Even and Odd Numbers: A Journey into The Algebraic Thinking Practice of Justification
By Isil Isler, Ana Stephens, and Hannah Kang, posted January 18, 2016 –
The notion that elementary school students can and should
engage in algebraic thinking is being increasingly accepted and advocated. We have
found that the four algebraic thinking practices of generalizing, representing,
justifying, and reasoning with mathematical relationships (Blanton et al. 2011;
Kaput 2008) provide rich experiences to engage elementary school students in
early algebra. In this post, we focus in particular on the practice of
justifying. Asking students to justify their responses can help us assess their
understanding as well as introduce them to an important part of what it means
to do mathematics. The earlier students become accustomed to justifying their
responses, the sooner they get into the habit of thinking that mathematics
should make sense. Constructing viable
arguments and critiquing the reasoning of others is one of the Common
Core’s Standards for
Mathematical Practice (SMP 3) that is closely related to the
algebraic thinking practice of justifying.
As an example of how students can engage in the practice of
justifying, consider the following activity that we implemented with
Our goals with this activity were to have students explore
even and odd numbers, develop conjectures based on the sums of even and odd
numbers, and justify their conjectures using numbers, pictures, cubes, or words.
Problems involving the exploration of even and odd numbers can provide a good
context to start engaging students in justification and proof. Having students
work with manipulatives and record the patterns they notice in tables can
support their development of definitions of even and odd numbers, which can in
turn help students construct generalizations about these numbers. In the
elementary grades, students usually define even numbers as “numbers without
leftovers when broken into pairs,” and odd numbers as “numbers with one leftover
when broken into pairs” (see the related Common Core
One particular item that we have used to assess students’ abilities
to justify in grades 3–5 is the sum of three odd numbers task (adapted from
Knuth, Choppin, and Bieda 2009):
Brian knows that anytime you add
three odd numbers, you will always get an odd number. Explain why this is
Before sharing sample responses from our students, we invite
you to think about expected student responses for this task as well as what
kind of response you would consider acceptable and why. Carpenter, Franke, and
Levi’s (2003) justification framework provides us with a way to think about
levels of sophistication in students’ justifications. They propose that
students’ justifications tend to fall into three broad categories:
1. Appeal to authority
As you may have already observed in your class, many
students, when asked to justify their responses, say, “I know because it is in
the book” or “My teacher (or parents) told me.” This type of response suggests
that the student accepts a given argument without question and that he or she
believes it is true because it has been said so; these responses are
categorized as “appeal to authority.”
2. Justification by example
This category is characterized by students’ reliance on
examples to justify that a conjecture is true for all numbers. Students usually
try a few examples to test the conjecture, and on the basis of what they find,
they claim that it works for all numbers. For instance, when asked why the sum
of two even numbers is even, a student might say, “The sum of two even numbers
is always an even number because 4 + 2 = 6 and 6 is even.” Although examples
can be useful in helping students “test” whether the conjecture is true for
those cases, they cannot prove that it is true for all numbers. (Examples
cannot prove that a conjecture is true for all numbers; however, a
counterexample is enough to disprove that a conjecture is false.) Students
might use examples because they don’t have the necessary “tools” to go beyond
examples. Therefore, we should help students see the limitations of examples, and
produce general arguments.
3. Generalizable arguments
In this last justification level, proposed by Carpenter et
al. (2003), students provide general arguments, which are usually based on the
definitions of even and odd numbers or built on already-justified conjectures
about sums of even and odd numbers. An example is “The sum of two even numbers
is always an even number because even numbers can be divided into pairs with no
leftovers, and if you add two numbers with no leftovers, the sum does not have
any leftovers.” Students may use manipulatives or pictures to explore and
justify the conjectures.
We encourage you to consider the ideas above and try them
with students if you are able. In the next blog post, we will look at some
examples of students exploring the idea of adding odd numbers. We want to hear
from you! Post your comments below or share your thoughts on Twitter
@TCM_at_NCTM using #TCMtalk.
P., Megan Loef Franke, and Linda Levi. 2003. Thinking Mathematically: Integrating Arithmetic and Algebra in the Elementary
School. Portsmouth, NH: Heinemann.
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.
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,
Jo, Deborah Schifter, and Virginia Bastable. 2011. Connecting Arithmetic to
Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades.
Portsmouth, NH: Heinemann.
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, firstname.lastname@example.org,
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, email@example.com,
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, firstname.lastname@example.org,
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.