# «This paper examines the design of tasks for developing and assessing mathematical knowledge for teaching, in particular the role of pedagogical context. ...»

4) and would fail to assess the distinctive knowledge and skill known to be associated with increased learning. In other words, the doing of mathematical work (such as comparing fractions) while keeping in mind a purpose (of choosing a representational model) and attending to what is involved (in using a model, as one uses it or talks about it) is common in teaching, but uncommon in the discipline of mathematics. For instance, a disciplinary impulse can lead one to focus on the mathematics problem given to students or to explore variations or generalizations of a mathematical problem (a distraction that played out in many of the interviews with mathematicians), losing track of the need to interpret the mathematical validity of a student’s confusing approach or generate a mathematical problem with a solution satisfying specific criteria. These latter tasks typify MKT, and it is pedagogical context that allows for their expression and that thus makes visible the articulation of the task of teaching, such as shown in Figure 3.

Our analysis is limited to sampling from items that have been produced to date, with a set of features of pedagogical context that is likely narrow. For instance, student background is not a prominent feature and plays a minor role in the items analysed.

This is likely a result of narrowness of existing items and likely to change as scholars continue to expand work in this arena. For instance, Goffney (2010) has pointed out the mathematical demands of equitable teaching and Wilson (2016) has explored the development of assessment items to measure such knowledge in relation to dual language learners. Despite these limitations, we propose that the role of pedagogical context is important in the development of tasks to support the development of equitable teaching and that lessons from the above analysis can provide valuable guidance.

We close by offering a suggestion about how MKT might be articulated in the work of specifying the design of MKT tasks, in line with an approach developed by Illustrative Mathematics. Their approach requires not only writing a problem, but providing a commentary (and sample solutions). Consider the item in Figure 5.

Choosing examples item Ms. Seidel is introducing the distributive property. To motivate her students, she wants to give them an example that will focus their attention on how using the distributive property can simplify computation. In which of the following examples will the use of the distributive property most simplify the computation?

a) 12 x 29 + 12 x 38 = ___ b) 17 x 37 + 17 x 63 = ___ c) 13 x 13 + 15 x 15 = ___ d) 16 x 24 + 16 x 24 = ___ Figure 5: Choosing examples item Based on a narrative for doing the task (Figure 6), a commentary might be written (Figure 7), where the commentary characterizes the MKT that the task is intended to develop or assess, intended use or the task, and the pedagogical context provided in the scenario. The production and review of such a commentary provide powerful tools for collaborative efforts to develop MKT tasks, where explicit statements about rationale for pedagogical context significantly enhance development, review, and professional sanctioning.

1. Tracking on the fact that the instructional purpose for the example is to focus students’ attention on how using the distributive property can simplify computation.

2. Considering different ways of evaluating the expressions and of using the distributive property and what these imply about what it means to simplify the computation, including recognizing the following: the most reasonable way of using the distributive property in (a) yields 12(29 + 38) = (12)(67), which reduces the computation from two to one application of multiplication; the most reasonable way of using the distributive property in (b) yields 17(37 + 63) = (17)(100), which reduces the computation from two non-trivial applications of multiplication to one simple one; it is not clear how to use the distributive property in (c); and although there are numerous quantities that could be factored out of the two terms (to similar effect as in (a)), none significantly simplifies the complexity of the multiplication to be done (use of doubling can be made with or without the use of the distributive property).

3. Recognizing that in problems such as these the distributive property does not avoid multiplication, but does allow for regrouping quantities into powers of 10, which greatly simplifies multiplication in a base ten system, and that (b) is the only one that affords this opportunity.

Figure 6: Narrative for the MKT reasoning involved in choosing examples item.

Examples shape instructional opportunities, however crafting and choosing good examples requires mathematical dexterity and skill in doing mathematical problems while tracking on instructional goals. This task asks for an example in which the distributive property can be used to simplify computation significantly. The purpose of this task is to see whether teachers flexibly consider different ways of evaluating the expressions using the distributive property and, simultaneously, what these imply for efficiency of the computation. It requires recognizing that the distributive property does not avoid multiplication, but does allow for regrouping quantities into powers of 10, which greatly simplifies multiplication in a base ten system. The task is currently written as a multiple-choice item for assessment. But it also can be used for launching a discussion about the nature of examples for which the distributive property is useful.

The mathematical task of teaching is choosing examples, but the teaching scenario needs to create a realistic need for choosing an example that requires the distributive property. In this scenario, the pedagogical purpose is to motivate learning of the distributive property. In particular, the scenario proposes motivating the distributive property by giving an example that will focus students’ attention on how using the distributive property can simplify computation. This means that the example needs to provide a sharp contrast in the extent to which the computation is simplified by using the property relative to not using it. The examples given in the options in this task are selected to create such a contrast, where only option (b) significantly reduces the complexity of the multiplication. The instructional setting of introducing the distributive property contributes to a sense that the scenario is realistic.

Figure 7: Commentary for choosing examples item.

Through this process, task developers can encode implicit hypotheses about what matters about the pedagogical context when teachers face particular content problems 7/14/16 Hoover 1 of practice. Teachers who are able to use the pedagogical context in tasks as a resource for responding to tasks demonstrate knowledge in a way that simulates teacher knowledge use in teaching; their reasoning with the pedagogical context can be used to scrutinize and make visible the implicit hypotheses, iterate item development, and refine articulations of mathematical tasks of teaching and MKT assessed.

Bringing together content and pedagogy has been a persistent theme in conversations about the content-knowledge education of teachers over the last 50 years. However, taking stock of scholarship on content knowledge for teaching, Graeber and Tirosh (2008) remind us that, while the concepts of pedagogical content knowledge and content knowledge for teaching are useful, the union of content and pedagogy remains elusive. Beyond introducing complexity and challenge for writing MKT tasks, pedagogical information plays a non-trivial function in tasks designed to develop and assess professionally situated mathematical knowledge by articulating constrained instances of the relationship between content and teaching that is at the heart of the notion of MKT. Ball (2000) characterizes the “intertwining of content and pedagogy” as a continuation of Dewey’s (1964/1904) effort to find the “proper relationship” between theory and practice. Our growing understanding of the role of pedagogical context in the design of and reasoning within MKT tasks is beginning to give us a better understanding of the “proper relationship” between content and pedagogy in characterizations of content knowledge for teaching.

NOTES

1. This work is based on research supported by the National Science Foundation under grants DRL-1008317, RECEHR-0233456, and EHR-0335411 and conducted in collaboration with Eric Jacobson and Laurie Sleep. We want to acknowledge the significant contributions of these colleagues in conceptualizing and conducting the research from which this paper is drawn. We also want to thank members of the Mathematics Teaching and Learning to Teach and Learning Mathematics for Teaching projects who helped with data collection and analysis: Deborah Ball, Hyman Bass, Arne Jakobsen, Kahye Kim, Yeon Kim, Minsung Kwon, Lindsey Mann, and Rohen Shah. The opinions reported here are the authors and do not necessarily reflect the views of the National Science Foundation or our colleagues.

2. We were denied access to COACTIV items and release of TEDS-M items occurred after we completed interviewing.

REFERENCES Ball, D. L. (2000). Bridging practices: Intertwining content and pedagogy in teaching and learning to teach. Journal of Teacher Education, 51, 241-247.

Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V.

Richardson (Ed.), Handbook of research on teaching (4th edition, pp. 433-456).

New York, NY: Mcmillan.

Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A.,... & Tsai, Y.

M. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133-180.

Dewey, J. (1964/1904). The relation of theory to practice in education. In R.

Archambault (Ed.), John Dewey on education (pp. 313-338). Chicago: University of Chicago Press. (Original work published 1904.) Goffney, I. M. (2010). Identifying, defining, and measuring equitable mathematics instruction. (Unpublished doctoral dissertation). University of Michigan, Ann Arbor, MI.

Graeber, A., & Tirosh, D. (2008). Pedagogical content knowledge: Useful concept or elusive notion. Knowledge and beliefs in mathematics teaching and teaching development. The international handbook of mathematics teacher education, 1, 117-132.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371–406.

Hill, H. C., Sleep, L., Lewis, J., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge: What knowledge mat- ters and what evidence counts. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111-156). Charlotte, NC: Information Age Publishing.

Hill, H. C., Umland, K., Litke, E., & Kapitula, L. R. (2012). Teacher quality and quality teaching: Examining the relationship of a teacher assessment to practice.

American Journal of Education, 118(4), 489-519.

Hoover, M., Mosvold, R., Ball, D. L., & Lai, Y. (2016). Making Progress on Mathematical Knowledge for Teaching. The Mathematics Enthusiast, 13(1), 3-34.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States.

Mahwah, NJ: Lawrence Erlbaum.

Phelps, G., Weren, B., Croft, A., & Gitomer, D. (2014). Developing content knowledge for teaching assessments for the Measures of Effective Teaching Study (ETS Research Report 14-33). Princeton, NJ: Educational Testing Service.

**doi:**

10.1002/ets2.12031.

Saderholm, J., Ronau, R., Brown, E. T., & Collins, G. (2010). Validation of the Diagnostic Teacher Assessment of Mathematics and Science (DTAMS) instrument. School Science and Mathematics, 110(4), 180–192.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.

Educational Researcher, 15(2), 4-14.

Wilson, A. T. (2016). Knowledge for Equitable Mathematics Teaching: The Case of