What Do Math Teachers Need to Know?
The outside observer can be forgiven for thinking that the teaching of mathematics is a simple endeavour that requires nothing more than a narrow range of content knowledge. After all, many of the decisions made and actions taken by effective math teachers are subtle and would escape the notice of those outside of the teaching profession. Additionally, many such individuals learned mathematics from teachers whose knowledge bases featured deficits that limited their effectiveness. To answer the question that forms the title of this article, it is first necessary to define effective mathematics teaching and to contextualize this definition in the reality of the level of requisite knowledge that teachers, in general, have today. With that context in place, it will then be appropriate to examine the many domains of knowledge that are required for effective mathematics teaching.
The Work of the Mathematics Teacher
To understand the work of truly effective math teachers, one must understand something about their position in relation to mathematicians. Over centuries and millennia, mathematicians have discovered seemingly unrelated math ideas, abstracted them to find points of commonality, and used these to create generalizations that have resulted in rules and procedures that we use today. Davis and Simmt (2006) describe this process as “‘compressing’ information into increasingly concise and powerful formulations” (p. 299). The results of this process are used by adults in their day-to-day lives. However, “teachers work with mathematics as it is being learned” by children (Ball & Bass, 2003, p. 11). A large body of research shows that merely teaching students mathematical rules and procedures does not lead to true understanding. The math teacher, then, must unpack the mathematical knowledge, break it down into the components with which the mathematicians started, and lead the students to reenact the mathematician’s process of abstracting and generalizing. The goal is for students to develop rules and procedures on a solid foundation of conceptual understanding. Ball and Bass (2003) consider this unpacking “a distinctive feature of knowledge for teaching” (p. 11).
Davis and Simmt (2006) see the role of teachers as opposite to that of mathematicians: while mathematicians compress mathematical ideas together for convenience of use, teachers “must be adept at prying apart concepts, making sense of the analogies, metaphors, images, and logical constructs that give shape to a mathematical construct” (pp. 299-300). Ball and Bass (2003) reveal that the actions taken by teachers during this process (see Figure 1) feature high levels of mathematical work (p. 11). Clearly, the knowledge that is required to do all these things goes beyond simply applying mathematical rules and procedures.
Research shows, however, that many teachers have significant gaps in their mathematical knowledge (Ball, Hill, & Bass, 2005). These deficits may prevent them from performing some or all of the functions shown in Figure 1 to an acceptable level. In an era when many educational jurisdictions are experiencing declining student achievement in mathematics, “skilled teachers who understand the subject matter” are required if the trend is to be reversed (Ball et al., 2005, p. 14). So, what knowledge is required by mathematics teachers? Research suggests that mathematics courses alone are not the answer. Both Davis and Simmt (2006) and Ball and Bass (2003) report a weak relationship between student performance in mathematics and the number of advanced math courses taken by their teachers. Ball and Bass (2003) even describe that, in some cases, higher numbers of such courses taken by teachers are accompanied by negative effects on student achievement. These findings lead Davis and Simmt (2006) to conclude that more math is not required and that it “might be that teachers require more nuanced understandings of the topics in a conventional curriculum” (p. 294). A more pertinent question to consider at this point may be, “What types of knowledge contribute to a more nuanced understanding of the topics that comprise mathematics curricula?”
Mathematical Knowledge for Teaching
Several research teams have studied the nature of the mathematical knowledge that teachers require. One of the most comprehensive frameworks that has resulted from these studies was developed by Deborah Ball and Hyman Bass. They studied math teachers in action to determine which skills enabled them to unpack mathematical content and guide students toward the construction of understanding. They then used these observations to extrapolate the types of knowledge that are required for this task (Ball & Bass, 2003; Ball et al., 2005). Their framework, known as Mathematical Knowledge for Teaching (MKT), with its varied domains of knowledge, is shown in Figure 2.
The Work of the Mathematics Teacher
To understand the work of truly effective math teachers, one must understand something about their position in relation to mathematicians. Over centuries and millennia, mathematicians have discovered seemingly unrelated math ideas, abstracted them to find points of commonality, and used these to create generalizations that have resulted in rules and procedures that we use today. Davis and Simmt (2006) describe this process as “‘compressing’ information into increasingly concise and powerful formulations” (p. 299). The results of this process are used by adults in their day-to-day lives. However, “teachers work with mathematics as it is being learned” by children (Ball & Bass, 2003, p. 11). A large body of research shows that merely teaching students mathematical rules and procedures does not lead to true understanding. The math teacher, then, must unpack the mathematical knowledge, break it down into the components with which the mathematicians started, and lead the students to reenact the mathematician’s process of abstracting and generalizing. The goal is for students to develop rules and procedures on a solid foundation of conceptual understanding. Ball and Bass (2003) consider this unpacking “a distinctive feature of knowledge for teaching” (p. 11).
Davis and Simmt (2006) see the role of teachers as opposite to that of mathematicians: while mathematicians compress mathematical ideas together for convenience of use, teachers “must be adept at prying apart concepts, making sense of the analogies, metaphors, images, and logical constructs that give shape to a mathematical construct” (pp. 299-300). Ball and Bass (2003) reveal that the actions taken by teachers during this process (see Figure 1) feature high levels of mathematical work (p. 11). Clearly, the knowledge that is required to do all these things goes beyond simply applying mathematical rules and procedures.
Research shows, however, that many teachers have significant gaps in their mathematical knowledge (Ball, Hill, & Bass, 2005). These deficits may prevent them from performing some or all of the functions shown in Figure 1 to an acceptable level. In an era when many educational jurisdictions are experiencing declining student achievement in mathematics, “skilled teachers who understand the subject matter” are required if the trend is to be reversed (Ball et al., 2005, p. 14). So, what knowledge is required by mathematics teachers? Research suggests that mathematics courses alone are not the answer. Both Davis and Simmt (2006) and Ball and Bass (2003) report a weak relationship between student performance in mathematics and the number of advanced math courses taken by their teachers. Ball and Bass (2003) even describe that, in some cases, higher numbers of such courses taken by teachers are accompanied by negative effects on student achievement. These findings lead Davis and Simmt (2006) to conclude that more math is not required and that it “might be that teachers require more nuanced understandings of the topics in a conventional curriculum” (p. 294). A more pertinent question to consider at this point may be, “What types of knowledge contribute to a more nuanced understanding of the topics that comprise mathematics curricula?”
Mathematical Knowledge for Teaching
Several research teams have studied the nature of the mathematical knowledge that teachers require. One of the most comprehensive frameworks that has resulted from these studies was developed by Deborah Ball and Hyman Bass. They studied math teachers in action to determine which skills enabled them to unpack mathematical content and guide students toward the construction of understanding. They then used these observations to extrapolate the types of knowledge that are required for this task (Ball & Bass, 2003; Ball et al., 2005). Their framework, known as Mathematical Knowledge for Teaching (MKT), with its varied domains of knowledge, is shown in Figure 2.
Subject Matter Knowledge
The left half of the framework explores the purely mathematical domains of knowledge that are required for effective teaching. Davis and Simmt note that Subject Matter Knowledge is not “a watered down version of formal mathematics, but a serious and demanding area of mathematical work” (p. 295).
Common Content Knowledge (CCK). This first body of knowledge is the easiest to understand and obtain. Rowland (2014) defines it as “‘school mathematics,’ applicable in a range of everyday and professional contexts demanding the ability to calculate and solve mathematical problems” (p. 236). Sullivan, Clarke, and Clarke (2013) refer to CCK as “mathematics needed to solve a task” (p. 16). The verbs in these definitions (i.e., “calculate” and “solve”) signify that CCK revolves around the rules and procedures that enable one to do (but not necessarily understand) mathematics. Being able to determine that 25 x 35 equals 875 is an example of Common Content Knowledge. Of course, knowledge of this nature is essential to teaching insofar as it allows teachers to distinguish correct answers from incorrect ones. However, it is not nearly enough on its own as it does not allow teachers to perform the actions shown in Figure 1 (Ball et al., 2005). Other types of knowledge are required.
Specialized Content Knowledge (SCK). This body of knowledge takes teachers beyond simply knowing how math works (which is CCK) and allows them to understand why math works the way it does. It involves a deep conceptual understanding that enables “teachers to represent mathematical ideas accurately, provide explanations for common rules and procedures, [and] examine and understand unusual methods” (Sullivan et al., 2013, p. 16). Whereas CCK permits a teacher to identify a wrong answer, SCK allows a teacher to determine the thinking behind the wrong answer and develop explanations and representations to help the student understand the mistake and use it to develop correct conceptual understanding. Figure 3 demonstrates three different solutions to the same multiplication question. Student B’s solution models the use of the most common multiplication algorithm. A teacher with strong CCK would recognize it as correct. However, that same teacher may not understand the thinking employed by Student A and Student C (and may not give them the credit they deserve). A teacher with strong SCK would be able to understand the procedures used by these two students, credit them appropriately, and, perhaps, use these lesser known approaches to model different ways of thinking about multiplication to the rest of the class (Ball et al., 2005).
Specialized Content Knowledge has been described as “knowledge of mathematics content that mathematics teachers need in their work, but others do not” (Rowland, 2014, p. 236). Some scholars have taken exception to this definition. Gadanidis and Namukasa (2007) argue that students should have access to SCK so that they, like teachers, would be able to analyze errors, identify the thinking that may have produced such errors, and recognize which representations and/or manipulatives might be best at a given moment. Davis and Simmt (2006) make a similar argument within the specific conceptual context of multiplication. Their point is well-taken; specialized content knowledge should be passed on to students. Students need, for a variety of reasons, to understand the conceptual foundations of the mathematical procedures and rules they use. However, the fact that Deborah Ball and her varied colleagues do not explicitly call for the teaching of SCK to students does not mean that they are opposed to the idea. The focus of their body of work on Mathematical Knowledge for Teaching is on what teachers do and need to know about mathematics. They do not address student knowledge directly; in fact, in one article, they call direct attention to the fact that their focus on teacher knowledge leads to a certain exclusion of students (Ball et al., 2005). It can be inferred, however, that Ball and her associates are in favour of the spread of specialized content knowledge to students. As teachers employ SCK to develop mathematical explanations and representations for students, and interpret and share the thinking behind student errors and unexpected solutions, they invariably transfer some of this knowledge to their students. Ball et al. would know that teachers cannot use SCK for all the purposes listed in Figure 1 without this transfer happening. While Gadanidis and Namukasa, and Davis and Simmt, make a good point, it is a point with which Ball, Hill, and Bass would agree.
Horizon Content Knowledge (HCK). The final domain of Subject Matter Knowledge “entails knowing what mathematical experiences precede those in a given level and what will follow in the next, and subsequent, grades” (Rowland, 2014, p. 236). Teachers with this type of knowledge are able to trace the evolution of mathematical ideas through multiple years of curriculum. They can see that concepts are repeated through the years, but are added onto, so that each new iteration, in addition to containing everything that came before it, features new complexity and sophistication (a structure that Davis and Simmt (2006) call recursive elaboration).
The left half of the framework explores the purely mathematical domains of knowledge that are required for effective teaching. Davis and Simmt note that Subject Matter Knowledge is not “a watered down version of formal mathematics, but a serious and demanding area of mathematical work” (p. 295).
Common Content Knowledge (CCK). This first body of knowledge is the easiest to understand and obtain. Rowland (2014) defines it as “‘school mathematics,’ applicable in a range of everyday and professional contexts demanding the ability to calculate and solve mathematical problems” (p. 236). Sullivan, Clarke, and Clarke (2013) refer to CCK as “mathematics needed to solve a task” (p. 16). The verbs in these definitions (i.e., “calculate” and “solve”) signify that CCK revolves around the rules and procedures that enable one to do (but not necessarily understand) mathematics. Being able to determine that 25 x 35 equals 875 is an example of Common Content Knowledge. Of course, knowledge of this nature is essential to teaching insofar as it allows teachers to distinguish correct answers from incorrect ones. However, it is not nearly enough on its own as it does not allow teachers to perform the actions shown in Figure 1 (Ball et al., 2005). Other types of knowledge are required.
Specialized Content Knowledge (SCK). This body of knowledge takes teachers beyond simply knowing how math works (which is CCK) and allows them to understand why math works the way it does. It involves a deep conceptual understanding that enables “teachers to represent mathematical ideas accurately, provide explanations for common rules and procedures, [and] examine and understand unusual methods” (Sullivan et al., 2013, p. 16). Whereas CCK permits a teacher to identify a wrong answer, SCK allows a teacher to determine the thinking behind the wrong answer and develop explanations and representations to help the student understand the mistake and use it to develop correct conceptual understanding. Figure 3 demonstrates three different solutions to the same multiplication question. Student B’s solution models the use of the most common multiplication algorithm. A teacher with strong CCK would recognize it as correct. However, that same teacher may not understand the thinking employed by Student A and Student C (and may not give them the credit they deserve). A teacher with strong SCK would be able to understand the procedures used by these two students, credit them appropriately, and, perhaps, use these lesser known approaches to model different ways of thinking about multiplication to the rest of the class (Ball et al., 2005).
Specialized Content Knowledge has been described as “knowledge of mathematics content that mathematics teachers need in their work, but others do not” (Rowland, 2014, p. 236). Some scholars have taken exception to this definition. Gadanidis and Namukasa (2007) argue that students should have access to SCK so that they, like teachers, would be able to analyze errors, identify the thinking that may have produced such errors, and recognize which representations and/or manipulatives might be best at a given moment. Davis and Simmt (2006) make a similar argument within the specific conceptual context of multiplication. Their point is well-taken; specialized content knowledge should be passed on to students. Students need, for a variety of reasons, to understand the conceptual foundations of the mathematical procedures and rules they use. However, the fact that Deborah Ball and her varied colleagues do not explicitly call for the teaching of SCK to students does not mean that they are opposed to the idea. The focus of their body of work on Mathematical Knowledge for Teaching is on what teachers do and need to know about mathematics. They do not address student knowledge directly; in fact, in one article, they call direct attention to the fact that their focus on teacher knowledge leads to a certain exclusion of students (Ball et al., 2005). It can be inferred, however, that Ball and her associates are in favour of the spread of specialized content knowledge to students. As teachers employ SCK to develop mathematical explanations and representations for students, and interpret and share the thinking behind student errors and unexpected solutions, they invariably transfer some of this knowledge to their students. Ball et al. would know that teachers cannot use SCK for all the purposes listed in Figure 1 without this transfer happening. While Gadanidis and Namukasa, and Davis and Simmt, make a good point, it is a point with which Ball, Hill, and Bass would agree.
Horizon Content Knowledge (HCK). The final domain of Subject Matter Knowledge “entails knowing what mathematical experiences precede those in a given level and what will follow in the next, and subsequent, grades” (Rowland, 2014, p. 236). Teachers with this type of knowledge are able to trace the evolution of mathematical ideas through multiple years of curriculum. They can see that concepts are repeated through the years, but are added onto, so that each new iteration, in addition to containing everything that came before it, features new complexity and sophistication (a structure that Davis and Simmt (2006) call recursive elaboration).
Ball and Bass (2009) point out that horizon content knowledge is not often used directly in teaching, but that its main function is to guide instruction. It helps teachers notice mathematical significance in student work and use it to create new teaching/learning opportunities, make connections between mathematical ideas, identify past sources of current misconceptions, and present mathematical ideas in ways that minimize the possibility of future misconceptions. A teacher with strong horizon content knowledge might choose to use the grid model of multiplication (shown in Figure 4) to teach multi-digit whole number multiplication based on its applicability to more complex multiplication contexts in subsequent years (Davis & Simmt, 2006, pp. 305-307).
Pedagogical Content Knowledge
The right half of the framework focuses primarily on aspects of pedagogy. Mathematical ideas inform these three domains, but they are brought in greater contact with the acts of teaching and learning.
Pedagogical Content Knowledge
The right half of the framework focuses primarily on aspects of pedagogy. Mathematical ideas inform these three domains, but they are brought in greater contact with the acts of teaching and learning.
Knowledge of Content and Students (KCS). Hill, Ball, and Shilling (2008) define KCS as “teachers’ understanding of how students learn particular content” (p. 378). They justify keeping this domain outside of Subject Matter Knowledge with the explanation that a teacher can have a deep understanding of mathematics but not know how students learn it, and that this deficit can have a profound negative impact on student achievement. Teachers with strong knowledge of content and students know how students will respond to particular tasks and what they generally find easy or difficult. Such teachers will also be able to interpret student thinking, even if it is only partially formed, more accurately (Sullivan et al. 2013). KCS provides teachers with knowledge “about typical errors in advance, thereby by enabling them to be anticipated” (Rowland, 2014, p. 236). Teachers who have this knowledge are better able to create lessons that will mitigate such errors and misconceptions before they manifest themselves in student work. Figure 5 shows an error that is commonly made in two-digit multiplication. This student has mistakenly multiplied 2 by 35 (instead of 20 by 35) in the second line of multiplication. This error exhibits a misunderstanding of the true meaning of that 2. A teacher who knows that this is a common error will be able to address it explicitly as part of the teaching of multiplication.
Knowledge of content and students, as envisioned by Ball et al., focuses on generalizations of how students learn mathematics. Insofar as this allows teachers to predict possible responses and points of difficulty, this is a very important fund of knowledge. However, these generalizations may manifest themselves in specific classrooms to varying degrees. Davis and Simmt (2006) describe how a teacher’s knowledge of his/her specific students contributes to effective mathematics instruction. They say that the knowledge that is needed to develop an understanding of a specific mathematics idea is often present in the minds of the students in the classroom. As such, it is important for teachers to know their students well so that they can draw out “what is already known (albeit often tacit) in a manner that allows it to be knitted together into more broadly available conclusions” (p. 309). It is suggested that the framework of Ball et al. could be amended to add this more specific knowledge into the KCS domain.
Knowledge of content and students, as envisioned by Ball et al., focuses on generalizations of how students learn mathematics. Insofar as this allows teachers to predict possible responses and points of difficulty, this is a very important fund of knowledge. However, these generalizations may manifest themselves in specific classrooms to varying degrees. Davis and Simmt (2006) describe how a teacher’s knowledge of his/her specific students contributes to effective mathematics instruction. They say that the knowledge that is needed to develop an understanding of a specific mathematics idea is often present in the minds of the students in the classroom. As such, it is important for teachers to know their students well so that they can draw out “what is already known (albeit often tacit) in a manner that allows it to be knitted together into more broadly available conclusions” (p. 309). It is suggested that the framework of Ball et al. could be amended to add this more specific knowledge into the KCS domain.
Knowledge of Content and Teaching (KCT). This domain of Pedagogical Content Knowledge provides teachers with the knowledge that is required to make sound instructional decisions. Knowledge of mathematical content and teaching practices are blended to allow teachers to sequence content appropriately, select effective representations for concepts, and decide which “student contributions to pursue and which to ignore or save for a later time” (Sullivan et al., 2013, p. 16). Returning to the example of multiplying two-digit numbers, a teacher with strong KCT might use a rectangular area model such as the one shown in Figure 6 to represent the conceptual meaning of the standard algorithm.
Effective task selection is also dependent on a teacher’s knowledge of content and teaching. Ball & Bass (2003) describe a task that requires students to put the decimal numbers .60, 2.53, 3.14, and .45 in order. At first glance, these numbers may appear appropriate to the task. However, a student whose understanding of decimals is weak could ignore the decimal points entirely and still order these numbers correctly. A teacher with strong KCT would understand this and create this task around a more effective set of numbers.
Knowledge of Content and Curriculum (KCC). There is a lack of agreement on the exact nature of the final domain of knowledge in Ball et al.’s framework. Ball and Bass (2009) do not describe it in depth; instead, they offer a short list of things that might fit into this category (“educational goals, standards, state assessments, grade levels where particular topics are taught, etc.”) before moving on to another topic (p. 5). It is not surprising, then, that KCC has been referred to as an “unconceptualized” domain, and broadly defined as encompassing “all knowledge interconnected with curriculum, pedagogy, and psychology for teaching and students” (Kim, 2013, p. 31). Sleep (2009) describes KCC as “underspecified” and offers significant elaboration (p. 225). Much of what she describes involves learning goals. This domain of knowledge, she says, allows the teacher to detail different types and “grain sizes” of mathematical learning goals for various curricular purposes, and to understand how the instructional tasks work to move the students toward these goals (p. 226). This type of knowledge also allows the teacher to assign different levels of priority to mathematical learning goals and determine “the appropriate depth of treatment and expected level of understanding for a given instructional activity” (p. 226). Sleep also places “the ability to develop a coherent mathematical storyline across instructional activities and lessons” under the heading of knowledge of content and curriculum (p. 225).
Discussion and Conclusion
The work of Ball and her varied associates, informed by the work and interpretations of other scholars, reveals that mathematics teachers require high levels of knowledge from many distinct but related domains. The importance of teachers’ Mathematical Knowledge for Teaching cannot be overstated. Ball et al. (2005), during a study of 700 elementary teachers and their students, found that the teachers’ Mathematical Knowledge for Teaching had a profound impact on student performance in mathematics. Over the course of a year, the researchers had access to student scores on the mathematics section of the Terra Nova, “a reliable and valid standardized test” (p. 44). They calculated gain scores to determine the amount of student growth over the year. They also issued the teachers a questionnaire that contained several items about common and specialized content knowledge. There was a strong correlation between the two sets of data. “The students of teachers who answered more items correctly gained more over the course of the year of instruction” (p. 44). Clearly, it is imperative that teachers have deep knowledge in all MKT domains if they are to enable their students to reenact the work of mathematicians to develop strong conceptual understandings of mathematics.
Effective task selection is also dependent on a teacher’s knowledge of content and teaching. Ball & Bass (2003) describe a task that requires students to put the decimal numbers .60, 2.53, 3.14, and .45 in order. At first glance, these numbers may appear appropriate to the task. However, a student whose understanding of decimals is weak could ignore the decimal points entirely and still order these numbers correctly. A teacher with strong KCT would understand this and create this task around a more effective set of numbers.
Knowledge of Content and Curriculum (KCC). There is a lack of agreement on the exact nature of the final domain of knowledge in Ball et al.’s framework. Ball and Bass (2009) do not describe it in depth; instead, they offer a short list of things that might fit into this category (“educational goals, standards, state assessments, grade levels where particular topics are taught, etc.”) before moving on to another topic (p. 5). It is not surprising, then, that KCC has been referred to as an “unconceptualized” domain, and broadly defined as encompassing “all knowledge interconnected with curriculum, pedagogy, and psychology for teaching and students” (Kim, 2013, p. 31). Sleep (2009) describes KCC as “underspecified” and offers significant elaboration (p. 225). Much of what she describes involves learning goals. This domain of knowledge, she says, allows the teacher to detail different types and “grain sizes” of mathematical learning goals for various curricular purposes, and to understand how the instructional tasks work to move the students toward these goals (p. 226). This type of knowledge also allows the teacher to assign different levels of priority to mathematical learning goals and determine “the appropriate depth of treatment and expected level of understanding for a given instructional activity” (p. 226). Sleep also places “the ability to develop a coherent mathematical storyline across instructional activities and lessons” under the heading of knowledge of content and curriculum (p. 225).
Discussion and Conclusion
The work of Ball and her varied associates, informed by the work and interpretations of other scholars, reveals that mathematics teachers require high levels of knowledge from many distinct but related domains. The importance of teachers’ Mathematical Knowledge for Teaching cannot be overstated. Ball et al. (2005), during a study of 700 elementary teachers and their students, found that the teachers’ Mathematical Knowledge for Teaching had a profound impact on student performance in mathematics. Over the course of a year, the researchers had access to student scores on the mathematics section of the Terra Nova, “a reliable and valid standardized test” (p. 44). They calculated gain scores to determine the amount of student growth over the year. They also issued the teachers a questionnaire that contained several items about common and specialized content knowledge. There was a strong correlation between the two sets of data. “The students of teachers who answered more items correctly gained more over the course of the year of instruction” (p. 44). Clearly, it is imperative that teachers have deep knowledge in all MKT domains if they are to enable their students to reenact the work of mathematicians to develop strong conceptual understandings of mathematics.
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References
Ball, D. L. (2016) Teaching mathematics to support the mathematical work of teaching [PDF presentation slides]. Retrieved from https://static1.squarespace.com/static/577fc4e2440243084a67dc49/t/58176141f7e0ab86f3d1f2af/1477927236048/102916_TeMaCC.pdf
Ball, D. L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In B. Davis & E. Simmt (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group, (pp. 3-14). Edmonton, AB: CMESG/GCEDM.
Ball, D. L., & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners' mathematical futures. Presented at the 43rd Jahrestagung für Didaktik der Mathematik, Oldenburg, Germany, March 1-4, 2009.
Ball, D. L., Hill, H. C, & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 30(3), p. 14–17, 20–22, 43–46. Retrieved May 21, 2017, from http://thelearningexchange.ca/projects/mathematical-knowledge-for-teaching-with-dr-deborah-loewenberg-ball/
Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) Know. Educational Studies in Mathematics, 61(3), 293-319. doi:10.1007/s10649-006-2372-4
Gadanidis, G., & Namukasa, I. (2007). Mathematics-for-teachers (and students). Journal of Teaching and Learning, 5(1). doi:10.22329/jtl.v5i1.277
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/62001225?accountid=15115
Kim, Y. (2013). Teaching mathematical knowledge for teaching: Curriculum and challenges (Doctoral dissertation). Available from ERIC. (1697500074; ED554175). Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1697500074?accountid=15115
Rowland, T. (2014). Frameworks for conceptualizing mathematics teacher knowledge. Encyclopedia of Mathematics Education, 235-238. doi:10.1007/978-94-007-4978-8_63
Sleep, L. (2009). Teaching to the mathematical point: Knowing and using mathematics in teaching (Doctoral dissertation). Available from Education Database. (Order No. 3392837). Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/304929452?accountid=15115
Sullivan, P., Clarke, D., & Clarke, B. (2013). Teaching with tasks for effective mathematics learning. New York, NY: Springer.
Ball, D. L. (2016) Teaching mathematics to support the mathematical work of teaching [PDF presentation slides]. Retrieved from https://static1.squarespace.com/static/577fc4e2440243084a67dc49/t/58176141f7e0ab86f3d1f2af/1477927236048/102916_TeMaCC.pdf
Ball, D. L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In B. Davis & E. Simmt (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group, (pp. 3-14). Edmonton, AB: CMESG/GCEDM.
Ball, D. L., & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners' mathematical futures. Presented at the 43rd Jahrestagung für Didaktik der Mathematik, Oldenburg, Germany, March 1-4, 2009.
Ball, D. L., Hill, H. C, & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 30(3), p. 14–17, 20–22, 43–46. Retrieved May 21, 2017, from http://thelearningexchange.ca/projects/mathematical-knowledge-for-teaching-with-dr-deborah-loewenberg-ball/
Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) Know. Educational Studies in Mathematics, 61(3), 293-319. doi:10.1007/s10649-006-2372-4
Gadanidis, G., & Namukasa, I. (2007). Mathematics-for-teachers (and students). Journal of Teaching and Learning, 5(1). doi:10.22329/jtl.v5i1.277
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/62001225?accountid=15115
Kim, Y. (2013). Teaching mathematical knowledge for teaching: Curriculum and challenges (Doctoral dissertation). Available from ERIC. (1697500074; ED554175). Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1697500074?accountid=15115
Rowland, T. (2014). Frameworks for conceptualizing mathematics teacher knowledge. Encyclopedia of Mathematics Education, 235-238. doi:10.1007/978-94-007-4978-8_63
Sleep, L. (2009). Teaching to the mathematical point: Knowing and using mathematics in teaching (Doctoral dissertation). Available from Education Database. (Order No. 3392837). Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/304929452?accountid=15115
Sullivan, P., Clarke, D., & Clarke, B. (2013). Teaching with tasks for effective mathematics learning. New York, NY: Springer.