Mathematics-for-teaching:
An ongoing investigation of the mathematics teachers (need to) know

Brent Davis, Elaine Simmt, & Dennis Sumara, University of Alberta

We report on a study of mathematics-for-teaching—that is, an investigation of the mathematical competencies that are enacted in the context of teaching. The research involves 24 practicing teachers whose mathematical backgrounds, teaching experiences, and curriculum responsibilities vary dramatically. The research method revolves around the study of established mathematics, during which we also investigate the development of mathematics that is new to the participants. We conclude with emerging implications for teacher preparation.

“Mathematics-for-teaching”
This study is aimed at a better understanding mathematics-for-teaching—that is, the sorts of mathematical knowledge that come into play when teaching. Following Ball and Bass (2000, 2003), we understand such competencies to be distinct from those employed in other “mathematically inclined” professions. Further, following Bass (2004), we see mathematics-for-teaching as a form of applied mathematics, and a core aim of the research is to map out some of the contours of the field. 
	We recognize that, by studying mathematics-for-teaching, we affect it. As such, our research is aimed not just at the articulation of necessary mathematics-for-teaching, but at the development of such knowledge. For this reason our research occurs in the context of in-service professional development sessions. In these sessions practicing teachers take part in a series of connected investigations of mathematical concepts, which include examinations of the physical experiences, the metaphors, the analogies, the applications, and other associations that are invoked when teaching and/or using a mathematical concept. As we discuss in the findings, teacher-participants often represent their understandings of these facets of mathematics in indirect and disconnected ways. Once represented, they can work collectively to knit together more sophisticated understandings.

Theoretical Framework
The mathematics-for-teaching research project is oriented by complexity science, s a transdisciplinary field of inquiry concerned with the sorts of phenomena that emerge in the co-specifying activities of agents whose collective activity exceeds the summed capacities of agents. Complex phenomena include neural activity, individual understanding, classroom collectives, the body of mathematical knowledge, and society. The research is also framed by the work of Lakoff and Núñez (2000), specifically their examinations of the bodily bases and linguistic entanglements of mathematical concepts.
	We find the use of complexity science useful in that it points to adaptive (or learning) systems. Such systems change themselves—and, through these changes, they come to embody their own histories. Phrased differently, for complex phenomena, the distinction between product and process is an artificial one. By way of immediate consequence to teacher education programs, this assertion renders problematic the common separation of established knowledge (often taught in departments of mathematics) and how knowledge is established (addressed within faculties of education). Complexity compels us to argue that, especially for teachers, an integrated understanding of the relationship between the established knowledge (collective cognition) and how knowledge is established (individual cognition) is critical to their mathematics-for-teaching.

Research Method
Currently in its third year, our study involves a cohort of 24 teachers, who teach children and youth from levels Kindergarten through high school, and whose mathematics teaching experience ranges from a few to more than 30 years. 
	The group meets every few months during the school year for full-day sessions. Though reflecting on our own teacher education practices, we have developed a research approach that enables us to attend to both explicit and tacit aspects of teachers’ knowledge. Our method is based, in part, on extended, group-situated engagements with tasks that may at first seem narrow with well-established answers—but that usually occasion rich learning experiences. Indeed the tasks serve to make explicit and challenge our assumptions about them. For example, the item used in this report was the question, “What is multiplication?”
	Usually, participants work together on such questions in small groups of 3 to 5. Within their conversations, teachers typically identify, compare, and critique the sorts of interpretations that they bring to bear on the study of the topic at hand and with respect to their own classroom practices. Once the participants feel that they have either generated an adequate response or have exhausted their knowledge of possible interpretations, the groups are asked to represent their ideas to one another with the intention of generating a summary, consensus response, as well as to provide the opportunity to use what is said in a generative manner. In every case so far, this strategy has contributed to an elaborated response that includes but exceeds the original understandings of all participants—including the researchers.
	Normally, teachers are also asked to examine and report on the relevance of these discussions for their particular grade level(s). The intention here is foreground how concepts are introduced and elaborated through a K–12 program of studies for mathematics. As the example below demonstrates, these discussions can prompt participants to become aware of aspects of mathematics—in particular, previously unnoticed associations among concepts.
	As co-researchers, we plan the sessions together and share the responsibility for leading varied activities. Sessions are audio- and videotaped. Along with the lead researchers, the two research assistants take observational field notes—based on their personal research interests. This feature compels us to take note of data and interpretive possibilities that we likely would not otherwise have considered.
	The principle data sources are the physical products of the teachers’ interactions—which, depending on the nature of the orienting task, can include video clips, transcripts of utterances, posters, images, models, proofs, and/or various other representations. For instance, in response to the prompt, “What is multiplication?”, the following summary list was developed on a chart:

Multiplication is:
•	repeated addition: e.g., 2x3=3+3 or 2+2+2
•	equal grouping: e.g., 2x3 can mean 2 groups of 3
•	number-line hopping: e.g., 2x3 can mean “make 2 hops of length 3”
•	series of folds: e.g., 2x3 can describe the action of folding a page in two and then folding the result in three
•	ratios and rates: e.g., 3 L at $2/L costs $6
•	array-making: e.g. 2x3 can be 2 rows of 3 
•	area-producing: e.g., a 2 cm by 3 cm rectangle has an area of 6 cm2
•	number-line stretching: e.g., 2x3=6 can mean that “3 corresponds to 6 when a number-line is stretched by a factor of 2”

Relevant portions of all records are reviewed and transcribed as we work through the multiple layers of data. 

Results
The most striking and consistent finding, across all the topics studied so far, is that these teachers have a broad knowledge of interpretations and applications of mathematics appropriate to the grade levels taught and the concepts addressed. They are able to move fluidly among the sorts of interpretations presented above, although they are often unaware they are doing so.
	This is not to say that they are always able to generate such explicit lists on demand, however. On the contrary, we find that participants’ first responses to a question, while usually appropriate, often represent just one of many possibilities—and usually a possibility that is automatized and requires little thought. In response to the “What is multiplication?” prompt, for instance, almost everyone answered “repeated addition” or “groups of”—and indicated surprise when we countered, “And what else?” 
	However, when the teachers have opportunity to share responses or explain for others, we usually note that their mathematics-for-teaching is much more sophisticated than these sorts of initial responses might suggest. As they interact around individual contributions, collectives often form within the group and the mathematical understanding itself begins to transform. In rather short order, we can no longer distinguish the mathematics of individuals, but find ourselves observing the emergence of the mathematical understanding of the collective itself. In these moments, it becomes difficult to attribute particular understandings to one person or another. Generally speaking the collective understandings that arise are observed to be connections among concepts, strategies, and explanations.
	Despite the emergence of this fluid knowledge, it is mostly grade- and topic-specific. Understandings of various interpretations are often disconnected from the knowledge of other concepts and the applications of concepts in grades other than their own. For instance, in the example above, secondary teachers were largely unaware of the array-based strategies of teaching multiplication used by teachers in earlier grades—and, for that reason, perceived no connection between the use of an array method to teach multiplication of binomials and high school students’ earlier experiences. Conversely, teachers in the early grades were generally surprised that their emphasis on “multiplication as repeated addition” or “multiplication makes bigger” could contribute to conceptual problems later on. As well, teachers expressed the view that a grid-based multiplication algorithm, as illustrated below, could be very useful to highlight the relationship between such images and symbolic processes of concepts that are recursively elaborated across the grades. 

Figure 1: Three rectangle-based interpretations of whole number  multiplication

Rooted in such realizations, teachers identified and developed interpretations and teaching methods for various concepts that span topics and grades. In the case of multiplication, for example, participants agreed that an emphasis on array-based interpretations across the grades enhanced their abilities to notice the elaborations that are made to topics from year to year—and, thus, their capacities to provide opportunities for students to make connections among topics.
We have gathered similar data on other topic areas, including subtraction, division, and exponentiation. Across these topics, a consistent conclusion has been that, in pragmatic pedagogic terms, there are no definitive answers to questions like “What is multiplication?” Put differently, what multiplication (or any other concept) is depends on context, purpose, and available tools and arises in action. No single interpretation will suffice for a robust and broadly useful understanding of the topic. This poses an interesting dilemma for teacher educators and certification agencies. How can we assess teacher knowledge if indeed there is no single answer and if any answer is context, purpose and tool dependent?
	Following a tenet of complexity, we understand this teaching approach in terms of recursive elaboration. We argue that effective mathematics pedagogy is a recursive elaborative process that is attentive not just to the range of possible interpretations, but to the collective dynamic that allows those interpretations to be presented and knitted together into more sophisticated understandings. The research design foregrounds the manner in which new possibilities for interpretation are presented in the juxtaposition of established ideas. 

Educational Significance
As the multiplication example demonstrates, teachers’ knowledge of specific concepts is largely tacit. They are able to bring such knowledge to bear in a specific teaching situation, but may identify it as a “necessary evil” rather than a legitimate aspect of the topic at hand. We thus argue that courses in mathematics for teachers should foreground both established mathematics and the manners in which that mathematics is established. More generally, with regard to the mathematics studied by teachers, this research suggests a number of points. For example, we conclude that research focused on exclusively “questions of mathematics” or “questions of learning” is inadequate in efforts to understand teachers’ mathematics-for-teaching. Further, teachers must be adept at ‘translating’ among available symbol systems and at recognizing when they are engaging in such translations. Inquiries into these matters must be undertaken in the context of doing mathematics that is genuinely new to the doers—noting that such mathematics can be structured around surprisingly familiar topics. The actual topics addressed can but need not involve manipulations or calculations—some discussions can and should be focused on, for example, the role of language in the development of mathematics. At the same time teachers must have opportunities to become aware of how concepts are presented and elaborated through the course of the K–12 curriculum—meaning that common divisions between elementary and secondary instruction may be problematical.
Our study is helping us better understand the sorts of mathematical knowledge that come into play when teaching. Teachers’ mathematics-for-teaching, as described above, is both collective and personal, established and newly generated, abstract and embodied—in brief, much more complex than typically presented in generic mathematics courses. Yet, despite the fact that teachers have little formal education in these aspects, they seem to have profound, robust, and readily elaborated understandings of mathematics-for-teaching.

References
Ball, Deborah L., & Hyman Bass. (2003). Toward a practice-based theory of mathematics knowledge for teaching. In E. Simmt & B. Davis (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group (pp. 3–14). Edmonton, AB: CMESG/GCEDM.
Ball, Deborah L. & Hyman Bass. (2000). Interweaving content and pedagogy in teaching and learning to teach: knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83–104). Westport, CT: Ablex.
Bass, Hyman (2004). Mathematics, mathematicians, and mathematics education. Paper presented at ICME-10, Copenhagen, Denmark, July.
Lakoff, George, & Rafael Núñez. (2000). Where mathematics comes from: how the embodied mind brings mathematics into being. New York: Basic Books.