Introducing Concepts in Calculus

Thanks to Jonathan Claydon for starting some great conversation about tough concepts in calculus, first with his post on avoiding magic tricks, and then starting work on a calculus chapter in Nix the Tricks. I just learned some cool new ways to introduce tricky ideas in calculus, and my teaching will be better for it. Check out this cool visual explanation of the Product Rule!

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Jonathan’s post also brought to mind a neat exchange I saw on Twitter recently on explaining where the Chain Rule comes from, also very cool!

Alright, now it’s time for a confession.

I don’t like introducing complex ideas like the chain rule by proving why they work.

I think this type of introduction-by-proof appeals to a subset of my students, but it tends to turn off others, and the kids who turn away are the ones I most want to engage.

Here is a different approach I’ve used for the Chain Rule:

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(doc) (Yes, I introduce transcendental functions before the Chain Rule. I realize many calculus teachers follow a different sequence, but this approach still works with different examples.)

I give students the handout and tell them that it shows functions on the left, and each function’s derivative on the right. I ask them what patterns they notice, and how they could use those patterns to find other derivatives in the future.

Students don’t usually figure everything out on their own. That’s not my goal. Instead, students have a chance to think about “inside functions” and “outside functions” and describe this funny derivative rule informally before we describe it formally. I build off of students’ language describing the types of functions they see and their connection with the derivatives. Then, after offering some explicit instruction in informal language, talking about inside and outside functions and multiplying by the inside derivative on the outside, I might offer them a more formal definition of the Chain Rule, like this one:

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I ask students to discuss in pairs or groups how this definition is connected to the examples they just looked at. After a bit of informal discussion on their own terms, I ask them to identify both f(x) and g(x) in several of the initial examples and I annotate a few functions on the board, using color to emphasize the different pieces and how they fit together.

The purpose of this sequence is to move from informal to formal, to give students a chance to make sense of an abstract rule on their terms before Leibniz’s, and to use worked examples to illustrate an idea while still putting the thinking on students. I have no illusions that this is sufficient to teach the Chain Rule, but hopefully at this point students are set up to be successful in engaging with some practice.

After this sequence, students are hopefully thinking, “ok, that kinda makes sense, but why is that the case?”. That’s where I think the conceptual explanations I referenced above come in. Once students have a basic grasp of a rule, I think they are in a much better position to grapple with the complexities of where it comes from. Even better, the initial exploration could happen on one day, stew overnight, and the next day I share a way of understanding where the rule comes from. Students’ informal understandings and experiences with a few concrete examples of a concept will hopefully help them better understand and make sense of an explanation of where that concept comes from.

I like this approach because I think it honors Jonathan’s desire, and my desire, to help math make sense to students, while also prioritizing informal thinking before formal thinking. I also like that, at every step, I can give students who struggle with calculus opportunities to engage on their own terms and feel like they can make sense of new ideas. I don’t think this is the right approach for every new mathematical idea, but in calculus lots of concepts have algebraically or computationally complex proofs, and this approach hopefully minimizes the challenges of that complexity.

Differentiation Strategies

I don’t like the word differentiation. Teaches assume it means “give students different work”, when I see differentiation as a set of tools used to make sure all students learn, regardless of academic background, prior experiences, or identity. I think setting ambitious goals for all students to work toward together is one of the best things about a classroom community. I think there are a range of strategies I can use to do that, and I don’t have a better word to lump these strategies together than “differentiation”.

Low-Floor, High-Ceiling Tasks 

A task with a low floor and a high ceiling doesn’t need to be anything fancy. It can be a pretty humble task that has features to provide access and extension to more students. One example I like is a card sort. I often use a lesson from the Shell Center in a unit on exponential functions, and have students complete this card sort:

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Students match graphs, tables, equations, and descriptions together, each describing a situation of either simple or compound interest. The sorting makes this task accessible for more students, as they can start to make associations even without much fluency with exponential functions. There are two blank cards for students to fill in missing equations, and blank spaces in the tables for students to fill in missing values. Some of the descriptions are also trickier than others, and lead to analysis comparing how often different situations take to double and other potential extensions to challenge students. The entire activity helps students to connect different ideas and make explicit the different ways that exponential structure plays out in compound interest while distinguishing exponential growth from linear growth. The lesson also includes thoughtful follow-up tasks for students to do more work with these ideas and apply them in different situations.

Teaching to Big Ideas 

Teaching to big ideas means exposing students to key understandings early in a unit, and returning to them over and over again. I just taught a unit on conic sections, and one of the big ideas is that replacing every x with an x – 2 moves a graph two units in the positive x direction, and replacing every y with a y + 3 moves a graph three units in the negative y direction. More broadly, these transformations can be used to translate a graph in any direction.

Students first see these ideas with circles, ellipses, hyperbolas, and parabolas. I like to give students an unusual example to further emphasize the big idea. This equation creates an interesting graph:

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While it’s not a conic section, it does behave in similar ways. I ask students to describe how to graph a general version of this equation using similar strategies to the ones we use with conic sections. Some students explore with vertical and horizontal translations; others get into different types of dilations, and some can be challenged to analyze changing the signs and exponents to see what happens. There’s lots of great structure here and connections to other ideas, and students can engage with the big ideas on multiple levels through the same problem.

One advantage of this approach is that a focus on this big idea extracts the essential mathematics from conic sections while putting less emphasis on things I care less about. It’s easy in that unit to get caught up in manipulating conic sections algebraically. While that can be a useful goal, it’s also one that requires much more background knowledge, and serves less of a purpose in the broader mathematics progression. Focusing on this big idea, important in conic sections and recurring in other topics, helps to make sure that what students are thinking about is the math that will be most helpful to them in the future.

Making Big Ideas Explicit

While the task I used above provides students with a useful opportunity to engage with a big idea, working with that big idea without making it explicit leads to a less equitable classroom. Many students will make connections on their own, and figure out that the structure of this function has a lot in common with other conic sections. But some won’t, and those who won’t are likely to be students who have struggled with math in the past, compounding gaps that already exist. Making the big idea explicit means pausing during the lesson to discuss what students notice and what strategies they are using, and reflecting at the end of the lesson on how these ideas are connected with ellipses, hyperbolas, and other objects. These discussions and reflections are great learning opportunities for everyone, whether students are practicing articulating their ideas using mathematical language, pausing to notice features they might have rushed past while focused on completing the task, or seeing a connection for the first time.

In the same way that I need to structure that task to make big ideas explicit, I need to make them explicit at every other step of the way as well, both leading up to this task and as this idea comes up in the future. It’s easy to forget that students see math differently than I do — that they don’t have the background knowledge I have and don’t see the broader structure of math as readily. I need to constantly remember to make essential ideas explicit so that every student has an opportunity to engage with them, rather than leaving it to chance. For more thoughts on big ideas, read David Wees’s piece here.

Aim for Relevance 

I don’t want to assume that every student finds math worth learning. I want to do what I can to help all students feel a sense of ownership and enjoyment in my class. One way of doing that is offering multiple perspectives on why students should learn math. I try to capture moments of wonder and curiosity, and help students have those a-ha moments that will make math class enjoyable. I frame math as a series of puzzles to be solved, practicing skills that will help them solve more puzzles in the future. I dive into applications of topics like exponential growth with examples of financial advice for students to evaluate based on what they’ve learned. I give students a window into how different ideas are relevant to higher math that they might study in the future. And I talk about how some math is worth learning just because it’s fascinating — mathematicians studied prime numbers largely out of curiosity for thousands of years before they became the building blocks of today’s web encryption standards. None of these arguments for learning math is sufficient on its own — each appeals to different students at different times. And not every student will be engaged every day; I need to be realistic in my goals. But offering a range of ways for students to make meaning of math class helps more students to come to class motivated and ready to engage.

Scaffolds 

Scaffolds are probably the strategy I think about the most, but they are also the hardest to master. I need to provide scaffolds that help students to access content, while also taking them away when possible. One strategy I’ve found useful is scaffolding different parts of a task at different times. For instance, after introducing the different parts of rational functions, I might ask students to analyze the end behavior of a group of functions. Then, I’ll give them this task, graphing rational functions when already given end behavior, vertical asymptotes, and intercepts:

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Then, I’ll ask students to find all relevant features of a function without graphing. Then they will put it all together. Varying the scaffolds helps students to focus their attention on different parts of the whole, and builds fluency in chunks that are manageable before attempting the entire task.

Building Relationships 

Building relationships is both important and subtle. A lot of relationship building happens in small ways, inside and outside of class. One way I approach relationships systematically is to pay attention to particular students that I know often feel bored in class, or often struggle and feel confused. Say I’m having students engage with an Illustrative Mathematics task in a unit on exponential functions and logarithms:

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Every student will experience this task differently. While students are working in groups, I am monitoring for different strategies to share with the class and making sure students stay on track. I’m also paying particular attention to students who often feel bored or students who make be bored in that task, trying to offer them an extension and doing what I can to keep them engaged. I’m also paying particular attention to students who often struggle and feel confused. My goal is that every student engages with the big ideas of the task to an extent where, when we go over it, they will be able to engage and learn from others’ strategies and understanding of the math. By timing student work around these students, I can make decisions deliberately that help every student engage and help every student feel like they can be successful. There’s no perfect solution, but by keeping tabs on those students’ experiences, I can make those decisions a little better.

Conclusion 

These strategies are the result of a paradigm shift for me. One paradigm is that students come into math class with different levels of knowledge and skill, and I need to offer them different experiences to meet them where they are. A different paradigm is that students come in with different levels of knowledge and skill, and if I look at all of my instructional decisions through that lens I can provide meaningful, common experiences that help every student engage with mathematical thinking.

None of these strategies alone solves the problem of students with different backgrounds and different experiences trying to learn the same concepts. I still think that is a goal worth working toward, and I think that these strategies together can make a difference in moving in that direction. None are easy to implement — I’m still getting better at all of them. But I’m optimistic that that improvement will make me a better teacher for the students who most need it.

 

Differentiation

I dislike the word differentiation. I think the goal is important — to teach so that all students learn, regardless of their academic background, prior experiences, identity, or any other factor. It’s really hard to teach lessons that effectively support learners who are struggling as well as learners who find the content easy.

But differentiation is often conflated with “give different work to different students”. When teachers describe a lesson as “fully differentiated”, they usually mean “every student did different work”. I worry that this implicitly lowers expectations, prevents students who are behind from catching up with their peers, and creates far more work for teachers than it’s worth.

I do give students different work at times, but as one of the last strategies I try. I instead try to find tasks with a low floor and a high ceiling, to teach toward big ideas that students can engage with on multiple levels, to make those big ideas explicit and ensure students engage with them multiple times in multiple contexts, to make learning feel relevant to students with a range of backgrounds, to incorporate scaffolds for tasks that allow all students to access them, to provide extra support and extension either inside or outside of class, and to build relationships so that students are more likely to engage with challenging ideas and buy into classroom routines.

I think that those strategies are incredibly important to my teaching, and I think that they should be grouped together one idea. But I don’t like calling them differentiation, because when I talk about these strategies as differentiation, other teachers just assume I’m talking about giving students different work.

Perspectives on Formative Assessment

Two quotes that are on my mind, on the topic of formative assessment:

Imagine what would happen if a pilot flew like many teachers assess. So I flew back from Seattle a few weeks ago. Just imagine what the pilot would have done. He would have flown east for nine hours. And then after nine hours he’d say it’s time to land so he’ll put the plane down and he’ll ask, “Is this London?” And of course even if it’s not London, he says “well everybody’s got to get off because I’ve got to get on the the next journey”. That’s exactly the way we’ve assessed in the past. We teach students material, and at the end of that teaching we find out if they’ve learned it or not, and if they haven’t we say too bad because we’re on to the next unit. So what formative assessment does is encourages teachers to take constant readings about where students are, just as a pilot takes constant readings about their position, and if the learning isn’t proceeding as you’d planned then you make adjustments.

-Dylan Wiliam (source)

Picture this scene: Dr. Gillupsie has grouped around him several of the young resident surgeons at Blear General Hospital.They are about to begin their weekly analysis of the various operations they have performed in the preceding four days.

GILLUPSIE: Well, Carstairs, how have things been going?
CARSTAIRS: I’m afraid I’ve had some bad luck, Dr. Gillupsie. No operations this week, but three of my patients died.
GILLUPSIE: Well, we’ll have to do something about this, won’t we? What did they die of?
CARSTAIRS: I’m not sure, Dr. Gillupsie, but I did give each one of them plenty of penicillin.
GILLUPSIE: Ah! The traditional ‘good for its own sake’ approach, eh, Carstairs?
CARSTAIRS: Well, not exactly, Chief. I just thought that penicillin would help them get better.
GILLUPSIE: What were you treating them for?
CARSTAIRS: Well, each one was awful sick Chief, and I know that penicillin helps sick people get better.
GILLUPSIE: It certainly does, Carstairs. I think you acted wisely.
CARSTAIRS: And the deaths, Chief?
GILLUPSIE: Bad patients, son, bad patients. There’s nothing a good doctor can do about bad patients. And there’s nothing a good medicine can do for bad patients, either.
CARSTAIRS: But still, I have a nagging feeling that perhaps they didn’t need penicillin, that they might have needed something else.
GILLUPSIE: Nonsense! Penicillin never fails to work on good patients. We all know that. I wouldn’t worry too much about it, Carstairs.

-Neil Postman & Charles Weingartner, Teaching as a Subversive Activity

I’ve come to bristle a bit at the phrase “teaching style”. I do have a style of teaching — different habits and structures I tend to use. But that phrase is also used as a justification for particular practices, regardless of whether they help students learn. “Well, that’s my teaching style, [Student X] will just have to adjust”. I’m happy I have my quirks and preferences in the classroom, but I need to balance my preferences with what works best for my students, and be willing to be wrong and try something new.

Task Propensity: Beyond Desmos

Task propensity refers to situations where students are so focused on the features of a specific task that they don’t generalize their thinking in a way that is useful to solve different problems in the future. In short, they lose the forest for the trees. I’m exploring how task propensity relates to Desmos activities and how this thinking could help me teach more thoughtfully with Desmos tools. I first learned about task propensity through this paper, and you can read the rest of my series on the topic here.


 

When I first read about task propensity, I was interested because it described one of my hesitations with Desmos activities. The activities are engaging and fun for students, but that engagement didn’t always lead to the thinking that I wanted it to. I have spent some time this fall thinking about strategies to counteract that phenomenon — keeping activities humble, designing focused follow-up tasks, and pausing the activity.

At the same time as I’ve been practicing those strategies when I use Desmos activities, I’ve seen task propensity in other areas of my teaching. Any rich task can fall victim to a focus on the task itself rather than the broader mathematical thinking that goes into solving it. I want to explore two examples of task propensity in activities I’ve used, and how I might modify those activities the next time I teach them.

Trig War 

I want students to practice evaluating sine and cosine functions, but I don’t want that practice to be any more soul-deadening than it needs to be. Inspired by Kate Nowak’s Log War, I put together Trig War. Students pair up and divide the stack of cards in half. They then each flip over a card, figure out whose value is larger, and that person keeps both cards. Wash, rinse, repeat. It’s pretty fun and gets a lot of practice in. At the same time, some pairs end up relying on one person to evaluate tricky values, especially those outside the unit circle, or they end up rushing and missing opportunities to think about the structure of the unit circle and sine and cosine functions, focused more on the War than on the Trig.

One idea I want students to take away from Trig War is a stronger intuitive understanding of where things are on the unit circle. Students might look at two values and, without evaluating them, know by visualizing that one value is positive and the other is negative. They might recognize that cos(x) = cos(-x) based on the structure of the unit circle and use that property to quickly evaluate negative values for the cosine function. They might compare two values that are very close together on the unit circle without evaluating by thinking about whether that function is growing or shrinking on the relevant interval.

But in most cases, students don’t do this thinking. They’re too wrapped up in the game, and don’t want to slow down and look for mathematical structure. I can instigate the thinking I want using the same strategies I identified for task propensity. I can pause the game, have students put their cards down, and pose a sample comparison that elicits strategies I’m interested in. By slowing down and focusing on one question as a class, I avoid leaving student strategies to chance, and share with them some of the thinking I’d like to see from them. I can do something similar with a follow-up task — reinforcing a strategy we discussed as a group, and provoking extended reasoning and generalization in a way that is hard in a game context in the moment. And finally, I can try to keep the game as focused as possible. The version I have used in the past is ambitious — it addresses both sine and cosine functions, including negative values and positive values beyond the domain of the unit circle. I could consider splitting this up into two games — the first focused strictly on the unit circle, and the second expanding the domain to other values and stretching student thinking, so that each game can be laser focused on the specific strategies I want to elicit and the goals I set for students.

Sequences and Skittles 

I often begin a unit on sequences and series with a Skittles activity adapted from Julie Reulbach. Each group gets a package of Skittles, a plate, and a cup. They put a certain number of Skittles in the cup, shake them, place them on the plate, and then remove Skittles according to a rule — remove all Skittles with the “S” up, remove all Skittles with the “S” up then add five back, and more. Julie uses this activity to focus on decay and recursive functions, but I’ve adapted it to address other ideas as well.

It’s a ton of fun. There’s plenty of management involved to avoid making a complete mess, but it gets at neat examples of recursive functions that provoke some useful mathematical thinking. At the same time, that engagement can mean students are thinking more about Skittles and the excitement in the room than they are about the math.

The follow-up task becomes particularly important with this activity. Students can record data while they play with the Skittles, but asking them to do much more thinking while playing with the cup is likely to lead to rushed work and shallow reasoning. Instead, I see this activity as having two stages — the high-engagement initial task of playing with the Skittles and recording data, and then a follow-up task, once the Skittles are away, where students analyze and make connections with the specific math concepts I want to get at. This is also a great example of a place where I need to stay humble. I’ve tried to run this with multiple different versions of growth and decay with different recursive rules, but I think Julie was onto something by focusing on just two rules, both decaying. The more focused the investigation is, the more likely I am to get all students to reach my mathematical goals, and the less is left to chance. Finally, there’s the potential for some really great thinking while students are playing with the second rule Julie used — removing Skittles with the “S” up and adding five back. But just letting the activity run and stepping back leaves those a-ha moments to chance. Instead, I can time the activity more deliberately, get the majority of groups starting that experiment at the same time, and pause as they start to realize that the number of Skittles will likely never reach 0 to discuss with the whole class. These are subtle changes, but they’re changes that do a ton to focus student thinking on mathematical goals and minimize the task propensity of an engaging activity.

In Closing 

If there’s one lesson I want to take away from this, it’s that I need to constantly ask myself, “what are students thinking about right now?” Memory is the residue of thought. Task propensity happens when student thinking is focused too much on a specific task, and less on the mathematics behind that task. This was something I missed for a long time. I thought, “students are engaged, and there’s a lot of god thinking that could come out of this task”, and left it there. Now, I’m trying to think more about how to harness that engagement to make sure all students do exactly the thinking I’m interested in.

Reading Research

I love to read research and learn about evidence-based ideas in education. I’m also often sad about the way research is used or misused in education debates, and I’ve been reflecting on how to better talk about research in ways that are productive. Here are three ways I try to use research, in my practice and in my writing:

Research About Learning 

I always want to better understand how students learn. Most research into learning happens outside of classrooms in controlled settings, but a broader and deeper knowledge of human attention, memory, and thinking helps me to better understand my students’ experiences. None of this research is prescriptive — it doesn’t tell me how to teach trig identities next week — but it does help me make informed decisions and better understand why some strategies work while others do not by giving me insight into the cognition and reasoning of students.

Single Studies About Teaching 

I often see single studies of teaching shared on Twitter or elsewhere. These are often interesting reads, whether they analyze the efficacy of homework, specific instructional practices, or the impacts of different school structures. Each study expands my perspective on teaching and learning and enriches my ideas of what rich classroom experiences can look like for students. At the same time, everything in education depends on the context. One study might offer useful conclusions for educators in the same situation, but it’s less likely they will generalize to any school or classroom in the world. New perspectives are useful, but it’s important to stay humble about what one study can reveal, and pay attention to the context as well as the results.

Broader Wisdom About Teaching 

Great research builds on others’ ideas to draw broader conclusions. When a large volume of work converges on a single idea, I try to think seriously about the impact on my practice. At the same time, larger trends in research tend to be far less specific. For instance, there’s a ton of research on the efficacy of formative assessment, but that research doesn’t tell me what to do in class tomorrow, it gives me some guidance as to how to incorporate one useful element of teaching into my practice. The more I know, the better decisions I’ll make, but there are no shortcuts and no easy paths to the “best” way of teaching.

How I Try to Read Research 

I read research is to continue my professional learning, not to tell me how to do my job. It’s hard to avoid the temptations of quick fixes and prescriptive ideas that fit what I want to believe. But the role of research is not to search out a stamp of approval for what I do in the classroom, or find definitive answers to the challenges of teaching. The role of research is to build a base of knowledge that helps me to better understand the choices I make and the actions I take in the classroom every day.

 

Teaching Math for a More Just World

 

In considering the kinds of risks this work requires and the rationales that effective teachers use to support such risk taking, they seem to be following the saying, “We act ourselves into new ways of thinking, not the reverse.” That is, much of this work requires deconstruction (unpacking the micro and macro issues that may be hidden in dominant practices) and deep reflection (knowing which principles we stand for). But more importantly, Creative Insubordination requires action on the part of teachers. Our actions, often leaps of faith, can lead to changes in how we think about a given situation in teaching. Luckily, learning how to advocate for our students can help us better advocate for ourselves (e.g., the right to have teacher collaboratives or common planning time). As teachers, we need to continue to look ourselves in the mirror each day and ask, “Am I doing what I said I would do in education when I entered this profession? And, if not, what am I planning to do about that?”

-Rochelle Gutierrez

I think about freedom as not only the absence of oppression but the presence of justice and joy. And so much of the work that we do focuses on the absence of oppression, but the reality is we will need to be as thoughtful about taking down all the bad things as we are about building up the beautiful future that we want.

-DeRay Mckesson

Many humans have different perspectives than I do on what a just society looks like. I’ve been reminded of that a few times this week. When I look at my classroom, my school, and the broader world, it’s impossible for me not to see inequities that create opportunity for some while marginalizing others. Surfacing those inequities is a necessary first step to taking action in ways that redress inequity and sustain that work over time. I’m grateful to many thinkers — Rochelle Gutierrez and DeRay Mckesson are on my mind right now, but there are plenty more — who have helped me to better understand inequity and equip me with a subset of the knowledge and skills to work against it.

It’s easy in that work to fall into a focus on the deficits and the problems, the structures to tear down and the obstacles in students’ paths. It’s much harder to step back when the going is hard and remember a vision of what a more equitable and just world would look like, both the microcosm in my classroom and the larger world. What do I want students to do with what they learn in my class? I want to help students gain knowledge of mathematics and mathematical thinking in ways that will help them access opportunity in the future. I also want to keep in mind more ambitious goals. What would it look like to teach students to reason from definitions and theorems in ways that help them interrogate assumptions in arguments outside of math class? What would it look like to teach students to look for and make use of structure in ways that allow them to discern surface-level features from the deeper intent and impact of political decisions? What would it look like to teach students to model with mathematics by constantly asking what the goal of the model is, and who benefits from it? I want the world to be full of people who speak and argue deliberately in ways that promote kindness, openness, and knowledge. I want the world to be full of people who are perceptive about the complexities of the world we live in and how that world impacts different humans. I want the world to be full of people who think of others’ well-being as quickly and readily as their own.

I can imagine countless ways that skills I teach could help students think and reason in ways that promote equity in an unjust world. But restricting my classroom teaching to “mathematics” as it is typically constructed prevents students from making those connections. It prevents students from seeing the reasoning they do as connected to a larger world and practicing that reasoning in relevant, impactful contexts.

I’m not that teacher right now. But I want to be, and it’s important that I remind myself of what I want mathematics teaching to mean, to my students and to the world. Math class is not the solution to the world’s problems, but mathematics is complicit if it pretends to be a neutral arbiter or conveyor of pure truth. I don’t want to pretend I can do more than I actually can, but I also don’t want to be paralyzed by framing math class as irrelevant or powerless in creating a more just world.