April 14, 2011

Alternative Assessment

Goetz (2005) believes cooperation plays an important role in learning, and that, we should assess what we value (p. 12). Ergo, his assessments in his precalculus classes contain cooperative activities in which students' grades are partially dependent on the interactivity and communication between the students. His tasks are posed as open-ended word problems with real-world applications. In one example, given a data set the students were expected to construct a mathematical equation that models that data. There was a diverse range of answers varying from polynomial to rational to exponential functions.

The students' grades were based on a grading rubric, which the students were aware of before the exam. One of the elements on the rubric took group participation into account, and the students were expected to explain the specific roles each member played in the cooperative activity. That way, each student would get the deserved amount of credit. Goetz believes in using assessment as a tool for learning, such that students will turn an exam into a learning experience. This corresponds with NCTM's view that assessment should enhance student learning.

Grading rubrics are essential for assessment. In my personal experience and from advice from my educators, I can report that students' scores on assessments without rubrics can become subjective and open to interpretation, and are a potential source of conflict between teacher and parents. In order to provide an explicit and objective grade, a grading rubric must be used. The higher resolution a rubric has, the more accurate students' score will be on that particular task. Sometimes, though, teachers create rubrics that are hard for students to understand, thus the students might score more poorly than if they had a clear indication of what is expected of them.

Brown-Herbst (1999) had her middle-school students construct their own grading rubric. Her class's rubric was based on a final draft submitted by teachers from twelve schools participating in a statewide project in Alaska. While constructing the rubric, her students had to interpret the language used by teachers to gain an understanding of the spectrums of performance. After three days of debate and discussion among the middle-schoolers, they finalized a rubric that was to be used on not only their end of year exam, but also on that project itself. In other words, the students were being assessed by their own criteria.

A project such as the one implemented by Brown-Herbst (1999) takes much time and planning, but the knowledge and skills gained by the students are worth it. Students reflected NCTM's (2000) Communication process standard: they translated mathematical teacher language into mathematical student language, and conveyed concepts and ideas to one another and refined them. Even previously implicit ideas have been made explicit by students who asked each other to clarify meaning; e.g., [a] seventh-grade girl spoke up: I need to know exactly what the math thing is (p. 453).

A final example as another form of assessment is offered by Bailey and Chen (2005). They introduced the graphing portfolio, in which students are expected to trace out a picture or graphic by using functions (either cartesian or a combination of cartesian and polar) to illustrate the lines in the graphic. This method is slightly related to Goetz's (2005) example, in that students are working backwards with functions: given a function's graph (or a curve of best fit), they need to find the equation. Graphing portfolios are useful for an artistic and creative touch in a mathematics course.


  • Bailey, E. C., & Chen, F. (). Graphing portfolios in calculus: Reinforcing concepts and inviting creativity. Mathematics Teacher, 98(6), 404–407.
  • Brown-Herbst, K. (). So math isn't just answers. Mathematics Teaching in the Middle School, 4(7), 448–455.
  • Goetz, A. (). Using open-ended problems for assessment. Mathematics Teacher, 99(1), 12–17.

April 7, 2011

Activating Prior Knowledge

Stillman (2000) elaborates on the different classes of, and the importance of, prior knowledge in the way students approach a task. Prior knowledge of the academic variety consists of knowledge gained through outside academic experiences, such as content learned in another class or study. Encyclopaedic prior knowledge includes general facts and trivia of the world, and episodic prior knowledge is that which is gained by a learner from personal experiences outside of an academic setting.

Sloyer (2004) offers a strategy he calls the extension-reduction strategy. This is a pedagogical strategy in which a teacher will present a problem that requires his or her students to use their prior knowledge to construct new knowledge. The goal of the strategy is to get the students to reduce the new problem down to a simpler and better understood problem (e.g., finding the area of a polygon by subdividing it into triangles). The teacher's role is to help and guide the students in activating their prior knowledge. Even though the students may have this prior knowledge, they may not always know how to use it productively. The example that Sloyer gives is a problem in which students try to find the volume of a segment of a cone. The prior knowledge here was that of finding a part of a whole. The value of the desired part is found by taking the value of the whole (whether that be a region's area, a finite series, a solid's volume, etc.) and subtracting the extra amount.

Hare's (2004) example is a bit more complex. In this study, students learned implicit differentiation by reinforcing and expanding their concept of a function. Students had varying misconceptions of the definition of function and thus could not take the implicit derivative of an equation properly. Through guided questions and activities, the students were not only able to use their prior knowledge but also improve on it, while at the same time gain new knowledge.


  • Hare, A. & Phillippy, D. (). Building mathematical maturity in calculus: Teaching implicit differentiation through a review of functions. Mathematics Teacher, 98(1), 6–12.
  • Sloyer, C. W. (). The extension-reduction strategy: Activating prior knowledge. Mathematics Teacher, 98(1), 48–50.
  • Stillman, G. (). Impact of prior knowledge of task content on approaches to applications tasks. Journal of Mathematical Behavior, 19, 333–361.