I want to use this occasion to reflect a bit on what the development of a consensual list of Student Learning Outcomes (SLOs) represents vis-a-vis the field of mathematics teacher education and, in particular, its curricular history. I want to suggest that GeT: A Pencil has helped the larger mathematics education community make progress in identifying how it is that knowledge from the field of mathematics education (particularly the empirical notion of mathematical knowledge for teaching) can be reconciled with the geometric content that has traditionally been curated by mathematicians for its use in designing the curriculum of geometry courses for teachers. 

For a long time, the question of what content should be covered in college geometry courses was one that involved the curricular organization of the mathematical domain of geometry. In this work of curriculum design, the considerations made concerned the history and scope of the subject and the different sequences in which the subject could be presented. In that sense, it was no different than how the content for other mathematics courses (e.g., calculus) might be experimented with. Facing the question of what geometry should be taught to teachers, this approach suggested the need to inquire within the mathematical domain of geometry and find topics and organizations of those topics that arguably would serve to educate future teachers. A premium was put on logical organization, aesthetic or historical value, and ease of understanding—not necessarily on readiness for use in teaching. 

As research in mathematics education started to inquire about mathematics teachers’ knowledge, one important insight that emerged was that when teachers engage in the work of teaching, they also do mathematics. Tasks of teaching, such as creating or modifying problems for students or determining the mathematical qualities of work that students do (e.g., consistency, correctness, generality) involve the teacher in doing mathematics while teaching. Appropriately, scholars have called these mathematical tasks of teaching (e.g., Ball et al., 2008). While some of this mathematical work could clearly be filed under topics within the curricular organization of a mathematical domain (e.g., making up problems about isosceles triangles surely draws from knowledge of isosceles triangles that could be covered in geometry courses), other mathematical work teachers do could be described as so specialized to the work of teaching that it might not be common among people who otherwise knew a mathematical domain. For example, understanding why point P in Figure 2, below, cannot possibly be the center of a circle tangent to line a at A is a problem a teacher would need to solve when confronting this proposed solution to a construction problem by a student; yet wrong solutions to problems are rarely part of what one learns in mathematics courses. 

This distinct mathematical knowledge for teaching is by no means as explicit as the knowledge documented in textbooks. One sees it manifested in actions (e.g., in teachers’ noticings or decisions), recognized by mathematically educated observers, and organized using resources from empirical research (e.g., typologies). One way in which researchers in mathematics education organized this knowledge–the domains of MKT offered by Ball et al. (2008)–was serviceable in demonstrating the diverse epistemological sources of the mathematical knowledge teachers use. The distinction between specialized content knowledge (SCK) and knowledge of content and students (KCS), for example, highlighted that while some knowledge special to teachers is purely mathematical (i.e., the truth or falsity of this knowledge could be established using mathematics alone; e.g., one can prove that point P can never be the center of a circle tangent to a at A), other knowledge also special to teachers might depend on blending mathematical and empirical knowledge (i.e., the incidence of particular error in a student population; e.g., how often does this idea come up when students are asked to construct a circle tangent to two intersecting lines? How hard is it for students to understand why the “solution” proposed in Figure 2 is incorrect?). This epistemological distinction among domains of mathematical knowledge for teaching has been useful for creating assessment instruments; it has helped create blueprints for the different types of items that need to be developed to tap the construct mathematical knowledge for teaching in all its aspects. 

When we started the GeT Support project in 2017, the GRIP team had already invested some effort in the development of an assessment instrument whose items were useful to measure the amount of MKT-G (mathematical knowledge for teaching geometry) a teacher has (Herbst & Kosko, 2014). I note that the instrument measured the amount of MKT-G, taking this as a single construct; the instrument does not verify that the individual knows any concept in particular. This instrument had been used to assess the amount of MKT-G in a national sample of high school mathematics teachers, and part of the idea of the GeT Support project was that the same instrument might be of use to inform instructors and the public about the contribution GeT courses could make to increasing capacity for geometry instruction. In the context of the GeT Support project we have indeed been able to gather data that shows that the same instrument can detect changes in MKT-G students’ experience during the time they take a GeT course (see Ion, 2020 ; Ko, Ion, & Herbst, this issue). The items in that instrument illustrate the construct (MKT-G) in general; each item is situated in the context of a task of teaching geometry and presents a problem that a teacher may need to solve in that context. The items are related to the general theory of MKT (Ball et al., 2008) in that each of them avowedly assesses knowledge of one of four domains of MKT (CCK, SCK, KCS, KCT), and the items sample content from the high school geometry curriculum. 

We showed examples of those items at our 2018 conference in Ann Arbor. Yet, not every GeT instructor recognized those items as examples of the knowledge that they taught their students in GeT courses. More importantly, it was clear to all, including ourselves, that the items themselves were not, by themselves, useful to think about the content for GeT courses. A reason for it is that the items are just very small bites of knowledge; they might suggest problems to solve, but they do not clearly point to the larger units in which the knowledge covered in a course gets structured. In that sense, they are hard to relate to the sort of curricular work described in the first paragraph. 

As a mathematics teacher myself, one way I think of this different granularity of knowledge is with the knowledge exchange diagram shown in Figure 1. On the one hand, concepts and theorems are relatively large objects of knowledge or mathematical ideas that might be at stake in a course. These objects of knowledge are instantiated in many smaller problems or in particular actions in solving such problems. The work on problems and tasks, on the other hand, makes room over time for a large number of small things–noticings, intentions, ways of seeing, tricks, etc. (e.g., representing b as (ab)/a) that are part of the ways of doing mathematics, part of the mathematical sensibility. Some of the mathematics done in the context of problems and tasks may never receive a name or be taught by itself. 

The exchange diagram opens up the possibility to realize that the same object of knowledge can have many different meanings depending on the various problems in which it is operational. For example, the theorem that says that a tangent is perpendicular to the radius of a circle at the point of tangency can be at stake in a number of tasks. It can be useful to come up with a method to construct a circle tangent to a given line; it can also be useful to argue why point P in the diagram below cannot possibly be the center of a circle tangent to line a at point A. In the first case, one uses the theorem to identify the locus of the center of the circle: It must be on the perpendicular to a at A. In the second case one uses it to feed a proof by contradiction: If PA was the radius of the circle tangent to a, then the angle PAO is right and the triangle OPA would be isosceles and with two right angles. The example illustrates how a given object of knowledge has various meanings that emerge in the context of different problems. This is not only the case for the knowledge we teach in mathematics courses, but I surmise it would be reasonably the case for mathematical knowledge for teaching. 

The exchange diagram can be useful to explain how the notion of mathematical knowledge for teaching could inform curricular work in GeT courses. Problems like the ones in MKT items are candidate examples for the mathematical work in the red square on the left. Consider, for example, problems like those shown in Figures 3 and 4.

Mrs. Miyakawa wants to assign a proof problem using the diagram below. If she asks students to assume (i.e., take as given) that ABCD is a rectangle, E is the midpoint of DC , and AE ⏊BE , what could she ask them to prove?

In Mr. Desimone’s geometry class, kites were defined to be quadrilaterals with two distinct pairs of congruent adjacent sides. He then asked his students to draw a kite that has congruent diagonals and two pairs of congruent opposite angles. Andrea came to him in distress after a few minutes saying that she’d tried all sorts of angle measures and diagonal lengths and all she could come up with were squares. Mr. Desimone told his student teacher that Andrea did not understand what the definition of kite means. What do you think Mr. Desimone means by that?

Figure 3. Problem 15102

Figure 4. Problem 15503

The two problems in Figures 3 and 4 describe situations in which a geometry teacher might need to do some mathematics while teaching. Using the exchange diagram, they both belong in the red square. But what are the green circles that those problems exchange for? In other words, what is the knowledge at stake in those problems? The mathematical domain of geometry has had expository treatises like Euclid’s Elements, Hilbert’s Grundlagen, or Moise’s Elementary geometry from an advanced standpoint within which one can locate the knowledge at stake in problems, but MKT-G problems have not had similar resources. The difficulty is not that problems like those in Figures 3 and 4 cannot be classified into topics; they could be classified in, perhaps, even more than one topic. But where would they make a difference? Because they can be classified in multiple ways, it is unclear how one would build critical masses of them to serve the creation of problem sets and units of study. Indeed, many such items could be created. How would we know what to emphasize?

With the introduction of the SLO, the picture gets a little clearer. The SLOs have two important virtues that recommend them as candidates to be the green circles in the exchange diagram. First, they have come from a thoughtful process of negotiation and argument among a group of instructors that includes mathematicians and mathematics educators. Second, in arguing for them, both considerations of their stature in the domain of geometry and in the teaching of high school geometry have been made. As a result, if one looked at them only with the eyes of a mathematician or only with the eyes of a teacher educator, these SLOs might appear heterogeneous. I tend to think that is a good thing. 

Having arrived at the SLOs and now having elaborations of the SLOs in various issues of this newsletter, our community has an important scaffold for the question of what content could or should be taught in GeT courses. The SLOs can, at a minimum, be a checklist; one could look through problem sets, notes, and syllabi and see where there are topics that match the SLOs. However, the SLOs can also be used generatively and along with other aspects of mathematical knowledge for teaching.

For example, one could see that problem 15503 (Figure 4) can be associated with the role of definitions in mathematics (SLO 5). One could also see that it deals with a mathematical task of teaching, interpreting what students do in response to problems, and that it involves specialized content knowledge about quadrilaterals. In fact, the topic of quadrilaterals might be a good host area to anchor the teaching of SLO 5; Usiskin (2007) wrote a monograph on definition using the classification of quadrilaterals that might support firming up these connections. An interesting issue, perhaps an idiosyncrasy, with quadrilaterals in high school geometry that reveals the value of this topic for the study of the role of definitions is that the HSG curriculum tends not to be consistent in how definitions of special quadrilaterals are read. While rectangles and rhombi are understood as defined inclusively (e.g., squares are rectangles), trapezoids are understood as defined exclusively (e.g., parallelograms are not trapezoids). One possibility afforded by this consideration is that instructors might use a unit on quadrilaterals to aim at the achievement of SLO 5 and, in that context, have their students work on a number of problems in which both quadrilaterals and definitions are addressed in the context of tasks of teaching. 

Overall, I believe that the SLOs provide us with a way into creating a curriculum for GeT courses that can bridge the mathematical topics that have been used to organize geometry instruction in the past and the particular occasions in which a teacher might have to do mathematics while teaching geometry. Without downplaying the value of collaboration and consensus in forming a community of colleagues, an important achievement of the SLOs is to have helped us figure out how the idea of mathematical knowledge for teaching can become part of curriculum making.

References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?. Journal of Teacher Education, 59(5), 389-407.

Ion, M. (2020). ​​Reporting on the MKT-G results from GeT Students. GeT: The News!, 1(3) 

Ko, I., Ion, M., & Herbst, P. (this issue). Does our MKT-G instrument measure the same knowledge in the same way for GeT students and for practicing teachers? GeT: The News! 3(3)