By Reid, David A.
Teaching Children Mathematics , Vol. 9, No. 4
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 teachers 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.
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 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.
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. …