Using a Journal Article as a Professional Development Experience

**Geometry**

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**Title:** Watch What You Say

**Author:** Sally K. Roberts

**Journal:** *Teaching Children Mathematics*

**Issue:** December 2007/January 2008, Volume 14, Issue 5, pp. 296-301

**Rationale/Suggestion for Use**

This article describes classroom conversations, appropriate models and representations, along with appropriate activities that support children in defining necessary and sufficient conditions for defining shapes and understanding the hierarchical nature (develop a family tree) of quadrilaterals .

One of the expectations in the Geometry Standard (PSSM) is that students classify two-dimensional shapes according to their properties and develop definitions of classes of shapes. This article gives support to this expectation.

**Procedure**

- The author introduces the article (page 296) by asking her students to describe over the phone how to draw specific quadrilaterals. Begin the session by asking participants working with a partner to do a similar activity. One person describes the characteristics and the other draws.

*Directions:*
- The person who will do the talking is secretly given the name of the figure to be drawn (Square) and without saying the name of the figure gives the other person the defining properties. The partner draws the figure based on the information given. They then discuss whether the figure drawn was the one intended and why or why not? Discuss what properties were necessary to articulate in order to draw a square? Did you give more properties than were needed?
- The partners switch roles. One person is secretly given the name of the figure to be drawn (Rectangle) and as above states defining properties. The partner draws the figure. They then discuss whether the figure is a rectangle and why or why not? What properties are necessary to articulate in order to draw a rectangle? Did you give more properties than were needed?
- As a whole group make a comparison chart outlining how squares and rectangles are the same and how they are different.
- As a group agree on definitions for a square and a rectangle and record on chart paper.

- Ask participants (allow time to discuss in small groups):
- “Is a rectangle a square? Why or why not?”
- What difficulties might your students have with determining whether a rectangle is a square? Why do you think this may be?
- “Are all squares rectangles?” Justify your answer. How do you think your students would answer this question?

- Analyze the Quadrilateral Family Tree (figure 4, page 298) and explain why each figure is placed where it is on the family tree. Allow time to share the discussion.
- In working with children, where do you think it is better to start building the quadrilateral family tree- at the bottom (square) or at the top (quadrilateral)? On what do you base your decision?
- Where would you place the trapezoid?

- The authors state that: “students who are introduced to the interrelated and inclusive nature of the quadrilateral family tree early in their education will be better prepared to encounter more advanced topics in geometry. Understanding the inclusive nature of definitions benefits students and makes their learning not only holistic but also more efficient.” (p. 298)
- Reflect on your own classroom experiences, and discuss in your groups why it is important for children to understand that squares are subsets of rectangles, which are a subset of parallelograms, which are a subset of quadrilaterals?
- What instructional strategies do you suggest to help build “the interrelated and inclusive nature of the quadrilateral tree”?
- Draw a Triangle Family Tree. Justify in your groups why you placed the triangles where you did on the Tree.

- Read the section, “The Lesson”, page 296 –298.
- In this section the author refers to properties of figures that are necessary and sufficient to define a figure. Discuss what is meant by necessary and sufficient when referring to definitions. Review the definitions recorded earlier on chart paper and discuss whether the properties defined for each are necessary and sufficient.
- Why do you think the author titled the article: “Watch What You Say”?
- Relate incidents from your own experiences where the title of the article might apply.

**Next Steps**

- At the beginning of the article the author refers to the van Hiele model of geometric thinking (see reference below for more information about van Hiele model). The model hypothesizes that young children begin their thinking about geometry at a visual level, i.e., they recognize shapes by an overall gestalt based on visual clues rather than defining properties.
- Where do you think your children are in relation to the van Hiele levels- visual or analytic?
- Based on what you read in this article, how can you help children move from the visual level to the next level where children use properties to categorize shapes and classes of shapes?

- If appropriate for the work environment of participants:
- Design a classroom ready lesson, along with instructional strategies, that would help move children from the visual to the analytic level of geometric thinking.
- Implement in your classroom, recording your student’s observations and your role in supporting the movement.
- Pose the question to your students, as was posed in the article, “Is a rectangle a square?” What questions could you ask your students to facilitate the mathematical conversation around whether a rectangle is a square, to get them to justify their reasoning, and to encourage all your students to participate in the lesson?

**Connections to Other NCTM Publications**

- Ambrose, R. C. & Falkner, K. (2002, April). Developing spatial understanding through building polyhedrons.
*Teaching Children Mathematics*, 8, 442-447.
- Angerame, S. S. (1999, October). Math-o’-lanterns.
*Teaching Children Mathematics*, 6, 72-76.
- Battista, M. T. (1999, November). Importance of spatial structuring in geometric reasoning.
*Teaching Children Mathem*atics, 6, 170-177.
- Battista, M. T. (2002, February). Learning geometry in a dynamic computer environment.
*Teaching Children Mathematics*, 8, 333-339.
- Brewer, E. J. (1999, December). Geometry and op art.
*Teaching Children Mathematics*, 6, 220-236.
- Clements, D. H. & Sarama, J. (2000, April). Young children’s ideas about geometric shapes.
*Teaching Children Mathematics*, 6, 482-488.
- Emenaker, C. E. (1999, December). Gingerbread-house geometry.
*Teaching Children Mathematics*, 6, 208-215.
- Granger, T. (2000, September). Math is art.
*Teaching Children Mathematics*, 7, 10-13.
- Koester, B. A. (2003, April). Prisms and pyramids: Constructing three-dimensional models to build understanding.
*Teaching Children Mathematics*, 9, 436-442.
- Lehrer, R. & Curtis, C. L. (2000, January). Why are some solids perfect? Conjectures and experiments by third graders.
*Teaching Children Mathematics*, 6, 324-329.
- Penner, E. & Lehrer, R. (2000, December). The shape of fairness.
*Teaching Children Mathematics*, 7, 210-214.
- Renne, C. G. (2004, January). Is a rectangle a square? Developing mathematical vocabulary and conceptual understanding.
*Teaching Children Ma*thematics, 10, 258-263.
- Robertson, S. P. (1999, May). Getting students actively involved in geometry.
*Teaching Children Mathematics*, 5, 526-529.
- Schoffel, J. L. & Breyfogle, M. L. (2005, March). Reflecting shapes: Same or different?
*Teaching Children Math*ematics, 11, 378-382.
- Sharp, J. M. & Hoiberg, K. B. (2001, March). And then there was Luke: The geometric thinking of a young mathematician.
*Teaching Children Mathematics*, 7, 432-439.
- Swindal, D. N. (2000, December). Learning geometry and a new language.
*Teaching Children Mathematics*, 7, 246-250.
- van Hiele, P. M. (1999, February). Developing geometric thinking through activities that begin with play.
*Teaching Children Mathematics*, 5, 310-316.
- Woleck, K. R. (2003, September). Tricky triangles: A tale of one, two, three researchers.
*Teaching Children Mathematics*, 10, 40-44.