So unfortunately, it turns out I’m not going to be able to go to all three days of the NYC mini-TMC; today was the only day it fits in my schedule. However, it was the perfect day to go for me, and I felt like I learned so much in all of the sessions I went to. I also was excited about making the connections with other math teachers, though I felt a little awkward because it seemed like some people already knew each other, either from the day before or from working together.

First thing in the morning, was Michael & Kristin’s talk on problem strings. I first was introduced to this instructional routine by Kara Imm and Janice from Math in the City at an MfA PD. Then I read Pamela Weber Harris’s book about building powerful numeracy, and she uses the strings too. But then I read the making number talks matter book by Ruth Parker, and I suddenly felt confused about the distinction between number talks and number strings.

Today’s session clarified that for me, and I really liked the way they explained the distinction. There are some definite similarities between the two routines in that they both involve math problems solved mentally to highlight mathematical strategies beyond the “traditional” algorithm.

However, the two number routines have different purposes. A number talk focuses in on one problem and all of the various strategies that students could use to solve it. It could be used at the beginning of a unit (what do they know? in formative assessment) or it could be at the end of the unit (what strategies are sticking? do they really reason about the numbers before choosing a strategy?). To contrast, a number string has a series of problems, each solved one-at-a-time (presented to the students sequentially). When the students share out their strategies, the teacher records the math visually (using the open array for today’s multiplication string), so that students can refer back to earlier problems in the string as they solve each subsequent problem in the string. This is because the series of problems is carefully sequenced, designed to start in a place where hopefully every student can do the first problem, and by the end, they’ll be introduced to a final, potentially new strategy.

In our string that we did together on the board, we looked at 10 x 20, 10 x 23, 13 x 20, and 15 x 23. I thought it was fascinating, and I loved listening to the talk moves involved in developing our thinking aloud and having someone else explain the reasoning. One of the concerns I heard a teacher voice in today’s session was about how difficult it is to get their students to really listen to each other, and I think that the revoicing technique (both, used by yourself and especially prompting the students to revoice each other) is an important one to get them to engage with each other’s ideas. I also thought the careful sequencing of asking students “Why did you decide to break it up like that?” is important to highlight the mathematical goal of the work. We also discussed trouble-shooting what to do if a student brings up a strategy that you didn’t want to share out or focus on, and how to acknowledge their contribution (especially if it’s valid), but to hold on to it for later and set it aside.

Then, we did something I found really fascinating and unexpected – we did four equations as a number string! We looked at some student work and discussed misconceptions (and we were also supposed to discuss what they did know, but my group had a harder time focusing on that, because it wasn’t as obvious! It made me realize how important it is to really have someone pushing us to follow a protocol to discuss that too, or we might only focus on the misconceptions). We then talked about how to design a problem string that would support these students in building on what they knew to tackle the problems that were there.

I was really struggling at this point with the idea of how to design the “perfect” number string. When I looked at what I thought the kids couldn’t do yet, it was obvious to everyone in my group that they were struggling with the notation for coefficient meaning multiplication (the student work showed some work that might’ve indicated thinking a coefficient is attached to a variable through addition), the idea of balance (with both sides of an equation needing to be operated on for the relationship to be maintained), the ideas of inverse operation, and the ideas of combining like terms. I didn’t know where to begin, and Michael made a comment to me that I thought was really valuable. He said that he thought elementary school teachers had a better view about designing problem strings than secondary teachers do, because we (6-12 teachers) want to come up with one technique/problem set that is going to address EVERY issue all at once, and help students do EVERYTHING they need to. Whereas elementary school teachers see the long-game better: they know students are going to be grappling with multiplication for a very long time, so they focus in on creating problem sets that focus on one little part of it. That helped me get unstuck in my thinking about the problems and how to create a string that would help students tackle the meaning of the coefficient as meaning multiplication, starting from where they were currently.

Here’s the string my group drafted today: (I’d love feedback)

6 + n = 12

m + m = 10

2m = 22

k + k + k = 24

3k = 18

n + n + n + 3 = 24

Each problem would be presented separately, one at a time, in order, and we would ask students to solve them mentally.

Anyway, we shared out our strings, and then we broke for lunch. I found the morning session SO INCREDIBLY helpful!

After lunch, they did a little 5 minute favorite share, where I learned about some statistical software from Amy, and some other websites that Heather shared, and Alla shared about the marble slides activities on Desmos. I took notes and I plan to check out the websites at some point in the future.

Next, we had a chance to do real math in groups. It was really interesting that they very purposefully had us work on “non-permanent vertical surfaces” and it sounded like there’s been some research about having students work in groups like that – I want to know more!! And I may need to spend some of my MfA grant money purchasing vertical white boards to use around my new classroom to copy this strategy if it’s so awesome!

The problem was a PCMI math problem from the number play column in the NY Times called “Tax Collected.” It was presented auditorily, but I have trouble listening unless I write it down, so I copied the important details into my evernotes. Essentially, the gist of the problem is that you will receive 12 paychecks of different amounts: $1, $2, $3, and so on until $12. However, the tax man is going to get paid. For every paycheck you choose to keep, he will keep all of the paychecks that are factors of that paycheck. Also, he always gets paid, so if you want to take a paycheck that has no factors remaining, you can’t! He’ll always take what’s leftover in the end. Our goal is to determine which paychecks and in which order do we want to take them in order to pay him the least amount of money and take the most amount of money.

My group found it obvious that taking the highest prime number first was important, because that meant we got 11 and he got 1. However, we also knew that in the end, it meant we were sacrificing 7, because it had no other factors – but 11 is still better than 8, so we moved on. We took 9 next, giving him only 3, but we had trouble articulating a generalized rule: was it about taking the highest odd perfect square whose square root was a prime? We weren’t quite sure yet. Then we knew we wanted to take 8, 10, and 12, but the order mattered – if we took 12 too soon, we’d lose the 8 (since there would be no factors left), but we decided 8 and 10 were interchangeable. So in all, we took 11, 9, 8, 10, and then 12 last, giving him 1, 3, 2, 4, 5, and 6 and sacrificing 7 in order. We were easily convinced that this was the largest possible amount for us to take because we had the largest 5 numbers in the set, and we had determined that because of the rule “taxman always gets paid,” it was at most possible to take half of a set of paychecks (you get paid 1, he takes 1), and since there were two primes in the top half (7 and 11), it’s only possible to take one, thus reducing the maximum down to 5 in this set.

Our next move as a group was to move to smaller groups to see if we could generalize our rules for which ones to take first. We did establish you should ALWAYS take the highest prime first, but I realized when (later, on my own) I extended the system to 24 paychecks, even if you take the highest prime, ultimately, that one winds up being a sacrifice, because at most, you get the highest prime (in this case 23), and the taxman takes 1, but you also sacrifice any primes over half of the total (in this case, primes over 12), since double those primes isn’t included in your set. I then came up with a system, and I think maximized my results, but I ran out of time since my next session was about to start. I was going to check in with a neighbor, but we never did go back to it! Whoops!

The next session was an interesting one about a math mistakes game called baldermath. It was basically play-testing a game for Michael. I wound up being judge, and finding it very hard (and totally choosing wrong!). It was interesting, and I was definitely engaged, but I didn’t feel like I took anything directly applicable to my classroom practice – other than I want to engage with more teachers in looking at ACTUAL student work more often so we have a better idea about the kinds of mistakes and misconceptions children really have! And also so that we have better ideas about the kinds of understandings that students with misconceptions can still have (i.e. just because they don’t understand this part, doesn’t mean they can’t get this other part!).

The last full session of the day was on statistical questions, and it was a perfect lead-in (for my first unit of the year is on statistics!). Amy Hogan led the session, and I was absolutely fascinated! We focused on a 5-step statistical process cycle (formulate good question, gather data, visualize data, analyze data, make conclusions, and ask new questions to begin again). In today’s session, we focused on a question I found fascinating, which was about the average commute time for US HS students – is it less than the 26 minute average adult commute? We used a cool website, Census at School, to draw a random sample, and ultimately convinced ourselves that, yes indeed, HS students had a shorter average commute.

She also introduced us to some good websites, which I think I’ll deploy to create histograms and box plots for my students to analyze in class (since some of the ones my colleagues shared with me aren’t great!).

There was definitely some topics that came up today in the class that felt a bit above my understanding, and I realized that while my college stats professor was horrible, the class was made infinitely harder because it’s actually a hard topic that’s somewhat counter-intuitive. I’ve got a lot more to delve into – but luckily, I just need to give students an introduction, as I teach middle school (and not HS or AP stats!).

Finally, they wrapped up with a flex session where David shared with us 4 projects. I loved all four of them, and there were at least two that I planned to use in my class with my students! I also liked his description of what made them worthy tasks: students are doing something they already know how repeatedly to gain fluency with it, but applying it to a novel situation to learn something new. I haven’t had time to finish working on all four projects yet, but I definitely will. At least one of them was related to some PCMI math I did a few years ago at an MfA PCMI online math session…

Anyway, I was absolutely amazed at the wonderful PD I attended today, for free. I went to a PD earlier this summer, and it was so useless that I left after 2 of the three days. Today’s PD was so helpful (especially the morning session about number strings vs. number talks and designing our own!) that I’m sad I can’t make it to all three days! I hope to continue to build relationships with all of the teachers who I saw present today and work with today as well. I hope this PD continues next summer!

Thanks so much for your reflection! I was really sad that I couldn’t make it and it’s helpful to have such a detailed description.

Also, if you’re interested in the research about vertical non-permanent surfaces, here is Peter Liljedahl’s most recent paper: http://peterliljedahl.com/wp-content/uploads/Building-Thinking-Classrooms-Feb-14-20151.pdf.