For people who want to learn or teach mathematics on the basis of reason …
Contents of "Reasonable Basic Algebra"
"Basic Algebra" is as undefined a course as can be inasmuch as its goal and contents would seem to depend on the question "Basic for what?".
On the other hand, a very strong case can be made that "Basic Algebra" should be defined as nothing more, nothing less than the investigation of (in)equations and (Laurent) polynomials. The case rests of course on the modern view of Arithmetic in which the various kinds of numbers, integers, rational, etc are tightly connected to the specification of outputs of (unary) operators (functions) by way of (in)equations and which thus intermingles notions from traditional Arithmetic and traditional Algebra. More precisely:
And, even if this should really, absolutely be the end of "Basic Algebra", the stage is in fact completely set for an ulterior systematic investigation of power functions, polynomial functions of degree 0, 1, 2, 3 and above, rational functions and transcendental functions by way of local (Laurent) polynomial approximations. Incidentally, this investigation is completely mathematically correct even though entirely without any recourse to limits. See Lagrange. In fact, this is precisely what the Integrated Sequence is about. (See the Epilogue in the Table of Contents below.)
- Using counting numbers, "Basic Algebra" first defines two basic (unary) operators, "addition to" and "multiplication by"---otherwise known as translation functions and dilation functions. (Think, for instance, of the multiplication tables each of which is the table of a dilation function.) So, as with any operator, the first thing one does is, given an input, to ask for the corresponding output. The two basic unary operators can then be combined into affine operators.
- Given an operator, the next thing one does is naturally to specify output(s) by way of (in)equation(s), and then to ask for the input(s), if any, for which the given operator will return the specified output(s). Unfortunately, more often than not, there are no counting numbers for which the given operator returns the desired output(s) and so the reverse operators, "subtraction from" and "division by", are of very limited use.
- On the other hand, signed decimal numbers, that is signed combinations of signed powers of TEN, make it possible to approximate the solutions of the above problems once the two basic operators have been extended to "algebraic addition to" and "algebraic multiplication by".
- One can then define two reverse (unary) operators, "algebraic subtraction from" and "algebraic division by" that give, though algebraic division usually only approximately, the solution of any basic problem. These operators can then be used to get the solution of any affine problem and even combined as the inverse affine operator which thus affords a sense of closure.
- The continuation into algebra is then obvious: substitute x for TEN with the incidental advantage that there is now no carryover/borrowing so that things are in fact simpler. Of course, the generalization of decimal numbers is really Laurent polynomials (i.e. including negative exponents) and in order to be in descending order of sizes, the exponents must be in ascending order when x is small and in descending order when x is large.
Reasonable Basic Algebra (RBA) is an implementation of the above view of "Basic Algebra", one that a reasonable number of developmental students can realistically be expected to complete in only one semester and that nevertheless offers a solid basis for ulterior studies.
However the interested teacher should note two very important points:
I have started to work on the change mentioned above but, as usual, things are not proceding as smoothly as I was hoping. These days, I have been working on:
- The ancillaries have now been recast in the new system already being used for RAF in which all the questions for all the ancillaries are in one single question base of "checkable items".
- The above view is not entirely implemented in the currently downloadable edition, namely that of the Table of Contents below. To that effect, in the next edition of RBA, there will be a new Chapter 3 - Functions and the current, rather short Chapter 3 - Addition and Chapter 4 - Subtraction will be merged into a new Chapter 4 - Translations. The reason this was not done in the current edition is only that the author suddenly got cold feet at the mere thought of introducing the word "function" in a developmental text. After repeated classroom use, though, he had to recognize that this had been a very, very bad decision. Fortunately, though, since he couldn't help but cast, if in a covert manner, the rest of the text in terms of functions, only minor editing should be necessary for the rest of the text. Other than that, the text is considered "stable" and the table of contents below "frozen".
However, this is not to be understood as meaning that the author considers that RBA cannot be improved but only that the author: i. is occupied with developing other parts of the Integrated Sequence, ii. does not feel strongly, at this time, about probably necessary changes while of course iii. he would be overjoyed to see them edited by users under the GNU Free Documentation License.
2 - (In)equalities - (In)equations
3 - Basic Problems
4 - Adding To - Subtracting From
However, I am running into a few problems:
- I am not sure which of (In)equalities - (In)equations and Adding To - Subtracting From should come first and which should come after. Both ways have their pro and con:
- Discussing (In)equations right after (In)equalities seems natural and has the advantage of providing immediately "practice" exercises with =, ≠, <, >, ≤, ≥.
- On the other hand, (In)equations arise naturally after Adding To - Subtracting From as "reverse problems".
- I am not sure where to cut between (In)equalities - (In)equations and Basic Problems. That though, should not be too hard to decide.
- At this time, in Adding To - Subtracting From, I am only dealing with the syntactic/linguistic aspect and the procedures themselves are specified only in the next chapter. But I am not sure that this does not look very artificial. On the other hand, incorporating the specification of the procedures for Adding To and Subtracting From in Chapter 4 seems overwhelming.
Finally, it should be stressed that, although I don't know of any other implementation of this view of "Basic Algebra" at this level, it is certainly not the only possible one. Even I can think of other ways to implement the above view, for instance by way of a purely syntactic approach---as opposed to the resolutely model-theoretic one that I adopted.
And, last but certainly not least, the use of technology might allow other radically more efficient implementations of this view of "Basic Algebra" than this paper-based one.
Current Table of Contents:
I. - ELEMENTS OF ARITHMETIC
- Counting Number-Phrases. What Arithmetic and Algebra are About, Specialized Languages, Real-World, Number-Phrases, Representing Large Collections, Graphic Representations, Combinations, About Number-Phrases, Decimal Number-Phrases.
- Equalities and Inequalities. Counting From A Counting Number-Phrase To Another, Comparing Collections, Language For Comparisons, Procedures For Comparing Number-Phrases, Truth Versus Falsehood, Duality Versus Symmetry,
- Addition. Attaching A Collection To Another, Language For Addition, Procedure For Adding A Number-Phrase.
- Subtraction. Detaching A Collection From Another, Language For Subtraction, Procedure For Subtracting A Number-Phrase, Subtraction As Correction.
- Signed Number-Phrases. Actions and States, Signed Number-Phrases, Size And Sign, Graphic Representations, Comparing Signed Number-Phrases, Adding a Signed Number-Phrase, Subtracting a Signed Number-Phrase, Effect Of An Action On A State, From Plain To Positive.
- Co-Multiplication and Values. Co-Multiplication, Signed-Co-multiplication
II. - INEQUATION AND EQUATION PROBLEMS
- Basic Problems 1: Counting Numerators. Forms, Data Sets And Solution Subsets, Collections Meeting A Requirement, Basic Formulas,
- Basic Problems 2: Decimal Numerators. Basic Equation Problems, Basic Inequation Problems, The Four Basic Inequation Problems.
- Translation And Dilation Problems. Translation Problems, Solving Translation Problems, Dilation Problems, Solving Dilation Problems.
- Affine Problems. Introduction, Solving Affine Problems,
- Double Basic Problems. Double Basic Equation Problems, Problems of Type BETWEEN, Problems of Type BEYOND, Other Double Basic Problems.
- Double Affine Problems.
III. - LAURENT POLYNOMIAL ALGEBRA
- Repeated Multiplications and Divisions. A Problem With English, Templates, The Order of Operations, The Way to Powers, }Power Language,
- Laurent Monomials. Multiplying Monomial Specifying-Phrases, Dividing Monomial Specifying-Phrases, Terms, Monomials.
- Polynomials 1: Addition, Subtraction. Monomials and Addition, Laurent Polynomials, Plain Polynomials, Addition, Subtraction.
- Polynomials 2: Multiplication. Multiplication in Arithmetic}, Multiplication of Polynomials,
- Polynomials 3: Powers of x0+h. }The Second Power: (x0+h)2, The Third Power: (x0+h)3, Higher Powers: (x0+h)n, Approximations.
- Polynomials 4: Division. Division In Arithmetic, Elementary School Procedure, Efficient Division Procedure, Division of Polynomials, Default Rules for Division, Division in Ascending Powers.
- Epilogue. Functions, Local Problems, Global Problems, Conclusion.
Like all titles to be made available on FMTo, RBA comes as a ready-to-use bundle that, in addition to the Text, includes the following Ancillaries:
- For each one of the eighteen chapters of the Text, there is a Homework involving a number of Exercises with:
- questions and open spaces to help the students investigate each Exercise,
- multiples-choice answers for each Exercise for the instructor rapidly to check the result of the students' investigations,
- an answer grid for the instructor to keep track of the students' progress.
- For each one of the three parts of the Text, there is:
- A Review Homework with open spaces for the students to investigate each one of twenty-five questions in the order of the Text.
- A Review Discussion in which the twenty-five questions in the Review Homework are being discussed at some length.
- A Review Pretest in which the twenty-five questions in the Review Homework appear, in text order or in random order, with multiple-choice answers.
- An Exam in which random multiple-choices variants of the twenty-five questions in the corresponding Review Homework appear in text order or random order .