| The
primary question to be answered by teachers is, How is each student
progressing in relation to the goals we have set and agreed on? |
Teachers monitor students’
progress to understand and document each student’s growth in
relation to mathematical goals and to provide students with relevant
and useful feedback about their work and progress. Goals for students’
learning may involve the formulation of long-range or short-range
performance criteria. These goals are set in collaboration with
students and others responsible for students’ learning. Effective
monitoring of a student’s progress toward those goals enhances
learning by clearly communicating what the goals are and the extent
to which students have met them.
Teachers have always monitored
their students’ progress. However, if students are to increase
their mathematical power, several related shifts in assessment practice
are warranted:
- A shift toward judging
the progress of each student’s attainment of mathematical
power, and away from assessing students’ knowledge of specific
facts and isolated skills
- A shift toward communicating
with students about their performance in a continuous, comprehensive
manner, and away from simply indicating whether or not answers
are correct
- A shift toward using
multiple and complex assessment tools (such as performance tasks,
projects, writing assignments, oral demonstrations, and portfolios),
and away from sole reliance on answers to brief questions on quizzes
and chapter tests
- A shift toward students
learning to assess their own progress, and away from teachers
and external agencies as the sole judges of progress
Judging Progress
toward Mathematical Power
The notion of mathematical
power, as described in the NCTM Curriculum and Evaluation Standards
for School Mathematics, includes a student’s ability to
"explore, conjecture, and reason logically, as well as the
ability to use a variety of mathematical methods effectively to
solve nonroutine problems" (NCTM 1989, p. 5). Judging progress
toward such a complex and broad target involves two components:
(1) setting goals in the form of performance criteria, and (2) assessing
students’ progress toward those goals.
Helping students set and
attain goals is at the heart of good teaching. Goals in the reform
vision of school mathematics involve collections of related ideas
(networks) within and across particular mathematical domains that
students are expected to know and use. Each mathematical domain
(e.g., addition and subtraction of whole numbers, common fractions,
geometry, statistics, functions) includes terms, signs, and symbols;
the rules for their use; the situations they commonly represent;
and the know-how to use these ideas to solve routine and nonroutine
problems.
To assess progress toward
the reform goals, performance criteria need to be publicly stated
and student performances judged in light of those criteria. There
are multiple possible paths toward the achievement of performance
criteria, and numerous possible benchmarks and sources of evidence
to indicate progress toward it. Furthermore, short-term expectations
for individual students may vary because students come to classroom
instruction with varying backgrounds, interests, and talents.
Example: A Middle-Grades
Statistics Unit
The following vignette illustrates
how technology can be used for the public dissemination of performance
criteria and how student performances can be judged in light of
those criteria.
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"In grades
5—8, the mathematics curriculum should include exploration
of statistics in real-world situations."
–NCTM
(1989, p. 105)
Openness Standard:
This assessment provides students with access to samples of other
students’ work and performance criteria.
Monitoring involves
examining evidence at different times.
Inferences Standard:
Using multiple sources of evidence can improve the validity of the
inferences made about students’ learning.
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What Does "Average"
Mean?
Ms. Lafleur teaches
mathematics in a middle school in Montreal, Quebec. One unit
of instruction during the year involves an introduction to
statistics with an emphasis on descriptive statistics. Her
goal for all her students is that they will use statistics
to describe and predict events in the real world. Last year,
Ms. Lafleur had her students design their own research experiments.
Teams of four heterogeneously grouped students worked at computer
workstations during the unit using a statistical package and
graphing software to carry out their work and design a presentation
of their findings.
The content she identified
as important to teach and assess included several components
of statistical problem solving: designing their own research
questions, collecting data to answer their questions, representing
the variables and data graphically, analyzing their data through
statistical methods, interpreting the data they collected,
and communicating their understanding through class presentations.
For each of these
components she selected examples to illustrate acceptable
levels of performance. The previous year she had video-taped
students as they presented their statistics projects to the
class. She selected examples from these tapes to show different
qualities of performance on the components identified. The
studnets were able to gain access to these tapes on their
computer screens by choosing a component, as the sample screen
(Lavigne 1994) in figure 4 shows.
If the studnets chose
"Data Presentation", for example, they would get
a screen that offered access to video examples of both average
and above-average performance (see fig. 5). (Although the
labels used here were the familiar labels of normative comparison,
they were being used to exemplify important performance criteria.
They represent "proficient" and "exemplary"
levels of performance.) The average example shows a students
presenting her group's data as raw scores in a table. In the
above-average example, the four students in the group have
each made individual pie graphs of different aspects of their
data.
In this
manner, the students could learn to identify the possible
factors that would distinguish an average from an above-average
performance for each component o their project.
Fig. 4 Table
of contents screen (Lavigne 1994)
Fig. 5. Data
Presentation screen showing video example icons (Lavigne
1994)
Ms. Lafleur also used
the computer to store and track evidence of an individual
student's progress over time by monitoring each studetn's
use of the computer for graphing and analyzing data. For example,
to exammine Sabrina's performance, Ms. Lafleur typed her notes
on Sabrina into the computer. She noted Sabrina's growth in
her ability to organize data. She used a computer program
that recorded every action Sabrina took with the computer.
She could reexamine these observations whenever she was interested
in monitoring a certain component of student growth. In the
first week Sabrina had difficulty organizing her data for
subsequent analysis, but by the end of the month she was very
proficient. Ms. Lafleur added to this computer information
other assesssment vidence: classroom observations, products
of Sabrina's work, Sabrina's responses to quizzes, and her
responses to questions about her work that were provided in
a written journal. All this evidence was stored in a computer
database for easy access; it was also used to build a case
for Sabrina's progress in statistical problem solving.
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This example (adapted from
Lajoie et al. [forthcoming]) illustrates the importance of setting
high expectations and monitoring the progress of all students if
specific goals in mathematical learning are to be accomplished.
Note that the computer can store observations, but it is the teacher
who must interpret such evidence in order to understand and document
students’ progress over time. Although the goals for all the
students (making sense of a batch of data and being able to communicate
findings to others using statistics) contained many components,
judgments of progress could be made by both the teacher and the
students through continual and recursive monitoring.
Communicating
with Students about Their Performance
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| Communicating
with students about their performance is part of a shift toward viewing
students as active participants in assessment. |
Monitoring students’
progress effectively depends on good communication between teachers
and students. This communication works in both directions: Teachers
gather evidence about students’ learning, and then provide
feedback to students about their progress. When gathering evidence,
teachers can make use of much more than occasional paper-and-pencil
tasks.
Example: Listening to
Students
One of the most powerful
sources of evidence about students’ learning comes from listening
to students explain their thinking during classroom instruction.
In the next example (adapted from Carpenter and Fennema [1992]),
a first-grade teacher listens to her students describe alternative
strategies for solving problems.
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Equity Standard:
This teacher encourages multiple paths to the solution of a problem.
Learning Standard:
Talking about solutions helps students become better problem solvers.
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How Many Peanuts?
Ms. Morris's first-grade
students are solving whole-number addition and subtraction
problems. Rather than relying on written tests or formal assessment
procedures, Ms. Morris continually asks her students to describe
the processes they used to solve a given problem, and students
are encouraged to describe alternative solutions. In the following
dialogue, the student solve a comparison problem:
Ms. Morris: The
African elephant ate 37 peanuts. The Indian elephant ate 43
peanuts. How many fewer peanuts did the African elephant eat
than the Indian elephant?
The children worked
on the problem for two or three minutes. Some of the children
used stacking cubes that had been joined together in stacks
of ten cubes. Others did not use any materials. After a minute
or so, several of the children raised their hands. After two
minutes, only one child, Mike, was working on the problem.
Ms. Morris asked him if he was done. When he shook his head,
she told him to keep working. After another thirty seconds,
he raised his hand.
Ms. M.: OK?
How many fewer peanuts did the African elephant eat? Mike?
Mike: 6
Ms. M. : Does
everyone agree with that? ...How did you figure it out, Mike?
Mike: Well,
I had 43 here [pushing out four stacks of ten cubes and
three additional cubes joined together], and I had 37
here [pushing out three stacks of ten cubes and a stack
of seven]. I put 30 on top of these 30. I took 3, and
I put them here; there were 4 left, so I took 4 off, and there
were 6 left.
As he
described what he did, he took three of the ten stacks from
the collection of 43 and put them on top of the three ten-stacks
in the collection of 37. Then he took the three single cubes
from the original set of 43 and put them on top of the seven
cubes in the set of 37. Then he took the remaining stack of
ten cubes form the original 43 and broke off four cubes. He
put these four cubes on the four cubes in the set of 37 that
were not covered. He was left with six cubes form the set
of 43 that did not match up with cubes in the set of 37.
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Used with permission
from the Wisconsin Center for Education
Research, School of Education, University of Wisconsin-Madison
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Ms. M.: What
do you think of Mike's solution?·Did anyone do it a different
way?
Marci: I took
37, and I needed 43. So I counted up 3 more. That was 40.
Then I took 3 more to 43.
Ms. M.: Good.
Does her way work out well?·It sure does. Did anyone do it
differently?
Linda: Well,
first I got 37. Then I got 43 [pushes out collections of
37 and 43 cubes joined together in stacks of ten, with the
extra cubes also connected together]. See, I know it couldn't
be 10, because if you had 10, it would be 47 instead of 43.
So I realized that it had to be less than 10. So what I did
was I imagined 3 more cubes here [points to the top of
the stack of seven cubes in the set of 37], and I imagined
3 more right here [pointing to a space next to the collection
of 37 that corresponds to where the three cubes are in the
collection of 43].
Ms. Morris gave each
child in the group time to complete the problems, and she
gave children who had a different solution an opportunity
to explain their solutions. The children all listened attentively
to other children's solutions, so they had the chance to learn
from on another. Ms. Morris also learned what each child could
do, and she learned more than whether a child got the correct
answer. The different solution strategies reflected quite
different levels of understanding. Mike had to model the problem
directly, whereas the solutions of Marci and Linda showed
more flexibility in operating with numbers. While the children
were working on the problem, Ms. Morris made notes about the
solution processes she observed. These notes helped her in
monitoring the students' progress.
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This
example could be extended to reflect growth or change with respect
to Mike’s ability to solve such comparison problems. His initial
strategy, direct modeling with cubes, may be replaced with counting
strategies such as Linda’s, by writing a sentence (e.g., 37 +
q = 43) and "counting on from smaller" to find the answer,
and so forth. Monitoring Mike’s progress in learning to solve
a variety of addition and subtraction problems would involve tracking
the strategies and procedures he uses to solve such problems over
an extended period of time. |
| Openness
Standard: Communicating with students about their progress helps them
understand expectations. |
Setting goals and gathering
evidence of a student’s progress in achieving them are unproductive
if judgments by teachers or others of his or her performance are
not regularly communicated to the student. Students need to understand
clearly what is expected; whether their work is of acceptable quality
(e.g., the assessment is based on the reasons they give and strategies
they use as well as on whether their answers are correct or not);
and the effectiveness of the draft, feedback, and revision cycle
for the production of large pieces of work. For assessment to be
equitable and valid, each student must receive feedback over time
on multiple occasions and in multiple formats on tasks that address
the breadth of important mathematical content. To satisfy the Coherence
Standard, feedback must be part of an assessment system that gives
students consistent messages about what mathematics is valued and
legitimate ways to demonstrate that knowledge.
The best feedback is descriptive,
specific, relevant, timely, and encouraging. It is immediately usable.
The feedback may be oral or written, formal or informal, private
or public, geared toward an individual or a group. The focus of
feedback may be a single assessment activity or multiple activities.
Providing effective feedback in a continual and recursive manner
will help each student become an independent learner.
Example: Providing Written
Feedback on Students’ Work
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| Openness
Standard: This teacher promptly provides useful information to this
student about the quality of his work |
In
this example, a seventh-grade teacher provides feedback intended to
help a student interpret his own work according to specific criteria.
The teacher allowed his students time in class to explore, working
with pattern blocks, the different ways to increase the size of squares,
triangles, and trapezoids. After several days of working with the
blocks, students were asked to formulate and write down what they
had learned. |
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Stewart's Work
Stewart summarized
his work in a diagram and statement [see fig. 6].
Fig.
6. Stewart's work [contributed by Ruth Cossey, Mills College]
Stewart has clearly
identified a pattern of squared numbers but has not expressed
his conjecture, "All squared numbers are the sum of odd numbers,"
precisely. The teacher wrote the following note to Stewart
and placed it on his report:
Stewart, your work
indicates that you know special odd numbers that sum to 16,
1+3+5+7, not just any odd numbers [e.g., 11+5]. You need to
be more convincing that your pattern will always work.
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Learning Standard:
Explicit feedback about performance not only helps students understand
what they know and can do but also helps ascertain what they have
yet to learn.
Inferences Standard:
Teachers use cumulative knowledge about students when giving feedback
on individual tasks.
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Such feedback helps the
student learn that generalizations apply to broad mathematical concepts
that may be abstracted from patterns. This feedback also communicated
the levels of thinking and completeness of explanations that were
expected of Stewart. What would you have said to Stewart about his
progress toward making and explaining his mathematical conjectures?
What suggestions would you give him to encourage his progress?
It is possible that your
answer to the last two questions is that you do not have enough
information. Teachers can make some inferences based on a single
activity if they can place it within a context of ongoing performances,
but students are better served if most feedback is cumulative and
based on many performances.
Performance
Tasks, Projects, and Portfolios as Assessment Tools
To demonstrate real growth
in mathematical power, students need to demonstrate their ability
to do major pieces of work that are more elaborate and time-consuming
than just short exercises, sets of word problems, and chapter tests.
Performance tasks, projects, and portfolios are some examples of
more complex instructional and assessment activities. The earlier
example of designing an experiment in statistics in Ms. Lafleur’s
eighth-grade class illustrates the use of projects for assessment
purposes.
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| Large
pieces of work, like performance tasks, projects, and portfolios,
provide opportunities for students to demonstrate growth in mathematical
power. |
Still another useful assessment
tool is a student portfolio. During a school year or course (or
even several years or courses), each student produces a large amount
of work. This material may be kept in a working folder. A portfolio
is created by selecting examples from that folder to demonstrate
the quality of that student’s work in mathematics. The process
by which students select what they consider to be their best work
is an important means by which they learn to reflect on their own
work.
One criterion that teachers
take into consideration when designing an assessment is whether
a specific activity allows students equal opportunities to demonstrate
their knowledge. In fact, one issue teachers face in monitoring
student progress is whether the performance tasks they use to judge
progress are equitable. Is there a sufficient variety of tasks and
do the conditions of the tasks allow students to demonstrate what
they know and are able to do?
Example: Judging Progress
Equitably
The following example illustrates
how a teacher worked with her colleagues to create a geometry assessment
task that was also a learning opportunity that gave all students
a way to demonstrate their knowledge.
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"In grades
9—12, the mathematics curriculum should include numerous and
varied experiences that reinforce and extend logical reasoning skills."
–NCTM
(1989, p. 143)
Learning Standard:
How does the assessment engage students in relevant, purposeful
work on worthwhile mathematical activities?
Mathematics Standard:
Is the mathematics significant?
"For some
students, the issue in mathematics is not the learning of mathematical
topics and procedures but rather the ability to produce solutions."
–Maria
Marolda and Patricia Davidson (1994, p. 97)
"Assessment
must provide opportunities for teachers or evaluators … to
determine the students’ effectiveness in dealing with the inherent
demands of the mathematical topics themselves."
–Maria Marolda
and Patricia Davidson (1994, p. 86)
Equity Standard:
Does the assessment help students demonstrate their best work?
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Letting Everyone
In
At the end of an analytic
geometry unit in tenth grade, three teachers, Ms. Lee, Mr.
Jackson, and Ms. Romario, were deciding on an assessment task
that focused on -
- making and testing
conjectures;
- deducing properties
of geometric figures using concepts of functions;
- translating between
geometric and functional representations;
- using dynamic geometry
software appropriately.
They decided that
the most appropriate assessment would be an investigation
similar to those students had done during the unit. They would
have students use the dynamic geometry software they had been
using during the unit to investigate a new situation. They
wanted the assessment to furnish evidence of what students
had learned from exploring geometric situations and making
conjectures. Students would continue their learning while
doing the assessment.
Ms. Lee suggested
the following task as one that might fit these conditions;
she argued that it was open ended, yet by focusing on the
sum of the distances of various points from a triangle's sides,
it required that students work in a geometric function context:
On your computer
make a sketch of a triangle ABC, with an interior point D
and the shortest segments from D to each of the triangle's
sides. To answer the following questions, consider your sketch
on the computer.
What conjecture[s]
can you make about the sum of the distances from D to the
triangle's sides? Do you think your conjecture will apply
to any triangle? Make a convincing argument for your answers
to these questions. Support your arguments with data you have
collected. Use tables or graphs to present your data.
Mr. Jackson commented
that he liked the general direction of the task but was concerned
that it was too open, that students might not get to the rich
mathematics that it was possible to explore in the problem.
He thought it likely, in fact, that many students would not
see much in this context beyond a relationship between the
sum of the distances and the triangle's largest and smallest
altitudes. He argued for providing more mathematical guidance
up front - to increase access, to point in several mathematical
directions, to preserve choice. Mr. Jackson revised Ms. Lee's
task accordingly:
Take an acute triangle
with an interior point P. Consider the perpendiculars from
point P to the sides and the triangle formed by the three
feet of these perpendiculars on the three sides. This is the
pedal triangle of pedal point P. [See fig. 7.]
1) Measure
- The sum of the
perpendicular distances to the three sides of the original
triangle from P.
- The sum of the
distances from P to the three vertices of the original triangle.
- The area of
the pedal triangle.
- The perimeter
of the pedal triangle
2) Explore how
these measures change for different locations of P inside
the triangle.
3) What conjectures
can you make about the sums, areas, and perimeters found in
your explorations? Do you think your conjecture will apply
to any triangle?
- Make a convincing
argument for your answers. Your argument can be written
or oral.
- Support you
argument with the data you collected.
- Use tables or
graphs to present you data.
- Explain a situation
where someone would want to know this information.
Fig.
7. An acute triangle with an interior point P
There was general
agreement that this task would increase access, but Ms. Romario
raised another equity issue: whether the criteria for "make
a convincing argument" and whether the term conjecture were
well understood by the students. It was true that all their
students had been exploring geometric situations and discussing
their findings, but how carefully had they been monitoring
the quality of arguments or the phrasing of conjectures?
Ms. Romario proposed
that they first engage the students in an activity before
giving Mr. Jackson's revised task as part of an end-of-unit
assessment, and then reconvene to discuss their observations.
The others agreed and selected an observation task for equilateral
triangles as a special case of Mr. Jackson's task [see fig.
8]. They would carefully observe the students, listen, give
feedback, then reconvene to discuss their data on access and
on the students' understanding of the criteria. On that basis,
they could elect to give their end-of-unit assessment, modify
their criteria, or do some further instruction.
In their subsequent
meeting, the teachers pooled their observations - especially
regarding students' understanding of the criteria and access
to the mathematics represented in the task. Each thought the
feedback on the criteria was satisfactory. The teachers also
expressed confidence about access: There appeared to be wide-spread
facility with the use of the software; the investigations
facilitated the use of supporting conjectures; and the choice
between written and oral presentations allowed students sufficient
latitude. Still, they recognized that there was a potential
difficulty for students for whom English was not a first language.
In order to address this inequity, students could be given
opportunities to respond in the language in which they felt
most confident and be encouraged to use multiple methods to
communicate. Alternatively, these students could be given
more teacher support in English to verify their understanding
of the task and to clarify the meaning of their conjectures.
Fig.
8. An equilateral triangle with an interior point P
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Equity Standard:
Do the modes of response invite each student to engage in the mathematics?
Learning Standard:
Self-assessment provides a valuable learning opportunity.
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This example describes how
teachers handled one equity concern, access–that is, whether
there were adequate provisions for allowing each student to exhibit
his or her best work. For this to occur, each student must understand
the task and be sufficiently familiar with its contexts and implicit
assumptions to be able to apply and communicate his or her mathematical
responses. Such concern about the challenges to the students posed
by chosen tasks allows teachers to feel more confident that they
are making valid inferences about their students’ progress.
Students
Can Learn To Assess Their Own Work
Students learn to share
responsibility for the assessment process as they come to understand
and make judgments about the quality of their own work. The shift
in teaching toward helping students increase their capacity for
analysis and their ability to formulate problems and communicate
correct mathematical work is supported when students become adept
at judging the quality of their own work and that of others, as
when, for example, selecting work to be included in portfolios.
Students also learn to look for ways in which the complex mathematical
situations they explore can provide information that will help them
determine whether their solution path is reasonable in comparison
to other possible strategies they may choose.
Example: Learning to
Judge One’s Own Work
The following vignette explores
the place of self-assessment and peer review in the learning process.
Ms. Harris, a fourth-grade teacher, attempted to structure an assessment
environment in which students believed they were responsible for
one another’s learning–that is, they had a right to receive
assistance and an obligation to give help when asked.
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Mirror images
Ms. Harris presented
this problem to the class: Find a way to figure the area of
any triangle if you know its height and width.
The homework assignment
was to write a report and to critique their own work. Rya
remembered using geoboards the previous week to find area.
Turning to Cam, her partner, she said, "Before, we used geoboards
to find areas of all sorts of weird shapes. Do you think we
can use them now?"
Cam looked at the
attempts he had made at drawing some triangles on his paper
and said, "Ms. Harris said we could use anything we thought
might help us solve the problem, so let's try using them.
They're fun, anyway."
"It will be easier
to make lots of triangles," responded Rya, as she got out
the geoboards.
"Yeah, and it's easier
to see what their areas are because of the pegs," Cam added.
Cam cleared the rubber
bands off his geoboard. Rya grabbed a red rubber band and
built a right triangle with a base of 2 units and a height
of 2 units on her geoboard. "Look, if you count the little
pieces, the area of this triangle is 2 squares." Cam concurred.
They investigated
many other triangles and began to make a chart of the height,
base, sides, and area of the triangles they built on their
geoboards. They noticed that the areas of the right triangles
they built were half the area of the rectangle with the same
base and height. They came to the conclusion that they could
multiply base and height and divide by 2. Obtuse triangles
caused them to question their visual representation of doubling
the area, but they eventually generalized their formula to
include those triangles, too.
The next day, Rya
and Cam got together to discuss how to fill out their "self-assessment"
sheets.
The class, like every
fourth-grade class in the state, was using specific criteria
to assess mathematical reasoning and communication (Petit
1992). Part of the sheet that Rya and Cam were working on
is shown in figure 9.
Sitting cross-legged
on the gray rug surrounded by benches, Cam and Rya exchanged
papers. Ms. Harris, who had not talked with them as they worked
on their investigation, decided to listen in on the conversation.
"How did we get
that formula?" inquired Cam.
Fig.
9: Vermont mathematics criteria (adapted from Petit 1992,
and used with permission)
"Well, Cam …
what did we do? … Oh, yeah, we saw that there were two
triangles in the rectangle."
"You didn’t
put that in your write-up (see fig. 10). Remember, Ms. Harris
wants to know why we came up with our answers."
"Right, that’s
what she means by explaining our decisions? OK, I’ll
say why you get the area if you multiply and then divide by
2. I think I’m done."
Rya started to get
up to leave, but Cam continued to question her.
"Last night I
decided not to put side in my chart. I never used it for an
area. I wasn’t even sure what it meant."
Rya got on all fours
to explain. "Well, it’s not the height, and it’s
not the length, it’s, you know, the other side."
"Oh, yeah! But
look, I don’t think this slanty side of the triangle
is only one unit long."
Fig.
10. Rya’s report (contributed by Clare Forseth, Marion
Cross Schools)
"You don’t
think so? Here, give me my paper. Let’s see. Oh, you
may be right. If I swing that side so it’s straight up
and down, it would go above the peg."
Cam further convinced
himself that they had made a mistake about the length of the
hypotenuse of the right triangles they had sketched. "And
do you see that slant line is also a diagonal of that little
square, so it has to be longer than the side of the square."
"I see what you
mean. Since we didn’t use it in our formula, maybe we
should just get rid of that column in our reports." Glancing
at Cam’s report (see fig. 11), Rya noticed the diagrams.
"Oooh! You have lots of pictures of triangles in your
report. That’s a good idea, ’cause that way the
drawings help you explain and you don’t have to write
so much. I’m going to add more pictures to mine. Oh,
and I like the way you labeled your triangles, but I think
that’s not how you spell obtuse."
The next day as Rya
read over her revised solution, she thought to herself, "I
bet this is my best piece so far." She scanned the criteria
sheet. She felt more confident in assessing herself now. Cam
had helped her see how to explain her reasoning more clearly.
Ms. Harris noted that
Cam used his pictures as models when he explained the diagonal
lines. Rya seemed to be moving beyond a just-get-something-down
approach to really wanting to learn and to share her thinking
in a way others would understand. Ms. Harris was happy to
see that their discussions and reports had moved beyond simply
noting a procedure to include justifications for their approaches.
The students knew that she would use the same assessment sheet
to give them feedback on their finished product. She thought
that this time her assessment and theirs would be pretty close,
which would be another indication that the students were beginning
to understand the quality of their work and would be in a
good position to select the best pieces of work for their
portfolios.
Fig.
11. Cam’s report (contributed by Clare Forseth, Marion
Cross Schools)
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Inferences Standard: When students learn to assess
themselves, they become independent learners.
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Students in this vignette
were being encouraged to monitor themselves and their peers. By
providing opportunities for students to evaluate, reflect on, and
improve their own work, teachers help students become independent
learners.
Summary
Monitoring students’
progress in classrooms enhances each student’s learning to
the extent that it facilitates and encourages continued learning
and helps each student become an independent learner. Effective
monitoring of students’ progress requires clarity about–
- the mathematics to be
learned;
- the kinds of evidence
necessary to describe students’ progress in learning that
mathematics;
- the variety of equitable
assessment methods and tools available to collect evidence of
students’ learning;
- the criteria for interpreting
that evidence and making valid inferences about what students
are learning;
- how those interpretations
are to be communicated to students in ways that support the achievement
of instructional goals.
Well-planned monitoring
of students’ progress provides meaningful information to both
teachers and learners. Continual and comprehensive monitoring allows
teachers to make more informed decisions about students’ growth
in mathematical power. Students who are clear about their learning
goals and the progress they are making toward them are more likely
to be reflective and confident learners of mathematics.
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