The assessment of students' knowledge of procedures should provide evidence that they can--

• recognize when a procedure is appropriate;
• give reasons for the steps in a procedure;
• reliably and efficiently execute procedures;
• verify the results of procedures empirically (e.g., using models) or analytically;
• recognize correct and incorrect procedures;
• generate new procedures and extend or modify familiar ones;
• appreciate the nature and role of procedures in mathematics.

Focus

In the context of school mathematics, procedures generally mean computational methods. But not all procedures in the mathematics curriculum are computational. Geometric constructions, such as bisecting an angle and constructing the perpendicular to a line at one of its points, are procedural but not computational. The various aspects of procedural knowledge identified in this standard apply equally well to noncomputational procedures.

Although it is important that students know how to execute mathematical procedures reliably and efficiently, a knowledge of procedures involves much more than simple execution. Students must know when to apply them, why they work, and how to verify that they give correct answers; they also must understand concepts underlying a procedure and the logic that justifies it. Procedural knowledge also involves the ability to differentiate those procedures that work from those that do not and the ability to modify them or create new ones. Students must be encouraged to appreciate the nature and role of procedures in mathematics; that is, they should appreciate that procedures are created or generated as tools to meet specific needs in an efficient manner and thus can be extended or modified to fit new situations. The assessment of students' procedural knowledge, therefore, should not be limited to an evaluation of their facility in performing procedures; it should emphasize all the aspects of procedural knowledge addressed in this standard.

Discussion

It should be evident that procedural knowledge is intertwined with conceptual knowledge. For example, one cannot extend or modify a procedure for finding the least common multiple of two numbers unless the concept of common multiple is itself understood. Thus, the examples that follow concern aspects of both conceptual and procedural knowledge. They focus on both computational and noncomputational procedures across grade levels and describe tasks for assessing the various aspects of procedural knowledge.

Grades K-4

1. Recognize when to use a procedure.

Divide the class into small groups. Direct each group to create story situations containing two-digit numbers with some involving multiplication. The groups then can exchange problems and identify those that require the multiplication of two-digit numbers.

The problems may differ with respect to the cleverness of the story or the context of the multiplication. Some possibilities include additive situations that suggest multiplication--Mike ate 11 strawberries for 23 days. How many did he eat?--or multiplicative situations--Wanda had 12 skirts and 15 blouses. How many possible outfits did she have? Solving more sophisticated problems can require more than one step: Curtis ran 6 miles every day and Valerie ran 5 miles every day. How many total miles did they run in two weeks? Assessment can focus on such considerations as whether the problems call for multiplication as requested, the richness of the situation, the meaning of multiplication in the problem, the ability of students to discriminate between problems that call for multiplication and those that do not, and whether the problem makes sense regardless of the procedure involved.

2. Verify the results of a procedure.

Solve 62-35. Use multibase blocks or other materials that can represent two-digit numbers to show that your answer is correct.

Each student should verify the subtraction process individually. Verification can involve regrouping the blocks or counting them. The purpose of the activity is to have students use a procedure to find the answer and then show empirically that their procedure works. Assessment can focus on the students' ability to interpret regrouping or to demonstrate how they used a counting technique; in any event, students should be encouraged to offer a more complete explanation than a simple description of the mechanical process of subtraction.

3. Generate a new procedure.

Anita is trying to find a way to solve two-digit subtraction problems like 75-26 without regrouping. How can she change the problem so that the answer will be the same and she will not have to regroup 75 into 60 + 15?

One method is to change the problem to 79-30 (add 4 to both numbers) or to 69-20 (subtract 6 from each number). Whatever method students use, it is important that they explain how the new procedure changes the numbers in the original task without changing the difference. Some students might try to process the problem by using different counting mechanisms. Assessment should focus on the accuracy of the selected procedure and the student's ability to explain why it works.

Grades 5-8

1. Recognize when to use a procedure.

The lockers in Pythagoras Middle School are numbered from 1 to 500. Starting with locker 1, we find that every sixth locker has a blue decal, every ninth locker has a yellow decal, and every tenth locker has a green decal. What is the number of the first locker to have all three decals?

Solving this problem requires finding the least common multiple (LCM) of 6, 9, and 10. Hence, students will need to use a procedure or rely on their conceptual understanding to produce a set of multiples and then select the least of those. In either event, the solution of the problem has two parts: (1) recognizing that an LCM must be found and (2) correctly finding the LCM of 6, 9, and 10. An assessment should consider both parts of the problem.

2. Reliably and efficiently execute procedures.

Find the least common multiple of the following numbers:

a. 12, 18   b. 7, 21      c. 8, 9



d. 1, 6     e. 6, 9, 10   f. 5, 6, 20

Note the variety of numbers given. In set b, one number is the multiple of the other. In set c, the numbers are relatively prime. In set d, one of the numbers is 1. In sets e and f, the three numbers might require some modification of the procedure that students usually use. The assessment should focus on whether students can arrive at the correct answer with reasonable proficiency.

3. Recognize correct and incorrect procedures.

Hershel was given the problem 2/5 < ? < 4/7. He said that 3/6 would be between 2/5 and 4/7. The teacher asked Hershel to explain how he got his answer and why he thinks his method works. Hershel said that he chose a numerator of 3 because 2 < 3 < 4 and a denominator of 6 because 5 < 6 < 7. Hershel claimed his method always works and gave the following examples:

1. The fraction 2/4 is between 1/3 and 3/5.
2. The fraction 4/9 is between 2/5 and 6/11.

Are Hershel's examples correct? Does his procedure always work? Explain your reasoning.

Small groups of students can generate other examples to test whether Hershel's procedure works. A key step in the problem is considering various pairs of fractions and a wide array of possibilities for the "new" numerator and denominator. Assessment should focus on whether students can generate examples that fit Hershel's model and on their cleverness in selecting those examples. If they continually choose examples in which the numerators and denominators differ by 2, they might incorrectly conclude that the procedure works. The identification of a fraction that doesn't work, such as 3/8 in example b, is a valid response. A more insightful response is to consider what the procedure means when it is applied to fractions like 1/3 and 2/4.

Grades 9-12

1. Give reasons for the steps in a procedure.

Justify each of the following steps in multiplying (x + 4) by (x + 2):

(x + 4) (x + 2) = x(x + 2) + 4(x + 2)



                = x2  + 2x + 4x + 8



                = x2  + (2 + 4)x + 8



                = x2  + 6x + 8

Reasons can be explained orally or in writing. Assessment should focus on how articulate the students are in providing mathematical reasons (axioms, definitions, theorems) for each of the steps.

2. Verify the results of a procedure.

1. Find the inverse of A =.

   How can you verify that your new matrix is the inverse of A?

2. Draw a line segment and trisect it using a compass and straightedge. Use paper folding to verify that the segment has been trisected.

In example a, verification is achieved by numerical methods. In example b, verification is determined empirically. The main idea here is that students can verify a procedure independently rather then rely on the teacher or the textbook for verification. Of particular importance is the situation in which a student discovers that an answer does not check out and hence reexamines the execution of the original procedure. Assessment should focus on whether students know how to verify a result and whether they can complete the verification process.

3. Generate new procedures or modify familiar ones.

In figure 9.1, use only a compass to find a point X such that PX (if drawn) would be parallel to line m. Describe and justify your procedure.

Fig. 9.1

A ruler should not be used to solve this problem. Assessment should focus on whether the construction is complete and accurate, on the student's description of what was done, and on the rationale for the procedure. The descriptions and explanations can be written or oral.

4. Appreciate the nature and role of procedures in mathematics.

A single item or task offers insufficient evidence to assess a student's appreciation of the nature and role of procedures in mathematics. A valid assessment must occur over time and take into account students' remarks and actions in a variety of mathematical activities or tasks that call for the use of well-known procedures or the generation of new ones. For this aspect of procedural knowledge to be realized, it is essential that the instructional program provide opportunities for students to generate procedures. Such opportunities should dispel the belief that procedures are predetermined sequences of steps handed down by some authority (e.g., the teacher or the textbook). Important questions to be considered in assessing this aspect of students' procedural knowledge include the following:

• Do students see that procedures are generated for a purpose or to meet a specific need?
• Do students value participation in the generation or extension of procedures?
• When students cannot recall a particular procedure, do they attempt to reconstruct the procedure or generate a new one, rather than seek help in recalling the forgotten procedure?
• Do students see that alternative procedures can meet the same need?
• Do they judge the relative merits of alternative procedures on the basis of their efficiency?

Furthermore, when a new procedure is introduced, the following questions should be assessed:

• Do students attempt to make sense of the sequence in which the steps are carried out?
• Do they question the logic in the sequence of steps?
• Do they question why a given procedure produces the desired results?
• Do they try to verify their results?

These behaviors can be indicative of students' understanding of the nature and role of procedures.