In grades K-4, the study of mathematics should emphasize reasoning so that students can--

• draw logical conclusions about mathematics;
• use models, known facts, properties, and relationships to explain their thinking;
• justify their answers and solution processes;
• use patterns and relationships to analyze mathematical situations;
• believe that mathematics makes sense.

Focus

A major goal of mathematics instruction is to help children develop the belief that they have the power to do mathematics and that they have control over their own success or failure. This autonomy develops as children gain confidence in their ability to reason and justify their thinking. It grows as children learn that mathematics is not simply memorizing rules and procedures but that mathematics makes sense, is logical, and is enjoyable. A classroom that values reasoning also values communicating and problem solving, all of which are components of the broad goals of the entire elementary school curriculum.

A climate should be established in the classroom that places critical thinking at the heart of instruction. Both teachers' and children's statements should be open to question, reaction, and elaboration from others in the classroom. Such a climate depends on all members of the class expressing genuine respect and support for one another's ideas. Children need to know that being able to explain and justify their thinking is important and that how a problem is solved is as important as its answer. This mind-set is established when children have opportunities to apply their reasoning skills and when justifying one's thinking is an expected component of problem discussions.

Discussion

This standard's descriptor, "Mathematics as Reasoning," was purposely chosen. Mathematics is reasoning. One cannot do mathematics without reasoning. The standard does not suggest, however, that formal reasoning strategies be taught in grades K--4. At this level, mathematical reasoning should involve the kind of informal thinking, conjecturing, and validating that helps children to see that mathematics makes sense. Consistent use of such questions as "Why do you think that's a good answer?" or "Do you think that you would get the same answer if you used these other materials?" conveys to the children the importance of critical thinking and establishes a spirit of inquiry.

Children should be encouraged to justify their solutions, thinking processes, and conjectures in a variety of ways. Manipulatives and other physical models help children relate processes to their conceptual under-pinnings and give them concrete objects to talk about in explaining and justifying their thinking. Observing children interact with objects in this way allows teachers to reinforce thinking processes and evaluate any possible misunderstandings.

Creating and extending patterns of manipulative materials and recognizing relationships within patterns require children to apply analytical and spatial reasoning. See figure 3.1.

Fig. 3.1

The kindergartner who created the attribute block pattern in figure 3.2 proudly announced that she had four patterns in one.

Fig. 3.2

Pointing to each element in turn, she said, "See, there's triangle, triangle, circle, circle, square, and square. That's one pattern. Then there's small, large, small, large, small, and large. That's the second pattern. Then there's thin, thick, thin, thick, thin, and thick. That's the third pattern. And the fourth pattern is blue, blue, red, red, yellow, and yellow. The triangles are blue, the circles are red, and the squares are yellow."

Children also reason analytically when they identify valid arguments. When the class considers 35 - 19 = [_], the teacher can ask questions like these: "Do you think it would help to know that 35 - 20 = 15?" "How would it help to think of 19 as 15 + 4?" "Would it help to count on from 19 to 35?" It is also important for children to recognize invalid arguments, such as "Would it help to count backward from 19?"

Many problems can be solved by the process of elimination, in which children systematically select the items that satisfy one or more given conditions by eliminating those that do not. "Who Am I?" and "What Am I?" games require this kind of thinking in both creating and solving problems. See figure 3.3.

Fig. 3.3

These activities also give students a chance to encounter informally several important ideas, such as the language of logic, the use of a counter-example, and distinctions between relevant and irrelevant information. The use of and, or, and not in these activities illustrates the language of logic. The reasoning in the "What Am I?" game is based on a counter-example involving known properties of an equilateral triangle. "If I have three sides, I am a triangle. But I can't be a triangle because all triangles with equal angles have equal sides and I do not have equal sides." The clue "I am not 25" in the "Who Am I?" game is irrelevant because another clue identifies the number as even: Clearly the number cannot be 25, and this information is of no value in solving the problem.

Applying reasoning skills to discover a relationship they have not recognized before can be an exhilarating experience for children, as a group of third graders learned. They were using a calculator to explore number relationships when they noticed that if one addend is decreased by any amount and another addend is increased by the same amount, their sum remains the same. After checking their conjecture with a variety of numbers, they recorded it as a discovery so that it could be shared with the rest of the class. See figure 3.4

Fig. 3.4

Our Discovery: When you add, if you make one part bigger and the other part gets the same amount smaller, you always get the same answer.

One member of the group thought the relationship should "work" for subtraction, too, until a partner showed several cases for which it did not work.

These children applied analytical reasoning and developed and tested conjectures, one of which they rejected on the basis of counterexamples.

An informal introduction to proportional reasoning is appropriate at the K-4 level. The problem-solving context of the following example also reinforces many of the reasoning processes already discussed. See Figure 3.4a

Fig. 3.4a

I have a shape that can be covered with twelve of these triangles. How many of these parallelograms would I need to cover my shape? How many of these trapezoids will cover my shape?

Since the problem concerns physical objects, the students can recognize visually that two triangles cover a parallelogram, that three triangles cover a trapezoid, and that the entire shape can be covered by six parallelograms or by four trapezoids. To justify their conclusion, the children can use twelve triangles to make a shape and check to see whether it can be covered by six parallelograms or four trapezoids. Students should also realize that some shapes composed of twelve triangles cannot be covered by parallelograms or trapezoids.

Mathematical reasoning cannot develop in isolation. As illustrated in this discussion, the ability to reason is a process that grows out of many experiences that convince children that mathematics makes sense.