The assessment of students' mathematical knowledge should yield information about their--

• ability to apply their knowledge to solve problems within mathematics and in other disciplines;
• ability to use mathematical language to communicate ideas;
• ability to reason and analyze;
• knowledge and understanding of concepts and procedures;
• disposition toward mathematics;
• understanding of the nature of mathematics;
• integration of these aspects of mathematical knowledge.

Focus

In mathematics, as in any field, knowledge consists of information plus know-how. Know-how in mathematics that leads to mathematical power requires the ability to use information to reason and think creatively and to formulate, solve, and reflect critically on problems. The assessment of students' mathematical power goes beyond measuring how much information they possess to include the extent of their ability and willingness to use, apply, and communicate that information. The assessment should examine the extent to which students have integrated and made sense of information, whether they can apply it to situations that require reasoning and creative thinking, and whether they can use mathematics to communicate their ideas. Additionally, assessment should examine students' disposition toward mathematics, in particular their confidence in doing mathematics and the extent to which they value mathematics.

An assessment of students' mathematical power is broad in scope and should include all the aspects identified in this standard and determine the extent to which they are integrated. The assessment of mathematical power should not be construed as the assessment of separate or isolated competencies. Although one aspect of mathematical knowledge might be emphasized more than another in a particular assessment, it should remain clear that mathematical power concerns all aspects of mathematical knowledge and their integration.

Discussion

To have assessment practices in mathematics that reflect these standards, all important aspects of mathematical knowledge must be assessed. Student-assessment standards 5-10 address each of these aspects and how they are interrelated and give guidelines for assessing them. This discussion focuses on determining the extent to which these aspects are integrated.

Problem situations from different areas of study offer a rich context in which to assess students' mathematical power. Problems derived from such situations typically require students to apply a variety of mathematical concepts and procedures and engage in some form of mathematical reasoning. Understanding these concepts and their interrelationships is essential to interpreting a situation and deriving an appropriate plan of action. Knowing what procedures are appropriate or necessary and how to execute them is essential to carrying out the plan successfully. Furthermore, these problems require students to use various forms of reasoning to arrive at a solution. Consider, for example, the following task derived from a social studies lesson on the commerce between North America and Hong Kong:

Pretend you are a pilot for a major airline-transport company. You have been assigned for the first time to a trans-Pacific flight from New York to Hong Kong. You are curious about the shortest route between the two cities, but all you have is a regular globe and a piece of string. You know that the distance around the Earth along the equator is 25 000 miles. With only these two items, how can you figure the shortest distance? What is this distance?

The task requires students to interpret the shortest distance between two points on a sphere, find a way of measuring that distance, and use proportional reasoning inventively and creatively. The task is suitable for junior high school students. Students can work individually or in small groups while the teacher observes their interactions. These observations can yield information about the students' ability to apply their knowledge in solving the problem. This task illustrates how mathematics can be integrated into other areas of the school curriculum.

The assessment of students' mathematical power is appropriate at all grade levels and should not be delayed on the grounds that students must know a great deal of mathematics before they can integrate this knowledge. Group tasks are particularly useful in the lower grades for assessing the integration of students' mathematical knowledge. At the K-4 level, a group task like the following can be devised:

Materials Required	Task
Large box of raisins Containers of different sizes Balance Calculator	Estimate the number of raisins in the box. Use any of the materials to make a better estimate. Check your estimate by different methods. Record your results and give an oral account of your work.

The assessment of students' performance on this task focuses on their choice of an overall solution strategy. A successful approach entails counting the number of raisins in a small unit and relating the smaller unit to the whole (the large box) in some way. Students can use a balance to halve successively the number of raisins until a manageable number is obtained and then multiply to obtain the total in the large box. Or they can count the number of raisins in a small container and find the number of small containers necessary to fill the large box. Or they can measure the height of a small proportion of the raisins in the large box and the height of the full box and relate the two heights. Whatever their strategy, students engage in counting, computing, measuring, and communicating to perform the tasks.

Reporting on the students' strategy can be done using a rating scale. Evidence of students' application of their strategy is obtained from the recorded results and oral accounts of what they did. Levels of mathematical argument can be similarly evaluated. An assessment of the mathematical concepts and procedures used can focus on the accuracy of counting and calculation; the correct choice of operation; the understanding and use of balance; and such calculator skills as accurate keying, recognition of the order of operations, and correct interpretation of the display. Finally, the assessment of communication skills can focus on the students' recorded and oral accounts of their work.

Problems and group tasks are not the only means of assessing the integration of students' mathematical knowledge. Written tasks can be used effectively. A written task can consist of multiple subtasks, encompassing various aspects of mathematical knowledge and their integration. Figures 4.1, 4.2, and 4.3 contain an example (Swan 1987, pp. 28-30) that illustrates this approach:

Fig. 4.1. Map showing distance from school to home

Fig. 4.2. Length and time of journey

Fig. 4.3. Speed by distance for Peter's journey

The graph in figure 4.2 describes each pupil's journey to school last Monday.

1. Label each point on the graph with the name of the person it represents.
2. How did Paul and Graham travel to school on Monday?
3. Describe how you arrived at your answer in part b.
4. Peter's father is able to drive at 30 mph on the straight sections of the road but has to slow down for the corners. Sketch a graph on the axes in figure 4.3 to show how the car's speed varies along the route.

The assessment of students' performance on this task focuses on their ability to apply several mathematical concepts, skills, and processes simultaneously. The task entails reading, interpreting, and selecting information from the map and combining it with information in the problem to estimate the children's relative travel times and establish the correspondences between points on the graph and the children. The selection and use of information necessary to label the points on the graph entails an understanding of the relationship among distance, time, and speed. Students must apply their knowledge of coordinate graphs and the variables represented to establish a correspondence between points and children. Additionally, they must infer from the graph the mode of travel used by two of the five children. The students' description of their reasoning in labeling those points offers evidence of their ability to communicate mathematical ideas. Finally, the task calls for the graphical representation of a car's speeds along the route depicted in the map, which requires the simultaneous consideration of the route, the additional information in part d, and the variables represented in the graph.

Students' responses to this task can be evaluated by rating their ability to read and interpret information in the map, combine this information with that in the problem statement, and translate and summarize it in a graph. Marks can be given for correctly identifying each point in figure 4.2. The students' description of their reasoning in locating those points can be assigned marks if Paul and Graham are identified as cycling or running or if the explanation matches the graph drawn in figure 4.2. The response to figure 4.3 can be assigned marks on the basis of the inclusion of the main features of the graph--correctly locating endpoints, indicating two minima in the correct position, showing the speed as 30 mph for at least 1 mile in the middle section of the graph, and correctly recording all other features of the graph. The score for the tasks would be the sum of the marks given.

Written tasks such as these can be useful in determining the extent to which students' knowledge of mathematics has been integrated. However, such assessment cannot be based solely on students' performance on a single task, regardless of how valid or appropriate the task might be. To assess the integration of mathematical knowledge, information must be obtained from several tasks performed over time in a variety of contexts.