I was rambling about my small group intervention to my partner, Charlie, tonight, and it was suggested that I should blog about this because it’s important to capture these ideas in writing. So here goes.
I’ve been thinking A LOT about designing a small group intervention in math that really support kids in the transition from elementary school to middle school. In particular, I’m focused on my “cerebral diversity” group – essentially, a group of 12 students who I meet with three times a week for about 35 – 40 minutes for additional math support. I’ve been trying to think about the best ways to support them in developing their mathematical reasoning.
So that brings me back to Pam’s diagram from my last post (this is clearly stuck in my mind!):
In middle school, I want students to develop their proportional reasoning (particularly in 6th grade), so that means they need a solid foundation of additive and multiplicative thinking strategies. I’ve already started to make connections to this diagram in my observations of my students’ struggles. Yesterday, I had a student do division by drawing circles and counting (one by one!) 64 tick marks into the 8 circles (and he only knew to draw 8 circles because he saw that number on his multiplication facts handout because he doesn’t have fluency with his multiplication facts). I realized he was stuck in a counting strategy for a multiplication problem – he didn’t even have additive strategies in that moment!
So this brings me to what I did with my small group intervention this week:
On Tuesday, we previewed the idea of common multiples by looking at the 2, 3, and 5 times tables. We talked about the numbers that appeared in MORE THAN ONE table and named them common multiples.
On Wednesday, we worked on a problem string to develop a strategy for doing “9 times anything.” We started by using money to talk about place value and multiplying by 10 and 100. I asked students to consider how much money I had if I had 14 $10 bills. And then I asked them how I could trade that in so that I had fewer bills. After a few people said “and two twenties” I clarified that the bank I went to ONLY had 1s, 10s, and 100s. We then filled in our place value with 1 hundred, 4 tens, and 0 ones. We did the same thing for 27 tens and then for 27 hundreds. Some of the kids had never heard the phrase “twenty-seven hundred” to describe “two thousand, seven hundred” so we discussed that language.
Once it seemed clear they understood how to multiply by ten (and had a better sense of WHY we shift the place value than the often heard “add a zero” explanation), I decided to move into this string. We started with 6 x 10, 6 x 9, and connected the two. Then we did 8 x 10 and 8 x 9. We did 7 x 10, then 7 x 9, and then made a connection between the two. I recognize now that I wasn’t using a good model to help the students think because I was just writing multiplication sentences up there. I also wasn’t grounding us in a context, so when I jumped to 14 x 10 and 14 x 9, they got a bit stuck. There were two answers in the room for 14 x 9: 131 and 126. We used the context of gum packets and sticks of gum (which they were all familiar with from class number strings I’d done with them) to reason that it must be 126. We continued to 13 x 10 and 13 x 9 and then 17 x 10 and 17 x 9. When this problem came up, one student shared how he did the subtraction as 170 – 10 – 7, and I asked if I could represent his thinking on a number line. He agreed, and my other students saw how his strategy was helpful.
I tried to scale up this strategy to anything times 99. I had the kids multiply by 100 and then one of them even predicted we would do x 99! Only one kid was actually able to use the strategy of x 10 – x1 successfully, and got 1683.
I also noticed that some of the kids were easily doing the x10 but getting stuck on the subtraction – and I realized they might not have subtraction strategies that were helpful. So I decided to focus on supporting that with the “I have, You need” game that Pam introduced me to.
On Thursday, I tried to revisit the strategy from the previous day for multiplying by 9, and most of the students had forgotten it! I asked them how to do 19 x 9 using the strategy from yesterday and only after much prompting a few kids were able to remember we did 19 x 10 and 190 – 19. I then asked them if they could write the “helper problems” for 62 x 9 (getting 62 from one of the students) and some of them could – but not all. I decided to try to have the kids generalize, but they struggled to say the two steps were “multiply the number by 10, then subtract that number.” At one point, I said “let’s say we have ANY number, let’s call it N” and I totally lost a few of them. I realized I was trying to shove something on them they weren’t ready for, so I pivoted. Instead of doing my original plan (which was going to be to build on the x9 strategy and try out x19), I decided to play the game with them instead.
We got in a circle, and I told them I was going to start small so we could learn the rules of how this game was played. I wrote “Target = 10” and I drew the following on the board: _____ + _____ = 10. I then told them I was going to give them a number (and pointed at the first blank) and they were going to give me the pair that we could add to it to make 10. I said the numbers in a somewhat random order, 9, 8, 6, 7, 5, 3, 1, 2, and I pointed at each of the different kids. I noticed two kids were struggling with this. One of them even said he wasn’t quite sure how to play, so I filled in the first blank with the number I gave him (7), and asked him for the second number and pointed at the blank. He had to count it on his fingers.
He shook his hand “so-so” when I asked if everyone understood how to play. I said ok, let’s try it with 100, and I gave the kids ONLY multiples of ten. Most of them were pretty good, but it became clear two of the kids were counting (I hope by tens, but it was silent, so I wasn’t sure). I moved into the multiples of 5 (so giving a kid 85 and expecting 15 back), and when I tried to give one student 25, she couldn’t get 75. I tried to relate it to money for her, but while she knew there were 4 quarters in a dollar, and if I had one quarter, she would have 3, she didn’t know how much money that was worth. She could say one quarter was 25, and two quarters was 50, but froze up when asked for three quarters.
This gave me great formative assessment about where the kids were. Since we were out of time, I told the kids that they should play this game with the types of facts they didn’t know yet (I mentioned to my two counters that they needed to work with the pairs to 10 first).
As an aside: in talking with my principal & AP, they’ve talked about how the kids don’t “move” in math – in other words, their test scores are stagnant. And I felt like this experience really showed me why. In 6th grade, the students are being assessed on their multiplicative reasoning and their proportional reasoning – but if they don’t even have additive strategies yet, they’re going to struggle with that level of reasoning! So I need to support my students in getting OUT of counting and getting to higher levels of thinking.
This leads me to my ideas about designing this support curriculum. I am thinking through the three days a week I have with these 12 kids, and I’m thinking about how to arrange it so that we have one day that’s supporting their work in class (either previewing vocabulary or a skill or reinforcing something, etc.), one day that’s a problem string with multiplication strategies, and one day that’s supporting their additive reasoning and moving them out of counting. I think we can play this game with target to 100 and once I get to know the students, I can differentiate the level of complexity I can give them based on which skills I see they need to work on.
As an aside: tonight, in the conversation with my partner, it became apparent why she’s so much faster with mental math than I am: she “owns” all of the addition pairs from 1 – 100 and 1 – 1000! Whereas I own 1 – 10, and 1 – 20, I’m shaky on my 1 – 100 pairs. I own the multiples of ten (i.e. 10 + 90 or 40 + 60), but when it comes to my multiples of 5, I don’t “own” the facts close to the middle. For example, I know the 95 + 5, 85 + 15, and 75 + 25 forwards and backwards without thinking. But 65 + 35 and 55 + 45 trip me up every time. I want to say 65 + 45 and 55 + 55, and I don’t want to use 35 at all there! When I get into the “ones,” it’s even worse! I don’t know them with fluency at all!
So I started practicing them tonight! And I’m going to keep practicing them, building the level of complexity until I have all of the numbers to 100 mastered! I’m wondering whether I should share with my kids that I’m working on these too. I think there’s something powerful for them to recognize that even adults might have gaps in their fluency, but that once you recognize a gap, you can practice that skill and improve.
Additionally, I now know that next year, I want to use this “game” earlier in the year to assess who’s still counting and who’s got some of these facts mastered.