In which absurdity is declared underrated

With the holiday’s coming I start thinking about how to handle two weeks of almost non-stop interacting with people who I only get to see once a year or so. How do you recap a year’s worth of adventure in just a few hours? What stories will you tell? How do you get others to tell their stories?

Perhaps it’s odd to some to even be thinking about some sort of strategic battle plan for conversations, but my current life in the woods with occasional flutterings into civilization to do/attend professional development has skewed my perceptions toward human interaction. And I’ve been learning French.

I’d never noticed it in English before until I saw that French-speakers do the same thing with respect to opening pleasantries. I know that a normal respond to “Hey, what’s up?” is “What’s up?” I know this. I have followed that script as long as I can remember. And, granted, there is some tonal work involved to indicate a level of current expressed happiness, but still, what’s up with the parroting? This happens in French as well where responding to “Ça va?” with “Ça va” a thing. And how many student interactions do you have on a daily basis that boil down to those two words? As you stand at the door while students file in how many “what’s up”s pass through your lips?

It’s normal human interaction: polite, superficial, rarely remembered. It’s a script. Follow it, and you don’t need to think.

And yeah, I know, TIME. It’s not possible to have unique conversations beyond the generic pleasantries with every kid in the class. The math just doesn’t work out if you actually want class to start before the bell rings again.

But what if you tossed out some crazy to a few random students as they walked in? How many students hear “what’s up?” and continue into the room on cruise control? And then through the class in the same setting? Engaging, but never all the way there. Why should they? Routine has been established by two words. The status is quo.

I believe in disequilibrium as a powerful force. How much more alert are you when something odd or unexpected happens? Kate’s a fan of instigating arguments. I especially like her moral that “confusion and mistakes are necessary for learning.” But why wait until the math to sow confusion? Why not start as they walk in the door?

Due to a childhood spent reading a lot of Far Side and Douglas Adams, I have a healthy love of the absurd which causes me to love the TED Ideas article on turning small talk into smart conversation that came across my dash today. This bit caught my eye first:

We stagger through our romantic, professional and social worlds with the goal merely of not crashing, never considering that we might soar.

Not crashing. Auto-pilot. Only partly engaging. How many students do you have in that pattern? I live in that pattern far more than I’d like to admit–doing what needs doing but not pushing hard because I might fail or be uncomfortable. Shaking that mentality is a work in progress and will probably be something I always keep an eye toward. But this isn’t about my personal fears of mediocrity.

This is about a challenge to myself and for you: break the parroting. The TED article has some nice ideas on this. I personally like their framing of not giving the expected response, such as

Beverly: It’s hot today.
Gino: In this dimension, yes.

What sorts of responses could you give to the “what’s up”s and “hi”s students toss your way as they walk in? Current news? Ponderings of world domination? Sneaking in odd comments that actually relate to a problem you are working on that day with the class? Hah, I can imaging doing that daily and once the kids are on to you they start dissecting your responses for clues to that day. They start engaging with the class before you’ve even begun.

Getting students to question the perceived norm existing in their heads about math class, about their peers, about society, and about themselves will always be a focus in my teaching practice. Working to do so through my typical lens of the ridiculous is just a bonus. Though I know of at least one other teacher that’s taking a similar tack.

So I ask you this: what’s up?


In which a blog is recommended

There are a lot of blogs out there but one I actually email subscribe to is One Good Thing. If you don’t read this one, it’s teachers posting about one good thing that happened in their day. Some are big things but most are the small things that happen to make us remember why we teacher.

News is inherently biased toward the ugly and the depressing and the horrifying because that is what sells and generates clicks. This is also why I think sites like One Good Thing are critical for retaining sanity and a balanced outlook on life.

A post this morning from Mr. Dardy made me want to write a quick note to myself her as a reminder when I get back into the classroom. Specifically this bit:

She told me ‘I thought my job in a math class is to know what formulas to use and how to solve equations with them.’ I explained to her that this was certainly part of her job, but that success in a math class should involve more than that.

So, note to self: Sometime early in the year make sure to ask students what their job is in a math class and what qualities are needed to be successful. Use this to start conversations about growth mindset and what mathematics really is as early as possible and maintain those conversations throughout the year.

foothold situation

I’m reading 5 Practices for Orchestrating Productive Mathematical Discussion and ran across the term foothold. Now, I watch a fair amount of sci-fi so this typically means alien possession within an organization to me so I did have a moment of confusion. That’s definitely not how it’s meant here.

While laying out the case for good task selection as a way to promote equity in the classroom, on p19 Smith & Stein note that

“Once a student has a foothold on solving the task, the teacher is then positioned to ask questions to assess what the student understands about the relationship in the task and to advance students beyond the starting point.”

I really like the term foothold used this way. “Will this task allow all my students to gain a foothold?” Isn’t that a nice question to ask as you plan out tasks for your kids?



in which I learn a new term

I find that the ability to remember names of things is not a forte of mine. I’m reading an article from Mathematics Teacher on The Circle Approach to Trigonometry and got to a section where they kept using the word ‘subtended’ and for the life of me I was not picking up what that meant in context (“an angle measure of 1 radian implies that the angle is subtended by an arc 1/(2pi) of a circle’s circumference.”) and I couldn’t pull up a definition from the ol’ memory banks. My brain is a bit slow after a day of reading all the things, so thank goodness for wikipedia and it’s graphics.

The other fun term in this article I think I’ve seen before but never really dug into was covariational relationships. Google search popped up a study from 2002 that defines covariational reasoning as “the cognitive activities involved in coordinating two varying quantities while attending to the ways in which they change in relation to each other.” (p354, Carson, M., et al, Applying Covariational Reasoning, Journal for Research in Mathematics Education, 2002). I like this term a lot as I think it describes the type of reasoning that is very challenging for mathematics students as they are confronted with more and more types of functions.


on transformations and compositions

It was brought to my attention that thinking about transformations of functions as compositions is odd. I think I already knew this, but I wanted to put it out here to see if anyone else ponders them in this way.

To be clear, I’ve never explicitly taught it that way, but I did spend considerable time working to get students to read functions in such a way that they could describe what what happening to x before the function did it’s thing and then what happened to the output of the function (ex: with asin(bx+c)+d b and c are ‘before’ while a and d are ‘after’).

This was my semi-random thought for the day.


The kids that make us into the teachers we want to be

Mathy McMatherson’s most recent post reminded me of a student I hadn’t thought of in a while.

This student made one of my classes a bit of hell for me. And using the words ‘a bit’ is a massive understatement. To the point that I would feel sick to my stomach before the class would start from nerves. I didn’t know how to work with her most days. She swore at me, other students, life. Frequently late and disruptive when she did show. Some days would be all rainbows and sunshine–focus, niceness, working on assignments and with others. But always that lightning storm right in the corner of my eye remained and as the tallest thing in the room my nerves never went away that class.

I don’t deal well with highly emotional situations and I avoid drama as much as possible. If you’re the type of person that thinks yelling arguments are an acceptable for of communication, we’re probably not going to get along. But this was my student. I couldn’t avoid her and I couldn’t understand the choices she was making. I could barely see past my own heart palpitations when she would walk in and my nerves over what could happen the next 50 minutes.

If I am better now at dealing with highly-emotional situations in the classroom, then this student is a big reason for it. Teaching kinda forces you into dealing with others yelling at you and taking out their anger on you and that is exactly what she was doing. It wasn’t me. It wasn’t her classmates. Over time I learned more about her. About what she was dealing with at home. About the stress in her life. Was she responding to the stress well? No. Of course not. She was a kid. One that had had a lot of adults bail on her.

I wouldn’t let myself be one of them.

My nerves never quite settled over the course of the semester, but they got better. When the school year ended and that class finished I exhaled fully for the first time in a long while. I suspect my blood pressure also dropped back to normal levels.

The next year I would see her in the halls and she would stop by my classroom in the mornings or after school once in a while. She would say hi and be all smiles. Sometimes she’d ask a math question and get homework help. I was left completely confused the first few times. In one semester she did more to push me toward being the teachers I want to be than any other student in the past. At the time I didn’t see it being too busy just trying to not break down in reaction to her rage at the world.

I miss her. I hope she’s doing well.


Today’s chart that has me thinking

[This whole thing is a ramble, but it’s an oddly accurate representation of what my brain does as I try and assimilate new information and ideas into the current structure. You’ve been warned.]

I mentioned a few posts back (and yes, I’m still thinking about elevator questions–trying to finish a book first that’s helping me thinking about them) that I’m reading The Art of Explanation by Lee Lefever. Throughout the book so far he’s been making use of the following diagram as a way to plan explanations:

p78. I'm reading an ebook so it may not match up.

p78. I’m reading an ebook so it may not match up.

One analogy he uses that I like is that if someone needs to change a tire, they probably already have the big idea down and are over on the right side of the chart needed an explanation of how. On the other end of the spectrum I think about the students I’ve had in my class. How many times did I start with “how” before establishing why? And I’m good at “how”! I can lay out steps like a pro with three examples and then practice time but if that kids is still over at the start of the alphabet, then I suspect that  “how” is just going to cause them to tune out or teach them how to mimic.

The “why” bit means a lot of things to me that I’m still poking at. Why this method? Why this problem? Why should I care? How do I, as a teacher, present problems in such a way that those “whys” are answered? Off the cuff, I think different tasks are going to do this in different ways. Some will, to borrow Dan‘s phrase, perplex the kids. Others will be topics with human interest (“real world” applications). Whatever we choose, we need to get an answer to “why” that a student will follow down the alphabet into the weeds of “how” and that’s where all sorts of mathematical fun can be had.

So I’m still processing, but I’m liking the image as a way to think about where students will start before designing lessons around specific problems. Yes I’m recognizing that it’s like a fancy way to say the kids need to buy into what you are doing, but I like thinking about ‘givens’ from new frame works and the why/how understanding spectrum depicted in the picture is new for me.

Lastly, I’ve worked with a lot of struggling kids who had a hard time getting to “how to be an active, engaged math student” because they were stuck on why they would want to be one and why they would want to worry about doing well in a math class when there were other things going on in their lives. I can’t address every why my kids come in the door with, but hopefully I can not add to their list and give them a place where more understanding can happen and give them something to be proud of.


Excerpt from my homework

Because I make poor (and awesome) life choices involving my free time, I signed up for a Math Forum online course ( Differentiated Math Instruction: Using Rich Problems to Reach All Learners). Max is running the show and so far thinking about the problems and the readings and reading what others are thinking is really great. Burning a candle at both ends is stressful but for fulfilling things it’s worth it and Math Forum stuff has always been worth it to me.

One of the assignments for this week was about reflecting on some articles and a problem we ‘adopted’. I wrote a bunch of paragraphs, but I wanted to share just one today that has me thinking a lot about meeting Students where they are at with regards to speaking in a math classroom:

I’ve read about ‘noticing and wondering’* in the past, but reading that article now has me thinking about how hard it was to get my own students started on problems without clear paths marked with neon signs for directions. I’m believing more and more that getting started is one of the biggest hurtles. Putting the first words on paper or the first brush stroke on canvas is so intimidating when I’m alone and here we are as teachers trying to get kids to put down their thoughts and ideas in front of a classroom of their peers. I think early-career me didn’t spend enough time thinking about how hard that first step really is so it’s something I’m keeping close now.

*If you are not familiar with the ‘Noticing and Wondering’ strategy employed at the Math Forum, head over to Suzanne’s blog post to check it out. Suzanne goes at it from a MS perspective and there are links at the top for this strategy through HS and ES lenses.


We need an elevator question, not an elevator speech.

I think teaching has an explanation problem. I don’t think this is a new thought by any means, but more than a lot of professions teaching suffers from the Curse of Knowledge. Most everyone teachers meet has done school. Often for 12+ years. This leads people to believe they know what teachers do since they’ve been around them so much.

The other side of this problem that I see is that people often respond to inquiries about what they “do” with a job title. This is especially true in widely known professions (I suspect polysomnographic technologists don’t have this problem). As a side note, I actually try not to ask that question as I prefer “what do you love?” or “what lights your fire these days?” when meeting new people. Way more interesting answers. Try it.

Here is my proposal. We need elevator questions. I don’t think I’m capable of explaining what I do as a classroom teacher in 5 minutes or less in a way that is comprehensible and actually encapsulates the work that goes into herding cats, so I’m not going to go there. What I am trying to figure out is an answer that starts something like this:

Random Person: “And what do you do?”
Me: “I teach. Hey, you’ve done school. Could I get your opinion on this …”
RP: “Huh, I’ve never thought about that. What an interesting/challenging problem!”

where the “…” is an actual classroom scenario that is short, comprehensible to non-educators, and presents a true problem in teaching were multiple options are being weighed. I want to give the general population something to chew over in a way that helps elevate teaching beyond “advanced show-and-tell”. I want everyone to associate teaching with thoughtful and complex and wonderful.

I’m pondering my own “…” and once I work it out I’ll post it. For extra credit reading, I recommend checking out the book The Art of Explanation. If you want a quick overview, check out the author’s 5 minute Ignite! Seattle talk from a few years back.

So what’s your elevator question that will give me a glimpse of the nature of your work?



in which I recommend classroom origami

My old classroom held relics from the students I taught each year. The day before winter break was one that I always took to talk about origami, show some clips from an awesome documentary, and teach the kids a few basic things. Recently on twitter some folding pictures happened which spawned me to fold which lead to classroom origami talk. I’ve shared this a bit before over at Global Math, but thought I would document a bit here.

PHiZZ 105 torus

Exhibit A

Every piece in this was done by a student of mine. Not all the kiddos signed their pieces, but many did. There are 5 of these in my old classroom–one for every year I taught at the school except for the first one due to a storm canceling school the day before winter break (I know, right?). I did, however, put it together. I’ve found few kids have the manual dexterity + patience needed to assemble a large one of these so I do that myself over the break. Here’s what it looked like before I started:

box of PHiZZ pieces

box of PHiZZ pieces

If this project looks huge (100+ pieces?!), never fear. Part of the beauty of the PHiZZ model is that there are various sizes you can make with the same module. Kiddos who get into origami often like to start with the 12-unit (cube!). You can also do 30-unit or 90-unit versions. And yes, you can go higher then those.

where to learn
I learned how to make the PHiZZ Units from this website back in college. Now we have the benefit of Tom Hull himself instructing how to make the unit in a youtube video.  I don’t show videos on how to fold in class since I like to go at the pace the class needs (and the pace a precalc class needs is much different than the pace of an algebra 1 class) and use the document camera. Now, the above torus is a variation on the typical balls people make.

Many of the books I have are well out of print and inherited from my grandmother who taught me how to fold when I was very young. I mostly do modular origami and I like pretty much anything by Tomoko Fuse. There are lots of rubbish websites, but OrigamiUSA has some free patterns and links to local origami organizations.

One of my favorite things about the PHiZZ model beyond the lovely creation at the end is that you can use paper from Staples Memo Cubes to make them. I actually don’t like PHiZZ models made with origami paper as most origami paper is either too slick, to thin, or both. Good washi paper would work, but it’s not worth the cost. I would typically purchase 2-3 of these cubes at the start of the school year and use them to teach some basic origami as well as using them for student reminders (they are bright).

other thoughts
As i was working on this last night30 pieces

I started wondering how do you teach this type of patience? In college I thought nothing of spending the 2-3 hours focus and make a 90 piece PHiZZ model. I find the process completely absorbing and satisfying in the way working a big math problem and coming up with an elegant solution is. How much of the origami side of my nature has influenced the math? How much more accepting am I that good things (beautiful things!) take patience, trial-and-error, puzzling over directions (sometimes in Japanese, which I do not read even a little bit), struggling and putting them down for a breather lest I rip them in half so I can come back later with fresh eyes.

More importantly, how much can teaching students origami help them build up those traits? Listening carefully. Immediate feedback. Attend to precision.

Start with the figures that only take a dozen folds. Do the ones that move or jump because those are so satisfying. Then do harder ones. Make origami a warm up one day a week. Smile with thanks at the kids leaning over to help their neighbors. Congratulate the ones that struggled last week but get this weeks’ figure. Encourage the ones growling in frustration to keep trying and slip them an extra sheet before they leave in case they want to try again later on their own without peer eyes on them.

Tonight I’ll be working on this module pattern. Happy folding :]