Introduction

Recall that GeT assessment item 15301 asks:

Mr. Gómez taught students the usual procedure for constructing a perpendicular bisector for a segment. Veronica asked Mr. Gómez to explain why the construction works, meaning how they can be sure that the line that is constructed is indeed perpendicular to the segment and passes through the midpoint. How could Mr. Gómez explain that?

As was discussed in the introduction to Part 1 of this article, because this item is situated in the practice of teaching, it contains multiple nested questions. First, there is the “internal question” posted by Veronica to Mr. Gómez; second, there is the “external question” posed to the GeT students themselves. In the first part of this article, I considered certain “foundational questions” about constructions: what it means to “construct” a geometric object and what we mean when we ask if a construction “works.” In Part 2, we consider the internal and external questions: Why does the construction work? How could Mr. Gómez respond to Veronica?

The internal question: How can we respond to Veronica?

We begin by placing ourselves in Mr. Gómez’s position and ask: How can we respond to Veronica? We start with isolating the internal question (i.e., the question posed inside the narrative framework of item 15301):

Veronica asked Mr. Gómez to explain why the construction works, meaning how they can be sure that the line that is constructed is indeed perpendicular to the segment and passes through the midpoint. 

We observe that Veronica’s actual words are not provided but, instead, paraphrased; moreover, the initial paraphrase (“Why [does] the construction work […?]”) is then interpreted in another, more explicit formulation (“How can they be sure that the line that is constructed is indeed perpendicular to the segment and passes through the midpoint [?]”). Although the second formulation is intended to be merely a clarification of the first, they are ostensibly different questions, potentially calling forth different kinds of responses:

1. The first question calls for an explanation of why the construction “works” (whatever that means).
2. The second question calls for a verification that the result has the desired properties.

Although explanation and verification often go hand in hand, they are not the same thing (see de Villiers, 1990). 

What kinds of responses should these two questions elicit? The second question reduces to the first if, and only if, we assume that a construction that is logically valid will, in fact, produce the desired outcome. This assumption, usually tacit in Geometry, suppresses concerns about precision and accuracy, and regards mathematical proof as both necessary and sufficient to ensure a correct product. But we need not make such an assumption, and we perhaps should not ascribe such a position to Veronica. It is entirely possible to recognize that a construction is logically valid and yet still have concerns that the product may be flawed due to errors in execution.

So we consider the two questions separately. The first question can be restated as: if we had a perfect compass, a perfectly steady hand, and were performing the construction process on an idealized diagram with perfectly small points and perfectly straight lines—if we imagine performing the construction in such a perfectly idealized world, free of human and mechanical error, how do we know that the result would indeed be a perpendicular bisector? Framed this way, it is clear that empirical considerations are neither relevant nor useful. The question is not about the actual construction performed on actual paper with an actual pencil but rather about a construction performed in the mind, an operation on generic conceptual objects, not on material objects. Verifying the accuracy of such an abstraction requires a similarly abstract validation system; that is, it requires a mathematical proof.

What could such a proof look like? A typical argument might begin by drawing segments from A and B to the two intersection points C and D, drawing segment CD, and letting P denote the intersection of CD with AB. The resulting figure (see below) then contains four small triangles (△ACP, △BCP, △DAP, and △DBP), sharing a common vertex at P; the figure also contains four larger triangles (△ACB, △ADB, △CAD, and △CBD), each composed of two of the smaller triangles. The four small triangles are all congruent; this is evident from the symmetry of the figure. The four larger triangles are not all congruent, but they are congruent in pairs (△ACB with △ADB and △CAD with △CBD). All of these congruences could, presumably, be proved using the given information and might potentially be useful in arriving at our goal. How do we navigate through this web of possibilities?

One plausible line of argument runs as follows: first, use the Side-Side-Side congruence criterion (SSS) to prove that the left-right pair of large triangles, △ACD and △BCD, are congruent. Having established this, use the definition of congruent triangles (often called “CPCTC” for “Corresponding Parts of Congruent Triangles are Congruent”) to conclude that ∠ACP and ∠BCP are congruent. Now use the congruence of AC with BC, the congruence of ∠ACP and ∠BCP, and the self-congruence of CP to conclude (by Side-Angle-Side) that △ACP is congruent to △BCP. From this, we again use CPCTC to extract both the conclusion that AP = BP (and thus that P is the midpoint of AB) and that ∠APC is congruent to ∠BPC (and therefore the four angles formed at P are all right angles).

Even in summary form, the proof is quite complex, as it requires first proving one pair of triangles is congruent, then using that fact (and consequences of it) to prove that a second pair of triangles is congruent. If this were written out in accordance with the usual norms of proving in secondary geometry, it would be even more complicated. (For example, using the fact that ∠APC is congruent to ∠BPC to infer that the angles at P are right angles may itself take several steps, depending on how “right angles” are defined.)

Is a different kind of proof available? Perhaps, depending on the axiomatic structure in use in Mr. Gómez’s classroom. If, for example, his classroom makes use of a transformation-based geometry, the proof could be radically different (and perhaps quite a bit shorter). On the other hand, if the class is used to assigning Cartesian coordinates to the plane in order to write analytic proofs, the whole problem reduces to finding the coordinates of C, D, and P in terms of those of A and B.

The brief discussion above outlines some of the ways that the logical validity of the “usual” construction procedure for a perpendicular bisector can be established, responding to Veronica’s initial question: “Why [does] the construction work […?]” However, this would only satisfy Veronica if she, like the mathematician, regards logical validity as sufficient to ensure a correct output. If, instead, the re-paraphrase of Veronica’s question (“How can they be sure that the line that is constructed is indeed perpendicular to the segment and passes through the midpoint [?]”) signals a student who is worried that a valid construction may nevertheless produce inaccurate results due to human error or mechanical failure, a different kind of response may be in order; the way you make sure that the results come out correct is by (i) being very careful with your use of construction tools and (ii) checking the accuracy of the output of the algorithm with measurement tools after you are finished. Of course, our ability to check the accuracy of the output is inherently limited by the precision of our measurement tools, which can potentially work both to confer legitimacy where it does not belong (see the discussion of angle trisection in Part 1) and to cast doubts on a fundamentally sound method.

The external question: What should Mr. Gómez say?

We now turn to the external question of item 15301: What could, or should, Mr. Gómez say? This is the actual question of the item, the one directed to the GeT student whose knowledge we wish to assess.

We observe that this is a question of mathematical knowledge for teaching (MKT), one that draws, at least potentially, on multiple domains of knowledge (Ball, Thames, & Phelps, 2008), not only subject matter knowledge (SMK), as may be embodied in a mathematical proof, but also various aspects of pedagogical content knowledge (PCK). We list here just some of the domains of knowledge that Mr. Gómez may wish to or need to think about as he considers his response to Veronica:

1. Knowledge of Content and Curriculum (KCC)

Mr. Gómez may consider the fact that his classroom follows a particular curriculum, one in which the axioms and theorems of Geometry are arranged in a particular logical order. Knowing that other curricula exist, and are widely used, opens up a wide range of possibilities for his explanation, while the fact that one specific curriculum is in use in his classroom narrows down the possibilities. In a curriculum that is based on geometric transformations, the entire argument may come down to a single observation about the reflection symmetry in the construction. In a curriculum that consistently makes use of “analytic” methods (that is, the use of coordinates and algebra to represent geometric objects), an entirely different method would be appropriate.

Moreover the curriculum in Mr. Gómez’s class is arranged in a specific sequence, one that may or may not correspond to the logical structure of the theory. For example, at some point in the course students are likely to learn that the diagonals of a rhombus are perpendicular bisectors of one another—a theorem whose proof is essentially identical to that of the construction under discussion. This theorem is a corollary of two others; the diagonals of any parallelogram bisect each other, and the diagonals of a kite are perpendicular. Since a rhombus is both a kite and a parallelogram, we immediately obtain the desired conclusion. When during the curriculum in use in Mr. Gómez’s classroom are these properties of quadrilaterals taught? If they precede the current discussion of constructions, they could be appealed to as part of the response to Veronica. Whether this would be beneficial or not depends not only on Mr. Gómez’s knowledge of the scope and sequence of topics in his curriculum but also on what he knows about Veronica.

2. Knowledge of Content and Students (KCS)

Knowledge of Content and Students (KCS) refers to a teacher’s knowledge of how students may understand (or misunderstand) the content that is being taught—the conceptions and misconceptions they may hold. For example, many students fail to understand that a logical proof establishes the truth of a property for all figures satisfying the conditions of the problem; instead, they “view deductive proofs in geometry as proofs for a single case, the case that is pictured in the associated diagram” (Chazan, 1993, p. 362). Does this misconception underlie Veronica’s question, “how can [we] be sure that the line that is constructed is indeed perpendicular to the segment and passes through the midpoint [?]”. In other words, does her question stem from a concern that the construction, although proved valid for this figure, might not work for a different one? If so, Mr. Gómez’s response should stress the generic nature of the argument, emphasizing words like “always” and “for every” in his argument.

On the other hand, perhaps Veronica, like many students, might be operating under a conception of “proof” rooted in empirical verification. If this were so, then Mr. Gómez might wish to try to reinforce a more deductive proof scheme, emphasizing that the purpose of a proof is to give us certainty without the need for measuring the results, while, perhaps, simultaneously reassuring her that if she wants to she can always measure the angles and segments, despite the fact that the result is a foregone conclusion.

3. Knowledge of Content and Teaching (KCT)

This domain of professional knowledge includes different ways of representing mathematical content. Some options that Mr. Gómez might wish to consider are:

• Different ways of representing a proof: for example, two-column form, paragraph proof, flowchart proof.
• Different ways of developing the proof with Veronica: as a presentation, as a Socratic dialogue, as an outline whose details are left for Veronica to fill in, etc.
• Different ways of representing the figure: on a chalkboard, on paper, in a dynamic geometry environment.

Related to the last of these, Mr. Gómez might also wish to consider the tools used to construct the figure. Should he draw the construction freehand? Should he use compass and straightedge? Should he have Veronica use the compass and straightedge? Would a DGS environment work better and if so, which one to use? None of these questions have a single correct answer, but Mr. Gómez must make decisions, whether consciously or not.

4. Horizon Content Knowledge (HCK)

Horizon Content Knowledge concerns Mr. Gómez’s knowledge of mathematics beyond the scope of the course he teaches. This might include questions like: does every segment have a unique perpendicular bisector in geometries other than that of the Euclidean plane? For example, in three-dimensional Euclidean geometry, there are infinitely many lines that are perpendicular bisectors to a given line; however, all of these lines lie on a single plane, so that the proper generalization of “perpendicular bisector” in three dimensions is a plane, not a line. In non-Euclidean geometry (for example, certain lattice geometries, as may be represented on a Geoboard), a perpendicular bisector may not exist at all because the “midpoint” of a segment may fall between two lattice points.

HCK may also include additional sources of knowledge, such as knowledge of the history of mathematics. Does Mr. Gómez know that Euclid’s Elements does not include “perpendicular bisector” as an object of study and that the construction of a midpoint to a given segment is organized differently than the “usual construction” he teaches his students? If so, does that knowledge suggest different ways he might respond to Veronica?

Does Mr. Gómez understand the role of constructions as furnishing “existence proof” in Geometry? That some objects cannot be constructed using compass and straightedge? That knowledge may elevate the significance of Veronica’s question; it is particularly important to know why this construction is possible because other constructions are not. The impossibility of trisecting an arbitrary angle, of doubling a cube, and of squaring a circle surely fall within the realm of HCK, and yet they can help motivate the significance of the constructions that are within the secondary Geometry course.

Concluding thoughts

In the discussion above, we complete our a priori analysis of GeT assessment item 15301. Having analyzed the foundational questions (What does it mean to construct? Is a construction possible?), the internal question posed by Veronica to Mr. Gómez (How do we know this construction works?), and the external question posed to the GeT students (What could Mr. Gómez say?), it remains to examine how GeT students responded to this item and what GeT instructors and other workshop participants saw in those responses. In the third and final part of this article, I will turn to those data sources and discuss what they suggest we can learn about GeT students’ knowledge of geometry for teaching and the usefulness of these assessment items for promoting discussion among practitioners. 

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.

Chazan, D (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics 24(4), 359-387.

de Villiers, Michael (1990). The role and function of proof in mathematics. Pythagoras 24, 17-24.