Chapter 3:
Shortly after posting Big Idea 2, I decided that it would be useful to add another example that featured how learning is about making connections.
“Boys and girls, here is our next Big Idea. Multiplication and Division are connected. This idea reminds us that learning is about making connections. It also reminds us to look for those connections. When we find new connections, we own them. They are ours because we discovered them. Our learning is stronger when we make the connections ourselves.
“Here is an example.”
On the board I wrote another graphic that has long been used in many classrooms.
Again, I launched into a series of questions designed to cause us to seek connections and relationships.
“Learning is about making connections. Let’s see what we can learn from this image. We know that multiplication and division are connected, but how can that lead us to deeper, more powerful connections?”
Then, as so often happens to all of us, the unexpected interrupted the anticipated.
“Mr. Wyborney, you also told us that repeated addition is connected to multiplication. If addition is connected to subtraction, and it’s connected to multiplication, then does that mean that subtraction is connected to multiplication?” A powerful question! I began adding layers to the image, not because I knew the answer, but because I was exploring, trying to learn.
As we explored together, we found interesting connections, but we did not know if what we found were examples of something that was always true. We had made the beginning of a connection but were still grappling with how to make sense of the connection, how to attach it to our current learning.
Instead of finding an answer, we had reached a point of wondering about what might be true. We were in the process of testing a connection and wondering what it might mean. We needed more examples to help us wonder more clearly.
Notice how we began developing more connections and eventually began tying those connections to adding — rather than subtracting — one to each of the factors. We were primed to wonder why. We were ready to make more connections. I wanted my students to understand that connections were readily available if we pursued them. The pursuit of learning, the search for connections, and the hunt for understanding were becoming prized in our classroom.
I posted the first graphic again and restated the question: “Is subtraction related to multiplication? Let’s look for connections. I want you to find the difference between all of the numbers within the triangle. Now tell me what you notice.” Soon, our classroom began to look like this:
Finally, I announced: “I think this relationship may always be true. But I don’t know. I think that, when we look at a multiplication fact family triangle but determine the three differences, two of those differences will always add up to the third difference.” Quickly, I sketched three more examples on the board.
Connections are not always apparent. They do not always come quickly, and sometimes they require a great deal of wondering.
We had not reached a conclusive point of learning. Instead, we had arrived at a better place — a place of wondering.
Again, the students saw the connections between multiplication and division, but much more importantly, they saw that many, many connections were waiting just below the surface. Those connections were within reach if we pursued them.
Perhaps most importantly, the teacher was beginning to make connections. I had no particular expertise with any of these ideas. I simply had a willingness to dabble and discover. My most empowering instructional move at this point was simply that I openly looked for connections myself, and I was determined to allow those connections to help me understand more about learning.
I believe the students also sensed that, while I was going to point out some examples of connections, I wasn’t going to make all of the connections for them. Learning is not solely about discovering connections other people have made. It is much more than that. It is about reaching for and establishing your own connections.
Learning is an act. It is an effort. It is a deliberate choice. For us, a key part of learning was choosing to actively seek and build those connections.
This particular Big Idea grew vastly over time. In fact, its growth was part of what inspired Big Idea 17, which I’ll detail later. Big Idea 3 was meant to be an example that allowed us to gain a foothold on the idea of seeking connections at deeper and deeper levels. Little did I know that it would provide such rich opportunities all on its own.