8.2: Evaluating Students' Learning
"Will this be on the test?" The bottom-line-question we hear most from our students. A question that by its nature provides teachers with insight about the level of understanding our students posses and their tendencies to rely upon memorization of content--a stark contrast to the teacher's strive for inculcation and process-drive mathematics. Keeping our finger on the pulse of learning in our classrooms means that everything has an instructional purpose.
Assessment is defined as the process of gathering information about students' understanding. The process includes both formal methods (tests, portfolios, projects, interviews, homework) as well as informal ones (questioning, supervising practice, group activities, journals). It allows teachers to build and pace appropriate curriculum and balance student needs with standards. This process called assessment must be a continuous and underlying stream of consciousness for all teachers. The classroom setting, activities, assignments and evaluations must all have instructional purposes that enhance understanding, whether the goal is remediation or deeper exploration.
The need for instructional assessment goes beyond the students' performance. Teachers must also use the assessment process as a reflective measuring stick for their teaching. As stated above, the speed and direction of curriculum should, in part, be determined by the level of understanding of the students. Teachers must have the adaptability to adjust the pace and determine when their methods or practices are effective. Through professional development opportunities, licensing, educational conferences, peers, etc. teachers should commit to remaining resourceful and keeping an "authentic" bag of tricks and stay mindful of the current interests of students to make the connections to content. A newer perspective on the process, authentic assessment (Meaningful Assessment, Johnson & Johnson) recognizes the practical applications of content, placing emphasis on understanding and creative or individualized expression.
Evaluation, on the other hand, occurs when teachers make professional judgement about student work and assign value to students' understanding. Evaluation comes in many formats and levels of importance and aims to maintain a delicate balance of validity and reliability. The validity of an evaluation compares performance directly to objectives. In teaching, objectives tie most directly and frequently to standards, but whatever the objectives for an evaluation, they must be clearly communicated and agreed upon prior to the evaluation. Students and teachers must have a common understanding about the level of expectation and clear methods for measuring performance. Reliability of an evaluation ensures that a test or other format is fair and will ensure that similar levels of understanding receive similar scores, taking into account multiple representations of the content.
Evaluations are most commonly interpreted as the components of grades, but not necessarily always. Remember, evaluations make judgements and assign value, but those judgements are not required components of the grading system. Diagnostic evaluations, for example, are used primarily to determine prior knowledge, practice skills, gain insight about motivation, attitude or interest, find out what students still do not know, and uncover common mistakes to adjust or reinforce teaching. Diagnostic evaluations will include such things as daily homework or quiz assignments, inquiry-type journal entries, or reviews of content from prior courses. As math teachers, we like our students to show their work. At this diagnostic stage, I am more concerned with the process of the math than the accuracy. If a student has the correct answer, but has flaws in the mathematical process, that indicates entirely differently learning troubles than for the student who follows a flawless process but has an inaccurate answer due to a mechanical arithmetic error.
My mentor teacher gives each new Geometry student an Algebra Skills assignment at the beginning of the year to determine how proficient the students are with basic skills such as squares and square roots, solving linear equations in one or two variables, graphing lines in a coordinate plane, and factoring and "FOILing" simple quadratic equations. This evaluation is critical for planning the curriculum and determining the readiness of the students. Readiness is important in mathematics, because the content builds from course to course and year to year. Basic skills should be mastered before moving to more difficult material, but social promotion and other factors often undermine this important growth. This is especially true in the Honors Geometry classes of my mentor teacher. Placement has no set criteria. A student can register for Honors Geometry with no real qualifications--their own desire, a parent's or counselor's is all it takes to qualify administratively. The Algebra Skills assignment is purely diagnostic, and has no bearing on a student's grade. It is highly effective in raising red flags and modifying teaching or redirecting students to more a appropriate level class if more evidence of skill deficiencies continues to mount.
Formative evaluations provide feedback about the progress being made toward the objectives. Examples of formative evaluations might include quizzes or challenging homework problems that require connecting knowledge from recent lessons. In math, it is particularly important to continually make those connections--to reinforce the building blocks and keep well-understood skills from getting rusty This phase requires students to demonstrate some independence and learning while still allowing teachers to maintain a low-stakes level of anxiety and get a feel for the progress the class is making toward the stated objectives. This phase is a very reflective one for teachers, because it is here that the opportunity exists for refinement of teaching strategies and student skills. At this formative stage, we also begin to transfer learning forward by implying or relating future content.
Summative evaluations provide most of the data for grades. Student must demonstrate the sum of their learning, revealing their true understanding and mastery of content. Summative evaluations most commonly occur in the form of exams, but are more frequently appearing as projects, portfolios, etc. Many teachers are having success with open-ended assessments that more fully address the "why?" of the mathematical content. If you have ever observed my teaching, you know that I intentionally lead students down a chain of "why's" to get to emphasize the connections and deepen the understanding. Regardless of the format, summative testing should reflect frequent high performance that demonstrates lots of mastery and keeps the stakes relatively low.
Testing and other summative assessments must be carefully prepared and ask clear, specific questions that are appropriate to demonstrating a mastery of skills, but are not overly challenging or intimidating. Standard well-written questions have clear meaning, appropriate use of vocabulary and rarely address more than one objective. This ensures that questions cannot be answered out of context or cause confusion or over-simplification of the content. This excerpt from the National Council of Mathematics Journal is a good example of such clarity.
So we come to grading. When the grade is called for by the administration, how will it be determined. I think there are several keys to managing this process successfully. First, teachers must communicate early and often with students about grading policies. A defensible system must be implemented at the beginning of the term to motivate students and guide teaching and learning. Rubrics that outline specific levels or performance may need to be developed to make certain students are aware of expectations in both product and process. In math, as in many other classes, grades are computed mostly arithmetically. In my class, students will track their own grades. With access to grades on the Web, this will be little challenge, but I hope to enhance the experience by requiring them to know what impact future performance will have on that final grade.
Second, I believe that mathematical grades should be supported by summative means that show thoughtful understanding and mechanical precision. Anything less is an indication of difficulty in future math courses. I do, however, believe that there is much we can do in math class to support that understanding and precision. While student teaching, I plan to include Study Habits as a component of the final grade. It will be relatively low-stakes, but hopefully checking notebooks, recommending summative-type exercises, preparing and connecting new lessons, etc. will provide valuable diagnostic practice and formative direction in the classroom. Modeling, encouraging and teaching effective study habits is just one way to help students develop mastery of content.
Additionally, my gradebook will include opportunities for self and peer evaluation. Since my eyes and interpretation may become acute from time to time, both teachers and students should practice learning assessment strategies from one another. Students are often harder on themselves than instructors or peers might be, so self-evaluation creates good opportunities for open discussion about performance. Peer evaluation has proven effective in my CoT seminar. I recently developed an assignment that required students to "trade-and-grade" expectations. This assignment forced students to look more closely at the criteria and rubrics in assessing one another's work. It also opened the instructor's eyes to how we may interpret that criteria a little differently. The assignment proved valuable enough during the Assessment semester that it is now a continuing required assignment for the seminar members each semester.
Expectation 9.4: Examining One's Practice is not directly relevant here, but I think there are points worth including. I only want to say that I think it is important to consider what methods, questions, objectives, effective communications, etc. are helpful or discouraging when the evaluative tables are turned upon you as a teacher. How do you feel when unfair questions are posed, objectives you were not aware of are not met, and no recommendations for improvement are offered? Always keep this perspective in mind when evaluating students. The critique is best when it is used fairly as an integral part of the instruction. Whether evaluating teacher performance or student performance, it is widely accepted that there is an inherent subjectivity to all assessment and evaluation strategies. Teachers should be aware of this and find ways to minimize the impact on student grades.
Incidentally, what does this inherent subjectivity say about the practice of standardized testing? Perhaps that is a good research project for future assignments or expectations. But for now, let us just say that we recognize the balance that must occur between standardized tests and classroom assessment. In contrast to virtually everything above, standardized tests usually have the highest stakes yet are the least instructional. Assessment tools with the highest stakes should be the most adaptive, should they not? They should create a platform that reaches every student in the most "authentic" way. Since that is unlikely to change anytime soon, teachers must force a balance that closely monitors student progress daily and uses a variety of classroom assessment tools from supervised practice to tests and projects that "will enable students to understand mathematics and perform well on standardized assessments" (NCTM News Bulletin, May/June 2005).
