)
81
Middle (59)
45
Secondary (713)
41
Tertiary
3
Across levels
20
Regions
Africa
7
Asia
27
Europe
22
Latin America
3
North America
112
Oceania
15
Across regions
4
Hoover, Mosvold, Ball, & Lai
We observe that many studies are small-scale, and a large number of the studies
apply non-standardized instruments or no instruments. In the studies where standardized
instruments were used to measure teachers’ knowledge, the instruments developed to
measure mathematical knowledge for teaching in the Learning Mathematics for Teaching
(LMT) project were most common. An abundance of studies focuses on primary teachers,
and most studies were carried out in North America.
Table 2 provides the fourteen categories developed for coding the research
problem. We have grouped these into three domains and use these groups to discuss the
literature in the following sections.
Table 2. Research problems addressed.
Problems
Number
of papers
%
Nature and composition of SM
55
28.9
What is SM?
34
What relationships exist among aspects of SM or with other variables?
21
Improvement of SM
81
42.6
What professional development improves teachers’ SM?
28
What teacher education improves teachersSM?
28
What curriculum/tasks improve teachers’ SM?
10
What teaching practice improves teachersSM?
0
How SM develops?
15
How to scale up the teaching and learning of SM?
0
Contribution of SM
33
17.4
Does SM contribute to teaching practice?
6
What does SM contribute to teaching practice?
12
Does SM contribute to student learning?
15
What does SM contribute to student learning?
0
Other
What SM do teachers know?
21
11.1
How policy influences teachers’ SM?
0
0
Total
190
100
In order to make table 2 more readable, we use the abbreviation SM to signify any
of the variety of ways in which mathematical knowledge for teaching might be
conceptualized and named. The intention is not to introduce yet another term or acronym
TME, vol. 13, no. 1&2 p.
9
for such knowledge. In this article, we have adopted more generic language to express an
inclusive notion of such knowledge and we avoid the use of any specific acronym label.
For the purpose of this introductory article, we used patterns evident in the review
described above to inform our extended reading of the field. Together, these efforts led us
to identify three broad themes. First, a number of studies investigate the nature and
composition of teacher content knowledge. Given that foundational research into
teaching-specific mathematical knowledge pointed to its elusiveness and complexity, it is
not surprising that scholars continue to investigate what it is its components,
measurement, features, and related constructs. A second group of studies, which
constitutes the majority of published articles, investigates approaches to increasing
teacher knowledge, in both the context of pre-service teacher education and the
professional education of practicing teachers. A third group of studies, fewer in number,
investigates effects of teachers’ knowledge on both teaching and student learning. In the
following sections, we use these three broad themes to organize our comments on
selected highlights from the literature. Following these, we provide suggestions about
possible next steps for further research in this field.
Nature and composition of mathematical knowledge for teaching. Current
studies continue to probe ideas about the nature and composition of teaching-specific
knowledge of mathematics. Some studies consider the construct in broad terms. They
may identify or elaborate aspects or frameworks, characterize or critique the construct,
compare different representations or sub-domains, or compare such knowledge with other
kinds of mathematical knowledge. Others examine a constrained area of knowledge:
some in relation to specific mathematical topics; some in relation to specific practices of
teaching, or at specific levels (such as interpreting and responding to student thinking,
curriculum use, or proving in high school geometry); and some in relation to specific
qualities (such as connectedness). However, these studies do not build on each other in
obvious ways and clear lessons are hard to identify. The one avenue of work that
represents progress for the field is the development of instruments, and we focus our
discussion there.
Instruments provide a crucial tool for investigating the nature and composition of
mathematical knowledge needed for teaching. They serve to operationalize ideas about
mathematical knowledge for teaching and test assumed models of the role it plays. They
are used to investigate the teaching and learning of such knowledge, relationships with
other variables, and other questions important for practice and policy. On the one hand,
rigorous instrument development is expensive relative to budgets available for most
studies and many instruments are used in a single study and limited in the extent to which
they meet psychometric standards and establish validity. On the other hand, several larger
efforts have invested in building instruments for large-scale studies and wider use in the
field. The Learning Mathematics for Teaching (LMT) instruments for practicing
elementary and middle school teachers (Hill, Schilling, & Ball, 2004) include nearly
1000 items on over a dozen different instruments and have been used in numerous
program evaluations and studies of relationships and effects. They have been extensively
validated (Schilling & Hill, 2007) and adapted internationally (Blömeke & Delaney,
2012). The Diagnostic Teacher Assessment in Mathematics and Science (DTAMS)
Hoover, Mosvold, Ball, & Lai
instruments for practicing middle school teachers (Saderholm, Ronau, Brown, & Collins,
2010) include 24 forms in four content areas, have been administered and rigorously
analyzed with a sample of several thousand teachers, and are currently being expanded.
The Teacher Education and Development Study in Mathematics (TEDS-M) instruments
for pre-service primary and lower secondary teachers (Tatto et al., 2008; Senk et al.
2012) include over 100 items and were originally administered to 23,000 pre-service
teachers in 17 countries.
These instruments represent an important contribution to the field. Extensive
cross-professional-community review and the building of agreed-on formulations of
important content knowledge have played a major role in the development of these
measures. The synthesis of ideas and the integration of expertise from multiple
professional communities have helped to clarify and improve ideas about mathematical
knowledge for teaching. In addition, the availability of common instruments has enabled
meaningful comparison and interpretation across programs, countries, and studies in
ways that contribute to the maturity of research on mathematical knowledge for teaching.
Several other efforts have developed instruments with less focus on broad
consensus or widespread use. The COACTIV instrument for practicing secondary
teachers (Kunter, Klusmann, Baumert, Voss, & Hacfeld, 2013) produced items of a genre
similar to those described above and used these to investigate relationships to other
variables and to understand issues of practice and policy related to the mathematical
education of teachers. Some instruments have been developed to focus on mathematical
knowledge related to a specific topic, such as fractions (Izsak, Jacobson, de Araujo, &
Orrill, 2012), geometry (Herbst & Kosko, 2012), algebra (McCrory, Floden, Ferrini-
Mundy, Reckase, & Senk, 2012), and continuous variation and covariation (Thompson,
2015). Others have focused on specific aspects of teaching, and the mathematical
knowledge required in these specific teaching practices, such as choosing examples
(Chick, 2009; Zodik & Zaslavsky, 2008) and scaffolding whole-class discussions to
address mathematical goals (Speer & Wagner, 2009). Many instruments have been
developed in relation to specific lines of research and often in response to perceived
issues with more established instruments. A number of researchers are concerned about a
potentially narrow interpretation of knowledge as declarative or about a possible
discrepancy between knowledge and knowledge use. These concerns have led some
scholars to explore different conceptualizations of the mathematics teachers need and to
look for alternative formats for measuring it (e.g., Kersting, Givvin, Thompson,
Santagata, & Stigler, 2012; McCray & Chen, 2012; Thompson, 2015).
Although the development of instruments is an important step toward building a
robust conception of teaching-specific knowledge of mathematics, these efforts also
reveal a lack of shared language and meaning of foundational concepts. Differences in
meaning for the construct PCK have been noted in the past (Ball, Thames, & Phelps,
2008; Depaepe, Verschaffel, & Kelchtermans, 2013; Graeber & Tirosh, 2008). These
differences persist, yet they are often overlooked with regard to instruments.
For example, Kaarstein (2014) examined whether the LMT, TEDS-M, and
COACTIV instruments, each referencing Shulman and stating that the respective
instrument measures PCK, measure the same thing. To study this issue, she constructed a
taxonomy of the different levels of categories in Shulman’s initial framework as well as
TME, vol. 13, no. 1&2 p.
11
the frameworks that were used to develop the three instruments. She then selected three
items one from each instrument — and categorized them according to each of the
three frameworks. Her main argument is that content knowledge and pedagogical content
knowledge are supposed to be distinct categories, and therefore three projects that use the
same basic categories should categorize items in the same way. However, from her
analysis the items would be placed in different basic categories using the criteria reported
by the projects. As an example, an item that was categorized as a specialized content
knowledge item (measuring content knowledge) in the LMT project would probably have
been categorized as a PCK item in TEDS-M and COACTIV. Kaarstein’s argument does
not necessarily threaten the validity of the measures from each of the three projects, but
her observation deserves further attention.
Similarly, a study by Copur-Gencturk and Lubienski (2013) echoes this concern.
In order to investigate growth in pre-service teacher knowledge, they used two different
instruments: LMT and DTAMS. When comparing groups of teachers who had
participated in different kinds of courses, they concluded that the LMT and DTAMS
instruments measure aspects of mathematical knowledge for teaching that are
substantially different. Teachers who participated in a hybrid mathematics
content/methods course had the most significant increase in their LMT score, and this
score remained stable although they took an additional content course. Teachers’
DTAMS score also increased during the hybrid course; during the content knowledge
course, only the content knowledge part of their DTAMS increased. This study thus
supports the idea that there is specialized mathematical content knowledge not influenced
by general mathematics content courses. That different instruments measure different
aspects of knowledge is not necessarily surprising, but it is worrying if instruments
ostensibly designed to capture the same construct in fact measure significantly different
facets of that knowledge, with little clarity about these differences.
The concerns raised by Kaarstein (2014) and Copur-Gencturk and Lubienski
(2013) suggests that the limited specification of the construct and the different ways of
operationalizing it makes it difficult to interpret results. This limits the extent to which
results from these instruments, taken individually or together, can inform the
conceptualization of mathematical knowledge for teaching or practical decisions needed
to design learning opportunities.
Developing teachers’ mathematical knowledge for teaching. With a growing
sense of the mathematics important for improving teaching and learning, practitioners
have turned their attention to increasing teachers’ knowledge of professionally relevant
mathematics and scholarly work has followed suit. A large number of studies make it
clear that the design and evaluation of teacher education and professional development
programs in developing teachers’ mathematical knowledge for teaching are top priorities.
From several decades of research, we propose what we see as a few related emerging
lessons:
Teaching teachers additional standard disciplinary mathematics beyond a basic
threshold does not increase their knowledge in ways that impact teaching and
learning.
Hoover, Mosvold, Ball, & Lai
Providing teachers with opportunities to learn mathematics that is intertwined
with teaching increases their mathematical knowledge for teaching.
The focus of the content, tasks, and pedagogy for teaching such knowledge
requires thoughtful attention to ways of maintaining a coordination of content and
teaching without slipping exclusively into one domain or the other.
These lessons are rooted in early efforts to document effects of teachers’
mathematical knowledge on student learning and are reinforced by current research on
the design and implementation of teacher education and professional development. We
begin by briefly reflecting on that early work and then tracing these lessons into current
research.
Much of the impetus for the surge in research on teaching-specific knowledge
began with reviews of several decades of large-scale research that found surprisingly
little to no effect of teachers’ mathematical knowledge on their students’ learning (Ball,
Lubienski, & Mewborn, 2001). The studies reviewed were often conducted with large
datasets but very coarse measures. Taking Shulman’s (1986) suggestion that the content
knowledge needed by teachers was characteristically different from that needed by other
professionals, researchers began to look more closely at the measures used in those
studies and at the findings. The clearest finding that emerged was that methods courses
consistently showed positive effects while content courses did not (e.g., Begle, 1979;
Ferguson & Womack, 1993; Guyton & Farokhi, 1987; Monk, 1994). The second was that
positive effects were more likely when the content taught to teachers was more closely
related to the content they subsequently taught. For instance, several scholars found
effects when using student exams to measure teachers’ knowledge (Harbison &
Hanushek, 1992; Mullens, Murnane, & Willett, 1996). Reinforcing these results, Monk
(1994) found that coursework in calculus influenced the achievement of secondary
teachers’ students in algebra classes, but not in their geometry classes. In general, when
the mathematics taught or measured is meaningfully connected to classroom materials or
interactions, it is modestly associated with improved teaching and learning.
For some practitioners and policy-makers, the implication of these empirical
studies, combined with logical arguments for teaching-specific professional knowledge,
has been enough to lead to prioritizing mathematical knowledge for teaching in the
mathematical education of teachers. Nevertheless, many policies continue to press for
increases in the number of mathematics courses required of teachers, regardless of their
connection to teaching, despite abundant evidence that such policies are unlikely to
improve teaching and learning (e.g., Youngs & Qian, 2013). Such policies have probably
been less the result of lingering doubt about empirical results and more the result of
overextending the notion that knowing content well is key to good teaching, even in the
face of disconfirming evidence. Of course, a certain threshold level of knowledge of the
subject is essential, but preparing teachers by requiring mathematics courses that are not
directly connected to the content being taught or to the work involved in teaching that
content is misguided.
More recent studies continue to reinforce these established lessons. One recent
line of inquiry is the investigation of features of innovative, well-received professional
development programs. To us, the most compelling result emerging from these studies is
that professional development requires designing pedagogically relevant movement
TME, vol. 13, no. 1&2 p.
13
between mathematical and pedagogical concern both to motivate teachers’ investment in
mathematical issues and to keep the mathematical attention on mathematics that matters
for the work of teaching. To elaborate, we offer several examples that contribute to this
claim.
With deep regard for the limited effects of decades of substantial national
investment in professional development, several research groups have organically
developed approaches informed by thoughtful reflection and attention to disciplined
observation of teachers’ engagement with and actual uptake of ideas and practices. One
important insight emerging from these decades-long investments is that cycling through
mathematical considerations, pedagogical considerations, and reflective enactment is
vital to the design of professional development. For instance, Silver, Clark, Ghousseini,
Charalambous, and Sealy (2007) set out to provide evidence for whether and how
teachers might enhance their mathematical knowledge for teaching through monthly
practice-based professional development workshops designed to cycle from activities of
doing mathematics, to examining case-based pedagogical and student-related issues, to
planning, teaching and debriefing lessons collaboratively (all related to a common
mathematics task or set of tasks). Examining the interactions of one teacher, they
document ways these activities provided opportunities for teachers to build connections
among mathematical ideas and to consider these ideas in relation to student thinking and
teaching. They do not measure teacher learning. Nor do they disentangle effects of what
they refer to as a professional-learning-task cycle from a number of other important
features of their professional development program. However, they document dynamics
in which the teacher, from an initial experience solving a nontrivial mathematics
problem, supported by mathematically sensitive facilitation, successively engages in
mathematical issues and pedagogical issues in ways that visibly build connections among
mathematical ideas, pedagogical practice, and growing mathematical knowledge for
teaching. In addition, they argue that their cyclic design increased teachers’ motivation
for learning mathematics, both in the workshops and in their daily practice.
Through successive opportunities to consider mathematical ideas in relation to the
activities of classroom practice, our participants came to see their pedagogical
work as permeated by mathematical considerations. (p. 276)
Similarly, in working to close the gap between a reform vision and the actual
practice of mathematics teaching and learning, Koellner et al. (2007) implemented a
model of professional development designed to help teachers deepen their mathematical
knowledge for teaching through a cycle of solving a mathematics problem, teaching the
problem, and analyzing first teacher questioning and then student thinking in videos of
their teaching. In order to understand the learning opportunities afforded by what they
refer to as a problem-solving-cycle design, they analyzed artifacts from two years of a
series of monthly, full-day workshops with ten middle school mathematics teachers,
including workshop videos and interviews with facilitators. The researchers used the
knowledge domains identified in Ball, Thames, and Phelps (2008) to analyze several
teacher interactions. They found that different learning opportunities were afforded by
different activities: specialized content knowledge was developed by comparing,
reasoning, and making connections between the various solution strategies; knowledge of
Hoover, Mosvold, Ball, & Lai
content and teaching was developed by analyzing teacher questioning in the video clips
from the teachers’ lessons; and knowledge of content and students was developed by
analyzing students’ solution methods (interpreting them and considering their
implications for instruction). More importantly, the researchers found that reflecting on
and discussing the nature of student thinking and teacher questioning of students evident
in videos of their own teaching led teachers to extend their mathematical knowledge for
teaching as they re-engaged with the mathematics problem and reconsidered how they
might teach the problem in light of their new regard for how students might approach the
problem. Throughout the analysis, the authors found that specialized content knowledge
interacts with pedagogical content knowledge in interpreting student thinking and
planning lessons. The authors argue that the workshops developed teachers’
mathematical knowledge for teaching by supporting teachers’ current knowledge, while
gradually challenging them to gain new understanding for the purpose of their work as
teachers.
The lessons from studies such as these are subtle. The movement between
mathematical study and pedagogical practice is central, but attention needs to be given to
dynamics regarding teachers’ motivation, the timing of different activities, and specific
mathematical opportunities arising from specific pedagogical activities. In reading these
reports, one gets the sense that really smart enactment of the professional development
was key to success and that replicating effects might be challenging. From this work, it
would seem important to discern the essential design features and elaborate the necessary
character of facilitation.
One effort along these lines is a study by Elliott, Kazemi, Lesseig, Mumme, and
Kelley-Petersen (2009). In the context of supporting facilitators’ enactment of
mathematically focused professional education, they analyzed facilitators’ learning and
the use of two frameworks provided as conceptual tools: (i) sociomathematical norms for
cultivating mathematically productive discussion in professional development, adapted
from Yackel and Cobb (1996) and (ii) practices for orchestrating productive
mathematical discussions, adapted from Stein, Engle, Smith, and Hughes (2008). In their
study, Elliot and colleagues collected extensive documentation and analyzed the learning
of 5 of the 36 facilitators trained at two sites in 6 two-day seminars across an academic
year. They found that although facilitators responded positively to the frameworks, they
experienced tensions in using the frameworks to ask questions about colleagues’
mathematical thinking and they struggled with the fact that teachers positioned
themselves and others as better or worse in mathematics. These dynamics got in the way
of productive mathematical discussions and frustrated facilitators. The analysis revealed
that one way to mitigate these tensions was by helping facilitators to identify
mathematical ideas that teachers would readily see as worth developing. This led the
researchers to see a need for developing more nuanced and detailed purposes for doing
mathematics in professional development in ways that teachers would see as relevant to
their work.
This then led the researchers to realize that they needed a way to focus the
purpose and work of professional development on connections between mathematics and
the work of teaching. To accomplish this, they added a third framework to their design.
The authors argue that the mathematical-knowledge-for-teaching framework engaged
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facilitators in understanding the ways in which specialized content knowledge (SCK)
connects mathematics to teaching and that the framework provided a meaningful
articulation of the purpose of the professional development and a helpful focus for the
mathematical tasks and discussions that took place.
By understanding how a SCK-oriented purpose for PD is tied to classroom
teaching and being able to articulate that understanding to teachers in accessible
ways, leaders will be able to begin to address the pressure they felt to assure
relevance in their PD. (Elliott et al., 2009, p. 376)
Again, the dynamics between mathematics and the motivation and use of that
mathematics is key to effective teacher learning of professionally relevant mathematics.
The field is also beginning to see evidence that these insights have measurable
yield. For instance, Bell, Wilson, Higgins, and McCoach (2010) argue that it is the
practice-based character of the nationally disseminated Developing Mathematical Ideas
(DMI) mathematics professional development program that best explains participating
teachers’ learning of mathematical knowledge for teaching. The researchers examined
pre and post teacher content knowledge for 308 treatment and comparison teachers across
10 well-established sites. They found significantly larger gains for treatment teachers’
scores and that these gains were related to breadth of opportunity to learn provided by
facilitators. Methodically considering a number of alternative explanations for treatment
teachers’ improvement, the researchers emphasize the classroom-practice feature of the
professional development, where teachers move back and forth between seminars and
their own classrooms, receiving written feedback from regularly observing facilitators.
Referring to Ball and Cohen’s (1999) argument that teacher learning needs to be
embedded in practice, they point out that connecting to practice can leverage teacher
learning in and from their daily work, greatly expanding overall capacity for teacher
learning and improvement. They argue that the practice-based nature of their design
contrasts with professional development that takes place apart from teachers’ practice.
DMI is quite different in this regard, for it encourages teachers to take their
nascent SCK, KCS, and KCT into their classrooms and try things out. Repeatedly,
teachers told us of their revelations — both in seminars and in their own
schools — as they drew on their growing knowledge of and enthusiasm for
mathematics and teaching mathematics in their classrooms. This anecdotal
evidence aligns with results from S. Cohen’s (2004) yearlong study of changes in
teachers’ thinking and practices over the course of their participation in DMI
seminars. (Bell et al., 2010, p. 505)
These different studies compellingly add to the arguments that teachers need
mathematical knowledge that is connected to the work they do and that situating the
learning of mathematical knowledge in teachers’ practice supports the learning of
mathematical knowledge for teaching. Bell et al.’s (2010) large-scale study of the effect
of professional development on teacher learning corroborates the qualitative, small-scale
findings of the other studies. The professional development models highlighted set
teachers up to learn in and from their practice. Together, the studies discussed above
point to the coordinated nature of mathematical knowledge for teaching and the ways in
Hoover, Mosvold, Ball, & Lai
which the coordination between mathematics and pedagogy is essential to teaching and
learning mathematical knowledge for teaching.
Impact of mathematical knowledge for teaching. Whereas more studies have
investigated the nature and composition of mathematical knowledge for teaching and
developing teachers’ knowledge, fewer studies have investigated the impact such
knowledge has on teaching and learning. As mentioned earlier, several studies report
positive effects of mathematical knowledge for teaching on student learning. Crucial to
this research has been the development of robust instruments assessing mathematical
knowledge for teaching. The field has found evidence linking mathematical knowledge
for teaching to student achievement using the LMT instrument (e.g., Hill, Rowan, & Ball,
2005; Rockoff, Jacob, Kane, & Staiger, 2011), the COACTIV instrument (e.g., Baumert
et al., 2010; Kunter et al., 2013), and the Classroom Video Analysis (CVA) instrument
(e.g., Kersting et al., 2010, Kersting et al., 2012). A fewer number of studies have
investigated links between teaching practice and mathematical knowledge for teaching
and/or student achievement (e.g., Hill, Kapitula, & Umland, 2011). In these studies,
student learning is mostly measured by standardized test scores, and the studies vary in
how they measure teaching quality. These studies indicate that, generally speaking,
mathematical knowledge for teaching impacts teaching and learning.
We acknowledge the importance of studies that identify an influence of teachers’
mathematical knowledge on teaching and learning, but are particularly excited about
studies that unpack the dynamics of how mathematical knowledge for teaching impacts
teaching and learning. In their study of 34 teachers, Hill, Umland, Litke, and Kapitula
(2012) demonstrated that the connection between mathematical knowledge for teaching
(measured with the LMT instrument) and the quality of instruction is complex. While
weaker mathematical knowledge for teaching seemed to predict poorer quality of
instruction, and stronger mathematical knowledge for teaching seemed to predict higher
quality of instruction, teachers who performed in the midrange on the LMT measure
varied widely in the quality of their instruction. Student achievement also varied widely
for teachers with mid-range mathematical knowledge for teaching. Furthermore, Hill et
al.’s (2008) study of 10 teachers found that although use of supplemental curriculum
materials, teacher beliefs, and professional development are factors of potential influence,
these factors might all cut both ways depending on the teachers’ mathematical knowledge
for teaching. These two studies underscore that simply establishing impact of knowledge
on teaching is not enough to make decisions about teacher education or policy.
To frame a fuller consideration of impact, we reflect briefly on the nature of
teaching and learning. Teaching mathematics involves managing instructional
interactions, including everything teachers say and do together with students focused on
content, where teacher knowledge is a resource for the work (Cohen, Raudenbush, &
Ball, 2003). This observation suggests that in addition to general effect studies on
teaching and learning, it would be helpful to know more about which specific aspects of
teaching and learning are influenced by teacher content knowledge, which specific
aspects of teacher content knowledge are influential, and how the influences impact
interactions among teacher and students around content. In other words, we propose that
Cohen et al.’s conceptualization of teacher content knowledge as a resource impacting
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instructional interactions is important for framing an investigation of mathematical
knowledge for teaching suited to informing the improvement of teaching and learning.
A promising direction in recent work has been initial investigation of the specific
influence that mathematical knowledge for teaching has on teaching. One example of this
kind is Speer and Wagner’s (2009) case study of one undergraduate instructor’s
scaffolding of classroom discussions. Using Williams and Baxter’s (1996) constructs of
social and analytic scaffolding as a frame, Speer and Wagner argue that aspects of
pedagogical content knowledge are important for helping students find productive ways
of solving particular problems and for understanding which student contributions —
correct or incorrect are important to emphasize in a discussion. They trace ways in
which particular knowledge of students’ understanding aids teachers in assuring that the
lesson reaches intended mathematical goals and in understanding the role of particular
mathematical ideas in students’ development.
In a similar vein, an exploratory study by Charalambous (2010) investigated
teachers’ knowledge in relation to selection and use of mathematical tasks. He
investigated the teaching of two primary mathematics teachers with different levels of
mathematical knowledge for teaching and found notable differences in the quality of their
teaching. He used Stein and colleagues’ mathematical tasks framework to examine the
cognitive level of enacted tasks, and he formulated three tentative hypotheses about
mechanisms of how mathematical knowledge for teaching impacts teachers’ selection
and use of mathematical tasks. First, he hypothesizes that strong mathematical knowledge
for teaching may contribute to a use of representations that supports students in solving
problems, whereas weaker mathematical knowledge for teaching may limit instruction to
memorizing rules. Second, he proposes that mathematical knowledge for teaching
appears to support teachers’ ability to provide explanations that give meaning to
mathematical procedures. Third, he proposes that teachers’ mathematical knowledge for
teaching may be related to their ability to follow students’ thinking and responsively
support development of understanding.
These two studies exemplify potential analyses of mathematical knowledge for
teaching in relation to frameworks of teaching and learning. They leverage findings about
teaching to probe the contributions of mathematical knowledge for teaching in ways that
begin to unpack the specific role such knowledge plays. They are not the only studies to
do so, but to date studies in this realm are rare. Building on these ideas, further
conceptualization of distinctly mathematical tasks of teaching might provide even more
focused contexts for studying mathematical knowledge for teaching as a resource for
teaching. Establishing agreed-upon conceptualizations of mathematical knowledge for
teaching related to well-studied components of the work of teaching and using these as a
common ground for instrument development would provide a solid foundation for
advancing the field.
From this brief review of recent progress on identifying, developing, and
understanding the impact of mathematical knowledge for teaching, we now turn our
attention to proposing directions for future work.
Hoover, Mosvold, Ball, & Lai
Next Steps for the Development of Mathematical Knowledge for Teaching
As described above, compelling examples of mathematical knowledge for
teaching and evidence associating it with improved teaching and learning have sparked
interest in making it a central goal in the mathematical education of teachers. However,
various impediments exist. The lack of rigorous, shared definitions and the incomplete
elaboration of a robust body of knowledge create problems for meaningful measures and
curricula development. Underlying these challenges are competing ideas about how to
conceptualize the knowledge, questions about the relationship among knowledge,
knowledge use, and outcome, and the need for ways to decide claims about whether or
not something constitutes professional knowledge.
We suggest three priorities for research and development of mathematical
knowledge for teaching: (1) focused studies that together begin to compose a more
coherent, comprehensive, and shared understanding of what it is, how it is learned, and
what it does; (2) innovation and reflection on method for investigating it; and (3) studies
of mathematical fluency in teaching and the nature of mathematical knowledge for
equitable teaching. Below, we argue that each of these is vital to long-term progress in
improving the mathematical education of teachers and the mathematics teaching and
learning that depends on it.
Investigating focused issues while contributing to a larger research program.
Scores of articles in the previous decade have argued for particular ways of
distinguishing and conceptualizing important knowledge, and many others have sought to
establish its presence and overall impact. With a sense of the importance of mathematical
knowledge for teaching, additional studies explored the teaching of such knowledge.
However, on the whole, conceptual work has been exploratory, measures have been
general, and studies of the mathematical education of teachers have been limited by
under-specification of the body of knowledge. We suggest that the field would benefit
from focused studies that build on each other in ways that begin to put in place the
machinery needed to develop an overall system for educating teachers mathematically.
Such a system would include clear content-knowledge standards for professional
competence, comprehensive content-knowledge course and program curricula, robust exit
or professional content-knowledge exams, and rationale for what is to be taught in pre-
service programs and what is better addressed in early career professional development or
later on. To get there, we propose collectively pursuing several focal areas of study.
First, mathematical knowledge for teaching needs to be elaborated — for specific
mathematical topics and tasks of teaching, across educational levels. Some of this work is
underway, but we suggest that more needs to be done in ways that research studies, taken
together, define a body of professional knowledge and provide a basis for curricula,
standards, and assessments. One area of need that stands out is the investigation of the
mathematical knowledge demands associated with particular domains of the work of
teaching, such as leading a discussion, launching students to do mathematical work, or
deciding the instructional implications of particular student work. This is a particularly
challenging area of study because the field lacks comprehensive, robust specifications of
the work of teaching. It is also a potentially promising area of study. Where initial
decompositions of teaching are available, such as for orchestrating discussions,
awareness of the mathematical knowledge entailed in the teaching can position teachers
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to learn both the domain of teaching and the mathematical knowledge more productively
(Boerst, Sleep, Ball, & Bass, 2011; Elliott et al., 2009). Nonetheless, domains of teaching
need additional parsing before they can be fully leveraged.
A second proposed area of study is determining meaningful “chunking” of
mathematical knowledge for teaching and practical progressions for teaching and
learning it. In considering the mathematics that students need to learn, topics are typically
decomposed into a sequence of small-sized learning goals. In contrast, teachers’
mathematical knowledge for teaching is not simply a mirror image of student curriculum.
Teachers need knowledge that is different in important ways from the knowledge
students need to learn. Mathematical knowledge for teaching is related to student
curriculum, but it is not clear what this relationship implies for how it is best organized.
In contrast to the mathematics that students need to learn, the specialized mathematics
that teachers need to learn appears to be constituted in ways that span blocks of the
student curriculum.
For instance, a teacher who learns how to model the steps of the standard addition
algorithm using base ten blocks might still need to think through modeling subtraction,
but as a minor extension of what is already learned, not as a new topic, requiring a new
program of instruction. The question deserves more careful examination, but our
experience is that teachers who participate in professional development related to a
particular strand of work on place value exhibit significantly increased mathematical
knowledge for teaching more generally across whole number computation, but with little
to no impact on their mathematical knowledge for teaching topics related to geometry,
data analysis, or even rational number computation. This is just a conjecture, but we offer
it as a way to indicate an area of study that would contribute to improved approaches to
the mathematical education of teachers. How big are these chunks? What are possibilities
for structuring the chunks? Which have the greatest impact for beginning teachers? Some
of these questions could be investigated as part of the elaboration research described
above. Our point is that beyond the important goal of identifying knowledge for specific
mathematical topics and tasks of teaching, across educational levels, research on how
best to organize that knowledge might usefully inform the mathematical education of
teachers.
This discussion leads to a third proposed line of investigation, one that explores
mathematical knowledge for teaching along a professional trajectory from before teachers
enter teacher preparation, through their training and novice practice, and into their
maturation as professionals. This would require navigation among questions about what
teachers know, what might be learned when, what is essential to responsible practice, and
what can be sensibly coordinated with growing professional expertise. For this, the field
would need to know more about the mathematical knowledge for teaching that
prospective teachers bring to teacher education and whether there are things that might
more readily be learned in the program and others that might be more productively
required before admission. The field would need to know more about mathematical
knowledge for teaching that is readily acquired from experience, as well as the supports
needed to do so. Researchers would need to investigate how to distinguish between the
mathematical knowledge for teaching that is essential to know before assuming sole
Hoover, Mosvold, Ball, & Lai
responsibility for classroom instruction and the knowledge that can be safely left to later
professional development. We suggest that such studies would contribute to developing
coherence, efficiency, and responsibility in an overarching picture of the mathematical
education of teachers.
Another proposed area of study would extend work that examines effects of
specific mathematical knowledge on specific teaching and learning in ways that identify
underlying mechanisms and informs views of when and how mathematical knowledge is
used in teaching. We noted above a need for more studies that unpack relationships
among mathematical knowledge for teaching, teaching practice, and student learning.
Such studies might examine the nature of student learning gains resulting from specific
teacher knowledge or they might investigate the mechanisms by which teachers’
mathematical knowledge for teaching has an impact. They would provide a better
understanding of the nature and role of mathematical knowledge in teaching, informing
both its conceptualization and validating underlying assumptions about its significance.
Finally, we suggest that the field would benefit from more studies of effects at a
mid-range level, above that of idiosyncratic, individual programs and courses and below
that of large-scale, international studies. In their 2004 International Congress on
Mathematics Education plenary, Adler, Ball, Krainer, Lin, and Novotna (2008) observed
that the majority of studies in teacher education are small-scale qualitative studies
conducted by educators studying the teachers with whom they are working within
individual programs or courses. The TEDS-M study and the development of some of the
instruments described above have supported an increase in large-scale and cross-case
studies, but as Adler and her colleagues point out, the study of courses, programs, and
teachers by researchers who are also the designers and educators of those programs and
teachers creates both opportunities and risks. From our review, our sense is that many
small studies are driven by convenience and reduced cost, but at the expense of rigorous
design and skeptical stance. Mid-sized studies would be enhanced by efforts such as
developing collaborative investigations across remote sites with either similar or
contrasting interventions. This is consistent with arguments about research on
professional development made by Borko (2004).
Next, we argue that the agenda sketched above will require explicit development
of methods for conducting such research efficiently and effectively.
Innovating and reflecting on method.2 We propose that a central problem for
progress in the field is a lack of clearly understood and practicable methodology for the
study and development of mathematical knowledge for teaching. First, many researchers,
including graduate students, seem eager to conduct studies in this arena, but choices
about research design and approaches to analysis are uncertain. In our review of the
literature, we found that methods vary widely, are relatively idiosyncratic, and are in
general weak in some cases attempting to make causal claims from research designs
poorly suited for such claims and in others providing thoughtful claims but from unclear
2 Material in this section is based on work supported by the National Science Foundation under
grant number 1502778. Opinions are those of the authors and do not necessarily reflect the views of the
National Science Foundation.
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processes and underdeveloped logical rationale. We suspect that a lack of clarity and
rigor of methods, including in our own work, are a result of several factors: unresolved
and underdeveloped conceptualization of the terrain; competing purposes of research
(often within a single study); and uncertain grounds for making claims about whether
something does or does not constitute professional knowledge. Struggles to design robust
studies and to articulate methods used suggest a need for increased attention to method.
This should not be surprising. The vitality of research in areas still in early stages
of theory development requires a concomitant consideration of method. Importing
method from other arenas is appropriate, but regard for the theoretical foundations of the
object of study and their implications for all aspects of method is also important. We
propose that reflective innovation of method, grounded in emerging theory of teaching,
can better account for confounding variables that are relevant to teaching and can inform
the alignment among research questions, design, analysis of data, claims, and
interpretations. To ground this proposal, we reflect on two approaches that have been
evident in efforts to study the nature and composition of mathematical knowledge for
teaching (interview studies and observational studies) and then suggest directions for
potential innovations.
Early investigations of teacher content knowledge were mostly limited to
correlational studies (e.g., Begle, 1979). Correlational studies remain prominent in the
field (e.g., Baumert et al., 2010 Hill, Rowan, & Ball, 2005; Kersting et al., 2010), but in
the 1980s and 1990s studies began using teacher interviews to investigate teacher
knowledge (see Ball, Lubienski, & Mewborn, 2001). This early work was limited in two
ways. First, it tended to focus on identifying deficits in teachers’ mathematical
knowledge instead of clarifying the mathematical knowledge requirements of teaching.
Second, although some good interview prompts emerged and supported a surfeit of
studies, generating additional high-quality prompts has not been easy. The strength of
these early interview prompts was that they were focused, specific, and offered
compelling examples of specialized mathematical knowledge that would be important for
teachers to know. The weaknesses were that they focused the conversation on teachers’
lack of knowledge, while providing little insight into how to rectify these lacks, and they
left the difficult work of generating good prompts invisible.
Similarly, methods for observational studies have often been weakly specified and
hard to use by other scholars as the basis for complementary study. For example, because
of the shortcomings of teacher interviews, the research group at the University of
Michigan developed a practice-based approach to the study of video records of
instruction (Ball & Bass, 2003; Thames, 2009). This approach requires simultaneously
conceptualizing the work of teaching together with the mathematical demands of that
work. It is empirical, interdisciplinary, analytical-conceptual research that involves
developing concepts and conceptual framing by parsing the phenomenon and
systematically testing proposals for consistency with data and with relevant theoretical
and practice-based perspectives. The approach is time-intensive and expensive, requires
skillful use of distributed expertise, and is sufficiently underspecified to make broader
use challenging. For instance, early on, these researchers wrote about ways in which
inter-disciplinary perspectives were central to their analyses (e.g., Ball, 1999; Ball &
Hoover, Mosvold, Ball, & Lai
Bass, 2003), but this characterization, although it captures an important feature of the
approach, is inadequate as a characterization of their approach and as a method for others
to use. It is underspecified, relies more on experienced judgment than on independently
usable criteria or techniques, and leaves key foundational issues in doubt (Thames, 2009).
Reflecting on our own use of these approaches, we offer several, somewhat ad
hoc, observations.
Teaching is purposeful work and, as such, imposes logical demands on the
activity, and these logical demands play a role in warranting claims about the
work of teaching and mathematical knowledge needed for teaching.
Mathematical knowledge for teaching is professional knowledge, and central to
its development and articulation is professional vetting or consensus building
based on cross-community professional judgment.
The pedagogical context provided in crafted items and prompts entails
engagement in the work of teaching and in the use of mathematical knowledge
and, as such, provides crafted instances for the study of specialized knowledge for
teaching.
There is an iterative process among the development of instructional tasks,
assessment items, and interview prompts and our increasing capacity for eliciting
and engaging mathematical knowledge for teaching.
Analysis of mathematical knowledge as professional knowledge for teaching,
whether in situ or in constrained instances, is fundamentally an empirical,
conceptual-analytic, normatively informed process, not a strictly descriptive one.
We believe that the first two observations have important methodological
implications that are as of yet unrealized. Key to understanding teaching and its
knowledge demands is understanding its contextual rationality. In other words,
meaningful study of teaching must account for the directed and contextual nature of the
work. We suspect that such study will require the development and use of methods fit to
the work of teaching and that this means greater reliance of underlying theory of teaching
in designing studies and choosing methods of analysis. As Gherardi (2012, p. 209)
succinctly summarizes in her writing about conducting practice-based studies, “Hence,
empirical study of organizing as knowing-in-practice requires analysis of how, in
working practices, resources are collectively activated and aligned with competence.”
She argues for thoughtful consideration of how methodological approaches are positioned
in relation to the nature of practice and its constant reconstitution in the context of
professional work. We agree with this position and suggest that it is exactly these issues
that need to be taken up in an investigation of method for studying mathematical
knowledge for teaching.
The second observation in our list above raises additional concerns for the
development of new methods. Mathematical knowledge for teaching is professional
knowledge, in the sense that it is shared, technical knowledge determined by professional
judgment (Lortie, 1975; Abbott, 1988), but it is distinctive as a body of knowledge in that
it requires the coordination of mathematics with teaching, which are different areas of
expertise resident in different professional communities (Thames, 2009). Thus, the study
of such knowledge requires coordination across different professional communities with
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different disciplinary foundations. In other words, the study of mathematical knowledge
for teaching requires, or is at least enhanced by, collective work across distinct
professional communities with different expertise and different professional norms and
practices, and such work requires special consideration and support (Star & Griesemer,
1989).
With the call for cross-professional coordination, the study of mathematical
knowledge for teaching involves much more than an assembly line model where different
professional constituents inject their specific expertise into a product handed down the
line. It calls for specification of processes for collectively considering whether a proposed
claim of professional knowledge is warranted. It is about establishing protocols for
merging and melding different expertise in the midst of improvement work that attends to
overall coherence and practical merit. It requires specific ways of working together, tools
for organizing the scholarly work, and boundary objects that provide meaningfully
bridges among communities (Akkermann & Bakker, 2011). Each of these adds to the
need for new methodology.
Innovation and reflection on method can be carried out in numerous ways.
Researchers can simply attend more closely to decisions of method and explicit reporting
of method. Alternatively, they can deliberatively develop, implement, and study methods.
In order to provide a sense of the kind of innovation and reflection that might be done, we
discuss some of the ways we have begun to explore methodological approaches for the
study of mathematical knowledge for teaching.
An emerging approach we find promising is to use sites where professional
deliberation about teaching are taking place as sites where we might productively
research the work of teaching and its mathematical demands. In recent studies, we have
designed interview protocols as a tool for generating data useful for studying the
mathematical work of teaching. For instance, to investigate the work involved in
providing students with written feedback, Kim (this issue) provides a strategic piece of
student work and asks interviewees to provide written feedback and to explain the
rationale for the feedback. Here, instead of videotaping classroom instruction and
analyzing the mathematical demands of teaching, Kim analyzes those demands as they
play out in a constrained slice of the mathematical work of teaching as evidenced in
responding to a teaching scenario provided.
We see this approach as an instance of a more general phenomenon, one of using
sites of professional deliberation about teaching as research sites for studying teaching.
For example, a group of mathematics teachers and mathematics educators in a
professional development setting might discuss responses to a particular pedagogical
situation in ways akin to the pedagogical deliberations of a teacher engaged in teaching.
Thus, this professional development event can be useful for studying professional
practice. It may even have the advantage that professional action and reasoning are more
explicitly expressed, yet of course, with certain caveats in place as well, such as
recognizing that real-time demands of teaching are suspended. Similar opportunities can
arise in other settings where pedagogical deliberations take place, such as teacher
education or the development of curriculum or assessment. For instance, recent
investigation of the design process for producing tasks to measure mathematical
Hoover, Mosvold, Ball, & Lai
knowledge for teaching suggests that writing and reviewing such tasks can provide
insight into teaching and its mathematical demands, even to the point of serving as a site
for investigating mathematical knowledge for teaching (Jacobson, Remillard, Hoover, &
Aaron, in press; Herbst & Kosko, 2012).
We propose that such an approach is distinctively different from general interview
techniques that have teachers reflect on their teaching. Crucial to this difference is that
the prompts are designed to provide authentic pedagogical contexts with essential, yet
minimal, constraints for directing targeted pedagogical work (such as a crucial
instructional goal, a key excerpt from a textbook, or strategically selected student work).
Good pedagogical context needs to be based on initial conceptions of key aspects of the
work, and constraints need to be designed to engage initial ideas about the nature and
demands of the work. Otherwise, the pedagogical context of the tasks is unlikely to
engage people in authentic pedagogical work.
Our recent experience with interview prompts of this kind has convinced us of
their potential for studying teaching and teacher knowledge. Several advantages are
evident: constraints provided can be manipulated; different professional communities can
be engaged; and bounded instances of work examined. The development of this approach
would support new lines of research that specify teaching and its professional knowledge
demands in ways that can better inform professional education and evaluation. They are
also easy to use and require only modest time and expense.
Such innovations begin to suggest the development of a “laboratory science”
approach for studying mathematical knowledge for teaching that takes advantage of the
tools of constrained prompts, the generative analytic techniques of instructional analysis,
and the multiple sites available for such study. By a “lab science” approach we mean
direct interaction with the world of instruction or slices of instruction using tools, data
collection techniques, and models and theories of teaching. Analogous to the ways in
which experimental psychologists isolate phenomena under controlled conditions in a
laboratory setting or biochemists manipulate protein processes at the bench, we propose
that the study of specialized teacher content knowledge can isolate activities of teaching
and the use of resources, examine those activities and resources in detail, and
systematically manipulate constraints to better understand phenomena. This work can be
done deductively, to test specific hypotheses, inductively, to discern functional
relationships, or abductively, to refine current understanding. Such an investigation of
method should be intimately tied to underlying foundational issues, both shaped by
theoretical commitments and giving precise definition and form to underlying theory.
In conclusion, we suggest that the development of usable, practical, and
defensible method, whether along the lines we have sketched here or along other lines,
will be critical to carrying out the extensive agenda described earlier for building a
understanding of mathematical knowledge for teaching adequate for sustainable
improvement of the mathematical education of teachers. We now sketch two areas of
study largely missing from the literature on knowledge distinctive for teaching
mathematics and argue that both of these are essential to viable progress on building a
theory and practice of mathematical knowledge for teaching.
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Addressing Two Key Issues: Mathematical Knowledge for Fluent and
Equitable Teaching. Although there has been substantial progress in conceptualizing
and understanding the mathematical understanding needed for the practice of teaching,
significant issues remain. We focus here on two aspects that seem to us to be particularly
critical to progress on mathematical knowledge for teaching. One centers on the
communicative demands of teaching, the other on what is involved in teaching to disrupt
the historical privileging of particular forms of mathematical competence and
engagement, resulting in persistent inequity in access and opportunity. We argue that
both of these are key to the long-term viability of efforts to improve the mathematical
education of teachers.
Teaching is inherently a communication-intensive practice. Teachers listen to
their students, explain ideas, and pose questions. They read their students’ written work
and drawings, and provide written feedback. Throughout these communications, they use
mathematics in a range of specialized ways. They must hear what their students say, even
though students talk and use mathematical and everyday language in ways that reflect
their emergent understanding. Similarly, they must interpret students’ writing and
drawings. When they talk, teachers must attune their language to students’ current
understanding, and yet do so in ways that are intellectually honest and do not distort
mathematical ideas to which they are responsible for giving their students access.
What is involved in this sort of mathematical communication in the context of
teaching? Because teaching is fast-paced and interactive, the demands are intense. Talk
and listening cannot be fully scripted or anticipated. A special kind of mathematical
fluency is required, tuned to the work of teaching. Asking a question in the moment;
explaining in response to a student’s puzzlement; listening to, interpreting, and
responding to a child’s explanation –– each of these involves hearing and making sense
of others’ mathematical ideas in the moment, speaking on one’s feet while seeking to
connect with others. Although much of the work on mathematical knowledge needed by
teachers is situated in relation to what teachers do, including using representations and
interpreting students’ thinking, as yet little of it has focused on the mathematical fluency
needed for the work teachers do in classrooms, live, in communicating with students. As
compellingly argued by Sfard (2008) and others (e.g., Resnick, Asterhan, & Clarke,
2015), it is this communicative work that is central to the practice of education. Failing to
investigate and squarely address communicative mathematical demands of teaching may
result in an impoverished theory of mathematical knowledge for teaching in ways that
sorely limit its utility and impact.
Another major area of work centers on the need to address the persistent
inequities in mathematics learning both produced and reproduced in school. Goffney and
her colleagues have begun to identify a set of practices of equitable mathematics teaching
(Goffney, 2010; Goffney & Gonzalez, 2015; Goffney, 2015), and several of the articles
in this volume explore the measurement of mathematical knowledge for equitable
teaching. The driving question is what do teachers need to appreciate and understand
about mathematics in order to be able to create access for groups that have been
historically marginalized? Part of this has to do with a flexible understanding of the
mathematics that enables teachers to build bridges between mathematics and their
Hoover, Mosvold, Ball, & Lai
students. One aspect of this is to represent mathematics in ways that connect with their
students’ experience. Another is to be able to recognize mathematical capability and
insight in their students’ out-of-school practices. Each of these entails a flexibility of
mathematical understanding, particularly of mathematical structure and practice. But it
also involves the ability to recognize as mathematical a range of specific activities,
reasoning processes, and ways of representing. Being able to do this can enable teachers
to broaden both what it means to be “good at math” as well as what can be legitimated as
“mathematics.”
Equity is not a new focus in mathematics education (e.g., Schoenfeld, 2002), and
there have been studies on the effect of gender and language on mathematics teachers’
knowledge (Blömeke, Suhl, & Kaiser, 2011) as well as the distribution of teacher
knowledge in different populations of teachers (Hill, 2007). In our review of the
literature, we observed that most studies on equity were focused on aspirations and
imperatives (i.e., arguments for teaching for equity). Few studies focused directly on
specific practices of equitable mathematics teaching or knowledge for equitable
mathematics teaching. We argue that increased focus in this area is crucial for three
reasons. First is the underlying principle that extant inequity in mathematics teaching and
learning is morally reprehensible in a civilized society (Perry, Moses, Cortez, Delpit, &
Wynne, 2010). Second is our contention that, while certainly not in itself a solution,
teacher content knowledge is both an indispensible and an untapped resource for
disrupting the historical privileging of particular forms of mathematical competence and
engagement. Third, as with nearly all achievement measures in early stages of
development, current instruments are significantly biased because of the contextual
features of where, as well as for and by whom, they are developed. The field needs good
instrumentation, for research and for practice. Overly delaying the development of
unbiased instruments may well undermine the political viability of well-meaning efforts
to improve the mathematical education of teachers. Such development will require solid
research in this difficult yet important arena.
In proposing these two areas of study, we acknowledge the conceptual and
methodological challenges each presents. We suspect that research in these areas has
been underdeveloped in large part because these foci involve subtle social dynamics less
readily captured in print and in more conventional measures. These challenges simply
add to our concern that concerted attention be given them. Our argument here is that
these two areas of study are not merely our favored topics, but that that they are essential
to long-term success.
Articles that Develop Measures and Measure Development
The agenda sketched above is both a reflection of emerging work in the field and
a proposal for future work. In many ways, the articles in this special issue, though
specifically addressing measurement, resonate with themes above. For instance, the
discussion about focused studies that contribute to a larger research program suggests
some benefits of creating a common framework for describing mathematics teaching. In
their article, Selling, Garcia and Ball (this issue) present a framework for unpacking the
mathematical work of teaching that is promising in this respect. Whereas other
frameworks often start with what teachers do, they focus first on the mathematical objects
involved in the work of teaching and then follow up by describing actions that teachers
TME, vol. 13, no. 1&2 p.
27
do on these objects. This idea builds upon and extends the notion of mathematical tasks
of teaching that has been highlighted in previous publications on the practice-based
theory of mathematical knowledge for teaching (e.g., Ball et al., 2008; Hoover, Mosvold,
& Fauskanger, 2014), as well as in previous efforts to conceptualize the work of teaching
(e.g., Ball & Forzani, 2009). A main aim with this framework is to inform and assist
future development of items and instruments for measuring mathematical knowledge for
teaching.
Phelps and Howell (this issue) discuss the role of teaching contexts in items
developed to measure mathematical knowledge for teaching. Given that mathematical
knowledge for teaching is understood as knowledge applied in the work of teaching, a
teaching context that illustrates a certain component of this work is typically included in
items. Phelps and Howell discuss different ways in which context can be critical to
assessing mathematical knowledge for teaching. They argue that attention to the role of
context might provide better understanding of the knowledge assessed in particular items
and might also inform further development of a theory in which teaching context is used
to define knowledge.
Whereas both of these first articles point to core issues regarding the
conceptualization of mathematical knowledge for teaching in the context of item and
instrument development the next two articles deal more directly with measurement.
Kim (this issue) focuses on designing interview prompts for assessing mathematical
knowledge for teaching. In particular, her discussion focuses on the task of providing
written feedback to students. To model this task, she combines a decomposition of the
task with aspects of the pedagogical context involved and sub-domains of mathematical
knowledge for teaching.
Where Kim’s study is more qualitative and conceptual in nature, Orrill and Cohen
(this issue) draw on psychometric models in their work. Their study hinges on the issue
of defining the construct measured, and they use a mixture Rasch model to analyze
different subsets of items to support an argument that the domain definition has strong
implications on the claims one tries to make about teachers’ performance. In light of our
observations about the lack of consensus about how to define the constructs that are being
measured and discussed across studies, a focus on careful construct definition and
implications is particularly relevant.
In the international literature on teaching and learning, a focus on equity is
prevalent. In research on mathematical knowledge for teaching, the discussion of
knowledge for teaching equitable mathematics also receives some attention although
issues of equity and diversity have not been emphasized in frameworks of mathematical
knowledge for teaching. In this connection, Wilson’s (this issue) and Turkan’s studies of
mathematical knowledge for teaching English language learners draw attention to this
missing area of research. Both involve design and application of measures. Wilson
proposes a new aspect of pedagogical content knowledge that is connected specifically to
the work of teaching mathematics to English language learners. Turkan addresses
practicing teachers’ reasoning about teaching mathematics to ELLs. Based on analysis of
data from cognitive interviews, she argues that there is a unique domain of knowledge
Hoover, Mosvold, Ball, & Lai
necessary for teaching ELLs thus supporting Wilson’s argument and she calls for
further investigations to identify and assess this knowledge.
Finally, this special issue includes two articles that investigate teachers’ views.
Koponen, Asikainen, Viholainen and Hirvonen investigate the views of teachers as well
as teacher educators about the content of mathematics teacher education. Results from
their survey indicate that teachers as well as teacher educators in the Finnish context
emphasize the need for courses in content knowledge that is distinctive for teaching
not just more advanced. They argue that the mathematical content of teacher education
needs to be tightly connected to the mathematics being taught, and even pedagogical
courses need to include knowledge connected with mathematics, in particular focusing on
knowledge of teaching and learning of mathematics. In the last article of the special issue,
Kazima, Jakobsen and Kasoka investigate Malawian teachers’ views about mathematical
tasks of teaching and the potential usefulness of adapted measures of mathematical
knowledge for teaching among Malawian pre-service mathematics teachers. The
measures as well as the applied framework of mathematical knowledge for teaching were
developed in the United States. Despite the significant cultural differences between
Malawi and the United States, the authors argue that the framework as well as most of the
items function well in the Malawian context.
Together, this collection of articles on the development and use of measures lies
at a transition from the lessons of past studies of mathematical knowledge for teaching
into vital arenas of research needed for systemic improvement on the mathematical
education of teachers.
TME, vol. 13, no. 1&2 p.
29
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... In mathematics, Hoover's review of research articles (Hoover et al., 2016) that focus on the distinct mathematical knowledge needed in the work of teaching mathematics and published in English in international peer-reviewed journals between 2006 and 2013, revealed 190 published articles. In 99 of these articles, researchers used an instrument to measure teachers' knowledge, and in 56 of these cases, the instrument employed was non-standardized. ...
... One problem with commonly used quantitative methods is that item parameters in different settings refer to different scales and can sometimes falsely flag items as problematic (false positives) or miss potentially problematic ones (false negatives). Furthermore, as our review of literature shows, even though the LMT instrument is the most frequently used measure of MKT around the world (Hoover et al., 2016), no studies-to our knowledge-involve collecting and analyzing data using the same form in two different cultural contexts. ...
... Among these frameworks, the MKT conceptualization of teacher knowledge, with the associated LMT instrument, tends to be predominantly used in studies of teacher mathematical knowledge for teaching, at least when it comes to studies focused on measuring such knowledge (Blömeke & Delaney, 2012;Hoover et al., 2016). As MKT is a practicebased framework, the knowledge domains are defined in relation to the work of teaching . ...
Article
Full-text available
The measures of mathematical knowledge for teaching developed at the University of Michigan in the U.S., have been adapted and used in studies measuring teacher knowledge in several countries around the world. In the adaptation, many of these studies relied on comparisons of item parameters and none of them considered a comparison of raw data. In this article, we take advantage of having access to the raw data from the adaptation pilot studies of the same instrument in Norway and Slovakia (149 practicing elementary teachers in Norway, 134 practicing elementary teachers in Slovakia) that allowed us to compare item parameters and teachers’ ability estimates on the same scale. Statistical analysis showed no significant difference in the mean scores between the Norwegian and the Slovak teachers in our samples and the paper provides further insight into the issues of cross-national adaptations of measures of teachers’ knowledge and the limitations of the methods commonly applied in the item adaptation research. We show how item adaptations can be refined by combining robust quantitative methods with qualitative data, how decisions on adaptation of individual items depend on context and purpose of the adaptation, and how comparability and heterogeneity of samples affects interpretation of the results.
... Following Shulman, research on content knowledge for teaching has flourished in educational research. In mathematics education specifically, numerous studies have investigated mathematical knowledge that is distinctive to teaching (for a review, see Hoover, Mosvold, Ball, & Lai 2016). Some studies have focused on the nature and composition of this knowledge (e.g. ...
... In their qualitative synthesis of studies that investigate the influence of mathematical knowledge for teaching on teaching practice, Mosvold and Hoover (2017) conclude that although some correlational studies indicate a potential influence, there is still a need to investigate how mathematical knowledge might influence teaching. Relatedly, Hoover et al. (2016) suggest that, "[e]stablishing agreed-upon conceptualizations of mathematical knowledge for teaching related to well-studied components of the work of teaching and using these as a common ground for instrument development would provide a solid foundation for advancing the field" (p. 17). ...
... Researchers indicate that teachers' subject matter knowledge influence their teaching practices (Copur-Gencturk, 2015;Hoover et al., 2016;McCrory et al., 2012;Spangenberg, 2021) and therefore affect student acquisition in algebra (Even, 1993;Huang & Kulm, 2012;Tchoshanow et al., 2017;Wasserman, 2015). Subject matter knowledge for teaching mathematics includes (a) common content knowledge (i.e., mathematical knowledge and skills that are not specific to teaching) and (b) specialized content knowledge (i.e., knowledge of different forms of representation of mathematical contents and explanations of mathematical rules and procedures that are specific to teaching mathematics) (Ball et al., 2005;Ball et al., 2008;Hill & Ball 2004;Hill et al., 2008). ...
... Lack of teachers' content knowledge negatively affects the pedagogical content knowledge and thus teaching practices (Ball, 1990;Charalambous et al., 2020;Copur-Gencturk, 2015, Hoover et al., 2016Kelcey et al., 2019;Howell, 2012;McCrory et al., 2012;Stephens, 2004;Welder & Simonsen, 2011). The findings of this study suggest that developing teachers" knowledge about algebra in practice is and should be an essential component of teacher professional development. ...
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Teacher knowledge is particularly important in middle school mathematics to enhance the meaningful transition from arithmetic to algebra. Developing a solid understanding of algebra requires teachers to unpack the meaning underlying the basic concepts, make clear and explicit distinctions among concepts, and facilitate students to connect these concepts with their prior knowledge and understandings in arithmetic. This study focused on the content knowledge (i.e., decompose, trim, and bridge) of three middle school mathematics teachers in practice during the introduction of basic algebra concepts (i.e., variable, algebraic expression, equality and equations). Case study was used as research design in this research. Individual interviews and observations regarding the instructions were used to gather data. Results showed that the participating teachers’ conceptions regarding the variable, algebraic expression and equation were rather narrow and it resulted in several constraints when unpacking their meaning in the classroom. While bridging was a commonly observed instructional practice among the participating teachers, the quality of bridging, did not always provide opportunities for a meaningful transition. Trimming, on the other hand, was not observed as a common teaching practice.
... The research reveals that pre-service teachers (PSTs) often do not have adequate subject matter knowledge as well as mathematical knowledge for teaching when they start teaching (Santagata & Lee, 2021). Hoover et al. (2016) assert that this mathematical knowledge involves an explicit conceptual understanding of the principles and the meaning underlying mathematical procedures, and connectedness rather than compartmentalization of mathematical topics and definitions. She states that improvements in academic and professional educational coursework should be a focus of current educational reforms. ...
Article
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In this paper, we report the analysis of thought processes used by Pre-Service Teachers' (PSTs') through clinical interviews as they solved an algebra task involving a linear pattern. The PST's were asked about a mathematical model they had constructed to describe a pattern problem. Our analysis suggests that conflict factors arise due to incompatibility in participants' personal concept definition and the formal concept definition of continuity. We identified how personal concept definitions of the participants differed and how this difference affected their decision on whether a graph was continuous or not.
... The study of the knowledge and competencies necessary for teaching mathematics, as well as the means and tools to develop them, is very complex; thus, there is no single way to define or operationalize them. As Hoover et al. (2016) stated, it does not exist a "theoretically grounded, well defined, and shared conception" (p. 3) of mathematical knowledge for teaching. However, many frameworks developed on this issue take into consideration the categories of content knowledge for teaching defined by Shulman (1986). ...
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Socializing Intelligence Through Academic Talk and Dialogue focuses on a fast-growing topic in education research. Over the course of 34 chapters, the contributors discuss theories and case studies that shed light on the effects of dialogic participation in and outside the classroom. This rich, transdisciplinary endeavor will appeal to scholars and researchers in education and many related disciplines, including learning and cognitive sciences, educational psychology, instructional science, and linguistics, as well as to teachers, curriculum designers, and educational policy makers. https://ebooks.aera.net/catalog/book/socializing-intelligence-through-academic-talk-and-dialogue
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'The variety of approaches that claim to constitute practice-based research are several and varied. Silvia Gherardi cuts through the various approaches to address practice-based research as itself a practice in an invaluable guide for organization and management researchers. Written in a characteristically accessible style, this volume is an indispensable guide.' - Stewart Clegg, University of Technology Business School, Sydney, Australia.
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