Category Archives: Uncategorized

A Model of Progression: Geometric Series

I’m still thinking about the idea of a model of progression — breaking down a complex activity into a series of manageable components.

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!

A Model of Progression

From Daisy Christodoulou and Dylan Wiliam:

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:

  1. Get confident with angles
  2. Start visualizing side lengths
  3. 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.

Hidden Lives: Student Thinking

I’m reading Graham Nuthall’s book The Hidden Lives of Learners. It’s a fascinating summary of a career of research working to take the student’s perspective and learn more accurately what students experience in the classroom.

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.

Methods, and How Research Informs My Teaching

Henri Picciotto:

According to Merriam-Webster:

Eclectic: selecting what appears to be best in various doctrines, methods, or styles

That 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”:

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.

Pitching the Common Core

A few years ago, Andrew Stadel wrote a number of “elevator speeches” about Common Core Math. I really enjoyed reading them and have used many of the ideas in conversations with other math teachers about the Common Core.

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.

Learning Distributions

I wrote recently about learning distributions, thinking about which of the following graphs best represent my class.

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.

Reaching Every Student

I’m reading John Hattie’s book Visible Learning, and he suggests three questions for schools to ask themselves: “What is working best?”, “Why is it working?”, and “Who is it not working for?”.

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?