I’ve been working to develop what I’m calling a meta-curriculum: a set of supplemental activities that push students to think about the why and how of learning math. Why is math worth learning? How do students learn best? Right now I have four different parts to the curriculum. None of these are meant to be one-offs; instead, these are resources I’ve put together to help me more consistently weave this ideas throughout my regular curriculum. These are all ideas I’ve used before, but in much more haphazard ways, and I used them much less consistently in my lower-level classes due to the pressures of time. Part of the reason for formalizing this is to hold myself more accountable in all classes, and to become more articulate and coherent in my communication of these messages.
This is a work in progress. I’ve piloted much of it this fall, and will pilot more before the end of the fall semester, and then go live with a more formal scope and sequence in the spring semester. In the meantime, please pass along feedback or suggestions to help make this a little better. Also, a big thanks to many, many folks who have already given feedback to improve different pieces of this project.
Why math is an elevator speech and series of anecdotes around why math is worth learning, and what the purpose of math learning is. The goal is to be both proactive and reactive — to make clear from the start of the year why we do math in my class, and reinforce this message when the opportunity comes up throughout the year. I have two “canned” responses ready for the inevitable “why do we have to learn this?” and “when am I ever going to use this?” I’ve also collected a few favorite essays, some suited for teachers, some for students, meant to both remind and reinspire over the course of the year.
I don’t teach math because you’re going to use it in the “real world”, or to get a job. Those are happy outcomes of a broader goal that’s much more important to me. I don’t much care if you remember how to graph a rational function in five years. I do care about whether you can reason — whether you can think abstractly, construct a logical argument, find patterns, make generalizations, evaluate a model, and use technology strategically. Those are skills that you will use in that place called the “real world”, and the best way we know to teach them is through mathematics.
Canned response to “when are we ever going to use this?”
Maybe never. You might never use what you learn about the Monroe Doctrine, or cellular respiration, or the motifs of A River Runs Through It either. Maybe you will, I don’t know. But that’s not the point. The point is to reason, to write, to argue, to figure things out, and to be a little more prepared for whatever life might throw your way.
Observation about student engagement
I had a student who really didn’t like my class. He was late literally every day. He avoided actually doing any math, sat back during group work, rarely did homework. But one of the times he was most engaged was working with complex numbers, looking at patterns in consecutive powers of i. It was literally a problem about imaginary numbers — but the pattern was perplexing for him, it was at the just right level of difficulty, and suddenly he was engaged, helping his peers, and making conjectures. That’s the point of math class — to find questions that interest you, and to answer them.
Readings for teachers and students
If We Talked About Other Subjects the Way We Talk About Math, Ben Orlin
A Mathematician’s Lament, Paul Lockhart
What Is Mathematics For, Underwood Dudley
[Fake World] Conjectures, Dan Meyer
Habits of Mind
These habits of mind are my attempt to distill the standards for mathematical practice into more concise, student-friendly language. This will be a work in progress, but the purpose is to set aside times for students to reflect on the habits of mind and think about times that they have applied them in tasks we’ve worked on. This is not meant to be evaluative or prescriptive — I’m not grading their habits of mind, or telling them they have to use certain habits for specific problems, just setting aside time for students to reflect on what they will (hopefully) bring away from my class, and put into words some of the goals of math class. This is not a one-off, and I think it’s unlikely to be effective at the start of the year. In my class, students are doing math from day one. After two or three weeks, I plan to have them look back on what they’ve done and find examples of these habits of mind, and to continue this reflection over the course of the year as we work together to define what it means to do math.
Habits of Mind:
- Numbers can represent objects, symbols can represent numbers, functions can represent relationships. Giving these mathematical objects a life of their own and making inferences based on them is reasoning abstractly, and allows us to solve problems with new tools.
Find patterns and make generalizations
- Patterns are everywhere; seeing them, and finding general rules based on those patterns helps us solve new problems in the future. A generalization represents a truth about a large class of problems; instead of having to solve a problem anew every time, we can use generalizations to find a solution.
Construct a logical argument
- Math provides unique opportunities to reason based on what we know and prove a statement beyond doubt. Mathematical arguments are not always made in words; knowing when mathematical symbols can make an argument more clearly or concisely is of equal importance to eloquence in writing.
Probe for deeper structure
- Two problems that look different on the surface often require the same mathematics to solve, and one object can often be looked at from multiple perspectives and connect to a larger set of objects that have the same structure. Looking for and making use of this structure organizes learning into a mental framework and suggests new solutions to problems.
Use technology strategically
- Technology can solve some problems much more easily than humans; others require the human intellect. Technology streamlines computation and calculation, and minimizes errors, but it is not a substitute for understanding math, and technology is only as intelligent as the human interpreting it.
Attend to precision
- Mathematics requires consistent attention to detail, even while working with a larger number of variables or engaging in complex problem solving. Being able to attend to these details and express them clearly is esential to formulating, solving, and communicating solutions.
Model with mathematics
- Modeling is, simply, taking the world and turning it into math; using that math to solve a problem, then translating the math back into the world. Doing so requires formulating a model, identifying relevant information, solving, then validating the model against the reality of the world.
How We Learn
How we learn is a series of key ideas, excerpts, and studies from the book Make It Stick framing how students learn, and reinforcing how those structures are reflected in my class. The goal is to justify some of my teaching practices that might be unusual, encourage students to persevere in the face of struggle, and build in students an understanding of how they learn most effectively. These resources already exist in many forms, with thanks to Julie Reulbach and Meg Craig who helped inspire this, but I want to disaggregate key ideas and reinforce them over the entire year, rather than try to teach all of the principles at once.
I have a quick summary of each principle in my own words, longer excerpts from the book describing the principle in more detail, and a few studies excerpted from the book that further support these ideas. The purpose is both to teach students something about their own learning and to justify some relevant things we do in class — most important to me are justifying spiraling and interleaving topics, and introducing the idea of desirable difficulty and its benefits for learning.
The principles I chose to focus on from the book are retrieval practice, desirable difficulty, spaced practice, interleaved practice, elaboration, generation, and structure-building.
I won’t drop everything in this post because there’s so much material; instead, read up on this google doc which has the summaries and excerpts, and will be updated as I find more resources.
I hate encouraging participation for the sake of participation, or attaching a grade to how often a student raises his or her hand. This participation rubric is meant to be a low-stakes way for students to reflect on their participation and engagement in class. It is ungraded, and meant to focus on a few high-leverage actions that make a difference in learning. Students will have a chance to reflect and identify areas they’d like to work on every few weeks, and the language here provides a jumping off point for conversations or interventions with students who are struggling.
I don’t want you to participate for points. I want you to participate to help you learn, and support the learning of students around you. These are ways you can take ownership of your learning and make the most of time in class.
- Learn by engaging with problems, asking questions, and doing math.
Advocate for yourself
- Speak up when you need help, and be proactive about addressing what you don’t know.
Share successes and failures
- Share unique ideas and own your mistakes to help everyone learn from them.
Give others a space to learn
- Share the floor, be open to new perspectives, and help others without thinking for them.
- How do you think you are doing with participation so far in this class? Why?
- What area do you most want to work on the next few weeks? Why?