The assessment of students' knowledge and understanding of mathematical concepts should provide evidence that they can--

• label, verbalize, and define concepts;
• identify and generate examples and nonexamples;
• use models, diagrams, and symbols to represent concepts;
• translate from one mode of representation to another;
• recognize the various meanings and interpretations of concepts;
• identify properties of a given concept and recognize conditions that determine a particular concept;
• compare and contrast concepts.

In addition, assessment should provide evidence of the extent to which students have integrated their knowledge of various concepts.

Focus

Concepts are the substance of mathematical knowledge. Students can make sense of mathematics only if they understand its concepts and their meanings or interpretations. For example, if students are to make sense of the procedure for subtracting whole numbers with regrouping, they must understand the concept of place value. Likewise, if students are to recognize that a given situation calls for subtraction, they must understand the concept of subtraction and recognize that the action depicted in the situation corresponds to one of its meanings (e.g., take away, comparison, partition). Because conceptual understanding is fundamental to doing mathematics meaningfully, an assessment of students' knowledge must examine their grasp of mathematical concepts.

An understanding of mathematical concepts involves more than mere recall of definitions and recognition of common examples; it encompasses the broad range of abilities identified in this standard. Assessment, too, must address these aspects of conceptual understanding. Assessment tasks should focus on students' abilities to discriminate between the relevant and the irrelevant attributes of a concept in selecting examples and nonexamples, to represent concepts in various ways, and to recognize their various meanings. Tasks that ask students to apply information about a given concept in novel situations provide strong evidence of their knowledge and understanding of that concept. Problems designed to elicit information about students' misconceptions can provide information useful in planning or modifying instruction.

Discussion

The assessment of students' understanding of concepts should be sensitive to the developmental nature of concept acquisition. Students' grasp of mathematical concepts develops over time. Many concepts introduced in the early grades are later extended and studied in greater depth. A fraction, for example, is introduced as a part of a whole in the primary grades, as a measure in the intermediate grades, and as a ratio in junior high school; finally, algebraic fractions are taught in secondary school. This progression is accompanied by the development of the language and notation of fractions and extended further through the exploration of the relationships among fractions and other concepts, such as decimals, and through the applications of fractions in various contexts, such as in proportional reasoning. Tasks and situations used to measure the understanding of concepts should change over the grades to determine whether students' notions of the concept are maturing.

This standard suggests the kinds of tasks needed to assess the various aspects of students' conceptual understanding and knowledge. As the following examples illustrate, it is unnecessary to assign separate tasks to assess each aspect of understanding; it is feasible--in fact, advisable--to design a single task that covers several aspects. Nor is it necessary to assess all aspects of all concepts or all students at all times. The aspects of conceptual understanding to be assessed should be selected according to the mathematical content and the level of the students. Furthermore, assessment tasks should be consistent with the methods of instruction. For example, children in the early grades, whose understanding of fractions is closely tied to the use of physical materials, should be encouraged to use such materials to demonstrate their conceptual knowledge. The examples are arranged by grade level, but many are appropriate for students in grades other than the ones specified. Similar problems can be created for different levels of involvement (whole class, small groups, or individual) and mode of response (written, oral, performance, or computer).

Grades K-4

1. Identify examples and nonexamples of concepts. (See fig. 8.1.)

Fig. 8.1

Which figures show that exactly 1/2 of the region is shaded?

Here, students are to identify examples and nonexamples of one-half of a region. The task is constructed to determine whether they can discriminate between the relevant and irrelevant attributes (i.e., the area of the shaded region must be equal to that of the unshaded region, but the two parts need not be contiguous or congruent).

2. Use models to represent concepts.

A yellow hexagonal pattern block represents one whole. Use blocks to represent 1/2, 1/3, 1/6, 2/3, and so on.

Students should be asked to explain their responses and encouraged to show a given fractional part in more than one way, for example, showing 1/2 using one trapezoidal block or three triangular blocks . Various blocks or combinations of blocks can represent one whole. In this context, students who consistently show fractional parts of a given whole demonstrate a good understanding of fractions as parts of regions. Students can complete this activity individually, in small or large groups, or in a multiple-choice format in which they select the block that represents a particular fraction.

3. Identify properties of given concepts; compare and contrast concepts.

Give students a sheet containing drawings of various quadrilaterals. (Include drawings of shapes in various orientations. Drawings should be numbered for easy reference.) Ask students to cut out the shapes and hold up those that fit the following descriptions:

Hold up shapes with two pairs of parallel sides. What are they called? (Parallelograms; responses should include squares and rectangles.)

Hold up shapes with four congruent sides. What are they called? (Rhombuses; responses should include squares in various orientations.)

This task requires students to identify shapes that have specified characteristics. From the shapes displayed, it can be determined quickly which children recognize the properties of various quadrilaterals and their interrelationships. For example, children who display only parallelograms that are not rectangles in response to the first question do not have a full understanding of parallelograms, rectangles, and squares. It is important that children be asked to explain their choices as a further means of assessment.

4. Integrate knowledge of concepts.

Direct students to work in groups of four or five. Give each group three paper "pizzas." Ask them to determine how much pizza each group member will receive if they share their pizzas equally.

This task can yield information about students' understanding of fractions as quotients and about their ability to use what they know about fractions in a novel situation. Students who can describe the amount of pizza each person receives as a fractional part of a whole pizza demonstrate a good understanding of fractions as parts of a region.

Grades 5-8

1. Recognize various interpretations of concepts; use diagrams to represent concepts; translate between modes of representation.

Complete each diagram in figure 8.2 so that it shows 2/5.

Fig. 8.2

This task assesses students' recognition of the various meanings and interpretations of a given fraction. It also requires them to translate between symbolic and pictorial modes of representation and to use diagrams to represent concepts. Students who can represent a given fraction in all these contexts demonstrate a broad understanding of the meaning of fractions. For seventh- and eighth-grade students, this task can be extended to representations of fractions as decimals, percents, and ratios.

2. Identify examples and nonexamples of concepts; compare and contrast concepts.

In figure 8.3, put a Q on each shape that is a quadrilateral; a P on each parallelogram; an R on each rectangle; an RH on each rhombus, and an S on each square. You can put more than one letter on a single shape.

Fig. 8.3

Students who correctly identify each figure are demonstrating that they know a single shape can be classified in several ways. This evidence shows that they understand class-inclusion relations among types of quadrilaterals (i.e., that all squares are rectangles and rhombuses and that all rectangles are parallelograms). This activity can be extended by asking students to justify their responses, orally or in writing, by completing the statement, "A quadrilateral is a shape that ......." Similar activities can determine whether students can recognize relevant and irrelevant attributes of types of shapes (e.g., that orientation is irrelevant).

3. Integrate concepts.

On a journey from Pittsburgh to New York, Pat fell asleep after half the trip. When she awoke, she still had to travel half the distance that she traveled while sleeping. For what part of the entire trip had she been asleep?

Assuming that the shaded part in each diagram in figure 8.4 shows when Pat was asleep, which diagram best depicts the answer to the problem?

Fig. 8.4

This task assesses whether students can use their knowledge of fractions to interpret the problem and identify the correct representation of its solution. It requires that they consider a fraction in relation to different units. They must first think of the whole trip as a unit, then consider a different unit (the portion of the total trip during which Pat was asleep).

Grades 9-12

1. Identify examples and nonexamples of concepts.

Which of the following represent rational numbers?

2/3           0            1.3434        -5.6







1.121121112. . .            7/(9-32)      -6/-2     25%

Again, the task is designed to determine whether students can discriminate between relevant and irrelevant attributes of rational numbers. Students who identify 2/3, , 7/(9-3²), and -6/-2 as rationals might be confusing fraction with rational number; those who classify 1.121121112..... as rational might not be distinguishing repeating decimals from nonrepeating decimals containing patterns. This task can be presented in many formats, including a written exercise in which students are asked to justify their selections.

2. Recognize conditions that determine a concept.

Who am I?

I am an equiangular quadrilateral. What special kind of quadrilateral am I?

Here, students must distinguish between properties that some quadrilaterals possess and properties that are sufficient to define them. For example, to answer this item correctly, students must recognize that although the angles of squares are equal, all rectangles have this property, so the correct answer must be rectangles, not squares.

3. Translate from one mode of representation to another.

What is the equation of the line shown in figure 8.5?

Fig. 8.5

This task requires students to translate a graphic representation of a line into a symbolic representation. To do so efficiently, students must be able to determine the slope and intercept of the line from its graph, then represent them symbolically to create the desired equation, y = 2x - 3. Students who are able to do such re-representations with ease demonstrate a solid understanding of the concept of line in analytic geometry and of the related concepts of slope and intercept.

4. Integrate concepts.

Connect the midpoints of the four sides of an isosceles trapezoid. What kind of figure do you get? Justify your answer.

This task yields information about the extent to which students have integrated their knowledge of geometric concepts. To solve this problem, students must be able to draw an isosceles trapezoid, find the midpoints of the sides, and then recognize the figure that results from connecting the midpoints. Students must then be able to apply additional knowledge about the conditions that determine whether a figure is a rhombus to justify their answers. Students might be able to identify the desired figure yet be unable to justify their responses.