Illuminating Understanding: Performance Assessment in Mathematics

Tricia Donovan

Observation of a student's actual performance a on task has been a
fundamental tool of assessment throughout history...But mathematics students fill in a bubble or a blank to indicate that they can understand somebody else's solution to a problem." (Mathematics Assessment: Myths, Models, Good Questions and Practical Suggestions, NCTM, p. 13) Too often this is the case in mathematics classrooms. You either have the answer right or wrong, and who cares how you figured it out. Using a performance assessment means, to the contrary, that you do care how a person arrived at an answer.

Performance assessments are designed to reveal a learner's understanding of a problem/task and her/his mathematical approach to it. The task can be a problem or a project; the task might even be a performance — demonstrate the balancing of equations (and, therefore, the essential nature of the equal sign). It can be an individual, group or class-wide exercise.

What the task purports to measure should be clear. Furthermore, it should emerge from classroom curriculum, for the object of any performance assessment assignment is to determine what learners know and how they use what they know. Performance tasks are not about good guessing, and usually not about single right answers. When we teach measurement for instance, we might want to know if learners are able to convert smaller units to larger units and visa versa, so we create an assessment task that requires finding the lengths of various objects and reporting those lengths in several units. Learners demonstrate what they know and their method of solution as they undertake the task. As teachers, we use this information to set the academic agenda (and, in some cases, the social agenda — working together more effectively as a group, etc.) for the individual, group and/or class. Therefore, any task not related to the anticipated or implemented curriculum is inappropriate for our purposes.

Finding a task that illuminates a person's knowledge and application of skills is no easy search: Is it the summative assessment task, or an emergent one embedded in the instruction that we seek? Are we creating a pre-assessment to determine prior knowledge of a subject? Whatever our intent, we should communicate it clearly to the learners. Whatever our intent, we need a task that is valid; that is, one that reveals levels of understanding regarding particular learning objectives addressed in the classroom, and for which criteria regarding what constitutes performance from entry to excellence have been articulated.

A good performance task usually has eight characteristics (outlined by Steve Leinwand and Grant Wiggins and printed in the NCTM Mathematics Assessment book). Good tasks are: essential, authentic, rich, engaging, active, feasible, equitable and open. In adult education, we might add that they should connect to participants' goals.

An essential task represents a 'big idea' and aligns with the core of the
curriculum. To be authentic, a task must use processes appropriate to mathematics practice and learners should value the outcome of the work. A rich task is one that has many possibilities, raises other questions, and can lead to other problems. An engaging task is one that challenges the learner to think, yet encourages persistence. Active tasks allow the learner to be the worker and decision-maker, and allow students to interact as they construct meaning and deepen understanding. Feasible tasks are safe, developmentally appropriate, and able to be completed during class time and as homework. Equitable tasks promote positive attitudes and develop thinking in a variety of styles, while open tasks have more than one right answer and offer multiple entry points and solution paths. Of course, to have all these qualities, a task must be near perfect. Good tasks hit most of the characteristics.

Examples of performance tasks follow. In a class working on fractions, for instance, the teacher might assign a task that asks groups or individuals to design an activity that will help the class understand how small 1/10 is. S/he might seek a broader task, too, by asking learners to list everything they have learned about fractions so far. If the class has been studying averages, s/he might ask learners to write an explanation that proves the statement "median is always the middle number" is either true or false. In addition, s/he might ask learners to look at some real estate listings in which a median house price is listed and discuss, given the range of houses listed, why the realtor chose to look at the median as opposed to the mean. The task might be extended by asking learners, "Who might want to know the mean in this case and why?" If studying geometry, learners might be presented with a diagram of a right triangle with a 45¡ angle and one leg that measures 5cm and be asked to list out everything they can tell about this triangle. For a class in which percents are the focus of study, a teacher might present an 'eating out' situation and ask how to figure out how much to leave, tip included.

A good performance task provides a lens through which to view student understanding. However, it's important to have a clear vision of what's being assessed, and the criteria should be transparent to all, including the learners. Not sharing the criteria for assessment has been compared by some to asking someone to take a driver's license test without telling them what's being tested. How do you prepare for such a test? How do you know if you're doing what's expected?

Most performance tasks are scored using a 'rubric'. A rubric can be divided into sections such as: understanding the problem; planning a solution; getting an answer. Points are then awarded for various levels of performance, such as "no attempt to plan a solution" or "completely inappropriate solution" or "partially correct plan" or "workable plan" that could result in correct answer.

A sample rubric set from the fall 1996 edition of The Problem Solver (Problem Solver Special Edition: Assessment of Mathematics Understanding, vol.4, No.1, Western Mass. SABES) was devised for a performance task that involved investigating rents in town, graphing them and finding averages. There was a 'Skills Assessment' rubric and a 'Habits of Mind' rubric. They looked something like this:

Top of page

Skills Assessment:

3-mastery | 2-demonstrated use | 1-unused or misused

Habits of Mind

Obviously, the nature of the performance task being used to make assessments as well as the purpose of the assessment determine the rubric form. We may want to assess only one competency — simplifying fractions, for instance - and in that case we can look at a problem that learners are working on, one that requires adding fractional amounts, and choose to look only at the work done regarding simplifying fractions. In such a case, we might look to see if the computations are done mentally or with pencil and paper, and if done with paper and pencil, we could then ask if the fractions are being simplified by the largest factors possible or by 2's, etc.

Perhaps the most difficult work with performance assessments, as with any assessment, lies in the final act. What recommendations do we make based
on what's been illuminated? At least with a performance assessment, there is
a clearer idea as to where the problems in understanding or skill exist. We can tell if careless computation or total lack of place value understanding is at play; we can tell if the concepts of perimeter and area are clear, but a person is using counting or adding as opposed to formulas to determine each. It's easier to be a good teacher if you know what's understood and what isn't. Performance assessments in mathematics make teachers wise in the ways of their learners.

Top of page

Tricia Donovan taught GED classes in Western Mass for 12 years before joining TERC in Cambridge as a curriculum developer/writer on the EMPower math project. She worked on the original ABE Math Standards and on the current ABE Math Frameworks. In addition, she
is editor of The Problem Solver, an ABE math newsletter funded by DOE and SABES West, and a doctoral candidate in the Teacher Education Curriculum Studies Department at the University of Massachusetts, Amherst.

Originally published in Adventures in Assessment, Volume 14 (Spring 2002), SABES/World Education, Boston, MA, Copyright 2002.

Funding support for the publication of this document on the Web provided in part by the Ohio State Literacy Resource Center as part of the LINCS Assessment Special Collection.