So Nancy Buck and I decided to enter the Chalkbeat’s Great American Teach Off. Voting started yesterday, and it feels a bit like a popularity contest. Worse, voting seems to be limited to people with facebook accounts – and even people who’ve told me that they’ve voted for us aren’t showing up as having liked the video. I don’t know why there are so many technical difficulties on Chalkbeat’s part, but it’s a bit frustrating to have people sending me emails saying they’ve voted, and then when I check on the video, their name isn’t listed as having liked it!
If you haven’t already, go to facebook, and like our video to help us get votes!
So when I first learned about Smith and Stein’s Five Practices, it was in the context of facilitating productive discussions in the math classroom. In case you’re not familiar with their book, here’s a link: 5 Practices Book on amazon
Last Spring and this Fall, I co-facilitated a PLT on the 5 practices, and I recognized that they permeate much more of teaching practice now than just when I’m considering a rich task for facilitating a whole class discussion.
The 5 practices are:
This year, I’ve begun facilitating PDs on CMP3 for NYC teachers. I’ve been noticing how the 5 practices support effectively implementing a Launch-Explore-Summary style lesson: Do the Math before the lesson (i.e. anticipate student mistakes and student strategies), monitor student work during the explore, so you can select students to share during the summary. Sequence the work during the summary so that you are able to draw the connections to the big ideas necessary.
I’ve also been trying out the math instructional routines from Amy Lucenta and Grace Kelemanik. I’ve been noticing the parallels between how I plan for a Contemplate then Calculate or Connecting Representations – the anticipation (including considering how to annotate the student work during class) is the pre-work where we have to do the math in advance. During the routine, we need to monitor student work to select students to share their strategies or their connections, which need to happen in a particular sequence. Then, we’re pushing students to connect to other areas of math and asking them to do a reflection about those connections.
Is it just me who’s noticing the depth to which these 5 practices permeate effective math teaching? Or is there research/articles making these connections explicit beyond just discussion? I’m not sure what’s out there.
On December 19th, I lauched my expressions unit in 7th grade through using the Contemplate then Calculate (CthenC) routine I’ve been learning from Amy Lucenta and Elisabeth V at MfA. Based on Amy’s suggestion that it takes three times of doing a routine to get really good at it, I decided to try out three days in a row of contemplate then calculate – and was pleasantly surprised by day 3 when in one class, my students jumped right into writing an expression (skipping “stage 10” altogether!).
We launched on day one with a relatively easy growth pattern of 2n + 2, with the squares arranged in a rectangular array. I flashed this image at my students (for the noticings), and then it disappeared.
My students noticed the following things:
named term 1, 2, 3
objects got larger to the right (then the student clarified: the size or number of squares got larger)
Adding 2 from left to right
4 squares, 6 squares, 8 squares
1st stage had 4 squares
2nd stage had 6 squares
It seemed like it was increasing by 2 squares
Term 1, 2, 3 on top
2 columns consistently
I then told students to work with their partners to figure out how many squares would be in the 11th term. And then I did the hardest part of this routine – tried to scramble around the room, eavesdropping enough to choose students to share their thinking (and confirm with them that they’ll share the ideas I heard), but not long enough to get stuck in any one conversation. I sometimes had to ask students to tell me how they or their partner was seeing it because I walked up when they weren’t talking – but then I felt bad walking away from students who I asked to talk when they weren’t done, but I knew I didn’t need to hear the rest (because I had already chosen someone to share their way of thinking about it, or because they weren’t finished with their thinking).
In class A, I chose two students to share their ideas. The first student saw the columns as one more than the term number each time, so she predicted that term 11 would have two columns that were each 12 long (which she described as 11 + 1), and then she doubled it to get the total.
The other student noticed there were 4 in the first stage, and it was increasing by 2 each time, so to get to term eleven, he said we would’ve added 2 ten times, so he did 2 x 10 and got twenty, which he added to the initial four.
At this point, another student who I hadn’t chosen said she found something similar, but different, and I encouraged her to share out. She said she saw two at the top and then a rectangle of 2 by the term number, so the 11th term would be a 2 x 11 rectangle (with 22) plus the two at the top for a total of 24. I had been searching for this method and hadn’t seen it, so I was excited when she volunteered it.
I then asked the class to consider how each student would’ve thought about the 100th stage and had them share first with their partners. Then I drew it out and wrote the annotations identically to the 11th term. Finally, I pushed my students to generalize and asked them what the “nth” term would look like, explaining that the nth term was an expression for “any term.” (somehow, connecting “any term” to “nth term” has helped my seventh graders in the past make sense of this somewhat confusing notation!). All in all, my students were able to get the three algebraic expressions matching the ideas exactly: 2n + 2, 2(n – 1) + 4, and 2(n + 1). We didn’t discuss the equivalence here.
Class B had 2n + 2 and 2(n + 1) come up, and we had similar discussion around it. No one shared the other expression, and since it wasn’t “necessary” for my main idea, I didn’t push it.
I’m going to pause in describing my CthenC roll-out to go to my final session at MfA about CthenC!
I’m finding that there’s still a bit of a disconnect for me in the theoretical knowledge of how the routine is supposed to be enacted and the practical application. I find that I don’t have enough working memory (or good enough ears/skills of observation) to find all of the students I want to select to share. I have to find better ways of “cheating” – because my students don’t gesture as much as I’d like them to, so I have very little window into their minds except to eavesdrop and I have 27 and 28 kids in my two 7th grade sections!
More on CthenC coming soon!
So last year was my first year teaching fraction division, and we did a rushed job of it. It was the last (mini) unit of 6th grade math before the state exam, and I basically taught it as the inverse of multiplication, with some ideas of scaling using ratio tables (from Kara Imm’s number talks on paint cans on her website, numberstrings.com). Many of my students already knew the algorithm from their fifth grade classes, so we didn’t do a great job at teaching that topic. And it was sufficient, but not enough for me. This year, I rearranged the order of our units with my colleague to make the fraction/decimal operations unit come right after our second/third unit, so that after our students have done integers (first), coordinate plane (second), and factors/multiples (third), we would go into fraction/decimal operations. This will allow us to reinforce those calculations within the units on expressions/equations and ratios/proportional reasoning (though, I’m honestly thinking for next year, we might want to do ratios and proportional reasoning FIRST and then fraction and decimal operations – whoops!).
Anyway, since I’ve never taught this topic to the level of depth I wanted to, I decided not to reinvent the wheel, and instead, to look at known resources. While I used CMP3 to launch this unit on fractions and decimals, they go more in-depth into addition and subtraction than we wanted to spend. Plus, their division problems in problem 3 didn’t go into detail about the two different ways of thinking about division (quotative and partitive) and that was something I wanted to try out with my students this year.
I really have been enjoying the lessons from Illustrative Math’s open up resources. I do find that I don’t have enough time to go into quite the level of depth that they do (there’s 17 lessons on just developing the ideas about division of fractions! each of which are anywhere between 40 and 50 minutes depending on optional activities). I’ve had to cut out some of the activities to make it work, but I REALLY really like it. I think next year, I might go straight into this investigation, and not do the other lessons from CMP3 first (it’ll depend on how we rearrange the units and whether we need those ideas of equivalent fractions to be developed in this unit or if they’ll have already been developed in a ratios and proportional reasoning unit first).
On the Friday before break, I had my students do a set of problems with fraction strips. Some of the students used the diagrams, some of the students wrote equations and used the algorithm, and some of the students thought about it in other ways (that I didn’t highlight). What came up was that some students said they were using an algorithm they didn’t really understand and I asked if they could connect it to what the person who used the diagram had done. We ultimately decided that the reciprocal represents the number of the divisor in one whole (which is basically the definition of reciprocal), and we multiply it by “how many wholes we have” (because we were doing whole number divided by a fraction first). I had never realized that interpretation before of fraction division – and I was impressed! I know this is tricky stuff (Liping Ma’s book, the Knowing and Teaching of Elementary Mathematics made me realize how tricky fraction division is – and it’s part of what inspired me to become a math teacher!). But our discussion around these problems was what led them to this conclusion!
This week, I started with trying to get a bit too much accomplished… I wanted to squish ideas from 4 different lessons into just today. In my double period, I managed to go through three handouts and do an exit ticket – and the amount of struggle they had with the exit ticket made me realize I really rushed it too quickly. I’m struggling with the balance of the kids who “already know the algorithm” and the kids who have limited prior experience with division of fractions, and I’m not sure yet how to balance that.
I am also struggling with how to go over some of the topics/skills in the work from Illustrative Math. I’m doing a lot of partner talk (which I normally do), but not as much whole class conversation – because I’m feeling like some of it is tricky to hash out in whole class conversation without looking at their diagrams – but I don’t always find them to be using the diagrams in a way that actually makes sense to solve the problems. So I wonder if I should’ve had more direct-instruction about it? But I think that defeats some of the purpose. I think it went a bit better in the second class I taught today because I knew which things to point out to them in the warm-up (where the diagrams were already half drawn for them) and how to help them connect it to the diagrams in the second handout.
I also made some strategic choices about which items to focus on and which items to cut. I used 4.6 warm-up and created a handout where they were drawing the tape diagrams. I pulled some other questions from the IM curriculum (in the practice sections, etc) and made a two-sided handout. The students got a chance to create the diagrams for a number of problems, some referring explicitly back to the warm-up of “How many groups of 2 1/2 in ten?” and “How many groups of 2 in 7?”
I had the students discuss in partnerships the questions of how they began to draw their diagrams and how they saw the answer in the diagrams (how many groups). I decided not to go over each diagram individually as a whole class, but rather to put up my answer key, share that theirs might not be identical, but they could edit/revise or ask questions about mine. I saw some pencils/erasers moving as I put up my answer key, but no one had any questions they wanted to share with the whole class. I struggle some times – I’m not sure that it was the best way to go over these problems, but no student’s work had stood out to me sufficiently during the work time as being one I wanted to highlight under the document camera, and sometimes, I struggle with what to have students do… Now I wonder if I could’ve had them do a “notice/wonder” discussion with a partner about my answer key instead! (Darn! Next time!)
I also had the students share out their answers to the following question (also from IM, though I’m not sure if it was in the same lesson or not). “Diego said that the answer to the question ‘How many groups of 5/6 are in 1?’ is 6/5 or 1 1/5. Do you agree with his statement? Explain or show reasoning.”
I chose this question because one of my students who was “done early” had this question incorrectly answered with “No, he’s wrong. It should be 1 1/6 because he has 1/6 remaining.” This student didn’t share her thinking with the class and I didn’t want to call her out, so I just tried to play dumb/confused when the student responded that it should be 1 1/5. I asked the class, “Wait, I’m not really sure what she means by it should be 1/5. I thought there was 1/6 left over? Talk with your partner about which one it should be.” I had a few students share out and rephrase/restate each other’s reasoning. Then that first class (with only a single) ended, and I left them to think about making sense of the “remainders.”
My second class did this same problem, and we had a much richer discussion about it. I shared with them “One student in my other class thought it was 1 1/6. What might they have been thinking?” and they were able to articulate that they just took the extra 1/6 and put it on the answer. Then I asked them to think about about what they would say to that student to help them understand why it’s 1/5, and a student used the equation 1/6 x 5 = 5/6 therefore 1/6 / 5/6 = 1/5 (because you want to know “how much of the 5/6 the 1/6 is”). I felt more satisfied with their reasoning, but I’m not sure that every student was there yet, despite the fact that most students gave a thumb vote agreeing with Diego. I struggle sometimes to get whole-class QUICK formative assessment that doesn’t require me to grade their work, but also gives me a view into their thinking. I’m not sure it’s possible – either I need to look at their work, or I don’t get a view into their actual thinking.
Anyway, in that second class, we then went on to 4.7’s fractional batches of ice cream. I skipped over the ropes section (even though I included it in the handout) due to timing, and also because I wanted to stick with the same kind of diagrams and I felt like ice cream would be something my students understood well. I was quite happy with the way I did this section of the lesson, and I think I’ll repeat it similarly with tomorrow’s class.
I gave them the handout and some independent think time. Then we transitioned into “table talk” time where students can continue working alone OR working with partners. When most students had finished Monday, Tuesday, Thursday and Friday (only one table hadn’t), I asked students to turn to their elbow partners and compare their diagrams – discuss how they saw the numbers in the diagrams and how they could use the diagrams to answer each question. The one group where no one had finished all four, I gave my answer key, and I encouraged them to compare their answers with that one – it wasn’t ideal to me, but I didn’t know how else to manage the fact that that group was SO FAR behind everyone else. I don’t think it was an understanding issue, but maybe more of an organization issue? Or a visual-spatial issue of making sense of the diagrams? I’m not sure.
Anyway, I then showed the four diagrams from IM’s “answer key” and I asked students to share a division equation for each day. We got all four equations up on the board (though we almost had one mishap – one student said 9 / 6 = 2/3, but students saw that he had reversed the divisor and dividend, and we corrected it – while that mistake can be valuable, it wasn’t the focus I wanted to go with, so I quickly dealt with it, rather than exploring it more in-depth).
Then I used the alike/different routine and I asked students to discuss how Monday/Tuesday (both had quotients greater than 1) were different from Thursday/Friday (both had quotients less than 1) and how they were alike. We discussed that they all involved dividing by 9 (which made sense to them because 9 cups of milk was required for each batch, and we were really asking the question “how many times does 9 go in?” My students also noticed that none of the dividends were divisible by 9). We also discussed that they were different because of the quotients being greater or less than 1 – and then I asked them how they could predict whether the quotient would be greater or less than 1 without doing the calculation – and they told their partners and then we shared out that they could compare the dividend the the divisor – if the divisor was greater, the quotient would be less than 1, whereas if the dividend was greater, then the quotient would be greater than 1.
At this point, I should’ve made a different instructional decision than I did. I was feeling rushed because I’d only had the students work on 2 of the 4 handouts I’d prepared for our double today, and I was feeling the pressure of time. Unfortunately, I decided to forge ahead with another handout, but it was really pointless to give out. Students barely had any time to finish reading the instructions before I was asking them to share with their partners. Luckily, it was a matching handout, so they still could discuss something, even if they hadn’t finished – but I don’t think it was as productive as it could’ve been. I then showed the answer key, and saw a bunch of kids asking questions – which I didn’t give us time to answer. 😦 Tomorrow, I will go back to this handout, revisit the front, and ask the students to actually work on the back – and then make sense of it together!
The reason I rushed that handout was because I wanted to give an exit ticket. Unfortunately, I don’t think the students were ready for this exit ticket – it made some leaps above where we were in class, and went to the finish line. It’s the cool down from 4.9 (and we had barely even skimmed the surface of 4.8 in class!):
Noah fills a soap dispenser from a big bottle that contains 2 1/3 liters of liquid soap. That amount of soap will fill 3 1/2 dispensers. How many liters of soap fit into one dispenser?
Use the diagram below to answer the question. Label all relevant parts of the diagram.