Supporting Rigorous Mathematics Teaching and Learning · Supporting Rigorous Mathematics Teaching and Learning SAS Math Summit August 7, 2014 Middle School Mathematics - Grade 8 learning

Supporting Rigorous Mathematics Teaching and Learning

SAS Math Summit August 7, 2014Middle School Mathematics - Grade 8

learning research and development centerinstitute for learning

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

© 2012 University of Pittsburgh LEARNING RESEARCH AND DEVELOPMENT CENTER

Session Goals

● Outline CHCCS instructional planning process using the Understanding by Design Framework

● Understand how Performance-Based Assessments (PBAs) assess the CCSS for both Mathematical Content and Practice.

● Understand the ways in which PBAs assess students’ conceptual understanding

● Understand how PBA’s stimulate high-quality Essential Understandings

2


Overview of Activities

• Analyze PBAs in order to determine the way the assessments are assessing the CCSSM.

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

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

• Discuss the ways in which PBAs can initiate Essential Understandings and lesson planning

3


Action Plan

Support high quality teaching and learning and work toward consistent expectations and outcomes aligned with standards across the district.

Essential Components…

● Instructional Planning teams for mathematics

● Professional development for teachers and administrators

● Instructional Coaches


Instructional Planning Teams

Starting in the spring 2013..

8 planning teams…. o Elementary Grades 3-4-5o Middle School Grades 6,7,8 o High School Math I, Math II



Starting in the spring 2014..

Continue 8 planning teams and add 4 teams…. o Elementary Grades K-1-2 o High School Math III


Stages in UbD

Stage One:Identify Desired

ResultsWhat long term goals are

targeted?

What meanings should students make to arrive at important understandings?

What essential questions will students keep considering?

What knowledge and skill will students acquire?

Stage Two:Determine Acceptable Evidence

What collection of assessments will provide evidence that students

learned the essentials in Stage One?

Stage Three:Plan Learning

Experiences and Instruction

What activities, strategies and instructional methods will help students learn the

essentials in Stage One and prepare students for the

assessments in Stage Two?



What each team will do....use UbD model

1. Create year-long scope and sequence documents aligned to Common Core for their grade level. Outline a pacing guide.

2. Develop Performance Based Assessments and scoring guides for the first three quarters.

3. Align scope and sequence documents to instructional materials.

https://sites.google.com/a/chccs.k12.nc.us/chccs-elementary-math-site/


Stages in UbD

Stage One:Identify Desired

ResultsWhat long term goals are

targeted?

What meanings should students make to arrive at important understandings?

What essential questions will students keep considering?

What knowledge and skill will students acquire?

Stage Two:Determine Acceptable Evidence

What collection of assessments will provide evidence that students

learned the essentials in Stage One?

Stage Three:Plan Learning

Experiences and Instruction

What activities, strategies and instructional methods will help students learn the

essentials in Stage One and prepare students for the

assessments in Stage Two?


Analyzing aPerformance-Based Assessment

10


Grade 8Focus Clusters

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

• Define, evaluate and compare functions.

11


Analyzing Assessment Items(Private Think Time)

Four assessment items have been provided: Typing Speeds Task

Olympic Pool Task

Two Different Graphs Task

Buying Widgets Task

For each assessment item:• Solve the assessment item.

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

12


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 speed 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:

13


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.

14

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500 3000 3500N

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.

15


4. Buying Tools Task §

16

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 in 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.

17


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?

18


The CCSS for Mathematical Content

19Common Core State Standards, NGA Center/CCSSO, 2010

Expressions and Equations 8.EEUnderstand the connections between proportionalrelationships, 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

20


The CCSS for Mathematical Practices1. 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.

21

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


Discussing Practice Standards(Small-Group Time)


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

22


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

Which mathematical practices do you better understand?


23


Assessing Conceptual Understanding

24


RationaleWe have now examined assessment items and discussed their connection to the CCSS for Mathematical Content and Practice. 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.

25


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 ateacher learn about a student’s understanding by looking at the student’s performance across the set of items as a whole?

What is the variance from one item to the next?

26

© 2012 University of Pittsburgh LEARNING RESEARCH AND DEVELOPMENT CENTER 27

Developing and Assessing Understanding

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


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 Number and Operations − Fractions?

28


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

29


The CCSS on Conceptual UnderstandingIn 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.

30

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


Assessing Concept ImageTall (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

31


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?

32


Representations of Proportional Relationships

33

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


Showing Structure in Tables and Graphs

34

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


1. Typing Rate Task §

35

Common Core State Standards, 2010, NGA Center/CCSSO




2. Olympic Pool Task §

36





3. Two Different Graphs Task §

37


Expressions and Equations 8.EEUnderstand the connections between proportionalrelationships, 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 Widgets Task §

38




learning research and development center institute for learning

Supporting Teaching and Learning of

Rigorous Mathematics Instruction

A Performance-Based Assessment: A Means to High-Level

Thinking and Reasoning

PARTICIPANT HANDOUT

SAS Math Summit

Middle School Mathematics – Thursday, August 7, 2014

Eighth Grade

© 2012 University of Pittsburgh 2

2012 – 2013 Focus Clusters Grade 8

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

• Define, evaluate and compare functions.

Analyzing Assessment Items

(Private Think Time)

Four assessment items have been provided:

Typing Rate Task

Olympic Pool Task

Two Different Graphs Task

Buying Tools Task


• Solve the assessment item.

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


1. Typing Rate Task

While working on a term paper, Donald noticed that he had typed 2400 words in 1 hour. After a little research, 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.

Time (in minutes)

Number of Words

Donald Professional

Typist

1 40 50

60 2400 3000

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.

Number

of

Words

Explanation: D

Time


2. Olympic Pool Task As shown in the graph below, an Olympic-sized swimming pool is being filled with 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 using 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.


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.

The following are two graphical descriptions of linear equations in systems with different scale units. Choose the equation that each line describes. Explain your choice for each case.


4. Buying Tools Task

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

Company B Cost of Tools

Number of Widgets

Price in dollars

200 8 400 16 600 24

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.


Discussing Content Standards (Small-Group Time)


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

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?



The CCSS for Mathematical Content – Grade 8

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.


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

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

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

Mathematical Practice.

The CCSS for Mathematical Practice

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


Discussing Practice Standards (Small-Group Time)

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

Mathematical Practice.


Discussing Practice Standards (Small-Group Time)


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

assessment item.

Deepening Understanding of the Practice Standards via the

Assessment Items (Whole Group)

Which mathematical practices do you better understand?



Developing and Assessing Understanding

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

in these ways?


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?


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

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.

Common Core State Standards for Mathematics, 2010, p. 9

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

Analyzing a Student’s Performance Analyze Nicole’s performance on four tasks.

What do you notice? What does Nicole know?


1. Typing Rate Task: Nicole’s Work


2. Olympic Pool Task: Nicole’s Work


3. Two Different Graphs Task: Nicole’s Work


4. Buying Tools Task: Nicole’s Work


Assessing for Conceptual Understanding

The set of PBA items are designed to assess student understanding of multiplication and division.

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 the variance from one item to the next?


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?


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 Practice?

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

Mathematical Practices?

Documents

Supporting Rigorous Mathematics Teaching and Learning · Supporting Rigorous Mathematics Teaching and Learning SAS Math Summit August 7, 2014 Middle School Mathematics - Grade 8 learning