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"The teacher
of mathematics should engage in ongoing analysis of teaching and
learning by–
- observing,
listening to, and gathering other information about students to
assess what they are learning;
- examining effects
of the tasks, discourse, and learning environment on students’
mathematical knowledge, skills, and dispositions …"
–NCTM
(1991, p. 63)
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When teachers have a good
understanding of what their students know and can do, they are able
to make appropriate instructional decisions. Such decisions may
include identifying appropriate content, sequencing and pacing lessons,
modifying or extending activities for students’ particular
needs, and choosing effective methodologies and representations.
The NCTM Professional Standards for Teaching Mathematics emphasized
the role of assessment in providing the information needed for analyzing
teaching and learning and asserted that "assessment of students
and analysis of instruction are fundamentally interconnected"
(p. 63). The primary question to be answered in using assessment
to make instructional decisions is, "How can I use evidence
about my students’ progress to make instructional decisions?"
The process of using assessment
to make instructional decisions involves using evidence of learning
from the students in the classroom. Evidence is used in three ways:
(1) to examine the effects of the tasks, discourse, and learning
environment on students’ mathematical knowledge, skills, and
dispositions; (2) to make instruction more responsive to students’
needs; and (3) to ensure that every student is gaining mathematical
power. Although evidence of progress originates with individual
students, as indicated in the "Purpose: Monitoring Students’
Progress" section, teachers also sample and collect such evidence
to provide information about the progress of the groups of students
they teach. They make instructional decisions and adapt their teaching
to respond simultaneously to the needs of individuals and of groups.
The quality of teachers’ instructional decisions depends, in
part, on the quality of their assessment and their purposeful sampling
of evidence during instruction. When teachers use appropriate evidence
of what their students understand and can do in making instructional
decisions, their teaching responds to individual and group needs.
The Assessment Standards
suggest shifts in assessment practices to support more responsive
instructional decisions. These include shifts–
- toward integrating assessment
with instruction (to provide data for moment-by-moment instructional
decisions) and away from depending on scheduled testing (generally
useful only for delayed instructional decisions);
- toward using evidence
from a variety of assessment formats and contexts for determining
the effectiveness of instruction and away from relying on any
one source of information (often, in the past, paper-and-pencil
tests);
- toward using evidence
of every student’s progress toward long-range goals in instructional
planning and away from planning for content coverage with little
regard for students’ progress.
Integrating Assssment
with Instruction: Moment-by-Moment Decisions |
| Observing,
questioning, and listening are the primary sources of evidence for
assessment that is continual, recursive, and integrated with instruction. |
Integrating
assessment and instruction in the classroom means precisely that:
assessing students’ learning to inform teachers as they make
moment-by-moment instructional decisions about students’ work
in the classroom. Such a blurring of the boundary between instruction
and assessment is consistent with the Learning Standard, which presents
assessment as an opportunity for learning rather than an interruption
in the learning process. Observing, listening, and questioning are
the most common methods for gathering evidence of learning during
instruction. |
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"The teacher
has a central role in orchestrating oral and written discourse in
ways that contribute to students’ understanding of mathematics."
–NCTM
(1991, p. 35)
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Posing appropriate questions
to students is important for moment-by-moment assessment so that
the progress of the group can be monitored and used in planning
and teaching. Careful questioning and probing are often needed to
determine what students are thinking during a lesson. Professional
Standards for Teaching Mathematics provides a list of useful suggestions
for questioning and talking with students (pp. 3-4). Teachers can
learn to be perceptive in interpreting student talk, to make quick
decisions about the flow of discourse, and to adapt their instruction
appropriately.
The six Assessment Standards
furnish a framework for helping teachers monitor their instruction
and assessment practices as they gather evidence during instruction.
Each standard could be interpreted with questions like the following:
- How does the mathematics
of the lesson fit within a framework of overall goals for learning
important mathematics?
- How do these activities
contribute to students’ learning of mathematics and my understanding
of what they are learning?
- What opportunity does
each student have to engage in these activities and demonstrate
what he or she knows and can do?
- How are students familiarized
with the purposes and goals of the activities, the criteria for
determining quality in the achievement of those goals, and the
consequences of their performances?
- How are multiple sources
of evidence used for making valid inferences that lead to helpful
decisions?
- How do the assessment
and instructional content and processes match broader curricular
and educational goals?
Questioning a few students
often leads to a redirection of the lesson for all. The justification
for changing the direction of a lesson could come from the confusion
ensuing as third graders attempt to measure their heights with metersticks
or from the excited conversation of high school juniors exploring
exponential functions on a graphing calculator. Similarly, a teacher
might adjust plans for the week as a result of recognizing that
students’ responses to a problem are far richer than expected,
musing, "Terrific! I hadn’t thought of so many ways to
do this problem. I wonder if there are even more. Maybe we should
spend some more time on it tomorrow."
Example: Changing Plans
in Mid-Lesson
The vignette that follows
illustrates how the interactive nature of assessment and instruction
in the classroom can enhance learning. In this story, a sixth-grade
student made the incorrect conjecture that all shapes with equal
perimeters have equal areas. His teacher decided the misconception
was worth exploring in class and changed her lesson plan accordingly.
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Mathematics Standard:
Encouraging students to make conjectures and seek verification involves
them in doing significant mathematics.
Students should
make connections between area and perimeter in meaningful contexts
instead of learning those topics as isolated, formula-based skills.
Equity Standard:
Perceptive questioning helps all students explain what they know
and can do.
Inferences Standard:
Careful questioning and probing provide evidence for making valid
inferences about students’ understanding.
Learning Standard:
Together, assessment and instruction can build on students’
understanding, interests, and experiences.
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Inside, Out, and
All About
Janna McKnight perched
on the edge of her desk at the back of the classroom. Her
sixth-grade students were elbowing each other and chatting
while parading to the front of the room to place their yellow
stickies on a bar chart. They had been finding the area of
a salt marsh on a map, a rather irregular shape. Each stickie
represented the area (in thousands of square meters) determined
by one student.
The portion of the
chart from about 85 to 95 was beginning to look like the New
York City skyline, with highrise towers huddled close together.
Close in, to the left and right of the city, were several
lower towers. Off to the right at 196 was one lonely stickie
with the initials TP, and down to the left were two other
stickies at 55 with the initials BT and AK.
Used
with permission from the Wisconsin Center for Educaion
Research, School of Education, University of Wisconsin-Madison
Ms. McKnight called
the class to order. "All right, let’s see what you’ve
found. Who’d like to make some observations about our
data?"
"Ninety-two thousand
square meters got the most, but 87 was a close second,"
Jeremy answered. "It seems like lots of people got answers
between 86 and 94."
"The lowest answer
was 55–two people got that–and the highest was 196.
Those people must have done it wrong, because those answers
are, like, too different to be true," Angel offered.
"Bernice, is
the 55 with BT on it yours?" Wallace blurted.
"The 196 is mine,
and it isn’t wrong!" Tyler offered in a defensive
tone.
Anthony interrupted.
"Bernice and I worked together, but I think our calculator
gave us the wrong answer. I just added it up again, and ours
should be 85 instead of 55."
"Tyler, would
you like to explain how you found 196 as the area of the marsh?"
"Well, you remember
how we put string around those circles the other day? You
didn’t give us any string today. But my sweatshirt has
a string in the hood, so I pulled it out and wrapped it around
the marsh. Then I straightened it out into a square on top
of the plastic, and it was about 14 units long and 14 wide.
So its area was about 196. The square wasn’t exactly
14, but it was pretty close!"
"So, Tyler found
the area of a square he built from reshaping a string that
fit around the perimeter of the marsh. I noticed most of the
rest of you doing something quite different. Amy?"
Amy explained that
her group had put clear plastic graph paper on top of the
marsh and counted squares. Dyanne explained how her group
counted partial squares as halves or fourths for more accuracy.
"Did anyone approach
the problem differently? What do you think about Tyler’s
procedure?"
"I think it’s
a lot better," said Richard, "because Tyler didn’t
have to do all that counting. I wish I’d thought of making
it into an easier shape. It would have saved a lot of work."
"But why is Tyler’s
answer so much bigger?" asked Nancy. "I don’t
know why it’s wrong, but Tyler’s answer is way too
big."
"If the distance
around the marsh is a lot, then the area is a lot–wouldn’t
that be right? Like when one is big, the other is big?"
asked Cindy. Numerous heads nodded in agreement, amid a few
dissenting frowns.
Pointing to the bulletin
board, where dot-paper records of a geoboard activity from
the previous week were displayed, Dyanne reminded the class
that shapes with equal perimeters could have very different
areas.
"All right,"
said Ms. McKnight, glancing at the clock over the chalkboard.
"We’re not going to have time today to get to the
bottom of this mystery. Let’s just write Tyler’s
conjecture on the board and see whether we can investigate
it further tomorrow. Who can use some good mathematical language
to say what Tyler’s been thinking?"
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Janna McKnight wisely did
not dismiss Tyler’s answer as wrong but probed to discover
the source of his confusion. In doing so, she uncovered a common
misconception about area and perimeter and made some decisions to
adjust her instruction.
What will Ms. McKnight do
tomorrow in class? The decisions that she made today have led her
to a point where she will have to revise her short-range plans for
tomorrow’s lesson. She will need to think about ways to follow
up on Tyler’s conjecture. One of her goals now is to help her
students explore the conjecture and decide its validity for themselves,
so she wants to plan an activity to make that happen. One idea would
be to have them measure the area and perimeter of their own handprints
(with fingers spread and with fingers together) and explore the
relationship, hoping they will begin to understand why Tyler’s
conjecture is false. This short-range plan also fits some of her
long-range goals for the class. She wants her students to have a
sense of what it really means to do mathematics, and she intends
to communicate the principle that making conjectures and testing
them for validity are important components of mathematical power.
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"It is students’
acts of construction and invention that build their mathematical
power and enable them to solve problems they have never seen before."
–Mathematical
Sciences Education Board (1989, p. 59)
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Janna McKnight’s understanding
of the importance of conjecture and verification in doing mathematics
helps her recognize the value of exploring Tyler’s conjecture
rather than quickly pointing out his error. Teachers are more effective
in assessing students’ understanding of mathematical ideas
when they are knowledgeable about performance standards, familiar
with the "big ideas" of mathematics (such as number sense,
proportion, and equivalence), confident of their abilities to deal
with important mathematical processes (such as problem solving,
reasoning, communication, and making connections), and convinced
of the importance of dispositions (such as motivation and confidence).
Ongoing professional development activities–attending conferences,
reading professional journals, and collaborating with other educators–help
teachers become aware of, and confident in, their understanding
of effective ways to help all students become mathematically powerful.
Using Multiple
Sources of Evidence for Short-Term Planning
In making short-range plans,
teachers use assessment evidence to make decisions about tomorrow,
the day after, and next week. As they plan a unit of instruction,
they review their goals and long-range plans and reconsider those
plans in light of information they have gathered about their students’
learning. As teachers consciously try to integrate instruction and
assessment, they identify places in the unit at which they will
check for specific types of understanding, or ask certain questions,
or collect certain work to inform themselves as they make instructional
decisions.
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Equity Standard:
Differences in students’ work may be better understood through
evidence from observations and questioning.
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Short-term instructional
decisions are improved when teachers are informed by evidence from
observations, questioning, and students’ products. This evidence
provides views of learning at varying levels of complexity that
can be compared to performance criteria and expectations. However,
a systematic approach to gathering complex data from a variety of
formats requires additional planning for the specific data to be
gathered. Observations and questioning offer opportunities for understanding
the influences of students’ unique prior experiences. Unexplained
differences in written work may be the result of inequities that
become apparent from questioning and observing. Combining documented
evidence, from observations and questions, with written products
gives a more complete picture of students’ mathematical power
than can be obtained from tests, quizzes, and assignments that focus
primarily on the mastery of procedures.
Numerous useful assessment
formats–such as journals, portfolios, writing, projects, and
extended investigations–require advance planning and extra
time to communicate expectations and criteria for scoring to students.
Because they are more complex than quick-answer questions on quizzes
and tests, these formats can furnish evidence of learning not often
captured by simpler formats.
Example: Using Evidence
to Plan Tomorrow’s Lesson
In the next example, Mr.
Ernest uses observation to determine the extent to which his first-grade
students are making sense of the important mathematical concepts
involved in nonstandard measurement. While planning for subsequent
lessons, he reflects on what he has learned about his students’
conceptual understandings and how their understandings are reflected
in their work.
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"It Took 8 Perpl Rods"
Sitting at his desk after school, Charlie Ernest quickly
recorded some anecdotal comments about what he had observed
during the rod-measuring activity his first graders had done
that day with sets of colored rods of various lengths. He
paused as he thought about what the class should do tomorrow
and how tomorrow’s work would fit into his long-range
plan. Certainly the children had enjoyed the work today. There
had been lots of laughter, especially when Trey tried to measure
John’s smile with white rods. And Terrance had been triumphant
when he and Jenise successfully measured her lunch bag.
Jenise had placed a black rod adjacent to the closed end
of her lunch bag and furrowed her brow as she dipped her head
to table height, squinting a sightline along the rod to make
sure that the rod and bag were even. Terrance had a plan.
"You do one black one, and I’ll do one. We’ll
take turns. Then we can do the purple ones." He had quickly
plopped another black rod down, touching the end of Jenise’s.
She placed a black rod next to his and straightened his out.
After placing one more black rod, Terrance lifted his hands.
"Four," he shouted, "that’ll do it. FOUR!
That’s all we can do. Write it down!"
Jenise wrote in large neat letters: We mesured my lunch
bag. For the blak one we got 4 and haf becus there was some
bag left over. We mesured the longest way.
Terrance was lining up purple rods while Jenise was writing.
"Eight," he said. "It took eight purple rods."
Jenise passed the paper to Terrance. "Naw, you write
it," he said.
"No, it’s your turn," Jenise replied. Terrance
never wanted to write. Draw maybe, but write–never. Still,
Jenise convinced him that it was his job to continue.
He dashed off this sentence: It took 8 perpl rods.
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Inferences Standard: Mr. Ernest recognizes the need
for multiple sources of evidence in making inferences about his students’
understandings of measurement and fractions. |
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After school, Mr. Ernest thought about what his children
were learning about measurement. He had expected that they
would use different-colored rods to repeatedly measure the
same objects, and he had hoped that he could see evidence
that they understood how the size of the unit made a difference
in the measurements they obtained. Indeed, most of the students
had indicated on their papers what units they were using.
Although Sara and Tran had not written down the color of the
rods with their measurements, when Mr. Ernest asked them to
explain, Tran knew which colors went with which measurements.
What else had Mr. Ernest noted during his observations? Well,
he had not anticipated Jenise’s use of a fraction. Now
he was curious to learn how the other students were dealing
with the issue of measurements that were not whole numbers.
It was late. Mr. Ernest reviewed the questions he had written
in his plan book for tomorrow’s class:
- Yesterday, Trina measured the side of her table with
rods. She said it was about 12 rods long. Did anyone else
get a different answer? (Write all answers on the board.)
- Discuss with a neighbor why you think we came up with
different measures for the length of Trina’s desk.
(Allow discussion time.) Who would like to report back what
you and your partner came up with?
- If it takes about 12 blue rods to measure the desk,
what color rods would you need more than 12 of? How do you
know?
Mr. Ernest knew the first question would get a set of data
on the board and prompt the students to begin questioning
the data. As usual, he would get the whole class involved
with the problem by asking them to talk with a partner. This
would provide more opportunities to listen to and observe
his students. He would also probe their thinking further by
asking "Why?" and "How did you know?"
He was very interested in finding out which students were
making sense of the concept that it takes fewer longer rods,
and how they would show that understanding as they justified
their actions and answers to his questions.
Mr. Ernest considered how he might modify or add to tomorrow's
questions as a result of his curiosity about his students'
estimation abilities when confronting fractional measurements.
Maybe he would ask, "Did anyone measure something that the
rod-train didn't fit exactly? What measurement did you report?
How did you decide on that number?" And he might start planning
a few lessons that would involve his students in an informal
exploration of simple fraction concepts.
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As this vignette illustrates,
assessing students’ learning in more complex ways than simply
checking for correct answers on written work is a difficult challenge
for teachers, a challenge requiring an approach to both teaching
and planning instruction that is different from traditional practice.
Teachers must choose appropriate activities, pose good questions,
listen carefully to students’ responses, and follow up with
questions that help students communicate what they think and can
do. An understanding of how students learn specific mathematical
concepts is important, including an awareness of common misconceptions
and a familiarity with strategies for helping students describe
and reconsider their understandings. Students’ potential strategies
can be considered and anticipated during short-term planning.
Using Evidence
of Learning in Long-Range Planning
In addition to making decisions
during instruction and planning for the next day’s lesson,
teachers plan over longer time spans. Before the start of a new
school year, they plan the broad topics with which they will engage
their students. These plans are often altered during the course
of the school year because evidence that students are developing
mathematical power enables teachers to make more informed plans
and alterations as the class progresses.
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This is an example
of using multiple sources of evidence.
Inferences Standard:
Valid inferences require that opportunities be provided for responses
in a variety of modes. Equity can be addressed by English-enhancing
preparatory activities such as–
- using pictorial
and manipulative materials;
- distinguishing
between the language used in daily communication and the language
of mathematics;
- controlling
the range of vocabulary and the use of idiomatic expressions.
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The key to ensuring that
every student is learning important mathematics and becoming mathematically
powerful is to make valid inferences about each student’s learning
as it proceeds. Such inferences require a balance of information
across a number of dimensions. Therefore, it is important that instructional
plans include methods for assessing a broad range of components
of mathematical power (e.g., each student’s confidence as a
mathematics learner, success in solving problems in various contexts,
conceptual understandings, and experience in communicating effectively).
These varieties of formats, contexts, and occasions must be considered
during long-range planning.
Making valid inferences
about students’ learning requires familiarity with every student’s
responses in a variety of modes, such as talking, writing, graphing,
or illustrating, and in a variety of contexts. When instruction
includes group work, then group work needs to be assessed. Cultural
considerations are also important; however, care should be taken
not to make assumptions based on cultural stereotypes, because each
student has unique responses to experiences in and out of school.
For non-native speakers of English, encouraging the use of the native
language might be appropriate, as well as using English-enhancing
preparatory activities. As teachers become familiar with their students–through
the careful collection and analysis of relevant assessment data–appropriate
instructional decisions can be made.
Example: Selecting
Appropriate Instructional Experience
Sometimes long-range
plans are open to change. A teacher may have a clear picture of
the major mathematical concepts she wants to engage her students
in, but as the year progresses she may need more detailed evidence
of students’ understanding in order to decide which units are
appropriate. In the next example a high school teacher uses one
particular task to help her decide which of two units to teach.
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Postal Patterns
Ms. Naman, an eleventh-grade
teacher, planned an assessment activity specifically to help
her decide which of two units of study would be more appropriate
for her students. In looking over the core curriculum for
the eleventh grade, Ms. Naman asked herself whether the units
she had already planned would adequately meet the standards
on algebraic representations and functions. She was concerned
about providing instruction at the appropriate level of difficulty
and setting an appropriate pace. Among her goals for her students
were two of the Curriculum Standards for grades 9—12.
She wanted them to learn to "represent situations that
involve variable quantities with expressions, equations, inequalities,
and matrices" (NCTM 1989, p. 150) and to "represent
and analyze relationships using tables, verbal rules, equations,
and graphs" (NCTM 1989, p. 154).
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"In grades
9—12, the mathematics curriculum should include the continued
study of algebraic concepts and methods so that all students can–
- represent situations
that involve variable quantities with expressions, equations,
inequalities, and matrices …"
–NCTM
(1989, p. 150)
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Ms. Naman had eight
teaching units available, but from her previous experiences
she expected to have time for only seven. Which unit should
she omit? Two of the units heavily emphasized statistics,
which she believed to be practical as well as motivating material.
However, the mathematical modeling unit seemed to furnish
more opportunities for students to learn about algebraic representations
and functions.
Ms. Naman decided
that, depending largely on how well her students could work
with equations and graphs, she would choose either the second
statistics unit or the mathematical modeling unit. As a result,
one item on her assessment agenda early in the year was to
get a sense of her students’ facility in recognizing
and expressing mathematical relationships through algebraic
and geometric modeling.
In mid-September,
after her students had done some work with patterns and graphing,
Ms. Naman presented them with the Postal Rate History assessment
task (adapted from Connecticut State Department of Education
1991) shown in figures 12 and 13. Two of her students, Chris
and Kelly, produced the work shown.
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"In grades
9—12, the mathematics curriculum should include the continued
study of functions so that all students can represent and analyze
relationships using tables, verbal rules, equations, and graphs."
–NCTM
(1989, p. 154)
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Quite a few of Ms.
Naman’s students did work that was similar to Chris’s
(fig. 12). Chris looked for patterns in the differences between
the years and the postage costs. However, perhaps because
he did not graph the data, no discernible pattern emerged.
Chris then oversimplified the problem by finding the average
postage increase per year–in essence assuming that the
overall relationship was linear. Using this assumption, he
calculated predictions for future costs that were far from
realistic.
Fig.
12. Chris’s work (task adapted from Connecticut State
Department of Education 1991)
Kelly’s solution
(fig. 13) was one of the most sophisticated ones produced
by the students in Ms. Naman’s class. Her graph showed
clearly that the overall relationship was nonlinear, although
for larger values on her y-scale, it appeared that
a line might approximate the curve rather well. In fact, Kelly
observed that postal costs increased at almost a constant
rate between 1971 and 1991 (about $0.10 every ten years) and
used this information to predict future costs. However, Kelly’s
work showed that she, too, had limited experience with graphing.
Her x-scale was inconsistent (30, 50, 70, 90, 92, 94,
96 . . .), and she failed to extend the graph to make estimates
of future costs.
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| Inferences
Standard: Inferences about students’ mathematical knowledge,
understanding, and thinking processes inform teachers as they make
decisions to adjust instructional plans. |
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Ms. Naman used the
fact that the typical work of her students was like Chris’s
as one piece of evidence that her students probably needed
more experience with analyzing relationships using tables,
equations, and graphs. The fact that even some of the strong
responses showed a lack of experience with graphing further
supported this view. She decided to teach the mathematical
modeling unit rather than try to cover both statistics units.
If most students had demonstrated an understanding similar
to Kelly’s, the second statistics unit might have been
more appropriate.
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