I have trouble talking about my teaching

I have trouble talking about my teaching.

Part of the reason for this is that teaching is so damn complicated. This makes it hard for me to have conversations about how my students learn — especially to teachers that I don’t know. Sometimes I just avoid talking about myself because I fear the incoherent answers that I’ll provide to the questions about what I do every day with my students. This got much worse last year when I started teaching through problems.

To get better at talking about myself, my classroom, and my students, I’m going to simulate such a conversation. I’m speaking to a fellow algebra 2 teacher. The scene opens when I make a comment about how they’ve set up their units.



Me: I like how you’ve structured your units. After your review unit, you start the year with exponential functions? That’s interesting.

Teacher: Yeah, it’s only the third year we’ve taught the course with the new standards, so our math team likes starting with exponentials and then diving into logarithmic functions. We then go into polynomial and rational functions and end with stats, probability, and trig.

Me: I like that. The standards place a huge emphasis on exponential functions now…it’s good to get that off the ground early. This is also my third year teaching with the new standards. There’s so much content in algebra 2 that I’ve run out of time each of the last two years. I couldn’t teach it all.

Teacher: There’s so much! How do you do your units?

Me: [Feels uneasy] Umm…I don’t have units.

Teacher: [Confused look on face] What do you mean?

Me: I suck at doing it, but I teach through problems.

Teacher: 

Me: It’s confusing, even to me. All the topics that were in my traditional units are now all mixed up…but it a way that helps bring them closer together. Instead of having discrete units where topics are isolated from one another, the problems allow for the concepts to be easily interleaved, spiraled, and married in ways that I found hard to do when I had units. I’ve realized that a lot of what and how students learn in math class can be studied nonlinearly…and that’s what my classroom reflects.

Teacher: So, wait, are the kids just solving random problems? How do they learn?

Me: Sort of, but I think a lot about how I sequence the problems. I’m very intentional about which problems kids do and when they do them. So while on the surface the problems may look random, underlying themes and concepts from algebra 2 emerge for students through the problems over time.

Teacher: … [still confused]

Me: Here, let me show you what I mean. In a typical math class, the units are sequenced and taught linearly. [Gets paper and begins drawing] For example, take four units from the school year. Traditionally, when we finish with one unit, we move on the next. [shows drawing below]

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Instead of using that model, I interleave the topics, skills, and vocabulary from each unit to span the entire school year. My old units are now parsed. Think of the first unit in purple as broken up into smaller pieces and spread out over the course of the school year. [Shows drawing below]

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Then the same for the 2nd unit in gold. [Updates drawing]

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And so on with the remaining two units. Notice that some pieces are bigger than others. In the end, it might look something like this: [Updates drawing again]

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One of my goals in using this model is that, since students are frequently revisiting key ideas from the units, it helps with retention. With all the units mixed up, it makes it harder for the students to remember what they’ve learned…but that’s the point. It’s messy by design. That said, I build coherence by thoughtfully sequencing problems.

Teacher: Hmm…I’m wondering how your lessons look?

Me: Well, I typically assign them 3-5 problems for homework. The problems aren’t “practice,” as homework is traditionally viewed. They are more like puzzles or explorations that I ask the kids to do before class. It’s not expected that they understand and speak to all of the problems when they walk into class…I fully expect them to have questions. I also expect them to do individual research to help them figure out the problems. And because of how concepts are interleaved, the problems are usually all on different concepts — and have roots in different units. We don’t typically study one idea per day as is customary in math class. Instead, we study several ideas — and sometimes they are not directly related.

Anyway, we’ll spend the entire period discussing the homework problems in small groups and as a whole class. I have large whiteboards all over the walls that help with these discussions. Students are fully responsible for putting up problems and trying to gain a better understanding of them together. If they cannot (or do not) put up meaningful work to drive our thinking for the day, then they don’t learn. Also, I put few constraints on how the discussions look and feel. The kids typically move about the room freely.

Teacher: So where do you come in?

Me: Most days I help students make sense of the problems while in small groups. I also sequence student presentations of solutions for the whole class discussion. Equity of voice is important here — I keep track of who presents and how often. I also step in with direct instruction on the problems when it’s needed.

On other days, usually 1-2 times per week, things will look more like a traditional lesson where the problems focus only on one key concept. I consider these my anchor experiences that usually focus on high-leverage concepts (like sequence notation or logarithms). I also bring in Desmos Activities all the time.

Teacher: I wonder, where do you get the problems that you use?

Me: All over the place. I steal most of them from other teachers online, but I do write some myself. Those suck. I use Regents problems, too.

Teacher: How did you learn about all this?

Me: Two summers ago I attended the Exeter Math Institute. It blew my mind. As an immersive PD experience that pushed me beyond my comfort zone, it helped me completely reimagine what math teaching and learning can look like. It was different and challenging. It was led by a teacher from Phillips Exeter Academy who used one of their problem sets with us for a week. Exeter has pioneered the problem-based model that I’ve adopted…and they are well known for their problems — they’re tough, but they’re rich. I have included a couple of them in the problems that I give my students.

Teacher: This sounds interesting…I would love to see it in action.

Me: You are welcome any time. I must say, though, there are tradeoffs to using this model. Lots of them. First, students generally don’t like it…at least initially. Giving them so much control and disrupting what they know to be “math class” causes plenty of frustration and discomfort. And they are regularly confused and don’t always leave each day with a “clean” answer or understanding of a problem or concept. This can be hard for everyone — them, me, their parents. Last year, I wasn’t prepared for the amount of dislike and pushback I got. Second, since students learn content nonlinearly, it’s a mess for me to plan and sequence. Also, each day can be somewhat unpredictable because what we do each day is largely dependent on students’ independent work before class and the motivation to drive learning during class. Our discussions can suffer as a result of kids not doing their part…which happens A LOT. What makes this worse is the fact that I’ve never met another public school teacher using this approach…so I haven’t been able to critically bounce ideas off anyone. This makes it very hard to improve. I miss co-planning. There’s more, but, yeah…[awkwardly changes the subject]

 

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Factoring trinomials by first rewriting them

So back in December, I gave this problem to my students:

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To my surprise, it got a lot of traction with the kiddos. We spent the entire period talking about it. The idea was for them to see how rewriting a trinomial with four terms helps us to factor it. I’ve used this approach, often called the “CAB method” and used with a large “X” to organize the product and sum of the A and C terms, to factor trinomials for the last several years and I really like it for two reasons:

  • It doesn’t matter if a is greater than 1.
  • It naturally integrates factoring by grouping. Traditionally, grouping is learned after factoring trinomials. But with this approach, I teach grouping before we even see trinomials. Yeah:

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So, yeah, this is all great, but as I was explaining this approach to a colleague, she asked me why it works. It was in that moment that I realized that I had no idea.

Well, it turns out that later that day she went ahead and wrote up a proof of the method.

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I read a quote somewhere or heard someone say that the real usefulness of algebra is the ability it affords us to rewrite things in order to help reveal their underlying structure. This method surely epitomizes that idea.

 

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Engaging tasks for students that I’ve never met

Anyone who teaches in New York City Public Schools knows that from time to time you get asked to cover a class when a teacher is absent. Personally, despite the hectic nature that is a school day, I typically enjoy these coverages mainly because I get to meet and interact with lots of students that I either don’t know or don’t teach.

Lots of times, there is an absentee lesson plan, but many times there’s not. What’s the result then? Me in front of a class of 30 adolescents for 45 minutes with nothing to offer them. Last year I realized that was tired of this. Here is an attempted remedy.

I’m want to compile mathematics tasks meant engage students that I’ve never met before. Since I teach high school, I will assume nothing about prior knowledge or motivation besides the students being in grades 9-12. The tasks should be highly accessible. Also, in terms of materials, I’ll have nothing but a whiteboard and/or Smartboard at my disposal. I love mathematics, but since I’m not the best at coming up with stuff on the spot, this page will be a necessary resource for me. It’ll be updated regularly whenever I come across worthwhile ideas.


  • The Four 4’s. Express the numbers 1-20 using only four 4’s and any set of operations. Additional challenge: express the numbers 21-???)
  • Similar to The Four 4’s: Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, x , ÷, and parentheses), express all of the integers from 1 to 25. Source.
  • Sprouts | A fun game that involves nothing but a pencil and paper. Get’s deep.
  • The password riddle | Connect the computer to Smartboard to show video. There are loads more like this from Ted-Ed.
  • What comes next? O, T, T, F, F, S, … | A clever little sequence.
  • Which One Doesn’t Belong? (Numbers and Shapes) | These give every student an opportunity to show off their mathematical perspective.
  • Add seven subtract one | A great problem to promote numeracy.
  • Are there any operations that make the equation 5   5   5   5  = 19 true? (Source)
  • Variable analysis game.
  • Find as many patterns as you can in Pascal’s Triangle.
  • Various problems from the Man Who Counted (book) by Malba Tahan

 

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End behavior of functions via Connecting Representations

As the second and final assignment for the Structured Inquiry course I’m taking with New Visions, I was asked to create a task using the Connection Representations instructional routine (#ConnectingReps). More on Instrutional Routines here.

The instructor, Kaitlin Ruggiero, mentioned that multiple choice questions are good starting points for developing these tasks. Adopting her suggestion, I used #4 from the June 2016 Algebra 2 Regents Exam. The question focuses on roots and end behavior of a function. (F.IF.8). I chose to narrow my focus to strictly end behavior.

Here are the first set of representations, graphs of several polynomial functions:

Here are the second set of representations, statements about the end behavior of each graph:

During rehearsal, I showed graphs A, B, and D and their corresponding end behavior statements. We followed the routine to match the representations. I then revealed graph C and had them come up with the statement, which is 4. Lastly, the class reflected on what they learned using meta-reflection prompts.

I don’t have a formal write-up of the activity, but here are the above images.

Reflections:

  • I designed this for my algebra 2 class. My gut is telling me that it may fit in well at the beginning of my rational and polynomial functions unit. I may also consider using it if/when we review domain and range.
  • Initial noticings about the graphs had more to do with the “inner behavior” rather than the end behavior. In other words, the class was drawn to the minima, maxima, and roots.
  • There was some blank stares when I revealed the statements. This will most likely happen with students, too. The mapping symbol (i.e. function arrow) can be confusing if you’ve never seen it before. But that was the point.
  • Most of the class chunked all of the “x approaches…” statements and realized that they were the same in each representation. Since two of the given graphs (A and B) had both ends going to either positive or negative infinity and the other graph (D) didn’t, this led them to conclude that graph D had to match with statement 2. From there they reasoned that since graph A is going up on both ends, it should match with statement 1. Similar reasoning was used to match graph C with statement 4.
  • By giving the class three graphs and three statements, the third match (C and 4) was kind of boring. I still made them justify why C and 4 matched, but it didn’t feel as meaningful.
  • In retrospect, I wouldn’t change any of the representations, but I would revise what I give the class and what I have them construct on their own in order to help them move between representations more fluidly:
    • Give only A and B and their matching statements (1 and 3). Students reason through the matches.
    • Then give graph C and have them construct the corresponding statement (which is statement 4).
    • As an extension, I would give statement 2 and have them sketch a graph that goes with it. All student graphs will be similar to graph D, but the “inner behavior” will be all over the place. Because of the infinite number of possible correct responses, we could show several graphs under the Elmo to guide this part of the routine. I could save these student-generated graphs for later analysis on other properties, including even/odd, roots, maxima/minima, etc.
  • I didn’t use the words chunk, change, or connect at any point during the rehearsal. This is somewhat disappointing since I want my students to use these terms to describe their reasoning during this routine. Mental note taken.
  • Instead of me selecting the next presenter, sometimes I should allow the student that just presented to choose. This is student-centric and I like it (when appropriate).
  • Compared to Contemplate then Calculate, I feel that Connecting Representations is a slightly more complex in nature. With that said, Connecting Representations uses matching, which is really user-friendly. Both routines emphasize mathematical structure, but it seems to me like Connecting Representations emphasizes structure between different representations while Contemplate then Calculate focuses on structure within a representation. Dylan Kane and Nicole Hansen hinted at this during TMC16.
  • Though I didn’t use Connecting Representations, this past spring I foreshadowed this work with my Sigma notation lesson. Given two representations (sigma notation and its expanded sum), students used reasoning to connect the two.
  • This in-depth experience with both of these routines will allow my students to surface and leverage mathematical structure through inquiry like never before. So exciting!

 

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Contemplate then Calculate

I’m taking a Structured Inquiry course this summer offered by New Visions for Public Schools. As part of the course, we are required to design a Contemplate then Calculate instructional routine and rehearse it with the class. David Wees and Kaitlin Ruggiero have both done some inspiring work with #CthenC routines and much of my motivation comes from them.

If you’re unaware of Contemplate then Calculate, the goal is to get students thinking about mathematical structure. The routine leverages student observations, honors their approaches, and highlights reflections on their own thinking. There is a public collection already created by David, Kaitlin, and the New Visions team.

Here was my task.

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Reflections:

  • I initially planned for this to come at the end of an exponentials unit, but after some discussion, I think it would fit much better at the beginning of the unit – or at least before we discuss exponential equations. One thing I’m great at is having students get overly consumed with procedures, so my hope is to get them thinking instead about the structure of the equation and the relationships that exist within it before any mention of the term “exponential equation.” I want their mathematical insights to drive how we solve these equations…and the entire curriculum.
  • To this end, I thought of four strategies that students may use, ranging from guess and check to equating exponents (excluding the one involving logarithms since the task would be presented before they appear). Interestingly, guess and check was not an approach that was adopted much during rehearsal (possibly because they were all math teachers).
  • Since my phrasing of the number of solutions was ambiguous, Robin mentioned that students could use a graphical representation to address this. They could reason that only one solution exists because f(x) = 10^(x+5) -1 is one-to-one.
  • I also considered how to represent the equation. Specifically, I thought about 100 = 10^(x-5),  10^(x-5) = 100, 100 – 10^(x-5) = 0, or using any other set of constants in place of 99 and 1. All of these alternatives have consequences for how students may approach the problem.
  • During the rehearsal, I did a fairly poor job at annotating, which is a critical aspect of the activity because it models student thinking for the rest of the class. Although I anticipated all the strategies while planning, I rushed myself during the process and the result was unclear and unorganized.
  • The reflection prompts should be tailored to the goals of the activity. The more specific they are, the more beneficial the reflection.
  • I should omit the question (e.g. “find all values of x”) during the flash of the task. This move opens things up for more diverse observations and student thinking.
  • An interesting extension was asking what if the 1 wasn’t there? How would this impact our strategy? This is a nice prelude to logarithms.
  • Designing this activity made me think about mathematical structure like I never have before. Often times I take student thinking related to structure for granted. This activity helped me better value student understandings of mathematical structure – and how to leverage those understandings to enhance learning.
  • One of the most beautiful aspects of this routine is that it goes beyond mere discovery learning. The goal isn’t for students to end up at the same strategy. The idea is to develop fluency and flexibility in their numeracy. It echoes the theme from my current book, Building Powerful Numeracy for Middle and High School Students by Pam Weber Harris.
  • All of my planning materials can be found here, which include the anticipated strategies students will use, my planned annotations, the annotations I actually did during the rehearsal, and the slides.

 

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