Because “Real World”

I’m tired of arguments against a particular practice in education, using the logic that “this would never happen in the ‘real world’, so why are we doing it in classrooms?”. Classrooms, and learning environments more generally, should not be designed to imitate the activities students will do in the real world. They should be designed for learning.

I try to teach my students that the goal of math class is learning, not performing. I caution against focusing on right answers at the expense of what can be learned from a problem or series of problems. In the real world, it is the quality of the work produced — the performance — that matters. In an environment designed for learning, the performance is far less important and can distract from learning.

Research in cognitive science has explored “desirable difficulties” — situations where performance is worse in the moment, but the cognitive demands and deeper processing as a result of those difficulties actually optimize learning. This research suggests that optimizing learning environments for performance in the moment can actually detract from learning, and points to only one of many examples where improving learning can actually be counter-intuitive.

A focus on performance can be more insidious. Anyone who has witnessed teachers taking problems from department- or school-wide assessments and giving them to students the day before a test with a few numbers changed can attest to this. The goal of the workplace is to produce high-quality work, but in that instance, a focus on producing high quality work degrades learning.

I don’t mean to say that education can learn nothing from the workplace. I just mean to say that, as arguments, “students will never experience [x thing] in a job, therefore schools shouldn’t do [x thing]” or “in the workplace, [x thing] happens, therefore schools should do [x thing]” seem to me insufficient to guide teaching and learning.

What’s Next?

When I started teaching, I didn’t enjoy my time in the classroom very much. During hard classes I could barely wait for the bell. And when I sat down to plan my next class, I would stare into space, stumped, stuck thinking about everything that went wrong with the last lesson and unable to plan what to do next.

Now, by the time I get back to my desk after a class the next lesson is often mostly planned. There’s lots of reasons for that, but one shift I’ve noticed is that when I teach, I spend a lot more mental energy thinking about the future. When an activity falls flat or I realize that students understand less than I thought, I respond both by figuring out what to do next in the moment, and thinking about what I’d like to do the next day or the day after to build from where students are based on what I’m seeing.

Noticing this shift reminded me of Lani Horn’s writing on an asset orientation. When thinking or talking about students who struggle, a deficit orientation focuses I can focus on shortcomings and blames the student’s past or experiences out of their control. An asset orientation focuses on their strengths and how they can move them forward from where they are. An asset orientation is forward-looking and solution-oriented. In the same way, I can think during class about what I wish I had done differently, what I wish students had known, or what I wish I had changed. Or, I can focus on what I’m learning about my students in that moment and how I will use that to move forward with their mathematical learning. This doesn’t mean hiding from my mistakes or pretending that a lesson went better than it did. It just means looking at my classroom and students with the perspective of “what’s next?”.


Note to self:

Some words take on meaning as pejoratives, used to denigrate a different perspective without engaging with the substance of that perspective. “Worksheet”, “traditional”, “rote”, many more. Without context, these words aren’t very useful. If I give my students a piece of paper with three problems on it, I might be labeled as giving them a worksheet devoid of understanding. If I cut those problems up and tape them around the room for students to work on in groups, suddenly I’m at the pinnacle of progressive pedagogy. The substance is the same, but the surface features are used to label one approach as better than the other, regardless of the specific context or my goals in that moment.

It’s easy to put someone else down by labeling their teaching with words that have been weaponized by a particular ideology. It’s much harder to speak in specifics about how different pedagogies play out in different classrooms — who they support, who they leave behind, how they build off of each other, and what they actually look like on a minute-by-minute and day-by-day basis. To do so means to value reality over rhetoric, and substance over style. I want to do more of that.

Scientific Knowledge for Teaching

Daniel Willingham writes in a recent article:

[R]esearchers have argued that teachers’ learning should not be limited to practical classroom strategies. The teacher who understands the psychological principles undergirding the recommended strategies will presumably find them more sensible and will see ties between seemingly disparate strategies. Perhaps most important, that teacher will also generalize strategies to novel situations. Teachers need what might be called a mental model of the learner: knowledge of children’s cognitive, emotional, and motivational makeup.

So teachers will benefit from knowing some things about how students learn. But what exactly, would teachers benefit from knowing? Willingham distinguishes between three types of knowledge in science: empirical observations, theoretical statements, and epistemic assumptions. The whole paper is worth a read, but I’d like to focus on one element of his argument that was particularly compelling for me. My paraphrase:

Teachers can benefit from bottom-up knowledge that is built on concrete observations, whether those observations are from research or their own practice. Top-down knowledge that begins with broad theories or generalizations about learning are likely to be less useful.

If teachers focus on bottom-up knowledge, they have the opportunity to buttress their everyday observations in classrooms with additional examples from research into how humans learn, building a rich and experience-based model of how their students learn. This type of knowledge is more likely to be humble; learning is fickle, and a certain practice may work in one context but not in another. Teachers are always learning, observing, and developing a more robust understanding of learning. If teachers focus on top-down knowledge, they come to their experience with broad statements about how students learn that can be adapted to justify a range of practices, separate from any evidence that they are effective for students. Willingham writes:

A statement like “learning is social” could be taken to mean “children learn best in social situations,” which is actually a very different statement–it is a statement about how children behave. But confusing it with “learning is social” could easily lead to thinking that because group discussion is more social than teacher instruction, it is a settled matter that it is more effective for learning, whereas the empirical reality is far more complicated.

In short, top-down knowledge about teaching can be twisted to support a range of ideas, and distances itself from everyday experience. Two examples of where I see this happen:

Cognitive Load Theory 

From a top-down perspective, cognitive load theory is a theory that, in problem solving situations, students’ working memory becomes overloaded, preventing them from learning. This statement could be used to justify any number of teaching practices, and to foreclose entirely any problem-oriented learning. But from a bottom-up perspective, cognitive load theory is a set of experiments in which, under certain conditions, students become overwhelmed by the demands of problem-solving and unable to learn from that experience. This perspective, while only subtly different, sets teachers up to incorporate that knowledge with their own experiences and to approach new situations with some humility — cognitive load theory doesn’t prescribe a course of action, but offers evidence for a practice that seems not to work in some contexts.


From a top-down perspective, constructivism is the perspective that learners actively construct new knowledge. It is often used to support inquiry-oriented approaches to learning, with the justification that if learners are to construct their own knowledge, they need to be active participants in that process. From a bottom-up perspective, constructivism is the recognition that what students learn depends on what they already know, and incorporates both teachers’ experience and research into how prior knowledge can have a large influence on future learning.

In both cases, the shift is away from prescriptive theories that try to make broad statements about best practice, and toward a focus on building a broad base of knowledge for teachers to draw from in their instructional decisions. Rather than coming into the classroom with preconceptions about what learning must look like, a bottom-up approach emphasizes concrete experience, the importance of context, and humility.

Task Propensity

I’ve really enjoyed writing about task propensity over the last few months, and I’ve learned a ton about teaching and learning in the process. The series of posts was inspired by a paper on efforts to reform mathematics curriculum in the Netherlands. Here is the an excerpt from the introduction of the paper that got me started on this path:

In each case student performance did not match expectations, and in each case textbooks analysis revealed a focus on teaching effective procedures for specific tasks, instead of developing mathematical insights at a more advanced conceptual level as the reform intended. We will argue that this is at least in part due to what we call, “task propensity,” which we define as the tendency to think of instruction in terms of individual tasks that have to be mastered by students. This task propensity entices teachers and textbook authors to capitalize on procedures that can quickly generate correct answers, instead of investing in the underlying mathematics while accepting that fluency may come later.

The authors focus on reform curriculum — curriculum that was meant to support a more conceptual mathematical focus. I’m using task propensity more broadly; in the same way that teachers and curriculum writers fall into the trap of focusing on individual tasks rather than the underlying mathematics, students working on rich, engaging tasks fall into that same trap. In short, it is easy to lose the forest for the trees, and it’s hard to step back from individual problems to see the bigger picture.

I was inspired to write this series because I see task propensity in my teaching all the time, both in the ways I structure tasks and activities and the ways students engage with them. I focused initially on Desmos activities because I think they can be particularly prone to task propensity, but I now see it in more and more places. I think task propensity is a natural human instinct, in which the practical concerns of solving the immediate problem supersede the learning that could be gained from stepping back, taking a broader perspective, and considering how the thinking in a task could be applied to new situations. I also think it’s inevitably a part of any ambitious and engaging curriculum.

I’ve explored three strategies for addressing task propensity that I think are also three useful design principles in any curriculum that focuses on students learning math by doing math — staying humble, follow-up tasks, and the pause.

Stay Humble 

When I stumble across a great lesson on the internet, it’s easy to focus on the allure of a fun activity that students will enjoy rather than the substance of the learning under the hood. That’s not to say student engagement is unimportant, just insufficient for meaningful learning. Staying humble means constantly calibrating my lens for what a great lesson is, focusing on substance over style. It means taking an ambitious lesson that is broad in scope and trimming it down to focus on a smaller number of well-defined mathematical goals. It means putting aside the big picture at times to zoom in on the building blocks that students need to support their larger understanding. Staying humble doesn’t mean teaching boring classes, but it does mean avoiding the temptation of sleek and sexy lessons when they’re just not the right tool for the job.

The Follow-Up Task 

Student engagement is great, but inevitably leads to a focus on the present rather than the future. In lots of tasks, that’s what I want, and I embrace the energy in the room. It also creates an opportunity for a deliberately designed follow-up task, where students return to a previous activity and consider its implications in a new problem. The initial task acts as an anchor to contextualize student thinking, whether they refer back to technology or manipulatives, borrow a bit of engagement from a fun experience, or reuse a useful problem type. Often during an engaging activity students are particularly engaged with the more “fun” elements of the task and not the underlying mathematics. Follow-up tasks take a step back from the incentives in the moment while returning to essential ideas that students can learn from.


When students are caught up in an engaging activity I don’t instinctively want to stop them. I want to enjoy the moment and watch them have fun. At the same time, they’re also likely caught up in the activity in a way that prevents them from slowing down and thinking about how the math they’re doing might help them solve new problems in the future. While pausing them might elicit some groans, it also provides a great opportunity for students to think metacognitively about the connections they’re seeing and the math under the surface, rather than getting lost in the sauce of the moment-to-moment tasks. Pausing an activity is fundamentally about harnessing energy in the room to advance specific goals rather than leaving student thinking to chance.


Spending this time exploring task propensity has helped me to think about teaching and learning in new ways. It’s an important reminder that kids learn what they spend time thinking about, and I want to plan my lessons deliberately to promote the type of thinking that will support new mathematical knowledge. Engagement is not the same as learning, but I can use student engagement and well-structured activities to create opportunities for students to do the thinking I want them to do. And student thinking should not be left up to chance — if I have a goal for students, I should modify or restructure the student experience to make sure they meet it.

Illusions of Learning

“Have you heard of IXL? I love IXL, it’s so easy, it makes me feel so smart.” – Student

IXL is a computer-adaptive website that many teachers use for skills practice. I have nothing in particular against it. I do think that, more broadly, computer-based personalized learning platforms and the way they are used can fall into the trap of chasing what students like, rather than what’s best for their learning.

Here is an excerpt I often come back to on the science of desirable difficulties in learning:

Not long ago, the California Polytechnic State University baseball team, in San Luis Obispo, became involved in an interesting experiment in improving their batting skills.

Part of the Cal Poly team practiced in the standard way. They practiced hitting forty-five pitches, evenly divided into three sets. Each set consisted of one type of pitch thrown fifteen times. For example, the first set would be fifteen fastballs, the second set fifteen curveballs, and the third set fifteen changeups. This was a form of massed practice. For each set of 15 pitches, as the batter saw more of that type, he got gratifyingly better at anticipating the balls, timing his swings, and connecting. Learning seemed easy.

The rest of the team were given a more difficult practice regimen: the three types of pitches were randomly interspersed across the block of forty-five throws. For each pitch, the batter had no idea which type to expect. At the end of the forty-five swings, he was still struggling somewhat to connect with the ball. These players didn’t seem to be developing the proficiency their teammates were showing. The interleaving and spacing of different pitches made learning more arduous and feel slower.

The extra practice sessions continued twice weekly for six weeks. At the end, when the players’ hitting was assessed, the two groups had clearly benefited differently from the extra practice, and not in the way the players expected. Those who had practiced on the randomly interspersed pitches now displayed markedly better hitting relative to those who had practiced on one type of pitch thrown over and over. These results are all the more interesting when you consider that these players were already skilled hitters prior to the extra training. Bringing their performance to an even higher level is good evidence of a training regimen’s effectiveness.

Here again we see the two familiar lessons. First, that some difficulties that require more effort and slow down apparent gains — like spacing, interleaving, and mixing up practice — will feel less productive at the time but will more than compensate for that by making the learning stronger, precise, and enduring. Second, that our judgments of what learning strategies work best for us are often mistaken, colored by illusions of mastery.

When the baseball players at Cal Poly practiced curveball after curveball over fifteen pitches, it became easier for them to remember the perceptions and responses they needed for that type of pitch: the look of the ball’s spin, how the ball changed direction, how fast its direction changed, and how long to wait for it to curve. Performance improved, but the growing ease of recalling these perceptions and responses led to little durable learning. It is one skill to hit a curveball when you know a curveball will be thrown; it is a different skill to hit a curveball when you don’t know it’s coming. Baseball players need to build the latter skill, but they often practice the former, which, being a form of massed practice, builds performance gains on short-term memory. It was more challenging for the Cal Poly batters to retrieve the necessary skills when practice involved random pitches. Meeting that challenge made the performance gains painfully slow but also long lasting.

This paradox is at the heart of the concept of desirable difficulties in learning: the more effort required to retrieve (or, in effect, relearn) something, the better you learn it. In other words, the more you’ve forgotten about a topic, the more effective relearning will be in shaping your permanent knowledge (Make It Stick, excerpted from 79-82).

Part of my role in the classroom is to engage students in thinking about challenging ideas, monitor their learning minute by minute, day by day, and beyond, and connect concepts over time as we revisit them in more and more depth. I try to do all of that through the lens of a scientific understanding of how students learn. In 2017, too much personalized learning colors perceptions with the illusion of mastery and relies on making content feel easy as a substitute for substantive engagement, trading durable, transferable learning for hollow confidence-building and short-term skill retention.

I am interested in computer-based platforms for supplemental practice if they make my life easier, but personalized learning is far from where it needs to be to take on a primary role in the classroom.

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!


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:

Screenshot 2017-11-07 at 3.44.23 PM.png

(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:

Screenshot 2017-11-07 at 3.50.40 PM.png

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.