H.O.T. I have been guided by Lauren Resnick's list of Higher-Order Thinking, since the early 90s. Most relevant here: H.O.T is Non-algorithmic. We already have machines that can think like machines. No reason then to hire people who only think like machines.
Symbolic Reasoning -- Not symbolic manipulation. This isn't a game with rules for moving playing pieces. The focus here is on communicating concepts symbolically. Not reciting lines from a phrase book, but actually speaking the language.
which is...Right Brain stuff...
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LANGUAGE ARTS
Outline/Rough Draft - We're making mathematical arguments; the same tools apply: Outline and rough draft are linguistic versions of 'top-down' design (See CS section).
Structure - Students are asked to identify, classify, extract mathematical fragments (ala sentence fragments) and clauses. Recognize (and correct) run-on equations. Noun-adjective issues are reinforced when thinking fractions: Three fifths. Seven xths (my spell checker has fits).
Identify the noun and adjective in these expressions: 3/5 Dw f(x)
Multilingual: Note that Spanish and French speakers have little trouble using the variable Cr for Car_red. More French may help: in 'y=mx + b'...Why is 'm' slope? ('monter'="to climb"?)
Parsing- Currently, students are poorly trained in parsing math expressions and word problems. For 'word problems', the training is initially restricted to working fragments, and processed
...as poetry:
Two trains D= RT D= RT
...leave Omaha...
….one heading east, and one west... De =ReTe Dw=RwTw <-------o------>
or trigger words:
….rectangle.... A▄ = LW P▄ = 2L + 2W
(Note the similarity to CS -- calling Libraries of functions).
Syntactic consistency
In spite of the mathematician's penchant for precision, many of the symbolic and linguistic elements are sloppy. Two immediate issues here:
Instruction overloading We say 'solve for x' for both 4x + 8 = 12 and 4x + 8y = c
Though mechanically similar, students find them conceptually very different. Initially, "Solve" is retained for the former, and "Isolate x" used for the latter. Conversely, there are many operations with mechanical and conceptually similarities, that COULD benefit from overloading. (a variation on 'reusability of code'.)
2(a+b) → 2a+2b is Distributing.
(ab)² → a²b² is a kind of distributing.
(See also Operator Overloading in the CS section)
Natural language programming -- Too often, concepts that COULD be connected fail because of linguistics. We use the words distributing and factoring, to describe a pair of operations, each the reverse of the other. You'd never know it from those words, where the first describes the action (operation), the second the results (operands). Minor adjustment: distributing and Undistributing; FOILing, and UnFOILing; adding and Unadding fractions.
Classes explore and create various word structures to convey their connections. "Why call it a half-rectangle?" "What's the reverse of exponentiation?" Wordplay encouraged: RATIOnal, LINEar. Only after the concepts are soundly embedded, are conventional terms introduced.
Vestigial terms. - Many aspects of mathematics instruction are vestigial. Not the least is relic terminology that needs to be excised, e.g. the cryptic, insidious 'undefined'; SS's and modern calculators have left that behind.
Epigrams - Artful quotes illuminate the 'habits of mind' we're trying to cultivate. Often, chapters will start with these.
The purpose of computing is insight, not numbers. - R.W. Hamming
You should know the answer ...before you start the calculation. - J. Wheeler, E. Taylor
"When you follow two separate chains of thought, Watson, you will find some point
of intersection which should approximate to the truth." -Sherlock Holmes, in
"The Disappearance of Lady Frances Carfax"
Use ordinary, everyday language when you analyze or debate a concept. When the image that emerges becomes so clear that anyone can understand it, that is the time to start
thinking of an applicable formula for it. - W. Heisenberg
Other times, a cartoon says it best. Hoping to obtain permissions for Far Side, Calvin & Hobbes, etc.
Which is really more...
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VISUAL ARTS