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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. 3–34

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.

7

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 (10–29)

51

Medium 2 (30–70)

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 (K–8)

81

Middle (5–9)

45

Secondary (7–13)

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 teachers’ SM?

28

What curriculum/tasks improve teachers’ SM?

10

What teaching practice improves teachers’ SM?

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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