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Article: Describing Reasoning in Early Elementary School Mathematics

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 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 leaming activities for their usefulness in fostering reasoning. All the episodes of mathematical activity classroom teacher and the research assistant for described in this article were recorded as grade two their potential to encourage reasoning, although students worked in small groups at their classroom not all of them turned out to do so. The regular mathematics center. Each group of students classroom teacher supervised the other centers, worked daily at a different center for about 45 min- which focused on art, reading, technology, and utes, usually at the end of the day. An experienced games. For more details on the project, see Reid teacher, working as a research assistant on the pro- (2000) ject that I was conducting, supervised and inter- The type of reasoning focused on in this article acted with the students at the mathematics center, is deductive reasoning. Deductive reasoning is and made video and audio recordings and usually described as drawing a conclusion from field notes. During the three months of premises, which are principles that are already David A. Reid observations, the mathematics center activi- known or hypothesized. For example, to reason ties included playing games such as Set, that "Bill will attend the party" because "Bill never Connect Four/Tic Tac Drop, and Master- misses an event with balloons" and "there will be mind; reading and discussing stories such as The balloons at the party" is a deduction from the two Doorbell Rang and The 512 Ants on Sullivan premises "Bill never misses an event with bal- Street, and engaging in mathematical activities loons" and "There will be balloons at the party." with base-ten blocks, pattern blocks, paper folding, Such deductions can be strung together into chains, and geoboards. I chose these activities with the and mathematical proofs are simply that: long chains of deductions. The examples of deductive reasoning given David Reid, david reid@ acadiau.ca, teaches prospective teachers at Acadia University in here differ according to the number of premises Wolfville, Nova Scotia, Canada. He is interested in mathematical reasoning at all ages, involved, the nature of those premises, and school-based research, and teacher professional development. whether only a single deduction or a chain of deductions is involved. 234 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.Specialization The kind of deductive reasoning that teachers are perhaps most likely to encounter is specialization. Specialization 1s 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 mle \"All penguins have feathers \" In the classroom that I studied, Maurice pro- vided 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 spe- cialization for a general rule for winning that Mau- rice 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 leamed the general mile for the game there. The general mile 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, 50 Ira can win.\" 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 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 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 leamed anything new after receiving the two white pegs for his second guess. Maurice's response an be re- expressed as \"Blue is comect 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 cor- rect.\" He had taken three premises about the situa- tion and drawn a conclusion that follows logically from them. Simple one-step deductions are the aldmg blocks of proving but need to be assembled mto chams to make a proof. Reasoning with chains of deductions is called simple multistep deductive reasoning. DECEMBER 2002 The Connect Four board as It appeared when Maurice demonstrated speclallzation. Ira had just placed a black plece In the position marked with an asterisk. The object of the game Is to place four pleces In a line. Pleces can be added only at the top of a column. The Mastermind board as It appeared when Maurlce made his sim- ple 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 Indlcates that one of the pegs In the guess Is the right color but In the wrong place. Hidden Pattern Guess Score Q00 Because of the emphasis on anthmetic m 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 amived to ask how twelve cockies 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 235 236 threes would be six, and because three plus three 1s 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 cockies: \"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 The deductions that we have seen so far ivolve reasoning from something that is known. In math- ematics 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 rea- soning. 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. 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 | An example of a hypothetical multistep deduc- tion 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. T never got a black one night there,\" she said, pointing to the blue peg m the second tum. 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 fum one must be m the comect 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 15 in position three, the green peg is in position four, and the orange peg 1s In position two. After each of these hypothe- ses was contradicted, Kyla concluded that the one remaining case, the yellow peg In position one, must be comect. The Role of the Teacher The teacher's presence was important to this study not only because she was able to observe the chil- dren'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 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 children who have been encouraged to explain their think- ing 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 sug- gest that teachers of younger students also should ask their students to explain their reasoning and should listen carefully to the kinds of reasoming that the students use. Conclusion The reasoning described in this article can be dis- tinguished in two ways. Some deductive reasoning nvolves only a single step, but some involves mul- tiple steps in a chain. Differences also exist in the nature and number of the premises. Specialization 1s always a single step from one premise, a general mule of some kind, to a specific conclusion. Simple deductions go from two or more inown premises to a conclusion. Hypothetical deductions go from a premise that is Aypothesized to be true to a TEACHING CHILDREN MATHEMATICS conclusion. Both types of deductions might reach STATEMENT OF OWNERSHIP conclusions in one step or multiple steps. These Statement of ownership, management, and circulation (Required by 39 U.S.C. 3685). 1. Publication kinds of reasoning can be ranked by sophistication, title: Teaching Children Mathematics. 2. Publication number: 0004-136X. 3. Filing date: September with specialization being the simplest and multi- 2002. 4. Issue frequency: September-May, monthly. E. Number of issues published annually: 9. 6. Annual subscription price: $28. 7. Complete mailing address of known office of publication: National step hypothetical deduction being the most com- Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502, Fairfax County. plex. Observing the kinds of reasoning that stu- Contact person: Sandy Berger, (703) 620-9840, ex. 2192. 8. Complete mailing address of headquarters or general business office of publisher: same as #7. 9. Full names and complete mailing addresses of dents use tells us something about them and the publisher, editor, and managing editor. Publisher. National Council of Teachers of Mathematics, 1906 tasks in which they are involved. By choosing Association Drive, Reston, VA 20191-1502. Editor: non e. Managing editor. Sandy Berger, 1906 Asso- ciation Drive, Reston, VA 20191-1502. 10. Owner: National Council of Teachers of Mathematics (non- tasks that encourage more sophisticated reasoning profit organization), 501 (c)3, 1906 Association Drive, Reston, VA 20191-1502. 11. Known bondhold- and asking questions that elicit such reasoning, ers, mortgagees, and other security holders owning or holding 1 percent or more of total amount of bonds, mortgages, or other securities: none. 12. Tax status. The purpose, function, and nonprofit status teachers can create effective environments for of this organization and the exempt status for federal has not changed during pre- learning mathematical reasoning. ceding 12 months. 13. Publication title: Teaching Children Mathematics. 14. Issue date for circulation data below: September 2002. 16. Extent 's each issue during preceding 12 months. A. Total number of copies: 33,778. B. Paid and/or requested circulation: (1) paid/requested outside-county mail subscriptions stated on form 3541: 29,395; (2) paid in-county sub- References scriptions stated on form 3541: none; (3) sales through dealers and carriers, street vendors, counter sales and other non-USPS paid distribution: none; (4) other classes mailed through the USPS: none. C. Total Lampert, Magdalene. "When the Problem Is Not the Question paid and/or requested circulation: 29,395. D. Free distribution by mail: (1) outside-county as stated on and the Solution Is Not the Answer. Mathematical Knowing form 3541: 3,900; (2) in-county as stated on form 3541: none; (3) other classes mailed through the USPS: none. E. Free distribution outside c. F. Total free distribution: 3,900. G. Total dis- and Teaching." American Educational Research Journal 27 tribution: 33,295. H. Copies not distributed: 483. I. Total: 33,778. J. Percent paid and/or requested cir- (Spring 1990): 29-63. culation: 88%. 15. Extent and nature of circulation. No. ingle issue published nearest to fil- National Council of Teachers of Mathematics (NCTM). Cur- ing date. A. Total number of copies: 31,000. B. Paid and/or requested circulation: (1) paid/requested riculum and Evaluation Standards for School Mathematics. outside-county mail subscriptions stated on form 3541: 29,305; (2) paid in-county subscriptions stated on form 3541: none; (3) sales through dealers and carriers, street vendors, counter sales, and other non- Reston, Va.: NCTM, 1989. USPS paid distribution: none; (4) other classes mailed through the USPS: none. C. Total paid and/or -. Principles and Standards for School Mathematics. requested circulation: 29,305. D. Free distribution by mail: (1) outside- de-county as stated on form 3541: Reston, Va.: NCTM, 2000 1,600; (2) in-county as stated on form 3541: none; (3) other classes mailed through the USPS: none. E. Reid, David A. "The Psychology of Students' Reasoning in Free distribution outside the mail: none . F. Total free distribution: 1,600. G. Total distribution: 30,905. H. Copies not distributed: 95. I. Total: 31,000. J. Percent paid and/or requested circulation: 95%. 16. School Mathematics: Grade 2." 2000. http://ace.acadiau.ca Publication of statement of ownership will be printed in th ecember 2002 issue of this publication. ~dreid/publications/PRISM-2/index .html. 17. Signature and title of editor, publisher, business manager, or owner: Sandra L. Berger, managing edi- Zack, Vicki. "Everyday and Mathematical Language in Chil- tor, September 26, 2002. I certify that all information furnished on this form is true and complete. I dren's Argumentation About Proof." Educational Review 51 understand that anyone who furnishes false or misleading information on this form or who omits mate- rial or information requested on the form may be subject to criminal sanctions (including fines and (1999): 129-46. A imprisonment) and/or civil sanctions (including civil penalties). DECEMBER 2002 237