Using a Journal Article as a Professional Development Experience

**Geometry**

PDF

**Title:** Key Ideas and Insights in the Context of Three High School Geometry Proofs

**Author:** Manya Raman and Keith Weber

**Journal:** *Mathematics Teacher*

**Issue:** May 2006, Volume 99, Issue 9, pp. 644-649

**Article:** Sherlock Holmes, Geometry Proofs, and Backward Reasoning

**Author:** Andy M. Gole

**Journal:** *Mathematics Teacher*

**Issue:** November 2003, Volume 96, Issue 8, pp. 544-546

**Rationale/Suggestions for Use**

The study of proofs is one topic that that most students encounter in Geometry with varying degrees of success. Teachers often search for new and exciting ways to help students generate informal and formal proofs. This article may be used with pre-service teachers or by in-service teachers interested in exploring ways to connect students’ intuitive sense making with formal proofs.

This article provides mathematics teachers an opportunity to reflect on practice by exploring:

- How interactive geometry software can be used to help students develop intuitions and conjectures
- The connections between an exploration and the formal proof

This reflection guide will take at least two sessions to complete.

**Materials**

- Copies of the article
- Chart paper for session 2
- Interactive geometry software on computer or graphing calculator

**Procedures/Discussion Questions**

*Session One*

*Goal:* Participants will explore how connections can be made between exploratory activities and formal proofs.

*Note:* It is intended that participants not read the article beforehand.

- Provide participants with the first task (Raman & Weber, example 1) to complete the proof with a partner.
- Lead a preliminary discussion of the various methods used to complete the task. Ask participants where they would predict students struggling with the development of the proof.
- Have the participants read the article (Raman & Weber) and stop at example 2 on page 646.
- Have interactive geometry software available on computers or calculators for each of the participants to use to complete the exploration. Project the completed exploration.
- Discuss the process and concepts used to complete the exploration.
- Ask participants to compare the proof they created in #1 to the one in the article and/or to others.

- Discuss the connection to the proof in terms of the three central parts outlined in the article (Raman & Weber) on p. 646: the key idea, an insight, and the details. Discussion questions can include:
- How might the exploration help in the development of a proof?
- What ideas do you have about helping students at each of the three parts?

- Divide the participants into two groups – assign Example 2 to one group and Example 3 to the other group.
- Use interactive software to explore the relationships in the example assigned
- Discuss with a partner the relationships you explored and how they are connected to the key idea-insight-details structure from the article.
- Discuss the proof presented in the article and how the exploration makes the proof more accessible to students. What supports would students need to understand the components of the key idea-insight-details structure?

*In between sessions*

- All participants should read “The importance of key ideas and insights” and “Concluding comments” from the article.
- Ask participants to bring one task/proof that they currently use in their class in order to develop a possible exploration in the next session.

*Session Two*

*Goal:* Participants continue to explore the connection between activity and proof

- Using computer software, ask participants to work in partner groups to develop explorations for the proof they selected.
- Two pairs get together, exchange their explorations in order to “edit/revise” them for classroom use.
- Discuss as a whole group:
- What was your professional learning from this process?
- Why does the key idea seem to connect the exploration and the formal proof?
- What is the importance of the role of the insights to in the process of producing a formal proof?
- Does there seem to be a flow between the key idea to an insight to the details?

- Have the participants read the first part of the article, Sherlock Holmes, Geometry Proofs, and Backward Reasoning stopping at “Although geometry proofs offer…” on page 545.
- Discuss how the use of Backward Reasoning can be used to help students as they connect the key idea, an insight, and the details for each explorations and formal proofs.

**Next Steps/Extensions**

- Have teachers use strategies shared and discussed during the sessions with their students. During a session share how these strategies are impacting student understanding and ability to develop formal proofs.

**Connections to Other NCTM Publications:**

- Besteman, N., & Ferdinands, J. (2005, February). Another way to divide a line segment into
*n* equal parts. *Mathematics Teacher*, *98*, 428-433.
- Contreras, J. N. (2003, April). A problem-posing approach to specializing, generalizing, and extending problems with interactive geometry software.
*Mathematics Teacher*, *96*, 270-276.
- Cox, R. L. (2004, January). Using conjectures to teach students the role of proof.
*Mathematics Teacher*, *97*, 48-52.
- Edwards, M. T. (2005, March). Activities for students: Using overhead projectors to explore size change transformations.
*Mathematics Teacher*, *98*, 498-507.
- Gluchoff, A. (2006, April). Hands-on fractals and the unexpected in mathematics.
*Mathematics Teacher*, *99*, 570-575.
- Groth, R. E. (2005, August). Linking theory and practice in teaching geometry.
*Mathematics Teacher*, *99*, 27-30.
- Hurwitz, M. (2002, October). Sharing teaching ideas: cos² x + sin² x and the trigonometric sum and difference identities.
*Mathematics Teacher*, *95*, 510-514.
- Johnson, I. D. (2006, March). Grandfather Tang goes to high school.
*Mathematics Teacher*, *99*, 522-526.
- Linn, S. L., & Neal, D. K. (2006, March). Approximating pi with the golden ratio.
*Mathematics Teacher*, *99*, 472-477.
- Posamentier, A. S., & Patton, L. R. (2008, December). Delving deeper: Enhancing plane Euclidean geometry with three-dimensional analogs.
*Mathematics Teacher*, *102*, 394-398.
- Stanton, R. O. (2006, March). Proofs that students can do.
*Mathematics Teacher*. *99*, 478-482.
- Worrall, C. (2004, October). Delving deeper: A journey with circumscribable quadrilaterals.
*Mathematics Teacher*, *98*, 192-199.