The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.
Victor’s Scrubbing Calculator offers an elegant solution to certain kinds of mathematical problems. But does it help students formulate them? (Like Victor did, turning this into this?) Maybe I’m just whining that it won’t poach an egg or iron the drapes, but if students struggle to formulate problems, is the Scrubbing Calculator any use to them?
josh g.June 1, 2011 - 3:51 pm -
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.
The VirtuosiJune 1, 2011 - 4:51 pm -
I don’t think it helps students formulate problems directly. However, having better tools means that you can formulate more interesting problems. If less time is spent on the actual computation, it seems you can work in some more-complicated-to-calculate-but-conceptually-just-as-easy problems.
I’d be excited to see some more tools like the Scrubbing Calculator, only in so much as they allow instruction to focus more on the formulation itself.
If in using more innovative computational tools you can get rid of half the time spent in class calculating, that suddenly gives you half a class more time to focus on formulating.
MyleneJune 1, 2011 - 5:02 pm -
Carl MalartreJune 2, 2011 - 7:43 am -
Dan, would it help you formulate your problem (as a student) if the system would accept english sentences, like this:
“I have a top margin of 60 pixels, add 140 pixels for the bottom margin, add 8 gap of 20 pixels, plus 9 bars of (let’s say) 45 pixels = 1260”
Note1: the equal appears automagically and tools allow the students to link numbers together on multiple rows.
Note2: maybe the student should declare what he is looking for, and in what unit.
In a second operation, the student could simplify
60 top margin + 140 bottom margin + 8 x 20 gap + 9 x 45 bar height = 1260
In a third operation, he could choose which is the variable and get this:
Also, as a student, I could be confused between units and labels (pixels vs top margin).
Dan MeyerJune 2, 2011 - 10:30 am -
It’s the “let’s say” moment that needs to be practiced and habituated before students will have need of the calculator. Setting up that relationship is a non-trivial task. Along those lines, I like Bowen Kerin’s writing on the guess / check / generalize sequence. That helps a lot.
Carl MalartreJune 2, 2011 - 1:48 pm -
Nice post by Kerin. My first feeling is that this is just a tool, a step in the workflow. What about if other tools were helping the students in transforming what they found in something with variables.
We could setup “traps” where students would have to discover if an answer is… a fraction, to solve Kerin’s problem:
“What if the correct answer to the equation is square(2) or even 2/3?”
And maybe Bret could explains how he sees these cases.
BretJune 4, 2011 - 6:53 pm -
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.)
Carl MalartreJune 4, 2011 - 7:02 pm -
This is getting interesting!
Site note: I’m also thinking that it could… scrub fractions!
BreedeenJune 4, 2011 - 9:04 pm -
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.
BretJune 4, 2011 - 9:12 pm -
Have you ever tried multiplying roman numerals? It’s incredibly, ridiculously difficult. That’s why, before the 14th century, everyone thought that multiplication was ridiculously difficult, and only for the mathematical elite. Then arabic numerals came along, with their nice place values, and we discovered that seven-year-olds can handle multiplication just fine.
I consider myself cognitively and developmentally ready to set up a multiplication problem. But to solve it, I need the right tool.
louiseJune 5, 2011 - 6:33 am -
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?”
Sue VanHattumJune 5, 2011 - 6:43 am -
Thanks, Bret (comment #7), for the link to the Thurston paper. It’s great!
Dieter VerhofstadtJune 7, 2011 - 5:00 pm -
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.