By Claire Riddell, posted July 18, 2016 —

In my previous post, I shared the story explaining the beginning of my journey on the importance of a vertical content understanding. If you haven’t read that piece, you can check it out here. As conversations with my fifth-grade counterpart continued, we started to discuss some of the questions I had posed the previous week:

• What concept, big idea, standard, or domain is the most challenging for my students?
• What can I do to find out more about what my students already know about this concept?
• What specifically about this concept is challenging for students?

As we worked collaboratively to answer these questions, one of the big ideas that came to the forefront was fractions. We began by looking for standards that addressed fractions in first grade. I noted that although first grade does not have the Number and Operations–Fractions domain, a standard within the first-grade Geometry domain, 1.G.A.3., conn ects to the idea of fractions.

1.G.A.3 was not my favorite standard. Within the Geometry domain, our classroom explorations about attributes and the composition and decomposition of shapes ( 1.G.A.1, 1.G.A.2) typically led to rich discussions and generalizations about the nature of these geometric ideas. 1.G.A.3 seemed disconnected from the other Geometry standards. It seemed to be a very skill-driven standard that did not allow for the depth of mathematical thinking that my students had become accustomed to.

We decided to take a look at how this seemingly simple standard in first grade grew and transformed as students move through the grade levels. We deliberately and methodically examined each standard that contributed to students’ understanding of fractions from first to fifth grade. In particular, we turned our attention to what in the standards represented new learning about fractions and which standards were building on an already-established idea or understanding. For more details, see table 1, where we list the all the standards we identified that were directly related to fractions. The text in blue represents a standard or part of a standard that we felt represented new learning from the previous grade level.

2016-07-18 Riddell-Table1

We analyzed which grade levels made the biggest jumps in demand from the previous grade level, and we took the time to discuss and even practice what each standard was asking students to do. We talked about what a student’s work would look like and how their explanations might sound if they understood the standard. We also examined what errors or misconceptions a student may have with regard to a particular standard. We tried to think of how this progression of ideas is meant to demonstrate depth and coherence around a single idea.

As we worked through these ideas, my own understanding of fractions evolved, and I uncovered a new-found appreciation for the mathematics in my own grade-level standards. Consequently 1.G.A.3 was not longer a dreaded standard. I now viewed 1.G.A.3 as a cog in some great machinery.

These conversations convinced me to curb my overreliance on pizzas and cakes, and incorporate more rectangles (roads, granola bars, pieces of wood, etc.). By incorporating more linear contexts, I hoped to give my students a smoother transition to fractions on a number line. I was also able to consider which essential understandings were on the horizon for my students and incorporate more explorations and conversations that emphasized the importance of equal shares, equal wholes, and comparing fractions based on the number of equal shares of the identical whole. The possibilities for our discussions of halves and fourths became endless, and my ability to question students to push their thinking forward was brought to new levels.

I talked about 3D glasses in my last blog post. With the work of tracing and analyzing a mathematical idea or concept through the grade levels, the depth and connectivity of the standards became apparent to me and my instruction transformed. This may seem like a daunting task, but for me, it was one that was steeped in rewards for teaching and learning. Let’s no longer settle for teaching an isolated set of standards and instead embrace our role as teachers of mathematics!

For a related perspective on these ideas, check out Graham Fletcher’s ShadowCon16 talk. 

Claire Riddell is an assistant in research for the Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) at Florida State University. Currently, she is serving as the Region IV Director for the Florida Council of Teachers of Mathematics (FCTM) and is an active member of the Duval Elementary Mathematics Council (DEMC), FCTM, and NCTM. She is interested in young children’s counting experiences and investigating instructional frameworks for mathematics.