Assessing the means by
which a teacher assesses students' understanding of mathematics
should provide evidence that the teacher
uses a variety of assessment methods to determine students' understanding
matches assessment methods with the developmental level, the mathematical
maturity, and the cultural background of the student;
aligns assessment methods with what is taught and how it is taught;
analyzes individual students' understanding of, and disposition
to do, mathematics so that information about their mathematical
development can be provided to the students, their parents, and
pertinent school personnel;
bases instruction on information obtained from assessing students'
understanding of, and disposition to do, mathematics.
The process of assessing
teaching should determine whether and how the teacher uses evidence
of students' understanding of, and disposition to do, mathematics
in making instructional decisions. The assessment of students' understanding
of mathematics should include methods used on a daily basis as well
as those used on a less frequent basis. These methods include evaluating
journals, notebooks, essays, and oral reports; evaluating students'
homework, quizzes, and test papers; evaluating classroom discussions,
including attention to students' mathematical problem-solving, communication,
and reasoning processes; and evaluating group work, clinical interviews,
and performance testing administered individually or in small groups.
Such a variety of student assessment techniques reflects a sensitivity
to the developmental level, maturity, and cultural diversity of
the students and should provide a sound basis for creating mathematical
tasks and directing mathematical discourse.
The student assessment standards
of the Curriculum and Evaluation Standards for School Mathematics
provide a basis for designing tasks to assess student understanding.
Student assessment methods should also be aligned with instruction.
For example, if calculators are used throughout the instructional
program, then they should be allowed in testing situations as well.
should be based on information obtained from assessing students'
mathematical understanding and disposition to do mathematics. A
teacher ought to be able to determine from an analysis of evidence,
for example, why a student cannot use a particular algorithm with
a reasonable degree of proficiency. Does the student lack a conceptual
basis for the algorithm? Is the student confused about the sequence
of steps to be followed? Does the student have a sense of when to
apply the algorithm, or is the algorithm applied in inappropriate
The oft-used phrase, "Are
there any questions?" cannot be reliably used to determine
whether students understand. However, if a student claims that a
parallelogram is a quadrilateral with two sides parallel and two
sides congruent and the teacher asks other students to produce a
counterexample if possible, then there is an indication that the
teacher is engaging students in tasks to check understanding. Another
assessment technique consists of asking students to react to and
evaluate another student's process for solving a problem.
As a result of student assessment,
a teacher should be familiar with a student's confidence level,
willingness to persevere, and other characteristics of disposition
noted in the previous standard. A teacher should be able to describe
individual students' mathematical dispositions beyond the general
descriptions of whether a student is motivated.
7.1 The elementary
principal, Barbara Moore, has been very impressed with the way the
second-year teacher Ed Dudley conducts his first-grade class and,
in particular, with the way that he makes extensive use of manipulatives
when teaching mathematics.
notices that assessment methods are not aligned with instruction.
The teacher is
willing to consider a variety of assessment methods.
notices, however, that when he evaluates students' progress, he relies
on paper-and-pencil tests that appear to emphasize computational outcomes.
When she talks with Mr. Dudley about this, he indicates that it just
seemed like a reasonable way to evaluate students - a very efficient
method. He indicates, however, that he is willing to try different
methods of assessing students' understanding of mathematics. Ms. Moore
offers several suggestions.
The teacher is
interviewing students to assess their mathematical understanding.
notes a technique that could be used for assessment purposes.
A week later, Ms. Moore
drops by Mr. Dudley's class to see if he has had an opportunity
to try any of the suggestions. She is pleased to note that in one
part of the classroom Mr. Dudley is interviewing students while
in another part of the room pairs of students are playing a numeration
game with cubes that can be linked together and a spinner that determines
the number of cubes each student receives. The students are to link
the cubes together whenever they have a group of ten cubes. The
students take turns spinning until each has had five turns. After
the five spins they are to write the number of cubes down on a sheet
of paper. Each student checks to see whether the other student has
written down the correct number of cubes. Ms. Moore is quite impressed
with the game. She plans to suggest to Mr. Dudley that the activity
could provide him with an excellent means of assessing students'
understanding of place value.
Ms. Moore decides to observe
one of the interviews that Mr. Dudley is conducting. She observes
the following exchanges:
Mr. Dudley gives two students,
Jo and Annette, a different number of counting sticks. Each student
is to bundle her sticks into groups of ten.
JO: I have 3 tens
and 4 ones.
teacher assesses the student's understanding of place value and notes
a possible difficulty.
Mr. D: Is that the
same as thirty-four? (JO hesitates. Mr. Dudley observes that she
seems unsure whether the representation of 3 tens and 4 ones also
Annette: I think
they are the same.
JO: I don't know.
Let me count them. (JO unbundles the sticks and begins counting
Annette: I have 4
tens and 2 ones. That's forty-two.
Mr. D: How do you
Annette: Look. (Points
to bundles of ten.) Ten, twenty, thirty, forty, now (pointing to
single sticks) forty-one, forty-two. See!
teacher notes that the student has a good grasp of place value.
Mr. D: That's very
good, Annette. Let's see if we can help JO JO, how are you coming?
JO: I counted and
got thirty-four. They must be the same.
principal compliments the teacher on aligning his assessment with
After Mr. Dudley finishes
his interviews, he discusses the class with Ms. Moore. She compliments
him on assessing students' understanding in much the same way he
teaches mathematics. He indicates that the interviews did take some
extra time but that generally they took less time than he had imagined.
He is very pleased with how much he learned about each student's
thinking about place value during the interviews. Ms. Moore helps
Mr. Dudley develop a chart to make his assessment more systematic.
Value and Counting
suggests a chart to help the teacher organize his assessment of
individual students' progress.
by ones and bundled them into tens. Not sure what number was
represented, however. Needs more work on recognizing number
when given representation. (Jan. 21)
Has good command of translating between written number and representation
using sticks. (Jan. 21)
supports the teacher's efforts to try new techniques.
The teacher indicates
he will base his instruction on what he learned during the interview
Moore compliments Mr. Dudley on his willingness to try something new
and how he organized the class so that all the students were involved
in learning activities. She points out that the game could also serve
as an excellent vehicle for assessing students' understanding. Mr.
Dudley indicates that he wants to provide JO with more opportunities
to count objects and to do activities to develop her understanding
of place value.
The teachers are
empowered with the authority and responsibility to shape programs
The teachers recognize
that problems related to assessment and grading may be contributing
to a broader problem.
In the mathematics department at West High School "teacher
teams" have the responsibility for the development and monitoring
of program and curricular changes. The "algebra team" of
Art Washington, John Nystrom, and Katie Cusciaro teach all ten of
the first-year algebra classes at West. Through a paid summer curriculum
project, their principal, Simone Richardson, has asked them to develop
a plan to reduce the high number of students who discontinue taking
mathematics after first-year algebra. The teachers are concerned as
well, since they have noted the high number of D's and F's that students
have been receiving in algebra. Ms. Richardson asks the team to pay
particular attention to the diverse backgrounds of the students at
The school district
provides funds during the summer for teachers to meet and address
The teacher realizes
an inconsistency between his informal assessments during class discussions
and the more formal assessments using tests and quizzes.
In July the
three teachers meet to discuss the problem. They agree that there
are many factors that contribute to poor student performance and
their high rate of dropping out of mathematics. They decide to focus
on their teaching and assessing techniques as a first step to improving
the situation. With respect to assessment, the teachers share their
tests, quizzes, and the means by which they assign grades. Art indicates
that many students do not complete his tests - they skip many items.
He suspects that since English is not the first language for many
of the students, they have difficulty reading some of the questions.
He says that about 80 percent of his grades are based on tests and
quizzes and usually the students do not do well on tests and quizzes.
Yet, he feels that they demonstrate a reasonable understanding during
Katie is frustrated as well.
She tries to create items in which the students are required to
"explain" or "draw and label." She shares the
1. Draw and label altitude
Explain why it is an altitude.
2. A student claims that
x2 is always larger than x. Is she correct?
Explain your reasoning.
3. Draw and label a rectangle
whose area is (x2 + x) cm2.
The teacher emphasizes
communication but may not be aligning assessment methods with students'
states that her students have a great deal of difficulty in writing
mathematics - perhaps because of the language problem. Katie indicates
that she has allowed students to work together when solving problems
but not when taking tests or quizzes.
The teacher tends
to see the problem as the students' problem rather than as an instructional
or curricular problem.
indicates that the absentee rate in his classes is very high - sometimes
approaching 50 percent. How can he teach them if they don't come to
class? He questions whether Katie is expecting too much when she wants
her students to "explain" and "draw and label."
After all, many students are not proficient with the basic skills.
He claims that he keeps it simple by sticking strictly with the tests
in the book, one test every Friday, with makeups on Monday. If students
don't show up for the makeup, that can't be his fault, he argues.
The teachers decide
to evaluate students on activities other than test and quiz scores.
The teachers hope
that the journal entries will increase students' disposition to
The teachers continue to
discuss the problem. They recognize that although there are some
things beyond their control, there are some things that they can
control, for example, how they evaluate students. They also recognize
that the cultural diversity of the students may require them to
adjust how they have been testing and grading students. They decide
on the following means of evaluating students:
1. Journals. Every
student will keep a journal. The journal will count the same as
a test grade. The daily entries will include examples worked out
in class and various methods presented in class for solving problems.
In addition, they will focus on the following items:
- What they learned that
- What they did in class
that helped them learn.
- Why they think it is
important to learn it.
- How they felt about
the class that day.
The teachers recognize
that changing methods of evaluation has implications for their teaching
team felt that these questions would also keep them on their toes
when preparing lessons. For example, question c will serve as a constant
reminder to provide reasons why a topic is important to learn.
The teachers will
place a greater emphasis on students communicating mathematics.
Class discussions. Greater emphasis will be given to class discussions
in evaluating students, thereby encouraging students to attend class.
Students will be given more responsibility to present solutions and
to explain procedures during class discussions.
The teachers decide
to try an alternative method of giving quizzes.
Quizzes. All quizzes will be taken in pairs; students will
be able to discuss their solutions with their partners. This will
facilitate communication and help the students feel less tension when
taking a quiz.
The teachers are
making an effort to match assessment with students' backgrounds.
Tests. All tests, including makeup tests, will be shared among
team members. They will maintain an emphasis on "explaining"
and will try some open-ended items as well, but they will give the
students greater latitude in responding. They will also make the tests
shorter so that students will have more time to respond.
of the program will be cyclical in nature.
The teachers and
the principal will monitor the program's effectiveness.
The three teachers
agree to reevaluate the effect of the program next December. Simone
is impressed with what the teachers have done. She says that she
will be interested in their assessment of the effectiveness of the
program next December. She reminds them that the ultimate goal is
to increase learning and to encourage more students to be successful
in doing mathematics and to continue their study of mathematics.
The high failure rate is counterproductive to achieving this goal.
The teachers and the principal decide to meet informally once or
twice a month to share ideas and make any modest revisions. They
hope that by this time next year, they can point to the program
as a model of success in increasing achievement and disposition
to do mathematics. Simone is hopeful that the program will also
help reduce the absentee rate among students.