When I was in 5th grade, I was asked to find the quotient of 8,024 ÷ 17.
Would you be surprised to learn that I have never been asked to solve that same problem again -- not even once! (Probably not surprised, are you?). Back when I was learning about multi-digit division, I could have practiced and rehearsed to memorize the solution, but that would have used up precious storage space in my brain that could have been available for something else.
The truth is, we do not ask students to solve, memorize, and store things like the quotient of 8,024 ÷ 17 because good mathematics instruction focuses on the patterns and relationships of numbers, not just isolated specific elements. Good mathematics instruction is not about specializing, it is about GENERALIZING.
Let's connect this idea to something a bit more universal in nature to illustrate the importance of generalizing even more: Every day, we turn on lights and open doors without much thought or effort. Opening the front door to your home is much like opening the front door to your friend's home, or the door at your favorite bookstore, or even the refrigerator door. We recognize value in generalizing the process of opening a door in order to apply that information to other doors that we encounter. The doors do not have to be in the same location or even open in exactly the same way. If we had to learn, memorize, and store information about how to open each individual door we encountered each day, we would spend most of our time just figuring out how to get inside. The importance of generalizing mathematical concepts is often an undervalued lesson. We either expect our students to automatically generalize information or we simply overlook the essential role that generalizations play in learning mathematics. We should introduce students to mathematical concepts beginning with things that are simple and then move toward the underlying generalizations in order to help our students better understand the patterns and relationships behind what they are learning. My own 5th grade teacher recognized (so many years ago) that the value of the lesson was not simply knowing that the quotient was 472, but rather, the value was in understanding the relationships of the numbers and generalizing both the process and my conceptual understandings so I could go on to divide any pair of values.
Before I share specific examples illustrating the importance of mathematical generalizations, I’d like to thank my wonderful thought partners and share a bit about our recent idea sharing session:
Every now and then I hear people talking about how they had an idea and scribbled their thoughts on a bar napkin. Well, my notes for this month's post are scribbled on a cardboard coaster with mathematical models and notes weaving in and out among the water rings caused by the condensation of my drink -- actually from my colleague's ice tea. I am at an educational conference this week and was thrilled to be surrounded by like-minded colleagues who were willing to talk about instruction as we tossed around ideas over a three hour dinner. Ideas were flying and I didn't want to lose a single one of them -- the napkins were cloth, so I reached across and grabbed Julie's drink coaster 😊 Thank you for being thought partners with me. I am grateful for the specific examples that stemmed from the wonderful academic conversations I had with Julie, Jason, Candace, Kristin, and others at the table.
So let’s call this next segment "The Coaster Notes”
Below are a few of the ideas we discussed as we shared each of our ideas about the critical importance of mathematical generalizations.
When we teach early skills in decomposing numbers, we are actually preparing students for subtraction with regrouping in later years. For example, students learn that 47 = 40 + 7 but 47 also equal 30 + 17. It is this second decomposition of 47 that will be essential when we are later asked to solve 47 – 28. The model shown below shows the progression of how 47 – 28 might be approached across various grade levels until students are simply using what we've come to know as the standard algorithm. Why didn't we just start with the standard algorithm? Well, it is important that student have a conceptual understanding of the process, so they can work more flexibly and fluently when using these types of calculations for real-world applications.
The work we do at the elementary level has far-reaching implications as students enter higher levels of mathematics in middle school, high school, and beyond.
With every lesson we teach, it is critical that we help our students discover the generalizations that create the patterns and relationships of the mathematics they are doing. When a student notices that the sum of an even and an odd integer always results in an odd integer, for example, that student is generalizing. Generalizations allow students to think about computations independently of the particular numbers that are used. Without this, and many other generalizations made in mathematics from the early grades, all work in mathematics would be cumbersome and inefficient [CA Digital Chalkboard].
Assess how much you talk with your students about the generalizations of their math work. If you are spending a large chunk of instructional time discussing specific solutions, you are most definitely missing the proverbial boat! Mathematics is about the patterns and relationships of numbers - to see these, we must look for the generalizations that create them. Set a goal to increase the amount of time you dedicate to discussing the generalizations and less time focused just on correct solutions.
