Big Teaching Questions

Compiling A List Of Posts

Hi all,

I need some help, if you have a few minutes. I am looking for some quality blog posts and/or websites which offer the following:

Stories from the Front: On the ground experiences of teachers teaching problem solving in the math… the good, the bad, the ugly

War Strategies:  Different ways teachers actually do problem solving in the classroom, and maybe some hints/tips/technqiues (e.g. whiteboarding, Moore Method, Harkness Table, problem sets, grouping ideas, hint tokens, etc.)

Weapons:  Good websites (or books) which contain good math problem solving problems (e.g. Exeter problem sets, AMC questions, etc.). My personal thought on questions is that they don’t need to be hard to be problem solving… In fact, the harder the problems are, the less accessible and fun the problem solving will be, and the more my kids will be turned off.

What I’m not really looking for is Polya’s How To Solve It, which is great reading but lacks in the day-to-day practicality and concreteness I’m looking for. I don’t need to know what problem solving is (like Potter Stewart, I know it when I see it), or read philosophical exhortations about how important it is in promoting meaningful and deep learning. I want practicality. Stories, resources, tips, etc.

If I get some responses in the comments, I will compile them into either a comprehensive post, or if there are a lot, I’ll make a new page (a la the Virtual Filing Cabinet) for it.

The reason behind this is selfish, but I’m hoping the output could be collectively useful. My department is thinking seriously about how to integrate problem solving into our curricula… and I wanted to show them: “hey, there are a ton of good ideas from teachers who do it!”

So if you could help a teacher out…

PS. Not to make you jealous, but yesterday I designed and ordered these buttons! (You have to recognize I don’t know what I’m doing with Photoshop, so the pictures aren’t all that great. And these buttons have a large bleed area, so the text will actually be just near the outer rim of the button instead of with all that blank space between the text and the outside of the pin.)


A Positive and Healthy Approach to Learning

A few weeks ago in Algebra II I had students fill out a series of questions… questions which was going to lead to a discussion about mathematics and intelligence. I cribbed this sheet from my friend and teacher extraordinare Bowman Dickson.

I didn’t capitalize on it immediately, but I think I can still get some good mileage out of this. The thing that brought me back to this sheet was that yesterday in Algebra II, I gave an assessment that students didn’t fare as well as thought they would.

With one section, today, I had a heart to heart with them about what I saw, and this disconnect, and I talked a lot about the difference between active learning and passive learning. I think I got through to them. And I said: take what I said to heart. Be an active learner. And I’m going to give you an assessment on the same material next week. Show yourselves that you are capable. Because I know you are — but you just need to learn and implement the right strategies to be able to do it and make a lasting change.

Or something like that.

So we’ve had the talk about the concrete things… and I think next week, after kids take the reassessment (and hopefully — HOPEFULLY! — do much better), it would be worth it to have a talk about the more abstract side of things… attitude. The way students approach math, think about math, think about intelligence.

I’d love for any ideas about how to structure/have this discussion. I’ll throw my class data below, but if I’m going to do this, I want whatever I plan to be as powerful as possible. I want it to really get kids to think about what learning is, and how important having a growth mindset is. I have a few thoughts, but nothing great. So any brainstorming you might have, awesome.

As linked to on Sonata Mathematique:

Yeah, I want that DOUBLE POSTER SIZE in my classroom.

Without further ado, here’s my (fascinating) class data…



The Messiness of Trying Something New

It’s now more than halfway through the first quarter, and things are … messy.

I’m pretty much going through Calculus like I did last year, except for the fact that everything is so much easier because I have standards based grading down. [1] I know what works. While Calculus was hell for me first quarter last year, it’s cake for me now. So calculus is not messy. [2]

So while Calculus is going smoothly, I’m finding Algebra II to be messy. Not in terms of my kids. I love my Algebra II classes. But like last year — when I vowed to really focus on Calculus and leave my other courses well-enough alone — this year I vowed to focus on Algebra II and leave my other courses alone.

Specifically, I’m working on two major things: making groups and groupwork a norm, and having problem solving be a regular (and non-special) part of the curriculum. (As you can guess, the two go hand-in-hand.)

I haven’t written much about my inclusion of problem solving into the curriculum, but right now we’re doing a day of problem solving before each unit (related to the unit), I have slowly started including problem-solving problems in our home enjoyment (our supremely corny term for homework), I have been putting simple problem-solving problems on each assessment, and we so far have had a single problem set (something which I may or may not continue with). Still, I should be clear that most of my curriculum and my classes are traditional.

Now, if you’re a teacher who teaches more traditionally and uses a standard curriculum, you know that this a huge change. Because there’s a huge activation energy involved in switching teaching modes. For me, I kept on saying “next year, next year” and I never did. It’s daunting! And why screw around with something that works well?

And if you’re a teacher who teaches with lots of groupwork, and uses problem solving regularly, you probably remember the year you went through the transition. And how it got easier each subsequent year, as you picked up more tricks of the trade. Tacit knowledge.

And if you’re not a teacher, what the heck are you doing reading this blog? Seriously?!? GET OUTTA HERE!

Switching to this mode has played havoc with my emotions. You see, it’s not healthy and I try to avoid it, but my self-worth is tied up with how well I think I’m doing in the classroom. When I feel like I’m doing things well, I walk around like I own the world. I have confidence. My head is held high. And when I feel like I’m doing a poor job, my head hangs low. I question my desire to teach. I wonder what I’m doing in the classroom. And I’m depressed.

This year, I’m playing emotional ping-pong.

There are times when I feel like I’m killing it in Algebra II. These are usually days before each unit, where we spend the entire period working in groups and problem solving. I love watching kids think and discuss, and they’ve gotten how to work well in groups down. I’ve never had it work so seemlessly. It’s amazing. They’re independent. They’re identifying their own misconceptions and fixing them. I leave these classes wondering why it took me so long as a teacher to get to this point… I feel like my kids are finally and truly grappling, and I love that. (And I’m starting to do this successfully when we’re not problem solving… I made an “exponent lab” which was just 20 “simplify this” problems… and I was seeing great things when they worked together.)

And then there are times when I feel like I’m being killed. I have classes where I want to crawl under my desk and hide. Some of these classes happen the day after kids problem solve, and they present their solutions. Kids put their work on the board, or under the document projector, and present. Or if we don’t have time, I’ll have them put their work up, and I’ll talk through it. These classes have never worked for me. It’s like pulling teeth. Kids don’t know how to present. They don’t know how to engage if they’re in the audience. It takes forever. I don’t think anyone is getting much out of these days. [3] Or there are the more frequent regular classes (where we’re not doing problem solving), and I find I’m standing at the front of the classroom the entire class, cold calling and explaining. And it’s ugh. I feel ugh. There’s no spontaneity. It’s not fun. I don’t mix things up or have different ways of introducing/practicing material to break up class.

What’s interesting is that I feel my kids think that I’m doing a crappy job. I know they — in actuality — don’t think our classtime sucks. (I had my kids anonymously answer some questions, including the what two or three adjectives would you use to describe our classtime question.)

But even though intellectually I know that my kids don’t think I’m doing a crappy job teaching, it doesn’t change the fact that I feel they think I’m doing a crappy job teaching. It’s a slight distinction, but maybe others of you out there know what I’m talking about.

So as I said changing things is messy. Because you don’t know what works yet, and what doesn’t. It’s taking a risk. It requires more work. And you feel like you’re constantly flailing and failing. And that’s not a good feeling. Here’s a recent Facebook “convo”:

I know this is sort of rambling. I’m just trying to work through some things, but I still don’t know where things are going. Which is why there isn’t a real point to this. Just a state of affairs, from an emotional vantage point. I’m not looking for sympathy or advice. I just wanted to try to get my thoughts down — and just let you know that if you’re going through a similar transition, you’re not alone.

[1] I have a list of standards I can choose from, I have good exemplars of problems for each standard, I learned how to effectively introduce it, and I know how to set it up so I don’t die with all the extra work that comes along with reassessments.

[2] But yes, there are lots of things I could do to improve it. Always, always…

[3] I’ve talked with a teacher who does a lot of group work and presentations, and she gave me some excellent suggestions (revolving around using giant whiteboard) which I’m going to take on board.

Our Experience with Understanding by Design

This post was written by both Sam and his guest blogger, Bowman.

At the Klingenstein (Klingon) Summer Institute, we (the illustrious Sam and Bowman) participated in planning a unit in the style of a Performance Based Assessment, which is very similar to method of planning advocated in Understanding by Design. Normal people just call it backwards planning. Whatever you name it, the core of this philosophy is that Enduring Understandings should be the focus of curriculum design, and not skills or learning activities. With a third (also illustrious) Calculus teacher, we put together a rough draft of a unit planned in this manner.The big idea of the unit was the Relationship between Limits and Rates of Change.  We decided that by the end of the unit, we wanted students to understand that: curves can be conceptualized as a joining together of almost linear pieces; and an infinitely fine approximation of a quantity is often needed to yield the exact value.

This was a difficult process. We thought it might be helpful to share some thoughts about planning this way and share some of the ways that we approached backwards planning at KSI. Here is the completed, though still rough rough rough, product [update: see the bottom of this post for another group’s work]:

What do you think an Enduring Understanding is?

SJS: It’s funny. When I was first introduced to UBD at my school for curriculum mapping (barf, BARF, BARF), it was exactly at “enduring understanding” that everyone threw up their hands and gave up. Partly because we’d show it to the consultant, and she’d say “no” and then give us no guidance from there. Partly because it forced us to grapple with exactly what it is we wanted the kids to learn, and dig down to the core of what we really valued about what we taught.

At the Klingon Institute, our math leader (also illustrious) said something that really spoke to me. He said “an Enduring Understanding is something you want your kids to remember 5, 10 years from now.” It sounds lofty, even corny. But when I took a moment to really think about it, it struck me: I need to know what it is I truly care about, and this is it. But thinking in this way — what truly is at the core, mathematically, of what you’re teaching? — is terrifying and hard. Especially if you’re something who has always focused on units and skills. It is also exciting, because you get to come up with Big Ideas and use those as your lesson/unit/yearlong themes. But honestly, more terrifying.

The other half of what an Enduring Understanding is, is that “it has to be something general, but not vague.”

If a student asks you “why are we learning this?” and the best you can do is say “well, you’re learning that so you can learn calculus [or X]…” and then they get to calculus [or X] and you say “well, you’re learning this so you can do engineering and open doors…” I’ve been known to do this. BARF. How unsatisfying for a student. And if I’m not mistaken, every time you say something like that, you yourself get a sick, guilty feeling.

A really good enduring understanding should put a stop to this infinite regression, and those guilty pangs you feel. Because you know exactly what it is you want a student to take away — and you can tell ‘em, loud and proud. Okay, it may not always be sexy, but it is something fundamental they can latch onto now.

BD: I think that everything that Sam just said is perfect, and considering we learned about these at the same program, I’m not all that surprised that we have a similar enduring understanding about what enduring understandings are. The only piece I would add is that it helped us to start all Enduring Understandings with the sentence “I want students to understand that…”

SJS: I’m going to emphasize that “I want students to understand that…” is needed when creating your enduring understandings, but you cannot lazily make it into a skill. Something like “I want students to understand that when solving a radical equation, there may be extraneous roots” sucks. That’s too specific, and is a single skill. It should be something that applies more generally, like “I want students to understand that sometimes the best way to count in math is to not count.” (That might be for a unit on combinatorics.)

BD: One additional point I would like to make though is that, though the final product (your list of enduring understandings) is hugely helpful, I found the most helpful part of curriculum design like this was collaboratively going through the process of trying to figure it all out. Attempting to articulate ideas and sift through the wide world of math to find the meat forced me to think so deeply about my curriculum. To be honest, I’m not sure how useful a list of someone else’s enduring understandings would be to you. It’s like taking someone else’s lesson plans – unless you think about it and modify it to be your own, it’s hard to implement in your own classroom.

SJS: I also think the process was valuable, but I’d disagree with Bowman about not finding others’ enduring understandings useful. I have limited time, am sometimes (often) lazy, and I can get on board with someone else’s enduring understanding if I buy into it. Like, for example, Bowman is going to come up with a whole host of enduring understandings for calculus, and I’m going to steal them. Right, Bowman? Right? Why reinvent the wheel when Bowman will carve it for you?

BD: If you’re happy with slightly-to-horribly-misshapen wheels…

How can Big Ideas and Enduring Understandings help you organize your curriculum?

BD: It’s easy to get caught up in using the book’s sequencing of content, but thinking about big ideas and enduring understandings can help rearrange everything else to help promote those enduring understandings above everything else. For example, next year we will explore solids of a known cross section before solids of revolution, because the enduring understanding in solids is that if you stack up a bunch of infinitely thin cross-sectional areas, you can create a solid. Solids of revolution are really solids of known cross section too, just with circular cross sections – the revolution is just a way to construct the solid, not the main idea behind the integral. By talking about solids of known cross section first, it might be a good way to highlight the deeper idea without getting caught up in a multitude of smaller ones.

SJS: You might think textbooks give us Big Ideas — quadratics, conics, etc. But Big Ideas are not topical, but transcend topics. As for how they can help me organize my curriculum, I don’t know yet. I do think they are going to be the anchors of a class.

BD: A list of big ideas in mathematics that we generated with a group of 13 awesome math teachers at KSI: models, functions, dimension, relations, transformations, estimation, comparison, distributions, measurement, operation, conjecture, representations, rates of change, logic, proofs/reasoning, inference, mathematical objects, classification, systems and structure, definitions, inverses, algorithms, patterns, symmetry, equivalence, infinity/infinitesimal, and discrete vs. continuous.

SJS: We like this list, but it seemed too birds eye view for us. When we worked on it, we found it made more sense to just zoom in a wee bit. Our big idea again was “The Relationship between Limits and Rates of Change”. It’s not like there’s a right answer to how to do this. You have to do what’s useful to you.

BD: So what’s the different between a BIG IDEA and an ENDURING UNDERSTANDING then? The point of big ideas is to give you thematically lynch pins around which to organize your curriculum instead of the typical CH 1.4, CH 2.3 – i.e., what ties all these topics together? The same big idea can occur across many different math courses. Then the enduring understandings are the learning outcomes that you want to come out from exploring these big ideas (see above for a much better description).

How do you assess Enduring Understandings in SBG?

SJS: Right now I honestly have no idea. Right now I’m thinking of making SBG 70% of the grade, and Big Things (projects, enduring understanding assessments, problem solving) 30% of the grade. Or something like that.

BD: The big thing that I am going to add to my class next year is writing for informal assessment. Even if it doesn’t count for standards grades in SBG, I think that I might just keep a list of enduring understandings for my own purposes and informally assess the students as I go through the semester. Then, when larger assessments come around, I will explicitly focus review around enduring understandings. Since standards in my SBG-hybrid system only account for around 40% of the grade, I think I will keep my SBG standards to be skills and focus my summative assessments around larger ideas, though I will make sure to be explicit about this with my students. This of course is not perfect, but like Sam, this is something I’m wrestling with.

How can you do this sort of work without having Noureddine (who was our curriculum group leader) giving you feedback?

SJS: That’s why we have blogs — for feedback! But I suppose the some questions you can ask yourself (regarding if you have a good Enduring Understanding):

1) Is it general, without being vague?
2) Do multiple “skills” fall under the mathematical principle/idea your Understanding encompasses?
3) Of all the content related things, is this something you’d want a student to remember 5 or 10 years from now? Honestly? REALLY? Okay now, really?

BD: One of my goals next year is going to be to more effectively utilize the resources at my school, i.e. the other teachers in my department. The more people that collaborate on something or check out your work, the better chance it has of being something valuable. I know this is a general principle, but I would have the temptation to not go to other members of my department because they don’t already plan like this – I am definitely not going to fall into that trap next year.

Does this sort of thinking re-orient (re-frame?) the way you look at teaching, or the meaning of what math class is?

SJS: For me, it’s helped me see the value of looking for a bigger picture. It’s complicated,  the question of “what do you teach?” Right now I teach skills, and I can do that pretty well. But skills for what? That’s the real question I’ve done a bang up job of dodging. So when I worked on this, it forced me to countenance that head on. What do I really want to give to my students, mathematically? [1] Example: It’s not completing the formula, say, but it’s the idea that you can transform a non-linear equation (x^2+6x-3=0) into a linear equation to help you solve it (by reducing it into x+3=\pm \sqrt{12}). I suppose it makes you think more about the larger themes of a class, or something.

[1] I’m not talking about habits of mind, or those sorts of things. What I’m talking about here is purely mathematical content.

BD: When you have a group of unruly students who will be sitting in front of you for 45 minutes every day, it is easy to get caught up in the day to day of lesson planning. The first thing I always jump to is the learning activities. To give myself a bit of credit, I think I often had an idea, though subconscious, of the bigger picture, but by never spelling it out for myself, I could never really spell it out for my students either. It’s hard to take the time when you start planning to think about big ideas, but I found in just the one unit we planned together that once we had identified the enduring understanding, solid learning activities were so much easier to come up with. This hierarchy has helped me see that I can’t really implement learning in my classroom until I frame what learning really means in terms of big ideas and enduring understandings.

SJS: Here’s a gedankenexperiment. If you asked your kids at the end of your course what the big mathematical takeaways were, what answers would you get? If I asked my kids that question at the end of the year… well, it would be a crusty hodgepodge of things. They don’t know my mathematical goals for the course, and clearly that’s because I myself don’t know my goals. Not broadly, not meaningfully.

What was the most frustrating part about curriculum design like this?

BD: I am someone who is good at working within a framework and tweaking that, but this involves rethinking the whole conceptual framework of your class. Also, it was frustrating to realize that all of my SBG standards were skills, and that I didn’t ever explicitly identify the big ideas. Being self-critical without being self-deprecating (and not in the funny way) is tough for me, but that’s partly what this process is for.

SJS: So many things, so many things. The gads and gads of time it took. The supreme annoyance when we couldn’t come up with a good Enduring Understanding or Big Idea. Our inability to easily come up with good assessments to check exactly what it is we wanted the students to learn. But mostly, it was what Bowman noted: realizing that even though I’ve taught Algebra II and Calculus for four years, I don’t really have a sense of what it is I truly want students to get out of it.

Are you going to change your teaching because of this?

SJS: I want to say yes, but I don’t think it’s something I’m going to be able to do wholesale. I think I’m going to try to do only one unit using this sort of planning — but do it really well. (That’s what our illustrious Klingon curriculum leader suggested.) And build up from there, each year. This is all a little lofty for me, and it’s no magic bullet for student understanding.

BD: Even if I don’t formally plan units this coming year with this method, I am happy to have my thinking shifted to be more in terms of big ideas and enduring understandings. Like, after you spend forever looking for new shoes, all you can notice about other people is their shoes – I’m hoping that even if I teach the way I did last year, I will be able to pick out the big ideas in the process and focus on those. Then codifying and formalizing unit plans into grand designs like this will be much easier in the future.

SJS: Kudos, sir, kudos. We can use this year to brainstorm these enduring understandings, as we teach and ask ourselves, forlornly, “what the heck is the takeaway from the rational root theorem?”

BD: Good luck with that.

Update: Another Klingon group said we could share their unit planning (for Algebra I) which is below.

I’m alive, I’m alive

I’m alive, I’m alive // And I’m sinking in.


First off, thank you very much to Bowman for his amazing, thoughtful, well-written guest blog posts. I told you he was a tour-de-force and I can only say that I hope you’re finding his ideas as inspiring as I have. I’m stealing everything I can from him. I hope you are doing the same. I’m all about the concrete, and he gives me the concrete. Inspirational, he is.

Personal Update

So I’m now back in New York City. Home. I attended 5 weeks of professional development. Two weeks at the Klingenstein Summer Institute in Lawrenceville, New Jersey, followed almost immediately by three weeks at the Secondary School Teacher’s Program at the Park City Math Institute in Park City, Utah.

Yes, I’ve gone from this to this:

Current Status of My Thoughts

I have to say: I am burned out. Five weeks is a long time. I am also inspired, and hope to soon sort through all that I’ve taken away to make some serious changes in my classroom. And next year, I am only teaching two preps (Algebra II and Calculus, but not the AP Curriculum). So I will have the breathing room to make changes, I hope.The changes will involve intentional group work and formative assessments, coupled with much more intentional atmosphere building of a place where mathematical thinking (right or wrong) is valued and errors are celebrated and not something to be feared.

Yeah, I know. These are small changes and you think I need to be more ambitious.

JK. I know these are huge. It will take a lot of thinking to figure out how concretely to enact them. It’s easy to say these ideas, but it’s way harder to actually visualize them happening, if I close my eyes. I have some ideas, but not nearly enough.

I’m also worried about finishing the curriculum (especially in Algebra II) next year I try to go for depth and misconceptions and mathematical thinking, rather than try to go at those things but then succumb to the expediency of the moment and don’t allow time for grappling and productive struggling and discussion. But I’m less worried than in previous years, for some reason, and I’m ready to just go for it and see what happens. I suppose it’s because I’ve taken a vow to not underestimate my kids and their thinking abilities. Which I think I’ve done, unintentionally, and now I have to correct that. So if any of you have experiences of making the transition from teaching procedures to teaching thinking, any want to share any advice, puh-leese help me out here in the comments. (I don’t only teach procedures, to be fair to myself, but if I had to put myself in a camp, I would put myself more in a procedural camp than the thinking camp.)

I promise I’ll share my thoughts about changes I’m going to make in the classroom next year, as I sort through things, just like I did with my maybe-too-extensive blogging about standards based grading last summer.(That being said, I also suppose I have to talk about how I’m going to revise SBG for next year in calculus. Which means I have to figure out how I’m going to revise SBG first. Hu-uh. Feeling daunted now.)

Last year I was timid about making changes. I did Standards Based Gradings, and I felt that was “enough.” I think that was a good start. But it was like a bandaid on a bigger problem. I need to work on my craft in the classroom, and SBG didn’t change that too much. And so this year: I’m going for a sea change. No more glacial change, I’m jumping in whole hog, and mixing metaphors like similes are to analogies. Or something.


I praised Bowman for being specific and concrete, and look at me here, being all musing. Sorry. It almost feels like I’m trying to psyche myself up for next year, and committing myself to change by announcing it publicly. Yes, I suppose that that’s exactly what this is.

I hope to be more concrete soon. It’s just that, well, this here blog has always been for me, partly to archive what I do (the concrete) and partly for me to sort through what I’m thinking and get some ideas down… because when they slosh around in my head: 1. I can’t sleep 2. I get a headache 3. I get paralyzed with the overwhelming sense that I need to do something but I don’t know what. It’s the paralysis that I hate the most. So I’m hoping to avoid that by starting to put thoughts to page. But I know: I hate reading these kinds of posts too. So if you got to this point: sorry.

Frankensongs and Frankenfunctions: Using Mashups to Teach Piecewise-Defined Functions

After a riveting session about brain science at the summer program I attended (where I met Sam!), I wanted to read a little bit more about epistemology. I chose a few books that the presenter suggested: I just finished reading “Made to Stick” by Chip and Dan Heath (about why some ideas stick in our mind better than others and how to turn your ideas into some of those better ones) and am about halfway through “Brain Rules” by John Medina (twelve basic rules about how the brain works). Both were fascinating and will absolutely influence my teaching.

One thing that I have really latched onto is the idea of working with students’ previous knowledge about everything and anything in order to guide and improve learning (both books kind of harp on this). Take this example from Made to Stick where they define a Pomelo (an example which the lecturer also talked about at the summer program):

A pomelo is the largest citrus fruit. The rind is very thick, but soft and easy to peel away. The resulting fruit has a light yellow to coral pink flesh and can vary from juicy to slightly dry and from seductively spicy-sweet to tangy and tart.

If you already know what a pomelo is, that should make sense, and if you don’t, you can still get a pretty good picture of what’s going on. But compare that definition to this one:

A pomelo is basically an oversized grapefruit with a very thick and soft rind.

Both define a pomelo, but the second one uses the crazy ideas in your head to build new knowledge, making a much more descriptive and much stickier idea – not only is it easier to learn what a pomelo is, you will remember it much better. AAAND, the big bonus, it’s more efficient! [Here is a picture of a pomelo, by the way, if you need one. They’re kind of gross, but I am still partial to pomelos – in Arabic, pomelo is “bomaly,” and at first the guards at school couldn’t understand my weird sounding name (“Booooowman”), so they chose to hear the closest familiar thing, and started calling me “bomaly.” The origin of one of my many nicknames.]

Using Schemas in Math Education

I was thinking back to my year to see if I used anything like this in to teach math. I thought of one example, which I wanted to share, and then decided to put out a call for others. Can you think of a specific instance where you used anything from students’ prior knowledge to effectively and efficiently make a mathematical concept stick?

Piecewise-Defined Functions and Music Mashups

When reviewing at the beginning of the year in my Calculus class, I found that a lot of students were surprisingly stymied by the idea of piecewise-defined functions, which kind of blew my mind (this was in my first two weeks teaching math, and I was not expecting this to be a tricky concept for seniors in high school). It dawned on me that piecewise functions (which I call “Frankenfunctions”) are a lot of like Music Mashups, like this awesome mashup of the Top 25 songs from 2009 by DJ Earworm:

I played the song for the class and before connecting it to math, we broke down what we were hearing – like, actually had a brief conversation about what a mashup is (basically, one song constructed from segments of many others, though we went into more detail). Then we talked about the piecewise functions with this context:

  • A piecewise-defined function is one function made up of pieces of many others.
  • Each segment on a piecewise function is just a little part of a much bigger function.
  • The segments are broken down into intervals based on the x-axis (or time axis).
  • In piecewise functions, only one “song” can be playing at a time for it to be a function.
  • Piecewise functions can capture more interesting situations where the relationships between the variables in play changes.
I still got some of the craziest graphs I have ever seen on the following quiz, but the metaphor gave me a way to talk through their mistakes with them and hopefully gave them a way to connect something that is easily comprehensible to the slightly more abstract idea here. Now, this maybe isn’t the best example because it feels a bit cheap and may not get at deep understanding of some of the “whys,” so I will repeat my call again…  Can you think of a specific instance where you used anything from students’ prior knowledge to effectively and efficiently make a mathematical concept stick?

from @bowmanimal 

Virtual Conference on Core Values: The Heart of my Classroom

The conference is here.

The question of what’s at the rapidly beating heart of your classroom is a tough one. Let me rephrase that: for me, it’s tough, because it is totally evolving. Also whatever is at the heart of your classroom is your hidden curriculum — something that isn’t content, but just as important (if not more so) for kids to take away. So it’s pretty hard to get a handle on. It’s values.


In my first three years, I would have said the heart of my teaching revolved around three words:




Yes. Those three words drove me. The thing about having a core philosophy is that: everything revolves around it.  Every assignment. Every interaction. Every expectation. And although there are hard decisions that have to be made, when I struggled through them, I found I eventually turned back to my core beliefs, and I saw the light. Do I let that kid, that sweet sweet kid, take a re-test? Do I really need to create a super involved rubric with benchmarks, or can I just outline the project? If everyone in the class bombs an assessment, what do I do? [1] When holding core beliefs, every choice has to be intentional. Because these are what you value, and you need to enact those values. If you can only “say” your values, but you can’t “see” your values… then you’ve failed. [2] [3]

This philosophy has helped me out a lot with classroom management. It has helped me gain the respect of at least a good number of students. But I have started to see that philosophy as a baseline, now, of what I am doing. I believe in more.

Current Status

In the past year, the heart of my classroom has expanded to include more than clear, consistent, and fair. Thanks to the philosophical reorientation that Standards Based Grading has given me, it now includes metacognition and proactivity. [4]

I want my kids to be aware of what they know and what they don’t know. I want them to aware of the process of learning, and strategies to help them along the way. And I want them to be able to act on that knowledge. This is my hidden curriculum.

In Calculus, I used Standards Based Grading, which is all about kids getting a handle on their own learning. It forces them to understand what they know, and what they don’t know, and really articulate it! [5]

Dismantling the course into individual skills allowed me to have a specific breakdown of what the student knows and what the student doesn’t know. A student might have mastered how to apply the product rule, but struggle with explaining in words where the formal definition of the derivative comes from.  With SBG, I know this. In a school newspaper article written about my calculus class, one student was quoted: “The fact that the material is broken down into very specific skills as opposed to chapters or sections means you can focus on what you don’t know and figure out what you need to improve.”  More than me knowing where my students’ strengths and weaknesses are, my students themselves can recognize them.

I talk about metacognition, but that’s only half the battle. Who cares? Kids knowing about their learning habits, that’s great. But it doesn’t help them unless they believe they can grow from it. This is something I’ve been thinking a lot about since reading Carol Dweck and her notions of growth mindset. If a student — especially my students who tend to come to class never really appreciating math — thinks they suck at math, that they aren’t a “math person,” they’ve already stabbed themselves in the eye, shot themselves in the foot, whatever. There is blood everywhere, and it sucks. My kids come in with a fixed mindset. To get them engaged, to act on the “metacognitive” work, to see that doing well in math isn’t a matter of being “born with it,” I need them to see themselves as people who can change through hard work. Because really, if they don’t believe that, they won’t be doing hard work. They’ll simply continue to try to get by in math.

The thing is, we’re human beings. We suck. It’s hard to alter our own perceptions of ourselves. It’s also hard to say “we suck” and then decide to move on from there to say “let’s do things to suck less”!

This year I’ve been trying to do some good work in getting kids to be proactive, and to build their confidence. It involves a lot of individual communication with students. It involves me showing them that I care. It involves me avoiding ever comparing a student to another. It involves me demonstrating passion which occasionally translates into passion in them. It involves me talking explicitly about how math is a process, a journey, and how anyone can do it. It involves me not falling into the trap of thinking of certain kids as “smart.”

Standards Based Grading has helped me get kids to be proactive. My favorite example of this is a student reflection I’ve blogged about before:

1. I like the way that even though I was falling rapidly into a hole, and it felt almost impossible to get out, once you talked to me I became proactive and tried my best to do better. I like to continue meeting with you. I also like to continue to participate in class and asking questions. I think asking questions in class was the biggest way for me to better understand the topics.

2. I wish I would have started from the first day of school in this attack math mentality. I was acting very passive and like ‘oh I don’t get it now, but I will later,’ which honestly was the worst thing I could have done. I also wasn’t used to the class setting and the grading system. But once you emailed me and I met with you and I know that this is a class that I have to be in it 100%, and that your method is one that helps us actually learn, it was just beneficial. I needed that scare and wake up class because I was in serious denial. I became more on top of things. However, I had to dig myself out of a huge hole that I put myself in, but eventually the rhythm has become one that I used to. And I’m almost in a weird way glad that I learned the hard way because now I truly understand Math.

But that’s just one example. For as many kids as I might have helped, I know the struggle of SBG was enough to turn some of my kids off to math. I couldn’t get them to act. I don’t think it was laziness on their part, but despair. They hadn’t fully embraced the growth mindset and realized they could do it. I failed to be able to counter this.
I value a growth mindset, and I try to promote it through my actions. That is the current central core of my classroom. I’m still working on it, but here’s where I stand now.

[1] Of course I don’t mean “clear, consistent, and fair” to mean everyone gets the same treatment. Context matters, and what’s fair is not always “treat everyone the same way.”

[2] See Sizer and Sizer’s The Students are Watching (my review here)

[3] A grand experiment would be to have someone watch a video of your class, and try to suss out your values, and where they are expressed through your actions and words.

[4] SBG also has helped me remember the point of teaching: student learning. And now I have a razor sharp focus on that goal.

[5] In Algebra II, I deal with metacognition also, but not as well. I do this by talking to my kids explicitly about categorizing what they know and what they don’t know.

I tried to make homework more meaningful, by creating a full feedback loop. If a student got something wrong, they were asked to re-do the work and correct it. Otherwise they would have practiced the skill incorrectly, or illuminated the concept poorly, and never fixed it. (The “ill-leave-it-to-learn-before-the-test” syndrome.) I did this using binder checks (and redux), which had the added benefit of keeping (most) students organized.