Integrating Assessment and Instruction

Integrating Assessment and InstructionAuthor(s): Judith Sowder, Larry Sowder, Megan M. Loef, Deborah A. Carey, Thomas P.Carpenter and Elizabeth FennemaSource: The Arithmetic Teacher, Vol. 36, No. 3 (November 1988), pp. 53-55Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193504 .

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Rcyzcirch into Practica

Integrating Assessment and Instruction Research . . . Research indicates that young children can solve a variety of story problems before any formal schooling occurs (Carpenter and Moser 1983). Children also use a variety of solution strategies in solving story problems. These strategies range from modeling the action in the problem with counters or fingers, through counting techniques, to derived facts and recall of facts (Carpenter, Carey, and Kouba, in press). Children's abilities to solve story problems appear to change naturally as they systemati- cally move through a hierarchy of problem types (Carpenter and Moser 1983; Riley, Greeno, and Heller 1983).

Within this hierarchy of problem types, a story problem's difficulty is determined by its underlying structure. Problem structure refers to the kind of action or the relationship among the quantities described in the problem and the nature of the unknown in the problem. For example, all three of the problems shown in figure 1 can be represented by "8 - 3 = ?" and are typically described as subtraction problems. Yet in each of

Edited by Judith Sowder and Larry Sowder

San Diego State University San Diego, CA 92182 Prepared by Megan M. Loef, Deborah A.

Carey, Thomas P. Carpenter, and Elizabeth Fennema

University oj Wisconsin Madison, WI 53706

these problems the structure differs, and the pupil is being asked to find a different unknown. The natural strategies that children use to solve each of these three problems reflect the problem structure and consequently differ noticeably. In solving problem A, a pupil may count out an initial group of eight objects, remove a group of three objects, and then count the remaining objects. In problem B, a pupil may count out a set of three objects and then continue to add on to that set, keeping the added objects separate until a total of eight is reached. Then the pupil will count the objects in the added-on set. In problem C, a pupil may count out a group of eight objects, count out a group of three objects, line the two groups up next to each other, and count the objects without partners. Although each problem can be solved by subtracting 3 from 8, the solution strategies are different and reflect the structures of the story problems being solved. In effect, the structure of a story problem determines not only the difficulty of the problem but also the solution strategies that pupils are likely to use.

A knowledge of problem types and of solution strategies enables the teacher to design instruction that builds on children's knowledge. In a recent research project teachers first participated in an in-service program in which they learned about problem types and solution strategies (Carpen- ter, Fennema, Peterson, and Carey, in press). The teachers then used this knowledge to design lessons that fo- cused on problem solving, to plan assessment activities, and to develop a curriculum that could be adapted to their pupils.

At the end of the year, the teachers from the in-service program showed different classroom behaviors and more extensive knowledge of their pupils than teachers who were not given the in-service training. Teachers in the in-service group posed problems, listened to pupils' processes, and worked in smaller groups more often than the non-in-service teachers. The in-service teachers were bet- ter able to predict the strategies that their pupils would use to solve a variety of story problems and number facts. They spent 20 percent of the mathematics class time working with number facts, in contrast with the non-in-service teachers, who spent 40 percent of their mathematics time on number facts. Nonetheless, results of an achievement test on number facts showed no difference between the performance of the pupils of the two groups of teachers! In interviews, in fact, pupils from the in-service teachers' classrooms demonstrated a higher level of recall of number facts than did pupils from the other teachers' classrooms (Carpenter, Fenne-

Fig. 1 Examples of three problem types for subtraction (Carpenter and Moser 1983)

A. Separate, result unknown Connie has 8 marbles. She gives 3 marbles to Jim. How many marbles does she have left?

B. Join, change unknown Connie has 3 marbles. How many more marbles does she need to collect to have 8 marbles all together?

C. Compare Connie has 8 marbles. Jim has 3 marbles. How many more marbles does Connie have than Jim?

November 1988 53



ma, Peterson, Chiang, and Loef 1988).

Although knowledge of problem types and children's solution strategies is not usually taken into account when teachers make instructional decisions, a pupil's progress through the mathematical sequence can be fa- cilitated by the teacher's structuring of the learning environment to match the child's natural development (Secada, Fuson, and Hall 1983; Car- penter and Moser 1984). The knowledge of a variety of problem types and of children's strategies can thus influence children's growth in the ability to solve addition and subtraction problems.

. . . Into Practice Assessing children's thinking and knowing the processes that individual children use to solve problems are essential aspects of a program that focuses on problem solving. Once teachers know how pupils solve problems, they can then use this information to determine the sequence of their instruction. The assessment of pupils' cognitions is a formidable task for a teacher faced with a classroom of twenty-five to thirty pupils. Indi- vidual interviews may be the most effective way to gather information on pupils, but a more practical approach is to integrate assessment into the regular instruction. Activities can be designed for a whole-class setting and can be used for both instruction and assessment purposes.

The following vignette portrays Ms. Silver assessing her pupils' knowledge early in the school year. Ms. Silver was not so much concerned with knowing which pupils could re- spond quickly or accurately to story problems as she was interested in the strategies that the children used to solve particular types of problems. Her participation in the research project had given her content knowledge of addition and subtraction word problems. As a result, Ms. Silver ex- pected her pupils to use a variety of successful strategies to solve different problems, even though neither the problems nor the strategies had been presented to the pupils in a formal mathematics lesson.

Ms. Silver's knowledge of the content also helped her decide which problems would give her the best information about the majority of her pupils. For example, problem В in figure 1 will give Ms. Silver information about the different levels of problem-solving abilities among her pupils because it is more difficult to solve than problem A, yet it can still be directly modeled. Problem A would probably be solved by many of the pupils and as a result would not offer an efficient assessment (Carpenter and Moser 1984; Riley, Greeno, and Heller 1983). In this lesson segment the teacher gave the pupils counters and began the activity with a story problem.

Mathematics becomes a subject that is shared and discussed among students.

Ms. S.: This morning I collected five dimes for milk money. Bob gave me some more dimes, then I had eight dimes all together. How many dimes did Bob give me? (The teacher gave the pupils time to work on the problem until most of the class had a solution. As the pupils worked she observed and recorded the names of those who were using a modeling or a counting strategy, plan- ning to ask them to explain how they figured out their answers. If they re- sponded with only the answer, she would ask them to tell the class what numbers they were thinking about when they were solving the problem.)

Ann: I know, 3. 1 counted 1, 2, 3, 4, 5, ... [on her desk she had made a group of 5, then added on, keeping the second group separate] 6, 7, 8. See [pointing to the second set], 3.

Nick: No, it's 8. 1 counted 1, 2, 3, 4, 5 [pause], 6, 7, 8. (He made a group of 5 with the counters, then added 3 more, and identified the total quantity as the solution.)

Ms. S.: Did anyone else solve the problem like Ann or Nick? (For the next minute or so the children discussed both strategies. The

class agreed that they were similar but that Ann had kept the second set of three separate from the first set of five as she added on her counters. Ms. Silver listened as the children discussed the problem.)

Ms. S.: Did anyone solve the problem a different way than Nick or Ann? [Pause] Jan, how did you solve the problem? (It was not apparent from observing how Jan had solved the problem.)

Jan: Well, I know that 4 plus 4 is 8. If you change one of the 4s to a 5, then the other number is changed to a 3, so 5 plus 3 is 8. Bob gave you three dimes.

Ms. S.: Everyone is doing such a good job of thinking about how to solve this problem. Would anyone else like to share another solution?

Pat: Well, I thought ... you had 5 to start with, so I counted 6, 7, 8. [For each count Pat extended a finger to keep track of the number that was added to 5 as he counted up to 8.] So you got 3 more.

Ms. S.: Nice job. Let's try another problem.

During this activity the teacher gained information about the problem-solving abilities and strategies of a number of her pupils. From her observations, Ms. Silver decided that the majority of the pupils could solve the problem. The discussion gave her information about the strategies that specific pupils used to arrive at the correct solution. For example, Ann's solution involved direct modeling with physical objects; Pat was able to use a more advanced counting strategy; and Jan figured out the answer using a known fact. From the re- sponses of these pupils, Ms. Silver was confident that they could solve easier join-and-separate problems in which the result set is unknown. She noted that Nick successfully per- formed the counting procedure but was unable to identify the quantity of the second set. She decided that it would be necessary to check Nick's performance on the easier join-and- separate problems. The information gathered through observation and discussion could be used for instructional decisions on grouping, problem selection, and activities to encourage

54 Arithmetic Teacher



the development of more advanced solution strategies.

Key Aspects of Assessment • Effective assessment can be infor- mal and should be continual. Because a great deal of variability occurs in children's problem-solving strategies, it is necessary to integrate assessment into each lesson. Finding out how a few pupils are thinking about one or two story problems each day gives the teacher information on individual pupils' thought processes that can imme- diately influence instructional decisions.

• The decision to select certain problems for instruction is influenced by the teacher's knowledge of pupils' thinking. For example, if some pupils are solving story problems by direct modeling with counters, then the decision can be made to select problems that facilitate progress toward more advanced solution strategies. Pupils who usually count all the objects in both sets in addition story problems, say, may be encouraged to count on from the larger number when the second addend is 1,2, or 3.

• Instruction that focuses on problem solving emphasizes the process rather than the product of the problem-solving activity. Mathematics becomes a subject that is shared and discussed among pupils. As a result, teachers gain new insights about their pupils and the pupils become aware of their peers' different ways of thinking about problems. Although pupils arrive at the same solution to a problem, it becomes apparent that no one correct procedure exists for solving the problem. All pupils can share in the discussion at their own level of problem-solving development.

Teachers can design as individual curriculum focusing on problem solving by basing it on their knowledge of the subject matter and their knowledge of their pupils. Curricular decisions are then based on what the learners know and how they think about the mathematics. This article comes from our research in the pri- mary grades, but focusing on children's thinking and understanding

and listening to students can be imple- mented with any subject matter in any grade. Teachers, as decision makers, can find effective ways of assessing children's mathematical knowledge and of understanding how that knowledge changes.

References

Carpenter, Thomas P., Deborah A. Carey, and Vicky Kouba. "Developing Understanding of Basic Operations." In Teaching and Learning Mathematics for the Young Child, edited by Joseph Payne. Reston, Va.: Na- tional Council of Teachers of Mathematics. Forthcoming.

Carpenter, Thomas P., Elizabeth Fennema, Penelope L. Peterson, and Deborah A. Carey. "Teachers' Pedagogical Content Knowledge of Students' Problem Solving in Elementary Arithmetic." Journal for Re- search in Mathematics Education 19 (No- vember 1988) :385^01.

Carpenter, Thomas P., Elizabeth Fennema, Penelope L. Peterson, Chi-Pang Chiang, and Megan Loef. "Using Knowledge of Children's Mathematics Thinking in Class- room Teaching: An Experimental Study." Paper presented to the American Educational Research Association, New Orleans, Louisi- ana, April 1988.

Carpenter, Thomas P., and James M. Moser. "The Acquisition of Addition and Subtrac- tion Concepts." In Acquisition of Mathemat- ics Concepts and Processes, edited by Richard Lesh and Marsha Landau. New York: Academic Press, 1983.

. "The Acquisition of Addition and Sub- traction Concepts in Grades One through Three." Journal for Research in Mathemat- ics Education 15 (May 1984): 179-202.

Riley, Mary S., James G. Greeno, and Joan G. Heller. "Development of Children's Prob- lem-solving Ability in Arithmetic." In The Development of Mathematical Thinking, edited by Herbert Ginsburg. New York: Aca- demic Press, 1983.

Secada, Walter G., Karen C. Fuson, and James W. Hall. "The Transition from Counting All to Counting On in Addition." Journal for Research in Mathematics Education 14 (Jan- uary 1983):47-57. W

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Integrating Assessment and Instruction