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