The assessment of
students' mathematical disposition should seek information about
- confidence in using
mathematics to solve problems, to communicate ideas, and to reason;
- flexibility in
exploring mathematical ideas and trying alternative methods in
- willingness to
persevere in mathematical tasks;
- interest, curiosity,
and inventiveness in doing mathematics;
- inclination to
monitor and reflect on their own thinking and performance;
- valuing of the
application of mathematics to situations arising in other disciplines
and everyday experiences;
- appreciation of
the role of mathematics in our culture and its value as a tool
and as a language.
Learning mathematics extends
beyond learning concepts, procedures, and their applications. It
also includes developing a disposition toward mathematics and seeing
mathematics as a powerful way for looking at situations. Disposition
refers not simply to attitudes but to a tendency to think and to
act in positive ways. Students' mathematical dispositions are manifested
in the way they approach tasks--whether with confidence, willingness
to explore alternatives, perseverance, and interest--and in their
tendency to reflect on their own thinking. The assessment of mathematical
knowledge includes evaluations of these indicators and students'
appreciation of the role and value of mathematics.
This kind of information
is best collected through informal observation of students as they
participate in class discussions, attempt to solve problems, and
work on various assignments individually or in groups. Such assessment
procedures as attitude questionnaires fail to capture the full range
of perceptions and beliefs that underlie students' dispositions.
From their first encounter
with shapes or numbers, children begin to form a conception of mathematics.
Teachers implicitly provide information and structure experiences
that form the basis of students' beliefs about mathematics. These
beliefs exert a powerful influence on students' evaluation of their
own ability, on their willingness to engage in mathematical tasks,
and on their ultimate mathematical disposition.
is much more than a liking for mathematics. Students might like
mathematics but not display the kinds of attitudes and thoughts
identified by this standard. For example, students might like mathematics
yet believe that problem solving is always finding one correct answer
using the right way. These beliefs, in turn, influence their actions
when they are faced with solving a problem. Although such students
have a positive attitude toward mathematics, they are not exhibiting
essential aspects of what we have termed mathematical disposition.
The assessment of students'
dispositions requires information about their thinking and actions
in a wide variety of situations and should consider all aspects
of disposition and the degree to which they are exhibited. Disposition
has many components, each of which a particular student exhibits
to a greater or lesser extent. For example, a student might be very
willing to try alternative methods of solving problems but be less
inclined to reflect on the solutions. Another student might be fairly
uninterested in routine exercises and yet work diligently to solve
nonroutine problems. An adequate assessment of these students' disposition
requires information on their willingness to engage in all aspects
of solving problems, including learning through problem solving.
In the classroom students'
dispositions are continuously reflected in how they ask and answer
questions, work on problems, and approach learning new mathematics.
As a result, teachers are in an excellent position to gather useful
information for assessing disposition. In addition, teachers benefit
from this assessment because it provides information for instructional
planning. If an assessment indicates that most students in a class
rarely attempt problems independently and frequently ask to be shown
the solution method, a teacher can choose to reexamine classroom
instruction to evaluate whether students are being encouraged to
solve problems and develop alternative solutions. In short, the
assessment of students' dispositions provides information about
changes needed in instructional activities and classroom environments
to promote the development of students' mathematical dispositions.
Because evidence of students'
mathematical dispositions is apparent in every aspect of their mathematical
activities, observation is a primary method of assessment. When
presented with a problem, particularly one in a new and unfamiliar
context, a student exhibits his or her mathematical disposition
in a willingness to change strategies, reflect and analyze, and
persist until a solution is identified. Students' disposition toward
mathematics can be observed in class discussion. How willing are
students to explain their point of view and defend that explanation?
How tolerant are they of nontraditional procedures or solutions?
Are they curious? Are they willing to ask, "What if . . .?"
What kind of questions do they ask?
Although observation is
the most obvious way of obtaining such information, students' written
work, such as extended projects, homework assignments, and journals,
as well as their oral presentations, offer valuable information
about their mathematical dispositions. Such projects as individual
or small-group presentations of problem solutions or proofs of theorems
can serve as evidence of students' willingness to persevere at mathematical
tasks and test alternative methods in solving problems.
The assessment of mathematical
disposition of students in the primary grades should focus on the
number and quality of students' experiences. An inventory like the
one in figure 10.1, when kept for a class
or for individual students, can be used as a record of their experiences.
This record can be used for determining if a full range of experiences
has been offered and for reporting to parents, guardians, and other
teachers the experiences of the students.
In the middle grades, students'
mathematical dispositions become more apparent in their daily work.
A checklist of actions that demonstrate mathematical disposition
can be used to help record students' progress. Such a checklist
can be completed by a teacher while focusing on five students during
a class period. Over six weeks, each student's mathematical disposition
can be evaluated at least once. A sample checklist is given in figure
Essays and research papers
are particularly useful for both developing and assessing students'
appreciation of the role of mathematics in our culture. Judgments
about the quality of the work should be based on its conceptual
merit, thoroughness, and originality. Just as the assessment of
writing includes a consideration of grammar, spelling, fluency of
expression, force of ideas, and consistency of logic and narration,
so mathematical assessment should include considerations of quality
as well as accuracy. An innovative solution, a well-reasoned argument,
and a willingness to go beyond the constraints of the task are of
equal importance in obtaining a correct solution and should be acknowledged.
Sample topics are the following:
- The role of geometry
in our society today
- Mathematics and computers
- What if standard units
of measures had not been created
- Changes in the applications
of statistics in sports
No one of the preceding
considerations across all grades will yield a definitive picture
of a student's disposition toward mathematics. However, extended
observations of students' efforts and interactions in a variety
of mathematical contexts can give teachers the feedback and information
necessary to adjust their instructional methods and encourage students'
progress in attaining intellectual autonomy.
The Program Evaluation
Standards are an integral component of the overall vision of evaluation's
primary role in guiding change. The vision includes program change
so that all the pieces will fit together. This change will provide
the foundation in grades K-4 for students to know mathematics through
doing mathematics, will develop and expand the ideas in grades 5-8,
and will lead to self-directed learning in grades 9-12. Evaluation
can help determine a mathematics program's status in relation to
the Curriculum Standards and ensure that the pieces fit together.
It can indicate the steps that need to be taken so that a program
aligns with the Standards. In a certain sense, the Program
Evaluation Standards are a guide to creating a program that meets
the challenge of the Standards.
One purpose of program
evaluation is to obtain relevant and useful information for making
decisions about curriculum and instruction. The assessment of students'
capabilities, as described in the previous ten evaluation standards,
is one means of collecting information about a program. The principles
of general assessment apply as much to program evaluation as to
student assessment. But there are other indicators of program quality.
Does the curriculum include the range of topics and emphases described
in the Standards? Is the program articulated from kindergarten
through grade 12 so that students' knowledge and experiences build
continually? Does prior mathematics study provide the conceptual
underpinnings for additional study and for its application to real-world
problems? Does instruction actively involve students in the investigation,
exploration, and development of mathematical ideas and the relationships
among them? Are calculators and computers used appropriately? Do
all students have full access to the program? Do students know mathematics
as an integrated whole?
Any program can be implemented
by degrees. At one level, the language of the new program can be
adopted while day-to-day instruction remains unchanged. At another
level, minor changes in structure can be made by inserting a new
unit into a course or slightly modifying the scope and sequence.
At a third level, deep structural changes can be made that include
altering how people think about a program, how mathematics is presented,
and how students come to know mathematics. The Standards
speaks to this third level of change. It is the role of program
evaluation to facilitate this deep structural change.