Maybe if they’re comfortable with applying arithmetic, but uncomfortable with tossing x’s and y’s in the mix.

But, I dunno, I want kids to learn some software writing skills … understanding variables is essential.

I notice that Victor is promising to soon publish a JavaScript library for authoring “scrubabble” text. How does this change the equation (so to speak)?

Come on, Dan — you know that’s not what a calculator is for. :) The slide rule wasn’t intended to help people formulate problems. It changed the world because it enabled people to actually solve those problems, and thus made the problems worth formulating in the first place.

The value that I see in ideas like the scrubbing calculator is that, once a problem has been formulated, it can be explored while in that natural form. Solving the problem no longer requires transforming it into other meaningless representations. It’s that transformation, that weird symbolic manipulation stuff, that dominates most people’s conception of mathematics, and frightens them away.

As long as we’re bringing up Einstein, here’s a bit from a wonderful old talk by Alan Kay. http://www.archive.org/details/AlanKeyD1987 Kay is talking about Piaget’s progression of “doing -> images -> symbols” (ie, the stages of kinesthetic, visual, and logical understanding), and Bruner’s work suggesting how all three forms of understanding coexist. (A little like Howard Gardner’s thing, although that came later.) Kay says:

“Jacques Hadamard, the famous French mathematician, in the late stages of his life, decided to poll his 99 buddies, who made up together the 100 great mathematicians and physicists on the earth, and he asked them, “How do you do your thing?” They were all personal friends of his, so they wrote back depositions. Only a few, out of the hundred, claimed to use mathematical symbology at all. Quite a surprise. All of them said they did it mostly in imagery or figurative terms. An amazing 30% or so, including Einstein, were down here in the mudpies [doing]. Einstein’s deposition said, “I have sensations of a kinesthetic or muscular type.” Einstein could feel the abstract spaces he was dealing with, in the muscles of his arms and his fingers…

“The sad part of Hadamard’s diagram is that every child in the United States is taught math and physics through this [symbolic] channel. The channel that almost no adult creative mathematician or physicist uses to do it… They use this channel to communicate, but not to do their thing. Much of our education is founded on those principles, that just because we can talk about something, there is a naive belief that we can teach through talking and listening.”

Since my project is focused on reducing symbolic abstraction, and moving reasoning into the visual and kinesthetic channels, you might say that my goal is to enable people to think more like Einstein. :)

There’s a fantastic paper by William Thurston called “On Proof and Progress in Mathematics” http://arxiv.org/pdf/math/9404236v1 which (among many other things) similarly asserts that practicing mathematicians work with pictures and metaphors in their heads, and resort to symbols when communicating. (He goes on to describe the detrimental effect that has on mathematical communication.)

Which is more cognitively difficult: solving equations or setting them up?

Can students be developmentally ready to do one and not the other? Or should we expect them to be able to do both at the same stage?

The answers to these questions may (or may not) say something meaningful about what we might expect to get out of a tool like this. And when such a tool might be beneficial.

I always avoid guess and check. With a calculator, a kid can spend an entire hour happily plugging in number after number, and not advance in cognition at all. On the other hand, I encourage picture-drawing instead of traditional notation. Pictures are symbolic, unless you’re in Jumanji. So are words.
When you speak of people using imagery or figurative terms – that is symbolism. People with a higher level of literacy understand that words conjure imagery. But words take a long time to write out. Start with words, then explain that mathematicians are lazy ( students can relate), so pick something to replace the word. There’s nothing wrong with a little picture, especially now that we are not restricted by available printed letters. A smiley face can be an unknown variable.
By the way, we are being told that it is no longer necessary to teach solving, it is equivalent to have students use a graphing calculator to determine intersection of lines. 2000 years of knowledge subverted to a machine. I don’t know. Is that the same logic as “we should not use slide rules, it makes it too easy?”

It does require some intelligence to see that the “reimbursement problem” figuring in abovementioned paper requires the reimbursement to be substracted from resp. added to the overspender’s resp. the underspender’s amounts.

You cannot use that argument to denounce the proposed solving method. Victor suggests an alternative to symbolic math to solve the problem. He does not say it makes the formulation of the problem easier.

The formulation can be made easier, or rather more intuitive or peraps more familiar, by applying an accountancy spreadsheet.

Friend paid 2000 dollars received (reimbursement value).
Bret paid 400 dollars paid (reimbursement value).

Whatever the way to facilitate the problem setup, more energy can be spent on it if the student can draw their attention away from the puzzling symbolic math required to solve it.