By Jamie Duncan, posted November 7, 2016 —

So, have you attended any math parties yet? In the previous post, I described the value that teachers of primary-grade students gain from going to upper-grade math parties with the intention of stealing the learning about mathematical content and instructional practices. But how do you really do that? First graders aren’t going to solve systems of equations or use the quadratic formula! One idea, when attending professional development offered either above or below your own grade level, is that we must link what we are learning to the Common Core Learning Progressions. Keep your eye out for the big ideas of the mathematics you’re learning. Ryan Dent helped me understand this as, “What is it you want your students to understand about math?” versus “What do you want students to be able to do as a result from this understanding?” For example, if you attend a training on algebraic reasoning, you are likely to hear about the importance of relationships, equality, or equivalence.

How do these ideas translate into my first grade classroom? In what ways can I open up my students’ thinking that will allow them to build a strong foundation for algebraic reasoning? I have yet to see a publisher’s curriculum that really takes its time to develop the idea of equality with our young learners. In fact, the late Van de Walle said, “The equal sign is one of the most important symbols in elementary arithmetic, in algebra, and in all mathematics using numbers and operations. At the same time, research dating from 1975 to the present indicates clearly that “=” is a very poorly understood symbol.” (Teaching Student Centered Mathematics, Van de Walle et al., 2013, pp. 230-31). Van de Walle gave this example: 8 + 4 = __ + 5.

How do think your students would respond? Really, take a second to anticipate their responses.

In “Early Childhood Corner: Children's Understanding of Equality: A Foundation for Algebra,” “Falkner, Levi, and Carpenter state that “no more than 10 percent of students at any grade from 1 to 6 put the correct number (7) in the box. The common responses were 12 and 17. In grade 6, not one student out of 145 put a 7 in the box” (TCM 1999, vol. 6, no. 4).

As primary teachers, we have the power to change this, and yes, it is in our standards! Take a look at the first-grade California Common Core content standard 1.OA.7: “Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.”

California Mathematics Framework, a narrative of California’s Common Core State Standards for Mathematics (K–6), states it like so: “Students need to understand the meaning of the equal sign (1.OA.7s) and know that the quantity on one side of the equal sign must be the same quantity as on the other side of the equal sign. Interchanging the language of equal to and is the same as, as well as is not equal to and is not the same as, will help students grasp the meaning of the equal sign” (Grade 1, p. 100).

So, where would you begin? What if, instead of frontloading and telling kids that equal means is the same as, we designed a lesson in which they begin to learn and understand that equality really represents an equivalent relationship between two quantities, and then tie to it the symbol of the equal sign? I think Graham Fletcher gives us a great place to start.

How different would your students’ understanding of equality be if they had rich experiences with tasks like these in comparison with being told the meaning and then given facts to practice? Can you think of a way to use the example from Van de Walle that gives all students access? I created the visual below to use as a task to work on the idea of equivalence after my students had worked through Graham’s Balancing Numbers task.

I started this task with a “What do you notice? What do you wonder?” Then, I used the framework of the 5 Practices for Orchestrating Productive Mathematics Discussions to facilitate the lesson (you can read more about these in Zack Hill’s posts about this very topic). Here are a few student work samples from my lesson:

The common misconception at the top of figure 2 was a wonderful learning opportunity for the class, allowing us to talk about what equal is and also what it is not. Students had a great discussion disproving this common misconception and that led to wonderful metacognitive reflections in their math journals. I take Joe Schwartz’s advice when responding to math journals and use a Notice/Wonder format that encourages students to reflect deeply and strengthen their own understanding.

Your Turn

How do you help primary students build their understanding of equality? Please share your tasks! It would be great to create a bank of tasks around this big idea. What have you stolen from other grade levels? How did it impact student learning in your classroom? Please share in the comments section below or reach out on Twitter (@jamiedunc3 and @TCM_at_NCTM)

Jamie Duncan has served as a classroom teacher for fifteen years. She is a master learning facilitator in her classroom engaging all students in the Standards for Mathematical Practice through 3-Act Tasks, facilitating meaningful discourse, Number Talks, and building procedural fluency from a foundation of conceptual understanding. Jamie is a contributor to math educators around the nation through the Math Twitter Blogosphere (MTBoS). She writes at www.elementarymathaddict.com, where she shares her learning journey and works together with teachers from across the globe. Her passion for meaningful learning has led her to present for her school district, the California Math Council–South, and NCTM’s Annual Conference. She is interested in learning more about student thinking and how that grows to mathematical fluency.