The assessment of students' ability to reason mathematically should provide evidence that they can--

• use inductive reasoning to recognize patterns and form conjectures;
• use reasoning to develop plausible arguments for mathematical statements;
• use proportional and spatial reasoning to solve problems;
• use deductive reasoning to verify conclusions, judge the validity of arguments, and construct valid arguments;
• analyze situations to determine common properties and structures;
• appreciate the axiomatic nature of mathematics.

Focus

The types of reasoning identified in this standard are fundamental to doing mathematics but cannot always be observed in students' verbal answers or written work. It is natural for students to form conjectures on the basis of the examples they have seen or worked and to develop arguments that are based on what they know to be true. Students can also have intuitive notions about proportional reasoning and spatial relationships. All students should have explicit opportunities to engage in such intuitive, informal reasoning and, hence, any assessment of students' reasoning abilities should obtain evidence of these processes.

Accordingly, assessment techniques should specifically assess students' use of different types of reasoning. Although some aspects of reasoning might be more appropriate than others at a given grade level, all aspects can be used at any grade level. However, in the early grades, some aspects may be used only in an intuitive sense.

Discussion

The following examples illustrate tasks or activities for assessing students' abilities to reason. They can be used in the context of instruction, such as in class discussions, or as formal assessment tasks. Young students should be asked to discuss or explain their answers orally.

Grades K--4

1. Deductive reasoning using known facts

If 35 - 20 = 15, what is 35 - 19? Why?

Again, students should explain their reasoning processes. Of interest is the student's ability to use the first numerical equation to develop the second. A response that relies only on the recall of facts and concentrates solely on the second equation in isolation from the first does not demonstrate deductive reasoning.

2. Analyzing a situation to determine common properties and structures

Ask students to work in small groups with cutouts of squares and rectangles. Ask each group to consider questions such as these:

1. What properties do squares and rectangles have in common?
2. What properties do they not have in common?

In a strong response, students would compare the properties of both figures simultaneously, recognizing that both figures have four right angles. Less sophisticated students might first list the properties of a square and then of a rectangle but would fail to make comparisons between the two figures. Clearly the task requires conceptual understanding--in particular, the ability to compare and contrast concepts.

3. Spatial reasoning

Blindfold children and have them handle a cube and a square pyramid. Have each child consider questions such as these:

a. What figure are you holding? b. How many "corners" (vertices) does it have?

This activity can be extended by including other solids or by posing other questions about the cube and the square pyramid, such as, "How many lines [edges] do the solids have?" Students who can provide more detailed descriptions of the solids are demonstrating better spatial reasoning than those who seem to rely on a rather "mechanistic" counting of vertices, edges, or faces.

Grades 5-8

1. Inductive reasoning

Ask students to consider the following situation:

Five students have test scores of 62, 75, 80, 86, and 92. Find the average score. How much is the average score increased if each student's score is increased by--

1. 1 point?
2. 5 points?
3. 8 points?
4. x points?

Write a statement about how much the average score is increased if each individual score is increased x points. Develop an argument to convince another student that the statement is true.

Some students might need more than three instances to form a conjecture. This exercise focuses on a student's ability to generalize from specific cases. Students who can find the answer for each specific problem but who are unable to state the general case are less able to use inductive reasoning than those who can state the general case after three or four instances. Students can work in small groups. A computer can be used to investigate situations other than the three posed or to consider the scores of an entire class.

2. Deductive reasoning and developing a plausible argument

Ask small groups of students to construct models like those in figure 7.1 and develop the formula for the area of a circle or explain why the formula A = r is plausible.

Fig. 7.1. Model relating the area of a circle to the area of a parallelogram

Students who can use the relationship between the shape of the "parallelogram" and its area and the circumference of the circle to develop the formula for the area of the circle are demonstrating plausible and deductive reasoning. The argument is plausible if it makes common sense and is mathematically correct.

3. Proportional reasoning

Pose this question: How many students in the school are left-handed? Have students develop a procedure in which they examine a sample of students for left-handedness and use proportional reasoning to determine the number of students in the entire school who are left-handed.

Students would need to collect data and set up and solve a proportion to answer this question. Other topics can be investigated in a similar way. Small groups of students can gather information about the community at large--for example, how many people are left-handed--as part of a long-term project.

Grades 9-12

1. Spatial reasoning

What formula(s) can be used to find the area of each of the cross-sections of a cube containing the points indicated in figure 7.2?

Fig. 7.2. Points indicating cross-sections of cubes for which formulas are to be found

The evaluation process should consider several steps: First, the students need to visualize the plane containing the indicated points. Second, they must be able to describe the shape of the cross-section. Finally, they need to identify the desired area formulas. Students can work in small groups or individually.

2. Deductive reasoning

Roger doesn't believe that adding the same number of points to each student's test score will increase the average score by that same amount. Write a valid argument to convince Roger that this is true.

This argument should be deductive: A specific case or several cases are not sufficient. Some students might select a specific increase (say, five points) and argue that case. Because of its specificity, this argument is not as strong as selecting a general increase (n points) and showing that the average increases by n points.

3. Appreciating the axiomatic nature of mathematics

Students at all levels must develop an intuitive sense that mathematics is based on established rules and is not a "bag of tricks" familiar only to those who teach or develop mathematics. It is particularly important that advanced students understand that there is an element of arbitrariness in how the rules are selected but that the encompassing system is consistent. The following problems can be given to geometry students, since one of the desired outcomes of teaching geometry is that students develop a sense of what constitutes an axiomatic system.

a. Write an essay on the following topic: In what way did the mathematicians who developed non-Euclidean geometry contribute to the notion that postulates can be arbitrarily selected in mathematics?

The essay should focus on how the parallel postulate (Euclid's fifth postulate) cannot be proved on the basis of preceding postulates and theorems and that there are different options in defining a parallel postulate. The axiomatic nature of mathematics should be highlighted. Students can work in small groups to develop an essay, which can then be presented in class.

b. Suppose a mathematical system assumes the following statement as a postulate:

If two lines are parallel and cut by a transversal, the alternate interior angles are congruent.

Suppose it proves the following statement as a theorem:

If two lines are parallel and cut by a transversal, the corresponding angles are congruent.

Is it possible to assume that the second statement is a postulate and prove the first statement as a theorem? Why?

The student's response should emphasize that the selection of the postulate is arbitrary but that once selected, the second statement can be proved on the basis of the accepted postulate.