Secondary Geometry Curriculum Sequencing?

Question

I am currently student teaching, and the main class that I am focusing on is a secondary geometry class. I am currently following my classroom mentors curriculum sequence which looks something like:

• Intro to Geometry (points, lines, planes, shape basics)
• Logic (basic propositional logic to introduce the idea of proofs)
• Triangles (properties, congruence, special right triangles, centers) where we are now
• Quadrilaterals/Polygons
• Circles
• Symmetry
• Transformations
• Constructions
• Trigonometry

which I feel like doesn't have a super good flow. We are jumping around in the textbook and there is no real intuitive flow between subjects. As it is already a quarter of the way through the school year, I know there is only so much I can do to adjust this sequencing but I would like to improve this for when I am teaching geometry in the future. Does anyone have suggestions for a better way to sequence my secondary geometry curriculum?

Answer 1

While I am not sure if your mentor teacher is up for the discussion for this year, I will note that, in practice, it is important to continue to question the sequencing of topics in a particular course. Mathematics is organized logically, and when a school year forces a chronology, the order of topics will be something you continue to revisit depending on the needs of your students.

Before I address your concern for “intuitive flow” in sequencing a secondary geometry curriculum, I must acknowledge two constraints that might limit one’s opportunities to make adjustments to the sequencing puzzle:

1. If you are constrained by the school’s desire to make sure that students pass the state exam, then your sequencing might value the topics in geometry that are “new” or “most tested” over those that you find interesting. These topics—including circles, special right triangles, triangle centers, and precise definitions of quadrilaterals—may not have been introduced in K-8 mathematics, and it may be your responsibility to address these at least haphazardly before your students tackle the state exam. For instance, the state math exam that my students will take in Massachusetts places a heavy emphasis on computations using algebra. Topics such as logic, trigonometry, constructions, and proofs are not tested at all. Hence, the sequencing that I was handed when I started teaching skipped over the topics in geometry that would not be covered on the exam.
2. If you are constrained by the need for heavy Algebra I remediation because your geometry class serves as a bridge to Algebra II, then your sequencing might value time spent reviewing algebra explicitly over development of proofs. The latter relies on precision of language and nuance that may not necessarily be meaningful for students who do not intend to continue to study mathematics for too long after your class. Along a similar vein, if you are constrained by the needs of students with learning, behavioral, or language challenges, then your sequencing might focus on practical applications of geometry and skip proof-based geometry altogether.

Your need for “intuitive flow” indicates to me that you might not be too concerned about these two constraints during your student teaching, but keep in mind that these are realistic constraints that might affect how you make decisions in your own geometry class (or even in your mentor teacher's).

Here are three guiding principles to how I have sequenced your topics below:

• Encourage exploration and discovery. The high school geometry class that I teach uses an inductive approach based on Michael Serra’s Discovering Geometry. Students perform investigations and keep a running list of conjectures throughout the year. Students test their conjectures and apply them to problems that may require some skills from Algebra I. As indicated in Serra’s text, constructions play an important part in the discovery process, so I introduce them early. Each subsequent topic is the answer to a problem posed at the end of a previous investigation. For instance, “What symmetries do translations, reflections, and rotations create?” might link a unit on rigid transformations to a unit on symmetry.
• Build from students’ previous experiences with geometry. The K-8 curriculum exposes students to a basic understanding of geometric shapes and vocabulary. When introducing the next topic, the narrative of the sequencing should acknowledge that students have had some exposure to the topic before they entered your class, perhaps with less precise language and less reasoning. While it is in your course that these ideas are refined and extended, helping students make these connections to prior knowledge provides access points to build discussion.

• Save proofs for the end. Serra explains using the van Hiele model:

  The van Hiele research shows students aren’t ready for formal proof until they have had concrete experiences with geometric figures and have successfully mastered earlier levels of visual thinking. Research shows 70% of high school students enter geometry operating at level 0 or 1 on the van Hiele measure of geometric reasoning (Shaughnessy and Burger 1985; Senk 1989). Yet "traditional" geometry textbooks that begin with establishing postulates and proving theorems are expecting students to immediately begin their geometry experience at levels 3-4. When the teacher and the textbook are presenting geometry at van Hiele level 3 or higher, while the students are functioning at level 0 or 1, it should be no surprise that there is such a high failure rate in traditional geometry courses.

  In my class, students are still asked to provide reasoning for their answers throughout the year. However, it is not until the end of the course, after repeated experimentation with their conjectures, that students are introduced to “formal proof.” (Compare to Glencoe McGraw-Hill and Holt McDougal secondary geometry textbooks, which introduce proofs in Chapter 2.)

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With these guiding principles and possible constraints in mind, the following is one way to sequence the units in a secondary geometry course that may provide the “intuitive flow” that you are looking for. I have also included a few ideas for how the units may connect thematically and added some units. (Note of practice: this particular sequencing is only somewhat in effect and still undergoing testing/revision as the year progresses!)

Part I: Building Blocks

Understandings of geometry from primary school are accessed, refined, and formalized through precise definitions and notation.

1. Introduction to Geometry
2. Constructions
3. Parallel Lines and Transversals

Part II: Basic Shapes and Measurement

Triangles from primary school are further investigated to discover connections to polygons. Pythagorean Theorem serves as a foundation for distance in the coordinate plane. Primary conceptions of measurement are extended to area, perimeter, circles, and volume.

4. Triangle Properties and Centers
5. Polygons and Quadrilaterals
6. Pythagorean Theorem, Special Right Triangles
7. Length, Area, Perimeter
8. Circles
9. Solids and Volume

Part III: Transformations

Rudimentary understandings of “slide, flip, turn” become formally defined as translation, reflection, and rotation in the coordinate plane. These lead to an investigation of symmetry and tessellation. A contrast is drawn between isometries and dilation. The discussion of similarity extends to an introduction to trigonometry.

10. Rigid Transformations
11. Symmetry and Tessellations
12. Dilations and Similarity
13. Trigonometry

Part IV: Proofs

Logic and algebraic proofs are an entry point to an intuition about proofs in geometry. Circling back to rigid transformations, congruent polygons pave way for the need for triangle congruence shortcuts, which are then used to prove other conjectures investigated during the course. Finally, the year ends by connecting algebra and geometry through coordinate proofs.

14. Logic
15. Triangle Congruence
16. Proofs of Conjectures for Quadrilaterals and Circles
17. Coordinate Proof

Answer 2

Trying to "linearize" the mathematics in a course is among the hardest problems facing curriculum designers and teachers. This is especially true for geometry, where if one does not use an axiomatic approach it is quite hard to both select good topics and the order to teach them. In a high school geometry class the typical student will not go on to become a STEM major in college so, it seems to me, highly desirable to organize what is taught to show students the appeal, applicability, and power of using geometrical ideas. Logic and proofs though important for mathematics majors is rather dry to say the least for many other students.

When I teach geometry for undergraduate majors I often start with with Euler's traversability theorem for graphs which has the charm of being a lovely result and highly applicable. (e.g. snow removal, garbage collection) I try to use the theory of graphs as a unifying backbone for the geometry I teach. I realize this approach works better for a survey of geometry in a college level course than what can be done in lower grades given the poor choice of geometry in the K-12 curriculum.

Answer 3

I cannot answer your (good) question, but if I may proselytize a bit re your "where we are now": The challenge of cutting out a (generic) triangle from the center of a piece of paper with one straight scissors cut leads to an almost physical proof that the three angle bisectors of a triangle meet in a common point:

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        ^{(Figure from this book.)}

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I distribute paper and scissors (8th-grade geometry in the U.S.) and run through examples leading to this gem.

Answer 4

I recommend that you use the sequence designed into the textbook. Whether it is inspired or not, it will help you prepare classes and your students organize their revision. It is much easier to study for a test on chapter 5 than to study pieces of five different chapters.

I am inclined to introduce logic/proofs material with triangles (classical construction proofs are very tangible) and then keep on hitting it with each new topic.

The curriculum I teach (secondary school in the Netherlands) breaks trig into Tangents, which turn up naturally after similar triangles and the rest (Sin, Cos etc) which only appear after we have handled quadratic equations.