Describing Reasoning
in Early Elementary
School Mathematics
N
CTM’s Standards documents (1989, 2000) call for increased attention to the development of
mathematical reasoning at all levels. In order to accomplish this, teachers need to be attentive
to their students’ reasoning and aware of the kinds of reasoning that they observe. For teach-
ers at the early elementary level, this may pose a challenge. Whatever explicit discussion of mathematical reasoning they might have encountered in high school and university mathematics courses could have occurred
some time ago and is unlikely to have included the reasoning of children. The main intent of this article is to
give teachers examples of ways to reason mathematically so that they can recognize these kinds of reasoning
in their own students. This knowledge can be beneficial both in evaluating students’ reasoning and in evaluating learning activities for their usefulness in fostering reasoning.
David A. Reid
All the episodes of mathematical activity
described in this article were recorded as grade two
students worked in small groups at their classroom
mathematics center. Each group of students
worked daily at a different center for about 45 minutes, usually at the end of the day. An experienced
teacher, working as a research assistant on the project that I was conducting, supervised and interacted with the students at the mathematics center,
and made video and audio recordings and
field notes. During the three months of
observations, the mathematics center activities included playing games such as Set,
Connect Four/Tic Tac Drop, and Mastermind; reading and discussing stories such as The
Doorbell Rang and The 512 Ants on Sullivan
Street; and engaging in mathematical activities
with base-ten blocks, pattern blocks, paper folding,
and geoboards. I chose these activities with the
David Reid, david.reid@acadiau.ca, teaches prospective teachers at Acadia University in
Wolfville, Nova Scotia, Canada. He is interested in mathematical reasoning at all ages,
school-based research, and teacher professional development.
234
classroom teacher and the research assistant for
their potential to encourage reasoning, although
not all of them turned out to do so. The regular
classroom teacher supervised the other centers,
which focused on art, reading, technology, and
games. For more details on the project, see Reid
(2000).
The type of reasoning focused on in this article
is deductive reasoning. Deductive reasoning is
usually described as drawing a conclusion from
premises, which are principles that are already
known or hypothesized. For example, to reason
that “Bill will attend the party” because “Bill never
misses an event with balloons” and “there will be
balloons at the party” is a deduction from the two
premises “Bill never misses an event with balloons” and “There will be balloons at the party.”
Such deductions can be strung together into chains,
and mathematical proofs are simply that: long
chains of deductions.
The examples of deductive reasoning given
here differ according to the number of premises
involved, the nature of those premises, and
whether only a single deduction or a chain of
deductions is involved.
TEACHING CHILDREN MATHEMATICS
Copyright © 2002 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Simple Deductive
Reasoning
Another common form of deductive reasoning is
simple one-step deductive reasoning, in which the
reasoning is a single deduction from two or more
premises. It differs from specialization in that specialization involves only one premise.
When a grade two student makes a simple onestep deduction, it is not likely to be clearly stated.
For example, consider this statement made by
Maurice when playing the game Mastermind with
the teacher (see fig. 2): “It’s blue. ’Cause if there’s
three there. I changed the blue and I only got two.”
The teacher had asked Maurice if he had learned
anything new after receiving the two white pegs
for his second guess. Maurice’s response can be reexpressed as “Blue is correct because three of the
colors in my first guess are correct, and the only
relevant change I made in the colors from my first
guess to my second guess was leaving out blue,
and only two colors in my second guess are correct.” He had taken three premises about the situation and drawn a conclusion that follows logically
from them.
Simple one-step deductions are the building blocks
of proving but need to be assembled into chains to
make a proof. Reasoning with chains of deductions is
called simple multistep deductive reasoning.
DECEMBER 2002
FIGURE 1
The kind of deductive reasoning that teachers are
perhaps most likely to encounter is specialization.
Specialization is determining something about a
specific situation by applying a general rule that
pertains to the situation. An example is concluding
“This penguin has feathers” from the general rule
“All penguins have feathers.”
In the classroom that I studied, Maurice provided an example of specialization while playing
Connect Four with another boy, Ira. When Ira
placed one of his pieces in the position marked
with an asterisk in figure 1, Maurice put his arms
behind his head and said, “He got me.” The teacher
asked why, and Maurice whispered to her the two
possible ways that Ira could win, by playing in the
second or sixth columns. This demonstrated a specialization for a general rule for winning that Maurice stated later: “Get three either way.” Maurice
had said earlier that he was good at Connect Four
because he played it at home, and he may have
learned the general rule for the game there. The
general rule can be written as “If you have three
pieces in a row with both ends free, then you can
win.” The specialization is “Ira has three pieces in
a row [in columns three, four, and five] with free
ends, so Ira can win.”
The Connect Four board as it appeared when Maurice demonstrated
specialization. Ira had just placed a black piece in the position
marked with an asterisk. The object of the game is to place four
pieces in a line. Pieces can be added only at the top of a column.
1
2
3
*
FIGURE 2
Specialization
4
5
6
7
The Mastermind board as it appeared when Maurice made his simple one-step deduction. The object is to guess the colors and order
in a four-color pattern picked by one’s opponent. Colored pegs are
used to record the hidden pattern and the guesses. A white scoring
peg indicates that one of the pegs in the guess is the right color but
in the wrong place.
Hidden Pattern
Guess
Score
1
2
Because of the emphasis on arithmetic in early
elementary mathematics, children are most likely
to display simple multistep deductive reasoning
while solving problems involving arithmetic. The
following examples occurred as the teacher read
the book The Doorbell Rang by Pat Hutchins to the
students at the mathematics center. While she read,
she paused each time the doorbell rang and more
people arrived to ask how twelve cookies could be
divided among the people present.
When the number of people reached four, Laura
quickly predicted, before being prompted to do so
by the teacher, that each child would get three
cookies. Saul agreed. He explained, “Because three
plus three would be, um, six, and another two
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threes would be six, and because three plus three is
six, and another three plus three would be another
six. So it’s three.” Saul could add numbers only
two at a time, so his reasoning was broken into
steps: determining how many cookies two children
would get (three plus three), determining how
many the other two children would get (another
two threes), and finally determining that six plus
six would give the required twelve cookies. He did
not express his final step, but it is the same as the
single step expressed by Maurice when there were
only two children sharing the cookies: “There’s
twelve because six plus six equals twelve.” This
example is classified as a multistep deduction
because Saul used a sequence of addition equations
to solve the problem.
Hypothetical Deductive
Reasoning
FIGURE 3
The deductions that we have seen so far involve
reasoning from something that is known. In mathematics proofs, however, it is often necessary to
reason from a hypothesis, something that is not
known to be true. This kind of reasoning might be
done either to show that something cannot be true,
as in a proof by contradiction, or to show that if it
were true for one number it also would be true for
the next number, as in a proof by mathematical
induction. Such reasoning, because it involves a
hypothesis, is called hypothetical deductive reasoning. Although hypothetical deductive reasoning
is often thought to be more difficult than simple
deduction from known statements, it can be
observed in the reasoning of early elementary
school students, in both one-step and multistep
forms.
236
The Mastermind board after Kyla’s third guess. The black peg
indicates that one of the colors in her guess is in the right place.
Hidden Pattern
Guess
Score
1
2
3
An example of a hypothetical multistep deduction occurred during another game of Mastermind
(see fig. 3). After giving Kyla two white pegs for
her third guess, the teacher asked her which peg
she thought might have been in the correct place.
Kyla pointed to the blue peg in the first row and
then changed her mind. “I never got a black one
right there,” she said, pointing to the blue peg in
the second turn. She then indicated that the green
peg could not be correct in the first try: “’Cause on
this one [turn three] I didn’t get a black.” Kyla
stated that the orange peg on turn one must be in
the correct spot, but then she realized that it could
not be: “’Cause I got a black one right here—no!
Oh my! It’s yellow.” Kyla’s reasoning included
three hypotheses: The blue peg is in position three,
the green peg is in position four, and the orange
peg is in position two. After each of these hypotheses was contradicted, Kyla concluded that the one
remaining case, the yellow peg in position one,
must be correct.
The Role of the Teacher
The teacher’s presence was important to this study
not only because she was able to observe the children’s reasoning firsthand but also because of the
questions that she was able to ask. While playing
Mastermind and the other games, the children
never asked other players to explain why they
wanted to make a particular move or guess, even
when they played as a team. The teacher’s questioning was essential to their voicing their reasoning, which allowed the teacher and the other children to observe their thinking. For older children
who have been encouraged to explain their thinking to the teacher, the habit of explaining becomes
a part of their usual mathematical activity (see
Zack [1999] and Lampert [1990] for work with
grade five students). The previous examples suggest that teachers of younger students also should
ask their students to explain their reasoning and
should listen carefully to the kinds of reasoning
that the students use.
Conclusion
The reasoning described in this article can be distinguished in two ways. Some deductive reasoning
involves only a single step, but some involves multiple steps in a chain. Differences also exist in the
nature and number of the premises. Specialization
is always a single step from one premise, a general
rule of some kind, to a specific conclusion. Simple
deductions go from two or more known premises
to a conclusion. Hypothetical deductions go
from a premise that is hypothesized to be true to a
TEACHING CHILDREN MATHEMATICS
conclusion. Both types of deductions might reach
conclusions in one step or multiple steps. These
kinds of reasoning can be ranked by sophistication,
with specialization being the simplest and multistep hypothetical deduction being the most complex. Observing the kinds of reasoning that students use tells us something about them and the
tasks in which they are involved. By choosing
tasks that encourage more sophisticated reasoning
and asking questions that elicit such reasoning,
teachers can create effective environments for
learning mathematical reasoning.
References
Lampert, Magdalene. “When the Problem Is Not the Question
and the Solution Is Not the Answer: Mathematical Knowing
and Teaching.” American Educational Research Journal 27
(Spring 1990): 29–63.
National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics.
Reston, Va.: NCTM, 1989.
———. Principles and Standards for School Mathematics.
Reston, Va.: NCTM, 2000.
Reid, David A. “The Psychology of Students’ Reasoning in
School Mathematics: Grade 2.” 2000. http://ace.acadiau.ca
/~dreid/publications/PRISM-2/index.html.
Zack, Vicki. “Everyday and Mathematical Language in Children’s Argumentation About Proof.” Educational Review 51
(1999): 129–46. ▲
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