1. Equality Project Part A Observations
Core Math Partnership Project
Thursday November 6, 2014

2. Level 1, Rigid Operational
Most students demonstrated understanding of Level 1, however some inconsistencies when the unknown was in different positions.
5 + 7 = ¨ x 8 = ¨
¨ + 7 = ¨ x 8 = 56
5 + ¨ = x ¨ = 56
Conclusion: Students are comfortable when the answer is on the right side of the equal sign, and need more experiences with the unknown in various positions.

3. Level 2, Flexible Operational
Most students were at Level 2 and not beyond.
Many students showed acceptance of having only one number on the left side of the equal sign if there was an operation on the right side.
¨ = ¨ = 7 x 8
12 = ¨ = ¨ x 8
12 = 5 + ¨ = 7 x ¨
However, students often commented that the equation was “flipped” or “backward” implying something was not quite right.

4. Level 2, Flexible Operational
Grade 4
Grade 5

5. Level 2...
However, most students were perplexed when there was no operation sign for: 9 = 9 m = m One student said: “This can’t exist.”
Grade 4

6. T or F: 9 = 9
Grade 4

7. T or F: 9 = 9
Grade 4
Grade 5

8. Level 2... How about, y=x ? Yet, many students accepted:
½ = ¼ = 25% = 0.50
Wondering...
It seems “9=9” is more abstract and reveals a shift from Level 2 to Level 3 understanding.
It seems “m=m” requires an understanding of variables and equality....
How about, y=x ?

9. Level 3, Basic Relational (computational approach)
Few elementary students were able to work successfully with equations that had operations on both sides of the equal sign.
Gr 2: ¨ + 6 = “Six out of 21 students answered this correctly.” (29%) Wrong answers included: 21, 24, 22, 28.
Gr 3: “None of my students were able to consistently solve problems in which there was a number sentence on both sides of the equal sign.”

10. Level 3 True or False: 11 – 5 = 11 – 3 – 2
Students were more successful with true-false statements than when solving for an unknown. Gr 4: Correctly solved by 13/21 students (62%). Gr 5: Correctly solved by 12/25 students (60%).
Almost all students with correct answers used a computational approach.

11. Level 3: = 13 + ¨
Gr 4: Correctly solved by 7/21 students (33%).
Gr 5: Correctly solved by 6/25 students (24%).
Grade 4

12. Level 3: = 13 + ¨
Grade 5
When successful, students demonstrated understanding of equality, and demonstrated making sense of the operations, understanding relationships among the quantities, and use of reasoning and problem solving.

13. Level 3 w/variable: 45 + p = 44 + 23
Gr 4: Correctly solved by 9/21 students (43%).
8 students solved with a computational approach.
1 student saw a short cut (p is 1 less than 26).
Most students were confused by “p” as an unknown value.

14. Level 3 w/variable: 45 + p = 44 + 23
Computational approach, used lots of “try and revise.”
Fascinating. Ready for some discussion, easy to see one more and one less relationship.

15. Level 4, Comparative Relational (using short cuts)
Student reasoning based on using “short cuts” and relationships among the quantities was rare.
Grade 5
The only way to assess whether students are at Level 4 is by asking students to explain their reasoning.

16. Level 4
Grade 4
Grade 4

17. Level 4
Keep in mind. . .
Not all problems lend themselves to reasoning with short cuts (relational approach) and students need to be able to solve equations using a computation approach.
Overall goal is sense-making, flexibility in using strategies for solving problems, and confidence as mathematical thinkers.
Use of “short cuts” shows deeper conceptual understanding, use of structure, and quantitative reasoning.

18. Reflections: Post-Assessment
Pose fewer problems — “Less is more.”
It is fine to select about 4-6 tasks.
Ask students to explain their reasoning.
Consider asking students to write out in words what the equal sign means.

19. Grade 4

20. Instructional Thoughts
Ten Minute Math – Number Talks
Anchor Charts
Balance Scales – Use Physical & Visual Representations of Quantities, not just Symbolic.

21. = 13 + ¨

22. Next Steps... Keep Going! 1. Log: 8-10 entries.
2. Artifacts: Student work, video clips, photos of charts, etc.
3. Post-Assessment: Feb/March
4. Report/Portfolio/Binder: Due March 26

24. Misc

25. Levels of Understanding Equality
Level 1. Rigid Operational Level 2. Flexible Operational Level 3. Basic Relational Level 4. Comparative Relational

26. Levels of Understanding Equality
Level 1. Rigid Operational: Can solve equations or evaluate true-false statements successfully that only have operations on the left side of the equal sign. Level 2. Flexible Operational: Can successfully solve equations with operations on the right side of the equal sign or interpret statements that have no operations. Level 3. Basic Relational: Can successfully solve or evaluate statements with operations on both sides of the equal sign, and explain or give correct definitions of the equal sign. Level 4. Comparative Relational: Can successfully use short- cuts (e.g., compensation strategies) and properties of the operations to solve equations or evaluate statements.
Levels of Understanding Equality
15 – n = 12 – 2
T or F: 3 x 16 =

27. Equality Project

28. Project: Student Understanding of Equality (Part A due Oct 23; Part B due Mar 26)
Part A. Pre-Assessment
Implement your pre-assessment developed this summer to gauge your students’ current understanding of equality. Summarize the results of the pre-assessment on individual items and make connections to the levels of student understanding of equality studied this summer. Prepare a written report which may consist of tables and graphs along with a brief summary of your findings. Bring approximately 3-5 samples of student work that shows the range of student reasoning in your classroom to our project session on October 23, along with your written report. Class time will be provided for a structured discussion of the student work and your initial insights and findings.
Part B. Ongoing Log and Post-Assessment
Keep an ongoing log throughout the year related to developing an understanding of equality with your students. Some questions to consider for your log: How do you purposefully shift your writing of equations and questioning throughout the year? How are your students growing in their understanding of equality? The expectation is that you make at least 8-10 entries throughout the school year, approximately 1-2 entries per month. As appropriate, collect student artifacts (e.g., written work, video clips, photos of charts) that provide evidence of shifts in student understanding. Give the post-assessment (or perhaps interim assessment) in March to check in on student understanding. Compile the results, compare it to the pre-assessment, and prepare a final report. It might be easiest to compile the information in a binder with sections for the classroom log, student and teacher artifacts, pre- and –assessments, student assessment results, and final summary and reflection. More details will be discussed in class.

29. Disclaimer Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.