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.

* * *

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.


2 responses to this post.

  1. Hi Max,

    Re “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”.

    Here’s John Holt, in “How Children Fail”:

    “I gave Marjorie 2 rods, and asked how many differently shaped rectangles she could make by putting them together. She saw there was only one… With 4 rods, there were two possible rectangles, a 1 x 4 and a 2 x 2. And so we worked our way up to 20, finding the factors of each number along the way… At no time on the way up to 20 did it occur to her that she could solve the problem by making use of what little she knew about factors. Given 10 rods, she did not think, “I can make a rectangle 5 rods long and 2 wide”; she had to work by trial and error each time. But she did get progressively quicker at seeing which combinations were possible and which were not.

    “I did not see until later that this increased quickness and skill was the beginning, the seed of a generalized understanding. An example comes to mind that was repeated many times. When the children had 12 rods, they made a 6 x 2 rectangle. Then they divided that rectangle in half and put the halves together to make a 4 x 3 rectangle. As they worked, their attack on the problem became more economical and organized. They were a long way from putting their insights and understandings into words, but they were getting there. The essential point is that this sort of processs not be rushed.”

    Here’s another bit from Holt:

    “Knowledge, learning, understanding, are not linear. They are not little bits of facts lined up in rows or piled up one on top of another. A field of knowledge, whether it be math, English. history, science, music, or whatever, is a territory, and knowing it is not just a matter of knowing all of the items in the territory, but of knowing how they relate to, compare with, and fit in with each other… It is the difference between knowing the names of all the streets in a city and being able to get from any place, by any desired route, to any other place.”

    “Why do we talk and write about the world and our knowledge of it as if they were linear? Because that is the nature of talk. Words come out in single file, one at a time; there’s no other way to talk or write. So in order to talk about it, we cut the real undivided world into little pieces, and make these into strings of talk, like beads on a necklace. But we must not be fooled; these strings of talk are not what the world is like. Our learning is not real, not complete, not accurate, above all not useful, unless we take these word strings and somehow convert them in our minds into a likeness of the world, a working mental model of the universe as we know it. Only when we have made such a model, and when there is at least a rough correspondence between that model and reality, can it be said of us that we have learned something.”

    In Holt’s view, all education, even pre-college education, is a process of discovery. The “relatively simple material” is relatively simple for you, but not for someone who has not built up the mental frameworks that you have. Perhaps the “semiconscious, unarticulatable hunches” are the scaffolding that learners need to build up sturdy frameworks.


    • Hello Bret,

      First, thank you so much for reading and commenting! If it’s not painfully obvious, I’m a big fan of your work.

      As I reread that bit you quoted, I refer to the goal, not the process. The bulk of the post, written in a much more breezy and less definitive style than your writing, looks to explore tradeoffs of media meant to aid discovery. Where pre-college splits is that, eventually, these intuitions coalesce into clear, lingual rules simple enough to be handled mechanistically. The more complex systems you are interested may have formalisms too complex to work with, or none at all; it doesn’t matter.

      There are plenty of systems that are simple enough to be rendered in crisp language or symbols. Being able to manipulate these efficiently requires the use of language. Language is unforgivingly precise. Before college (roughly), this precision aids in removing extraneous information, not reinventing the wheel, and communicating unambiguously with others and reference sources. Beyond that, language becomes a hindrance, as you say. But, for these simple systems, the language must be based on nonlingual understanding. I offer some rather rudimentary ideas on how this understanding can be encouraged, with emphasis on avoiding information technology.

      “The ‘semiconscious, unarticulatable hunches’ are the scaffolding that learners need to build up sturdy frameworks.” Yes. Thinking back on a number of old posts on this blog, I’ve probably come up with this idea many times before, without quite putting it into words.



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