Last month, we introduced the topic of our next few posts: the Science of Math instruction. This is an area of increasing importance, not just here in Florida, but nationally. The most recent National Assessment of Educational Progress (NAEP) data showed alarming trends in math, including a continued lack of post-COVID recovery with the greatest declines among students already performing in the bottom quartile. As written in a recent New York Times article: “The country’s lowest scoring students are in free fall.”
As such, we’ve been thinking a lot about math as of late, considering: What does cognitive science tell us about learning math? And how can we make these takeaways applicable and actionable at the school and classroom level?
But what does it look like to make math concrete in older classes or with more complex math topics?
One practice we all know and love in the earliest grade levels is making math concrete. This likely conjures images of objects or manipulatives used to represent numbers or situations, or, a bit later, the use of visuals or pictures to concretize story problems. There is solid research backing these strategies. Several leading studies highlight the benefits of using manipulatives in math instruction. Gersten et. al. (2009) demonstrated their positive effects on comprehension, especially for struggling students. Witzel and Allsop (2007) showed how manipulatives can build conceptual understanding. More recently, a meta-study by Schoenherr et. al (2024) on visualization strategies crossed the 0.4 “hingepoint” for high-impact teaching practices.
I recently saw a powerful example of this while observing master teacher Jose Cobo of True North Classical Academy, and wanted to share it here to inspire even those working in upper grades to consider implementation.
First, some context. In the lesson I observed, Mr. Cobo was rolling out convergent series to his Calculus BC class. Before the words “BC Calculus” and “convergent series” convince you to stop reading, hear me out! I am no math expert – yet Mr. Cobo’s instruction was both beautiful and accessible, largely because of the way he launched this content by making it concrete.
So let’s see how he did this, by completing the Do Now activity as students did:
What do you think? Did you say something like, “the sum of the area approaches 1,” or, “as you continue to take one half of the previous number and add it to your sum, the sum gets closer to 1”? If so, you’re well on your way to articulating the key takeaway of this lesson:
| When you add an infinite set of consecutive numbers as described by the series, and the sum approaches a number, we say that the series converges. |
Imagine if, instead of the visual Do Now activity, Mr. Cobo had launched his lesson with the notes page he eventually revealed to students:
If you’re like me, you might have felt some combination of confused, overwhelmed, and ready to give up.
In Your Classrooms
So try it out! Maybe your focus is elementary, and this serves as a reminder of the power of leveraging concrete objects or pictures to initiate learning. Maybe you work with older students, and Mr. Cobo offers a reminder that even concepts that seem most abstract can be made more accessible – more concrete – for our learners. With a little pre-planning, the practice can have a profound effect.