I tried to put this idea into practice with the formula for the sum of a geometric series. Here is a proof of the formula, lifted from Purplemath.

I’ve tried to teach this proof before, always pretty unsuccessfully. The long division step seemed like hocus pocus and lost most kids, and they either got lost in the sauce or disengaged completely. Maybe a few kids got something out of it, but it also reinforced student ideas that math doesn’t make sense and is something done to them, rather than with them.

I want students to understand this complex idea, and one essential building block is to understand the following property of polynomial multiplication and division:

So the day before I planned to introduce the geometric series formula, we did an instructional routine drawn from Routines for Reasoning called Recognizing Repetition. I gave students these expressions:

First, I asked students to distribute and rewrite each expression and notice what was being repeated each time. They worked individually at first and then shared ideas at tables, and each table reached some informal ideas about every term cancelling except the first and the last. Then, I asked students to think about generalizations they could make to formalize their thinking into a broader rule. This led to a challenging discussion — again, full of informal ideas but without many resources to write a formal generalization. I ended up doing some of the work to formalize our collective thinking, and we finished with a short meta-reflection on the process of recognizing repetition and writing generalizations.

My goal with this task was to separate the polynomial operations from the rest of the mathematical thinking, get students comfortable with the essential ideas of polynomial operations, and set them up for better success with the proof the next day.

Working through the proof felt much more successful than I’ve ever been before, but with one additional challenge. There is a parity issue, where the difference between n-1 and n creates some challenges in figuring out exactly how many terms each expression represents and what the appropriate exponent should be in the final formula. It’s tricky — that if a series includes all terms from 0 to n-1, there are actually n terms in that series. While doing the polynomial operations bit ahead of time made a big difference, next time I need to figure out how to focus on that parity question to help the proof go even more smoothly.

More broadly, I really enjoyed doing this type of thinking, and want to figure out what else I can do to lead into challenging topics in a progression that is designed for student understanding. Next up, exponential functions and logarithms!

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The coach has to design a series of activities that will move athletes from their current state to the goal state. Often coaches will take a complex activity, such as the double play in baseball, and break it down into a series of components, each of which needs to be practised until fluency is reached, and then the components are assembled together. Not only does the coach have a clear notion of quality (the well-executed double play), he also understands the anatomy of quality; he is able to see the high-quality performance as being composed of a series of elements that can be broken down into a developmental sequence for the athlete.(Embedded Formative Assessment, p.122)Wiliam calls this series of activities ‘a model of progression’. When you break a complex activity down into a series of components, what you end up with often doesn’t look like the final activity. When you break down the skill of writing an essay into its constituent parts, what you end up with doesn’t look like an essay.

The key sentence for me is: “When you break a complex activity down into a series of components, what you end up with often doesn’t look like the final activity.”

Sam Shah wrote recently about what I think could be described as a model of progression for learning the unit circle. He breaks his progression down into three phases:

- Get confident with angles
- Start visualizing side lengths
- Putting it all together

Within these phases, Sam goes into more detail to look at the specific questions and tasks that will lead students through each phase of the progression. And the progression is only one element of a larger progression of trigonometric thinking.

It’s important that Sam’s progression for the unit circle takes time to reach complex tasks. The progression doesn’t ask students to figure out too much too soon, and unashamedly focuses on small building blocks in order to build toward larger goals.

I love this type of thinking, and while I’ve done it informally, I want to improve at making progressions a deliberate part of my planning. A template for backwards planning might look like:

- Select broader curricular topics for a course or portion of a course
- Develop models of progression for those topics
- Select day-by-day learning goals that lead through those models of progression
- Outline success criteria to see whether students have met learning goals

I want to try and put this into practice with several units during the latter part of this year and, if it feels useful, make thinking about models of progression a regular part of my planning.

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Rata: Yea, we did a chart on it, but I can’t remember what we put on it now…this big picture on this big piece of paper on the wall. And our group had to do something on weather, and you had to write these, the north, south, east, and west on it, and see, and put, which weather brings the hottest (laugh).

Interviewer: Right, and your group did that?

Rata: Yes, and you had to put it up on the wall.

Interviewer: Right, and do you remember which was the warm, dry one?

Rata: No (laugh).

Interviewer: Can you picture it in your mind, the one your group did? Who did the writing on the chart?

Rata: Bruce.

Interviewer: Did he? Did you help?

Rata: Um, no, the other two didn’t help us, only me and Bruce done it. I did some of the writing on it and he, he wrote it out, and I wrote ‘weather’, and he, um, we both thought it up, and looked on our chart [weather records] to see which one was warm.

The interviewer and Rata are discussing a question she couldn’t remember how to answer about the relationship between wind and weather in New Zealand. I’m fascinated by what she remembers and what she doesn’t. This exchange reminds me of an idea I try to think about in my teaching:

Students learn only what they think about.

If my class does not make students think about math, they’re not going to learn math. In this instance, for whatever reason, Rata was thinking about a lot of things making that poster but not about the relationship between wind and weather.

I wonder how many lessons I’ve taught where, looking back on it, a student could say, “Oh I remember, we were doing this on whiteboards, and Carter was writing it in green and we were on the side of the room by the door…”, but not actually remember the mathematical content I’m interested in. I would bet there are more activities than I would like to admit that don’t cause students to do the necessary mathematical thinking.

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According to Merriam-Webster:

Eclectic:selecting what appears to be best in various doctrines, methods, or stylesThat pretty much describes my stance as an educator.

During my four-plus decades in the classroom, I’ve seen many math edu-fads come and go: new math, individualization, manipulatives, problem-solving, group work, constructivism, constructionism (yes, that’s a thing), portfolios, complex instruction, differentiation, interdisciplinary-ism, backward design, coding, rubrics, problem-based instruction, technology, Khan Academy, standards-based grading, making, three acts, flipping, inquiry learning, notice-wonder, growth mindset… not to mention various generations of standards.

It doesn’t take long for a conversation between teachers to include something sarcastic about the fad

du jour. By being sarcastic, we put up an umbrella to try protect our sanity from the ideas raining on us from administrators, academics, and yes, even colleagues. I will go further, and boldly say to the proponents of the current pedagogical panacea: I’m sorry, but whatever “evidence-based” product you’re selling today, I’m not buying. The research it is based on is flawed. The anecdotes that support it only apply to specific circumstances which are not easy to replicate. In short, as I have written before: nothing works.

Graham Nuthall in The Hidden Lives of Learners:

The term “method” is a convenient shorthand for talking about teaching and about the things that teachers do. But it is dangerously misleading when people begin to think of teaching methods as the equivalent of medical treatments or agricultural fertilisers. It leads to the notion that we can compare teaching methods in the same way as we can compare the effects of different drugs of chemicals. It also leads to the recently popular demands that research on teaching should use randomised trials of the kind used in medical research.

In the realities of the classroom, methods do not exist. Every teacher adapts and modifies so-called methods. Research shows that teachers who believe they are using different methods may be doing essentially the same things, and teachers who believe they are using the same method may be doing quite different things.

John Holt in How Children Fail:

At that point Bill Hull asked me a question, one I should have asked myself, one we ought all to keep asking ourselves: “Where are you trying to get, and are you getting there?”

There are lots of things I use in my classroom that I might call “methods”:

- Three-act tasks
- Desmos activities
- Developing the question
- Number talks
- Vertical non-permanent surfaces
- Standards-based gradingStandards-based grading
- Five Practices discussions

I could name plenty more.

I believe methods are important. I would be a less effective teacher if I had fewer methods to choose from. But methods do not make me an effective teacher. Much more important are my choices of what methods to use, how I use them, and how to learn whether or not they are working.

There is, as Nuthall points out, plenty of research on methods. Much of it is conflicting. I can use those ideas to inform my teaching, but research on methods tends to be prescriptive: do this, and students will learn more. I’m skeptical of any dogmatic claims in favor of one method.

I do believe in research, but rather than research telling me how to teach, I’m interested in research on how students learn. Research can help me understand students’ beliefs about their learning, the relationship between content knowledge and problem solving, how students learn and retain new knowledge, the role of incentives and feedback, and more. None this research is prescriptive, and none of it tells me how to teach on Monday. But that body of knowledge can inform the methods I choose to use, how I use those methods, and how I understand whether or not those methods were effective, on that day, for those students.

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I ended up in a conversation recently with a local woman who had been an English teacher for thirty years and has been retired for twelve. She asked me what I thought of the Common Core. While Andrew’s arguments for the Standards for Mathematical Practice really resonate with me, they didn’t feel like the right tool in this situation. Here is roughly what I shared with this former teacher, which I think is my go-to elevator speech for folks outside of math education, arguing for the Common Core on simple, broad terms:

I think the Common Core is great. We’ve had standards for what students should learn for a long time, and I think we always should. Seems useful to agree on what those standards are. They aren’t perfect — some people have an issue the way the Common Core standards introduce algebraic thinking in math, or emphasize non-fiction in English. There’s plenty to disagree about. But the standards that came before weren’t perfect either. The Common Core standards are, in general, fewer and clearer than previous state standards. And they make life easier for me, because I can more easily use lessons from teachers all around the country who share the same goals I do.

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It was a fun thought experiment. I think that my actual classes usually play out something like this:

It’s just the easiest to teach students in the middle, without challenging high-achieving students or providing adequate support for strugglers. Not something I’m proud of, but it’s the reality most days.

I think my ideal class looks something like this:

I see this as a moral question — if kids have struggled in the past, they need to be my priority in the future. But I think that it can be practical as well. I don’t want to restrict my thinking to narrow learning goals. Hopefully every student moves their thinking forward with the content we’re looking at in class. But students on the left side of that graph often have other learning goals that are important. Social goals around how students learn together. Goals for students’ beliefs and mindset toward math. Foundational skills that those students are learning in addition to content goals for that day of class. With a large menu of possibilities, achieving that distribution seems more doable.

I think it’s also important to think about the difference between these two graphs:

In one, every student learns a lot, every class. I don’t want to shortchange high-achieving students, I just want to broaden what I am able to do to support students who need that support the most.

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That last question seems like the most important. Something I want to avoid is saying “that worked” or “that didn’t” when the truth is, “that worked for most students, but not for the students who often struggle the most”. This led me to a little thought experiment.

Let’s think about two variables. **Prior achievement** — how successful have students been in math class in the past? And **student learning** — how much did they learn today? Here are some possibilities:

(I realize that this representation has lots of flaws. Lessons can have multiple goals, simplifying these variables onto a single spectrum loses important information, and it would probably be more accurate to think of these as scatter plots or probability distributions. And more. But I still think this is a useful exercise.)

**Here’s the thought experiment.** Which of these, if any, represents the “ideal lesson”? Which most often plays out in your lessons? Which distributions are acceptable outcomes? Which distributions are never acceptable? How can different types of lessons complement each other? What other questions are worth asking here?

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The first four principles don’t seem particularly groundbreaking. Makes sense that to get better I need to push beyond my comfort zone, set specific goals, bring a great deal of focus, and receive some form of feedback. The fifth principle — develop a mental model of expertise — is one that I want to unpack here.

**Mental Models **

Mental models, also referred to as mental representations in Ericsson’s book, are one of the key building blocks of expertise. They’re tricky to nail down though, and they look different in various domains. For violinists, a mental model might mean an incredibly clear understanding of what their playing should produce. An expert violinist can imagine exactly what a piece should sound like before playing it, and uses that mental model to bring more life to their piece than if they were playing solely from the sheet music in front of them. A chess grandmaster might spend a great deal of time studying openings of other grandmasters and analyzing the specific decisions those players made. Through that analysis, they build a clear picture of what a great opening looks like and are better able to execute their own openings in different situations.

Experts in different fields use deliberate practice to build these mental models and improve their performance. The practice they engage in is also not simply a rehearsal of what they hope to reproduce. Violinists listen to other great musicians and work meticulously through a piece to visualize what each note will feel like. Chess players study games of great players to gain insights into their own play. Routine practice — playing the violin, engaging in a chess match — is one part of deliberate practice. But it must be accompanied by experiences that deliberately build a mental model of what expertise looks, feels, and sounds like.

So how does this happen for teachers? I’ve got three ideas that I think have made me a better teacher and helped me to practice purposefully when I enter the classroom.

**Cognitive Science **

I think an understanding of cognitive science is absolutely essential in creating an accurate mental model of teaching. If I want to be an effective teacher, I need to have a clear idea of both what I want to do in the classroom each day, as well as what I hope will be happening in the heads of my students. I don’t need exhaustive knowledge of the human brain, just some cognitive principles of learning that can help inform my thinking and teaching. I like Daniel Willingham’s comparison of teaching to architecture: engineering science doesn’t tell architects what a building should look like. But it does give architects some useful principles that influence the probability of positive outcomes. In the same way, principles of cognitive science provide useful guidelines in designing learning experiences, and the more insight I have into how students’ brains work, the better I’ll be able to make sense of and improve me own teaching. I can’t say enough good things about Make It Stick and Why Don’t Students Like School — two books that have taught me a ton about how learning actually happens.

**Observation **

I’ve been lucky to work in schools with welcoming open door policies toward observation. Beginning with my teacher residency year, I’ve observed more than 50 different teachers, across disciplines. Through this observation I’ve learned a great deal. Some of that learning is in little tricks — phrasing directions, or lesson structures I hadn’t considered. But much of the observation has been a gradual process, expanding my ideas of what great teaching can look like. I remember my first lesson of student teaching; I left that class thinking it had gone really well, and that I was pretty good at this teaching thing. I didn’t have a clear idea of what great teaching, or even half-decent teaching, looked like. As I observed more teachers and taught more classes, I developed a better idea of what I wanted my class to look like, and realized how students had learned from those first few lessons. In particular, observing teachers who I know have different strengths than I do has given me new insights into my teaching. While my gains were more dramatic early in my career, continuing to observe teachers helps me to expand my mental model of what teaching can look like and set the bar higher for my own teaching.

**Reading **

I’ve spent more hours than I care to admit reading other educators’ ideas in books, on blogs, and on Twitter over the last few years. Through that reading, I’ve expanded my ideas of what great teaching looks like through ideas like instructional routines and discussion structures for math class. I’ve built a better idea of what great curriculum can look like through materials teachers share and write about. I’ve reconsidered the role of feedback in my classroom and rethought the potential of feedback to support learning. And much more. Reading a new perspective in a book or a blog post doesn’t make me a better teacher instantly. But that knowledge accumulates to create a rich mental model of what I want to do in my teaching, and provides some tools I can use to improve.

**Deliberate Practice **

Using cognitive science, observation, and reading to inform my mental models is something I enjoy — I enjoy learning about teaching, sharing ideas with other educators, and pushing myself in new ways. But beyond my academic interest, building robust mental models improves the quality of the practice I engage in every time I walk into the classroom. These ideas help me to push beyond my comfort zone, suggest new goals to work toward, inspire me to focus on improvement, and provide a form of feedback when I compare my mental models with what actually happens in my classroom each day. I see this as a critical link between knowledge *about *teaching and skills *of *teaching. I often worry that, in my reading and writing and talking about education, I become better at talking about education but not at actually teaching. Focusing that effort on building effective mental models and putting them into practice in the classroom connects abstract learning to deliberate practice and practical improvement for me as a teacher.

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- If homework isn’t useful for learning, I shouldn’t assign it.
- Homework shouldn’t take any student more than 15 minutes. The longer homework takes, the more inequitable it is likely to be.
- Homework should never depend on how well a lesson goes — if we get sidetracked in class or have to slow down unexpectedly, homework should still be useful for students.
- Homework is the best opportunity I have to integrate regular spaced and interleaved practice.
- If a student can’t complete homework one night, for whatever reason, that should not prevent them from learning in class the next day.

I attempted to shift halfway through last year to a model I learned about from Steve Leinwand, which he calls “2-4-2 homework”. In this model, homework assignments consist of eight problems. The first two address the topic we’re currently working on. The next four are mixed practice from other topics in the course. The final two involve some extended reasoning or explanation.

While I really like the elegance of this model, I’ve found it hard to follow through with. Too often I am writing homework at the end of the day when I want to go home, or the period before a class I’m teaching, and with three preps it’s easy to leave writing homework until the last minute. I end up just throwing random problems onto a handout and handing it out. I also wasn’t explicit enough with students about my goals for homework and the purpose of the structure. I do think it had some value in the mixed practice it provided, but definitely also some room for improvement.

One change I’m making is streamlining the way I write homework. I have a little three-section notebook I use for lesson planning, one section for each prep. In that notebook I’m going to start keeping a list of topics that should be the focus of mixed review. This will include prior topics that I know students are likely to get rusty with quickly. For instance, in my Algebra II class I might regularly ask them to solve a simple system of equations, or graph a quadratic function written in vertex form. It will also include skills students will need for upcoming units. If I’m going to start graphing polynomials in two weeks, students should probably be factoring every night, with some other polynomial arithmetic thrown in. Having a list of these topics instead of making problems up off the top of my head every day should make me more efficient in writing homework and also more focused in what I ask and what homework reveals about student thinking.

I usually enjoy writing the last two problems. I might ask students to explain why factored form reveals the zeros of a function during a unit on polynomials, or to find a function that doesn’t intersect the y-axis in a unit on rational functions. These aren’t meant to be the hardest problems they’ve ever seen, but they will hopefully elicit several perspectives. I’ve also found the Exeter problem sets to be useful places to find creative questions; they’re definitely worth flipping through for high school teachers.

I also want to be much more consistent with the way I review homework. I think that I could start each day by having students discuss one of the last two questions at their tables, and pick one or two students to share their perspectives. Then, if I notice any quick hits that I can clarify on the first six problems I can address those. Major issues get filed away for later. Then on to the rest of the lesson.

Another change I want to make is to tell students what the 2-4-2 structure is and why I think it is important — and to bolster that argument with cognitive science. With a clear purpose, when we go over homework I can frame my choices around the purpose of that problem. If there is confusion on one of the first two problems, I may table it and revisit it later in the lesson where that idea fits better. If there is confusion on one of the mixed review problems, I can tell students that we will revisit that topic in the future rather than trying to fix it all right there. I want reviewing homework to be a quick 2-3 minutes whenever possible, and that means not going down every rabbit hole a student is interested in exploring. It’s often frustrating for students when I move quickly through homework review, but making explicit the goals of homework can help to place that learning in the larger context of our goals in math class.

Hopefully these structures will help me be more consistent and more purposeful with homework the second half of this year. Here goes.

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Worth noting that at the start of the year we spent some time discussing “Why math?”. I made an argument for learning math along the lines of Underwood Dudley’s in What Is Mathematics For, which probably influenced responses.

Mathematics is studying the meaning of the world and universe through numbers and patterns.

Mathematics is learning how numbers can interact with each other and how our world is filled with complicated problems that math can solve.

Mathematics is figuring out stuff using things. I don’t know if there even always has to be numbers in there, although that’s what I think of math as mostly being. In different math classes I’ve had to think in a lot of different ways, so I also think it’s that — learning to think in different ways. I always think math is about problem solving, but it’s also weird because all of the math we do was invented by dudes from Ancient Greece, or something like that. I don’t really know.

Mathematics is a form or problem solving in which you find values of certain things using different skills. It’s not about getting the right answer, but about how you got to it. You can do this by finding patterns and applying them to getting your answer.

Mathematics is the study of problem solving in the form of numbers and shapes. By solving equations and mathematical problems, our brains learn how to work through and figure out real world problems. Although we may never need to know mathematical facts again, the skills we learn from math are forever useful.

And, my favorite:

Mathematics is something that is useful up to about 8th grade, and then comes something taught to us to fill gaps in our school day and stress us out. It is utterly useless by this point and I feel we are only taught it to honor the mathematicians who discovered theorems and phenomenons so that they don’t feel like their whole career and life was a total waste of time. Although now they’re probably all dead so who really cares, unless their ghosts come back to haunt us. Even though I believe all of this I do appreciate your honesty when saying we probably won’t use this math again and your class seems legit. Cool bye.

Lots to work on here.

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