Session 111: Performance-Based Assessment: A Means to ...Supporting Rigorous Mathematics Teaching and Learning SAS Math Summit August 6, 2014 Elementary School Mathematics - Grade

Supporting Rigorous Mathematics

Teaching and Learning

SAS Math Summit

August 6, 2014

Elementary School Mathematics - Grade 5

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

Session Goals

• Deepen understanding of the Common Core State

Standards (CCSS) for Mathematical Practice and

Mathematical Content.

• 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.

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.

3


Analyzing a

Performance-Based Assessment

4


Grade 5

Focus Clusters

• Extend understanding of fraction equivalence and

ordering.

• Build fractions from unit fractions by applying and

extending previous understanding of operations of

whole numbers.

5


Analyzing Assessment Items(Private Think Time)

Four assessment items have been provided:

48 Gumdrops Task

Cutting Ribbon Task

Stew Recipe Task

Ellen’s Math Task

For each assessment item:

• Solve the assessment item.

• Make connections between the standard(s) and the

assessment item.

6


1. 48 Gumdrops Task

§Two children are sharing 48 gumdrops.

Jessica says, “I want 2/4 of the set of 48 gumdrops.”

Samuel says, “I want 2/3 of the set of 48 gumdrops."

a. Is it possible for Jessica and Samuel to each have a fraction of the

gumdrops that they want?

b. If you respond yes, use diagrams and equations to explain how you

know they can each receive the share of gumdrops they want. If you

respond no, use diagrams and equations to explain why the children

cannot receive the number of gumdrops they each want.

7


2. Cutting Ribbon Task §

Amber has a 12-foot-long piece of ribbon. She wants

pieces that are each 3/4 of a foot long.

How many 3/4-foot-long pieces can Amber cut from

the ribbon? Use words, diagrams, and multiplication

or division equations to explain your thinking

8


3. Stew Recipe Task§

Molly’s recipe for stew says to use 1/4 pound of potatoes per

person. Molly has 24 pounds of potatoes. Molly wonders if she can

make enough stew to serve all 36 people at her party.

a. Calculate 1/4 × 36. What information can Molly get from the

answer?

b. Calculate 24 ÷ 1/4. What information can Molly get from the

answer?

c. Will Molly have enough potatoes? If not, how many more

pounds will she need?

9


4. Ellen's Math Task §

Start with 9.

Multiply by 5. 9 x 5 = 45

The answer is 45, and 45 > 9

45 is bigger than 9.

It even works for fractions.

Start with 1/2. 1/2 x 4 = 2

Multiply by 4. 2 > 1/2

The answer is 2, and

2 is bigger than 1/2.

Ellen’s calculations are correct, but her rule does not always work.

Sometimes multiplication gives a product smaller than the number you started with.

Explain when this is true, and give an example using words, diagrams and equations.

10

Multiplying any number will always give you an answer bigger

than the number you started with. Look, I can show you.Ellen says:


Discussing Content Standards (Small-Group Time)


With your small group, discuss the connections

between the content standard(s) and the assessment

item.

11


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?

12


The CCSS for Mathematical Content: Grade 5

13Common Core State Standards, NGA Center/CCSSO, 2010

Number and Operations – Fractions 5.NF

Apply and extend previous understandings of multiplication and

division to multiply and divide fractions.

5.NF.4 Apply and extend previous understandings of multiplication to

multiply a fraction or whole number by a fraction.

5.NF.5 Interpret multiplication as scaling (resizing), by:

5.NF.5b Explaining why multiplying a given number by a fraction greater

than 1 results in a product greater than the given number

(recognizing multiplication by whole numbers greater than 1 as

a familiar case); explaining why multiplying a given number by a

fraction less than 1 results in a product smaller than the given

number; and relating the principle of fraction equivalence a/b =

(n × a)/(n × b) to the effect of multiplying a/b by 1.

5.NF.6 Solve real-world problems involving multiplication of fractions

and mixed numbers, e.g., by using visual fraction models or

equations to represent the problem.

5.NF.7 Apply and extend previous understandings of division to divide

unit fractions by whole numbers and whole numbers by unit

fractions.


The CCSS for Mathematical Content: Grade 5

14Common Core State Standards, NGA Center/CCSSO, 2010


Apply and extend previous understandings of multiplication and

division to multiply and divide fractions.

5.NF.7a Interpret division of a unit fraction by a non-zero whole number,

and compute such quotients. For example, create a story context

for (1/3) ÷ 4, and use a visual fraction model to show the quotient.

Use the relationship between multiplication and division to explain

that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

5.NF.7b Interpret division of a whole number by a unit fraction, and

compute such quotients. For example, create a story context for

4 ÷ (1/5), and use a visual fraction model to show the quotient.

Use the relationship between multiplication and division to explain

that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

5.NF.7c Solve real-world problems involving division of unit fractions by

non-zero whole numbers and division of whole numbers by unit

fractions, e.g., by using visual fraction models and equations to

represent the problem. For example, how much chocolate will

each person get if 3 people share 1/2 lb of chocolate equally?

How many 1/3-cup servings are in 2 cups of raisins?


Determining the Mathematical

Practices Associated with the

Performance-Based Assessment

15


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.

16

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.

17


Assessing Conceptual Understanding

18


Rationale

We 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.

19


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?

20


21

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?

22


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

23


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.

24

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

25


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?

31


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?

32


1. 48 M&M's Task§

33

Common Core State Standards, NGA Center/CCSSO, 2010


Apply and extend previous understandings of multiplication

and division to multiply and divide fractions.

5.NF.4 Apply and extend previous understandings of

multiplication to multiply a fraction or whole number by a

fraction.

5.NF.6 Solve real-world problems involving multiplication of

fractions and mixed numbers, e.g., by using visual

fraction models or equations to represent the problem.

5.NF.7b Interpret division of a whole number by a unit fraction,

and compute such quotients. For example, create a story

context for 4 ÷ (1/5), and use a visual fraction model to

show the quotient. Use the relationship between

multiplication and division to explain that 4 ÷ (1/5) = 20

because 20 × (1/5) = 4.


2. Cutting Ribbon Task§

34



Apply and extend previous understandings of multiplication

and division to multiply and divide fractions.

5.NF.4 Apply and extend previous understandings of

multiplication to multiply a fraction or whole number by a

fraction.

5.NF.6 Solve real-world problems involving multiplication of

fractions and mixed numbers, e.g., by using visual fraction

models or equations to represent the problem.


3. Stew Recipe Task §

35



Apply and extend previous understandings of multiplication and division to

multiply and divide fractions.

5.NF.4 Apply and extend previous understandings of multiplication to multiply a

fraction or whole number by a fraction.

5.NF.7a Interpret division of a unit fraction by a non-zero whole number, and

compute such quotients. For example, create a story context for (1/3) ÷ 4,

and use a visual fraction model to show the quotient. Use the relationship

between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because

(1/12) × 4 = 1/3.

5.NF.7b Interpret division of a whole number by a unit fraction, and compute such

quotients. For example, create a story context for 4 ÷ (1/5), and use a

visual fraction model to show the quotient. Use the relationship between

multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5)

= 4.

5.NF.7c Solve real-world problems involving division of unit fractions by non-zero

whole numbers and division of whole numbers by unit fractions, e.g., by

using visual fraction models and equations to represent the problem. For

example, how much chocolate will each person get if 3 people share 1/2 lb

of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?


4. Ellen's Math Task §

36





5.NF.4 Apply and extend previous understandings of multiplication to multiply a

fraction or whole number by a fraction.

5.NF.5 Interpret multiplication as scaling (resizing), by:

5.NF.5b Explaining why multiplying a given number by a fraction greater than 1

results in a product greater than the given number (recognizing

multiplication by whole numbers greater than 1 as a familiar case);

explaining why multiplying a given number by a fraction less than 1 results

in a product smaller than the given number; and relating the principle of

fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.


4. Ellen's Math Task (continued)

§

37





5.NF.7a Interpret division of a unit fraction by a non-zero whole number, and

compute such quotients. For example, create a story context for (1/3) ÷ 4,

and use a visual fraction model to show the quotient. Use the relationship

between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because

(1/12) × 4 = 1/3.

5.NF.7b Interpret division of a whole number by a unit fraction, and compute such

quotients. For example, create a story context for 4 ÷ (1/5), and use a

visual fraction model to show the quotient. Use the relationship between

multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) =

4.


Visual Models: Multiplication of Fractions

38Draft, 8/12/2011, comment at commoncoretools. wordpress. com .2


Area Model: Multiplication of Fractions

39Draft, 8/12/2011, comment at commoncoretools. wordpress. com .2

Documents

Session 111: Performance-Based Assessment: A Means to ...Supporting Rigorous Mathematics Teaching and Learning SAS Math Summit August 6, 2014 Elementary School Mathematics - Grade