A drama-based approach to calculus

I am in the classroom of Bryce Jandon in Wattson, UA. Before us stands a student wearing a curly wig and plastic, no-lens glasses. His name is Dale and, for today, he is a calculus teacher.

“This is a slope,” Dale explains while holding aloft a string manipulative that he has contorted into a curve.

“Good work,” Bryce smiles, his eyes glistening with pride.

After class, I take Bryce to one side to discuss the innovative calculus program that he has developed in partnership with the Extraordinary Learning Foundation™.  We start with a conversation about Dale.

“It’s not that Dale actually believes he is a calculus teacher,” Bryce explains, “I don’t think we often give kids enough credit. He knows it’s a dynamic drama-based experience.”

I nod sagely in agreement, “Is that a common misconception?”

“Yes,” replies Bryce, “people who don’t understand what we do are like, ‘how can a kid teach calculus?’, ‘doesn’t the kid know he’s not a teacher?’ and so on. It’s kinda like that. But that’s not the purpose. A drama-based pedagogy enables students to empathise themselves into what it would be like if they could do calculus or whatever. That’s a key point. It means that they see it as part of the seascape of potentialities.”

Of course, Bryce is right. I ask if he can explain a little more about the program and how it developed.

Modestly, he suggests, “I owe it all to my mentor, Principal Peters. I was teaching fifth grade social studies when she came to me and suggested I give calculus a go. Naively, I was kinda horrified. I explained that I didn’t think I was up to it; that I didn’t know any calculus. But she put me at ease – using a drama based pedagogy, we could all learn calculus together through co-deliveracy. And anyway, she explained, she couldn’t find a proper calculus teacher.”

I already knew the answer, but I asked Bryce to explain what his pedagogy involves.

“Well, I set the scene. We place it inside a narrative where one of the students has got to teach this class on calculus. This is the set-up, right? We know from cognitive science that narrative is held best in the brain – at the front somewhere – and so that’s the thinking. In Act I, the rest of the students are colleagues of the teacher who help him research his lesson before becoming the students in Act II.”

“Some people would ask how effective this method is for teaching calculus,” I suggested.

“Ah yes,” Bryce nods knowingly, “The guys who want to measure everything and calculate everything: The positivists.”

We both chuckle.

“Look,” Bryce explains, “I’d say to them that we are doing much more important work here than merely developing a student’s ability to regurgitate disconnected facts on a standardised test. The jobs of the future will not even exist. They will require collaborativity, evaluativity and the ability to work together. All of these crucial skills are developed through this unique pedagogical approach. In comparison, the idea of learning how repeat a few algorithms for the purposes of a 19th century test in an airless room is just over.”

‘Exactly,’ I think as I walk to my car at the end of our meeting: Learning calculus is not the point.


Developing a Thinkiness scale

At the Extraordinary Learning Foundation™ we have developed a number of ways of making Thinkiness visible such as this one:

And also this one:

These express Thinkiness as a provocative verb construct. A doing. However, as we continued to work with the complex and surprising concept of Thinkiness over time, we began to realise that there were different levels, with some tasks and conceptualisations being Thinkier than others.

It was while the team here at West Bay University were contemplating Deleuze’s concept of becoming that we came-up with what has come to be our powerful metaphor for how this kind of development comes about. Rather than going up like a tree does, we decided to express the increasing intensification of Thinkiness as a series of different depths.

I’m guessing you’ve not seen anything like this before and neither had we. The beauty and simplicity of the abyssian metaphor immediately struck us as a close correlate of going deeper and further into our students’ heads; precisely what Thinkiness is all about!

This clearly illustrates that Thinkiness is not dichotomous. The direct teaching of indigestible facts has a role to play in developing Thinkiness, it’s just that it’s not a very important one and is potentially a form of abuse.

Instead, all practitioners in their own myriad ways and using a full spectrum of pedagogical techniques can develop Thinkiness in an authentic and relevant manner that is true to themselves and their students, now that they have this diagram.

Apply neuroscience in your classroom!

I recently visited Barry Rubiou over at West Bay University’s Cognition Lab. There, they are working on a brand new form of instruction guided by the latest findings in neural imagining. They call this ‘Neuro-Scientific Pedagogy’ or NSP for short and it offers the potential to totally revolutionise the work of schools.

You can do this yourself with some brain clip art

Different sections of the brain highlighted in different colours.

Brain scanning studies conducted in Rubiou’s lab have demonstrated that we learn less efficiently when under extreme duress, when in physical pain or when intoxicated. We never knew this before and it totally explains why traditional forms of instruction are completely ineffective.

Rubiou’s team have also identified that our brains actually grow when we make a mistake! This happens even if we are not aware that we have made a mistake. It also happens before we have even made a mistake. This has massive implications. For instance, teachers should guide students to make as many mistakes as possible and assessments should credit mistakes more than correct answers.

But this isn’t all. Rubiou is set on developing a totally new and unique pedagogy. He thinks that because of neurons, children should not be taught artificial procedures. “We know from scans that the brain privileges narrative,” he explains. So, instead of teaching children maths, we should consider embedding a maths problem in a story about a duck and a briefcase. Students will then use their own strategies to solve the problem; a problem that will now be irresistible. After all, that’s what real mathematicians do – it’s how Newton made-up calculus!

So watch this space. The neuroscience revolution is set to utterly change the education landscape. Expect more ideas that you’ve never heard about before and that nobody can disagree with because brain scans.


For some time now, here at the Extraordinary Learning Foundation™, we have been working on Actionizing Thinkiness. This is why we developed the Think-it-out™ toolkit. We have been working with teachers to help engage more thinking in their otherwise thought-free lessons.

However, we have encountered a problem. Teachers typically use the toolkit to direct questions to students in class. In other words, the teacher maintains complete control over the learning episode. This is self-evidently undesirable so we wondered whether we could develop a model of co-deliveracy that was authentic, engaging and allowed learners to take control of the thinkiness.

This is the thinking behind the thinking that led to us thinking-up the idea of Thinkiballs™.

Mace Jakins is a fifth grade social studies teacher at Benington International School, UA. His chestnut hair shines as he describes the process of working with one of our Extraordinary Learning Foundation™ Associates on developing a pedagogy of Thinkiness that was also a pedagogy of authenticity and respect.

“We had the idea of writing out the Think-it-out questions on pieces of paper and then screwing these up into balls,” Mace explains.

“Each learner gets a ball but they don’t know what’s on it. At any point in the lesson a learner may shout ‘THINKIBALLS!’ and throw their ball at another student or the teacher who then has to unscrew it and answer the question.”

Mace takes a gulp of his double soy macchiato and pauses. He eyes glisten moistly as he continues.

“At first my control freak side came out. I just couldn’t bear the idea that the lesson could go off track or an important concept might be interrupted. When you’re teaching the skill of empathy then you often have to develop it over time. I remember the learners co-creating a personal response to a diary entry of a refugee visiting the local library for the first time. We had three ‘THINKIBALLS!’ in a row right at the start of that and I thought we’d never get going.”

So I asked Mace how he moved past that.

“I started to realise that I had given my learners voice and choice. They were telling me something and I needed to listen. I started to realise that this was bona fide authentic learning that is real and extant. I started to realise that this was where it was at: that this was the shizzle: that this was Thinkiballs!”

The skill of competence

It is becoming increasingly clear that none of the jobs that people do today will even exist in five years time (apart from maybe undertaking). Therefore, there is no point in teaching students to regurgitate rote, disconnected facts. We cannot predict which facts they might need because we don’t know what they will be doing and, even if we did, the jobs of the future will not require low-level cognitive skills like fact-knowing. Instead, these tasks will be done by computers – Google will know facts for us. Careers will require higher level cognitive processes. Neuroscience shows that these are the executive skills that coordinate the brain.

At the Extraordinary Learning Foundation™, we have been working on better ways of developing the higher level skills of comprehension and communication. When you look at the performance of experts and scan their brains in a scanner then different areas ‘light-up’ when they are performing these functions. Clearly, the ability to understand anything that you are told or that you read – comprehension – and the ability to communicate will both be highly valued in the job market of the future. Sugata Mitra is clearly right to focus on these skills.

However, another key feature of high-level performance is the ability to be good at things. I am always keen to take lessons from real-life to bring into the classroom and so I have intentionally observed a lot of skilled professionals. When you watch experts at work such as bar-tenders, burlesque dancers or police officers, you notice that they have the skill of ‘competence’ in abundance. They are really good at what they do. We have therefore been working on ways of developing this key metacognitive skill with students.

We envision competence to be like a tree because trees like conifers are narrow at the top, indicating that fewer people are competent in any particular sort of thing.


We have found this useful in helping students to envision what competence looks like. An important task in developing the skill of competence is to ask students to construct their own understanding of competence, perhaps over a number of lessons and whilst working with others.

The key questions to ask can be summarised as follows:


Our key finding is that these questions can be asked at any time whilst studying any topic. No longer do we have to restrict ourselves to ‘covering’ content. Thus, focusing on the skill of competence is a perfect partner to student-led project work. It really does not matter what they are doing because they can still work on this skill.

This is liberating. Students can work on things that they find really engaging and motivating and still be learning key 21st century skills at the same time!

Narrative Do-ology

I am in the classroom of Julian Malvolio at Bayswater Elementary in Kunnunna, UE. He is setting up for the day; about to teach his Grade 5 creative writing class. We take five to have a chat about what’s bugging him.

“I was finding that the kids were just replicating a procedure. They weren’t thinking about it. They were just going through it.”

I have been thinking about this problem for some time and, after much deliberation, I have decided that I am right. The crux-point is one of a lack of understanding. Sure, kids can go through some kind of procedure to write a story, select characters and so on. But the ways this is done can be learnt and practised rote. There is much more to writing than simply placing words together, one after the other. Often, the writing is derivative, focusing on wizards and dragons.

This is why here, at the Extraordinary Learning Foundation™, we have developed the process of Narrative Do-ology™.

Narrative Do-ology

By moving away from simply regurgitating prose, we ensure that students truly understand the plots that they have invented themselves. The process starts with “The Act” – a way of doing that may or may not involve the construction of formulaic words. It is about telling a story.

Then, we actionize thinkiness in order to metacognitivise and truly consider the thinking that we will be deploying in order to deal with the concepts that we intend to think about.

Finally, we move to the process of understanding the narrative. To do this, children draw diagrams of the narrative structure or express it with abstract symbols. A delta, “Δ”, might indicate a key character, an arrow could represent a journey and an equals sign might signify an equivalence. Dialogue can be tricky.

It is this process that I have been working on with Julian. He is grateful to me for my wise words and for the Extraordinary Learning Foundation™ for making it possible for him to actuate this advice.

“Writing is about so much more now,” Explains Julian, “It is about a vibrancy of pictures, symbols and different forms of representation and expression. It is about story. Gone are the days when writing was just about putting words on a page.”

If math is a 30cc 2-stroke garden leaf blower then where are all the leaves?

Leaf blowers are annoying, right? They make noise and wake people up on Sundays and even Thursdays. They emit harmful carbon dioxide, are heavy to carry and are hard work for the user. And you have to wear ear defenders which look kinda dorky.

Leaves are what make a leaf blower worthwhile. Only when you see the ease with which a leaf blower can gracefully and efficiently corral leaves does it start to make sense as a piece of garden machinery. Better still, ask someone to try to gather leaves with a simple table fork. After a few hours of this, see how readily they will accept the need for a leaf blower.

This whole metaphor – for it is a metaphor for something and we’ll see what that is in a minute – hit me the other day whilst I was watching a cool and zeitgeisty TV show. I immediately realized how profound it was. Clearly, we need to completely revolutionize the way that we teach maths and the leaf blower gives us a clue.

Ask any teacher what the biggest problem in education is right now and they’ll say that it’s motivating students. You see, kids get bored in math class. This is clearly not because they expect to be constantly entertained and lack self-discipline. It is because they can’t connect the math they are learning with their everyday experiences. Everyone says so. And this is where my metaphor comes in.

You see, math is like the leaf blower. The leaves are like authentic problems that the math can solve. Do you see it yet? I am saying that we have to present kids with authentic problems to solve, let them struggle a little – this is like the part where I suggested picking up leaves with a table fork – and then they will see the need for the math. This works because math only has any value inasmuch as it can be used to solve commonplace, everyday problems (that are slightly contrived).

The leaves

Let’s put this into practice by posing a problem. You work for your county painting fences (of course, here you should substitute the name of your actual local county). You need to paint a fence that’s 30 feet long by 6 feet high. You have to give it three coats of paint. It takes 30 minutes to paint a six-foot length of the fence and the paint takes 20 minutes to dry.

You can spark their curiosity – I call this ‘roping the mark’ – by showing a video like this:

By now, the students will be drawn in to the conflict inherent in the grand narrative that you have set-up. And so it’s time for the next stage.

The table fork

Ask your students for the solution to the problem. At first, most are likely to suggest painting the whole fence once, waiting 20 minutes, painting the fence again and so on. If you add these times together then you get an answer. However, some students are likely to realize that some of the fence paint will be dry before you reach the end of the fence and so, provided you have an infinite supply of labor like most counties do, you don’t have to wait that long.

The sensible approach here is to start partitioning the fence into lengths. However, these lengths will be essentially arbitrary. A student might work out what length of fence is painted after 20 minutes and this should then develop into a discussion about the thickness of the paintbrush and how long one stroke takes.

The leaf blower

By this point, your students will be ready to hear a ten-minute mini-lecture where you give them an algorithm for working out the fence-paint problem whilst introducing them briefly to differential calculus. This is like what they do in Japan.

The payoff

Instead of having your students learn abstract maths for which they cannot immediately spot a commonplace use, they will now have all of their maths taught to them in an exciting and engaging way. From henceforth, your students will be motivated.

All you have to do is keep looking for leaves which, in my experience, usually get stuck in garden beds or the entrances to drains.