All the episodes of mathematical activitydescribed in this article were recorded as grade twostudents worked in small groups at their classroommathematics center. Each group of studentsworked daily at a different center for about 45 min-utes, usually at the end of the day. An experiencedteacher, working as a research assistant on the pro-ject that I was conducting, supervised and inter-acted with the students at the mathematics center,

and made video and audio recordings andfield notes. During the three months ofobservations, the mathematics center activi-ties included playing games such as Set,Connect Four/Tic Tac Drop, and Master-

mind; reading and discussing stories such as TheDoorbell Rang and The 512 Ants on SullivanStreet; and engaging in mathematical activitieswith base-ten blocks, pattern blocks, paper folding,and geoboards. I chose these activities with the

classroom teacher and the research assistant fortheir potential to encourage reasoning, althoughnot all of them turned out to do so. The regularclassroom teacher supervised the other centers,which focused on art, reading, technology, andgames. For more details on the project, see Reid(2000).

The type of reasoning focused on in this articleis deductive reasoning. Deductive reasoning isusually described as drawing a conclusion frompremises, which are principles that are alreadyknown or hypothesized. For example, to reasonthat “Bill will attend the party” because “Bill nevermisses an event with balloons” and “there will beballoons at the party” is a deduction from the twopremises “Bill never misses an event with bal-loons” and “There will be balloons at the party.”Such deductions can be strung together into chains,and mathematical proofs are simply that: longchains of deductions.

The examples of deductive reasoning givenhere differ according to the number of premisesinvolved, the nature of those premises, andwhether only a single deduction or a chain ofdeductions is involved.

234 TEACHING CHILDREN MATHEMATICS

Describing Reasoningin Early ElementarySchool Mathematics

David A. Reid

David Reid, david.reid@acadiau.ca, teaches prospective teachers at Acadia University inWolfville, Nova Scotia, Canada. He is interested in mathematical reasoning at all ages,school-based research, and teacher professional development.

NCTM’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 rea-

soning 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 evalu-

ating learning activities for their usefulness in fostering reasoning.

This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Copyright © 2002 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

SpecializationThe kind of deductive reasoning that teachers areperhaps most likely to encounter is specialization.Specialization is determining something about aspecific situation by applying a general rule thatpertains 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 pro-vided an example of specialization while playingConnect Four with another boy, Ira. When Iraplaced one of his pieces in the position markedwith an asterisk in figure 1, Maurice put his armsbehind his head and said, “He got me.” The teacherasked why, and Maurice whispered to her the twopossible ways that Ira could win, by playing in thesecond or sixth columns. This demonstrated a spe-cialization for a general rule for winning that Mau-rice stated later: “Get three either way.” Mauricehad said earlier that he was good at Connect Fourbecause he played it at home, and he may havelearned the general rule for the game there. Thegeneral rule can be written as “If you have threepieces in a row with both ends free, then you canwin.” The specialization is “Ira has three pieces ina row [in columns three, four, and five] with freeends, so Ira can win.”

Simple DeductiveReasoningAnother common form of deductive reasoning issimple one-step deductive reasoning, in which thereasoning is a single deduction from two or morepremises. It differs from specialization in that spe-cialization involves only one premise.

When a grade two student makes a simple one-step deduction, it is not likely to be clearly stated.For example, consider this statement made byMaurice when playing the game Mastermind withthe teacher (see fig. 2): “It’s blue. ’Cause if there’sthree there. I changed the blue and I only got two.”

The teacher had asked Maurice if he had learnedanything new after receiving the two white pegsfor his second guess. Maurice’s response can be re-expressed as “Blue is correct because three of thecolors in my first guess are correct, and the onlyrelevant change I made in the colors from my firstguess to my second guess was leaving out blue,and only two colors in my second guess are cor-rect.” He had taken three premises about the situa-tion and drawn a conclusion that follows logicallyfrom them.

Simple one-step deductions are the building blocksof proving but need to be assembled into chains tomake a proof. Reasoning with chains of deductions iscalled simple multistep deductive reasoning.

Because of the emphasis on arithmetic in earlyelementary mathematics, children are most likelyto display simple multistep deductive reasoningwhile solving problems involving arithmetic. Thefollowing examples occurred as the teacher readthe book The Doorbell Rang by Pat Hutchins to thestudents at the mathematics center. While she read,she paused each time the doorbell rang and morepeople arrived to ask how twelve cookies could bedivided among the people present.

When the number of people reached four, Lauraquickly predicted, before being prompted to do soby the teacher, that each child would get threecookies. Saul agreed. He explained, “Because threeplus three would be, um, six, and another two

235DECEMBER 2002

FIG

UR

E 1 The Connect Four board as it appeared when Maurice demonstrated

specialization. Ira had just placed a black piece in the positionmarked with an asterisk. The object of the game is to place fourpieces in a line. Pieces can be added only at the top of a column.

1 2 3 4 5 6 7

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2

The Mastermind board as it appeared when Maurice made his sim-ple one-step deduction. The object is to guess the colors and orderin a four-color pattern picked by one’s opponent. Colored pegs areused to record the hidden pattern and the guesses. A white scoringpeg indicates that one of the pegs in the guess is the right color butin the wrong place.

Hidden Pattern

Guess

1

2

Score

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236 TEACHING CHILDREN MATHEMATICS

threes would be six, and because three plus three issix, and another three plus three would be anothersix. So it’s three.” Saul could add numbers onlytwo at a time, so his reasoning was broken intosteps: determining how many cookies two childrenwould get (three plus three), determining howmany the other two children would get (anothertwo threes), and finally determining that six plussix would give the required twelve cookies. He didnot express his final step, but it is the same as thesingle step expressed by Maurice when there wereonly two children sharing the cookies: “There’stwelve because six plus six equals twelve.” Thisexample is classified as a multistep deductionbecause Saul used a sequence of addition equationsto solve the problem.

Hypothetical DeductiveReasoningThe deductions that we have seen so far involvereasoning from something that is known. In math-ematics proofs, however, it is often necessary toreason from a hypothesis, something that is notknown to be true. This kind of reasoning might bedone either to show that something cannot be true,as in a proof by contradiction, or to show that if itwere true for one number it also would be true forthe next number, as in a proof by mathematicalinduction. Such reasoning, because it involves ahypothesis, is called hypothetical deductive rea-soning. Although hypothetical deductive reasoningis often thought to be more difficult than simplededuction from known statements, it can beobserved in the reasoning of early elementaryschool students, in both one-step and multistepforms.

An example of a hypothetical multistep deduc-tion occurred during another game of Mastermind(see fig. 3). After giving Kyla two white pegs forher third guess, the teacher asked her which pegshe thought might have been in the correct place.Kyla pointed to the blue peg in the first row andthen changed her mind. “I never got a black oneright there,” she said, pointing to the blue peg inthe second turn. She then indicated that the greenpeg could not be correct in the first try: “’Cause onthis one [turn three] I didn’t get a black.” Kylastated that the orange peg on turn one must be inthe correct spot, but then she realized that it couldnot be: “’Cause I got a black one right here—no!Oh my! It’s yellow.” Kyla’s reasoning includedthree hypotheses: The blue peg is in position three,the green peg is in position four, and the orangepeg is in position two. After each of these hypothe-ses was contradicted, Kyla concluded that the oneremaining case, the yellow peg in position one,must be correct.

The Role of the TeacherThe teacher’s presence was important to this studynot only because she was able to observe the chil-dren’s reasoning firsthand but also because of thequestions that she was able to ask. While playingMastermind and the other games, the childrennever asked other players to explain why theywanted to make a particular move or guess, evenwhen they played as a team. The teacher’s ques-tioning was essential to their voicing their reason-ing, which allowed the teacher and the other chil-dren to observe their thinking. For older childrenwho have been encouraged to explain their think-ing to the teacher, the habit of explaining becomesa part of their usual mathematical activity (seeZack [1999] and Lampert [1990] for work withgrade five students). The previous examples sug-gest that teachers of younger students also shouldask their students to explain their reasoning andshould listen carefully to the kinds of reasoningthat the students use.

ConclusionThe reasoning described in this article can be dis-tinguished in two ways. Some deductive reasoninginvolves only a single step, but some involves mul-tiple steps in a chain. Differences also exist in thenature and number of the premises. Specializationis always a single step from one premise, a generalrule of some kind, to a specific conclusion. Simpledeductions go from two or moreknown premisesto a conclusion. Hypothetical deductions gofrom a premise that is hypothesized to be true to a

FIG

UR

E 3

The Mastermind board after Kyla’s third guess. The black pegindicates that one of the colors in her guess is in the right place.

Hidden Pattern

Guess

1

2

Score

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3 ��

conclusion. Both types of deductions might reachconclusions in one step or multiple steps. Thesekinds of reasoning can be ranked by sophistication,with specialization being the simplest and multi-step hypothetical deduction being the most com-plex. Observing the kinds of reasoning that stu-dents use tells us something about them and thetasks in which they are involved. By choosingtasks that encourage more sophisticated reasoningand asking questions that elicit such reasoning,teachers can create effective environments forlearning mathematical reasoning.

ReferencesLampert, Magdalene. “When the Problem Is Not the Question

and the Solution Is Not the Answer: Mathematical Knowingand Teaching.” American Educational Research Journal 27(Spring 1990): 29–63.

National Council of Teachers of Mathematics (NCTM).Cur-riculum 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 inSchool Mathematics: Grade 2.” 2000. http://ace.acadiau.ca/~dreid/publications/PRISM-2/index.html.

Zack, Vicki. “Everyday and Mathematical Language in Chil-dren’s Argumentation About Proof.” Educational Review 51(1999): 129–46. ▲

237DECEMBER 2002

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