Grinding through testing season and I found a minute to get back in the classroom. Today, I was able to make it in a kindergarten class to wrestle with something Dan Meyer mentioned during his NCSM Webinar (my world rocked @ 7 minutes and 15 seconds). Dan said that “10 frames, blocks, or whatever is not modeling”. My immediate thoughts:
- what the ?!?
- how can using 10-frames and base-10 blocks not be modeling?
- if that’s not modeling math in elementary school, what on earth is modeling supposed to look like?
Illustrative Mathematics explains SMP 4: “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation.”
How did I miss this!? I have always considered manipulatives to be modeling BUT IT’S NOT (in the purest sense of the word).This is me trying to make sense of modeling at the K-5 level and I welcome any comments that might help or disagree:
I’ve come to understand that manipulatives are a means to the modeling of mathematics.
Holy cow, this understanding was/is huge for me! When kindergarten and 1st grade teachers are having students build numbers on a ten-frame and record the quantity, they are not modeling mathematics unless the model is describing a context. If this is the case, an equation can be written to explain the mathematical “phenomena”…and that is the golden ticket.
Immediately I thought of the C-R-A Model and where this new understanding of modeling fits in the spectrum of my understanding. I must continue to work conceptually through the concrete and representational models, but it is not until the math is formalized in the “abstractedness of an equation” that the true modelling of math occurs. That’s what’s great about modeling…it cannot happen without a context. All the more reason to keep pushing for problem based learning. So I used the Peas in a Pod Kindergarten 3-Act Task and put my new found knowledge/understanding to work.
Students immediately started firing out estimates for how many peas were in all 3 pods. Just the estimates alone gave the teacher and I insight to students’ understanding of quantity. After students made their estimates we had them draw a representation for how many peas they thought were in each pod. Students decomposed their estimate into 3 groups to represent each pod of peas (we asked them to show 3 different ways the sum of their pods could match their estimate).
After, we shared the Peas in a Pod (Act 2) with students to provide them with the correct amount of peas in each pod. The comparison of their estimates to the actually made for some great conversations about more and less, which is still a monster this time in the year with Kindergarten.
By the end of the lesson students had modeled the math on ten frames but it wasn’t modeling until the equation was written. This makes perfect sense because the equation is the formalization of the math. I could have this modeling idea all wrong but that’s where my mind is at right now. Completely vulnerable because I am grappling with this idea and my brain is malleable. All I know is this…if an end goal of high school is modeling mathematics, I need to do my part in developing that understanding for me, my colleagues, and students…12 years before graduation.
I heard modeling described by a CC advisor at a PARCC meeting in this way- modeling with mathematics means taking a situation and mathematizing it. Phil uses modeling in his work when he takes a physical situation and digitizes it into a system which can be used repeatedly. It especially helped me to think about modeling through the example given by the advisor- determing the volume of a large natural body of water. The quantities are unknown, and the situation is unwieldy enough to require the solver to mathematize the situation using estimation and some mathematics understandings. The solver might first determine a benchmark, create a way to estimate the size of the body of water, and begin working from there. I loved this way of thinking about modeling. I don’t however, think it a cardinal sin if folks use the word model to describe ten frames, etc. Model is a noun in that case, not a verb. When the student explains their thinking about a context using a ten frame and then “algebrafies” it (stealing your term!) with numbers and other symbols then they are using a model and modeling with mathematics, and frankly, when they’ve developed understanding to the point that they can take the context and go straight to working with symbols and numbers (with understanding) and skip the model (because it is internalized, perhaps?) they’ve become more efficient and elegant in their thinking.
I’ve tried to express this in a few webinars, but obviously not clearly nor frequently enough!
It’s interesting that you bring up the noun and verb idea as it relates to the word model. In a conversation with Graham we discussed that same conclusion. Adds to the point that engaging in mathematics should be purposeful.
What Turtle said.
Graham, I love this post. I think that we can only get better at what we do by challenging ourselves to think about our practice, and what it is we think we know. That’s been one of the greatest benefits (for me) of joining the MTBoS. Your work with kindergarteners is inspiring. I need to get in there more often and actually get my hands dirty.
Wow! All my favorite people!
Turtle and Mike-we continue to learn from you 2 each day and it’s inspiring! Jenise and I fed this gerbil in our brains the other day until it became a capybara! Your comments and insight have allowed me to put this monster on a diet and control it! Modeling doesn’t “need” to be an equation (but usually is) and I’m climbing out of that pigeon hole as we speak. The noun and verb reference paints a much clearer picture.
Jenise-you question my questions which helps us both grow. Glad I could throw some disequilibrium into your life…like you’ve done to me on numerous occasions.
Joe-when you moving to Georgia? We have a math coach position and it’s screaming your name. I’m sure it’s much warmer here than the New Jersey Turnpike in January:-). If my wife wasn’t a kindergarten teacher there is no way I would be “getting dirty”. She blows my mind with the things they accomplish and that’s where I find my inspiration. It can be scary down there with the “little people” but I remind myself that their not “tainted” with rules and procedural thinking…but they can partake in modeling.
Thanks all for continuing to inspire! Happy mathematizing and algebrafying!
Dan Meyer?!? Who’s he? 🙂
Love how this post runs through your grappling with the idea of modeling. Your connection to C-R-A is excellent.
My two cents:
Mathematical ideas can be represented in multiple ways, with diagrams, manipulatives, tables, etc. Problems start with a representation – a story problem, video, picture . . . They often end with another – an equation. Students may get to that final representation by working through some others: tables, diagrams, even other equations. It’s these representations that I think are most critical in building student understanding of the mathematics they’re studying because they help support and communicate students’ reasoning.
Ok, maybe just one cent.
I’m completely thrown off by this post. I’ve been referring to these same tools and others like it as models. I’ve read the SMPs several times and had made such a connection. As I read this post I couldn’t help but disagree. This post challenged me to look closer at SMPs 4 and 5. My new interpretation is the tools discussed in SMP 5 can be used to model the contextual situations discussed in SMP 4.
This poses an interesting perspective on the argument between using naked numbers and using numbers within a context.
I can hardly wait to see what my students will be able to do in 12 years! I know you are not the only teacher building these understandings (at least I hope not). I already incorporate as much of the problem-based, 3-act, hands on types of learnings that my spoon-fed students can handle. And I’ve seen them grow, and complain… And grow some more. Amazing how that works!
I think that’s why we do, what we do! To one day be inspired by the student that runs into us at a store and says “You’re the reason I became______.” That’s my golden ticket!!!
In terms of building that understanding…the people I have learned from the most are the ones that have never stopped building on their own knowledge. As long as I keep pushing my own understanding I know I’m doing whats best for kids!!!
Keep on pushing and sharing to makes us all smarter!