GeT: A Pencil

Author: Stephen Szydlik

  • Drawing the Line: Constructions in Hyperbolic Geometry

    Introduction: Hyperbolic Geometry and Its Models

    Hyperbolic geometry resides in a unique but challenging position within the GeT curriculum, providing powerful opportunities for an instructor to enrich student understanding of geometry. First, it provides experiences with a significant axiom system distinct from Euclidean geometry. Second, by changing a single axiom (the parallel postulate), it provides a meaningful contrast to Euclidean geometry, thereby offering worthwhile insights into that more familiar world. Third, it provides visible support for a variety of essential Student Learning Outcomes (SLOs). Finally, it adds historical context to the lively story of 2000 years of geometry, which began with Euclid. 

    However, hyperbolic geometry presents a conundrum for GeT instructors. It can be intimidating to both students and instructors because it contains strange and hard-to-visualize theorems. How can we have parallel lines that have but one common perpendicular? What do limiting parallel rays look like? How can there be more than two lines passing through a point parallel to a third line? Hyperbolic geometry can appear esoteric, unreal, and unimportant when viewed as an abstract axiom system. A GeT instructor, particularly one without experience in hyperbolic geometry, might naturally choose to downplay the subject in the classroom, especially given the course’s many other essential topics.

    For me, the solution to this challenge entails combining several aspects of the GeT course curriculum: axiomatic models of hyperbolic geometry, Euclidean straightedge and compass constructions, and dynamic geometry software. These three ingredients combine to provide a powerful pedagogical tool for understanding hyperbolic geometry.

    Models of an axiom system are interpretations of essential undefined terms that satisfy the axioms. In the case of hyperbolic geometry, these terms include “point,” “line,” and “congruence.” When interpreted properly in the models, all of the axioms of hyperbolic geometry are satisfied, particularly the hyperbolic axiom: there exists a line ℓ and a point P not on ℓ such that there are at least two lines through P parallel to ℓ. A logical consequence is that in models of hyperbolic geometry, all of its theorems are valid. The models help us make sense of these theorems by allowing us to experience hyperbolic worlds in concrete ways. As such, we can use them as laboratories to explore conjectures and visualize theorems.

    The three models we briefly consider here are the Beltrami-Klein disk, the Poincaré disk, and the Poincaré half-plane:

    • In the Beltrami-Klein disk (K-disk henceforward), we interpret “points” to mean points interior to a fixed circle γ. “Lines” are open chords of γ.
    • In the Poincaré disk (P-disk), “points” are points interior to a fixed circle γ, while typical “lines” are open arcs of circles, interior to γ but orthogonal to γ.
    • In the Poincaré half-plane (Half-plane), “points” are points above the x-axis in the usual Euclidean plane, while typical “lines” are open semicircles above the x-axis, centered on the x-axis.

    If we define a distance metric and “congruence” in terms of that metric, then each of these interpretations satisfies the axioms for hyperbolic geometry, making it a model. In each of the models, the hyperbolic axiom, with its multiple parallels, becomes transparent (see Figure 1, or a dynamic version at https://tinyurl.com/szydlik22).

    Figure 1: Visualizing the hyperbolic axiom in the (l-r) K-disk, P-disk, and half-plane models.

    Euclidean and Hyperbolic Constructions and the Role of Dynamic Geometry Software

    Classical straightedge and compass constructions have been a part of the geometry curriculum since the time of Euclid. Their pedagogical utility is clear; they encourage the use of precise mathematical language by the constructor. They drive mathematical arguments; “How do you know that your construction works?” is a natural question to ask. Constructions provide students with experience in problem-solving and preservice teachers with opportunities to practice the curriculum they can expect to teach (see CCSSM, 2010). Indeed one of the essential SLOs is that students should “be able to carry out basic Euclidean constructions and justify their correctness” (SLO 8). Early in the GeT course that I teach, we spend a day or so developing basic Euclidean constructions and arguing their correctness. We bisect and copy angles, raise and drop perpendiculars, construct parallels and perpendicular bisectors, and, as a homework assignment, construct a circle passing through three noncollinear points.

    The Euclidean constructions and the hyperbolic models are inextricably linked. Because hyperbolic models are defined as objects in the Euclidean plane, any geometric structures in the models are entirely Euclidean. The interplay between the models and Euclidean constructions is bidirectional. We rely on Euclidean constructions when we wish to visualize a hyperbolic theorem in one of the models. On the other hand, when we execute a construction of a hyperbolic object in one of the models, we enhance our understanding of Euclidean constructions. For example, consider the challenge of constructing a hyperbolic line through two points in the half-plane. In Euclidean terms, this amounts to constructing the semicircle centered on the x-axis that passes through two points A and B that are above the x-axis. The steps required to complete this construction include finding the point O where the perpendicular bisector of segment AB meets the x-axis and constructing the circle centered at O that passes through A (and B). This is not a particularly difficult problem, but one can see that it does require problem-solving on the part of the students, as well as a facility with basic Euclidean constructions (see Figure 2, or the dynamic geometry version at https://tinyurl.com/szydlik22).

    Figure 2: Constructing the “line” through A and B in the half-plane model.

    Of course, constructions in hyperbolic geometry may require a significant number of steps. Consider the process of constructing the “line” through two points, A and B, in the P-disk. This involves finding the inverse point A’ for A, then constructing the circle passing through A, B, and A’ (see Figure 3, or the dynamic geometry version at https://tinyurl.com/szydlik22). This construction necessitates the following Euclidean steps:

    1. Let O be the center of the given Poincaré disk γ. Construct ray OA.
    2. Construct the line through A perpendicular to OA. Let B and C be the intersection points of this perpendicular with circle γ.
    3. Construct the tangent to γ at C (or D). This is the line perpendicular to OC at C.
    4. Let A’ be the intersection of this tangent and ray OA. A’ is the inverse of A relative to circle γ.
    5. Construct the perpendicular bisectors of segments AA’ and AB. Let E be the intersection of those perpendicular bisectors.
    6. Construct the circle centered at E with radius EA. By construction, this circle passes through A’ and B as well. The part of this circle interior to γ is our Poincaré line through A and B.
    Figure 3: Constructing the hyperbolic line through A and B in the P-disk.

    The steps in the construction of a P-disk line are entirely Euclidean, and carrying out this construction will support a student’s understanding of the relevant Euclidean constructions. However, completing these steps every time we desire a Poincaré line is time-consuming and can distract from a higher-order task that might use that line. This is where dynamic geometry environments (DGEs) such as GeoGebra offer a convenient solution. GeoGebra automates basic Euclidean constructions, eliminating much of the low-level straightedge and compass strokes (note that while GeoGebra is my DGE of choice, many other options are available, including Geometer’s Sketchpad, Cinderella, and Cabri).

    One especially useful feature of GeoGebra (and other DGEs) is the ability to create custom “tools,” or macros that automate a series of steps in a construction. For example, rather than constructing all of the chords, perpendiculars, tangents, and circles required to create a hyperbolic line in the P-disk, a user might use a GeoGebra “Poincaré Line” tool to simply plot two points in the disk, with the tool automatically drawing the required circle arc through the two points.

    The Hyperbolic Toolbox

    GeoGebra offers quick access to construction tools for Euclidean geometry, and I find it useful for my students to have access to similar tools in the hyperbolic models, beyond just the hyperbolic line tool described above. So, together we develop a toolbox of “essential” hyperbolic constructions in each model. We have tools to construct hyperbolic segments, rays, and lines, bisect angles, raise and drop perpendiculars, construct circles, and measure lengths and angles. Measuring is not a construction, of course, but automating the measurement process in our models is helpful, especially when we want to confirm congruence of segments or angles. With this full suite of tools, we can execute complex hyperbolic constructions in our models. Moreover, utilizing the dynamic nature of the software, we can drag the free points in any construction and see the effects on subsequent constructions. This allows us to bring the full power of the constructions to bear.

    In my GeT course, we apply the hyperbolic tools in three distinct ways. First, though I provide most of the tools to students, I ask them to develop some of the tools themselves. Specifically, students have the facility to develop tools to construct lines in each of the models, especially if provided with some necessary background in the case of the Poincaré disk. Raising perpendiculars is both an accessible task and a great problem-solving challenge for students.

    Second, we use the construction tools as building blocks for more extensive hyperbolic constructions. One exercise is to explore whether common Euclidean objects exist in hyperbolic geometry and if so, what properties they might possess. For example, while it is easy to construct a rectangle in Euclidean geometry, can we do the same in hyperbolic geometry? Where does the construction go wrong? What about a rhombus? The tools offer us opportunities to better understand the history of geometry as well, specifically in the attempts to prove the parallel postulate. Each such attempt to prove Euclid’s fifth postulate was doomed to failure. While identifying a flaw can be challenging in the abstract, carrying out the constructions in a hyperbolic model can be illuminating. In particular, the flaw in each proof will always appear in our hyperbolic models. Figure 4 illustrates one proof attempt by Farkas Bolyai (see Greenberg, 2008, p. 229). The dynamic sketch at https://tinyurl.com/szydlik22 provides some of the flavor of such an activity, though the students would typically carry out all the necessary constructions, rather than having them provided in “checkbox” form.

    Figure 4: Finding the flaw in a proof of the parallel postulate using the Klein disk model.

    Finally, we use our hyperbolic toolbox to motivate and to visualize the major theorems in hyperbolic geometry. For example, among the strangest ideas in that world is that parallel lines have at most one common perpendicular, and when it exists, the common perpendicular segment offers the shortest distance between the parallel lines. Giving students parallel lines in one of the models, having them find the (hyperbolic) shortest segment between the lines, and identifying the key properties of that segment offers rich opportunities for exploration (see Figure 4, or the dynamic sketch at https://tinyurl.com/szydlik22).

    Figure 5: Finding the shortest segment between parallel lines in the P-disk.

    Supporting Student Learning Outcomes

    Hyperbolic constructions offer opportunities for an instructor to address many of the essential SLOs in a GeT course in meaningful ways while also supporting teaching practices endorsed by the NCTM (2020) and others (e.g. Abell et al., 2017). The connections to several of these SLOs are clear: working within the models using DGE helps emerging teachers to develop their abilities to “[c]ompare Euclidean geometry to other geometries such as hyperbolic or spherical geometry” (SLO 9), to “carry out basic Euclidean constructions and justify their correctness” (SLO 8), and to “effectively use technologies to explore geometry and develop understanding of geometric relationships” (SLO 6). Moreover, when exploring hyperbolic geometry and constructions in general, students naturally gain experience in proof (SLO 1). With the availability of the hyperbolic toolbox, additional activities become feasible that support student development addressing other learning outcomes. For example, working with models can increase student understanding of the role of axioms in geometry (SLO 4). Asking students to find flaws in proofs of the parallel postulate enriches their knowledge of the history of geometry and strengthens their appreciation for Euclid (SLO 7). Indeed, though many prospective teachers may never encounter hyperbolic geometry in the curricula that they teach, exposing them to the beauty of hyperbolic geometry is, nevertheless, a worthwhile experience.

    Resources

    There are a wide variety of resources for instructors interested in further exploring dynamic constructions in the hyperbolic models. On my personal website, you can find GeoGebra files that include the full suite of tools associated with each of the three models discussed above. There, you can also access links to GeoGebra sketches that describe the development of those tools as well as examples of ways to use the tools in the classroom. Alternatively, a search through the activities at geogebra.org, perhaps using the names of the models as keywords, produces a wealth of resources. For non-GeoGebra users, the Cinderella software includes built-in construction tools for the Poincaré disk, and NonEuclid offers a standalone package of Poincaré disk tools as well. 

    References

    Abell, M., Braddy, L., Ensley, D, Ludwig, L., & Soto-Johnson, H. (2017). Instructional practices guide. Mathematical Association of America. 

    GeT: A Pencil. (2022). Student Learning Outcomes. Available at https://getapencil.org/student-learning-objectives/. 

    Greenberg, M. J. (2008). Euclidean and non-Euclidean geometries: Development and history (4th ed.). Freeman.

    National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics [CCSSM]. Washington, DC: Authors. Available at http://www.corestandards.org/Math/. 

    National Council of Teachers of Mathematics [NCTM] (2020). Standards for the preparation of secondary mathematics teachers. Available at https://www.nctm.org/uploadedFiles/Standards_and_Positions/NCTM_Secondary_2020_Final.pdf. 

  • People and Clubs: An Axiomatic System

    “One must be able to say at all times – instead of points, straight lines, and planes – tables, chairs, and beer mugs. “David Hilbert [4]

    Axioms serve as fundamental bricks in the foundations of mathematics. Given a small collection of statements assumed to be true, a universe of subsequent truths may spring forth, grounded in those assumptions and constructed using mathematical logic. Change the axioms even slightly and you change that mathematical world.  In geometry, the collective reconsideration of Euclid’s Fifth Postulate and the alternatives offered up in its place led to the “strange new universe” of hyperbolic geometry created by János Bolyai and others. [2]

    Students at the onset of a GeT course typically have at least a passing understanding of proof, and many have taken a university course focusing on sets, logic, and proof structure.  However, students generally have little experience with axioms beyond the definition of the term.  Nevertheless, developing a sense of axioms and axiomatic systems is essential to understanding the nature of mathematics. A GeT course provides a natural place in the undergraduate curriculum to address this important concept.

    One problem that I have used in my GeT course for many years to introduce the idea of axiom systems was developed by a group of mathematicians at the University of Wisconsin in the early 1990s:

    The axioms here are essentially those for a finite affine geometry with “people” and “clubs” playing the respective roles of points and lines. Notably, Axiom H4 is a disguised version of Playfair’s postulate, logically equivalent in Neutral Geometry to Euclid V.  Disguising the axioms serves a pedagogical purpose.  Students’ familiarity with Euclidean notions of points and lines comes with some baggage;they often hold tacit assumptions about the relationship between points and lines (e.g., lines are “straight,” and if two lines look like they intersect, then they do, at a point).  These assumptions tend to impede their ability to grapple with the consequences of the axioms for geometry on a purely logical basis.  Framing the problem in terms of “people” and “clubs” counteracts that tendency and encourages students to prove statements by relying solely on the axioms.

    I should note that while the GeT course that I teach centers around axioms and the Euclidean archetype [3], this activity neither requires nor expects students to possess a deep understanding of axioms.  In fact, I typically use it as a “first day” problem to introduce them to axioms and the desirable properties of axiom systems.  I have also used the problem successfully in a lower-level geometry course for non-majors.  On the other hand, the problem allows for further deep investigation; the clubs in the town of Hilbert have a rich structure!

    “People and Clubs” offers students opportunities to model, to construct viable arguments, and to reason abstractly, processes highly valued by our community and advocated by educators. [1] Students model the problem in several different ways.  Some use letters (or numbers) to represent the n people in Hilbert, and sets of those letters as clubs (e.g. “BDE” as a three person club).  Others provide visual representations with people as dots, containing ovals showing club membership.   

    Regardless of representation, however, students are generally able to dispose of the n=2 population rapidly since determining its viability does not require the use of Axiom H4.  Students tend to find that fourth axiom thorny, in part because the statement involves both the existence and uniqueness of a particular club.  When they understand this language, they are better able to see the logical difficulties with a 3-person town.  In our class discussion, I challenge them with several different examples of impossible 3-person systems (see the picture below).  For example, the system consisting of strictly two person clubs illustrated in the third example violates the “existence” clause of Axiom H4: Given club AB and person C, there is no club that contains C that has no members in common with AB.  Challenging students with examples such as this encourages them to make explicit their use of the axioms—a skill that becomes useful when investigating larger Hilbert populations and that is critical when considering the extreme cases of n=0 and n=1. 

    Perhaps the most interesting situation that we consider in class is a 5-person Hilbert. A careful analysis of the smaller population situations pays off here because a direct proof of the impossibility of the n=5 population requires several cases.  In my classes, students are often initially uncertain about whether a town of this size is possible.  Many believe that a system consisting entirely of 2-person clubs is possible, as it is in an n=4 person town.  Others may think that at least one of the other 5-person systems is valid.  The identification and disposal of the cases offers opportunities for an intellectually satisfying discussion.  For example, a system of 2-person clubs violates the “uniqueness” clause of Axiom H4:  Given club AB and person C, there are two clubs (CD and CE) that contain C and have no members in common with AB.  The table below illustrates some (but not all!) of the possible 5-person club systems and where they go wrong:

    Potential Club System for n=5 Axiom Violation
    AB, AC, AD, AE, BC, BD, BE, CD, CE, DE(H4) “Uniqueness”
    ABC, AD, AE, BD, BE, CD, CE, DE(H4) “Uniqueness”
    ABCD, AE, CE, BE, DE(H4) “Existence”
    ABCDE(H3)

    Discussion of the “People and Clubs” problem typically takes about one class period, though I often ask my students to spend an evening considering the 5-person Hilbert situation.  The activity works well as a standalone problem, and GeT instructors who do not teach an axiom focused course may wish to stop here.  On the other hand, a deeper analysis of the problem in a GeT course can pay dividends throughout the semester in several ways.  First, it can help students identify important general properties of axiomatic systems.  For example, adding the axiom “(H5): There are five people in Hilbert,” quickly leads us to the notion of consistency.  On the other hand, “(H5): There are not five people in Hilbert,” creates a redundant axiom system since we can prove that statement.

    Second, the problem offers meaningful examples of models of an axiomatic system.  This is particularly helpful when discussing independence of axioms: an axiom “(H5) There are four people in Hilbert” is independent since we can find models of (H1)-(H4) where the (H5) holds (the n=4 Hilbert case), and other models (e.g., n=0, 1) where (H5) fails.  Finally, “People and Clubs” supports further mathematical exploration and conversation. The trivial case n=0, for example, provides an occasion to highlight the importance of precise language and interpretation of quantifiers.  For deeper exploration, I will often have students try to prove (or disprove) statements that rely heavily on Axiom (H4).  For example, if clubs X and Y have no members in common and Y and a third club Z have no members in common, then can we conclude that X and Z have no members in common?  A series of questions such as this, perhaps posed as true/false statements, affords opportunities for students to practice making arguments within the system and in refining general proof skills.  Other potential questions are included below.

    References

    1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
    2. Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries: Development and History. W.H. Freeman and Company, 2008. 
    3. Grover, B.W., Connor, J. Characteristics of the College Geometry Course for Preservice Secondary Teachers. Journal of Mathematics Teacher Education 3, 47–67 (2000). https://doi.org/10.1023/A:1009921628065.
    4. Reid, C. (1970). Tables, Chairs, and Beer Mugs. In Hilbert (pp. 57-64). Springer, Berlin, Heidelberg.
  • Member Highlight – Interview with Stephen Szydlik

    Member Highlight – Interview with Stephen Szydlik

    ​​Four questions with Stephen Szydlik, Professor of Mathematics at the University of Wisconsin Oshkosh

    • What is special about your GeT course? My GeT course is axioms-based, but I blend the rigorous mathematics with opportunities for exploration and active learning. I try to give my students opportunities to model authentic mathematical behaviors: investigation, conjecture, counterexample and logical argument. I emphasize proof, but we spend lots of time working with the hyperbolic models, especially using dynamic geometry software.
    • Who are your students? The students in my GeT course are almost entirely preservice secondary teachers. Most come from within Wisconsin, and many have never been outside the state. I try to encourage them to travel whenever possible, and just as stepping outside their culture provides them with a unique perspective on their homes, so too does our investigation of hyperbolic geometry offer new insights into the Euclidean geometry that they will teach. (At least that’s my goal!)
    • What are you most interested in learning/achieving through participating with the GeT: A Pencil community? I’ve already learned so much from the GeT: A Pencil community about the many different ways of structuring a GeT course! I am most interested in learning how well we prepare our future teachers and if there are ways that we can better serve them.
    • What is your favorite book you have read recently? I just finished Fredrik Backman’s novel Us Against You. At its most basic level, it’s a story of an economically depressed northern town struggling to heal after an act of violence. The characters are complicated and richly drawn, and the story, centered around the reemergence of the town’s hockey team, is riveting and emotional. It was a rewarding read!

  • A GeT Course “Classic”: The Euclidean Archetype

    We are all members of the Euclidean Archetype workgroup. As we summarized in our report, a GeT course organized around the Euclidean archetype will focus on the axiomatic development of fundamental principles of geometry. Informed by the spirit and organization of Euclid’s Elements, this course emphasizes mathematical precision, rigorous proof, and clear communication. We have all taught geometry with varying amounts of experience. We agree on many goals that a course should have but each of us prefers a different balancing of the ingredients. What follows is our discussion of the essential elements and the plusses and minuses of teaching a class using the Euclidean Archetype. 

    THE ESSENTIAL COMPONENTS 

    SS: The Euclidean archetype centers on axiom systems, and any GeT course following this framework should emphasize that structure: precise language, identification of agreed-upon undefined terms and axioms, and the development of the theorems of geometry from those foundations. A worthy highlight of this course is the independence of the parallel postulate. This requires some work, including a careful development of the concepts of models and independence and an exploration of alternative axiom systems for Euclidean geometry (including Euclid’s axioms and some other modern system).

    SC: I agree. I would add that the structure naturally leads to an emphasis on proof writing. I find it useful to spend some time in a simpler axiom system such as an incidence geometry to enable students to practice writing proofs with fewer subtleties and issues.

    NM: I think there are two key components here, that don’t necessarily have to be combined, but often are. This is sometimes referred to as the Euclidean Axiomatic archetype, and the two components are Euclidean Geometry and an axiomatic approach. You could have a course focused purely on Euclidean Geometry; you could have a purely axiomatic geometry course; and putting them together, you could have an axiomatic geometry trying to get at the main ideas of Euclidean geometry. There are certainly courses that mostly do one of these without the other. For example, some books focus on explorations of Euclidean geometry using dynamic geometry software without an axiomatic approach. On the other hand, some completely axiomatic courses don’t get very far into Euclidean geometry because it takes so long to prove elementary facts about incidence geometry and betweenness proceeding carefully from elementary axioms. Probably to be considered part of the Euclidean Axiomatic archetype, you need to explore some of both. Getting to both probably requires that we broaden both pieces, though. As SS notes, we will want to talk about models and independence, which will require us to work, at least a bit, with some non-Euclidean geometries; and to get to the interesting parts of Euclidean geometry, we will probably have to move away from the idea of proving absolutely everything from a purely axiomatic standpoint.

    OTHER TOPICS TO INCLUDE

    SS: Proving the independence of the parallel postulate opens up the world of non-Euclidean geometry, and exploring the seemingly strange world of hyperbolic geometry is a natural branching off point for this archetype. It provides students with an alternative axiom system to consider and by developing its major theorems students gain a stronger understanding of the more familiar Euclidean world. The archetype also provides an opportunity to study Euclidean straightedge and compass constructions. Careful development of these tools provides significant payoff if the instructor chooses to investigate models of hyperbolic geometry in some detail. Dynamic geometry software can be a powerful tool in this investigation.

    SC: Compass and straightedge constructions are foundational in Euclidean geometry. Students can use these to make conjectures, prove theorems, and develop geometric intuition. Students can also consider models where various axioms fail to hold, such as geometry on the sphere, or on the Cartesian plane using the taxicab metric to measure distance.

    NM: I agree with all of these ideas. Spherical geometry is also natural to look at in the context of parallel lines– with spherical, Euclidean, and hyperbolic geometry, we have cases with no, one, and more than one line(s) through a point parallel to a given point. I think spherical geometry is more accessible to students since they already know what a sphere is. There is also a sense in which spherical, Euclidean, and hyperbolic geometry are the building blocks for all 2 dimensional geometries. There is dynamic geometry software for each of these, and I also like to have students work with physical models.

    ADVANTAGES

    SS: I love the structure of this archetype. Building geometry from a set of axioms and undefined terms allows students to see a logical development of the subject. Even a short exploration of the Elements gives students an appreciation for the monumental achievement of Euclid while helping them recognize the need for precise language and rigorous proof. In addition to focusing on strengthening students’ logical reasoning abilities, the archetype also offers natural opportunities to build in a historical examination of geometers, from Thales to Saccheri to Bolyai to Riemann to Hilbert, as well as many others in between. I believe that a strong foundation in the axiomatic structure of geometry is an essential component of the preparation of future teachers of the subject.

    SC: Euclidean Geometry has long been a model of deductive reasoning and teaching students to write proofs. Teaching it also presents a great opportunity to incorporate the humanities (art, history, western civ.) into the math curriculum. Most exciting part of teaching it for me was following the long and technical journey through Neutral geometry not allowing students to assume familiar results such as 180 degrees in a triangle. When, finally we bring in the Euclidean Parallel postulate, the parallel projection theorem, similar triangles, the Pythagorean Theorem, and trigonometry immediately enrich the study. Finally, it is natural to discuss practical applications.

    NM: Geometry has long been a place in the mathematics curriculum where logic is discussed in a mathematical setting. I don’t think there is a better setting than a geometry class to get students thinking about the roles of axioms, definitions, and theorems, and to start thinking about metamathematical ideas about when statements are unprovable in a given system.

    DRAWBACKS

    SS: With its emphasis on an axiomatic development of geometry, this archetype does not as naturally lend itself to applications or pedagogical conversations as some other archetypes might. Moreover, Euclid’s axioms have little to say about geometric transformations, an important component of the Common Core State Standards for Mathematics. However, these topics could be included with careful planning by the instructor.

    SC: Preservice teachers need additional perspectives, extended time with transformational geometry, and opportunities to do the kind of exploration emphasized in the common core. It is possible but much more challenging to include these features in a Euclidean course.

    NM: One big drawback of a purely axiomatic approach is that there isn’t an axiomatization of Euclidean geometry that is fully complete and rigorous that is at the appropriate level for most undergraduates. If we use something like Hilbert’s axiomatization, we end up spending a lot of time giving fairly technical proofs of trivial results. Actually, Euclid’s treatment is still one that is at about the right level for most students, but it does make some unstated assumptions. The other piece that this approach usually leaves out is the opportunity for students to explore and make conjectures before trying to prove them, which is another giant piece of doing mathematics that geometry courses are especially well suited for. That’s why I tend to structure my courses around the experiencing geometry archetype, but for all the reasons we have discussed, I almost always include a section of the course structured around the axiomatic Euclidean archetype. One way to do this is to spend several weeks in the middle of the course having students prove basic theorems of neutral geometry from a simple four axiom system.

    Steve Cohen is Associate Professor of Mathematics at Roosevelt University.

    Nat Miller is Professor of Mathematical Sciences at the University of Northern Colorado.

    Steve Szydlik is Professor of Mathematics at the University of Wisconsin Oshkosh.