GeT: A Pencil

Author: Michael Weiss

  • What Can We Learn From an Assessment Item About Constructing Perpendicular Bisectors? Part 3:  Looking at student responses and expert anticipations of those responses

    Introduction

    This is the final installment of a three-part deep dive into a single assessment item the GRIP team designed to probe students’ knowledge of the student learning objectives (SLOs). Item 15301 was written for the purpose of investigating SLO 3, Secondary Geometry Understanding: Understand the ideas underlying the typical secondary geometry curriculum well enough to explain them to their own students and use them to inform their own teaching. The item 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?

    In the first two parts of this deep dive, I discussed this item from an a priori perspective. I observed that this item actually consists of multiple nested questions. First, there is the question posed to Mr. Gómez by Veronica within the situation of teaching; I refer to this as the “internal question.” Second, there is the question posed to the GeT students themselves about the situation of teaching; I call this the “external question.” In addition to these two questions, I also discussed various “foundational questions” about the mathematics of geometric constructions.

    In what follows, I briefly summarize some of the main ideas of that a priori analysis and use those main ideas to propose a collection of codes for themes that we might expect to find represented in students’ responses to item 15301. These codes will then be used to examine two sources of data: first, a collection of responses to item 15301 produced by a cohort of 47 GeT students; and second, a collection of comments on item 15301 from a group of experienced GeT instructors who participated in a workshop in Summer 2021. These data sources will allow us to explore the following questions:

    1. What themes are represented in students’ responses to item 15301?
    2. What themes did the GeT instructors anticipate would or could be present in students’ responses?
    3. What did GeT instructors notice when they were given the opportunity to examine students’ responses directly?

    We begin with a very brief review of some of the ideas developed in the first two parts of this analysis. In the first part, I sought to problematize the foundational question “Can we construct a perpendicular bisector?” as follows:

    1. First, I observed that the answer to the questions “Can we construct a given geometric object, and, if so, how?” depends, in a highly nontrivial way, on what axiomatization we use for our geometry. For example, in a “compass and straightedge” geometry (as in Euclid’s Elements), angle trisectors cannot be constructed at all; however, in more modern “ruler and protractor” geometries, their construction is trivial. Even for the case of the construction of a perpendicular bisector, which is possible in both systems, the question of how to enact such a construction differs significantly between the two systems.
    2. Second, from a rather literal point of view, no geometric construction ever perfectly produces its intended object. Any construction inevitably falls short because of two categories of limitations: (a) user error, such as an unsteady hand, a ruler that slips on the page while a line is being drawn, etc.; and (b) intrinsic limitations owing to the fact that the “points” and “lines” we draw (and draw on) are not really points and lines but merely symbolic representations of them. For this reason, the physical marks drawn on actual paper (or in a digital representation) are never more than approximations of the conceptual objects they represent.
    3. This last observation, regarding the mismatch between idealized geometric objects and actual drawings, cuts both ways—just as no actual construction in the real world can perfectly produce the object it is intended to generate, it is also the case that even a mathematically “incorrect” construction algorithm may, in some cases, produce a result that is close enough to be indistinguishable from what is sought.

    Veronica’s question is thus more problematic than it seems at first. In seeking to understand why a given construction is valid, she (potentially) raises deep questions about the structure of our geometric knowledge, its contingency on certain conventions of definition and axiomatization, and the relationship between the ideal world of mathematical abstraction and the real world of experience and measurement.

    Following this discussion, we turned to a close reading of item 15301 and noted that Veronica’s question (the “internal question”) is actually formulated in two not-quite-identical ways. It is first formulated as a request for an explanation of “why the construction works”, the item immediately reframes this as “How can we be sure that the line… is indeed perpendicular to the segment and passes through the midpoint?” Whereas the first question calls for an explanation, the second seeks a verification. Corresponding to these two questions, students might respond in one of two ways:

    1. Students might provide a mathematical argument for why the construction is a valid one, or
    2. Students might appeal to empiricism as a means of verifying that the product of the construction algorithm has the intended properties.

    These two broad themes may be further subdivided into sub-themes. A mathematical argument may make use of synthetic methods (as, for example, in a traditional two-column proof), analytic methods (as in an argument that uses coordinate geometry to transform the geometric question into an algebraic one), or transformational approaches. Any of these approaches could be presented as either a formal proof or a less detailed argument that indicates the main points of what could go in a formal proof.

    Likewise, an empirical approach to answering Veronica’s question could take different approaches, depending on whether the student perceives the underlying problem as a concern for user error or feels a degree of skepticism regarding the reliability of the method itself. In the former case, a response might emphasize the need to exercise caution when using the construction tools. In the latter case, a response might suggest using measurement tools to verify the result of the construction after the fact.

    Finally, in our discussion of the external question (“How could Mr. Gómez explain that?”), we observed that a GeT student’s anticipation of Mr. Gómez’s response could draw on multiple domains of knowledge within the construct of Mathematical Knowledge for Teaching (Ball, Thames, & Phelps, 2008). Such a response might call upon Knowledge of Content and Curriculum (KCC), Knowledge of Content and Students (KCS), Knowledge of Content and Teaching (KCT), and Horizon Content Knowledge (HCT).

    Thus, when we turn to the responses of the cohort of GeT students and instructors, we expect that we may find some or all of the following themes present:

    1. A theory-building disposition (Weiss & Herbst, 2015) —a sensitivity to the particular axiomatic structure in use for one’s theory of geometry, and an awareness that it is only one of many possibilities;
    2. A tendency towards skepticism—an awareness that constructions presuppose flawless operation with idealized tools that cannot be executed in the real world;
    3. An orientation towards pragmatism—a willingness to accept an imperfect construction as long as the result is close enough to the desired one;
    4. A mathematical argument, which may take the form of either a formal proof or an informal argument;
    5. An empirical disposition, which may take one of two forms:
      1. exercising caution in the use of tools and execution of the algorithm, or
      2. the use of measurement tools to verify the accuracy of the completed result;
    6. One or more knowledge domains within MKT.

    In the next section, we describe the data sets in more detail and use the themes above to classify the responses of our students and instructors.

    Data sources and methods

    Assessment item 15301 was pilot tested in Spring 2021 with a cohort of 47 GeT students associated with six different instructors and/or universities. Seven of those students provided no response to the item; another eight responded with “I don’t know,” “Unsure,” or similar responses. The present analysis is based on the responses of the remaining 32 students. The item was included in an assessment conducted online using a Qualtrics survey. Because the assessment platform only allowed responses in the form of typed text, many modes of communication that might otherwise have been called for (including not only diagrams but also mathematical notation) were not available to respondents. For this reason, it seems prudent to be somewhat skeptical of the data; it is likely that students’ responses may have been quite different (and, one suspects, richer) had the assessment been administered in a paper-and-pencil format.

    After the assessments were administered, a group of six college-level Geometry instructors with varying levels of teaching experience came together in Summer 2021 for a virtual workshop organized around examining students’ responses to the assessment items. Among these instructors were both mathematicians and mathematics educators; most of them had taught a course specifically targeted at Geometry for Teachers (GeT), although at least one taught a Geometry course that was not explicitly “for teachers.” In each meeting of the workshop, participants were asked a series of questions about four assessment items. Participants were asked to describe not only what they would consider to be a good response to the item but also what they expected a student might say in response. Participants were then shown the set of student responses and asked to describe what they saw in those responses.

    In analyzing the two sets of responses, I used the following methodology. First, each student response was tagged with one or more of the following codes: THEORY-BUILDING, SKEPTICISM, PRAGMATISM, ARGUMENT, CAUTION-TOOLS, MEASUREMENT, and MKT. These seven codes correspond to the themes enumerated at the end of the previous section. A response would be coded with a given tag if it could be construed as invoking or alluding to the corresponding theme. Thus, for example, the tag ARGUMENT was used for any response that seemed to be suggesting or calling for a formal proof, whether or not such a proof was actually provided. The same codes were then used to tag GeT instructors’ responses. In principle, a single item could be tagged with more than one code; in practice, none of the items were found to contain evidence of more than one of the themes.

    Student responses to Item 15301

    Of the 32 responses from GeT students, eight were not tagged with any of the codes above. Most of these amounted to nothing more than a restatement of the property in question: for example, “By showing the slopes are perpendicular and that the two segments are equal,” “Show that it creates a 90 degree angle,” and “A perpendicular line creates a 90 degree angle.” Such responses do not explain, or even hint at, how one would show those properties. Another response consisted simply of the two words “the center.” It is impossible to know what the respondent intended by this or if it was the result of an error in entering their response. The longest untagged response read:

    The circles we create help us visualize. Consider that these two circles would overlap. We can assume that inside this overlap is where the midpoint is. By constructing the perpendicular bisector in this manner, this is the best way to ensure that it is consistent.The circles we create help us visualize. Consider that these two circles would overlap. We can assume that inside this overlap is where the midpoint is. By constructing the perpendicular bisector in this manner, this is the best way to ensure that it is consistent. (Response A53)

    Although this student had quite a lot to say, I was unable to interpret exactly what was intended by this response.

    Of the remaining 24 items, 16 responses were tagged with the code ARGUMENT. None of these responses were fully-developed mathematical proofs, but some of these responses sketched out an argument that could be plausibly interpreted as the outline of one. For example, one response (A19) read, “He could use the perpendicular bisector theorem.” Although lacking in details, this does indicate an efficient method for proving that the construction is valid. Another student responded:

    Show how the two circles have the same radius meaning the intersections are the same distance away from each point and there are two points of intersection which are necessary for the creation of a line. (Response A29)

    The argument here is not fully coherent, but it does seem clear that the student was at least attempting to provide some sort of mathematical argument or informal proof. Most responses tagged with the code ARGUMENT were of this sort. Although they did not contain a fully correct or convincing mathematical argument, they contained evidence that the student at least understood the question to be calling for one.

    However, not all students understood the question as calling for a mathematical argument. Six responses indicated that the student understood the question as calling for some kind of empirical measurement, rather than a theoretical justification. Typical responses tagged with the MEASUREMENT code were:

    • “Measure the segment before bisecting it and then measure it after bisecting” (A55)
    • “Measure the angles around the intersection. If one of them is 90 degrees, then it is perpendicular. Then measure each side of the segment, and if they are equal then it is a perpendicular bisector.” (A51)
    • “Use a compass and a ruler to measure.” (A32)

    In addition to these, two responses were tagged with the CAUTION-TOOLS code:

    • “Making sure you accurately use your geometric construction tools and are precisely lining up to each point.” (A5)
    • “With the use of a straightedge it can be sure to be perpendicular” (A26)

    These three codes — ARGUMENT, MEASUREMENT, and CAUTION-TOOLS — were, in fact, the only three codes used. Thus, at a very coarse level of description, we can say that about half of the responses indicated that the student understood the question as calling for a mathematical argument of some sort; a quarter of the responses indicated that the student interpreted the question as calling for some kind of appeal to empiricism; and the remaining quarter contained either no meaningful content or none that could be classified. None of the student responses gave any indication that students were thinking about any of the foundational questions discussed in Part 1 of this essay or drawing directly on any of the specialized content knowledge domains discussed in Part 2.

    Instructor responses to Item 15301

    1. What do instructors think is necessary knowledge for Item 15301?

    In the summer workshop, the six instructors were initially asked the question “What is the knowledge needed to answer this item?” All six instructors either provided a mathematical proof or discussed what prior knowledge one would need in order to provide a mathematical proof; every instructor interpreted the item as calling for a mathematical argument. Examples of these responses are:

    • “The construction forms a rhombus and the diagonals of a rhombus are perpendicular and bisect each other. A point equidistant from the endpoints of a segment is on the perpendicular bisector of the segment.”
    • “Reflexive property (segments), then SSS, then CPCTC, then SAS, then linear pair of angles.”
    • “Knowing that the diagonals of a rhombus are perpendicular and bisect each other gives the result immediately (I’ll note that in my years of teaching constructions, few GeT students remember this since we haven’t discussed it in class). More often, they take an approach like the one [other instructor] mentions: using two triangle congruences to show constructed line is perpendicular to AB and that it bisects AB. So the argument requires experience/understanding of triangle congruence proofs.”

    As the three examples above show, responses varied significantly in how much detail was provided and how many alternatives the workshop participants entertained. However, each of these responses understands the prompt as an invitation to provide a mathematical proof or, at least, the outline of one.

    Two instructors included in their responses some reference to the fact that what counts as a proof may vary depending on the type of geometry being taught (synthetic, analytic or transformation-based). One such response, for example, included the following:

    The most efficient way to solve this problem is to know that a point is on the perpendicular bisector of AB if and only if it is equidistant from A and B. One either needs to quote the result, or to prove it… A transformation-based approach could work as well: the initial figure, AB, and the construction protocol, are both invariant under the act of reflection across the perpendicular bisector of AB, and therefore the line constructed must be as well. Therefore the line constructed is the perpendicular bisector of AB.

    None of the instructors made explicit reference to the situated aspect of the item prompt; they did not refer to the fact that Mr. Gómez’s response was to be addressed to a student in a high school classroom. We could imagine, for example, responses that referred to ways of knowing or misconceptions that are common among secondary students or that discussed the fact that what would work as an appropriate answer might depend on the curriculum being taught, etc. The fact that none of the GeT instructors, when asked “What knowledge is needed to answer this item?”, responded “They need to know something about how students learn proofs” or “They need to know that proving constructions is not common in many secondary curricula” suggests that they approached this problem primarily as a mathematical task, not as a task of teaching. In terms of the a priori analysis in the first part of this paper, we could say that they discussed the knowledge needed to answer the internal question but not the knowledge needed to answer the external question.

    1. How do instructors expect GeT students to respond to item 15301?

    Workshop participants were next asked to anticipate what type of responses they would receive from GeT students. Only four of the instructors responded to this question, but all four of them expressed an anticipation that students would provide a mathematical argument of some sort. The four responses were:

    • “The arcs intersect at two points that are equidistant from the endpoints of the segment (by construction). Therefore, when a line is drawn using those two points, a perpendicular bisector is formed.”
    • “One student might reference parallel lines followed with a rotation of 90 degrees from the midpoint makes perpendicular lines. To explain perpendicular bisectors. Another student might use the the curvature in the first quadrant in relation to the intersecting lines to find 90 degrees. Then conclude the intersecting lines are perpendicular. Another student may not want to use the curvature of the lines and reason with triangles.”
    • “I would guess that a few people might come up with a correct deductive proof. More if they were in a course that covered constructions and specifically emphasized the ‘proving why it works’ step of constructions. (It is interesting that this question is premised on a teacher teaching the constructions without that vital last step.) I would expect that many would say something like, ‘you can see this will always happen because they are the same distance away.’ Such an argument might actually have more merit than it would initially appear since it would relate to the symmetries.”
    • “I would expect most GeT students would start proving pairs of triangles congruent to each other. Quoting the perpendicular bisector theorem seems unlikely to me, as does recognizing this as a rhombus and citing the property of the diagonals of a rhombus.”

    While there are clear differences among these responses, particularly with respect to how much variation in student responses the individual instructors anticipate, they all share an interpretation of the problem as calling for a mathematical proof of some sort. None of the instructors expected that the item would call forth an empirical response from a GeT student nor that it would evoke a discussion of theoretical considerations, secondary students’ conceptions of geometry, variations in curriculum, or any other SCK-related topics.

    1. What do instructors notice in GeT students’ responses?

    At this point in the workshop, instructors were shown the 32 student responses described above and were asked to comment on which ones included evidence that the students did, or did not, have the knowledge needed to respond to the prompt. In the ensuing discussion, one instructor identified just four responses as containing at least some evidence that the student contained sufficient knowledge to answer the question. Another instructor described three responses as being “on the right track.” (It is, perhaps, noteworthy that these two instructors identified only a single student in common.) Yet a third instructor wrote:

    It is interesting that no one tried to answer this prompt! None of these are as good as I might hope. I wonder if any of them did justifications of constructions in their courses. It doesn’t look like anyone remembered having done this one. But, at the same time, some of these are not so far off. A19 is correct and is a reasonable answer. (The perpendicular bisector theorem says a point in on a perpendicular bisector of AB if and only if it is equidistant from A and B.) A7 is also close to this idea. A34 is close to an idea of the beginning of the symmetry proof.

    Two instructors zeroed in on an empirical conception of mathematical justification as being a significant feature of the GeT students’ responses. One of these instructors wrote, “There are (at least) two really important misconceptions that I see: (1) If it looks correct, then it is correct because we constructed it with straightedge and compass (A5, A25). (2) We can measure it with a ruler or a protractor to see that it’s correct (A15, A17, A55).” 

    1. What other kinds of knowledge do instructors see as potentially embedded in Item 15301?

    Finally, instructors were asked to comment on which Student Learning Objectives (SLOs) were potentially involved in answering Item 15301 and to add any additional comments on the item. Instructors identified SLO 8 (“Be able to perform basic Euclidean straightedge and compass constructions and be able to provide justification for why the procedure is correct”), SLO 1 (“Derive and explain geometric arguments and proofs in written and oral form”), SLO 5 (“Understand the role of definitions in mathematical discourse”), SLO3 (“Understand the ideas underlying the typical secondary geometry curriculum well enough to explain them to their own students and use them to inform their own teaching”), and,—at least potentially—SLO 7 (“Demonstrate knowledge of Euclidean Geometry, including the history and basics of Euclid’s Elements, and its influence on math as a discipline”). 

    Instructors also observed that the item contained two different statements of (ostensibly) the same question:

    The statement, ‘why something works’ and ‘how can we be sure it is correct’ are not the same thing. I think most GeT students responded to the second part of the question, not the first part of the question. And I do think that second question connects more to the CCSS-M math standards (i.e. using appropriate tools strategically).

    When prompted to consider what other kinds of knowledge instructors see as potentially embedded in the item, some instructors did identify issues related to student thinking and curriculum—issues that, as noted above, were not evoked by the prompt, “What do students need to know in order to answer this question?” For example, two instructors observed that the appeal to empiricism is somewhat unsurprising, given what is common in secondary Geometry classrooms. One wrote:

    Perhaps GeT students are trying to imagine themselves in the hypothetical situation of responding to a student who has not yet learned triangle congruence proofs etc. I think this is pretty typical for how constructions are introduced; students are taught the steps as they are learning the definitions for the geometric objects…  So the ‘how can we be sure it is correct’ regards what is available to those students at the time: getting empirical evidence using other tools they know how to use that the construction meets the requirements of the definition of perpendicular bisector.

    The other wrote:

    I also think this question is potentially very revealing of the different proof schemes that GeT students hold, in particular their tendency to default to empiricism in the context of construction problems. Whether we call that a ‘knowledge’ issue is another matter; I would prefer to describe this in terms of ‘different ways of knowing’, rather than ‘correct’ / ‘incorrect’. But we do know that justifying a construction with a proof is (bizarrely) not normal in the classroom, no matter how much we would like it to be.

    Discussion

    Despite instructors’ rather pessimistic appraisals of the GeT students’ work, it bears repeating that roughly half of all responses indicated that students interpreted the prompt as calling for some kind of mathematical argument. It is true that most of the arguments they offered fell far short of what most GeT instructors would accept as a correct proof; however, we should, perhaps, take some comfort in the evidence that a significant portion of the students have at least been enculturated into mathematical practice and its sensibilities enough that they understand that a question in the form of “How can we be sure that this works?” is supposed to be answered with a proof.

    In contrast, it seems significant that none of the instructors anticipated that GeT students would respond to the prompt with an empirical strategy, either one that focuses on controlling the means of production or on one that emphasizes measuring the output of the algorithm. This may indicate a substantial “expert blind spot” for GeT instructors; we are so thoroughly accustomed to approaching mathematics teaching from the perspective of what is mathematically correct that we forget that these cultural norms do not come naturally to students and should not be taken for granted. If we, as GeT instructors, think that a proof-centric approach to mathematical validation is the sine qua non of secondary Geometry instruction, it is vital that we recognize that our future secondary teachers do not automatically share that value and that other forms of validation (such as an appeal to authority or empiricism) need to be not only anticipated but also confronted directly in the GeT classroom.

    It also seems significant that issues related to the situated nature of the task—the fact that the external question is not just a mathematical problem but a problem of mathematical teaching—were not mentioned by any of the GeT students nor by any of the GeT instructors in their response to the initial question “What do students need to know in order to answer this question?” This suggests that students and instructors alike may tend to default to the role of mathematical problem-solvers, rather than consider other specialized domains of knowledge that are important for mathematical teaching. It may be worth considering whether those other domains of knowledge—knowledge of how students think about proof, of how different curricula do or do not establish relationships between geometric constructions and axiomatic systems, etc.—could or should play a larger role in the objectives of a Geometry for Teachers course.

    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.

    Weiss, M., & Herbst, P. (2015). The role of theory building in the teaching of secondary geometry. Educational Studies in Mathematics, 89(2), 205-229.

  • What can we learn from an assessment item about constructing perpendicular bisectors? Part 1: Problematizing the premise of the item

    Introduction

    Among the assessment items the GRIP team designed to probe students’ knowledge of the student learning objectives (SLOs), item 15301 was intended to target SLO 3, Secondary Geometry Understanding: Understand the ideas underlying the typical secondary geometry curriculum well enough to explain them to their own students and use them to inform their own teaching. 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 is the case with many items intended to assess mathematical knowledge for teaching (MKT), this item is situated in the context of the work of teachers. As a consequence of this situated nature, the item actually consists of (at least) two different questions nested within each other. The first question is the question posed by Veronica to Mr. Gómez: “Why does the construction work… How can [we] be sure that the line that is constructed is indeed perpendicular to the segment and passes through the midpoint?” In what follows, I will refer to this as the internal question—the question that is asked inside the situation of teaching. The second question is the one posed to the GeT students themselves: “How could Mr. Gómez explain that?” I will refer to this as the external question—the question that is asked about the situation of teaching. These two questions are obviously closely related, but they do not necessarily have the same answer. The internal question is a purely mathematical question, one that calls for a mathematical explanation (which may or may not be a proof). In contrast, the external question is about mathematics teaching and (at least potentially) might call for a different kind of response. In the language of the MKT framework (Ball, Thames, & Phelps, 2008), we might say that the internal question probes common content knowledge (CCK), while the external question draws on other knowledge domains within MKT, such as (to name just two) knowledge of content and students (KCS) and knowledge of content and curriculum (KCC).

    This description oversimplifies matters because the internal question is actually posed twice, in different terms. Initially Veronica asks “why the construction works”; almost immediately the question is explicated (by whom?) as meaning “how they can be sure that the line that is constructed is indeed perpendicular to the segment and passes through the midpoint”. However, this second question is more than a paraphrase or an elaboration of the first; it is an interpretation of it, one of many possible glosses of what Veronica might have meant by asking “why the construction works”. Are there other plausible interpretations?

    In addition to these layers, the internal and the external, let us consider a third, perhaps more fundamental, stratum of knowledge embedded in this assessment item: the level at which we ask “how do we construct a perpendicular bisector for a segment?” or, perhaps even more fundamentally, “what does it mean to construct a perpendicular bisector for a segment?” I will refer to these as foundational questions. Note that these foundational questions are presented by the item as settled matters; at the point at which the reader enters the situation of teaching, Mr. Gómez has already taught a method, one described as “the usual procedure.” There is, evidently, no question what perpendicular bisectors are, what it means to “construct” a perpendicular bisector, nor how one does that. All of these are unproblematic for Mr. Gómez, for the item, and for the GeT student who is asked to respond to the item. It is only for Veronica (perhaps) that these are unsettled questions!

    But perhaps Veronica is on to something. In what follows, I will attempt to problematize these foundational questions and show that they should not be taken entirely for granted. This a priori analysis will suggest a number of anticipated categories of response to the external question (“How could Mr. Gómez explain that?”), which in the second part of this paper I will use to categorize GeT student responses to item 15301.

    Moving outward from this a priori analysis, we also can consider two additional, more empirical approaches to the assessment item, based on two different contexts in which the problem has been used. First, we may take note of the fact that this item has been used with GeT students, generating a collection of student responses. Second, we note that this item was one of many discussed by a collection of GeT instructors and others in the 2021 Summer GeT Assessment Item Workshop. What do GeT students say in response to this prompt, and what can we infer from those responses about their knowledge of geometry for teaching? Additionally, what did participants in the GeT summer assessment item workshop notice about the item and in the collected student responses?

    I propose to describe item 15301 from five different points of view:

    1. The foundational questions:
      1. What is a perpendicular bisector? 
      2. What does it mean to construct a geometric object? 
      3. Can we, in fact, construct a perpendicular bisector for any segment? 
      4. How can we distinguish between a construction that actually works and one that only approximately works?
    2. The internal questions:
      1. Why does the “usual construction” work?
      2. How can we be sure that the line constructed is indeed perpendicular to the segment and passes through the midpoint?
    3. The external question:
      1. How could, or should, Mr. Gómez respond to Veronica?
    4. Questions about GeT students:
      1. What do GeT students say in response to this prompt, and what can we infer from those responses about their knowledge of geometry for teaching?
    5. Questions about GeT experts:
      1. What did participants in the summer workshop notice about the item and about student responses to the item?

    My discussion of these questions will be divided into three articles. In the remainder of this article, I will discuss the foundational questions. In the second article in this series (see elsewhere in this issue), I will discuss the internal and external questions. A discussion of what item 15301 evoked from GeT students and experts will be contained in the third article in the series, to be published in a future issue of this newsletter.

    Foundational questions about constructing perpendicular bisectors

    I here consider a number of basic questions regarding perpendicular bisectors and what it means to construct one.

    A. What is a perpendicular bisector?

    Given a line segment AB, a perpendicular bisector is a line that passes through the midpoint of AB at a right angle. In Euclidean geometry, every segment possesses a unique perpendicular bisector. This follows from two facts: (i) every segment possesses a unique midpoint; (ii) given any point P on AB, there exists a unique line perpendicular to AB through P.

    It may be surprising that the phrase “perpendicular bisector” does not appear in Euclid’s Elements nor does Euclid provide a method for constructing a perpendicular bisector to a segment. To be sure, the Elements discusses perpendicular lines at great length, and presents methods for both constructing the midpoint of a segment (Elements 1.10) and constructing a perpendicular to a given segment through a point on that segment (Elements 1.11). Indeed, the construction of 1.10 locates the midpoint of a segment AB by means of a line drawn through AB at a right angle, but the fact that this line is perpendicular to AB is neither noted in the proof nor stated explicitly. Similarly the construction of 1.11 constructs a perpendicular to AB through a point C on AB by first constructing, on AB, a segment DE of which C is the midpoint, so that the perpendicular line eventually constructed does pass through the midpoint of DE—but once again this fact goes without notice. One might well argue that “perpendicular bisector” as a concept is tacitly present in these two items, but the tacitness is precisely the point I wish to focus on. Euclid sees no need to name, define, or describe a method of construction for “perpendicular bisectors.” They are simply not visible in the text. In fact, a Google nGram search for the phrase “perpendicular bisector” shows that the phrase did not begin to appear in English-language textbooks until the latter part of the 19th century.1 

    Furthermore, even if we stipulate that Elements 1.10 and 1.11 are, at least implicitly, about constructing perpendicular bisectors, it is worth noting that the constructions presented there are not exactly the same as the “usual procedure” taught by Mr. Gómez in item 15301. Euclid’s construction of the midpoint for segment AB has essentially two steps: (i) construct an equilateral triangle ABC on segment AB, and (ii) bisect the vertex angle ACB. The point where the angle bisector intersects AB is its midpoint; this angle bisector also intersects AB at a right angle, although as mentioned above, Euclid does not note this latter fact. See the figure below.

    It may be argued, contrary to what I have written above, that Euclid’s construction is not, in any essential way, different in practice from the “usual procedure” taught by Mr. Gómez because the performance of Euclid’s second step (bisect the vertex angle ACB) requires the performance of sub-steps that produce the bottom half of the figure constructed by the “usual procedure.” This is certainly true. But structurally, the two construction algorithms differ in that Euclid’s construction is described and conceptualized differently from the “usual” one. We observe that, in keeping with Euclid’s incrementalist style, this construction relies on two prior constructions: constructing an equilateral triangle on a given base (Elements 1.1) and bisecting an angle (Elements 1.9). Just as a theorem, once proved, may be cited as a reason in a proof of a subsequent theorem, so too may a construction, once demonstrated, be invoked as a single step in a subsequent construction. We might make an analogy with software coding; the prior constructions function as subroutines, which can be “called” by the main program. Once a subroutine has been compiled, we do not peer “inside” it to see the individual lines of code of that comprise it. The “usual” construction, in contrast, contains no subroutines; the algorithm is structurally “flat” with each operation performed within a single main routine. So even though the “usual” construction of a perpendicular bisector contains the same sequence of moves as the Euclidean construction of Elements 1.10, the two are organized and conceptualized in distinct ways.

    B. What does it mean to construct a geometric object?

    In Euclid’s Elements2, a construction is a sequence of operations performed with a compass and an (unmarked) straightedge that, given some initial configuration of points and lines, produces a new configuration with some specified properties. Thus, for example, the first Proposition in the Elements is “to construct an equilateral triangle on a given finite straight-line.3” In fact, the first three of Euclid’s five postulates are nothing more than statements of the basic operation of a compass and an unmarked straightedge.

    In the Elements, constructions perform a function similar to what, in modern-day mathematics, we call an existence proof4. That is to say, if we wish to establish that there exists a line that (i) is parallel to any given line and (ii) passes through any specified point not on the line, the way we do that is by describing explicitly how such a parallel line can be constructed using a compass and straightedge (see Elements 1.31). In the absence of a construction, we have no way of knowing whether an object with a given set of properties exists at all, let alone whether it is unique. Euclid consistently avoids making use of any auxiliary objects in his proofs until the constructibility of the required objects has first been demonstrated. For example, Euclid’s proof of the Base Angles Theorem for isosceles triangles (Elements 1.5) cannot make use of angle bisectors because angle bisectors are not constructed until later (Elements 1.9).

    In a very real sense, an object that cannot be constructed using compass and straightedge does not exist in Euclidean geometry (or at least in the geometry of Euclid’s Elements). For instance, because there is no construction algorithm for trisecting a general angle5, Euclid never considers what properties angle trisectors might possess or deploys them in arguments. They are simply outside of his universe of discourse.

    Euclid’s universe of discourse is determined by his choice of axioms; other axiomatic schemes for Euclidean geometry are possible. Beginning in the mid-20th century, many textbooks began replacing the compass and straightedge axiomatic structure with a different one, known as ruler and protractor axioms. This approach was pioneered by Birkhoff and Beatley in their 1941 text Basic Geometry and subsequently adopted by both the School Mathematics Study Group’s 1960 textbook Geometry and by Moise and Down’s popular 1964 text of the same name. As their names suggest, the ruler and protractor axioms take as fundamental the possibility of measuring the length of any segment and the magnitude of any angle6. In such a system, “constructing an object” means using a ruler and/or protractor to determine the placement of specific points and draw rays so that segments and angles have the required measurements. For example, in ruler and protractor geometry, trisecting an angle is trivial: the process is (i) measure the given angle, obtaining a real number x; (ii) divide x by 3, obtaining another real number y; (iii) draw a new angle with measure y.

    This last example shows that answer to the question “What does it mean to construct a geometric object?” depends in a highly nontrivial way on the precise axiomatic framework in which we work. What counts as a construction in ruler and protractor geometry may not count as a valid construction in compass and straightedge geometry.

    C. Can we, in fact, construct a perpendicular bisector for any segment?

    As a practical matter, the answer to this question is no, or at least not exactly. Our ability to actually carry out an indicated construction algorithm is constrained not only by the mechanical qualities of the tools with which we work but also by the mechanical skills of the person wielding those tools.

    Some of these constraints are fairly superficial and can be classified as “user error”. When drawing a line with a straightedge or an arc with a compass, the user’s hand may slip, the tool may slip across the paper, or the paper may slip on the table; any of these can result in “lines” that are not straight, “arcs” that wobble, and so forth. Then there is the issue of lining up the tip of the pencil with marks on the page while simultaneously aligning the edge of the ruler with a printed image; often the result is lines that are slightly offset from where they should be. Whether we regard these errors as inevitable or ascribe them to a lack of proficiency on the part of the user, they are common experiences that any classroom teacher should be aware of. So, when Veronica asks how they can be sure that the line that is constructed is indeed perpendicular to the segment and passes through the midpoint, one reasonable interpretation of her question might be simply: How do we reduce the amount of user error in the handling of the tools, so as to make sure that the algorithm produces an output that is what it ought to be?

    Even if we imagine a student with a perfectly steady hand, there still remain constraints in the diagrammatic resources used to represent geometric objects—constraints that render any constructed figure only an approximation of what it is supposed to be. This is because the “points” and “lines” we draw (and draw on) are not really points and lines. Points, in Euclidean geometry, are understood as locations in space that have no spatial extent; they are located at a place, but do not occupy any space. In Euclid’s famous formulation, a point is “that of which there is no part” (i.e., an indivisible atomon with no interior substructure). The points that appear in printed diagrams, in contrast, are small black circles of ink, with a diameter of perhaps half a millimeter. Similarly, a line is supposed to be a purely one-dimensional object, “a length without breadth” in Euclid’s words, but a printed line is always made up of ink strokes that have some thickness and are (if viewed under a microscope) easily seen to be only approximations of the perfectly straight objects they are intended to signify. When we then consider that students may work with pens or pencils that are not particularly sharp or whose tips may break in the middle of a construction; with rulers whose “straight” edges may have nicks and dents; and with all the other imperfections that are endemic to well-worn tools, we see quickly that even under the best of circumstances, the most we can possibly hope for from a construction of a perpendicular bisector is a (picture of a) line that approximately passes through the midpoint of a (picture of a) line at an angle that measures approximately 90 degrees.

    Whether these sources of error or imprecision should be regarded as significant or not depends on many contextual factors, but one thing that should be clear is that these types of concerns—concerns about the disconnect between the idealized mathematical object and the physical representation of the object—cannot be entirely allayed by appealing to the mathematical rigor of the algorithm itself. If we are concerned not with logical justification but rather with imprecision that results from faulty tools or faulty execution, the only way to detect or mitigate that imprecision is through recourse to empiricism. That is to say, even if we know we have performed the algorithm correctly, we may still worry that the result is not sufficiently precise for our purposes (whatever those purposes may be); in that case the only way to check the accuracy of the result is by making measurements with a ruler and protractor.

    What is also true is that we cannot escape the limitations of our material world by seeking refuge in digital representations. Even dynamic geometry software requires some degree of approximation; points are stored as a pair of coordinates, and numerical data when stored in a digital device can only be stored to a finite number of digits. Any software system, no matter how sophisticated, must inevitably approximate, and such approximations invariably lead to breakdowns in the ability to resolve the difference between very close quantities. For example, this can be demonstrated in GeoGebra as follows. First, define a numerical value n controlled by a slider. Next, define two points: A = (0, 0) and B = (10-n, 10-n). Now type the commands f = Line(A, B) and g = PerpendicularBisector(A, B). As long as n is a relatively small integer the software has no problem either computing the equations of the two indicated lines or drawing their graphs. But as soon as n gets too large (say n > 8), both lines disappear from the graph, and their equations display as undefined. Zooming in on the display shows that the two points A and B are still resolvable, with sufficient magnification, as distinct points, but the software can no longer calculate the slope of the line through AB nor of the line perpendicular to it. The construction requires a level of precision in excess of that possessed by the tool.

    As soon as we recognize that dynamic geometry software is constrained by the precision with which it can store and display numerical values, we begin to question whether it actually shows us what we think it shows us. Do the angle bisectors of a triangle, constructed in a dynamic geometry diagram, really meet at a point, or do they merely intersect at points that are too close to each other to be resolved as distinct by the software? How can we possibly tell the difference? In the end, regardless of whether our tools are digital or mechanical, our only means of determining the accuracy of a construction is empirical, and empirical measurements are, inevitably and essentially, always approximate.

    D. How can we distinguish between a construction that actually works and one that only approximately works?

    We have seen that different axiomatizations of Euclidean geometry lead naturally to different notions of what constitutes a “valid construction.” We have also seen that, regardless of the axiomatization in play, a construction that has been validated on theoretical grounds—that is, one that can be “proved to work”—will nevertheless produce results that are only approximately correct once it is enacted with actual tools (whether mechanical or digital).

    We now consider the reverse situation: what of constructions that cannot be validated using a proof but that nevertheless produce results that are approximately correct—or even correct to within the limits of empirical verification?

    Consider, for example, the following construction protocol for “trisecting” an arbitrary ∠ABC:

    1. Construct an arc, centered on the vertex B and intersecting the two sides of the angle BA and BC in points P and Q, respectively.
    2. Subdivide secant PQ into three equal parts with points R and S.
    3. Draw rays BR and BS.

    Every step of this construction can be carried out using a compass and straightedge. However, the construction is not logically valid; it does not produce three angles of equal measure, each one-third the measure of the original ∠ABC. For small enough angles this algorithm produces results that are reasonably close to the desired ones. How small an angle is “small enough,” and how close is “reasonably close”? For example, if we begin with an angle ∠ABC measuring 20°, the “trisectors” produced by this method determine angles whose measurements differ from the correct values by less than 1%, or about 0.06°, much too small to discern with the eye or to measure with a protractor. Even for an angle measuring 40°, the difference between what is intended and what is actually produced is less than half a degree, too small to measure with precision, especially given the error inherent in our measurement tools.

    So does this construction “work” or not?  Can we be sure, to paraphrase Veronica, that the rays constructed by this method actually divide the angle into three equal parts? The logician and the mathematician object strenuously to this claim; surely we are not satisfied with “almost right” constructions. But all constructions are, at best, “almost right,” at least when they are translated from abstract descriptions of idealized objects into concrete marks drawn on paper with physical tools wielded by imperfect humans. If empiricism urges us to be skeptical about logically valid constructions, does it not also push us to be tolerant of logically invalid constructions that produce results that are so close to correct as to be indistinguishable from accurate ones?

    Conclusions

    The above consideration of what I have called “foundational questions” is intended not only to describe, in some detail, the backdrop to Veronica’s question to Mr. Gómez in item 15301 but also to bring to the surface some of the categories of perception and appreciation that might be brought to bear on answering the remaining questions articulated in the introduction:

    2. The internal questions: Why does the “usual construction” work? How can we be sure that the line constructed is indeed perpendicular to the segment and passes through the midpoint?

    3. The external question: How could, or should, Mr. Gómez respond to Veronica?

    4. Questions about GeT students: What do GeT students say in response to this prompt, and what can we infer from those responses about their knowledge of Geometry for teaching?

    5. Questions about GeT experts: What did participants in the summer workshop notice about the item and about student responses to the item?

    Some of the themes that have emerged from this discussion are (i) the role of constructions as existence proofs in Euclidean geometry, (ii) the way in which different axiomatizations of Euclidean geometry lead to different notions of what counts as a “valid construction,” (iii) that even within a particular axiomatic structure, what counts as the “usual” construction is a matter of convention, (iv) that whether a construction “works” may, depending on context, depend not only on the logical validity of the sequence of steps performed but also on the degree to which the final result matches our specifications of it when measured using realistic tools, and that (v) depending on how much accuracy we require in the final output, a valid construction may produce incorrect results (when performed by imperfect users of imperfect tools), while (vi) an invalid construction may produce results that are “correct enough” for our purposes.All of these observations serve to problematize the premise of item 15301: “Mr. Gómez taught students the usual procedure for constructing a perpendicular bisector for a segment…” In Part 2 of this article, I explore the internal and external questions contained within the item itself.

    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.

    Birkhoff, G. D., & Beatley, R. (1941). Basic geometry. Chicago, Illinois: Scott, Foresman and Company.

    Fitzpatrick, R. (2007). Euclid’s Elements of Geometry: The Greek Text of J.L. Heiberg (1883–1886) from Euclidis Elementa. Edidit et Latine interpretatus est I.L. Heiberg, Lipsiae, in aedibus B.G. Teubneri, 1883–1886. Raleigh, North Carolina: Lulu.com.

    Luby, T. (1825). An Elementary Treatise on Trigonometry: With Its Different Applications. London, U.K.: Hodges and M’Arthur. 

    Moise, E. E., & Downs, F. L. (1991). Geometry. Addison-Wesley. 

    School Mathematics Study Group. (1960). Geometry. New Haven, CT: Yale University Press. 

    Wright, R. P. (1868). The Elements of Plane Geometry. London, U.K.: Longmans & Company.

    _________________________

    1The first appearance of this phrase in a text intended for the study of plane (i.e. Euclidean) geometry appears to have been in The Elements of Plane Geometry for the Use of Schools and Colleges (1868) by Richard P. Wright, although an earlier appearance is found in an 1825 trigonometry textbook by Thomas Luby. The 1868 text by Wright also contains the earliest appearance in print that I have found of the “usual method” for constructing perpendicular bisectors.

    2Throughout this article, the translation used is that of Fitzpatrick (2007), based on the Greek text of J.L. Heiberg

    3Euclid uses the phrase “straight line” (or sometimes “straight-line,” as here) to denote what in modern parlance would be called a “line segment.” Lines, in Euclid, are always finite in extent, though capable of extension to arbitrary lengths (and hence infinite in potential).

    4More specifically, they correspond to what in modern mathematics would be called a constructive existence proof. Modern mathematics also includes nonconstructive existence proofs, which demonstrate the logical necessity of something existing without explicitly showing how it can be constructed.

    5The nonconstructibility of a 20° angle, and hence the impossibility of formulating a trisection algorithm for angles in general, was not established until 1837. It is unclear (and probably unknowable) whether Euclid regarded the trisection problem as unsolvable or merely unsolved; in any case, because it was unsolved, he never refers to it at all.

    6More precisely, the ruler postulate asserts the existence of a one-to-one correspondence between the set of equivalence classes of congruent segments and the set of positive real numbers, and the protractor postulate asserts the existence of a one-to-one correspondence between the set of equivalence classes of congruent angles and the set of real numbers greater than 0 and less than 180.

  • What can we learn from an assessment item about constructing perpendicular bisectors? Part 2: Anticipating responses to the item

    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.

    1. 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.

    1. 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.

    1. 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.