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I’ve always been a sucker for a good metaphor. A well-used, provocative metaphor can go a long way to helping to both deepen and broaden our understanding of the world in which we live. So to say that Cathy Fosnot had me at landscapes of learning would be a bit of an understatement. In fact, as we continue to move through this nine-week #notabookstudy project, it is her idea that mathematical learning takes place across a landscape, and not along a straight and narrow highway, that has breathed so much life into the way that I see my own mathematics story and my growing relationship with the numbers, patterns and connections that are part of the world in which we live.

Beyond my own personal journey, however, the landscape metaphor has opened my eyes to the learning lives of my two boys, Luke and Liam. I’ve always seen Luke, age 10, as a bit of a conformist. He likes to know the rules regarding many things. He loves puzzles, codes and mysteries. He’s a figure-it-out type of guy. Liam, on the other hand, is a little bit of an enigma. He’s imaginative and curious and loves to dwell in a world filled with questions, possibilities and…well…costumes. Together, they’ve provided me with the perfect “testing ground” for some of the ideas and concepts that we’re talking about in this #notabookstudy.

On the way home from school last Tuesday, a few hours before our weekly broadcast with Cathy, I decided to try out some of the things that I had been reading about in Chapter 6: Algorithm vs. Number Sense. As we walked along, I posed a simple addition question to both of them at the same time. “What’s 34 plus 26,” I asked, realizing that I might have taken more time to place the question in some sort of context. But I was curious about one thing—how they would proceed in their computation.

They both came up with the same answer at the same time and so I turned to Luke first, asking him to explain to me how he went about arriving at the answer ’60’.

“Well, it was easy. I added the 4 and the 6, which gave me 10. I knew that I had to carry a one which gave me 3 + 2 + 1. So that’s 6 and the answer is ’60’.”  It was clear to me that he was using a mental image of the algorithm that he had been taught in school.

Liam waited patiently for Luke to finish and then he began his explanation. “I got the same answer as Luke, but here’s how I did it. I knew that 30 plus 20 was 50. I also knew that 4 plus 6 was 10. So I just added the 50 and 10 and got 60!”

I tucked the story away until I had the chance to share it with Cathy during our Tuesday night conversation. But it wasn’t until our broadcast was nearly over that it struck me. I became curious about the way Luke and Liam actually imagined the computation question in the first place. When Luke thought about my question, did he naturally stack the numbers, preparing to work his algorithmic magic, or did he first think of the numbers in a horizontal line? And how did Liam picture the same set of numbers and the same question?

I began to wonder whether the way that we hold numbers, number sentences and equations in our mind has an effect on the way that we go about figuring things out. So, the next morning, I asked the boys the same question, but instead of doing it “in their heads”, I asked them to write things down. And here are the results:

 

 

 

 

 

 

 

 

 

There’s likely a lot that I could say about these two images. I could wonder about whether Luke recognizes the power of place value in the algorithmic approach. I could marvel at the progress Liam seems to be making towards understanding the distributive property. (I could even ask why Liam continues to reverse his “5’s”). For me, however, the real power of these two images has to do with the assumptions that I make (and have made for many years) about what is actually going on in the mind of our learners. What happens in the spaces that occur between my communication of a mathematical idea, question or concept and how it is perceived and interpreted? How could the way a simple equation is written affect the way students start to think about it? How do different minds construct mathematical models?

I have to admit that I tend towards algorithmic thinking in my own life. If you must know, I’m a stacker. But what would happen if, as a parent, I took Cathy Fosnot’s landscape of learning seriously and made a conscious effort to get out of my own mind—my own ingrained way of thinking—and stepped into the mind of another traveler on the road?