In grades 5-8, reasoning shall permeate the mathematics curriculum so that students can--

• recognize and apply deductive and inductive reasoning;
• understand and apply reasoning processes, with special attention to spatial reasoning and reasoning with proportions and graphs;
• make and evaluate mathematical conjectures and arguments;
• validate their own thinking;
• appreciate the pervasive use and power of reasoning as a part of mathematics.

Focus

Reasoning is fundamental to the knowing and doing of mathematics. Although most disciplines have standards of evaluation by which new theories or discoveries are judged, nowhere are these standards as explicit and well formulated as they are in mathematics. Conjecturing and demonstrating the logical validity of conjectures are the essence of the creative act of doing mathematics. To give more students access to mathematics as a powerful way of making sense of the world, it is essential that an emphasis on reasoning pervade all mathematical activity. Students need a great deal of time and many experiences to develop their ability to construct valid arguments in problem settings and evaluate the arguments of others.

The development of logical reasoning is tied to the intellectual and verbal development of students. Through grades 5-8, students' reasoning abilities change. Whereas most fifth graders still are concrete thinkers who depend on a physical or concrete context for perceiving regularities and relationships, many eighth-grade students are capable of more formal reasoning and abstraction. Even the most advanced students at the 5-8 level, however, might use concrete materials to support their reasoning; this is especially true for spatial reasoning. The 5-8 mathematics curriculum should pay special attention to the development of student's abilities to use proportional and spatial reasoning and to reason from graphs.

Technology can foster environments in which students' growing curiosity can lead to rich mathematical invention. In these environments, the control of exploring mathematical ideas is turned over to students. Both inductive and deductive reasoning come into play as students make conjectures and seek to explain why they are valid. Whether encouraged by technology or by challenging mathematical situations posed in the classroom, this freedom to explore, conjecture, validate, and to convince others is critical to the development of mathematical reasoning in the middle grades.

Discussion

The seeds of logical thinking are planted as students learn to describe objects or processes accurately and to elaborate their properties, similarities, differences, and relationships. Students should be encouraged to explain their reasoning in their own words. Listening to their peers and their teacher describe other strategies helps students refine their thoughts and the language they use to express their thoughts. Such questions as the following should abound in the mathematics classes: Why? What if ..... ? Can you give an example of ...... ? Can you find a counterexample? Do you see a pattern? Is this always true? Sometimes true? Never true? How do you know? Such questions prompt students to validate and value their own thinking.

Identifying patterns is a powerful problem-solving strategy. It is also the essence of inductive reasoning. As students explore problem situations appropriate to their grade level, they can often consider or generate a set of specific instances, organize them, and look for a pattern. These, in turn, can lead to conjectures about the problem. Students should be encouraged to validate these conjectures by constructing supporting arguments, which can be at many levels of sophistication.

Students at these grade levels should be exposed to problem situations that are challenging but within reach. For example, students can be asked to explore the numbers that occur between twin primes for primes greater than 3. They might first look for twin primes to find examples and then make, test, and validate conjectures.

5 6 7; 11 12 13; 17 18 19; 29 30 31

What do 6, 12, 18, and 30 have in common? Is this true for all twin primes? Why or why not? This problem presents an excellent opportunity for a class to use a computer to generate lists of primes.

Students can be introduced to many kinds of mathematical reasoning. To help them recognize one aspect of the beauty of mathematics, groups of students can each be asked to cut out a triangle of their choice and then see whether they can "tile" the plane using copies of their own triangle. Less sophisticated students might say, "It works!" on the basis of a single instance. Others might look at the group's examples and see that the pattern seems to work for every triangle; students eventually might reason from the angle-sum property of triangles that the tiling method always works. See figure 3.1.

Fig. 3.1. Tiling method

Students can use reasoning to illustrate when something never works. For example, students can be given a particular collection of numbers (3, 6, 12, 15, 21, 27, 42, 51) and be asked to find a set of these numbers that sums to 100. Once it is clear that no one will succeed, students can be challenged to reason why such a sum is impossible: Any sum of multiples of 3 is a multiple of 3, so the sum cannot be 100.

Students should also encounter situations in which reasoning from a counterexample is useful: Suppose one has two numbers that divide 72. Does their product also divide 72? (2 divides 72; 3 divides 72; 2 x 3 divides 72.) Is this always true? A counterexample, such as 4 divides 72 and 8 divides 72, shows that the product does not always divide 72.

The ability to reason proportionally develops in students throughout grades 5-8. It is of such great importance that it merits whatever time and effort must be expended to assure its careful development. Students need to see many problem situations that can be modeled and then solved through proportional reasoning. Such problems can range from simple to complex, as illustrated by the three problems below:

Students observe that their classroom has 16 windowpanes. If every room has 16 panes, how many windowpanes are in a 20-room school?

A shop sells special cookies for $1 each, or 10 for $9. Tom wishes to buy 30 cookies. How much should he pay? (Vergnaud 1988)

A group of 8 people are going camping for 3 days and need to carry their own water. They read in a guide book that 12.5 liters are needed for a party of 5 persons for 1 day. How much water should they carry? (Vergnaud 1988)

Geometric as well as number situations should be created. For example, similarity of figures and scaling--in fact, all scale-model-to-real-object problems--provide appropriate settings for proportional reasoning.

If given opportunities to reason from graphs about interesting situations, students can develop an appreciation for the problem-solving potential of making, using, and talking about graphs. The following example (Swan 1985) offers a flavor of the potential of graphical representations as tools for reasoning.

Students are given a carefully drawn picture of a roller-coaster track (fig. 3.2).

Fig. 3.2. Roller coaster

The challenge is to sketch a graph (with no numbers) to represent the speed of the roller coaster versus its position on the track.

Now, to reverse the problem, students are given a part of the graph of speed versus position for another roller coaster (fig. 3.3). The question becomes,

Fig. 3.3. Roller-coaster graph

What does the roller-coaster track look like?

Investigating graphical representations and their relationships to algebraic representations can give students a real sense of the dynamic relationship between the variables. Such problem settings also allow students to reason directly to, and hypothetically from, graphs.

Students can develop their spatial reasoning abilities in a variety of interesting settings. They can gather a collection of small objects, such as spools, golf balls, small footballs, small cans and bottles, and foam cups, and then try to draw what they think the shape of the shadow of each object might look like. Students can then test their conjectures by using the overhead projector to cast shadows; they can also be asked to identify the object solely on the appearance of its shadow. Relating the size of the shadow to the size of the object also allows the use of proportional thinking.

Reasoning about what is happening and why should be a constant part of the study of mathematics. Students in grades 5-8 should explore mathematical reasoning through problem situations that are appropriate to their ages and interest.