Assessing Problem Solving
A Look at Alternative Methods
For a Changing Mathematics Curriculum


Linda Duvall

EMT 705

I have been a classroom teacher of mathematics since 1962. During the last ten years of my teaching experience technology has impacted all areas of society including, even though very slowly, mathematics education. I became more aware of that impact when I became a Leader Teacher with Project LITMUS. Prior to that time I had been teaching in a small private school and was not encouraged by the administration to keep up with what was happening in mathematics education by attending conferences or taking summer college work . That first summer with LITMUS was really an eye opening experience for me. As I learned about and became familiar with the NCTM Curriculum and Evaluation Standards for School Mathematics I began to question the mathematics I was teaching in my classroom. The more I learned about the power of graphing calculators and computers as teaching tools, the more I questioned the methodology of my teaching. My personal goal in teaching had always been to help my students prepare for the future, whether it included post high school education, the work place, or the establishment of a home and family. I was no longer convinced that the experiences my students were having in my classroom were serving to prepare them adequately for that future. So I began to change the mathematics curriculum in my classes to include the use of technology and to provide my students with more opportunities in problem solving. But, my focus was on objectives and content and I neglected to consider how I would assess progress in problem solving. As a result I did not feel completely satisfied with what I was doing. My experiences this summer, classwork, readings, and discussions have led me to conclude that because of the changing curriculum, the mathematics that my students need to learn requires a deeper understanding. The tests I have used in the past to assess students work and justify a grade, most often were concerned, not with the problem solving process, reasoning, or mathematical communication(three of the primary standards that transcends the Standards K - 12), but only with the answer. Through my readings on assessing problem solving this summer, I have come to understand that the "why" behind my dissatisfaction was because the evaluation strategies I was using did not adequately assess the success of my students in the problem solving process. Therefore, the logical issue for me to consider in this paper is "How can I better assess the progress of my students in problem solving?".

The best resources for information on evaluating progress in problem solving as it relates to the Standards were the following: the problem solving Evaluation standard in the Standards, an NCTM publication entitled How to Evaluate Progress in Problem Solving, and the NCTM Assessment Standards . If I am going to attempt to teach in the spirit of the Standards then I must also attempt to assess my students progress in the spirit of the Standards. Because the problem solving process cannot easily be measured through a test, the teacher is required to make inferences about the students progress in problem solving based on tests or activities that require the student to perform. In order to attempt to make the inferences valid, many techniques of assessment should be used by the teacher. By giving a variety of assessments, the student will have more opportunities to respond and thereby reveal more about the problem solving strategies that are used by him or her. The Standards support the utilization of multiple assessment techniques. In my readings I have found many alternative assessment strategies. The following is a short discussion of some alternative methods I plan to use next year as I assess the progress of my students in problem solving.

The first is informal observation and questioning. This assessment technique is one in which the teacher observes the students while they are involved in problem solving situations. These can be individual, small group, or even whole class problem solving situations. It is suggested that when this technique is used, the teacher limit the attitude and performance attributes on which to focus in order to better assess student progress through observation.

Along with informal observation the teacher can employ informal questioning during the observation to gain deeper insight into the problem solving techniques of the student. Not every student needs to be evaluated through informal observation and questioning in every problem solving session. The teacher can stimulate student response with questions such as: "How did you .....?", "What do you think .....?", or "Why did you .....?". If a teacher is planning to use informal observation and questioning to assess progress in problem solving, a plan must be designed for recording notes so that progress can be traced over a period of time. Comment cards, rating scales, and check lists can be used to meet this need.

A second assessment technique is the structured interview. Using a structured interview to assess problem solving is more effective when limited to one or two students at the time. This very structured instrument includes questions and problems selected well in advance of the interview. This evaluation strategy might enable a teacher to identify specific problems students are experiencing in their problem solving.

A third assessment technique involves collecting data from students through student self assessment. This data represents the efforts of students to assess their own progress. The two specific types of self assessment I would like to use this year are the student report and the attitude inventory. In a student report the student is asked to reflect on a specific problem solving experience. The teacher provides questions that will help the student reflect but not influence the response. The report can take on a general nature or it can be more directed depending on the goals the teacher has for the assessment. When this method is used in the classroom, it is very important that the student feel comfortable enough with the learning environment to be honest with the teacher. If the student just wrote down words to satisfy the teacher then, of course, such a report would be little value in assessing that students progress.

An attitude inventory allows the student to give a self-appraisal of his or her own performance or attitudes. The inventory can be a valuable technique to assess attitudes students have towards problem solving in mathematics and even mathematics itself. A limiting factor of this technique is that it relies on the student and is therefore strongly influenced by the maturity and insights of the student and maybe what the student considers as the teachers expectations.

A fourth strategy for assessing students progress in problem solving is known as Holistic Scoring. This is a technique for evaluation students written work. There are two types of Holistic Scoring that I would like to include. They are Analytic Scoring and Focused Holistic Scoring. In analytic scoring each phase of the problem solving process is assigned a point value. The teacher selects the phases that are to be assessed and assigns a range of possible scores( 0 - 2 is suggested) for each phase. This method can be used when the teacher believes that assessing the problem solving process means more than just checking to see if the student came up with the correct answer.

The Focused Holistic Scoring technique assigns a single score for the entire solution. The specific criteria of a focused holistic scoring scale will take time to develop and will need to be adjusted frequently. But, when well planned, this assessment method can be used as an efficient strategy that focuses on the problem solving process not just the answer.

I will continue to use the assessment techniques that I have used in past years:open ended questions, multiple choice, short answer questions discussions, and explanations are some of these techniques . Just because methods of assessment have been around for a while doesn't mean that they are bad. The important thing is that every student be given many varied opportunities to demonstrate the progress that he or she is making in problem solving. In addition I plan to include informal observation and questioning, structured interview, student self-assessment, and holistic scoring. The big drawback that I foresee with these new evaluation techniques is the time factor. The teacher will need time to plan new assessment methods to fit the class and the activity, time to execute them, time to draw inferences from the results, time to share the results with the students, and time to consider the curriculum changes that might need to be made as a result of the results of evaluation.

I should not neglect to mention that students need to be informed about the evaluation strategies that will be used in the classroom. Every student should be informed when the work in which they are engaged will be assessed. As students experience the new assessment techniques and realize that they are designed and used to help them improve their problem solving skills, they will begin to understand that assessment is another tool that can be used by both the teacher and the student to facilitate learning.

As I begin to include these new assessment techniques in the coming school year, I will focus at first on the beliefs and attitudes my students exhibit about mathematics and problem solving. As the year progresses I will consider other aspects of problem solving that are more intrinsic to the problem solving process. I believe that the process followed in problem solving is equally as important as the answer in many problem solving situations. If a student's problem solving procedures are to be measured with accuracy and validity then it is necessary to utilize a variety of evaluation techniques.

In conclusion, I believe this paper is reflective of a change in my attitude toward assessment of problem solving and maybe assessment in all areas of my teaching . If assessment is to be effective and valid, it must be well planned concurrently as curriculum is planned and it must assess the curriculum that is being taught in the classroom. I realize that problem solving skills will not develop or improve overnight or in just a week or two. It then, is my responsibility as teacher to provide many, varied opportunities for my students to experience problem solving. Possibly an even more serious responsibility is mine as I attempt to assess the progress of my students in problem solving. As a classroom teacher of mathematics, I will work to establish an environment in which my students feel comfortable, not threatened. I hope that this will encourage my students to respond to problem solving with new energy. It is my vision that as I incorporate new assessment strategies into my teaching, they will be just other tools my students can use to become mathematically literate, empowered to solve problems, to reason, and to communicate mathematically. When students accomplish these goals, they will be better prepared to survive and succeed as lifelong learners in a technological world.

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