Practice Puzzles: Math Practice Three

Math Practice One and an explanation of the puzzles
Math Practice Two

There is nothing more charming (and alarming) to me than to listen to a child try to explain themselves mathematically. I truly do love every second of it...I never tire of it, and it never gets old. Having said that, I'm not the only teacher who has to reconcile the balance between pushing past, and lingering in, the nonsense. In all of these Math Practice Puzzles that I am sharing, there is way more to the practice than I have given rise to here in these captured moments. The point was to find ONE thing we could use as an entrance to the practice...I wasn't looking to frontload it in its entirety, I just wanted one shared, contextualized experience to draw out the practice. The rest has unfolded as it has, with some practices more easily expanded on than others. Which feels exactly as it should for working with the Littles. They have Kindergarten until Senior Year to sort it all out. I'm just doing my small part.

So, Practice Three....there are so many deep and important truths in this practice, but the one that we have cleaved to early on is the idea of revision. This practice is tightly wound with other practices, and as we see more patterns, learn more structures, take command of more precise language, and test more solutions and strategies, our communication and explanations will continue to evolve. But these are lofty aspirations indeed, if you consider how we start on this path. There is no primary teacher on the planet who hasn't heard such carefully crafted nuggets as: "I know it's five because that is my favorite number." Or, "I know because I knew it in my head" (which is MOST exciting when it comes with a wrong answer...."8 + 3 =12, because I knew it in my head"). Other oldies but goodies include "I counted on my fingers" and "I guessed".

Early on, I introduced the notion that the onus is on the explainer....if somebody doesn't understand you, it's your job to keep revising your explanation until they do. It's not because they aren't listening to you (they aren't though) and it's not because they aren't smart (they really are) it's because the explanation wasn't sufficient. The correlating piece to this is it doesn't have to be perfect to get started. Focusing on revision means that all you have to do is start. Just say SOMETHING. Then you'll get some feedback, and you can add on/change/delete to make it more clear. Get more feedback. Make more changes. And so on. You don't have to wait until you know exactly what to say or how to perfectly explain something. Just say anything.

From the classroom: Afoa had JUST told us that the two numbers we were going to use to solve our problem were written on the poster, 37 and 43. I turned to Natalia and asked, "So...which numbers do we use?" And she says, I swear, "I don't know." I want to cry haha but instead I tell Afoa, "Well, I'm afraid that explanation wasn't enough for her to understand" and bless his pea-picking-heart he immediately updates it to, "It's the purple numbers on the bottom, 37 and 43." I made SUCH a big deal about this to the whole class! "HOLD UP!! Did everyone SEE what Afoa JUST did???" and then I explained that what he had done was soooooo mathematical because he didn't tell Natalia that she wasn't listening, he didn't tell her oh well you just don't get it, he CHANGED what he said to make it a BETTER explanation." Then Afoa talked about what he had said before and how he changed it. Natalia confirmed that NOW she understands what to do.

Look. It was only marginally mathematical and even calling it an "explanation" is a stretch (it was more like a clarification) but it was all we needed. We have talked about this idea over and over and, in conjunction with a few other key moves, my second graders have made enormous leaps in their ability to explain and question each other. I'm taking it!

Practice Puzzles: Math Practice Two

Math Practice One and explanation of the puzzles can be found here.
 

Math Practice Two: Reason Abstractly and Quantitatively
This practice is when a mathematician makes sense of the problem in context, then ignores the context to do the calculation, then steps back into the context to make sure the answer makes sense. For example, a child may make sense of the following math story as a combining story:
          "There are 6 kids playing soccer. Some more kids come and now there are 10 kids playing soccer. How many kids came to join the soccer game?"
        In other words, they see this as combining the kids who were playing with the kids who came to join them, and ending up with 10 kids. The model is 6 + ____ = 10. Even though this is additive in context, a child may solve the problem by subtracting 10 - 6 = ____. Once they've made sense of a problem, they can solve it however they want. Then go back and see if it still makes sense.

 I watched my students do this a lot the first few weeks of second grade. They would build the two parts (contextualize) then put them together and count them by 1s, 2s, 5s....however they wanted (decontextualize). They started out building everything as single blocks, and then combined the quantities into one long train. (See the picture above) They were able to go back to the context and tell me the unit, for example that the answer was a number of stickers for this particular problem. But once they made the long train, the original parts were lost to them.

Look, here was the problem: "Diva had 4 stickers. She went to the store and got another 8 stickers. How many stickers does she have now?" So the problem was that they could answer (12 stickers)...but if I then asked, "So how many stickers did she buy?" they would say "12". Because they totally forgot about the parts once the parts were swallowed up into the whole amount that was the sum.

This is problematic because keeping track of those parts is what connects addition to subtraction and it's what is going to allow us to solve subtraction problems as missing-part addition problems (for example, seeing 12 - 8 = _____ and thinking 8 + ___ = 12).

As they got better at thinking about quantities, I started making a VERY big deal about students I saw who were making the parts and color-coding them or counting by 10s and then continuing with the ones without physically moving the cubes to be together, (See picture above) thus maintaining the two parts. When I asked them I would usually say, "Which ones are the stickers she had? Which ones are the stickers she bought?" and when they could answer it was a very good thing. I'd even reserve their trays intact, to use as a mini lesson to start our next workshop....."Friends, do you see what she did??? Like all mathematicians, she arranged this so we could tell which ones were the stickers she bought! Can we answer the question of how many stickers she has now? Of course we can! AND we can still see the two numbers she made....Who remembers what this first number means? What about this second one? And how many all together? Who thinks they can work on keeping their parts like Ariel does? Off you go."

Practice Puzzles: Math Practice One

 The first few weeks of this school year, I stumbled across a way of introducing and using the Math Practices that really worked for Room 29, so I wanted to share it. I've seen lots of examples of "kid friendly" language for the SMPs (Standards for Math Practice) but this is a little more organic. What I did was, I started our Problem Solving Workshop time on the second day of school. And as my students struggled to make sense of problems, to explain themselves, to model the mathematics and find viable solutions, I just looked for ways that they were already, intuitively, using the SMPs. Little kids are natural mathematicians....never once did I have to tell them how to be mathematical. I just had to open my mind to the possibilities of what each practice might look like.....as performed by a 6 or 7 year old.

Once I had collected all eight (and I'm not going to lie, a couple were a real stretch haha) I used blank puzzles and I drew the incident right on there, labeling it all up. I am nobody's artist, but they are easily impressed and it turns out hairstyle is an easily identifiable attribute among my students. So even though any picture was only marginally akin to the child, everyone totally GOT IT.

 I gave each group of four students a tray with the puzzle pieces for one practice ready to be put together. They had a blast putting it together and then they went NUTS when they realized it was THEM. Oh my gosh! SO GOOD. :D

 

Once they had the puzzle together, they read it to each other, and made sense of what it was saying. Since they were right there when it happened, and I had made such a big deal out of each one and even repeated it over and over, they had a built-in context for making sense of each one.

Math Practice One : Make sense of a problem and persevere in solving it

            All the kids were on the carpet. I had written a simple “put together” problem on a poster at the front on my easel. “Diva had  _____ stickers. She went to the store and bought another _______ stickers. How many stickers does Diva have now?” In this process, the students have acted out the problem with a variety of numbers I supply them. She had 4 and bought 7. She had 8 and bought 3. She had 12 and bought 4. Different students act out the building of the numbers and combining them. When it comes time to do the problem on their own, I give them bigger numbers they wouldn’t really be able to do in their heads, like she had 17 and she bought 18.

            After giving them the numbers 17, and 18, I ill-advisedly did one more check for understanding. That’s when I asked Janiya what was happening in the problem. After a tense 60 seconds of silence, she slowly said, “Divaaaaa….is….she has…..stickerssssss?” Yes! And how many does she have? Janiya stares off into space. She clearly thinks I will lose interest and ask somebody else. No way, Sister. We are at an impasse, until she absentmindedly swings her head around and looks toward the poster.

            “OH MY GOSH!” I practically yell, “DID EVERYBODY SEE WHAT JANIYA JUST DID????” The other 34 students (you heard me, it was a rough first month) look at me expectantly. That’s how I imagine them, anyway. And I make a VERY big deal out of THIS THING Janiya did….because when Janiya wasn’t sure what number to build, SHE LOOKED BACK AT THE PROBLEM! Isn’t she a good mathematician? That’s what mathematicians do, when they are making sense of a problem, they LOOK BACK TO FIND WHAT NUMBERS TO USE.”

            It was a stretch, but it’s a point we’ve made over and over….’Remember what Janiya did? She did what all mathematicians do….she looked back at the problem when she needed to remember which numbers to build.”