a performance-based assessment: a means to high-level ......supporting rigorous mathematics teaching...

Supporting Rigorous Mathematics Teaching and Learning

Chapel Hill – Carrboro City Schools LuAnn Malik, K-12 Math Coordinator Grade 8

learning research and development center institute for learning

A Performance-Based Assessment: A Means to High-Level Thinking and Reasoning

© 2012 University of Pittsburgh LEARNING RESEARCH AND DEVELOPMENT CENTER

Forms of Assessment

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Assessment for Learning Assessment of Learning

Assessment as Learning


Session Goals

•  Deepen understanding of the Common Core State Standards (CCSS) for Mathematical Practices and Mathematical Content.

•  Understand how Performance-Based Assessments (PBAs) assess the CCSS for both mathematical content and practices.

•  Understand the ways in which PBAs assess students’ conceptual understanding.

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Overview of Activities

•  Analyze performance-based assessments (PBAs) in order to determine the way the assessments are assessing the CCSSM.

•  Analyze and discuss the CCSS for Mathematical Content and Mathematical Practices.

•  Discuss what it means to develop and assess conceptual understanding.

•  Discuss the CCSS related to the tasks and the implications for instruction and learning.

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The Common Core State Standards

The standards consist of:

Ø  The CCSS for Mathematical Content

Ø  The CCSS for Mathematical Practices

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Analyzing a Performance-Based Assessment

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North Carolina Focus Clusters Grade 8

•  Understand the connections between proportional relationships, lines and linear equations.

•  Define, evaluate and compare functions.

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Analyzing Assessment Items (Private Think Time) Four assessment items have been provided:

Ø  Typing Speeds Task

Ø  Olympic Pool Task Ø  Two Different Graphs Task

Ø  Buying Widgets Task

For each assessment item: •  Solve the assessment item.

•  Make connections between the standard(s) and the assessment item.

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1. Typing Rate Task § While working on a term paper, Donald noticed that he had typed 2400 words in 1 hour. After researching, he found out that a professional typist can generally type 50 words per minute. In order to compare his typing rate to the typing rate of a professional typist, Donald made the following table.

a.  Find both Donald’s typing rate and the professional’s typing rate. State whether

Donald or the professional types faster, and why. b.  On the axes below, the line representing Donald’s typing rate has been drawn and

labeled D. On the same axes, sketch the line that represents the Professional typist’s speed and label it P. Use equations and/or words to explain why you sketched the line the way you did.

Explanation:

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2. Olympic Pool Task § In the graph below, an Olympic-sized swimming pool is being filled with water from a fire hose.

a.  After studying the graph, Nick claims the hourly rate shown in the graph is 1000 gallons per hour. Explain why Nick is correct in equations, tables, drawings, and/or words.

b.  Brian says, “Since the rate is 1000, that tells me the slope is 1000 so I can write an equation for the line. It would be y = 1000x.” Do you agree or disagree with Brian? Use equations, tables, drawings, and/or words to justify your answer.

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0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500 3000 3500 N

umbe

r of H

ours

Number of Gallons

Filling an Olympic Pool


3. Two Different Graphs Task § Selena is working on a problem in math class. She says, “I drew triangles on both lines and I can see the height is twice the base on all the triangles. So, because the ratio of height and base of the triangles tells me about the slope of the line, I picked y = 2x for both Graph A and Graph B.” Selena’s work is shown below. a.  Use information from the graph to explain why Selena’s reasons for

choosing the equation y = 2x make sense for Graph A. b.  Use information from the graph to explain whether you agree or disagree

with Selena’s reasons for choosing the equation y = 2x for Graph B.

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4. Buying Tools Task §

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Company A and B both sell tools. You want to buy tools from either Company A or Company B.

0

50

100

150

0 1000 2000

Pric

e in

Dol

lars

Number of Tools

Company A - Cost of Tools

a.  Which company sells tools at the lower price?

b.  Use the data and the graph and in the table to explain how you know which company sells tools at the lower price.


Discussing Content Standards (Small-Group Time) For each assessment item:

With your small group, discuss the connections between the content standard(s) and the assessment item.

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Deepening Understanding of the Content Standards via the Assessment Items (Whole Group)

As a result of looking at the assessment items, what do you better understand about the specifics of the content standards?

What are you still wondering about?

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The CCSS for Mathematical Content

15 Common Core State Standards, NGA Center/CCSSO, 2010

Expressions and Equations 8.EE Understand the connections between proportional relationships, lines and linear equations. 8.EE.5 Graph proportional relationships, interpreting the unit rate

as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance time equation to determine which of the two moving objects has a greater speed.

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.


Determining the Mathematical Practices Associated with the

Performance-Based Assessment

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Getting Familiar with the CCSS for Mathematical Practices (Private Think Time)

•  Count off by 8. Each person reads one of the CCSS for Mathematical Practices.

•  Read your assigned Mathematical Practice. Be prepared to share the “gist” of the Mathematical Practice.

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The CCSS for Mathematical Practices 1.  Make sense of problems and persevere in solving

them.

2.  Reason abstractly and quantitatively.

3.  Construct viable arguments and critique the reasoning of others.

4.  Model with mathematics.

5.  Use appropriate tools strategically.

6.  Attend to precision.

7.  Look for and make use of structure.

8.  Look for and express regularity in repeated reasoning.

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Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO


Discussing Practice Standards (Small-Group Time)

Each person has 2 minutes to share important information about his/her assigned Mathematical Practice.

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Discussing Practice Standards (Small-Group Time) For each assessment item:

With your small group, discuss the connections between the practice standards and the assessment item.

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The CCSS for Mathematical Practices 1.  Make sense of problems and persevere in solving

them.

2.  Reason abstractly and quantitatively.

3.  Construct viable arguments and critique the reasoning of others.

4.  Model with mathematics.

5.  Use appropriate tools strategically.

6.  Attend to precision.

7.  Look for and make use of structure.

8.  Look for and express regularity in repeated reasoning.

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Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO


Deepening Understanding of the Practice Standards via the Assessment Items (Whole Group)

Which mathematical practices do you better understand?

What are you still wondering about?

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Assessing Conceptual Understanding

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Rationale

We have now examined assessment items and discussed their connection to the CCSS for Mathematical Content and Practices. A question that needs considering, however, is if and how these assessments will give us a good means of measuring the conceptual understandings our students have acquired.

In this activity, you will have an opportunity to consider what it means to develop conceptual understanding as described in the CCSS for Mathematics and what it takes to assess for it.

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Assessing for Conceptual Understanding

The set of PBA items are designed to assess student understanding of expressions and equations.

Look across the set of related items. What might a teacher learn about a student’s understanding by looking at the student’s performance across the set of items as a whole?

What is varying from one item to the next?

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Conceptual Understanding

•  What do the authors mean by conceptual understanding?

•  How might analyzing student performance on this set of assessments help us determine if students have a deep understanding of Expressions and Equations?

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Developing Conceptual Understanding

Knowledge that has been learned with understanding provides the basis of generating new knowledge and for solving new and unfamiliar problems. When students have acquired conceptual understanding in an area of mathematics, they see connections among concepts and procedures and can give arguments to explain why some facts are consequences of others. They gain confidence, which then provides a base from which they can move to another level of understanding.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics.

Washington, DC: National Academy Press

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The CCSS on Conceptual Understanding In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice.

These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

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Common Core State Standards for Mathematics, 2010, p. 8, NGA Center/CCSSO


Assessing Concept Image Tall (1992) differentiates between the mathematical definition of a concept and the concept image, which is the entire cognitive structure that a person has formed related to the concept. This concept image is made up of pictures, examples and non-examples, processes, and properties. A strong concept image is a rich, integrated, mental representation that allows the student to flexibly move between multiple formulations and representations of an idea. A student who has connected mathematical ideas in this way can create and use a model to analyze a situation, uncover patterns and synthesize them to form an integrated picture. They can also use symbols meaningfully to describe generalizations which then provides a base from which they can move to another level of understanding.

Brown, Seidelmann, & Zimmermann. In the trenches: Three teachers’ perspectives on moving beyond the math wars. http://mathematicallysane.com/analysis/trenches.asp

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© 2012 University of Pittsburgh LEARNING RESEARCH AND DEVELOPMENT CENTER 30

Developing and Assessing Understanding

Why is it important, when assessing a student’s conceptual understanding, to vary items in these ways?


Analyzing a Student’s Performance

Analyze Nicole’s performance on four tasks.

What do you notice? What does Nicole know?

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1. Typing Rate Task: Nicole's Work

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2. Olympic Pool Task: Nicole's Work

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As shown in the graph below, an Olympic-sized swimming pool is being filled with water from a fire hose.

a.  After studying the graph, Nick claims the hourly rate shown in the graph is 1000 gallons per hour. Explain why Nick is correct in equations, tables, drawings, and/or words.

b.  Brian says, “Since the rate is 1000, that tells me the slope is 1000 so I can write an equation for the line. It would be y = 1000x.” Do you agree or disagree with Brian? Use equations, tables, drawings, and/or words to justify your answer.

0

1

2

3

4

0 1000 2000 3000 4000

Num

ber o

f Hou

rs

Number of Gallons

Filling an Olympic Pool


3. Two Different Graphs Task: Nicole's Work

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Selena is working on a problem in math class. She says, “I drew triangles on both lines and I can see the height is twice the base on all the triangles. So, because the ratio of height and base of the triangles tells me about the slope of the line, I picked y = 2x for both Graph A and Graph B.” Selena’s work is shown below.

a.  Use information from the graph to explain why Selena’s reasons for choosing the equation y = 2x make sense for Graph A.

b.  Use information from the graph to explain whether you agree or disagree with Selena’s reasons for choosing the equation y = 2x for Graph B.


4. Buying Tools Task: Nicole's Work

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Company A and B both sell widgets. You want to buy tools from either Company A or Company B.

a.  Which company sells tools at the lower price? b.  Use the data in the graph and in the table to explain how you know which

company sells tools at the lower price.

0 50

100 150

0 1000 2000 Pric

e in

Dol

lars

Number of Tools

Company A - Cost of Tools


Using the Assessment to Think About Instruction

In order for students to perform well on the PBA, what are the implications for instruction?

•  What kinds of instructional tasks will need to be used in the classroom?

•  What will teaching and learning look like and sound like in the classroom?

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Step Back

•  What have you learned about the CCSS for Mathematical Content that surprised you?

•  What is the difference between the CCSS for Mathematical Content and the CCSS for Mathematical Practices?

•  Why do we say that students must work on both Mathematical Content and the Mathematical Practices?

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Representations of Proportional Relationships♦

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http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf, pg. 5



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Showing Structure in Tables and Graphs ♦

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1. Typing Rate Task §

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Common Core State Standards, 2010, NGA Center/CCSSO


as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance time equation to determine which of the two moving objects has a greater speed. ♦


2. Olympic Pool Task §

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3. Two Different Graphs Task §

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Expressions and Equations 8.EE Understand the connections between proportional relationships, lines and linear equations. 8.EE.6 Use similar triangles to explain why the slope m is the

same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. ♦


4. Buying Tools Task §

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a performance-based assessment: a means to high-level ......supporting rigorous mathematics teaching...

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