TDG Seminar, February 2007Secondary Students’ Solutions 1 Comparing Levels of Sophistication of Secondary Students’ Solutions George W. Bright Professor

TDG Seminar, February 2007 Secondary Students’ Solutions 1

Comparing Levels of Sophistication of Secondary

Students’ Solutions

George W. Bright

Professor EmeritusUniversity of North Carolina at Greensboro

[email protected]


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.


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.


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.


Model for Classroom AssessmentLearningTargets

AssessmentMethods

Inferences about

Thinking

Instructional Decisions

Purposes

Communication

Documentation


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.


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.


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)


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 self-assessment, 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.


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.


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)


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 sense-making appropriate for the needs of the students and the purposes of the lesson.

(Weiss, Pasley, Smith, Banilower, & Heck, 2003, p. 103)


Proportional Reasoning Quiz

Solve each of the four problems in the Proportional Reasoning Quiz.


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.”


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 feetB. 7 feet by 10 feetC. 17 feet by 20 feetD. 27 feet by 30 feet


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 lapsB. 24 lapsC. 21 lapsD. 6 laps

Why is important to have this problem on the quiz?


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 shotsPeterson: 22 of 29 shotsWilliams: 15 of 19 shotsReynolds: 33 of 41 shots

Which player is the best free-throw shooter?A. Novak B. PetersonC. Williams D. Reynolds


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?


Proportional ReasoningStudent Item 1 Item 2 Item 3 Item 4 Score 1 C D C D 4 2 C D A D 3 3 C D A A 2 4 C D A C 2 5 A A C A 1 6 A D C D 3 7 C D C D 4 8 C D C C 3 9 C D C D 410 C B A A 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?


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.


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?


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.


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?


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.”


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?


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.


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.


Charlene’s Solution

photo: 3 x 5 = 15

copy: 6 x 10 = 60

60 is more so the copy is more square.


Sandra’s Solution

photo: 3 by 5

copy: 6 by 10

3/5 and 6/10 are equal, so they are equally square.


Video as Common Experience

• Video can help provide a common experience so wecan 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 somewhatdifferent experience


Listening to Students

Five students were interviewed on the same problem.

The students were enrolled in Algebra I.

The interviews were conducted in February.


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?


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?


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?


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?


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?


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.


George W. Bright

[email protected]

Activities and examples from:

Dynamic Classroom Assessment: Linking Mathematical Understanding to Instruction in

Middle Grades and High Schoolpublished by ETA/Cuisenaire

Documents

TDG Seminar, February 2007Secondary Students’ Solutions 1 Comparing Levels of Sophistication of Secondary Students’ Solutions George W. Bright Professor