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.

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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.