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This doesn’t feel true about mathematics. Much of the math I teach I would enjoy going down a similar rabbit hole with students, though it hopefully wouldn’t take as long.

But this comic also made me think about calculus. There are plenty of gaps in my calculus understanding — I’m not sure I could prove the product rule without some significant help, for instance. I’ve worked through proofs of Lagrange error before but I’m a long way from really understanding how that whole thing works. Not to mention the Fundamental Theorem of Calculus, which I can use pretty fluently yet don’t particularly understand why it’s true.

Maybe this is a reminder to deepen my own content knowledge. At the same time, my instinct is that there are times when it’s appropriate for a tool to remain an abstraction. I would like to verify that abstractions work — for instance, use Desmos to verify that a few product rule applications do, in fact, produce appropriate derivatives. I wonder if I could come up with criteria for when an abstraction is far more useful than understanding why that abstraction is mathematically correct.

I’ve seen the phrase “The four Cs” thrown around more and more recently — critical thinking, communication, collaboration and creativity. 21st century skills, etc etc. I’m going to zoom in here on collaboration.

I want my students to be more thoughtful and effective collaborators when they leave my class. What I know less about is how to structure experiences that will teach students to do so.

Students should collaborate, sure. I find purposeful partner and group work to help students better learn math. But does it also help teach them to collaborate?

I find it interesting that teachers, who are seemingly charged with teaching collaboration, work in a profession where there is little sustained collaboration between colleagues in many schools. In reflecting on my experiences working in groups it seems I have learned much less than I would like to think about collaboration, the general, all-purpose skill, and more about collaborating with those specific people in that specific context.

This person has great ideas but when I ask him to write something up it will take two weeks and three reminders, I should just do it myself. This person is great at taking the student perspective and thinking through how it will impact their experience; make space for her to share before we make any decisions. This person gives excellent, honest feedback; even when it stings, I know that it comes from the right place and is on the mark. Those are the types of lessons I think I’ve learned from my experiences working collaboratively.

I’m sure I’ve learned broader skills of collaboration along the way. My point is just that my practice of putting students in groups “because they need to learn how to collaborate” is probably insufficient to meet the goal. Seems likely that humans learn collaboration like any other skill: practice and reflection. Plenty of practice, spaced over time, and reflection that is mindful of how lessons learned may apply in new contexts in the future.

Now to figure out how to do that.

Summer is here (apologies to those final folks who are still in school). Every summer I try to spend some time doing math, focused on challenging myself and learning in ways that will support my teaching. I’ve got four different ways I’m doing that this summer. No big commitments for me, but instead a few different avenues to explore and learn when I have the time and inclination.

Exeter Problem Sets
The Exeter curriculum is online and free. It starts with the upper-middle school math that leads into Algebra I, and continues through calculus while exploring some lovely math along the way. Everything is problem-based, and the curriculum is really just problem sets that build high school mathematics piece by piece. Every time I have dug into the Exeter sets I have learned new math, made new connections, filed away different problems for future use, or had some inspiration about how to better sequence and teach the ideas in my curriculum.

Park City Math Institute Problem Sets 
The PCMI problem sets are similar, but are designed for math educators. They require little prior knowledge to dive into and explore some fascinating math topics, while also providing a great opportunity to play with ideas, make connections, and discover. This summer’s sets are being posted one day at a time on this website, and prior years can be found on the same site.

Brilliant 100 Day Challenge 
I had never heard of the Brilliant website before, but they are posting one challenging problem a day over the summer for 100 days. The problems are excellent, and I’ve had fun exploring some other ideas on their site as well.

GDay Math 
James Tanton’s GDay Math site  has a few different courses to work through.  I have previously explore Exploding Dots and did quadratics; in both cases I learned for more than I expected about topics I thought I already new. He also offers courses on fractions, combinations and permutations, and area.

Happy mathing!

The school year has ended and I’m on to summer vacation. I’m working on a few projects, some related to teaching and some not. I’m also thinking about next year — my priorities, my goals, and my commitments.

In that reflection, I realized that one commitment I haven’t even considered ending was this blog. Writing has become a central part of my identity as a teacher. I think things through in writing. I encounter a challenge in the classroom and start thinking about how I can write about it. I set goals for what I want to learn, and writing about that process holds me accountable and helps me cut through to the essential takeaways.

At the same time, I’ve built a written record of how my ideas have evolved over time. I’ve become someone who can sit down and write when I need to — I no longer put off writing tasks as long as possible. Writing has opened doors and created relationships in my professional life that I never thought would be possible.

This is all to say thanks for reading. And for anyone out there who has considered starting a blog — I don’t know if it’s for you, but it’s absolutely worth a try. Most of all, my advice is to write for you. Don’t write worrying about what others will think or how it looks to someone you don’t know. Write because it will help you learn, help you reflect, and help you grow. You can learn a lot by sitting down to write about your teaching, once a day, once a week, once a month, or once a year.

Since grad school, I’ve been using the same set of student survey questions to assess my teaching. The questions are linked here, though I have since moved them into a Google Form.

These questions are drawn from the Measures of Effective Teaching project, funded by the Gates Foundation. They write in their preliminary report about the design of the student survey:

The goal is not to conduct a popularity contest for teachers. Rather, students are asked to give feedback on
specific aspects of a teacher’s practice, so that teachers can improve their use of class time, the quality of the
comments they give on homework, their pedagogical practices, or their relationships with their students.

I think this is a great premise. If I ask students whether they liked my class or how good my teaching is, their answers are likely to be heavily influenced by whether or not they like me and how they are feeling that day. If I ask more specific questions about my instruction, I’m more likely to get useful and objective information about my teaching.

The survey above selects a subset of questions from the original study and were adjusted to use Likert scales. I’ve stuck with them because they are the questions I used first and I’ve found it helpful to gather comparative data over time.

It’s often a bit of a blow to my ego to hear what students have to say. At the same time, comparing different groups of students has led to valuable insights. For instance, at my current school, I initially did poorly on the question, “Our class stays busy and doesn’t waste time.” I got a 3 from my first cohort, which is an average of “Sometimes”. I then bumped up to a 3.4, then a 3.7, then a 3.9, which is nearly an average of “Usually”. This slow but measurable improvement in one aspect of my classroom management has been gratifying. This is not to say that one survey questions should define my teaching. There’s no reason I should expect a perfect score on that question, and I could do well and still not do much to support student learning. But I haven’t changed the fundamentals of my pedagogy in that time, and my students’ perception is that I have used their time more purposefully. For another perspective on the relationship between using student time effectively and classroom management, check out Matt Vaudrey’s thoughts here.

A frustration from the survey has been the question, “How clearly does this teacher explain things?” I have barely been able to budge this one, and after my improvements on two classroom management questions it has hovered at the bottom for my last two cohorts, somewhere just below “Usually”. Not that this is a terrible result, or that explaining things clearly is the only thing that matters in my teaching, but I think it does matter, and I would love to find ways to practice that skill and improve students’ perception that I explain things clearly.

I think that this survey is useful, but it also has limitations. I think it could be complemented effectively by some open-response questions that ask students to talk about one area I’m doing particularly poorly in, and to solicit broader feedback than is possible using Likert scales. A survey is just one way of assessing my strengths and areas for improvement. But it only takes a few minutes to have students fill out a Google Form, and I’m most happy that I’ve stuck with the same questions over time so that I can compare cohorts and try to measure my own improvement.

Final tip for anyone who wants to use this: the below formula might be useful to anyone converting Likert scale ratings into numbers to more easily average and find trends. You can just drop this into a new tab, name your tab with the raw information “Data” and drag to convert every rating to a number.

=IF(OR(Data!B2 = “Much better”,Data!B2 = “Much harder”,Data!B2 = “Much more”,Data!B2 = “Always”,Data!B2 = “Extremely well respected”,Data!B2 = “Strongly agree”,Data!B2 = “Extremely clearly”),5,IF(OR(Data!B2 = “Somewhat better”,Data!B2 = “Somewhat harder”,Data!B2 = “Somewhat more”,Data!B2 = “Usually”,Data!B2 = “Very well respected”,Data!B2 = “Agree”,Data!B2 = “Very clearly”),4,IF(OR(Data!B2 = “Similar”,Data!B2 = “Similar”,Data!B2 = “Similar”,Data!B2 = “Sometimes”,Data!B2 = “Somewhat well respected”,Data!B2 = “Neutral”,Data!B2 = “Somewhat clearly”),3,IF(OR(Data!B2 = “Somewhat worse”,Data!B2 = “Somewhat easier”,Data!B2 = “Somewhat less”,Data!B2 = “Rarely”,Data!B2 = “Slightly well respected”,Data!B2 = “Disagree”,Data!B2 = “Slightly clearly”),2,IF(OR(Data!B2 = “Much worse”,Data!B2 = “Much easier”,Data!B2 = “Much less”,Data!B2 = “Never”,Data!B2 = “Not at all well respected”,Data!B2 = “Strongly disagree”,Data!B2 = “Not at all clearly”),1)))))

I’ve been thinking recently about the difference between the principles of an idea and how that idea functions in the classroom; the difference between theory and practice. Conversations about education, especially those in popular media, tend to make broad generalizations on principle without ground-truthing to figure out how an idea plays out with real live teachers and students.

The principles of personalized learning, that one-size-fits-all education does not meet the needs of every student, are undoubtedly true. But in practice, that idea often functions to put students in front of computers for long periods of time, creating lifeless classroom where learning is reduced to spreadsheets and joy is sucked from the room.

The principles of Understanding by Design are useful to organize purposeful curriculum. But in practice, that idea often functions to require teachers to write an objective or enduring understanding on the board, without actually engaging with backwards design or creating a more meaningfully sequenced curriculum.

The principles of constructivism, that students must create their own meaning of new ideas, communicate something true about human cognition. But in practice, that idea often functions to ask students to figure everything out themselves and withhold necessary supports in the name of inquiry, a pedagogy that exacerbates inequities by hurting previously low-performing students the most.

The principles of mindset research, that growth or fixed mindsets have a large influence on future learning, are sound. But in practice, that idea often functions to reduce mindset thinking to platitudes about praising effort rather than ability, platitudes that are often hollow and certainly insufficient to the challenging task of changing mindsets.

These are just a few examples; I could go on. My point is that I have worked to practice this type of thinking; when I hear an idea in education, I try to stay just as curious about the broader principles as about how it functions in classrooms with the imperfections of teachers and the fickle nature of learning.