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?

Good questions! When students write what they’ve done it gives us a window into their thinking. As I read I was thinking about the large role working memory has in math. Both boys used some memory strategies to solve the math. I wonder if the strategies kids prefer is related to the strength of their working memory. When I only used the algorithm, I would often write the problem in the air as I pictured it. But I don’t need to do that when I picture a number line & use splitting.

You didn’t mention Liam’s age, but I am guessing he is younger.

I wasn’t there obviously, but more than likely their approaches are a reflection of their teaching up to this point.

The standard algorithm if well taught contains necessary knowledge re place value.

This is not a criticism of teachers, but a comment on the type of PD, resources available.

Standard algorithm is far more efficient and will work in all cases.

Liam’s method is the go to method of today. As a parent I would have some concerns.

Liam likely reverses his 5s because no one has insisted he stop. Just becomes a bad habit.

Guessing that Liam has just finished grade 2, I would be making sure he knows algorithm and has closure on this topic, rather than worrying about distributive property.

Standard algorithm which has been worked out by very great minds over centuries is crucial and independent of personalities of children

Sorry this sounds like a parent interview or sermon, it is simply a way of raising some points of math and math education.

It just happens to be about your children, so sounds too preachy.

Hi Stephen.

Both children produce fairly faithful demonstrations of procedures taught in standard resources and promoted in PD, though clearly one method is ascendant and prevalent these days and the other less in vogue, even in places denigrated or played down yet still known and taught — by experienced teachers at any rate.

This is further confirmed by the respective pencil-and-paper layout of their thoughts — these mirror exactly the typical representations of these procedures found in such resources for teachers.

It is therefore obvious to infer that you are not seeing the products of two child geniuses generating original mathematical processes, but the effective learning of two attentive pupils both well-taught and well versed in the respective methods favoured by their teachers.

Alternatively, they have perhaps selected preferred approaches among multiple such procedures they have learned. I see nothing in your anecdote to suggest invention on either child’s part. I see effective learning and I infer effective teaching in the backstory.

I do not see the basis on which you might infer that these “different minds construct mathematical models”. Perhaps different minds do so (I’m sure, in fact, that this is the case to some extent), but this anecdote provides no evidence of that.

If one posits that they had identical educational experiences perhaps one can say that it is evidence that the children are showing individual preferences by the choices they have made. This is not the same thing as what you said; or it necessitates interpreting your word “construct” in a sense different than used in current educational discourse.

With reference to our Twitter conversation the other day, I will point out that while Luke’s approach shows explicit use of place value, Liam’s does not — in the sense that matters the most.

The decomposition the latter uses is a return to the Roman model of representing benchmark values as unitary objects to manipulate, albeit clumsily, with ten being represented by a pair of symbols 10 rather than the singleton X (for example); and 60 representing six of these rather than the older XXXXXX or its abbreviation LX.

Observe where things end up on the page. The multiples of 10 are moved (arbitrarily, I argue) to the right and the single digit addition to the left. Arrows are required to show the flow of ideas and values — because the role of (ahem) *position* in a positional value is all but entirely eliminated as having anything to do with this calculation.

The former illustrates the modern concept of place value and its organizing effect on processes of calculation; the latter is a throwback to a 2000-year-old system, albeit expressed in a modern system of numerals. That does not mean it is invalid … indeed it is perfectly valid, and it served the Romans well enough (relative to their contemporaries) during the long ascendency of their civilization. But it also held them back (relative to later societies that rode the waves of enlightenment and scientific revolution).