​

Gr 5-6 Math Learning Progressions with Student Work Examples

This work is licensed under a Creative Commons Attribution 4.0 International License. Educators and others may use or adapt. If modified, please attribute and re-title.

Version 1.0 | Developed By:�Carla Evans & Caroline Wylie�National Center for the Improvement of Educational Assessment

​

Background on Learning Progressions

2

Learning Progressions

The stages of learning to walk include:

• Pre-walking baby steps (3-6 months old)
• Crawling (6-9 months old)
• Pulling up on furniture (9 months old)
• Walking with help (9-12 months old)
• Standing without help (12-18 months old)
• Walking (two years plus)

3

A Progression is helpful, not deterministic

• Not every baby hits every stage at the same time.
• Some babies never crawl.
• Some regress.
• But the patterns of typical progress are helpful.

Mathematics Learning Progressions

What are they?

• Built on cognitive science, learning theory, and research studies that focus on how students typically develop understanding of big ideas in mathematics over time.
• More than a scope and sequence.

Why Useful?

• Provide an interpretive framework:
• Support listening for what students do and do not understand, rather than focusing on correct/incorrect.
• Analyze student work for understanding.
• Support asset-based interpretations and next steps.

4

5

A Couple Notes about Learning Progressions

Learning progressions are NOT a curriculum. You will use your curriculum to elicit student thinking and then use the progressions to make sense of and interpret evidence of student thinking as you analyze their work.

• Sarama and Clements (2009) state that the value of learning progressions is that they can help teachers see “themselves not as moving through a curriculum, but as helping students move through levels of understanding” (p. 17).

​

Learning progressions are not evaluative but are intended to be used for descriptive evidence and instructional decision-making.

• Not all possible strategies are represented on the progressions.
• There is no one right path to move students towards more efficient and sophisticated strategies.

​

​

​

6

Learning Progressions

7

Gr 5-6 Multiplication & Division Progression

8

9

Individually, start from the bottom and work your way up to start to make sense of the progressions.

Multiplication Progression

Division Progression

Quick Turn and Talk

10

What do you notice in the progressions?

What are you wondering about?

Interact with the Progressions

1. For what grades are the progressions likely to be relevant?
2. Where are students in the grades 5-6 typically at this time of year on these progressions?
3. What is the range within each grade level of students on each progression?
4. How does this information impact teaching and learning?
5. What is the goal (based on the standards) for the end of each grade level?

11

Stand up and form Grade 5 & 6 pairs to discuss the questions for 5 minutes.

We will ask one group to share out and others to add in if areas of disagreement.

Share Out from the Jigsaw

Whole group conversation:

• One group volunteers to share out their responses to the questions.
• Does another group have anything significantly different to add?

​

• For what grades is the progression likely to be relevant?
• Where are students in the grades 5-6 typically at this time of year on this progression?
• What is the range within each grade level of students on the progression?
• How does this information impact teaching and learning?
• What is the goal (based on the standards) for the end of each grade level?

12

Heuristic for Looking at Student Work

1. What do you notice in the student work?
2. How does the student work relate to the learning progression (where does it fall; what can be built upon)?
3. What could you do next instructionally for the whole class, small groups, or for individuals?
1. What nudge might support this student’s thinking?
2. How do I group students together for small group instruction?
3. Are there students who can share different strategies with the whole class to show their thinking and explain their reasoning?

​

13

Example 1: Multiplication

Comparing Student Work Samples A-C

14

A

B

C

Max and Thomas each delivered vegetables to a store. Max delivered 8 bags of vegetables with 40 pounds in each bag. Thomas delivered 9 bags of vegetables with 35 pound in each bag. How many pounds of vegetables were delivered all together?

Examples were taken from Hulbert, E., Petit, M. M., Ebby, C. B., Cunningham, E. P., & Laird, R. E. (2024). A Focus on Multiplication and Division: Bringing Mathematics Education Research to the Classroom (2nd ed.). Routledge.

Example 1 Shareout

• What do you notice in the student work?
• How does the student work relate to the learning progression (where does it fall; what can be built upon)?
• What could you do next instructionally for the whole class, small groups, or for individuals?

15

16

Additive Strategies

Sample A: Uses repeated addition rather than multiplication. Unclear how added 40 eight times or 35 nine times. Then added 320 + 315 = 635 using traditional algorithm. Correctly labeled answer.

Multiplicative Strategies:

Sample C: Multiplies 8 x 40 -- it appears that they use powers of 10 and do 8 x 4 and then add on the zero. Use associative property to break 9 x 35 into 9 x 30 and 9 x 5 to get 315. Don’t show intermediate step of 270 + 45. Then add the two numbers together and correctly label.

​

Early Transitional Strategies:

Sample B: Skip counts by 35 and by 40, then adds and correctly labels answer.

​

Connecting Strategies

Moves Student Thinking Forward

Strategies should not be taught in isolation, but in a connected way intended to move student thinking forward.

It is ESSENTIAL to understand where students are and where they need to go.

​

​

Taken from Chantel DeNapoli’s Presentation, “Using Student Evidence to Inform Instruction”

Example 2: Division

Comparing Student Work Samples D-H

18

D

E

The fourth-grade class earned $132 from a bake sale. The class decided to donate an equal amount of the money to four different organizations. How much money did the class donate to each organization? Show your work.

Examples were taken from Hulbert, E., Petit, M. M., Ebby, C. B., Cunningham, E. P., & Laird, R. E. (2024). A Focus on Multiplication and Division: Bringing Mathematics Education Research to the Classroom (2nd ed.). Routledge.

19

F

G

H

The fourth-grade class earned $132 from a bake sale. The class decided to donate an equal amount of the money to four different organizations. How much money did the class donate to each organization? Show your work.

Examples were taken from Hulbert, E., Petit, M. M., Ebby, C. B., Cunningham, E. P., & Laird, R. E. (2024). A Focus on Multiplication and Division: Bringing Mathematics Education Research to the Classroom (2nd ed.). Routledge.

33

4⟌132

-12

12

-12

0

The class will donate $33 to each organization

Example 2 Shareout

• What do you notice in the student work?
• How does the student work relate to the learning progression (where does it fall; what can be built upon)?
• What could you do next instructionally for the whole class, small groups, or for individuals?

20

21

Early Transitional Strategies:

Sample E: Skip counting strategies; Equation is correct; error in the skip counting (there are 35 numbers in her list because she included 42 and 46, which are not multiples of 4); no units labeled so unclear if student understands answer refers to dollars

​

​

Multiplicative Strategies:

Sample G: Student uses traditional algorithm and labels units within context of answer

​

Transitional Strategies:

Sample H: Inefficient partial quotients; student uses multiplication and addition to solve rather than division; correctly interprets amount each class will donate and labels units

​

Additive Strategies:

Sample D: Student shows evidence of sharing out by groups of tens and then threes; repeated subtraction; no units labeled so unclear if student understands answer refers to dollars

​

Sample F: Student shows evidence of sharing out by ones; repeated addition; no units labeled so unclear if student understands answer refers to dollars

​

Gr 5-6 Fractions Progression

22

23

Individually, start from the bottom and work your way up to start to make sense of the progression.

Quick Turn and Talk

24

What do you notice in the progression?

What are you wondering about?

25

Quick share out:

• What did you notice?
• Wonders?

Interact with the Progression

• For what grades is the progression likely to be relevant?
• Where are students in grades 5-6 typically at this time of year on this progression?
• What is the range within each grade level of students on the progression?
• How does this information impact teaching and learning?
• What is the goal (based on the standards) for the end of each grade level?

26

Stand up and form Gr 5-6 pairs to discuss the questions for 5 minutes.

We will ask one group to share out and others to add in if areas of disagreement.

Share Out from the Jigsaw

Whole group conversation:

• One group volunteers to share out their responses to the questions.
• Does another group have anything significantly different to add?

​

• For what grades is the progression likely to be relevant?
• Where are students in grades 5-6 typically at this time of year on this progression?
• What is the range within each grade level of students on the progression?
• How does this information impact teaching and learning?
• What is the goal (based on the standards) for the end of each grade level?

27

Heuristic for Looking at Student Work

• What do you notice in the student work?
• How does the student work relate to the learning progression (where does it fall; what can be built upon)?
• What could you do next instructionally for the whole class, small groups, or for individuals?
• What nudge might support this student’s thinking?
• How do I group students together for small group instruction?
• Are there students who can share different strategies with the whole class to show their thinking and explain their reasoning?

​

28

Two Examples to Choose From:

Example 1: Comparing Student Work Samples A-E

Example 2: Comparing Student Work Samples F-J

29

Ashley bought 6 pounds of candy. She put the candy into bags that each hold ¾ of a pound of candy. How many bags of candy did Ashley fill?

Example 1: Comparing Student Work Samples A-E

30

B

Examples were taken from Petit, M. M., Laird, R. E., Ebby, C. B., & Marsden, E. L. (2023). A Focus on Fractions: Bringing Mathematics Education Research to the Classroom (3rd ed.). Routledge.

1 + 7

12 8

​

2 + 21 = 23

24 24 24

​

A

C

31

D

E

Example Shareout

• What do you notice in the student work?
• How does the student work relate to the learning progression (where does it fall; what can be built upon)?
• What could you do next instructionally for the whole class, small groups, or for individuals?

32

33

Fractional Strategies

Student A: Uses common denominators to solve.

​

Student D: Reasons through elimination of the multiple choice options and magnitude reasoning.

​

Transitional Strategies

Student B: Creates number line with fractions, but it is unclear how 1/12 figures into their reasoning.

​

Early Fractional Strategies

Student E: Creates two visual models, but doesn’t seem to understand the denominators are not the same so you can’t just add.

​

Non-Fractional Strategies

Student C: Adds numerators.

Example 1

Provide Additional Practice & Connect the Strategies

34

Concrete → Representational → Abstract (Symbolic Notation)

Example 2: Comparing Student Work Samples F-J

35

Ashley bought 6 pounds of candy. She put the candy into bags that each hold ¾ of a pound of candy. How many bags of candy did Ashley fill?

F

G

Examples were taken from Petit, M. M., Laird, R. E., Ebby, C. B., & Marsden, E. L. (2023). A Focus on Fractions: Bringing Mathematics Education Research to the Classroom (3rd ed.). Routledge.

H

6 ÷ ¾ =

6/1 x 4/3 =

24/3 = 8 bags

36

J

I

Example Shareout

• What do you notice in the student work?
• How does the student work relate to the learning progression (where does it fall; what can be built upon)?
• What could you do next instructionally for the whole class, small groups, or for individuals?

37

38

Fractional Strategies

Student H: Divided whole number by fraction showing correct interpretation of problem and then calculated answer using correct procedures.

​

Transitional Strategies

Student F: Created number line showing 6 pounds of candy and then divided each pound into fourths; used skip counting of fourths to identify 8 bags.

Student I: Created a table and subtracted ¾ starting with 6 pounds for each new bag.

Student J: Used repeated addition to calculate number of bags using unit fraction strategy.

​

Early Fractional Strategies

Student G: Created area models showing ¾ filled in, but wasn’t able to figure out how many bags or show understanding of 6 pounds.

​

​

​

​

Example 2

Provide Additional Practice & Connect the Strategies

39

Concrete → Representational → Abstract (Symbolic Notation)

Gr 5-6 Ratio & Proportions Progression

40

41

Individually, start from the bottom and work your way up to start to make sense of the progression.

Quick Turn and Talk

42

What do you notice in the progression?

What are you wondering about?

43

Quick share out:

• What did you notice?
• Wonders?

Interact with the Progression

• For what grades is the progression likely to be relevant?
• Where are students in grades 5-6 typically at this time of year on this progression?
• What is the range within each grade level of students on the progression?
• How does this information impact teaching and learning?
• What is the goal (based on the standards) for the end of each grade level?

44

Stand up and form Gr 5-6 pairs to discuss the questions for 5 minutes.

We will ask one group to share out and others to add in if areas of disagreement.

Share Out from the Jigsaw

Whole group conversation:

• One group volunteers to share out their responses to the questions.
• Does another group have anything significantly different to add?

​

• For what grades is the progression likely to be relevant?
• Where are students in grades 5-6 typically at this time of year on this progression?
• What is the range within each grade level of students on the progression?
• How does this information impact teaching and learning?
• What is the goal (based on the standards) for the end of each grade level?

45

Heuristic for Looking at Student Work

• What do you notice in the student work?
• How does the student work relate to the learning progression (where does it fall; what can be built upon)?
• What could you do next instructionally for the whole class, small groups, or for individuals?
• What nudge might support this student’s thinking?
• How do I group students together for small group instruction?
• Are there students who can share different strategies with the whole class to show their thinking and explain their reasoning?

​

46

Example 1: Comparing Student Work Samples A-F

47

B

Examples were taken from Petit, M. M., Laird, R. E., Wyneken, M. F., Huntoon, F. R., Abele-Austin, M. D., & Sequeira, J. D. (2020). A Focus on Ratios and Proportions: Bringing Mathematics Education Research to the Classroom. Routledge.

A

C

Donna runs at an average rate of 12 minutes a mile. At this rate how many miles does Donna run in 28 minutes?

Example 1: Comparing Student Work Samples A-F

48

E

Examples were taken from Petit, M. M., Laird, R. E., Wyneken, M. F., Huntoon, F. R., Abele-Austin, M. D., & Sequeira, J. D. (2020). A Focus on Ratios and Proportions: Bringing Mathematics Education Research to the Classroom. Routledge.

Donna runs at an average rate of 12 minutes a mile. At this rate how many miles does Donna run in 28 minutes?

D

F

Example Shareout

• What do you notice in the student work?
• How does the student work relate to the learning progression (where does it fall; what can be built upon)?
• What could you do next instructionally for the whole class, small groups, or for individuals?

49

50

Proportional Strategies

Student C: Student finds unit rate (min/mile) and then applies by multiplying by 28 to get solution. Solution is not simplified.

Student D: Student sets up ratio relationships (12:1 and 28: X) uses multiplicative relationships to get scale factor of 2 ⅓ so solve the problem.

​

Early Transitional Proportional Strategies

Student A: Student uses multiplicative relationships to identify that 28 minutes falls between 2-3 miles. Student then divides 28 by 12 to get 2.4 miles because interprets the remainder as 4/10ths of a mile. Student does not use ratio or rate reasoning to solve the problem and misinterprets remainder.

Student B: Student seems to notice that 12 minutes a mile x 2 = 24 minutes and 2 miles. They then try to figure out the extra 4 minutes (28-24). Rather than use rate or ratio reasoning, the student notices that 4 minutes is ⅙ of 24 minutes (24/4=6; though should be out of 12 minutes).

​

Early Ratio Strategies

Student F: Student notices 12:1 = 24:2, but then uses incorrect procedure to find solution.

​

Non-Proportional Strategies

Student E: Student uses multiplication and addition to solve. No evidence of ratio or proportional understanding other than perhaps in first multiplication.

51

Concrete → Representational → Abstract

Examples:

• Ratio table
• Visual model
• Graph on equivalent ratios on a coordinate plane (x,y)

Example:

Scaling a punch recipe with 2 parts ginger ale to every part juice.

Examples:

y=kx

​

sets up proportion

3 min = 8 min

14 gal = x gal

​

www.nciea.org

This work is licensed under a �Creative Commons Attribution 4.0

International License.

www.ccee-ca.org