Resume of Chapter 7 Assessment and Realistic Mathematics Education

The Safety-net Question-an Example of Developmental Research on Assessment (M. van den Heuvel Panhuizen)

1. Arguments and concerns regarding open-ended problems

The changing from closed to open problem or constructed response problem is one of the most striking characteristics of world-wide reform of mathematics assessment. This type of problem is an important feature of assessment within RME, that based on the concept of mathematics as a human activity (Freundenthal, 1973), in which the main goal is that students learn to mathematize, the students must be able to analyze problem situations by using mathematical tools, according to their thought process as their own answer.

Beside RME, arguments in favor of stressing the importance of open problems also made in some internationally researches. According to Clarke (1988), and Lemon and Lesh (1992), improving open-ended problems consists of a return to a more closed problem format. This result offers a perfect illustration of the tension that can exist between openness and certainty. To obtain more certainty about a particular aspect of the students’ understanding, the problem must focus on the specific aspect. Consequently, the problem might be less open, the result of the problem being less informative. Then, the tension between openness and certainty was confronted in the context of written assessment as the specific developmental research on assessment in this chapter.

2. The research context

The developmental research on assessment discussed in this chapter was conducted in the framework of the “Mathematics in Context” project that begun in 1991 and was expected to run through 1995. The aim of this project is to develop a new American middle school mathematics curriculum for grades 5 through 8. The project is being conducted at the National Center for Research in Mathematical Sciences Education at the University of Madison in collaboration with the Freudenthal Institute of the University of Utrecht. This philosophy of this project is that mathematics is the product of human inventiveness and social activities.

One of the forty teaching units that has been developed for this framework research project is fifth-grade unit entitled “Per Sense”. The goal of this unit is to help students make sense of percentage. The unit begins by exploring the students’ informal knowledge, they are introduced to a “qualitative” approach to percentage, in which estimating and making a link to simple fractions and ratio play an important role.

There are several different kinds of assessment in Per Sense: (i) an initial assessment at the beginning of the unit, (ii) assessment activities at the end of each chapter, (iii) assessment activities during the unit, and (iv) a final, more formal, assessment at the end of unit. The developmental research on assessment that was conducted in the design focused on the final assessment.

3. The first stage of developmental research

Based on the mathematical-didactical issues, Per Sense Test is a test that was designed to cover at least some of the key concepts and skills involving percentage. It consists of a total of ten problem, including the Best Buys problem.

First version of the Best Buys problem has a key feature of percentage that is a relation between two numbers or magnitudes that is expressed by special ratio, namely ‘so many out of the hundred’.

Then, the focus of research issues in the first stage is the experiences gained from administering the problem and the indications for improvement that were provided by the students’ responses.

Furthermore, context of data collection through field test in three seventh-grade classes from two schools in a town near Madison, Wisconsin, in May, 1992.

4. The second stage of the developmental research

The result of the first stage formed the starting point for the second stage of the research. This stage revised set of problems, developed into a new version entitled “show what you know Book on Percents”.

Talk about second version of the Best Buys problem, there are two issues needing resolution in the first version: the lack of clarity in what was meant by a best buy and the uncertainty or lack of understanding of the relative nature of percentages.

Furthermore, the research issues whether the question really succeed in identifying those students who, even though they had failed to demonstrate this understanding initially, in fact understand the relative nature of percentage.

Then, context of data collection in this stage is the participation classes were from schools in a town nearby Madison, Wisconsin, with a different teacher and involving a total of 44 students.

Then, result of the second version of the Best Buys problem is most of the students who had answered the first question correctly actually provided further elucidation in their answer to the second question.

Moreover, a second examination of the safety-net question was the Parking Lots problem to assess to what extent students apply percentages spontaneously when comparing two ‘how many out of the’ situations which can be transposed to ‘so many out of the hundred’. There are many strategies students used: calculating the percentages, approximating by using the percentage bar, converting a fraction into a percentage, using the ratio table, and global relative reasoning. And the result show that 24 out of 44 students solved the first part problem using relative reasoning directly.

5. Final remarks

It has long been believed that individual interview are the only possible means for obtaining true insight into students’ understanding, thought processes and strategies. But, the point of view in RME approach, a strong preference was expressed from the outset for observing and interviewing. Another matter is the feasibility of adopting a typical interview technique, like asking additional question, second chance question, and the standby sheet. The integration of interview techniques into written assessment is in accord with the change from static approach to written assessment to a more dynamic approach, which is characteristic of assessment both within RME and current reform movement in mathematics education and assessment.