By Isil Isler, Ana Stephens, and Hannah Kang, posted February 1, 2016 –

In the previous blog post we provided some ideas about algebraic thinking and mathematical justification. We also introduced the following task:

Brian knows that anytime you add three odd numbers, you will always get an odd number. Explain why this is always true.

In what follows, we share a sample of student responses to the sum of three odd numbers task and use Carpenter and his colleagues’ (2003) justification framework to identify the nature of their responses.

Jordan:

Alexis:

Peggy:

Before reading further, please think about the nature of these responses with regard to the justification framework. What do these responses tell us in terms of how students reason about the Sum of Three Odd Numbers task? Do these arguments show that the conjecture is always true? Which ones would you hope to see your students provide to show that the conjecture is always true?

Jordan’s work is an example of justification by example. This is, by far, the most common type of justification we observed in our study. After trying out a few examples, students usually conclude that a given conjecture—in this case that the sum of three odd numbers is an odd number—is true for all numbers.

Next, let’s focus on Alexis’s response. This response is different from Jordan’s argument. It uses specific numbers but makes use of the structure of the numbers rather than their values. Alexis represented specific numbers using circles, split them into pairs, and produced a general argument based on the definition of even and odd numbers and the structure of the representation. The specific numbers are thus used in a “generic” way. This response can therefore be categorized as a generalizable argument in Carpenter and her colleagues’ (2003) framework. This type of general argument is called representation-based proof and is accessible to elementary school students (Russell, Schifter, and Bastable 2011; Schifter 2009). During our intervention and on our assessments, we found that our students benefited from using representations because they helped students focus on underlying structure and reflect on why.

Finally, let’s focus on Peggy’s response. How is it different from Jordan’s and Alexis’s responses? Peggy’s argument is not dependent on specific numbers; rather, it is built on generalizations about sums of even and odd numbers. Therefore, this argument is also categorized as a general argument. The argument uses a chain of accepted arguments to prove that the conjecture is true. In other words, the student used statements that were already justified in the classroom (i.e., the sum of two odd numbers is an even number, and the sum of an even number and an odd number is an odd number) to justify another conjecture—that the sum of three odd numbers is odd. We encourage you to support your students to develop conjectures based on previously justified conjectures. As was the case with representation-based proofs, we found that elementary school students in our early algebra intervention were able to produce general arguments using justified conjectures.

We can help our students move beyond examples-based reasoning by asking them why they think a given conjecture is always true and encouraging them to look for and make use of structure (SMP 7). Here are some suggestions to help you engage your students in justifying and proving:

• Ask your students “What do you notice?” “Can you come up with a conjecture?” “Do you think it will work for all numbers?” “Why or why not?” Such questions will help them notice patterns, make generalizations, and justify their thinking.
• Encourage your students to work in pairs or small groups to share their thinking one another. This will help them communicate their ideas with their peers and clarify their own thinking.
• Pay attention to different categories of justifications while students work in small groups (perhaps using Carpenter et al.’s [2003] framework as a guide). This will help you capitalize on different student responses while orchestrating whole-group math discussions.
• Help your students see the limitations of examples. You can ask such questions as, “How do we know that it works for all numbers since we have not tried them all?” “How can we be sure that there isn’t any number that does not ‘work’ according the conjecture?” Asking these questions will help your students question the validity of using a few examples to justify the conjecture and appreciate the need for general arguments.
• Support your students in developing definitions that you would like them to use and refer to later. This will be important as they are encouraged to develop general arguments. Defining odd numbers as “numbers with a leftover when broken into pairs” helped our third graders justify that the sum of two odd numbers is always an even number.
• Encourage your students to use representations and pictures to explain their thinking and justify their arguments. This will help your students produce general arguments by focusing on the structure of the representations. Ask your students how their representations show that the argument is always true and why.

Your Turn 

We encourage you to try the Sums of Evens and Odds activity and/or the Sum of Three Odd Numbers task with your students and share with us what you learn! Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

References

Carpenter, Thomas P., Megan Loef Franke, and Linda Levi. 2003. Thinking Mathematically: Integrating Arithmetic and Algebra in the Elementary School. Portsmouth, NH: Heinemann.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

Kaput, James J. 2008. “What Is Algebra? What Is Algebraic Reasoning?” In Algebra in the Early Grades, edited by David W. Carraher and Maria L. Blanton, pp. 5–17. New York: Lawrence Erlbaum Associates.

Knuth, Eric J., Jeffrey M. Choppin, and Kristen N. Bieda. “Middle School Students’ Production of Mathematical Justifications.” 2009. In Teaching and Learning Proof across the Grades: A K–16 Perspective, edited by Despina A. Stylianou, Maria L. Blanton, and Eric J. Knuth, pp. 153-170. Studies in Mathematical Thinking and Learning Series. New York: Routledge.

National Council of Teachers of Mathematics (NCTM). 2011. Developing Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3–5, edited by Maria Blanton, Linda Levi, Terry Crites, and Barbara Dougherty. Essential Understanding Series. Reston, VA: NCTM.

Russell, Susan Jo, Deborah Schifter, and Virginia Bastable. 2011. Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. Portsmouth, NH: Heinemann.

Schifter, Deborah. 2009. “Representation-Based Proof in the Elementary Grades.” In Teaching and Learning Proof across the Grades: A K-16 Perspective, edited by Despina A. Stylianou, Maria L. Blanton, and Eric J. Knuth, pp. 87–101. New York: Routledge.

Dr. Isil Isler, isler@wisc.edu, is a recent graduate of the University of Wisconsin–Madison. She is interested in algebraic thinking, and reasoning and proof in the elementary and middle grades. Dr. Ana Stephens, acstephens@wisc.edu, is an associate researcher at the Wisconsin Center for Education Research at the University of Wisconsin–Madison. She is interested in the development of students’ and teachers’ algebraic reasoning and helping teachers focus on students’ mathematical thinking. Hannah Kang, hkang52@wisc.edu, is a Master’s of Science candidate at the University of Wisconsin-Madison. She is interested in equity and diversity issues within mathematics education, as well as students’ algebraic thinking. 

The research reported here was supported in part by the National Science Foundation (NSF) under DRK-12 Award No. 1219605/06. Any opinions, findings, and conclusions or recommendations expressed in this blog are those of the authors and do not necessarily reflect the views of NSF.