By Claire Riddell, posted July 5, 2016 —

While teaching first grade, I sought out a partnership with a fifth-grade teacher to co-teach some math lessons in both of our classes. This partnership was not required by an administrator or a district initiative; instead it came from our mutual desire to understand mathematics and pedagogy before and after our respective grade levels. After experiencing just a few lessons in fifth grade, I realized that this partnership was opportunity to see into the future. I was able to experience what my students would be doing in mathematics in just four short years. I couldn’t believe how much of what they were learning in first grade connected to what they were expected to do in fifth grade.

The next time I looked at the first-grade standards, it was if I had put on 3D glasses. The depth and connectivity of the standards became apparent, and the way in which I taught began to transform. During the fifth-grade lesson, I observed students making a line plot to display a data set of their fractional measurements (5.MD.2). I then thought about the work my first graders were engaging in by measuring objects with whole-number length units (1.MD.2), as well as organizing, representing, interpreting, and questioning the data we had collected (1.MD.4). Our first-grade conversations that occurred during the measurement unit became more precise, and the questions we asked of our data sets became more contextualized and rich. (And you had better believe the fifth-grade teacher’s instruction changed, too! But I’ll stick to my side of the story.)

Those of you who teach reading know how fortunate we are to have the Anchor Standards. If you want to know what your first-grade students will eventually be doing with their ability to “identify the reasons an author gives to support points in a text” (CCSS.ELA-LITERACY.RI.1.8), you can simply look at that exact standard number in every grade from kindergarten to grade 12. In doing so, you will ultimately see that students are going to be asked to “delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the relevance and sufficiency of the evidence” (CCSS.ELA-LITERACY.CCRA.R.8). This vertically aligned set of English Language Arts Standards allows teachers to see clearly the path before and after their grade-level standards.

In mathematics, our job is a bit more muddled. We are challenged to identify the salient ideas that our students must understand and then consider how these ideas grow and transform. After my time in the fifth-grade classroom, it felt unacceptable to know and understand only my own grade-level standards to teach students at the levels necessary to produce the strong, well-rounded young mathematicians I desired. It was time to embrace the exciting learning that elementary school mathematics has to offer.

So, you may be thinking, That’s nice, but this is a daunting task. How and where should I even start this journey of vertical content understanding?

I suggest first starting with three fundamental questions:

• 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?

Now here comes some good news! There are experts out there thinking about this who have created resources to jump-start our thinking. First and foremost are the Progressions Documents for the Common Core Math Standards made available through the University of Arizona. These narrative documents describe “the progression of a topic across a number of grade levels, informed both by research on children’s cognitive development and by the logical structure of mathematics.”

Additionally, you can access Dr. Jere Confrey’s Learning Trajectories for Interpreting the CCSSM that describes “how concepts, and student understanding, develop over time.”

With these two resources in hand, you are prepared to begin the work of thinking about preceeding ideas as well as what ideas are on the horizon as your students work through some really exciting topics in mathematics. By engaging in this work of examining how ideas vertically articulate, you position yourself to proudly claim your role as a teacher of mathematics.

Your Turn 

Please leave a comment (or reach out via twitter) with your experiences engaging in the mathematical ideas prior to and beyond your grade level. And check back in two weeks as we explore how I took the vertical understanding journey with a topic that transcends K–grade 6 mathematics: fractions!

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.