The goal of Algebra++ (APP) is a series of three algebra 'textbooks', written and disseminated as spreadsheets (SS), which couple the core tenets of computer science (CS) to our underused 'right-brain' powers (logical storage, not physical). Titles:
The ++ iconizes the CS orientation, symbolic richness, and incremental progression of the series.
A constant question to be addressed: When is/could/should the teaching of algebra be like programming computers? When should it NOT be?And when is it like teaching art?
NB: The SS is not meant to replace paper and pencil work; rather, to foster understanding and experience algebra -- numerically, verbally, graphically, and symbolically. The educational goal is a regimen for developing speed, power... and uniquely...flexibility in doing algebra, using all brains available.
The SS naturally handles the numeric/tabular, visual/ graphical, symbolic, verbal formats, SIMULTANEOUSLY, and in a way that is free-format, customizable, and personalizable, while still reusable and standardizable. The SS also teaches subliminally: recursion, sequences, series, and explicit and implicitly-defined functions are demonstrated with your very first spreadsheet.
As a bonus, the students will be learning the SS, the universal number-processor (cf. word-processor) and hopefully, some good programming practice. Even with an old computer and SS, students can crank out results, faster than any calculator. Good job skill. Ideally, we'll get a few more CS majors, too.
Note: Having extolled the virtues of the SS, the final test is paper and pencil, and the student's brain. In the first two algebras, I'm one of the many teachers and schools who don't allow calculators. The machines can help teach the concepts, but should not be a prosthesis for the human brain.
The Reunion of Broken Parts (al-jabr) (Oxford dictionary) -- Algebra, is to a great extent, assembling and disassembling equations. The same mechanical sense that we apply to fixing a bike or dismantling a toaster, apply here. Much therefore is 'non-algorithmic' (L. Resnick) and 'concrete random' (A. Gregoric). Students should not believe that 'recipe' thinking is problem-solving, and need to let go of left-to-right/top-to-bottom bias. In fact, the dreaded 'word problem' is akin to a jigsaw puzzle (concrete and random). Many students actually like them, when viewed properly.
Problem-solving as navigation - "The path is not apparent..."-- Resnick. Generalized problem-solving has many parallels to path-finding. Students should learn to orient, spot landmarks, pick a direction, mark their trail, recognize dead-ends. Weigh the trade-offs of four hard steps vs. ten easy ones.
Incremental - Following the methods of flight testing, studies by the National Science Teacher's Association, 'flow' (M. Csikszentmihalyi), and the pioneering Saxon math series, the progression here will be gradual, incremental, and parallel-processed. Topics will be interleaved, rather than grouped in chapters. For example, the rules of exponents are introduced gradually over a dozen or more sections, interwoven with graphing sections, fraction work, etc.
Clean - Many students are lost when mathematical rigor is imposed at the outset. Here, rigor and 'completeness' are end goals, through iterative process, initialized with a conceptual 'feel for the problems', graduating through successive refinement to the standards of rigor. To this end, 'natural language programming' will be first used to cement the concepts (e.g. input and output), while at the end standard terms (domain and range) will be taught, so that students can transition to subsequent math texts, with no glitches.
Clear - Signal-to-Noise S/N Many 'modern' books are loud: full of color, photos, diagrams, and "Mrs. Wong has to make jiao-tzes for her mahjong party...." To most students, this is noise, which drops the transmission rate. This series will minimize such noise, aiming for figurative colorlessness (e.g. "....train from A-Town to B-Town...") etc. Similarly, there will be limited use of color, other than gray-scale, which makes it fair to color-challenged people, and cheaper if you decide to print to paper, or make copies. The exceptions to this noise reduction will be for certain applications, especially real-world problems, where sorting through the noise and smoke is a real-world skill.
Uncluttered - Good craftsmen regularly clean up debris from their work, continuously as they go. Clear out the clutter. Keep numbers nice. Travel light.
Topics and Exercise
While this work involves many visual and linguistic innovations, the actual topics covered and exercises worked will be (or at least look) for the most part conventional. 1) This allows the series to interface with standard curricula, and thus cover standard outcomes. 2) It also appears less foreign to most teachers and students (which is critical for getting them to try it.)
Exercises still have drill elements, just like softball or piano practice; we drill to build automaticity. However, we avoid the carpet-bombing approach ("Let's drop 10,000 problems on 'em; we're bound to hit something"). Instead, long-term practice, dominated by continuous review, punctuated with the smart bomb (e.g. that real-world application direct hit) provides long-term imprinting. Similarly, though many new elements emphasize understanding over memorization (programming), certain kinds of memorization are still critical, e.g. times-tables, standard formulas, etc ( Look-up table, in CS parlance).
Real-world
applications will for the most part be realistic, not contrived:
Exceptions are made for certain funny, fanciful, exotic, dramatic or toy-model examples, yet still the real-world connections are explored
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. Not 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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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:
1. 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 'code reusability'.)
(See also Operator Overloading in the CS section)
2. 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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Our visual communication is far older than written or symbolic communication. We process imagery before anything else. This should be cultivated to develop algebra sense. A picture is still worth a thousand words and a dozen formulas (paraphrase of Einstein).
Here, whenever possible, we first approach a problem visually. The goal is a graphical sense of the problem: 'seeing' the nature of the question, the solution path, the solution(s!), and critically...the neighborhood of the solutions. Here, primary tools are our senses of size and motion ...
Analog sense - We have an innate analog sense...an analog computer (my own usage). We process intervals of distance, time, music, force, motion, saturation, etc...without a numeral in mind. Observe that military and surgical units prefer analog clocks and watches; that digital speedometers are very rare.
Like many teachers, I use various graphical techniques to demonstrate mathematical relationships and develop student's analog sense. Unexpectedly, even when later given the option to use calculators and/or symbolic manipulations, many still prefer to draw pies to solve fraction/percent problems; or get numbers from a tabulated curve, rather than calculator. Sample tasks:
Flow of the problem solving is diagrammed with various arrows styles. The kinesthetic brain senses and records the 'motion'. This doodling is encouraged, for marking your trail, which is important when backtracking, or reviewing the problem six months later. ( See also CS section: Commenting).
Visualizations in USEFUL dimensions. Surprisingly, to see the solution path, a problem or relation may be best viewed from a higher dimension than given (or occasionally lower), even if we don't actually solve for a number from that higher perspective. For example, this is easily done by hand in the case of linear or compound inequalities, going from the standard 1D view, to the clarity of the 2D view. For Linear Programming, only the 3D view that software can generate, can take us from 2D to the clarity of the 3D perspective (See last two parts of the video)
Spatial reasoning - Nested parentheses [{3-4(9^2-3)+5(6+2)}+15-54*2] - 16 are a necessary evil only to programmers and typesetters. Rather, when processing this by hand, or within the SS text cells, we can simply change font size.
Likewise, here we teach the programming practice 'whitespace'. "Have a contest with your classmates, to see who can use whitespace to best effect, on the previous expression."
[ {3 - 4(9^2-3) + 5(6+2)} + 15 - 54*2] - 16
This is of course a subjective call.
Bracketing - Lastly, I've integrated this term from photography, where we bracket exposures. In APP, we commonly start problems by finding upper and lower bounds...the right answer being somewhere in between. For example, the area of the circle is more than the square enclosed, and less than the square that encloses the circle.
OK, now Left Brain.....
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I. The Fundamental maxim of CS is modularity or top-down design.This is woefully lacking in current texts and will be emphasized throughout. More elaborate 'procedures' are first planned in broad terms, and then built from combining functions, or modules, analogous to electronic circuit components. Here, students first explore, validate and practice each component/module, before combining into procedures.
II. Functions, functional calls, inputs and output, are all core concepts. Early and concrete introduction of functions will aid this:
A(r)= pi r^2
Insurance(Model, year, age, air bags, defensive driving,....)
The arcane "f(x)" formalism has numerous sources of confusion for students, and will be avoided till students are far advanced in the series.
III. Descriptive variables, are likewise sacred to modern programming. Observe:
12 14 9#5 11#5 C C C C 6 6
Each pair is from radically different fields of study, yet shows powerful and compact notations, highlighting similarities and differences. In this series, while the canonical 'x' will still appear, at every opportunity we use descriptive variables, with greatest emphasis on subscript form. Students will also be well-versed in pseudo-subscript form (Ar), iconic form (A▄), programmer/SS form (A_rectangle).
This triad --- modularity, functions and variable thinking --- is central to the ability to orient, organize information, frame the question, and devise solutions. As an example:
A_total(b,h,l,w):= A_rectangle(l,w) + A_triangle(b,h)
By hand:
(Note: Outline...Rough Draft ...Final)
Reusability of Code/Algorithms - The standard algebra curriculum violates this principle, with a grab-bag of tools, recipes and algorithms that clutter the minds of new students, and too frequently never appear later---often, because they go obsolete almost immediately. Instead, a Swiss-army knife approach is cultivated, where a user is trained to think flexibly with a smaller set of tools.
Examples: For the two-parameter curve fit (e.g. y = mx + b), all advanced versions share a common algorithm (likewise for inequality problems). Yet, in both cases the standard curriculum tackles basic versions using solitary methods that bear no resemblance to the generalized tool, even though that tool works fine on the basic versions.
Likewise, 'Systems' are typically treated with tables, which fail for anything beyond the simplest systems. Rather than tables, the use of mind maps (which are naturally scalable), is demonstrated.
Commenting - Develops the habit of good annotation and documentation. Unlike calculator, spreadsheets have many options for annotation of planning, algorithms, and debugging.
Information Hiding = Black-box - "This is why the quadratic formula works...." Not needed. Just show me how to use it (safely). The typical math book would explain electric motor theory, to someone who wants to use an electric drill. (Instead: 1.Trigger 2. Forward 3.Reverse 4. Changing the bit. 5. Don't stand in water).
Tools like the quadratic formula, will be encapsulated, with emphasis on interfacing with the black-box - not how it was built. Proofs will tend to be omitted, unless they are accessible, teach sound reasoning, and/or allow students to reconstruct the method or formula at later time. (However, an "Under-the-hood" feature will be used for those students who want to delve deeper.)
Delayed Evaluation - Don't perform any calculations, until you know you must. Too often books teach students poor form, conducting unnecessary manipulations and calculations.
Example: 32*7/16 (Did you multiply first?)
Example: Solve m/4 = 7/10 (Did you cross-multiply? Review what you did with the 10)
Inheritance - Students will be conditioned to watch for and discuss Inheritance, among operations and functions. Since multiplication is a short-hand for addition, it should inherit most properties (e.g. commutativity and associativity.) Since exponentiation is a form of multiplication, inheritance is expected. Since radicals are equivalent to fractional exponents...we don't need yet ANOTHER set of rules!
Operator overloading - Standard curriculum uses operator overloading. For example, we use "-" as both subtraction AND negation. Better to distinguish these verbally, and symbolically: standard calculators show [-] for subtraction, but [(-)] or [+/-] for negation.
Far worse, we use "=" in 4-5 different contexts, and expect the beginning student to contextually resolve ambiguities. You could NEVER get away with this, in a programming language. Yet, somehow the standard curriculum thinks it can. Instead, with minor adjustment in notation (:=, ?=, ≡, etc) many ambiguities are resolved.
Validation - The techniques of software validation carry far more credence with new users than lengthy monologues and mathematical proofs:
3x + 2x =
Test with x = 0, 1, -1, 2
The same 'testing' habit, will often negate the need to ask "Is that right?", or look up the answer in the back of the book.
Parsing - With mathematical expressions, much of the blame is on the lamentable PEMDAS, which is taught as a checklist, rather than operator rank.
Parsing is conditioned by working fragments, in paired/grouped exercises:
a) ….. + 7 - 3 + 5*....
b) ….. + 7 - 3 + 5*2....
c) ….. + 7 - 3 + 5*2 -....
and by following PEMA locally, rather than globally.
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DISSEMINATION The textbooks will be free for individual users. Schools would pay a per-license fee.
EXPENDITURES
1. The bulk of this funds a sabbatical for me, sufficient to complete the first 'book'. 2. Will need services from an intellectual property lawyer. 3. Obtain permissions to use copyrighted materials. (Far Side , Calvin & Hobbes, etc)
CONCLUSION:
Thank you for reading this proposal. I hope the long incubation of this concept is evident. I first gave a conference talk entitled 'Algebra++' a decade ago. These are not things I 'hope to try'. These are methods I've tested and refined with thousands of students. The methods work, and resonate with students and teachers. The emphasis is on understanding: If you know WHAT to think, you will know what to do. Especially, ten years from now. The reverse of 'remember' is forget; there is no reverse to 'understand'.
Thanks again for your support.
Credits: Answer keys in the video use some problems from Saxon's Algebra 2, and Stewart's Precalculus. Music is Michael Jones "After the Rain".