This past March, I was invited to speak about high school geometry to a college geometry class, one that we might describe as a Geometry for Teachers (GeT) class insofar as future teachers were an important, even if not the only, constituency. I was asked to talk about why high school students need to study geometry—something that may be taken for granted in identifying geometry as a mathematics class that future teachers need to take but may make sense to ask from the perspectives of both college mathematics students and the mathematicians who teach them. In our proposal to NSF to fund the GeT Support project, we noted that contemporary mathematical research has little to do with the geometry content students learn in high school, and the same might be said about the mathematical experiences of undergraduates. Thus, if the geometry content does little to cement future learning or research, it is worth asking what role high school geometry plays in high school students’ mathematical development. 

This question of the purpose of the study of geometry is not new. At the end of the 19th century, as the high school curriculum was being formed in the United States, the idea that different scholarly disciplines were useful to train different mental faculties was in vogue. Geometry was then justified on account that it was expected to train logical reasoning. González and Herbst (2006) describe this logical argument along with others which emerged later during the 20th century. An intuitive argument was offered that described geometry as providing a language that students could use to relate to their everyday experiences in the material world. A utilitarian argument was offered during the second world war period that described geometry as providing useful resources for the world of work. And during the time of the New Math (in the 1950s and 1960s) a mathematical argument emerged that described geometry as providing an opportunity for students to experience what mathematicians do: define, conjecture, prove, and so on (González & Herbst, 2006). To some extent those four arguments (logical, intuitive, utilitarian, and mathematical) are still being made in different quarters; furthermore, we may come by a better answer to the question by integrating some of those arguments. While the logical argument was questioned early in the 20th century, both the intuitive and the mathematical argument have always felt complementarily compelling to me. 

A modeling approach 

In my talk to that class in March, I proposed that the geometry course provides students with opportunities to engage in the mathematical modeling of their experiences with space and shape. Herbst, Fujita, Halverscheid, & Weiss (2017) used this idea as the centerpiece of their graduate textbook on the teaching and learning of geometry. They offered Figure 1 below as a way to represent what could happen in high school geometry. The box in the center represents an envisioned modeling approach to high school geometry. The approach is informed by two sets of sources. On the one hand there is a source that we could associate with the intuitive argument named above: Real world objects and activities are often represented (i.e., named, described, depicted) using tokens that might be described as geometric. So, words—like line, square, and turn—and shapes are often used in describing how people manage space and shape. This does not mean that by virtue of their use of those tokens those people are doing mathematics in a way commensurate with that of mathematicians, but an argument that builds on embodiment and materiality could be envisioned to suggest engagement in physical activities and their enactive representation has some mathematical qualities. The diagram in Figure 1 suggests that high school geometry could build environments on top of those existing relationships between the real world and our cultural representations of it. These environments are described as “geometric models of representations of real world objects” which means at least two things. On the one hand, these models are particular interpretations of the primitive objects, relationships, and postulates of formal geometries such as Euclidean geometry. On the other hand, these models are environments for mathematical practice—environments in which the activities (e.g., calculation, construction) and dispositions (e.g., pondering whether a solution is unique) of mathematical practice can be mobilized to produce information that can be interpreted in terms of real world objects and activities. The arrow from Geometries to these models and the adjective geometric attached to models point to the aspiration that work within those models be guided by the mathematical sensibility that reigns in mathematical practice. Teachers and curriculum developers who are likely to know geometry as a mathematical domain can organize these environments in which students’ experiences with real world objects and activities and their representations are involved in the activities of mathematical practice and scrutinized with the sensibilities of mathematical practice. A quick example of these relationships is the modeling role the use of a straightedge plays in helping think of a straight line when confronted with concrete objects that might be described as forming a straight line—questions of incidence, betweenness, separation, parallelism, and so forth can be brought from the geometric theory of lines and specific practices with the straightedge may help interpret those questions in terms of concrete objects. The famous quote by Poincaré, “geometry is the art of reasoning well from badly-drawn figures” (cited in Bartocci, 2013), comes to mind along with the common practice of drawing in geometric problem solving. A more detailed example may help make the point clearer.      

The pool problem 

The pool problem serves as a more extended example of the modeling approach. This problem could be used relatively early in the high school geometry course, after students have learned some basic constructions (e.g., how to construct a perpendicular bisector to a segment) and how to prove triangles congruent. The problem can be used to develop students’ knowledge of a theorem about right triangles: that, in any right triangle, the midpoint of the hypotenuse is equidistant from the vertices. More importantly, the problem can be used to inscribe in high school geometry a simple example of the mathematical difference between sense and reference (Frege, 1997/1892): The same mathematical object (the same point in this case) can be the referent of two different ways of speaking (two different meanings or senses). This notion is pervasive in mathematics as two different procedures can construct the same mathematical object. The pool problem can be used to instill the disposition to ask whether two different construction procedures define the same object. 

The pool problem says

Three swimmers are going to jump into a rectangular pool and race toward a buoy. One of them will do so from a corner of the pool, a second from a side of the pool adjacent to that corner, and the third from the other adjacent side. Suppose the swimmers are in position and you have to position the buoy. Where should you position the buoy to make the race fair?

Note here that some geometric concepts are being used to describe the concrete situation: the pool is rectangular; the word adjacent is used to refer to the sides of the pool that make one of its right angles, etc. Finally, the notion that the race will be fair alludes to equidistance. That reading of the problem using geometry along with existing habits to study the geometry of the microspace (i.e., at a scale much smaller to that of the human body) might suggest an initial diagrammatic modeling to represent the problem: Students could draw a rectangle and plot three points for the swimmers. This initial modeling decision can be quite consequential in affording opportunities to project mathematical sensibilities onto the work; it can be advantageous, from a mathematical point of view, to have rectangles of different sizes and aspect ratios, rectangles drawn free hand, and rectangles constructed with straightedges. The expectation is that mathematical sensibilities and mathematical practice will help organize superficial variations among those representations. Poincaré’s proposition that “geometry is the art of reasoning well from badly-drawn figures” suggests that to reveal the geometry, it would be valuable if the diagram had some inaccuracies to begin with (such as the strokes representing the sides of the pool not being exactly straight or the angle representing the corner of the pool not being exactly 90 degrees; see figure 2).

The drawing in Figure 2 represents the pool situation using geometry. Can students use it and what they know in order to make inferences about the situation? First of all, there are concrete answers that can be made available immediately: The distances between any point picked at will inside the rectangle can be measured with rulers, and the point may be moved so as to make the distances closer to each other. These answers can be revealed useless by virtue of the modeling choices: The distance that will be equal for one set of swimmer positions will not necessarily work for a different set of swimmer positions, and even if those positions were controlled, the particular measure of the distances among the points representing swimmer positions and prize location would unclearly prescribe how to locate the prize position in the actual pool. Thus, early modeling choices, using the tokens of intuitive geometry (vocabulary and imagery), might reveal the need for something else. In what follows I adopt the position of a teacher anticipating how one could think of the problem with a class: I consider my audience in terms of what they are expected to know (hence timing when and how I bring in what I know).

If we started from considering only two swimmers first, specifically the swimmer at the corner and a swimmer at one of the sides, it would be easy to see that the midpoint of the segment determined by the two swimmers is equidistant from them (and the pool context also suggests that would not be a reasonable place to locate the prize as such midpoint would not be in the pool!). But many other points in the pool would be equidistant from those endpoints. As students have learned to construct the perpendicular bisector of a segment, they might also know that this line is the locus of all points equidistant from the endpoints of the segment. And if they didn’t yet know that, this might be a good time for them to come to know it. It might help to ask questions like: Since the midpoint of the segment does not work, what would be a point inside the pool equidistant from the endpoints? Students might or might not bring the perpendicular bisector as a resource to think about other points. If they didn’t, one could ask how they would pick the point they are looking for, aiming to get explicit instructions. In our work with this problem in actual high school classrooms, we have seen students gravitate to the perpendicular bisector directly or to ideas germane to its construction (e.g., using the compass to make an arc from each of the endpoints and picking the point of intersection of the arcs). In case the students had thought of the perpendicular bisectors, the question, “how could we prove that a point chosen on the perpendicular bisector is equidistant from the endpoints?” might get to a partial conclusion of this simpler problem. In case the students had not thought of the perpendicular bisectors and instead constructed a single point, the question, “is there any other point that would work?” might move the discussion toward the end of characterizing all the points that would be reasonable locations for the prize.

Once that simpler problem has been solved, we can bring in the third swimmer. The ways of locating possible places for the prize in the simpler problem could help find possible locations for a prize that would be equidistant from the second and third swimmer. The question, “is there a spot that is equidistant from the first and the second swimmer and equidistant from the second and third swimmer?” could get students to think of the intersection between the two perpendicular bisectors. The question, “[how] do we know whether the first and the third swimmer are equidistant from the point of intersection of these two perpendicular bisectors?” is also relevant here and could bring awareness of the transitivity of congruence. In that event, we have found a solution to the problem, but we have not arrived yet at the statement of the theorem as no attention has been given to the midpoint of the hypotenuse; not even a right triangle is visible. Indeed, the midpoint of the hypotenuse and the intersection of the perpendicular bisectors of the legs of a right triangle are two different meanings (two different senses in Frege’s terminology) of the same referent. A virtue of this problem, and of this way of approaching the problem, is that it delays attention to the referent, making room, as we discuss below, for proof.  

While one solution to the original problem has been proposed, it is mathematically sensible to ask whether that is the only one. Furthermore, the fact that we have found the one by doing some things with the diagram instead of others, it is also sensible from a diagramming action perspective to ask whether we would get a different point had we made other choices. In some cases, the question may yield simple, even trivial, answers, and in other cases, the question could lead to compelling stretches of mathematical practice. For example, if instead of considering first the two points on one side and second the two points on the other side, we altered that order, would the solution change? Students might be quick to note that the intersection of the same two perpendicular bisectors is a single point, no matter which perpendicular bisector is drawn first. The other possible choice is, or can be made to be, however, less immediate.

What if the first two points used for the simpler problem did not include a swimmer at the corner but were points on the sides of the pool? Because the segment determined by those two points has not been drawn yet, thinking about their perpendicular bisector might not be immediately obvious to students. The simpler problem warrants thinking about it, as the midpoint of that segment would be one reasonable place to position the prize if the swimmers jumped into the pool from the sides. One solution would be the midpoint of that segment. Yet, the real world situation also suggests some discomfort with it. In Figure 3, we can note that diving toward the midpoint of the segment between them would require swimmers to form an acute angle with the side of the pool and that it would feel more comfortable if that angle was larger. Thus, a practical reason might justify asking where, other than at the midpoint of the segment between the swimmers, are all the points equidistant to them. Based on what was considered in the first solution, students would likely gravitate toward thinking of the perpendicular bisector too, but this observation about the angle might importantly seed a preference toward a solution farther from the midpoint of the segment. Because we are not merely interested in solving the problem but in constructing the materials that matter in the theorem at stake (and more, as all of this matters in understanding why the circumcenter is unique), that preference is desirable; it helps create the conditions for students to understand that the referent of all these constructions is unique through rational means, by reasoning well about badly-drawn diagrams.

Now it would be reasonable to bring in the third swimmer, diving from the corner. Again, one could consider two different perpendicular bisectors. Considering only one would provide a solution but leave open the question of whether choosing the other perpendicular bisector would provide a different solution. Figure 4 shows what this could look like in a diagram deliberately chosen to favor the posing of these questions. A deliberately chosen, badly drawn diagram would be one in which one could see the three different meanings of the location of the prize (three different construction procedures) as if they pointed to three different referents. 

A superficial appraisal of figure 4a would avail the question, “are we saying that there are three different points equidistant from the three swimmers?” As the last two solutions were found on the perpendicular bisector of AC, the first point found (X; see Figure 4a) might appear as the odd one out and allow the questions: “Is X really equidistant from A and C? What would need to be true about triangles AMX and CMX if AX was congruent to CX?” If available, the reciprocal property of the perpendicular bisector (if a point is equidistant from two points, it is on the perpendicular bisector of the segment formed by the points) could come in handy to infer that X should be on the perpendicular bisector of AC. If not, the conclusion could be reached by noting that triangles AMX and CMX should be congruent by side-side-side (see figure 4b). 

If not only Z and Y but also X are determined to be on the perpendicular bisector of AC, the students could be invited to draw the figure again, paying attention to locating X, Y, and Z. This is another case in which sense and reference are different; while one could talk about those as potentially different points, there would be no way of drawing them as distinct points that lie on a single line, even in a badly-drawn diagram!

However, one could still ask, “What about M?” If one now considers triangles, say BMR and AMR, they would need to be congruent by side-angle-side, making BM ≅ AM. Similarly, because BMS and CMS are congruent triangles, BM ≅ CM. This all would suggest M would also be in the intersection of all three lines and the only point that can be used to place the prize. X, Y, Z, and M would be one and the same point.

Back to the modeling approach

Badly drawn diagrams, such as those in Figures 4a, 4b, and 5, are doing things that formal, axiomatic geometry cannot do. By offering a diagrammatic interpretation of geometric concepts and relationships, they activate a source of intuitive feedback to the questions, assertions, and predictions that may come from the mathematical sensibility. This mathematical sensibility is one capable of endowing a badly drawn diagram with some properties and looking for it to produce other possible properties. This mathematical sensibility is one that will be ready to question the consistency of both classes of properties; and it is essential in order for reasoning with diagrams to model mathematically the relationships between real world objects and activities (including the activities of drawing, folding, and moving about in space) and the geometric representations of those objects and activities. As the narration above suggests, for the high school geometry class to have access to that mathematical sensibility not only well-chosen tasks are needed but also a teacher who is disposed to both ignore publicly what they know about geometry while embodying the dispositions to know that are part of the mathematical sensibility. In such a context, one could see the high school geometry class as a place where students are apprenticed into the practice of mathematicians by working with the intuitive ideas of geometry. High school geometry could be the place in which one learns to resolve the logical contradictions that arise from drawing bad diagrams and thinking with badly drawn diagrams. This modeling approach builds on earlier arguments for why students should study geometry. The importance of engaging in the practices of mathematicians is affirmed as a way to solve problems by making representations whose attributes are ascribed and verified rationally as opposed to empirically. Geometric content is valuable not because geometry is a preferred example of a mathematical system of postulates, theorems, and proofs, but because experiences managing space at different scales provide sources for representations (embodied, iconic, symbolic) that can help pose problems and provide feedback on thinking. Logical reasoning is called upon to support mathematical practice, as well as to confront the feedback from various representations. And the capacity to mathematically model experiences with shape and space, while contextualized in geometry, might support modeling in work contexts, where it can be useful to anticipate rationally the solutions to problems of managing space and shape.

References 

Bartocci, C. (2013). “Reasoning well from badly drawn figures”: the birth of algebraic topology. Lettera Matematica, 1(1-2), 13-22.

Frege, G. (1997). On Sinn and Bedeutung In M. Beaney (Ed.), The Frege reader (M. Black, trans., pp. 151-171). Blackwell. (Original work published 1892)

González, G. and Herbst, P. (2006). Competing arguments for the geometry course: Why were American high school students supposed to study geometry in the twentieth century? International Journal for the History of Mathematics Education, 1(1), 7-33.

Herbst, P., Fujita, T., Halverscheid, S., and Weiss, M. (2017). The learning and teaching of secondary school geometry: A modeling perspective. New York: Routledge.