The Fallacy of the Right Answer

The Fallacy of the Right Answer is everywhere. With regards to education technology, it dates back at least to BF Skinner.

Skinner saw education as a series of definite, discrete, linear steps along a fixed, straight road; today this is called a curriculum. He referred to a child who guesses the password as “being right”. Khan Academy uses similar gatekeeping techniques in its exercises, limiting the context. Students must meet one criterion before proceeding to the next, being spoon-fed knowledge and seeing through a peephole not unlike Skinner’s machines. Furthermore, these steps are claimed to be objective, universal and emotionless. Paul Lockhart calls this the “ladder myth”, the conception of mathematics as a clear hierarchy of dependencies. But the learning hierarchy is tangled, replete with strange loops.

It is fallacious yet popular to think that a concept, once learned, is never forgotten. But most educated adults I know (including myself) find value in rereading old material, and make connections back to what they already have learned. What was once understood narrowly or mechanically can, when revisited, be understood in a larger or more abstract context, or with new cognitive tools. There are two words for “to know” in French. Savoir means to know a fact, while connaitre means to be familiar with, comfortable with, to know a person. The Right Answer loses sight of the importance, even the possibility, of knowing a piece of information like an old friend, to find pleasure in knowing, to know for knowing’s sake, because you want to. Linear teaching is workable for teaching competencies but not for teaching insights, things like why those mechanical methods work, how they can be extended, and how they can fail.

I feel like somewhere along the way, we decided we loved Speak & Spell more than Logo and that still informs ed-tech today
— Audrey Watters (@audreywatters) June 24, 2014

Symbol manipulation according to fixed rules is not cognition but computation. The learners take on the properties of the machines, and those who programmed them. As Papert observed, the computer programs the child, not the other way around (as he prefers). Much of this mechanical emphasis is driven by the SAT and other unreasonable standardized tests which are nothing more than timed high-stakes guessing games. They are gatekeepers to the promised land of College. Proponents of education reform frequently cite distinct age-based grades as legacy of the “factory line model” dating back to the industrial revolution. This model permeates not only how we raise children, but more importantly, what we raise them to do, what we consider necessary of an educated adult. Raising children to work machinery is the same as, or has given way to, raising them to work like machinery. Tests like the SAT emphasize that we should do reproducible de-individualized work, compared against a clear, ideal, unachievable standard. Putting this methodology online does not constitute a revolution or disruption.

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Futurists have gone as far to see the brain itself as programmable, in some mysteriously objective sense. At some point, Nicholas Negroponte veered off his illustrious decades-long path. Despite collaborating with Seymour Papert at the Media Lab, his recent work has been dropping tablets into rural villages. Instant education, just add internet! It’s great that the kids are teaching themselves, and have some autonomy, but who designed the apps they play with? What sort of biases and fallacies do they harbor? Do African children learning the ABCs qualify as cultural imperialism? His prediction for the next thirty years is even more troublesome: that we’ll acquire knowledge by ingesting it. Shakespeare will be encoded into some nano-molecular device that works its way through the blood-brain barrier, and suddenly: “I know King Lear!”. Even if we could isolate the exact neurobiological processes that constitute reading the Bard, we all understand Shakespeare in different ways. All minds are unique, and therefore all brains are unique. Meanwhile, our eyes have spent a few hundred million years of evolutionary time adapting to carry information from the outside world into our mind at the speed of an ethernet connection. Knowledge intake is limited not by perception but by cognition.

Tufte says, to simplify, add context. Confusion is not a property of information but of how it is displayed. He said these things in the context of information graphics but they apply to education as well. We are so concerned with information overload that we forget information underload, where our brain is starved for detail and context. It is not any particular fact, but the connections between them, that constitute knowledge.  The fallacy of reductionism is to insist that every detail matters: learn these things and then you are educated! The fallacy of holism is to say that no details matter: let’s just export amorphous nebulous college-ness and call it universal education! Bret Victor imagines how we could use technology to move from a contrived, narrow problem into a deeper understanding about generalized, abstract notions, much as real mathematicians do. He also presents a mental model for working on a difficult problem:

I’m trying to build a jigsaw puzzle. I wish I could show you what it will be, but the picture isn’t on the box. But I can show you some of the pieces… If you are building a different puzzle, it’s possible these pieces won’t mean much to you. You might not have a spot for them to fit, or you might not yet. On the other hand, maybe some of these are just the pieces you’ve been looking for.

One concern with Skinner’s teaching machines and their modern-day counterparts is that they isolate each student and cut off human interaction. We learn from each other, and many of the things that we learn fall outside of the curriculum ladder. Learning to share becomes working on a team; show-and-tell becomes leadership. Years later, in college, many of the most valuable lessons are unplanned, a result of meeting a person with very different ideas, or hearing exactly what you needed to at that moment. I found that college exposed to me brilliant people, and I could watch them analyze and discuss a problem. The methodology was much more valuable than the answer it happened to yield.

The hallmark of an intellectual is do create daily what has never existed before. This can be an engineer’s workpiece, an programmer’s software, a writer’s novel, a researcher’s paper, or an artist’s sculpture. None of these can be evaluated by comparing them to a correct answer, because the correct answer is not known, or can’t even exist. The creative intellectual must have something to say and know how to say it; ideas and execution must both be present. The bits and pieces of a curriculum can make for a good technician (a term I’ve heard applied to a poet capable of choosing the exact word). It’s not so much that “schools kill creativity” so much as they replace the desire to create with the ability to create. Ideally schools would nurture and refine the former (assuming something-to-say is mostly innate) while instructing the latter (assuming saying-it-well is mostly taught).

What would a society look like in which everyone was this kind of intellectual? If everyone is writing and drawing, who will take out the trash, harvest food, etc? Huxley says all Alphas and no Epsilons doesn’t work. Like the American South adjusting to an economy without slaves, elevating human dignity leaves us with the question of who will do the undignified work. As much as we say that every child deserves an education, I think that the creative intellectual will remain in an elite minority for years to come, with society continuing to run on the physical labor of the uneducated. If civilization ever truly extends education to all, then either we will need to find some equitable way of sharing the dirty work (akin to utopian socialist communes), or we’ll invent highly advanced robots. Otherwise, we may need to ask ourselves a very unsettling question: can we really afford to extend education to all, given the importance of unskilled labor to keep society running?

 

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If you liked this post, you should go read everything Audrey Watters has written. She has my thanks.

Brief Thoughts On Scratch

Previously, I’ve lambasted the children’s programming language Scratch for its cockpit’s worth of controls.  This encourages its users to try anything and see what works, rather than plan, predict, and understand exactly what each piece of code is doing. It’s instant gratification … and a tight feedback loop.

Scratch is not a tool to learn programming or metacognition; Scratch is a tool to create artistic displays that could not otherwise be created (by children). Scratch thus allows children to explore ideas not related to mathematics or programming. They have creative freedom, much like art class. And what elementary schooler produces anything particularly good, objectively speaking, in art class? So don’t judge the Scratch projects too harshly.

Scratch is a social platform, except that the socialization happens in real life. Get a few kids in a room using it, and they’ll share both creations and code,  motivate each other, and change goals on the fly. This differs from more mature programming, where one has a specific goal in mind. The other key difference is that most languages discourage straight up experimentation; one have to know what one is doing in order to do it. Scratch reverses this: a kid can learn what a command does through using it. This is because all the commands are displayed, ready to be used.

Not only displayed, but also labelled, unlike the Khan Academy programming language that drops down four numbers with no context. It’s a way to slyly introduce relative and absolute motion – move up by, move to – in a way that lets kids work out the rules. No, they won’t work out all the rules, but I think they’ll come to fewer incorrect conclusions (misconceptions) in a reactive medium than with marks on paper. They will figure it out later, much later.

Scratch is a way to put Lego bricks into the bucket. The kid will reassemble them into many different knowledge structures over the years before creating something strong and beautiful – an educated mind. It’s during that process, that struggle, that they can learn to program with planning and expressiveness, rather than tacking on bricks ad-hoc. It’s a stage everyone goes through, and Scratch can help a child make the most of it. But don’t confuse acquiring bricks with figuring out how to assemble them.

This isn’t to say that Scratch as it exists is perfect; far from it. We need to keep rethinking what tools are best to give to our children (and adults, for that matter). But I’m backing off my previous stance that guided minimalism is the answer. (Or is my new “wait and let them figure it out” view too fatalist?)

The Top 5 Things Done Wrong in Math Class

Sorry to jump on the top-n list bandwagon, as Vi Hart deliciously parodies, but that’s just how this one shakes out. Some of the reasons why these things are done wrong are pretty advanced, but if you’re a high school student who stumbled upon this blog, please stay and read. Know that it’s okay that you won’t get everything.

All of these gripes stem from the same source: they obfuscate what ought to be clear and profound ideas. They’re why math is hard. Like a smudge on a telescope lens, these practices impair the tool used to explore the world beyond us.

EDIT: This list focuses on notation and naming. There are other “things” done wrong in math class that any good teacher will agonize over with far more subtlety and care than this or any listicle.

5. Function Composition Notation

Specifically , which is the same as . No wait, . Probably. This notation comes with a built-in “gotcha”, which requires mechanical memorization apart from the concept of function composition itself. The only challenge is to translate between conventions. In this case, nested parentheses are ready-made to represent composition without requiring any new mechanistic knowledge. They exploit the overloading of parentheses for both order of operations and function arguments; just work outwards as you’ve always done. We should not invent new symbols to describe something adequately described by the old ones.

Nested parentheses lend themselves to function iteration, . These functions are described using exponents, which play nice with the parens to make the critical distinction between and . This distinction becomes critical when we say arcsine aka and cosecant aka are both the inverses of sine. Of course, things get confusing again when we drop the parens and get because . This notation also supports  first-class functions: once we define a doubling function , what is meant by ? I’d much rather explore this idea, which is “integral” to calculus (and functional programming), than quibble over a symbol.

4. The Word “Quadratic”

I’m putting “quadratic” where it belongs: number four. The prefix quadri- means four in every other context, dating back to Latin. (The synonym tetra- is Greek.) So why is called “quadratic”? Because of a quadrilateral, literally a four-sided figure. But the point isn’t the number of sides, it’s the number of dimensions. And dimensionality is tightly coupled with the notion of the right angle. And since equals itself, then we’re dealing with not just an arbitrary quadrilateral but a right-angled one with equal sides, otherwise known as a square. So just as is cubic growth, is should be called squared growth. No need for any fancy new adjectives like “biatic”, just start using “square”. (Adverb: squarely.) It’s really easy to stop saying four when you mean two.

3.14 Pi

Unfortunately, there is a case when we have to invent a new term and get people to use it. We need to replace pi, because pi isn’t the circle constant. It’s the semicircle constant.

The thrust of the argument is that circles are defined by their radius, not their diameter, so the circle constant should be defined off the radius as well. Enter tau, . Measuring radians in tau simplifies the unit circle tremendously. A fraction time tau is just the fraction of the total distance traveled around the circle. This wasn’t obvious with pi because the factor of 2 canceled half the time, producing instead of .

If you’ve never heard of tau before, I highly recommend you read Michael Hartl’s Tau Manifesto. But my personal favorite argument comes from integrating in spherical space. Just looking at the integral bounds for a sphere radius R:

It’s immediately clear that getting rid of the factor of two for the (theta) bound will introduce a factor of one-half for the (phi) bound:

However, theta goes all the way around the circle (think of a complete loop on the equator). Phi only goes halfway (think north pole to south pole). The half emphasizes that phi, not theta, is the weird one. It’s not about reducing the number of operations, it’s about hiding the meaningless and showing the meaningful.

2. Complex Numbers

This is a big one. My high school teacher introduced imaginary numbers as, well, imaginary. “Let’s just pretend negative one has a square root and see what happens.” This method is backwards. If you’re working with polar vectors, you’re working with complex numbers, whether you know it or not.

Complex addition is exactly the the same as adding vectors in the xy plane. It’s also the same as just adding two numbers and then another two numbers, and then writing i afterwards. In this case, you might as well just work in . (Oh hey, another use of exponents.) You can use the unit vectors and , rather than i and j which will get mixed up with the imaginary unit, and besides, you defined that hat to mean a unit vector. Use the notation you define, or don’t define it.

Complex numbers are natively polar. Every high school student (and teacher) should read and play through Steven Witten’s jaw-dropping exploration of rotating vectors. (Again students, the point isn’t to understand it all, the point is to have your mind blown.) Once we’ve defined complex multiplication – angles add, lengths multiply – then falls out as the square root of  completely naturally. You can’t help but define it. And moreover,  goes around the other way, and its alternate representation  goes around twice, but they all wind up at negative one. Complex numbers aren’t arbitrary and forced; they’re a natural consequence of simple rules.

Even complex conjugates work better with angles. Instead of an algebraic argument and a formula to memorize, we can geometrically see that we we need to add an angle that brings us back to horizontal, which is just the negative of the angle we already have. This is mathematically equivalent to changing the sign on the imaginary component of the vector, but cognitively it’s very different. You can, with clarity and precision, see what you are doing in a way numerals can never express.

1. Boxplots

Boxplots make the top of the list because they’re taught at a young age and never challenged. They are brought up as a standard way to visualize data, when the boxplot was a relatively recent invention of one statistician, John Tukey. Edward Tufte has proposed variants which dramatically reduce the ink on the page. They are much easier to draw, which is important when you want to convince children that math isn’t about meticulous marks on the page. They have no horizontal component, so in addition to being more compact, they also do not encode non-information in their width.

Boxplots infuriate me because they indoctrinate the idea that there is one way to do it, and that it is not up for discussion. More time is spent on where to draw the lines than why quartiles are important, or how to read what a boxplot says about that data. Boxplots epitomize math as a recipebook, where your ideas are invalid by default and improvisation is prohibited. Nothing could be further from the truth. Moreover, boxplots slap a one-size-fits-all visualization on the data without bothering to ask what other things we could do with them. Tukey’s plots don’t just obscure the data, they obscure data science.

Media for Thinking the Eminently Thinkable

Bret Victor has done it again. His latest talk, Media for Thinking the Unthinkable, gives some clues as to how scientists and engineers can better explore their systems, such as circuits or differential equations. He’s previously shown an artistic flair, demoing systems that allow artists to interact directly with their work, without the use of language. In both cases, the metric for success is subjective. What the novel user interface is doing is not determining the “right” outcome but allowing a human to better see and select an outcome. Victor is trying to help users develop intuition about these systems.

That strategy looks promising for a sufficiently complex systems, but when it comes to pre-college education, the goal is not to instill semiconscious, unarticulatable hunches. This relatively simple material demands full, clear, and lingual understanding. This decidedly different goal will, or at least should, guide the design of educational technology in a different direction than Victor’s demos.

What is an educational technology? In today’s world. you’re probably thinking of a game, video, recorded demonstration, or something else with a screen. Those all qualify, but so do manipulatives (teacher-speak for blocks and other objects for kids to play with the grok a concept) and pencil and paper. For the simplest of problems, say 2+2, there’s no need to get out the iPad, just grab four blocks. For advanced math, a computer is necessary. So where is the meeting point, where the see-saw of objects and devices balances?

What follows is a case study of educational technology applied to a specific mathematical idea, multiplication of negative numbers, which is pretty close to that balance point. We’re trying to explain why a negative times a negative equals a positive, a classic grade school trouble spot. (Instead of fixing a specific misconception, we could have to open-ended exploration of a slightly larger topic. Touch Mathematics has some interesting material on that front.)

The MIND Research Institute has developed a series of visual math puzzles that all involve helping a penguin traverse the screen. In their multiplication unit, the penguin stands on a platform in front of a gap. Multiplying by a number stretches the platform proportionally. A negative number will rotate the platform 180 degrees, so two rotations cancel each other. When a student interacts with this system, they gain a useful metaphor for thinking about negative numbers: go left on the number line. However, many of the more advanced puzzles seem contrived, and are difficult to do without resorting to pencil and paper because they don’t provide a cognitively useful metaphor. Moreover, adopting the unit, setting up computers for the students, and setting aside time has significant overhead. Can we go simpler?

Let’s take the rotation metaphor and find physical objects for it. We could use plastic spinners with telescoping arms that allow them to stretch. The arm should lock into place at regular intervals, and the rotation in two opposite directions. I don’t think this will work because the system no longer reacts, and the student has to provide the ideas, and it’s susceptible to mechanical breakage. We could have pentagonal pieces that look like squares with a slight arrow on one side that allows us to indicate direction. Then we place groups of them together, three groups of two for 2×3, and rotate them when we have a negative. But then we have to rotate all of them, which is time consuming. We could put blocks into “boats”, some way of grouping them and assigning a rotation, but that’s even more cumbersome. All of these methods require special manipulative to be purchased, organized, stored, and cleaned. To summarize, I can’t think of a good way to adopt the rotation metaphor into physical objects.

At even less fidelity, we can use square blocks. These are generic enough to justify keeping around for other lessons, and we can have magnetic squares for the teacher to put on the board and actual blocks for the kids to play with at their desks. We can use the grouping idea, and different colors for positive and negative blocks. Here’s how I envision it working:

So the blue squares are positive numbers, and the red squares are negative ones, except in the second two examples when we’re on the other side of the number lines, in Negative Land. In Negative Land, the blue squares are negative! So that means that the red squares that were negative are now positive.

That doesn’t make much sense. We’ve created an abstract visualization, so that even as the kids grasp the squares they won’t grasp the concept. The Negative Land mumbo jumbo winds up hand waving away the crux of the issue: why is a negative times a negative positive? We’ve illustrated the idea of reversing something twice but haven’t done much to justify it. Even worse, we’ve created two representations of the same number. The middle two lines are both -2, and the outer two lines are +2, but they don’t look the same.

Even more low tech: repeat the same exercise on paper with Xs and Os standing in for red and blue squares. This is eminently practical but the students now lose the ability to hold something concrete in their hands. We’ve pared the metaphor of flipping twice down to its essence, but lost a lot of its effectiveness in the process.

To quote Edison, “I haven’t failed. I’ve found ten thousand ways that don’t work.” Well, not quite so many, but same idea. Hopefully I’ve illustrated some of the pitfalls in making educational technology both practical and effective. And maybe you learned something about negative numbers in the pro — wait a sec. If explaining different ideas for explanations can actually work, then we may not have to come away empty handed after all.

The computer is assisted imagination. We can take the metaphors expressed most clearly in software and give them to the kids directly. Tell them to imagine themselves on a stretching, rotating platform. Better yet, line up groups of students and have them all rotate together, sensing the math with their own bodies.

The hard part of crafting a lesson plan, whether in person or over technology, is devising the metaphor, a new way to think about a topic so that it makes sense. Once we see negative numbers as rotation, systems of inequalities as unbalanced mobiles, complex numbers as spinners, then the students can explore them. That can be in spoken or written word, symbols, movement, sculpture, drawing, or on the computer. That’s the easy part.

This doesn’t bode well for educational technology companies. If their products are effective, their central metaphors can be easily expatriated to classroom exercises. At best, the metaphor is wrapped in unnecessary packaging; at worst, the packaging is all there is, hawked by cargo cults worshipping motion and interactivity as if these things only exist on a screen.

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An addendum, back to Bret Victor. In a note about the talk, he defines its subject matter as “a way of thinking. In particular — a way of using representations to think powerfully about systems.” (Emphasis his.) He is striving to create the next generation of tools that make unthinkable thoughts thinkable. Only these “powerful, general, technology-independent ways of thinking” will still be relevant a hundred years from now. It’s a daunting, open-ended task, especially considering how much trouble we got into just with arithmetic.

With the announcement of iOS 7, John Maeda criticized not just the OS but the debate it engendered. By phrasing interface design as a binary, of photorealism vs. flat abstraction, “To Skeu or Not To Skeu”, we lose sight of the possibilities that lie before us. Maeda writes, “What we need now is to move beyond the superficial conversation about styles and incremental adjustments to boldly invent the next frontier of interface design.” What do those new designs look like? “Something we haven’t even dreamed of yet.”

Reading visionaries like Victor and Maeda, it’s tempting to join the great quest to fundamentally alter how every person uses software, and by extension, how every person thinks. But part of me doubts the realism of their grandiose pronouncements. On the other side of the coin, Matt Gemmell is an iOS developer very much concerned with the present and its tools. He thoroughly deconstructs iOS 7, seeing it as an improvement, but not extravagantly so. It’s a needed update after six years. In another six we’ll be able to see the next UI paradigm, but he doesn’t waste breath trying to guess it now. Next century is of no concern. Write this month’s app, get this week’s paycheck, enjoy dinner tonight with friends and family. We’ll create the tiniest bit of the future in the morning.