Hypothetical Assessment Trajectory

We will see how to design a hypothetical assessment trajectory to fit the learning trajectory. Some of the ideas we describe have been suggested by Dekker and Querelle (1998).

Before. The first assessment activity should be when starting to teach a new concept or idea with some fundamental value (middle zoom level). The teacher wants to know whether the students have mastered the prior knowledge necessary to start successfully with a new unit. Already, this assessment activity will change the learning trajectory. Possible and suggested test formats for these goals are—

• Oral test. Questions are posed that involve basic knowledge and skills . This format is appropriate because it enables the teacher to recapitulate important topics with the whole group in a very interactive way. Although basic knowledge and skills should be stressed, the format also allows the teacher to check competencies in a relatively effective and fruitful way.

• Aural test. Questions are posed orally but answers are written down. This gives students who are not too fluent in English a second and probably fairer chance to express their ideas. This format also lends itself very well to checking whether students are able to express their informal knowledge about things to come; this is again relevant to designing the learning trajectory.

• Entry test. A short, written entry test consisting of open-ended questions.

• Other test formats. It goes without saying that the teacher is free to choose from any of the test formats described before or to design other test formats.

During. While in the trajectory, there are several issues that are of importance to teachers and students alike. One is the occurrence of misconceptions of core ideas and concepts. Because students in a socio-constructivist or interactive classroom get lots of opportunities to re-construct or re-invent their mathematics, the opportunities to develop misconceptions also abound. Because there is only one teacher but more than 30 students, the teacher needs some tools to check for student misconceptions. Dekker and Querelle (1998) recorded that cubes were mistaken for squares, Pythagoras theorem was remembered with a multiplication or “×” sign instead of a plus or “+” sign, and perimeters and areas were regularly mixed up. Possible assessment tools include:

• Production items. Students design a simple short-answer test. Of course answers should be provided as well (see discussion of this format), and all of the required content should be covered. The test could be graded but another, more rewarding possibility is to compose a class test using the student-designed items. Misconceptions will turn up and can then be discussed.

• Student-generated items. Students hand in a certain number of single-answer questions on the subject involved. These are used in a computer-based quiz for the whole group and are discussed afterwards.

As discussed previously in some detail, all assessment should result in feedback, and hopefully in feedback that goes far beyond grading a test. Feedback to the students is especially important when most students fail to solve a problem—a problem which the teacher thought fit nicely in the learning trajectory.

A very forceful way to get quality feedback is formed by the two-stage task. In this case, feedback on the first stage is given before the students start working on the second stage. In reality, this means that the teacher gets feedback from the students on how well the teacher’s feedback worked. Other information-rich formats include:

• Oral questions are asked when the topic is discussed in the classroom. In this case, the discourse is an assessment format.

• Short quizzes, sometimes consisting in part of one or more problems taken directly from student materials.

• Homework as an assessment format (if handled as described in our earlier section on homework).

• Self-assessment—preferable when working in small groups. Potential important difficulties will be dealt with in whole-class discussion.

Throughout the school year, the teacher will constantly evaluate the students’ individual progress and the progress of the whole classroom within the learning trajectory and thus evaluate the intended learning goals as benchmarks. 

This ongoing and continuous process of formative assessment, coupled with the teachers’ so called

intuitive feel for students’ progress, completes the picture of the learning trajectory that the teacher builds. The problem of a strongly interactive classroom environment is that for teachers and students alike it is difficult to know whether or not they contribute to the group learning process and what they learned individually. Formats that may be helpful to evaluate students’ progress include—

• Discussions with individual students about their understanding.

• Observation of students in groups and while working individually.

• Extended-response open questions, which require own productions, display of results for the whole group, or discussion by the whole class.

• Peer-assessment can be a tremendous help because students see the mistakes of their fellow students and then try to decide whether full or partial credit should be given for a certain solution.

After. At the end of a unit, a longer chapter, or the treatment of a cluster of connected concepts, the teacher wants to assess whether the students have reached the goals of the learning trajectory. This test has both formative and summative aspects depending of the place of this part of the curriculum in the whole curriculum. Different test formats are possible, while we see often that some formats with timed, written tests are the teacher’s favorite—most likely because of their relatively ease of design and scoring and the limited possibility of feedback in a qualitative way.