A numerical grade offers only a glimpse of a student’s knowledge. If the goal of assessment is to obtain a valid and reliable picture of a student’s understanding and achievement, evidence must come from a variety of sources. These sources may include oral presentations, interviews, written work, observations, or various combinations of these. Examples of written work include projects, homework assignments or activities, journals, essays, quizzes, and exams. Records of a student’s progress may include anecdotal records, portfolios, and mathematical journals. Rating scales and observation checklists are also helpful devices to record evidence of a student’s continued growth in understanding. The advantage of using several kinds of assessments is that a student’s understanding can be continuously monitored. In addition, because students differ in their perceptions and thinking styles, it is crucial to provide opportunities for students to demonstrate their individual capabilities. Continuous use of a single type of assessment can frustrate students, diminish their self-confidence, and make them feel anxious about mathematics.
The assessment of a student’s mathematical knowledge must address the student’s ability to solve problems, to use the language of mathematics, to reason and analyze, to comprehend the key concepts and procedures, and to think and act in positive ways. Assessment should also examine the extent to which students have integrated and made sense of mathematical concepts, and can apply understandings to situations that require creative and critical thinking.
Methods for assessing a student’s ability to solve problems include observing the student solving problems individually, in small groups, or in class discussions. Other methods include listening to a student discuss problem-solving processes and analyzing tests, homework, journals, and essays. These discussions, as well as the student’s work, can then be assessed using a rubric or rating scale. Some key components to consider when assessing a student’s ability to problem solve include willingness to engage in the problem, use of a variety of strategies, perseverance, finding of one or more solutions, and consistency in verifying the solution.
Assessment of a student’s ability to communicate mathematically includes the meaning he/she attaches to the concepts and procedures of mathematics. It also involves his/her ability in talking about, writing about, understanding, and evaluating mathematical ideas. In assessing a student’s ability to communicate, attention should be given to clarity, precision, and appropriateness of mathematical terms and symbols, as well as to the depth of understanding communicated. When assessing the student’s communication skills, the use of discussion can also inform the teacher regarding the student’s ability to function as a critical participant in small groups or within the class.
An understanding of mathematical concepts involves more than mere recall of definitions and recognition of examples. It also encompasses a broad range of abilities. Assessment must include these aspects of conceptual understanding. Observational checklists, anecdotal records, performance tasks, or written records can be used to assess such conceptual understanding.
Learning mathematics also includes developing a positive attitude towards mathematics. The assessment of a student’s attitude requires information about her/his thinking and actions in a wide variety of situations. A student’s attitudes are reflected in how he/she asks and answers questions, works on problems, and approaches new mathematical concepts and ideas. Observations, homework assignments, journals, and oral presentations are all excellent ways to assess a student’s mathematical attitude. Student self-assessments can also be used to provide information regarding a student’s attitudes toward mathematics, as well as information that informs the teacher’s insights into a student’s understanding of content.