1. Student Understanding of Equality
Core Math Partnership Project
Thursday October 23, 2014

2. A well-developed conception of the equal sign
A well-developed conception of the equal sign ... is characterized by relational understanding: realizing that the equal sign symbolizes the sameness of the expressions or quantities represented by each side of an equation.
There is general agreement that relational understanding of the equal sign supports greater algebraic competence, including equation-solving skills and algebraic reasoning.
Because algebra is an important gateway not only into higher mathematics, but also into higher education more generally, the importance of building high quality relational understanding of the equal sign is of critical importance.
(Matthews et al., 2012, p. 318)
Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3),

3. Proportion of Students Correctly Solving Algebraic Equations
One item required students to interpret the equal sign (see Figure 1), and two items
required students to determine the solution to an algebraic equation (see Figure 2).
What value of m will make the following number sentence true?
(a) 4m + 10 = 70
(b) 3m + 7 = 25
students who exhibited a relational understanding
of the equal sign were more likely than students who did not exhibit a relational
understanding to solve the equations correctly
The following questions are about this statement:
3 + 4 = 7 ↑
(a) The arrow above points to a symbol. What is the name of the symbol? (b) What does the symbol mean?
(c) Can the symbol mean anything else? If yes, please explain.
For example, 82% of the students with relational understanding of the equal sign solved the equations correctly, compared to only 28% of the other students.
Equations: 4m + 10 = 70 and 3m + 7 = ( 177 students: 47 sixth, 72 seventh, 58 eighth)
Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for research in Mathematics Education,

4. Article: Why the Common Core Changes Math Instruction by Valerie Faulkner
Faulkner, V. N. (2013). Why the Common Core changes math instruction. Phi Delta Kappan, 95(2),

5. Eliminate this old habit
Defining equality as “same as.”
For example, don’t say: “Remember students, the equal sign means same as.”
Adopt this new habit
Define equality as the “same value as.”
You might say this: “Remember students, the equals sign means the “same value as.” The two values do not have to look alike, but they will have the same value tells a different math story than 4 + 3, but we know that they will both yield the same value of 7, so they are equal. Are they exactly the same? No, but they are equal.”

6. Why?
The definition “same as” is mathematically incorrect and leads to misconceptions.
Equals means that two things are the same based on one attribute — their quantitative value.
Just as red or rough is an attribute, a thing’s quantitative value also is an attribute.
It is as absurd to say that is the same as as it is to say that a red truck is the same as a red lollipop. In the former, they have the same value; in the latter, they have the same color. In neither instance are they the same thing.

7. Shifting our Language as we talk about the Meaning of the Equals Sign
With a partner, practice your professional language:
It is incorrect to say, “the same as” because...
It is correct to say “the same value” because...

8. Equality Project Part A

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

10. 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 =

11. Project Part A: Check in on your students’ current levels of understanding equality.
1. Give your pre-assessment to your students (can be all or some items).
2. Prepare a brief report that shows the results in tables or graphs and as a written paragraph summary.
3. Bring 3-5 work samples that shows a range of student understanding.

12. Protocol for Structured Sharing
Step 1. Person 1 shares overall results (2 minutes).
This is what I gave to my students . . .
What I learned about my students . . .
Step 2. Person 1 shares student samples (2 minutes).
Here is an example of a student that lots of my students solved it this way.
This student showed solid relational understanding only some of my students reasoned this way.
This student’s work was really interesting to me because....
Step 3. Each colleague provides at least one comment, observation, or wondering. (2 minutes total)
Repeat with another person presenting.

13. Across the Grades
Prepare a short summary (2-3 points) of the major trends and findings across your grade band to report out to the whole group.

14. Now What: Next Steps
As you think about your students’ current level of understanding (or lack of), discuss your plans or brainstorm ideas for next instructional steps to get your students to a strong level of relational thinking about equality by March.

15. Equality Project Part B

16. Log and Post-Assessment
1. Log: 8-10 entries (e.g., twice a month) on students’ developing understanding of equality.
2. Artifacts: Collect student artifacts (e.g., written work, video clips, photos of charts) that provide evidence of shifts in student understanding.
3. Post-Assessment: Assess students in Feb/March.
4. Report/Portfolio/Binder: Analyze results, compile portfolio: log, artifacts, assessments, student results, and written final summary and reflection. (Due March 26.)

18. 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.

19. 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.