GeT: A Pencil

Author: Get: A Pencil Admin

  • GeT Course Student Learning Outcome #2

    Evaluate geometric arguments and approaches to solving problems.

    Geometry courses provide a natural setting for students to reflect on their reasoning, share their reasoning, and then critique the reasoning of their peers. If students only ever see correct arguments given by teachers and textbooks, they may not learn how to critically evaluate arguments. The ability to evaluate other people’s arguments is an important real-world skill that is related to, but separate from, the skill of proof writing. 

    Critiquing reasoning is a competency that needs to be practiced in order to improve and is an essential skill for future geometry teachers. Students should have opportunities to critique reasoning throughout a geometry course. This can take on many forms–critiquing their own or other students’ proofs, working together in groups to solve a problem, classroom discussions of problems or proofs, posing problems that lead to student disagreement, and learning about the geometric thought process in the Van Hiele Levels. 

    Providing opportunities for students to critique geometric reasoning is also important for understanding nuances in geometric definitions [see SLO 5] and in geometric notation. An essential opportunity for students to practice critiquing reasoning is when GeT students present their reasoning and proofs to the class, with the instructor modeling and moderating positive and negative feedback and depth of questioning. Some instructors have also found it valuable to introduce others’ arguments that could arise in high school geometry contexts, such as sample student proofs or video approximations of secondary teaching situations. Instructors can position GeT students in the role of a teacher, rather than another student, and invite broad and diverse observations about “students’” thinking. These discussions allow GeT students to practice critiquing reasoning while simultaneously deepening their understanding of secondary geometry content [see SLO 3].

    While some of this reasoning will be in the form of formal proofs, other reasoning will be much more informal. Class discussions and solving problems in groups can be great opportunities for students to practice critiquing the work of others. If groups are solving non-routine problems together, there are bound to be many opportunities for students to discuss their reasoning and to listen to and evaluate the reasoning of their peers. Likewise, whole-class discussions are further opportunities for students to hear and evaluate the reasoning of their peers. In an inquiry-based geometry classroom, one of the goals of the instructor is likely to be to create an environment in which students feel supported in sharing their reasoning and their thoughts about other people’s reasoning in a supportive way. 

    If a classroom environment is achieved in which people feel safe to share their reasoning and their thoughts about other people’s positions, then genuine disagreements may arise in the class. When this happens, this can be one of the situations that is most conducive to deep learning. When students feel that they have an emotional stake in the outcome of a discussion, they start paying deep attention to arguments on both sides. Instructors can look for problems to pose that are likely to lead good students to disagreements. Non-Euclidean geometries [see SLO 9] are particularly fertile for leading to productive disagreements, as everything there will be new and unfamiliar, and it will take some time to reach an agreement as to how to proceed. For example, students can try to decide which of Euclid’s Postulates and/or Propositions [see SLO 7] are true in a new geometry or what familiar definitions [see SLO 6] give rise to in other geometries. Any situation where students are making conjectures and trying to evaluate if they are true will lead to opportunities for students to come up with competing ideas that they will need to resolve. 

    Interestingly, GeT instructors anecdotally have reported that a growing number of college students are exhibiting gaps in their geometric understandings and convey that students in their college classrooms sometimes struggle with visualizing relationships among quadrilaterals and have difficulties characterizing them. The Van Hiele levels are levels of learners’ geometric thinking and understanding (Mason, 1998). The five sequential levels include Visualization, Analysis, Abstraction, Deduction, and Rigor. This model of geometric learning posits that students at all levels will move through these different levels each time they encounter a new geometric subject. Although we expect preservice teachers to reach a high geometric thinking level (level 4 or 5), students who enter a high school geometry class typically perform at the lower levels. Therefore, it is recommended that GeT instructors create opportunities for preservice teachers to critique reasoning at various thinking levels. While it is natural for group activities to provide opportunities for the analysis of reasoning, the use of individual assignments can also be useful. For example, the use of end-of-class self-reflection assignments can provide GeT instructors feedback regarding gaps in student understanding or provide evidence of creative thinking and insightful connections.

    Some other ways that instructors have implemented this standard in their classrooms include: 

    • Grading mock proofs on a test;
    • Having students create rubrics for an assignment and evaluate their own work;
    • Looking at possible K12 classroom activities and asking students to critique them and to discuss them in reference to their own future teaching; and
    • Doing a jigsaw, pair/share, or speed dating activities discussing proof.  

    Regardless of the form that critiquing takes, it is an essential aspect of a GeT course as it helps students think critically, improve their reasoning skills, learn how to develop solid mathematical arguments, and become better mathematicians.

    References

    Mason, M. (1998). The Van Hiele levels of geometric understanding. In L. McDougal (Ed.) The professional handbook for teachers: Geometry (pp. 4-8). McDougal-Littell/Houghton-Mifflin.

  • GeT Course Student Learning Outcome #7

    Demonstrate knowledge of the history and basics of Euclid’s Elements and its influence on math as a discipline. 

    Euclidean geometry is named after Euclid, the Greek mathematician who lived in Alexandria around 300 BCE. Euclid put together what was known at the time about Euclidean geometry into the thirteen books of The Elements. In The Elements, Euclid sets out a sequence of definitions, postulates (axioms for geometry), common notions (axioms common to all mathematical subjects), and propositions (theorems derived logically from the preceding materials). It is the “oldest extant large-scale deductive treatment of mathematics.” For most of the 2400 years since it was written, it was considered to be an essential text that any educated person would have studied; it is only in the last 150 years that this was no longer true. Likewise, for most of that time it was considered to be the gold standard of mathematical rigor; it is only in the last 150 years that its rigor has been surpassed.

    Many mathematical disciplines can be illuminated by considering their histories, and this is especially true of geometry. Students need to know this history to place modern ideas about proof into context. A geometry class is likely where students will first encounter the notion of axiomatic proof, and for some of them it may be their only encounter with it. (See SLO 4 on Axioms, Theorems, and Models.) Euclid’s first three postulates also set the stage for constructions using straightedge and compass. (See SLO 8 on Constructions.)

    One particularly interesting piece of this history concerns Euclid’s fifth Postulate: “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Over the years, this postulate was considered inferior to the others, and many mathematicians tried to prove it from the others. It was only in the early 19th century that it was discovered that there is a geometry in which the other postulates are true, but this one is false. This is how hyperbolic geometry was discovered. (See SLO 9 on non-Euclidean geometry.)

    Most students will not have seen Euclid’s 5th postulate in its original form but would likely know something more like Playfair’s postulate that given a line and a point not on the line that there is a unique line through the point parallel to the given line. Proving the logical equivalence to Euclid’s 5th postulate is a good homework problem.

    One question that remains relevant to modern geometry teachers is to what extent a geometry course should cover Euclid’s actual writings and methods versus to what extent they should be replaced by simplified treatments (as is done in many high school geometry books) or by more technical rigorous methods (such as those based on Hilbert’s Foundations of Geometry). As discussed above, for most of the time since it was written, almost all educated people studied The Elements. The debate about what treatments of geometry might be better than Euclid for modern students started in the mid-nineteenth century. There was enough debate that, in 1879, Charles Dodgeson, the mathematician better known as Lewis Carroll, published a book entitled Euclid and his Modern Rivals, which argued that Euclid’s treatment of geometry was superior for teaching students than any other treatment then proposed. This debate continues to this day.

    One reason to consider teaching Euclid’s original proofs, besides their historical interest, is that they are generally at the right level of sophistication for students; more modern treatments that are considered more rigorous might also be too complicated for students. One theory of cognitive development espoused by Jean Piaget among others states that “ontogeny recapitulates phylogeny,” that individual development often follows a similar path to historical development of a subject. Given the outsized role that Euclid’s work has played in the historical development of geometry, it is not surprising that it might be at just the right level of sophistication for many students.

    Looking at Euclid also makes it natural to consider where Euclid’s treatment may have gaps. For example, Euclid’s first postulate says that it is possible “to draw a straight line from any point to any point,” but if you look at the way he uses it, he really assumes that the line drawn is unique. Looking at his proofs through the lens of a geometry where any two points can be connected but not necessarily uniquely, such as on the sphere, makes this issue immediately clear.

    Euclid also introduces the idea of superposition to prove the Side-Angle-Side congruence condition for triangles. This argument was criticized because of the assumptions his proof makes about transformations without stating them explicitly. This can lead into a discussion of the transformations approach to Euclidean Geometry. 

    _________________________

    1Wikipedia, “Elements.” Retrieved January 27, 2022. Also found in other sources such as the preface to The Delphi Collected Works of Eucid (Delphi Classics, 2019).

  • GeT Course Student Learning Outcome #8

    Be able to carry out basic Euclidean constructions and to justify their correctness.

    Geometric constructions support the curriculum in a GeT course and the development of mathematical thinking in several essential ways: 

    • they provide a natural place for making a mathematical argument; 
    • they encourage practice in using precise mathematical language when describing a construction;
    • they provide the students a sense of where assumptions in building mathematical systems come from;
    • they provide openings for discussion of the historical development of geometry, especially the work of Euclid;
    • and they give students experience with the curriculum they will be expected to teach.

    For these reasons, constructions remain an essential part of any GeT course curriculum. It should be noted that within the context of geometry, the term “construction” most often refers to traditional straightedge and compass constructions, and this is generally the assumption that we make here. However, the term can be used more generally to include other tools and manipulatives such as paper-folding (e.g. using origami or patty paper) or MIRAsTM. In this context, using an external tool for the creation of an ideal geometric object whose success is argued rigorously might be reasonably considered a “construction.” Indeed, the Common Core Standards for School Mathematics includes the following in its “Congruence Standard”:

    G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

    A GeT instructor giving even a cursory treatment of constructions should consider including the constructions listed above in CCSSM G-CO.12 in their course. Having students discover the constructions themselves can be a powerful mathematical experience; having them present the constructions to the class offers opportunities for practice in using mathematical terminology and in making rigorous and logical arguments. The question, “How do we know that this construction is correct?” arises as a matter of course. In this way, constructions naturally reinforce SLO #1 (Proof).  

    Constructions (especially those using straightedge and compass) support SLO #7 (Euclid’s Elements) as well. Not only are many of the “standard” constructions listed above given in Book I of the Elements, they also illustrate the discipline of mathematics as put forth by Euclid. The standards Euclid used to build geometry form the basis for the current standards of rigor used in mathematics. In building geometry, Euclid set out to establish results, many of which were known during his time, by starting with as few assumptions as possible. By viewing geometric results as constructions from two simple tools, namely a straightedge and a compass, Euclid was able to create the foundation on which all his propositions rest. Because of his approach, Euclid had to rely heavily on the power of logical reasoning. Moreover, he needed a good grasp of concepts/definitions in order to convince himself and others that his constructions were able to do what they were purported to do. Employing Euclid’s approach to building geometry allows us to view a field of mathematics (geometry, algebra, analysis, etc.) as a unified whole, and it has provided us with numerous beautiful results, many of which have found practical applications in the world.

    Instructors looking for ways to incorporate transformations (SLO #10) into their GeT course will find that constructions offer a useful pathway, as the fundamental transformations (reflection, rotation, translation) can be represented as straightedge and compass constructions. In addition, for those GeT courses that specifically emphasize transformations, constructions offer opportunities for mathematically rich explorations. As an example, an instructor could ask students to construct the composition of two reflections of a triangle across distinct lines (either parallel or intersecting). Identifying the single rigid motion that has the same effect requires careful construction, precision of language, and the use of cases. Nevertheless, the problem is an accessible one, and it can lead naturally to a discussion of the group of isometries.

    Finally, for an instructor who intends to include the exploration of models of non-Euclidean geometry (SLO 9) in their course, work with Euclidean constructions is time well spent. Since many of our models of hyperbolic and spherical geometry reside in Euclidean geometry, constructions in those models rely on Euclidean counterparts. For example, constructing a “line” through two points in the Poincare half-plane involves finding a perpendicular bisector of a Euclidean line. With the benefit of dynamic geometry software (See SLO 6), even some of the complex constructions in non-Euclidean geometry become accessible, once one understands the fundamental Euclidean constructions.

  • GeT Course Student Learning Outcome #10

    Use transformations to explore definitions and theorems about congruence, similarity, and symmetry.

    Typically, there are three different approaches from which instructors can choose in order to embed transformation geometry learning outcomes into a GeT course. Instructors may choose to teach a GeT course using a formal transformation approach; they may choose to include a dedicated transformation unit within a course that has a predominantly traditional Euclidean focus; or they may integrate transformation approaches and concepts throughout the course offering. 

    There are two main types of transformations that arise in GeT courses: isometries, also known as congruence transformations (mappings that preserve both angle measure and segment length) and similarity transformations (mappings that preserve angle measures and proportionality of segment lengths). The set of isometries of the plane form an infinite non-Abelian group. Moreover, each element of the group can be classified as the identity transformation, a reflection, rotation, translation, or a glide reflection, and the composition of at most three reflections can be used to generate the other isometries. Depending on abstract algebra prerequisites, instructors may choose to highlight and formalize group properties and connections to finite symmetry groups. 

    It is recommended that GeT instructors build on familiar function notation when introducing transformations. This may lead to a better understanding of sequences of isometries as compositions of isometries. This notation may also make it easier for students to use rigid motions to express symmetry. GeT instructors can take advantage of looking at proofs through multiple approaches (Euclidean, analytic, transformational) to deepen students’ understanding of specific theorems. For example, students can be prompted to compare other strategies after proving the base angles of an isosceles triangle are congruent by using the concept of symmetry. Some GeT instructors may begin with an informal approach to the understanding of reflections and rotations and extend the concepts of symmetry to study geometric shapes using transformations (translations, rotations, and reflections) and combinations of them. The definition of congruence is then conveyed in terms of rigid motions and the definition of similarity in terms of dilations and rigid motions. The notion of a translation typically will be introduced with directed line segments and a rotation with directed angles, though some instructors might take the opportunity to more deeply explore the concepts in terms of vectors, matrices, and coordinate geometry.

    Instructors may instead choose to begin solely with sequences of reflections, which can be used to generate all other isometries of the plane. Each line of the plane is associated with a reflection that satisfies two properties: (1) every point on the line is fixed by the transformation and (2) the line is the perpendicular bisector of the segment connecting any point not on the line and the point’s image under the transformation. By exploring the images of points and figures resulting from sequences of reflections about parallel and intersecting lines, GeT students can discover and establish relationships with translations, rotations, and glide reflections. Instructors may choose to also have students explore, informally, orientation preserving/reversing properties and the aspects of the group structure (associativity of composition, existence of identity and inverses, and non-commutativity).

    GeT instructors have reported that students sometimes struggle with understanding fixed point properties of transformations. A case in example is when a segment is rotated a specified number of degrees about a center of rotation when the center of rotation (the fixed point) is not on the segment. Even though the center of rotation is specified, some students often choose one end-point of the segment as a center of rotation and use it as the fixed point. Using activities and technology that allow students to experience multiple examples of such properties can be included in a GeT course (see SLO 6 on use of dynamic geometry software) to clarify such misunderstandings. Douglas and Picciotto (2018) provide a guide and activities for transformational proof in high school geometry for teachers and curriculum developers. They note that The Common Core State Standards for Mathematics (2010) only require using transformations to justify the triangle congruence and similarity criteria (e.g., SAS, AA) and do not specify transformational proofs beyond this. However, Douglas and Picciotto recommend a high school geometry course using both traditional Euclidean and transformational approaches, beyond congruence and similarity criteria. It is important that prospective teachers are exposed to multiple approaches so that they not only are prepared with a solid background in geometry content but also able to make informed decisions regarding the choices they make in their own classrooms.

    To prepare GeT students for transformational proofs beyond triangle congruence and similarity criteria, the design principles of St. Goar and Lai (2021) may be useful. They suggest emphasizing the bidirectionality of the definition of congruence (or similarity), highlighting that transformations act on the entire figure (or plane), and encouraging GeT students to explain how and why a prescribed transformation necessarily maps one figure to another as part of their proving activity. Instructors may also choose to compare definitions based on properties of transformations with other definitions (see SLO 5 on role of definitions). For example, one could define a kite as a convex quadrilateral for which a diagonal is a line of symmetry and prove properties about its congruent sides and congruent angles that are used as definitions in a traditional approach. Other opportunities to learn transformational geometry include explorations of tessellations of the Euclidean plane as well as explorations of transformations in non-Euclidean geometries (see SLO 9 on non-Euclidean geometry).

    References

    Douglas, L. & Picciotto, H. (March 2018). Transformational proof in high school geometry: A guide for teachers and curriculum developers. Retrieved from https://www.mathedpage.org/transformations/proof/transformational-proof.pdf.

    National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards. Washington, DC: Authors.

    St. Goar, J., & Lai, Y. (2021) Designing activities to support prospective high school teachers’ proofs of congruence from a transformation perspective. PRIMUS. doi: 10.1080/10511970.2021.1940403

  • GeT Course Student Learning Outcome #3

    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.

    While there are students who do not plan to teach geometry in Geometry for Teachers (GeT) courses at most institutions, it is required for those who will become secondary math teachers. To be a good secondary geometry teacher, one must understand the content, know the best practices for teaching the content, and be able to reflect on one’s teaching. 

    Although high school geometry is described as “devoted primarily to plane Euclidean geometry, studied both synthetically (without coordinates) and analytically (with coordinates)” (NGA & CCSSO, 2010, para. 2), for many reasons, students in the U.S. often enter a GeT course with varying levels of knowledge of Euclidean geometry. As GeT instructors, it is our job to fill in the students’ knowledge gaps so they are prepared to teach secondary geometry. However, due to limited time in a GeT course (usually one semester) and GeT instructors’ varied preferences in content selection, it is not practical to suggest a list of geometry topics to be covered in a GeT course. Thus, the GeT course should focus on helping students understand essential mathematical practices and develop problem-solving skills that can be translated to any geometry topic. 

    The Standards for Mathematical Practices are the most important piece of the Common Core State Standards for Mathematics (CCSSM), which “describe varieties of expertise that mathematics educators at all levels should seek to develop in their students” (NGA & CCSSO, 2010, para. 1). Even though some states have moved away from using CCSSM and have developed their own state standards, their new state standards typically include these eight practices or something similar to them. These practices form the foundation for good mathematics teaching. They are:

    1. Make sense of problems and persevere in solving them. 
    2. Reason abstractly and quantitatively. 
    3. Construct viable arguments and critique the reasoning of others. 
    4. Model with mathematics. 
    5. Use appropriate tools strategically. 
    6. Attend to precision. 
    7. Look for and make use of structure. 
    8. Look for and express regularity in repeated reasoning. (NGA & CCSSO, 2010) 

    Furthermore, these practices provide the structure for mathematical problem solving, and any GeT student can benefit from becoming a better problem solver. We also want pre-service secondary geometry teachers to be able to model these practices in their future classrooms so GeT instructors should model these in our own classrooms.

    All GeT instructors need to be aware of teacher preparation standards that have been created to help prepare secondary geometry teachers (Table 1) and to incorporate them in their GeT course designs in a way that fits their teaching agenda. Many different professional organizations (e.g., AMTE and NCTM) contributed to these standards and suggested what faculty should be doing to prepare better secondary mathematics teachers. GeT instructors should also be aware of the national and state curriculum standards (Table 1) and introduce them to GeT students so that they can start to become familiar with the standards that they will teach. The majority of states in the U.S. have adopted the CCSSM for their K-12 schools (see this map), and if your state does not use CCSSM, it is best to Google “State K-12 Mathematics Standards.” 

    Table 1

    Resources for Standards

    StandardsIssuing Organizations
    Teacher Preparation StandardsThe Mathematical Education of Teachers II (2012)Conference Board of the Mathematical Sciences
    Standards for Preparing Teachers of
    Mathematics (2017)    		Association of Mathematics Teacher Educators
    Standards for the Preparation of Secondary
    Mathematics Teachers (2020)

    Standards for the Preparation
    of Middle-Level Mathematics Teachers
    (2020)    		National Council of Teachers of Mathematics
    Curriculum Standards
    for K-12 Schools    		Principles and Standards for School
    Mathematics (2000)    		National Council of Teachers of Mathematics
    Common Core State Standards for
    Mathematics (CCSSM) (2010)    		National Governors Association Center for Best Practices, Council of Chief State School Ofcers
    Technology
    Integration
    Framework    		Technological Pedagogical Content
    Knowledge (TPACK) (2012)    		tpack.org

    Because GeT courses are required for students who will become secondary mathematics teachers, GeT instructors must understand the needs of this group of students when in their course. It is not enough for these students to know the content, these students must gain specialized knowledge to teach effectively. Shulman (1986) describes this as pedagogical content knowledge; it includes, in part, an understanding of what makes learning some topics easy or difficult. To have this type of understanding, students must have opportunities to reflect upon and compare/contrast analogies, illustrations, and examples. Ball, Thames, and Phelps (2008) describe pedagogical content knowledge as a bridge between content knowledge and the practice of teaching. GeT instructors should foster the construction of this knowledge by sharing teaching techniques and through conversations about teaching geometry content. For example, by taking the time to discuss multiple approaches to solving problems or by examining different frameworks for writing proofs, the GeT instructor is providing students the opportunity to reflect on misconceptions and ways that make the content more understandable by others. This type of knowledge is necessary for future teachers.

    Another aspect that helps with secondary geometry understanding is technology. Many teacher education standards and curriculum standards have addressed the use of technology in some way. For example, one of the grade 8 CCSSM geometry standards specifically mentions geometry software. Therefore, the GeT course needs to utilize technology to help understand and explore concepts in geometry. More information about the use of technology will be addressed in the Technology SLO.

    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.

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

    Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

  • GeT Course Student Learning Outcome #5

    SLO #5  [Definitions] Understand the role of definitions in mathematical discourse:

    1. Understand the importance of precise definitions for geometric objects and that necessarily some geometric terms and relationships must remain undefined.
    2. Recognize that there are a variety of acceptable definitions for some geometric objects.

    The role of definitions in mathematics is a rich area for students’ exploration that is often overlooked. In many math classes, definitions are given by the textbook or teacher. However, in a geometry class, definitions can be a fruitful area for students to explore. Students can propose their own definitions for elementary concepts, such as a square, a triangle, circle, or even a straight line. They can engage in class discussions about verifiable mathematical definitions versus vague descriptive definitions, and they can compare and contrast definitions with different properties included.  For example, when asked to define what a rectangle is, one student might say it is a quadrilateral with four equal angles; another might say it is a quadrilateral that has at least three right angles and doesn’t have four equal sides; another might say it is a quadrilateral with reflection symmetry across the perpendicular bisectors of its sides; yet another might say it is a quadrilateral with four congruent angles and two pairs of congruent parallel sides.  

    Classes can have rich discussions regarding both the equivalence and the quality of proposed definitions. Criteria for the quality of definitions could include: (1) use of commonly understood words or previously defined terms, (2) accurately describing what is being defined, and (3) including no superfluous information. One strategy to convey the need for (1) is to “define” two “nonsense” words with definitions that refer to one another and hence have no meaning. Some definitions must involve undefined terms, to avoid infinite regress. A strategy to convey this is to ask students to come up with a definition of a familiar object and prompt them to define the terms they use in their definition.

    Choices for definitions necessarily set the context for proving activity. For example, proving that two lines are parallel because they do not intersect can be very different from proving that they are parallel because they are everywhere equidistant. GeT students can also consider how changes in the assumptions within a geometric definition can lead to changes in the interpretations of other terms involving that definition. For example, a circle is often defined as the set of all points in a plane that are equidistant to a given point. If a geometry adopted the Euclidean metric for distance, then the property that distinct circles have a finite number of intersections holds; however, this property is not maintained with the Taxicab metric (Krause, 1975). 

    The logical consequences of statements involving a definition include the assumed meanings for terms within a definition as well as the axioms of the system . For those who include significant non-Euclidean topics  there is opportunity to investigate the same definitions using different models. For example, there are no quadrilaterals with four right angles on the surface of the sphere or on the hyperbolic plane, but there are still quadrilaterals with reflective symmetry over the perpendicular bisectors of their sides. On the hyperbolic plane, there are lines that do not intersect but are not everywhere equidistant. On the sphere, there are not any lines that do not intersect, but there are still lines that make equal corresponding angles with a transversal. Some instructors have been surprised to discover that when students spend time in class exploring definitions and then are given a new space to explore on their own, they can productively spend weeks exploring the implications of potential definitions. For instance, what is a circle on the surface of a cone? If a circle is defined as the set of all points obtained by going a fixed distance from a given center in all directions, we get different circles than if a circle is defined as a figure with constant curvature, which is in turn different than if a circle is defined as a closed figure such that every straight line segment from the center to the boundary is the same length. Each of these types of circles have different properties that students can explore.

    The taxonomy of geometric objects is closely tied to definitions, and the exercise of classifying objects helps GeT students attend to the ramifications of adopting different definitions and become prepared to support prospective students’ reasoning at different Van Hiele levels (see, e.g., Burger & Shaughnessy, 1986). In elementary school, students are taught how to identify and classify different quadrilaterals as rectangles, rhombi, squares, or none of the above. As definitions become formalized in middle and high school geometry, they become associated with increasingly generic representations. It is in the GeT course that students consider the results of adopting alternative definitions for geometric terms. For example, trapezoid is typically defined inclusively in college geometry courses (a trapezoid is a quadrilateral with at least one pair of parallel sides) but it is sometimes defined exclusively in elementary and secondary courses (a quadrilateral with exactly one pair of parallel sides). Other terms commonly encountered in secondary geometry for which GeT students could discuss the consequences of adopting definitions supporting exclusive or inclusive meanings include: whether coincident lines are types of parallel lines, whether kites are types of rhombi, and whether the identity transformation is considered to be a type of rotation or a type of translation. Determining whether and when definitions have equivalent meanings and the consequences of adopting exclusive or inclusive definitions prepares GeT students for the varieties of geometric definitions they may encounter in teaching secondary geometry. 

    References

    Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for research in mathematics education, 17(1), 31-48.

    Krause, E. F. (1975). Taxicab Geometry. Addison-Wesley Publishing. Menlo Park, CA. 

  • GeT Course Student Learning Outcome #9

    SLO 9: Non-Euclidean Geometries.  Compare Euclidean geometry to other geometries such as hyperbolic or spherical geometry.

    Just as visiting another country can offer us a richer perspective on our own culture, so too can the study of non-Euclidean geometries help students to develop a deeper understanding of Euclidean geometry. While it is natural for students to be uncomfortable working in a geometry that varies from their intuition, non-Euclidean geometries afford an opportunity to explore and visualize novel worlds that can engage their imagination. In addition, learning the rules of these geometries puts students that plan to teach in the position of their students who may be learning Euclidean geometry for the first time. The choice of which different non-Euclidean geometries to consider might depend on the demands of the GeT course, but all offer new perspectives on familiar geometric objects and relationships.

    In our everyday experience we regularly encounter multiple geometries. Buildings tend to be Euclidean. We expect floors and ceilings to be planar. Outside, the horizon reminds us that we live on a sphere. Our visual field routinely processes distant objects as smaller than comparably sized things that are nearby, just as they could be represented in projective geometry. In the car, we measure distance with a taxicab metric. Fans of science fiction may even encounter images and ideas of hyperbolic geometry. Notably, non-Euclidean geometries can be viewed through two different lenses: geometrically, as spaces that are physically different from Euclidean space, or axiomatically, as spaces in which different axioms are true. 

    The amount of time devoted to non-Euclidean geometries can vary widely depending on factors including audience, instructor preference, and institutional expectations. For a class consisting primarily of pre-service teachers, a substantial amount of Euclidean content is necessary, though at least some non-Euclidean geometry is recommended. In a comparative geometries course it would be natural to consider several different non-Euclidean geometries, while a class that focuses primarily on Euclidean geometry might include a brief survey of some non-Euclidean examples or focus on one flavor for a longer period of time.  In any case, what follows are some of the learning opportunities offered by each.

    Incidence Geometries are useful for getting a sense of how theorems follow from a set of axioms. These involve a reduced set of axioms and perhaps make it easier to introduce some principles of proof writing in that context. Taxicab geometry is an easily described alternate geometry that can lead to rich mathematical exploration. Here we note that when we speak of “non-Euclidean geometry,” we mean this broadly, referring to geometries that are different from our usual notion of Euclidean two-or three dimensional space. In taxicab geometry, we change our usual definition of distance in the plane. Rather than using a Pythagorean measurement, we measure the distance between two points as the sum of the absolute differences of their Cartesian coordinates. This radically changes the form of objects that are defined in terms of distance. For example, a circle (the set of points at a given distance from a given point) no longer appears round. Ellipses, hyperbolas, and parabolas provide an even greater challenge!

    Spherical geometry offers the advantage of being (fairly) easy to visualize (or hold in your hand).  As a more accurate representation of the surface of the planet than a flat Euclidean world, it has relevance.  An introduction to spherical geometry immediately challenges our understanding of the undefined term “line” and our belief that between any two points there can be drawn a unique line. Other explorations might have students consider parallel lines on the sphere or the angle sum of a triangle.

    Taxicab and spherical geometry serve well as examples of non-Euclidean geometries that can be explored at any point in a GeT course.  Hyperbolic geometry can be as well, though its close relationship to Euclidean geometry is perhaps best appreciated when students are more experienced with axioms and axiomatic systems. Hyperbolic geometry differs from Euclidean geometry only in a parallel postulate.  In Euclidean geometry, we assume there is exactly one parallel through a given point not on a given line. In hyperbolic geometry, we adopt a different parallel postulate, so that there are multiple lines through a given point parallel to a given line.  Changing this axiom is, in fact, how hyperbolic geometry was first developed historically.  Moreover, we can simply remove that axiom altogether to end up with a third geometry, Neutral geometry.  Comparing the mathematical properties of these three geometries and their interplay leads to rich discourse. It is also worth noting that although hyperbolic geometry perhaps arises most naturally from this axiomatic change, it can also be viewed geometrically as a space of constant negative curvature. In this sense, it provides an instructive example of a non-Euclidean geometry having properties different from Euclidean geometry.

    The relationship between Euclidean, hyperbolic, and Neutral geometry can be made explicit by proving the equivalence of parallel postulates in Neutral geometry.  For example, transitivity of parallelism (“Two distinct lines each parallel to a third line are parallel to each other.”) is logically equivalent to Euclid’s fifth postulate in Neutral geometry.  Proving that equivalence, or one similar, can be a valuable experience by strengthening student understanding of proof.  These proofs are demanding but are generally relatively brief.

    Rectangles (quadrilaterals with four right angles), for example, are among our most familiar geometric objects.  However, while the existence of rectangles can be easily established in Euclidean geometry, it can be proven that they exist neither in hyperbolic geometry or in spherical geometry. In both of those cases, a pair of lines can share at most one common perpendicular line making a rectangle impossible.  In Neutral geometry, we can neither prove nor disprove their existence.  Likewise, we can show that similar, non-congruent triangles do not exist in hyperbolic geometry.  Examples such as these differentiate between the geometries and demonstrate the necessity of Euclid’s fifth postulate. In this way, they can help strengthen student understanding of axiom systems. 

    Hyperbolic geometry offers opportunities for students to strengthen their facility with geometric straightedge and compass constructions through an exploration of hyperbolic geometry models.  Since both the Poincaré and Klein models of hyperbolic geometry are situated within Euclidean geometry, constructions of “lines” and “perpendiculars” in these models translate to Euclidean constructions. These constructions can cover a range of complexity, from the trivial (e.g., constructing a “line” in the Klein disk) to the highly involved (“Dropping a perpendicular” in the Poincaré disk).  Dynamic geometry software is an excellent resource here, as there are a wealth of online tools available that automate some of the most difficult constructions.  This has the added benefit of encouraging students to engage in higher order thinking on constructions in ways that were not possible only a few years ago.

  • GeT Course Student Learning Outcome #4

    Understand and explain the relationship between axioms, theorems, and geometric models in which they hold (such as the plane, the sphere, the hyperbolic plane, etc.); understand how knowledge about different models can help us understand which theorems can be proven from a given set of axioms.

    a) Recognize and be able to communicate the distinction between an axiom and a theorem and describe how mathematical systems arise from axioms.

    b) Construct logical arguments within the constraints of an axiom system.

    c) Understand the roles of models in determining the independence of axioms and the validity of proofs.

    Geometry courses are one of the few places we talk explicitly about axiom systems with our students. Most teachers of college geometry classes will be familiar with the idea that it is possible for mathematical subjects to be reduced to a set of axioms and then a set of theorems proven (exclusively) from those axioms. However, even though this can be done in principle, it is something that working mathematicians rarely do outside of geometry and logic classes, so not everyone teaching a geometry class will have a lot of experience working with axiom systems.

    In building an axiom system, we begin with undefined terms as well as with statements, called axioms, that are accepted to be true without proof. In geometry, the undefined terms include “points” and “lines”. The axioms establish assumptions about undefined terms and the relationships between them.  A world in which we can give meanings to all of those terms is called an interpretation of the terms. For example, if we are talking about “points” and “lines,” we could interpret them as points and lines in the Euclidean plane, but we could also interpret them as points and great circles on the sphere. An interpretation is called a model of the axioms if all of the axioms are true in the interpretation. For example, if we have an axiom that says that two points lie on a unique line, the Euclidean plane would be a model of this axiom. The sphere would not because antipodal points like the north and south pole on the sphere can be connected by many different great circles  

    Axioms within a system are independent if no axiom in the system is a logical consequence of the others.  This means that for any axiom, we should be able to find an interpretation in which that axiom fails, but all of the others are true. Models and independence are intimately tied into the history of geometry.   Perhaps the biggest question about Euclid’s axiom system was whether his fifth postulate could be proven from his first four axioms.  Although an apparently consistent hyperbolic geometry was developed in the early 1800s, it was not until Beltrami presented a model for that system later in the century that the independence of the parallel postulate was established. 

    A theorem is a statement that has been proven from the axioms without regard to interpretation. In a college geometry class, proof can be thought of as a convincing deductive argument relying on explicit reference to axioms or previously proven theorems.  Since a model of an axiom system is an interpretation of the undefined terms that satisfies the axioms, every theorem translates to a true statement in a model. Therefore, if a mathematical statement turns out to be false in a model, then the statement cannot be a theorem, i.e. it cannot be proved from the axioms. Moreover, just as models can be used to show that a statement cannot be proven, they can show that a statement cannot be disproven. That is, demonstrating a model where a statement holds shows that the negation of the statement is not a theorem. Models, therefore, serve as a sort of laboratory for geometric conjectures and can be a powerful tool for exploring the properties of an axiom system.  

    Students need to understand what the axioms mean, and then they can try to convince someone that a theorem is true whenever the axioms are true.  As in other parts of the geometry curriculum, we see a trade-off between trying to be as rigorous as possible and trying to be developmentally appropriate. Because this is the main place for considering axiom systems in the college math curriculum, understanding of the elements of an axiom system (axioms, models, theorems, interpretations, undefined terms) should be an explicit learning goal. Considering interpretations where axioms do not hold is a good starting point. It encourages students to grapple with what the axioms actually mean, their distinction from other axiomatic elements, and their role as a foundation of mathematical systems. This is, therefore, a natural place to bring non-Euclidean geometries into the picture. 

    When choosing how to introduce an axiom system, instructors must balance the need to establish expectations for axiomatic proof with the need to understand significant, non-obvious geometric results in a reasonable amount of time. In many classes, students start out with a very simple axiom system of basic facts true in almost any geometry and then proceed to prove theorems from them. A common approach is to introduce an axiom system consisting of Euclid’s axioms without the parallel postulate, which is used to develop a “neutral geometry”. This has the advantage of making the axioms simple enough to focus on the logic of building deductive inferences using them. One strategy for motivating students to reason axiomatically about neutral geometry is to introduce problems with rules in which undefined terms “points” and “lines” have been replaced either with nonsense terms (“Every Fo has two Fes.”) or with some other context (“Every club has at least two members.”). This encourages students to reason from the stated rules rather than using their geometric intuition. 

    When considering an axiomatic development of geometry, it is important to consider the developmental readiness of one’s students for reasoning abstractly. Although GeT students should have learned some elements of the Euclidean geometric system in their K-12 geometry curriculum, it can be helpful if the instructor starts with lower Van Hiele level tasks to scaffold students’ development of geometric reasoning and proof, especially with the consideration that GeT students are often from different STEM majors with varied prior knowledge. Varying expectations of rigour and abstraction can also provide opportunities for assessing students’ development of deductive reasoning (e.g., their application of logic) that is not purely axiomatic. Highlighting differences in expectations for justifications can help to solidify students’ understandings of the elements of axiom systems and proofs.