Equality

The Grade 2- 10 MIPI+ question:

Every grade contains the exact same MIPI+ Equality question. Unlike the other MIPI+ questions, there is absolutely no variation from grade to grade.

Make this statement true: 8 + 4 = __ + 5
Show how you figured it out.

This was done deliberately for two reasons. First of all, it identifies misconceptions students have around the equal sign. Secondly, it allows for a longitudinal study of students’ understanding of the equal sign. If a student, group of students, class or school has a strong misconception of equality, how does that change over their schooling? We have an opportunity to determine which high leverage strategies are working to build understanding. If a student, group of students, class or school really understood the equal sign and then, at some point, demonstrates misconceptions, we have an opportunity to delve deeper into the conditions that may have caused that to occur.

A conscious choice was made not to include variables on the MIPI+ question in higher grades. We felt that once you included the variable, students thought about the question more procedurally, solving it using an algorithm, rather than conceptually. After administering this question, teachers are more than welcome to include another similar question, such as 6 + 9 = x + 3, to provide students with an opportunity to demonstrate their procedural understanding.

Students must have a relational understanding of the equals sign. “The equal sign represents an equivalence relation between two quantities – what’s on the left side equals the right side.” (Knuth et al, 2008) An incorrect operational understanding of the equal sign will interfere with students’ algebraic reasoning.

Students need to understand:

The equal sign represents an equivalence relation between two quantities. ie. The left side of the equals sign and the right side of the equals sign are the same value.

What is the misconception that will be addressed?

Students incorrectly view the equal sign as an operational symbol. ie. When students have an operational understanding of the equal sign, they believe that the equal sign means “Add the numbers” or “The answer comes next”. Incorrect responses to 8 + 4 = __ + 5 include: 8 + 4 = 12 + 5 or 8 + 4 = 12 + 5 = 17

How is the misconception developed?

Review the resources you use when asking students addition, subtraction, multiplication, and division questions. How often are they structured as follows? # + # = ?, # – # = ?, # * # = ?, # ÷ # = ?

When students only see equations written in the format shown above, they often develop the misconception that the answer is always stated on the right side of the equals sign.

Key Ideas around Equality from ARPDC

What is Conservation of Number?

What is “Conservation of Number?”

Conservation of Number is the idea that the number of objects remains the same even when rearranged.

Misconception:

Students may believe that moving objects by increasing or decreasing the space between them increases or decreases the quantity respectively.

 

The importance of this concept:  

Students who do not understand conservation of number will struggle with place value and operations.  Imagine a student using base ten blocks to subtract. When they have to exchange one ten for ten ones, they will have to recount the blocks each time. When adding 4 + 3, they may need to “count all”: 1-2-3-4…5-6-7

High Leverage Strategies:

Please note that the processes for implementing the strategies provided are scripted. This is NOT to script you but to provide you with a possible process for implementing this strategy. Please adapt to your style and the needs of your students!

High Leverage Strategy 1: Seeing “objects” to build Conservation of Number and Equality

 

Further Learning:

Read the article Understanding the equal sign matters at EVERY grade (Lorway, 2017) to dig deeper into answering the following questions:

  1. What is Equality?
  2. What does the equal sign mean?
  3. Why is Equality important?
  4. Why begin with Equality?
  5. What’s a teacher to do? There are great, simple strategies included here that focus on the use of dots and manipulatives. These can be used starting in Kindergarten, though you wouldn’t expect students to formal symbolic representation.
  6. Why should I be cautious about the “balance” metaphor?

Watch the webinar “Insight into Equality” (ARPDC) and use their Webinar Guide to think about Equality.

Explore ARPDC’s Elementary Mathematics Professional Learning site, focusing on its “Insight into Equality” section.

More Readings:

Behr, Merlyn, Stanley Erlwanger, and Eu-gene Nichols. (1980) How Children View Equality Sentences. PMDC Technical Report, no. 3. Tallahassee, Fla.: Florida State University, 1975. ERIC No. ED 144 802. Retrieved from https://gpc-maths.org/data/documents/doks/behr-howequal.pdf

Carpenter, T. P., Levi, L., & Falkner, K. P. (2000). Children’s Understanding of Equality: A Foundation for Algebra. (2000). Retrieved from http://ncisla.wceruw.org/publications/articles/AlgebraNCTM.pdf

Falkner, K. P., Levi, L., Carpenter, T. P (1999). Teaching Children Mathematics, Teaching Children Mathematics, v6 n4 p232-36 Retrieved from http://ncisla.wceruw.org/publications/articles/AlgebraNCTM.pdf

Knuth, E., Alibali M. W., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2008). The Importance of Equal Sign Understanding in the Middle Grades. Retrieved from http://learning.arpdc.ab.ca/pluginfile.php/30793/mod_resource/content/1/ImportanceOfEqualSignUnderstanding.pdf

Knuth, E., Stephens, A., McNeil, N., & Alibali, M. (2006). Does Understanding the Equal Sign Matter? Evidence from Solving Equations. Journal for Research in Mathematics Education, 37(4), 297-312. Retrieved from http://www.jstor.org/stable/30034852

Powell, S. (2012). Equations and the Equal Sign in Elementary Mathematics Textbooks. The Elementary School Journal, 112(4), 627-648. doi:1. Retrieved from http://www.jstor.org/stable/10.1086/665009 doi:1

References:

ARPDC. (n.d.). Elementary Mathematics Professional Learning. Retrieved from https://learning.arpdc.ab.ca/course/view.php?id=351

Carpenter, T. P., Levi, L., & Falkner, K. P. (2000). Children’s Understanding of Equality: A Foundation for Algebra. (2000). Retrieved from http://ncisla.wceruw.org/publications/articles/AlgebraNCTM.pdf

Lorway, G. (2017). Understanding the Equal Sign Matters at Every Grade. Delta-K,54(1), 39-43.