PROOF AND PROOVING IN A MATHEMATICS COURSE FOR
PROSPECTIVE ELEMENTARY TEACHERS
Jane-Jane Lo, Raven McCrory
Western Michigan University & Michigan State University, U.S.A.
In this study, we propose a fourth element of proof to supplement Styliandies’
(2007) definition and we define three levels of proving activities in mathematics
courses for prospective teachers. Class episodes from one instructor are used to
illustrate how these three levels are instantiated and how the four elements of
proof emerge in his attempts to teach mathematical argument and proving to
future elementary teachers.
INTRODUCTION
The mathematics education community worldwide is facing the challenge of
improving students’ ability to prove and reason mathematically at all grade
levels. Prior research has shown that young children can make legitimate
mathematical arguments and even formal arguments that count as proof (Maher
& Martino, 1996; Stylianides, 2007). Yet, studies have also shown that many
prospective and practicing elementary teachers hold a procedure-based view of
mathematics and to verify mathematical statements, they rely on external
authority (textbook or instructor) or accept a few examples as evidence of truth
(Simon & Blume, 1996). One way to address this weakness is through
mathematics courses specially designed for prospective teachers.
In his study of elementary classrooms, Stylianides (2007) defines proof as
follows:
Proof is a mathematical argument, a connected sequence of assertions for or against a
mathematical claim, with the following characteristics:
1. It uses statements accepted by the classroom community (set of accepted
statements) that are true and available without further justification;
2. It employs forms of reasoning (modes of argumentation) that are valid and known
to, or within the conceptual reach of, the classroom community;
3. It is communicated with forms of expression (modes of argument representation)
that are appropriate and known to, or within the conceptual reach of, the classroom
community. (p. 291).
Stylianides uses this definition to analyze instruction involving proof and
illuminate possible actions teachers may take to support proving activities in
their classrooms. We propose that in order to understand the proving activities in
mathematics courses, we need to include a fourth element in the definition:
4. It is relative to objectives within the context (context dependence) which determine
what needs to be proved.
This fourth element of proof is related, but not identical, to using a set of
accepted statements. In one case, it may be adequate to prove that the result of
adding 2 and 3 is 5 by counting objects; in another, the proof may require
showing that the addition operation is in fact the correct operation to use. An
example below will illustrate this point more clearly.
In this study, we build upon Stylianides work and seek to understand proving
activities in mathematics courses for prospective elementary teachers. We
propose a three-level hierarchy of proving activities in mathematics courses for
prospective teachers. Class episodes from one instructor will be used to illustrate
how these three levels are instantiated and how the four elements of proof
emerge in his attempts to teach mathematical argument and proving to future
elementary teachers.
THEORETICAL FRAMEWORK
Researchers have identified three major roles for proof: to prove, to explain, and
to convince (Hanna, 1990; Hersh, 1993). In the context of K-12 mathematics,
there is a growing consensus that proof should not be taught as a meaningless
exercise that is used only to establish formally the truth of a statement, but should
also be taught as a tool that can be used to explain why a statement is true. The
definition of proof proposed by Stylianides (2007) quoted earlier reflects this
viewpoint.
Research on mathematical knowledge for teaching (cf., Hill, Ball & Rowan,
2008) as well as the conceptualization of proof outlined briefly outlined above
suggest that prospective teachers need to learn more than just constructing a valid
mathematical proof. They also need to be able to connect such understanding to
issues related to their students’ conceptions, curricular materials, and
instructional techniques that might arise from proving activities in elementary
classrooms. A better articulation of the goals for learning about proof in
mathematics courses for future teachers is required.
GOALS FOR PROVING ACTIVITY IN MATHEMATICS COURSE FOR
FUTURE TEACHERS
We posit that future elementary teachers need to learn proof at three different
levels:
a. Proof as a mathematical tool or technique for showing or verifying that
something is true (or false). This is what we might teach in a high school
geometry class, or what we might expect a student in a number theory class to do.
b. Proof as a mathematical object with particular characteristics and
standards. This includes making explicit the steps of a proof, the representations
used, the assumptions on which the proof is based, and all three of Stylianides
elements of proof. It implies explicit discussion of proof itself, rather than only
using proof to show that something is true.
c. Proof as developmental with the level of assumptions, arguments and
representations depending on the students’ age and grade level. A future teacher
needs to think about what children at a particular grade level could be expected
to know that could be used in a proof; what kind of arguments they are capable
of making; and what kinds of representations they can use. This knowledge needs
to be developed explicitly, and it draws on knowledge of content, students,
teaching and curriculum, cutting across nearly all aspects of knowledge for
teaching. The developmental nature of proof is related to the 4 th part of the
definition of proof given above, but the context dependence of proof is not only
about developmental level; it is also about goals and objectives of any given
proving activity.
These three levels correspond roughly to doing proofs, understanding the nature
of proof, and adapting the concept of proof to different developmental levels, all
important for mathematics teachers at any level of education.
In the remaining part of this paper, we will use episodes from a lesson on
divisions of fractions taught by one of our case study instructor, Pat, to illustrate
the usefulness of the above conceptualization.
EPISODES FROM A LESSON ON DIVISIONS OF FRACTIONS
The teacher, Pat (a pseudonym) presented the following word problem to the
class:
A batch of waffles requires ¾ of a cup of milk. You have two cups of milk. Exactly
how many batches of waffles could you make?
In a classroom where the instructional focus is on procedure, an explanation such
as the following may be accepted as “proof” by the classroom community.
(1) This is a division problem. I divide 2 by 3/4. To do so, I multiply 2 by 4/3. 2
times 4 is 8. 8/3 is the same as 2 2/3. So the answer is 2 2/3 batches.
In his classroom, Pat gave explicit instructions for his students to draw pictures
to model their solution process, to explain every quantity, diagram, and step in
their reasoning with reference to the original problem, and to come up with a
number sentence that matched the given problem. These steps were routine in his
classroom, part of the modes of argumentation and modes of argument
representation that were common to this class as illustrated by the following
accepted proof for the correct answer 2 ¾ batches that was the result of a
collective classroom effort.
(2) 2 cups of milk is equivalent to 8/4 cups. Two ¾ cups make 6/4 cups. 8/4 –
6/4= 2/4. So I know I can make at least 2 batches. The remaining ½ cup is
equivalent to 2/4 cup and can make 2/3 of a batch. Thus the answer is 2 ¾
batches.
The following picture was drawn on the board to support this proof.
Figure 1: Student explanation
The explanation in Figure 1 was agreed upon as a correct explanation of the right
answer, but the discussion of the problem did not end here. Pat asked the class to
think about what was wrong with writing 2 1/2, the answer given by some
students and a common error made in this type of division problem. Pat’s
question illustrates the contextual dependence of proving. In this instance
showing that a wrong answer was wrong became as important as showing that
the right answer was right. Several issues emerged during the subsequent
discussion. For example, one student pointed out that the unit for 2 was batches
(of cookies) but the unit for ½ was cup (of milk), so they could not be put
together. Mathematically, only quantities of the same unit could be added
together or subtracted from one another. Furthermore, Pat drew attention to the
fact that in the picture, “two shaded squares” (at the bottom) were used to
represent both two “1/4 cups” and two “1/3 batches”. In other words, each square
could be conceptualized as ¼ cup or 1/3 batch depending on what was being
counted as the whole.
Pat pushed students further to write “a number sentence that matches the problem
you were asked to solve”. He aimed at getting students to justify their choice of
operation. He drew upon prospective teachers’ experience working with
elementary students on one-step word problem involving whole numbers:
Pat: When you worked with your kids, you said something like … you have 3 fish
bowls, each bowl has four fishes in it. How many fishes are there altogether?
The kids would take 4 blocks, 4 more blocks and 4 more blocks, and go 1, 2, 3,
4, 5… Okay 12, there are 12. When you said, write a number sentence, what do
you want the kids to write?
Students: 3x4=12.
Pat: 3x4=12 so that's what you want them to write.”
Students offered several number sentences for the cookie problem, including 2 x
4/3. Pat pushed for a sentence that included only the numbers in the problem,
resulted in the right answer, and could be justified within the context of the
problem. These were standard modes of argument representation in Pat’s class.
After the sentence 2 ÷ 3/4 was offered, he asked “Why is it a division? If it is
division, what is it about this problem to make it a division?” This was not a
trivial task for the students. The first attempt was essentially a description of
steps to come up with the answer 2 2/3, that did not address the “why division”
question. Another student set up an algebraic equation: ¾ × x = 2, as number of
cups per batch times the number of batches equaled to the totally number for the
cups. To solve for x, one had to divide ¾ from both sides that led to 2 ÷ 3/4.
This was not accepted by Pat because it was not a mode of reasoning accessible
to elementary students. This is another example of the contextual dependence of
proof: although this explanation undoubtedly worked for the students in Pat’s
class, it was not acceptable given the objectives of the lesson. Finally, the idea of
measurement division was offered: giving the size of the group (e.g. 3/4 cup),
how many groups of that size can be formed from the given quantities (e.g. 2
cups)?
The discussion could have ended right there if the sole purpose of the discussion
had been to justify that division was the correct operation to model the given
word problem. But Pat continued, drawing his students’ attention to a related
word problem: “A batch of waffles requires 3 cups of milk. You have 8 cups of
milk. Exactly how many batches of waffles could you make?” He pointed to
similarity in terms of the physical action one might take to solve these problems
(e.g. repeated subtraction) as well as the pictorial representations (e.g. 8
separated boxes to represent “8” or 2 sets of four connected boxes to represent
“8/4”). The lesson ended with Pat reminding the class that the challenge they
themselves faced when coming up with a number sentence to model a fraction
division word problem was of the same nature of the challenges elementary
students faced when they tried to abstract the operational sense of division from
their physical modeling activity in the whole number context.
DISCUSSION
In this paper, we built on the definition of proof proposed by Stylianides (2007)
in an attempt to characterize the proofing activities in mathematics courses for
prospective elementary teachers. We found that it was necessary to consider the
goal of the activity in order to determine what will be accepted as a
mathematically valid proof by the classroom community.
In Pat’s class we see examples of the construction of proof using all four of the
elements from Stylianides (as modified by our 4th element). We also see the three
levels of proof: At level 1, students constructed a proof that 2 2/3 was the correct
solution to the problem. At level 2, Pat drew explicit attention to the nature of
proof when he set up the requirements for an acceptable argument. In essence he
told his students in this episode and at many other times during the semester that
a proof must account for every number in the problem and solution, must make
sense within the context of the problem, and must result in a correct answer. At
level 3, Pat made explicit the developmental nature of proof, rejecting an
algebraic proof because it would not work for the future students of these future
teachers. Level 3 was again apparent when he used a simpler example to
emphasize what young students might experience in their proving activities.
Analysis of this episode has illustrated one instructor’s effort to achieve multiple
goals through proving activities. By drawing students’ attention to various issues
such as modes of representations, modes of reasoning, common student errors,
and the abstraction involved from whole number to fraction operations, Pat
provided his students with ample opportunities to develop mathematical
knowledge that would be needed for engaging elementary students in reasoningbased proving activities. Future studies are needed to examine the effect of such
explicit attention on students’ ability to do proof, to understand the nature of
proof and to connect their content knowledge of proof to the knowledge of
students, teaching, and curriculum as they continue their professional path.
REFERENCES
Hanna, G. (1995). Some pedagogical aspects of proof. Interchange, 21, 6-13.
Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in
Mathematics, 24, 389-399.
Hill, H. C., Rowan, B. & Ball, D. L. (2005). Effects of teachers’ mathematical
knowledge for teaching on student achievement, American Educational
Research Journal, 42, 372-406.
Hill, H. C., Ball, D. L. & Schilling, S. G. (2008). Unpacking pedagogical content
knowledge: Conceptualizing and measuring Teachers’ topics-specific
knowledge of students. Journal for Research in Mathematics Education, 39,
372-400.
Maher, C. A., & Martino, A. M. (1996). The development of the idea of
mathematical proof: A 5-year case study. Journal for Research in
Mathematics Education, 27, 194-214.
Simon, M. A., Blume G. W. (1996). Justification in the mathematics classroom:
A study of perspective elementary school teachers. Journal of mathematical
behavior, 15, 3-31.
Stylianides A. J. (2007). Proof and proving in school mathematics, Journal of
Research in Mathematics Education, 38, 289-321.
Thompson, P. W. & Saldanha, L. A. (2003). Fractions and multiplicative
reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds). A research
companion to Principles and Standards for School Mathematics (pp. 95-113).
Reston, VA: National Council of Teachers of Mathematics.