I have a question for all you math bloggers out there. HOW DO YOU DO IT?! How in the world do you find the time to sit down and write a post at least once a week? More importantly, how do you make it sound so *perfect?! *UGH! You’re like those people I went to school with– the ones who could whip out an flawless, on-demand five-page literary analysis in 3.2 seconds. How do you do it? How do your words just come out sounding smart and perfect? I honestly don’t know which one is harder for me– finding the time to sit down and write a post or the actual writing of the post. The former is near impossible, while the latter is simply excruciating. I wish I could make the ideas just flow and not have to rethink every second word.

You’ve probably used or at least heard of the voice-to-text app, right? Oh, what I wouldn’t give for a brain-to-blog app– an app that automatically records your thoughts into a blog. Yep, that needs to be invented yesterday. I’d have about 20 thousand posts by now because I’m constantly *thinking *about what I’d write in a post. Sitting down and actually translating those thoughts to coherent sentences is another story…

This is the second day (well, night, actually) that I’ve attempted to compose this post. Let’s hope that baby stays asleep long enough this time for me to finish. So a few nights ago I had the tremendous pleasure and privilege of learning from the brilliant Tracy Zager in her webinar, *How Do They Relate? Teaching Students to Make Mathematical Connections,* hosted by Global Math Department. Wow. That’s about all I can say. *Wow wow wow. *I am not articulate enough to describe the awesomeness of her presentation. I can, however, share a few humble thoughts.

In her presentation, Tracy emphasized the importance of relational thinking in mathematics. Oftentimes, students perceive math concepts as discrete topics rather than as a “landscape of interconnected concepts.” The term *landscape* immediately made me think of Fosnot’s landscape of learning (click here for an example). Similar, yet different. Anyway, Tracy asked us to think about two concepts that our students might have trouble relating i.e. fractions and division. Having just wrapped up a unit on developing multiplication with my 2nd graders, I thought perhaps addition and multiplication might be a good pair, although I felt like my students made a fairly strong connection between the two and wouldn’t really have trouble relating. Tracy then gave us the sentence stem, “What does ____ have to do with ____?” where the blanks are intended to be math concepts. In my opinion, this is a fabulous way to guage the students’ depth of understanding of interrelated concepts. I will definitely add this to my routine next year!

So Tracy challenged us to go back to our classrooms and pose this question to our students. Yikes! I felt both eager and apprehensive about how they would respond. Eager because, well, my kids have a way with words that can either bring you to tears with laughter or stun you to silence (in a good way). Apprehensive because…what if they didn’t know how to respond?! What if they didn’t see the relationship?! How would that reflect on my teaching???

So, a few days later, I posed the following question to my students– *What does addition have to do with multiplication? *I sent them off to reflect individually and record their thoughts on their iPads. Some wrote equations. Some drew pictures. One student told me, “You’re adding on for both.” Then he proceeded to draw an open number line, and illustrated ‘adding on’ by taking a jump from left to right. “See? When you add, you’re adding on. Multiplication, it’s like you keep adding on. So with both, you’re adding on.”

I brought the group back together to share their responses. Here’s what they came up with:

I thought the ideas of ‘remembering’* *and *‘*memorizing’* *were interesting, however my favorite response described addition as a ‘backup’ for when you don’t have a certain multiplication fact memorized. This student didn’t stop there. He went on to state matter-of-factly…

“…they’re kind of like continents…”

In my mind I was thinking, “WHAT?!” “Tell me more,” I said.

Slowly and thoughtfully, he continued, “So, addition is like a continent, and multiplication is a country in that continent…”

I couldn’t make this up if I tried. Read on.

“…and subtraction is the opposite, so it’s another continent…subtraction is like… Australia.”

I couldn’t help but laugh out loud. After all, I’ve never heard of a mathematical operation being likened to a massive landmass. Of course, I had to ask,”What about addition? Which continent is addition?”

He thought for a moment, then said,”Addition is like…like Asia.”

I was used to this student’s sagacious remarks, but regardless of his poetic analogy, I wondered if this is what Tracy was referring to when she spoke of students overgeneralizing, or attempting connections that don’t hold mathematically.

So what do you think? If subtraction was a continent, which continent would it be?

Mona, I feel I have to take the role of “mentor” (ha!) seriously, and be critical as well as appreciative. I wouldn’t normally do this of course. But in this case I have to be honest. You’re being a little disingenuous. You write about other bloggers, wondering how they do it and asking, “how do you make it sound so perfect?! UGH!” And then you turn round and write a perfect post yourself. Nobody is fooled. Don’t think we won’t notice, we who have been trying out “notice and wonder” for a while now.

The webinar sounds great, and I really like that “What does ____ have to do with ____?” question. I’m going to have to use that. I’m looking forward to Tracy’s book coming out!

I love the continent-country analogy. Partly just because it is an analogy. I think metaphor is such a powerful tool for thinking, for linking ideas, for holding structure in mind. (And I know that in English class they say if there’s a “like” there, it’s simile not metaphor, but I see metaphor as a continent, say South America, and simile is, like, Sao Paulo.) I always try to make my students conscious of metaphor as an all pervasive thing. Often I start with the shape of a bit of orange peel, and say “What is it for you?” The students always see it as so many things:

http://year4atist.blogspot.fr/2013/10/metaphor.html

In this case, it allows an idea to come into the open. It’s crying out to be drawn as a (map-shaped?) Venn diagram: “So is multiplication a part of addition like this…?” (This is an interesting one, for us as teachers, because almost all of the students’ experience of multiplication will be translatable into addition terms. In secondary it reveals itself as a different continent sometimes – how would you do pi x pi as addition? But that’s for later on…)

Thank you Mona for burning the midnight oil for us and crafting another great post. You’re brain must be a tidy place: a brain-to-blog would never work for me! In the meantime, keep using the old method!

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I hope you do find more time to continue blogging. Thanks for this great share!

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I love this post so much!

He said “So, addition is like a continent, and multiplication is a country in that continent…and subtraction is the opposite, so it’s another continent…subtraction is like… Australia.”

Whoa! So here’s what I’m wondering about. It sounds like he thinks addition and multiplication are more closely related than addition and subtraction. And that’s interesting to me because I think of inverse relationships as quite close. So now I’m wondering about how kids think about opposite as compared to inverse? When you’re in 2nd grade, opposite seems really far apart, and addition and subtraction seem like polar opposites, like night and day or high and low or black and white. Over time and with more relational thinking, students might think of inverse relationships as more familial. More like a do/undo idea. Or like a shirt that can be turned inside out or right side out, but stays the same shirt.

All that rambling is to say that your student has me thinking hard! 🙂

And as for the blog thing, if you haven’t been writing as part of your regular life, it is going to be hard at first. With a little time, it will flow more. That said, NOBODY fires off perfect blogs without working on their writing. All my favorite bloggers invest a good amount of time crafting their posts. That’s why their blogs are so good and so readable.

So please keep going!

Tracy

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That’s the best thing I’ve read all day!

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Hi Mona! Great post! I loved Tracy’s webinar as well! What she said about relational thinking continued to support what I learned from Van de Walle’s “Teaching Student-Centered Mathematics.” I guess I would think of subtraction as Canada (yes, I know a country) and addition as the United States because they are married together meaning kids should learn both together over and over again to develop the understanding of their relationship. I blog too, but only about ten percent as often as I would like to. I keep a list of the things I need to write about so when I have time I don’t forget. Good work! Thanks for sharing!

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Hi, Jamie! Thanks for taking the time to respond. Van de Walle has been on my “To Read” list for about 3 years now, especially since I’ve used Georgia’s Standards of Excellence Frameworks for math as a resource in my classroom. A lot of these units are based on Van de Walle’s work, but they also incorporate the ideas of Sherry Parrish and Dan Meyer. I love your idea of addition and subtraction being married and the need for students to continuously develop an understanding of their relationship. I keep a list of things I need to write about, too, but mine is a mental list so that produces zero results 99.9% of the time. You would think that I would have learned by now…

Thanks again for reading!

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A motivating discussion is definitely worth comment.

There’s no doubt that that you need to publish more on this subject,

it may not be a taboo matter but generally people do not speak about these issues.

To the next! All the best!!

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