Comparing Levels of
Sophistication of Secondary
Students’ Solutions
George W. Bright
Professor Emeritus
University of North Carolina at Greensboro
gbright45@comcast.net
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Secondary Students’ Solutions
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Acknowledgements
• Based on work completed as part of a project funded by a
grant from the National Science Foundation (#9819914);
George W. Bright and Jeane M. Joyner, co-Principal
Investigators.
• All conclusions and opinions expressed are those of the
authors and do not necessarily reflect the position of the
Foundation or any other government agency.
• Professional development materials published by
ETA/Cuisenaire as Dynamic Classroom Assessment.
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Goals for the Session
• Examine students’ solutions to proportional
reasoning problems and rank the level of
sophistication of those solutions
• Reflect on how to use rankings of level of
sophistication to make instructional plans.
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Classroom Assessment
• Classroom assessment is the process by which
teachers gather information about what students
know and can do and then use that information to
make more effective instructional decisions.
• Classroom assessment involves planning effective
mathematics instruction based on understanding
how students think about key ideas.
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Model for Classroom Assessment
Learning
Targets
Purposes
Instructional
Decisions
Communication
Documentation
Assessment
Methods
Inferences
about
Thinking
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Fundamental Belief
When teachers understand what students know and
can do, and then use that knowledge to make more
effective instructional decisions, the net result is
greater learning for students and a greater sense of
satisfaction for teachers.
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Direct Effects
Back and Wiliam (1998) conclude from an examination
of 250 research studies on classroom assessment that
“formative assessment does improve learning” -- and
that the achievement gains are “among the largest ever
reported for educational interventions.” The effect size
of 0.7, on average, illustrates just how large these gains
are.
(Wilson & Kenney, 2003, p. 55)
Wilson, L. D., & Kenney, P. A. (2003). Classroom and large-scale assessment. In J. Kilpatrick, W. G. Martin, & D. Schifter
(Eds.), A research companion to Principles and Standards for School Mathematics (pp. 53-67). Reston, VA: National Council of
Teachers of Mathematics.
Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education, 5, 7-74.
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What Teachers Might Do
If mathematics teachers were to focus their efforts on classroom
assessment that is primarily formative in nature, students’ learning
gains would be impressive. These efforts would include gathering
data through classroom questioning and discourse, using a variety
of assessment tasks, and attending primarily to what students
know and understand [emphasis added].
(Wilson & Kenney, 2003, p. 55)
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Residual Effects
[S]tudents learn more when they receive feedback about
particular qualities of their work, along with advice on what they
can do to improve. They also benefit from training in selfassessment, which helps them understand the main goals of the
instruction and determine what they need to do to achieve. But
these practices are rare, and classroom assessment is often
weak… Teachers must have tools and other supports if they are
to implement high-quality assessments efficiently and use the
resulting information effectively.
(Pellegrino, Chudowsky, & Glaser, 2001, p. 38)
Pellegrino, J. W., Chudowsky, N., & Glaser, R. (Eds.). (2001). Knowing what students know: The science and
design of educational assessment. Washington, DC: National Academy Press.
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Questioning as an Element of Instruction
Questioning is among the weakest elements of mathematics and
science instruction, with only 16 percent of lessons nationally
incorporating questioning that is likely to move student
understanding forward.
Lessons that are otherwise well-designed and well-implemented
often fall down in this area.
(Weiss, Pasley, Smith, Banilower, & Heck, 2003, p. 65-67)
Weiss, I. R., Pasley, J. D., Smith, P. S., Banilower, E. R., & Heck, D. J. (2003, May). Looking inside the
classroom: A study of k-12 mathematics and science education in the United States. Chapel Hill: Horizon
Research, Inc.
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Questioning Patterns in Instruction
By far, the most prevalent pattern in mathematics and science
lessons is one of low-level ‘fill-in-the-blank’ questions, asked in
rapid-fire, staccato fashion, with an emphasis on getting the right
answer and moving on, rather than helping the students make
sense of the mathematics/science concepts.
(Weiss, Pasley, Smith, Banilower, & Heck, 2003, p. 65-67)
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Rigor
Fewer than 1 in 5 mathematics and science lessons are strong in
intellectual rigor; include teacher questioning that is likely to
enhance student conceptual understanding; and provide sensemaking appropriate for the needs of the students and the
purposes of the lesson.
(Weiss, Pasley, Smith, Banilower, & Heck, 2003, p. 103)
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Proportional Reasoning Quiz
Solve each of the four problems in the
Proportional Reasoning Quiz.
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Quiz #1
1. Mrs. Allen took a 3 inch by 5 inch photo of the Cape
Hatteras Lighthouse and made an enlargement on a
photocopies using the 200% option. Which is “more
square”, the photo or the enlargement?
A. The original photo is “more square.”
B. The enlargement is “more square.”
C. The photo and the enlargement are equally square.
D. There is not enough information to determine which
is “more square.”
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Quiz #2
2. The Science Club has four separate rectangular plots for
experiments with plants (listed below): Which rectangle is
most square?
A. 1 foot by 4 feet
B. 7 feet by 10 feet
C. 17 feet by 20 feet
D. 27 feet by 30 feet
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Quiz #3
3. Sue and Julie were running equally fast around a track. Sue
started first. When Sue had run 9 laps, Julie had run 3 laps.
When Julie completed 15 laps, how many laps had Sue run?
A. 45 laps
B. 24 laps
C. 21 laps
D. 6 laps
Why is important to have this problem on the quiz?
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Quiz #4
4. At the midway point of the basketball season, you must
recommend the best free-throw shooter for the all-star game.
Here are the statistics for four players:
Novak: 8 of 11 shots
Peterson: 22 of 29 shots
Williams: 15 of 19 shots
Reynolds: 33 of 41 shots
Which player is the best free-throw shooter?
A. Novak
B. Peterson
C. Williams
D. Reynolds
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Students’ Responses
Examine the students’ responses.
1. Which students understand proportional reasoning well?
2. Which students need much more instruction?
3. Taken as a group, what do these students know?
4. What could you do to help the group move forward in
developing understanding?
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Proportional Reasoning
Student
1
2
3
4
5
6
7
8
9
10
Item 1
C
C
C
C
A
A
C
C
C
C
Item 2
D
D
D
D
A
D
D
D
D
B
Item 3
C
A
A
A
C
C
C
C
C
A
Item 4
D
D
A
C
A
D
D
C
D
A
Score
4
3
2
2
1
3
4
3
4
1
1. Which students understand proportional reasoning well?
2. Which students need much more instruction?
3. Taken as a group, what do these students know?
4. What could you do to help the group move forward in
developing understanding?
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Square Fields Problem
A farmer has three fields.
One is 185 feet by 245 feet, one is 75 feet by 114 feet, and
one is 455 feet by 508 feet.
If you were flying over these fields, which one would seem
most square? Which one would seem least square?
Explain your answers.
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Students’ Solutions: Square Fields
Examine the students’ solutions to the square fields problem.
• What strategies did students use to solve the problem?
• Which solutions are correct and which are incorrect?
Do these solutions confirm your conclusions about which
students understand proportional reasoning well?
How does this information change your perceptions about what
the group as a whole understands?
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Gathering Different Information
Different assessment methods can reveal different
information about students' thinking.
Student work on the square field problem gives
additional information about the students’
thinking.
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Next Task
How would you help these students understand
proportional reasoning?
Would you use manipulatives or models? If so,
which ones?
What is the next mathematical task you might pose
to help these students reorganize what they know?
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Reconsider a Problem
Look at the solutions that some students proposed for
the “Photocopier” problem.
Mrs. Allen took a 3 inch by 5 inch photo of the Cape Hatteras
Lighthouse and made an enlargement on a photocopies using the
200% option. Which is “more square”, the photo or the
enlargement?
A. The original photo is “more square.”
B. The enlargement is “more square.”
C. The photo and the enlargement are equally square.
D. There is not enough information to determine which
is “more square.”
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Debriefing Students’ Solutions
Which solutions would you want to have shared with
the class? Why?
In what order would you want those solutions shared?
Why?
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Paul’s Solution
photo: 3 by 5, difference is 2, and 2 out of 5 is 40%
copy: 6 by 10, difference is 4, and 4 out of 10 is 40%
They are both 40% away from being a square, so
they’re equal.
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Mario’s Solution
photo: 3 by 5, difference is 2
enlargement: 6 by 10, difference is 4
Photo has smaller difference, so it is more square.
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Charlene’s Solution
photo: 3 x 5 = 15
copy: 6 x 10 = 60
60 is more so the copy is more square.
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Sandra’s Solution
photo: 3 by 5
copy: 6 by 10
3/5 and 6/10 are equal, so they are equally square.
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Video as Common Experience
• Video can help provide a common experience so we
can better compare the ways we make sense of
students’ thinking.
• When we work with different students, it is somewhat
difficult to share. Each person has a somewhat
different experience
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Listening to Students
Five students were interviewed on the same
problem.
The students were enrolled in Algebra I.
The interviews were conducted in February.
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Interview Problem
From a shipment of 500 batteries, a sample of 25
was selected at random and tested.
If 2 batteries in the sample were found to be dead,
how many dead batteries would be expected in the
entire shipment?
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Interpreting Students’ Responses
• What was the most surprising student comment
that you heard?
• Do most of the students seem to understand
proportional reasoning? What is the evidence for
your inference?
• What misunderstandings are apparent? What is
the evidence for your inferences?
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Predicting
• How do you think these five students would
solve the “Rectangular Plots” problem (Quiz
Question #2)?
• Would they get correct answers?
• What strategies would they use?
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Instructional Planning
Once we have a sense of the level of thinking of a
group of students, the next challenge is choosing
the next task for those students to complete.
Would any of the Quiz problems be useful
instructional tasks for these students? Why or why
not?
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Proportional Reasoning and Algebra
Why is proportional reasoning important for
understanding algebra?
Must students understand proportional reasoning
before they can learn algebra?
How might you help students understand proportional
reasoning while they are also learning algebra?
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Student Learning is the Bottom Line
We all care deeply about how much and how well
students learn mathematics.
Understanding students’ thinking is an important first
step in planning instruction that helps students learn.
Using a variety of tasks can help generate more
accurate information about what students know so that
instructional decisions can be better aligned with the
needs of students.
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George W. Bright
gbright45@comcast.net
Activities and examples from:
Dynamic Classroom Assessment: Linking
Mathematical Understanding to Instruction in
Middle Grades and High School
published by ETA/Cuisenaire
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