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Studying Students' Reasoning in Writing Generalizations

Using a Journal Article as a Professional Development Experience
Reasoning and Proof

Title:          Studying Students’ Reasoning in Writing Generalizations
Author:      Angela S. Krebs
Journal:     Mathematics Teaching in the Middle School
Issue:        February 2005, Volume 10, Issue 6, pp. 284 - 287

Rationale/Suggestions for Use
The article asks students to generalize about patterns using two approaches: numerical/algebraically and geometrically/visually; to show the connections between the two approaches, and then to explain their reasoning as to why the generalizations make sense.

The Principles and Standards for School Mathematics (NCTM, 2000) states: “Reasoning is an important part of mathematics. Students should enter the middle grades with the view that mathematics involves examining patterns and noting regularities, making conjectures about possible generalizations, and evaluating the conjectures. In grades 6-8 students should sharpen and extend their reasoning skills by deepening their evaluations of their assertions and conjectures and using inductive and deductive reasoning to formulate mathematical arguments” (p 262).

Materials

• Toothpicks for each pair of teachers

Procedures

1. Present the Toothpick Task, #’s 1 to 3, as presented in Figure 1, page 284 of the article. Teachers work in pairs and are asked to share their findings and explanations with the group.
2. When teachers present solutions, note if they found the formula for any figure N by reasoning from the concrete/pictorial or numeric/tabular representations. In each case ensure teachers explain why their rule makes sense. (Note that the article does not share student solutions to finding a formula for the perimeter but see reference in Next Steps below for a detailed description along with student’s work of solutions to this part of the problem.)
3. Assign #4 of Task: “Write a formula you could use to find the total number of toothpicks needed to make any figure N. Tell what your variables represent. Explain how you figured this out.”
4. After teachers complete #4, have them
• Share their rules and explain why their rule makes sense.
• Discuss if generalizations were made using a table or concretely (pictorial).
• Did they and can they relate the generalization back to the problem presented?
• If a pair reasoned using geometric/concrete representation of the problem, ask if their rule also makes sense if they had represented the problem with numbers in tabular form and visa versa.
5. Have teachers read the article, noting, in particular, the strategies students used to make their generalizations about finding the total number of toothpicks and how they explained their reasoning. Do the recorded audiotape conversations and accompanying diagrams of the students on pages 285 and 286 of the article make sense?
6. Ask the teachers to share how the strategies and explanations of reasoning of the students compared to what they did.

Next Steps

• Present the problem presented on page 266, PSSM, Reasoning and Proof Standard for Grades 6-8, to the group to explore. It is a good follow-up to the problem done in the Toothpick Task above. It could be used as a reminder and a discussion that one needs to be cautious when generalizing inductively from a small number of cases, because not all patterns generalize in the way we may expect.
• Refer teachers to the Activity, Building with Toothpicks, p. 13 – 17 in Navigating through Algebra in Grades 6 -8, NCTM, 2001. Discuss the student work, in particular, the student’s explanations of how they found the formula for the perimeter.

Connections to Other NCTM Publications