Young students are just forming their store of mathematical knowledge, but even the youngest can reason from their own experiences (Bransford, Brown, and Cocking 1999). Although young children are working from a small knowledge base, their logical reasoning begins before school and is continually modified by their experiences. Teachers should maintain an environment that respects, nurtures, and encourages students so that they do not give up their belief that the world, including mathematics, is supposed to make sense.

Although they have yet to develop all the tools used in mathematical reasoning, young students have their own ways of finding mathematical results and convincing themselves that they are true. Two important elements of reasoning for students in the early grades are pattern-recognition and classification skills. They use a combination of ways of justifying their answers—perception, empirical evidence, and short chains of deductive reasoning grounded in previously accepted facts. They make conjectures and reach conclusions that are logical and defensible from their perspective. Even when they are struggling, their responses reveal the sense they are making of mathematical situations.

Young students naturally generalize from examples (Carpenter and Levi 1999), so teachers should guide them to use examples and counterexamples to test whether their generalizations are appropriate. By the end of second grade, students should be using this method for testing their conjectures and those of others.

Even though the student's wording—that shapes were "equal"—was not correct, he was demonstrating powerful reasoning as he used the blocks to justify his idea. In situations such as this, teachers could point to the faces of the two smaller blocks and respond, "You discovered that » the area of this square equals the area of this triangle because each of them is half the area of the same larger rectangle."

What should be the teacher's role in developing reasoning and proof in prekindergarten through grade 2?

Teachers should create learning environments that help students recognize that all mathematics can and should be understood and that they are expected to understand it. Classrooms at this level should be stocked with physical materials so that students have many opportunities to manipulate objects, identify how they are alike or different, and state generalizations about them. In this environment, students can discover and demonstrate general mathematical truths using specific examples. Depending on the context in which events such as the one illustrated by figure 4.29 take place, teachers might focus on different aspects of students' reasoning and continue conversations with different students in different ways. Rather than restate the student's discovery in more-precise language, a teacher might pose several questions to determine whether the student was thinking about equal areas of the faces of the blocks, or about equal volumes. Often students' responses to inquiries that focus their thinking help them phrase conclusions in more-precise terms and help the teacher decide which line of mathematical content to pursue.

Teachers should prompt students to make and investigate mathematical conjectures by asking questions that encourage them to build on what they already know. In the example of investigating patterns on a hundred board, for instance, teachers could challenge students to consider other ideas and make arguments to support their statements: "If we extended the hundred board by adding more rows until we had a thousand board, how would the skip-counting patterns look?" or "If we made charts with rows of six squares or rows of fifteen squares to count to a hundred, would there be patterns if we skip-counted by twos or fives or by any numbers?"

Through discussion, teachers help students understand the role of nonexamples as well as examples in informal proof, as demonstrated in a study of young students (Carpenter and Levi 1999, p. 8). The students seemed to understand that number sentences like 0 + 5869 = 5869 were always true. The teacher asked them to state a rule. Ann said, "Anything with a zero can be the right answer." Mike offered a counterexample: "No. Because if it was 100 + 100 that's 200." Ann understood that this invalidated her rule, so she rephrased it, "I said, umm, if you have a zero in it, it can't be like 100, because you want just plain zero like 0 + 7 = 7."

The students in the study could form rules on the basis of examples. Many of them demonstrated the understanding that a single example was not enough and that counterexamples could be used to disprove a conjecture. However, most students experienced difficulty in giving justifications other than examples.

From the very beginning, students should have experiences that help them develop clear and precise thought processes. This development of reasoning is closely related to students' language development and is dependent on their abilities to explain their reasoning rather than just » give the answer. As students learn language, they acquire basic logic words, including not, and, or, all, some, if...then, and because. Teachers should help students gain familiarity with the language of logic by using such words frequently. For example, a teacher could say, "You may choose an apple or a banana for your snack" or "If you hurry and put on your jacket, then you will have time to swing." Later, students should use the words modeled for them to describe mathematical situations: "If six green pattern blocks cover a yellow hexagon, then three blues also will cover it, because two greens cover one blue."

Sometimes students reach conclusions that may seem odd to adults, not because their reasoning is faulty, but because they have different underlying beliefs. Teachers can understand students' thinking when they listen carefully to students' explanations. For example, on hearing that he would be "Star of the Week" in half a week, Ben protested, "You can't have half a week." When asked why, Ben said, "Seven can't go into equal parts." Ben had the idea that to divide 7 by 2, there could be two groups of 3, with a remainder of 1, but at that point Ben believed that the number 1 could not be divided.

Teachers should encourage students to make conjectures and to justify their thinking empirically or with reasonable arguments. Most important, teachers need to foster ways of justifying that are within the reach of students, that do not rely on authority, and that gradually incorporate mathematical properties and relationships as the basis for the argument. When students make a discovery or determine a fact, rather than tell them whether it holds for all numbers or if it is correct, the teacher should help the students make that determination themselves. Teachers should ask such questions as "How do you know it is true?" and should also model ways that students can verify or disprove their conjectures. In this way, students gradually develop the abilities to determine whether an assertion is true, a generalization valid, or an answer correct and to do it on their own instead of depending on the authority of the teacher or the book.