The assessment of students' ability to use mathematics in solving problems should provide evidence that they can--

• formulate problems;
• apply a variety of strategies to solve problems;
• solve problems;
• verify and interpret results;
• generalize solutions.

Focus

If problem solving is to be the focus of school mathematics, it must also be the focus of assessment. Students' ability to solve problems develops over time as a result of extended instruction, opportunities to solve many kinds of problems, and encounters with real-world situations. Students' progress should be assessed systematically, deliberately, and continually to effectively influence students' confidence and ability to solve problems in various contexts. Giving students feedback about the results of this assessment, on both the processes used and the results attained, is critical to their development as problem solvers. In addition, an assessment can give helpful information to teachers regarding problem situations that are challenging, instructive, and interesting, and yet not defeating, for students.

Assessments should determine students' ability to perform all aspects of problem solving. Evidence about their ability to ask questions, use given information, and make conjectures is essential to determine if they can formulate problems. Assessments also should yield evidence on students' use of strategies and problem-solving techniques and on their ability to verify and interpret results. Finally, because the power of mathematics is derived, in part, from its generalizability (e.g., a two-space solution can be generalized to a three-space solution), this aspect of problem solving should be assessed as well.

Discussion

Methods for assessing students' ability to solve problems include observing students solving problems individually, in small groups, or in whole-class discussions; listening to students discuss their problem-solving processes; and analyzing tests, homework, journals, and essays. Feedback to students can have a variety of forms, including written or oral comments or numerical scores on a specific exercise. Scoring schemes include giving two scores (one for the answer and one for the strategies used); rating the student's work on a scale (e.g., 4--perfect, 3--nearly all correct with some computational errors, 2--right idea but poor execution, 1--tried the problem, 0--nothing was done); or giving points for such primary features as computation, pictures, tables, strategies, and verification. One way of reporting progress in problem solving is with a problem-solving profile. The profile can include ratings on a student's willingness to engage in problem solving, the use of a variety of strategies, facility in finding the solution to problems, and consistency in verifying the solution. The report to parents should include a sample of the student's problem-solving work.

The following are examples of situations appropriate for students at various grade levels that can be used to assess their abilities to solve problems successfully.

Grades K--4

1. Formulate problems.

You have this amount of change:

• 8 pennies
• 5 nickels
• 11 dimes
• 5 quarters

These items are for sale:

• Box of cereal for $1.60
• Glass of milk for $0.40
• A poster for $0.90
• One ball for $1.20

Use this information to make up a problem.

This situation can be given to students as a writing task. Students can be evaluated on how much of the given information they use, the reasonableness of their problem, and its mathematical sophistication. The problem "How many coins are there?" includes less of the given information than the problem "Do I have enough money to buy one of each item? If not, how much do I need?" The quality of the problem, however, is not necessarily dictated by the inclusion of all the information: For example, "If I buy a box of cereal, how many glasses of milk can I buy?" is a far more sophisticated problem but one that uses a minimum of given information. Students can work in small groups to generate problems. Calculators can be helpful.

2. Solve problems.

Read the following problem and answer the question posed:

Paula, Teresa, and Dale ran a race. Paula took three minutes and Dale took four minutes to finish the race. Who won the race?

This task can be used in instruction to determine if students recognize that essential information is missing. Once they determine that they need to know Teresa's time, other questions can be asked: Is it possible to give a time for Teresa so that she can win? Is it possible to give a time for Teresa so that Paula can win? It is important to observe if the information given is reasonable and if students can verbalize why a certain value for Teresa's time is given. This example shows how a routine exercise can serve as the basis for generating other tasks by deleting a condition, removing the question, or adding irrelevant information.

3. Apply strategies to solve problems.

With a calculator, find three numbers whose product is 2431. Keep a record of what you do to find the answer.

This task is suitable for guessing-and-testing strategies and can be used in a testing situation. Students should be encouraged to write down their guesses and explain what they did. It should be observed if students take a systematic approach to developing guesses, if they keep a list of guesses, and if they place limitations on what the numbers can be (e.g., only odd numbers will work). The calculator is essential for generating a number of guesses in a short period of time. Students should be given points both for the right answers and for the use of one or more appropriate strategies. Random guessing should not be awarded strategy points. Some time should be available for students to explain their approaches.

Grades 5--8

1. Solve problems.

Students' pulse rates vary. What would be considered the normal pulse rate for students in your class? You might want to consider various characteristics or conditions (e.g., exercise) and find out how they relate to pulse rate.

The evaluation of this activity should focus on the reasonableness of students' questions, the various forms of representations used to report data, whether the results are verified, and whether any generalizations are made. This activity can extend over several days and is appropriate for small-group work. A score or rating for the total project can be given to each group. Specific parts of the students' work can be emphasized by scoring them separately, such as the number of representations used (table, graph, equations, and written report). A calculator and a computer are essential.

2. Formulate problems; verify and interpret results.

Four of every five dentists interviewed recommended Yukky Gum. Write a question to go with this statement to make a problem. Solve the problem.

This task can be embedded in instruction and is appropriate for large-group discussion. The assessment of students' performances on this task focuses on their choice of questions. The question must be logically connected to the statement. Consider, for example, the question, If 1000 dentists were interviewed, how many recommended Yukky Gum? Students' explanations will give some indication of how they viewed the task and what features they focused on in developing their questions. Requiring students to solve their own problems gives an indication of their ability to solve problems and encourages them to ask reasonable questions.

The activity can be extended by asking students to formulate a problem whose answer is "160 dentists." Writing a question to fit an answer requires an interpretation of the result in light of the conditions stated. The question should be judged on whether the result, 160 dentists, matches the given conditions. Some note of students' posing of appropriate questions can be recorded.

3. Apply strategies; solve problems; verify and interpret results.

Keep a problem-solving journal in which you (a) record interesting problems (including some not yet solved), (b) describe what strategies you used or thought about, (c) explain how you verified solutions (e.g., checked the conditions, reworked the problem in another way, checked the reasonableness of solutions), (d) identify similar or related problems, and (e) record problems posed by other students.

A problem-solving journal requires students to reflect on what they do when they solve problems. It can provide information on all aspects of problem solving. Teachers will detect progress if students increase the number of strategies they report using, verify their solutions in different ways, and find relationships among problems. The development of a language to talk and write about problems also should be evident in students' increasing use of terms describing strategies and related problems.

Grades 9--12

1. Formulate problems; solve problems.

1. You have 10 items to purchase at a grocery store. Six people are waiting in the express lane (10 items or fewer). Lane 1 has one person waiting, and lane 3 has two people waiting. The other lanes are closed. What check-out line should you join?
2. You are considering purchasing one of two cars, both four years old. One car costs $3000 and gets 20 miles a gallon. The other costs $4500 and gets 35 miles a gallon. Which car is the best buy if you plan to keep it two years?

What additional information do you need to answer these questions?

One aspect of formulating problems is identifying whether additional information is needed. Neither of the problems above provides all the information needed to make a decision. Students need to identify the missing information and the likely estimates for the missing quantities. In question a, the number of items each person has and the speed of the checkers are considerations. In problem b the number of miles traveled each year, the price of gasoline, and cash available are considerations. If money has to be borrowed to purchase the more expensive car, it can make a difference. These problems are appropriate for individual or small-group work embedded in instruction. Notes can be kept on the variety of questions generated and what additional information is assumed. In class, the willingness of students to engage themselves in finding the necessary information can be observed. Calculators are important for question b.

2. Solve problems; generalize solutions.

Prove each of these statements:

1. The sum of two consecutive whole numbers is not divisible by 2.
2. The sum of three consecutive whole numbers is divisible by 3.

State what you consider to be the general case for the statements. Prove or give a counterexample.

This task can be included on a test or as a take-home problem. Evaluating this task includes assessing students' ability to prove the statements; the nature of the statement developed; and their ability to prove or disprove the general statement. A possible statement of the general case is this:

Is it true that the sum of an even number of positive consecutive whole numbers is not divisible by the number but that the sum of an odd number of positive consecutive whole numbers is divisible by the number?

This problem can be scored by giving points for the different parts of the problem solution--one point for accurately representing the first statement [(n + (n + 1))/2 = an integer?], one point for giving a convincing argument for two consecutive whole numbers, one point for accurately representing the second statement, one point for giving a convincing argument for that statement, two points for stating the general case, two points for providing an adequate proof, and two points for explaining what strategies were used.