Before anything else, I would like to express my gratitude to the “presumably sick and tired administrator” I already mentioned in earlier Notes for prodding me into trying to articulate and formulate my thoughts in the matter of common final exams and also to the colleague responsible for confronting me with the tangible issue of a very questionable common final exam.

There are two sides to “learning”: There is what the learning process is, in and of itself (and the ways it can be helped or hindered), and then there is the assessment of the learning process (from whatever viewpoint). While, as I will argue, the two are very far from being independent, here I will focus on the latter issue.

To begin with, it is essential to make several distinctions.

1. The most immediate one is that one does not learn everything the same way. And here I do not mean that “people have different styles of learning”. While this is true to some very limited extent, the differences in “learning styles” have mostly to do with what is being learned, as opposed to who is learning. For instance, one does not learn, say, foreign languages the same way one learns, say, psychology, accounting or algebraic topology. A subsidiary issue is the level and kind of proficiency being sought as, if nothing else, the amount and kind of “investment” one had to incorporate in the learning process are going to be very different: even if one is eventually to play the same music, one does not learn playing the piano to play in tearooms the same way as one would learn eventually to be able to give recitals at Carnegie Hall. For instance, in the latter case, one must invest heavily in solfège while one may perhaps not have to do so in the former case. Thus, before we can talk about how to assess a learning process, we must begin by specifying what the learning process is to be about.
2. Another distinction must be made, namely that between evaluating the learning process itself and evaluating the extent to which the result of the learning process works in the real world. For instance, I may have learned “all about” multiplication of counting numbers but, when in a given real world situation, I may not be able to perform multiplications in a way appropriate to this real world situation. For instance, the particular situation may call for multiplication of 8 digit numbers and while I may have no problem doing that, I may be too slow for what the situation requires. This is in fact exactly what led to the funding of the original development of digital computers: it was not that the people involved in computing artillery trajectories “by hand” could not do it, it was that there was no way they could do it fast enough.
3. However, perhaps the most important issue is that no matter what the learning process is about, we must distinguish the external evaluation of the learning process from the internal evaluation of what one is doing as one is learning. As I am learning how to cope with a situation in a chess game, I must evaluate various possible moves as to their likely consequences. The external evaluation might–but need not–be the extent to which I win games. Yet, a major issue is that the extent to, and the manner in, which the learning process itself can depend on its external evaluation is not at all obvious: since any kind of evaluation procedure ought at least to specify what it is that it is testing and what it takes to pass, the evaluation of the output of the learning process cannot but influence the learning process itself. Not to dwell on the very real danger of “learning to the test”: If I know that I am going to be tested on endgames, this will not encourage me to study openings and so I will not really have been learning to play chess. On the other hand, the particular way I am assessing various possible moves in a given situation might help an external observer assess the learning process I am undergoing.

Here I will concern myself with “developmental mathematics” since this has become a major issue in two-year colleges, if not beyond, and, of course, since this is what this site is entirely devoted to.

I will begin by further considering the above three distinctions in the case of developmental mathematics:

1. While the learning process in “developmental mathematics” must, per force, be the way mathematics has to be learned, in fact, in the case of developmental students, who
   • are not only adults but adults who have been severely maimed by their previous education in “math”,
   • tend not to come from the most economically favored part of the population so that not only does their social status put a considerable economic burden on them, it is also not exactly conducive to the complete trust in oneself that is absolutely necessary for memorization.

   this learning process must be even more based on logic and discussion and, to that effect, be given a lot of time and tender loving care: Indeed, while comfortable,”made” mathematicians can be expected to be able rapidly to take some things of faith, like “named” theorems, developmental students, having been betrayed so many times before, are absolutely not capable to do so.
   For a further discussion of this aspect of the problem, see Chapter Thirteen and “Math Anxiety”

2. Inasmuch as we are dealing with developmental students, the real world situation that is most … real for them is the college level course they hope to be taking afterward. This means that reliance on so-called “real world applications” to justify matters is not likely to appeal to them. Even more importantly, it also means that the one and only external evaluation of a developmental mathematics course is how students do in subsequent course(s).
   

   As for a predictive assessment of a developmental mathematics course, it should focus at least as much on the “developmental” as it does on the “mathematics”: We certainly cannot assess the extent to which the students will be able to operate in the subsequent college level courses by just measuring their proficiency on developmental mathematics items.

3. As a result of the particular characteristics of the developmental population mentioned in 3. above, the external evaluation of the learning process would seem to have to be very close to the internal evaluation by the students themselves of what they have to do in order to deal with the mathematical situation they are confronting. For instance, to evaluate a developmental student’s learning of chess, one would have to check how the student evaluates her/his possible moves in a given situation. Just to count how many correct moves the student has chosen will not tell anybody anything about whether or not the student will be able to play real games.
   
   In the same manner, to find what the student will respond to:

   Compute, if possible. If not possible, write “not possible”: 3–7

   is not going to help us assess whether or not the student understands the most important difference between counting numbers and integers.

   What is important here is to make sure that the student realized that the definition of the addition of integers involves several notions: the size-comparison of integers, the addition of counting numbers and the subtraction of counting numbers. Whether we like it or not, nothing short of that can prevent the students from making “common mistakes” and/or will shield them from “common misconceptions”.

   Nor will

   Determine whether the following represents an equation or an algebraic expression. Any time you find an equation, circle its left hand side: 4x+2 = 7

   help us decide whether or not the student understands the difference between an equation and an algebraic expression.

While one might well accept the impossibility for “non-mathematician” administrators to deal with such matters as I have mentioned above, it is already more difficult to see how “math ed” faculty can fail to recognize the nature of the difficulties encountered by students in developmental mathematics and how they can support approaches based, when all is said and done, on memorization. However, it is quite impossible to understand how faculty who teach both college level courses and developmental courses can fail to see how what they are doing when they teach developmental courses cannot possibly help their developmental students to succeed in college level courses. It certainly speaks a lot about their lack of self-knowledge, about their relationship with mathematics. But that is quite another story.

Finally, and to come to the point of this Note, there is the issue of “common final exams”. A major reason for them is that the faculty teaching developmental mathematics has to justify what it is doing to two very different bodies:

• To the administration inasmuch as the latter must enforce a legitimate societal demand,
• To the faculty who teach subsequent college level courses.

For the curious reader, my own response in the case of Reasonable Basic Algebra can be found at Three Exams.

In another installment of these Notes, I will discuss the pro and con of final exams versus make-up exams and of other, possibly more effective, ways to reconcile the faculty with the realities of developmental mathematics. One possibility for instance would be to have faculty teach college level courses to students coming from developmental classes that they themselves have taught. Perhaps difficult to implement but where there is a will … .

I will also discuss the issue of how to let multiple sections taught by different instructors remain consistent without infringing on academic freedom and, which is the least of the problems, how to have common exams acceptable to all the instructors.

Summary of Past Episode: After some forty years of not so benign neglect, and under somewhat clumsy pressure on the part of the Vice-President For Academic Affairs, the mathematics department, suddenly become aware that the immense majority of its students were in the developmental courses, appointed a committee to deal with the matter. Two or three years later, the state of developmental students had not improved one bit but it looked as if my own students who are using Reasonable Basic Algebra might be threatened by the Final Exams to come. And then, quite by accident, I happened to have another conversation with the “presumably sick and tired administrator” at my school whom I mentioned earlier in Memorandum For A Reasonable Sequence during which we discussed the shape of the learning curve in RBA.

I had of course been criticizing the output of CEMEC, the committee rather unfairly empowered by the department to solve the problem while business could continue as usual, as if nothing was remiss. While they are nowhere near what commercial textbooks are now down to, the texts CEMEC had developed were nevertheless not much more than collections of topics even if written in reaction to carefully catalogued “elementary” errors committed by students in subsequent courses.

The question, though, is whether merely saying “this is a mistake and here is the correct way to do it” has any chance of being successful. The telltale is the huge number of exercises even if not repetitive and I cannot see that this is anything but memory-based. My own response, expounded in Chapter Thirteen, is that not making such mistakes can only result from the students understanding what they are doing and the necessity of communicating what they were doing to others without ambiguity: “No, I don’t know what you mean”.

The question the “presumably sick and tired administrator” then asked was of course how my students were doing and I had to admit that I didn’t really know. My excuse was, equally of course, that one semester was too short really to have a measurable impact. The administrator, though, persisted: what was my feeling? I then explained that my students essentially go through three phases.

1. During a first phase, which roughly coincides with Part One - Elements of Arithmetic, students flatly refuse to believe the characterization of mathematics as something in which one is more interested in trying to make sense of what is going on rather than “just learning how to do” a bunch of items to be on the exam. As a result, they try to recall, more or less successfully, whatever they remember of Arithmetic. However, with questions such as
   

   On Monday, your balance was two hundred thirty four dollars and fifty six cents in the red. On Tuesday, you deposit sixty eight dollars and eighty three cents. What is your new balance?

   which require to be read, the students are not too successful even though they have been aware of the kind of questions all along. They thus tend to be angry.

2. During a second phase, which roughly coincides with Part Two - Problems, the students start taking the stuff a bit more seriously, in part probably because the stuff is not so “familiar” but the level of anxiety rises correspondingly because they keep making errors in problems such as
   

   Find the solution subset of the double problem:
   
   either x<+21.46 or x≥-53.03 but not both.

   while realizing now that the problems are not difficult.

3. However, in the third phase, which roughly coincides with Part Three - Laurent Polynomials, students begin to accept the fact that they really could “do all the work” without me telling them what to do but now they see that time is now running out. Nevertheless, and perhaps surprisingly, they are not angry anymore.

So, what I was of course deploring was that the students in the next semester will be back to square one and likely disgruntled while there was really no reason for not building on this change of attitude.

And then, it struck me: since the department had cut down polynomials from the Basic Algebra course, Part Three - Laurent Polynomials, could become Part One of a Reasonable Intermediate Algebra (RIA). This would also have the advantage that Part One - Elements of Arithmetic, in addition to the introduction of the notion of function already advertised, could be developed a bit to include decimal numbers and the notion of approximation.

In view of the fact that nothing more really was needed for Reasonable Algebraic Functions, the way to Differential Calculus was paved.

And then, it seemed as if not only ought the Basic Algebra course be linked with an English Reading oourse, but that the Intermediate Algebra course could be similarly linked with an English Writing course in which students would learn how to write their cases on mathematical problems. The argument was that just as with reading mathematics, writing mathematics ought to transfer rather easily to writing “convincing arguments”.

The question remained, though, as to what the rest of RIA ought to consist of. The “presumably sick and tired administrator” let me have the course description for the Intermediate Algebra course which I annotated as follows:

A critical reading by Schremmer of the 1991 course description of Intermediate Algebra.

I. Real Number System (CLO I)
A. Identify real numbers, natural numbers, integers, rational numbers and irrational numbers
B. Determine the order of real numbers
C. Identify some of the properties of real numbers

Notes.
This is sheer make believe. How can definitions be provided in the absence of a serious conceptual grasp of equations. As a matter of sheer nomenclature to be memorized, it is not very desirable as a beginning. For example, how is one to compare √15 and π in current Intermediate Algebra?

In alternate Basic Algebra, RBA chapters 1 to 6 as to be modified plus two chapters on decimal numbers and decimal approximations, things make “perfect sense”as decimal approximations of solutions of equations:

From x^2 =15, we get with the help of (x_0+h)^2 = x_0^2 + 2x_0•h + h^2 (RBA chapter 17) that the approximate solution—called √15—is, depending on the level of approximation
√15 = 3 + […]
√15 = 3.8 + […] or 3.9 + […]
and given that π = 3.1 + […]
we have that π is less than √15

II. First Degree Equations and Inequalities (CLO II)
A. Solve equations including fractional forms
B. Solve literal equations
C. Solve inequalities
D. Solve absolute value equations and inequalities
E. Solve word problems using first degree
equations and inequalities

Note. At best, this is an ill-defined grab bag.
A, B and C are done properly in RBA chapters 7 to 11.
D is not done in RBA nor will it done in the alternate Intermediate Algebra for two reasons. RBA does discuss the fact that one can compare signed numbers from either the signed point of view or the point of view of “size” but at no point in the development does there arise a need for a systematic investigation of (in)equations involving sizes. The other reason is that although it could be easily done on the basis of what is already there, it seems more reasonable to allocate what little time there is to what cannot be dispensed with.
E was very fashionable in the nineties but has started to be discredited by somewhat serious research.

III. Polynomials (CLO III)
A. Determine the degree of a polynomial
B. Classify a polynomial by number of terms
C. Perform factorization of polynomials including multiple step factoring
D. Translate word problems into equivalent quadratic expressions
E. Solve equations of the form (A) (B) = O where A and B are polynomials
Notes.
A and B are both jokes. Students have no problem with degree and number of terms once they have familiarized themselves with polynomials.
C is downright criminal inasmuch as there is NO procedure to factor polynomials even of degree 2.
D. See above
E. This is nothing more than the fact that if you know that two numbers multiply to 0, then you know that one at least has to be 0. This has nothing to do with algebra.

IV. Algebraic Fractions (CLO IV)
A. Reduce algebraic fractions
B. Multiply and divide algebraic fractions
C. Add and subtract algebraic fractions
D. Simplify complex algebraic fractions
E. Reduce algebraic fractions after adding or subtracting
F. Solve word problems using algebraic fractions
G. Solve literal equations with algebraic fractions
Notes.
A to E make sense only once the relevant arithmetic ideas have been understood. And then, the transfer to algebra poses little difficulty. Moreover, while there is really no need for algebraic fractions by themselves, they do not pose any difficulty in the investigation of Rational Functions.
F and G. See above
LAST BUT NOT LEAST: It is difficult to see how “addition, subtraction, multiplication, division—both in ascending and descending order of exponents—of Laurent polynomials, (that is including negative exponents) together with the expansion of (x_0+h)^n (binomial theorem)” can be neither in “redefined” Basic Algebra nor in current Intermediate Algebra !!!!

V. Exponents and Radicals (CLO V)
A. Simplify expressions with rational exponents
B. Simplify radical expressions
C. Add, subtract, multiply and divide radical expressions
D. Solve equations containing one or more radical expressions
E. Solve word problems using radical expressions
Notes,
A. Rational exponents is a very sophisticated notation for which there is no use at this level: For instance, 17^(3/5) is the name of the solution of the equation x^5 = 17^3, not exactly very likely to be encountered at this level.
B and C are quite surprising both in view of the absence of their counterpart for “polynomial” expressions and of the lack of context here.
D is beyond the pale
E See above.

VI. Second Degree Equations and Inequalities (CLO VI)
A. Solve second degree equations
B. Solve second degree literal equations
C. Solve second degree inequalities
D. Solve word problems using second degree equations and inequalities

Notes. This is more than a bit glib as the obstruction to solving second degree equations is that there is one more term than the = sign has sides so that it is a far cry from solving affine equations. In any case, they are best left out of Intermediate Algebra.for the same two reasons as in CLO II: There is no need for the investigation or quadratic equations until quadratic functions are investigated—which is where everything makes perfect sense—and the time is better employed elsewhere. This properly belongs to Precalculus One.

VII. Rectangular Coordinate System (CLO VII)
A. Determine the slope and intercepts of a line
B. Determine the equation of a line
C. Determine parallel and perpendicular lines
D. Graph linear equations
E. Graph parabolas
Notes
A is misleadingly worded: The slope of a line is most important. The y-intercept is the output for input 0 and thus of little interest. The x-intercept on the other hand is the input for which the output is 0 and therefore the solution of an affine equation.
B usually consists of:
—the “point-slope formula” which is better seen as a prototypical “Initial Value Problem” (IVP) in the case of affine functions. This properly belongs to Precalculus One.
—the “two-points formula” which is better seen as a prototypical “Boundary Value Problem” (BVP) in the case of affine functions. This properly belongs to Precalculus One.
Because both types of problems are extremely important—but also extremely difficult type of problems—it is certainly worth looking at these prototypes but this requires the context of affine functions.
C is of secondary importance at this stage and should be omitted from alternate Intermediate Algebra for the usual two reasons.
D is out of context as “graphing linear equations”—presumably in two variables—belongs to 2-dimensional geometry and should be omitted from alternate Intermediate Algebra for the usual two reasons.. This is in contradistinction with the graphing of affine functions.
E. This makes sense only once parabolas have been defined which they don’t seen to have been here. This properly belongs to Precalculus One.

VIII. Relations and Functions (CLO VIII)
A. Define relations and functions
B. Evaluate a function
C. Graph relations and functions
D. Translate word problems into equivalent functional expressions
E. Solve word problems using functional notation
Notes
A, B. This makes little sense at this late stage inasmuch as it cannot lead anywhere here. See Epilogue in RBA.
C. This is completely misleading inasmuch as it uses the word “graph” instead of the word “plot”. What happens of course, is that other than with affine functions, there is absolutely no way to turn a plot into a graph by just “joining smoothly” the plot points. A lot of qualitative information must be dug out of the input-ouput rule of the function before this can be done. This is in fact the theme of RAF, the text I developed for Precalculus One.
D. See above
E All mathematicians fervently wish this were possible.

IX. Systems of Equations (CLO IX)
A. Solve systems of linear equations in two variables:
1. Using elimination
2. Using substitution
B. Solve word problems using systems of linear equations in two variables
C. Solve nonlinear systems in two variables
Notes
A comes rather late since “solving systems of [presumably two] linear equations in two variables” is a necessity for VII-B. In fact it is dealt with in RBA chapter 12.
B See above
C is a howler in vagueness. Besides, it makes no sense until functions beyond affine functions have been introduced. This properly belongs to Precalculus One—if at all. Should be omitted from alternate Intermediate Algebra for the usual two reasons.

So, on prima facie evidence, it seemed that it should not be too hard to develop RIA

[To be continued]

Summary of Past Episodes: We left me after I had submitted an “expression of interest” in providing a chapter “dealing with two important agendas for mathematics education in societies around the world, namely quality and equity” after which common sense had prevailed and I had forgotten all about it.

A couple of months later, though, and to my great surprise, I was advised that “your proposal was Conditionally Accepted by the group of Editors and [that] you are asked to develop the full first draft of the chapter by 30 July, 2009.“

That it was “conditional” of course immediately turned me off: Without knowing the conditions, I surely had neither the time nor the patience to write a chapter, a lot of work, to find it later rejected in the old MAA Monthly manner mentioned at the outset of this adventure in academia. Sure, I would have liked to advertise my views in a “scholarly” book that might impress administrators, but the price was likely much too steep. Nevertheless, I kept on reading.

However, if I hadn’t already been essentially turned off, the next paragraph alone would have done it: it read in part: “We are particularly pleased that the list of authors includes both some highly accomplished authors with wide experience and international reputation and some newer researchers.” Gone was the “authorship [being] requested from”, inter alia, ” mathematics teachers” of the Call for Chapter Proposals. It now looked like we were back to the usual: pure, useless mutual educando in which at least half of any paper consists in quoting other papers.

But, since the message continued with “some comments on your chapter”, I got curious and read on. I am quoting the two paragraphs that followed in full:

Some comments on your chapters were: This is an interesting proposal that is potentially related to the theme of the book. It promises to discuss quality as an alternative to rigor within mathematics or the utilitarian use of mathematics. Two comments are suggested to the author. One is that the focus of the book is on the interaction of the quality and equity agendas. Hence the expectation is that every chapter need to address each concept. In this case, what is the implication of basing quality on reason and common sense on equity issues? Secondly, the author may need to define reason and common sense that posit them as outside cultural and social influences - or is he arguing that the quality of mathematics education is geographically and temporally determined?

and

This proposal explores a proposal for the betterment of the teaching of mathematics at university level. It seems that the author only takes on the issue of quality and assumes that equity comes as a result of the first. It is not clear to me how this proposal is a contribution to the concerns of the book.

The comments in the first paragraph were not entirely untrue. For instance, I could see why the reviewer would ask the first question, “what is the implication of basing quality on reason and common sense on equity issues?“. In a way, I had not made my position clear: I had only said that the reason there was no equity in the current situation was that mathematics education was based on memorization and I had not spelled out why this would automatically work against the “disadvantaged”. But the second question got me more than a little bit worried: Did the reviewer really think that a mathematician would not hold that “reason […] is outside cultural and social influences”?

The comment in the second paragraph on the other hand jarred me as, indeed, I had “only tak[en] on the issue of quality and assume[d] that equity comes as a result of the first.”

So, I sat down at my Mac and in the course of five days wrote Chapter Thirteen which I dispatched attached to the following:

Upon reading that my proposal was only “Conditionally Accepted by the group of Editors” you will not be overly surprised that I was getting ready to decline … until I read one of the comments which caused me to sit down at my desk and stay put until the attached stuff was done. Which took me a bit longer than I had counted on.

The stuff is coming at, er … 7130 words [the maximum had been set at 6000], but then, after all, we are talking magnum opus, radical departure much apt to be misunderstood, etc. Besides, there should have been a chapter 0 analyzing all that was wrong with the current non-education that I could have referred to instead of spending over half my allowance doing just that. Oh well!

Anyway, under the GNU Free Documentation License that this was written under, and should they happen to be at all interested by this piece of truly an-academic writing, the editors have complete latitude to do whatever rewriting/editing/academising/cutting/digesting/expanding/translating/whatevering they may wish. In other words, if I am ready to put the stuff in whatever reasonable format the editors wish and, of course, do a little bit of polishing here and there, e.g. references, as far as I am concerned the chapter is written.

What happened is that I had long felt the need to carefully delineate and explain what I am trying to accomplish with the magnum opus and that I had already made numerous unsuccessful attempts, e.g. in in the Preface to Reasonable Basic Algebra and in my Notes From the Mathematical Underground. See freemathtexts.org. Somehow, I never felt I was really describing what I was trying to do and why. It was only reading that one sentence from the reviewer that suddenly gave me the key to what I had missed all that time and that caused everything finally to fall in … one long place.

But now, after having already spent too much time writing this manifesto, I need to return to the magnum opus and must leave it entirely to the group of Editors to decide whether “reasonable” mathematics as a political act is an idea worth pursuing.

But please, no matter what the decision, keep in mind that this will have been more than worth it to me and that I am very grateful both to you and to the anonymous reviewer.

Best regards—apologetic should events require it.
–Schremmer

Within 24 hours, I got the following unsurprising response:

I […] read chapter 13.

I am afraid it is not in a style or content that we are interested in in this publication.

It perhaps fits in more into a separate book on your views of
mathematics education but it does not have sufficient material on issues
of quality and equity.

Please let me know if you are interested in trying for another
contribution to the book to be written is the standard academic style.

to which I responded:

I wrote what I thought was relevant to quality and equity but when the book comes out I will be interested in taking a look, so could you let me know? I will get my school to buy it.

In any case, I am totally unable to write in the standard academic style.

And, as I already said, I am glad this made me write something I had long wanted to.

Best luck
–Schremmer

And thus ended the dream: A lesson to anyone with still any hope in academia.

It would seem that I may not have been entirely fair to the “presumably sick and tired administrator” at my school whom I mentioned in Memorandum For A Reasonable Sequence. Things seem to be perhaps moving but, at best, with very, very deliberate speed and what is going to happen remains totally unclear.

But first let me go back and provide a bit of background even though the story is rather unlikely to be unimaginable to people in the profession.

A few years ago, the Vice President for Academic Affairs took notice of the fact that the Mathematics Department hadn’t exactly gone out of its way to facilitate access to mathematics for the “great unwashed masses”. We do have the required Arithmetic and Basic Algebra non-credit courses as well as the usual Intermediate Algebra credit course from which students are supposed to be able to get into PreCalculus One and then PreCalculus Two and then, if they are still alive, Calculus One—Differential. (There are of course other “Paths To Possibilities” but this is the one I am interested in.) What the VP may have noticed was that precious few students make it through that path: Less than a quarter of one percent of the students registering in Arithmetic pass Differential Calculus. What the VP actually complained about, though, was the passing rates in the first two courses of the “path”: Arithmetic and Basic Algebra.

To remedy this unfortunate situation, the VP—who is nowhere near being a mathematician—thought to order the Department to hire people with a degree in Math Ed instead of in just Mathematics. The underlying reasoning, though, was not clear since a couple of “Educologists” could not possibly make any difference given the very large number of developmental sections to be taught. Still, that would be good for the moral of the few “educators” in the Department whom the VP is said to approve of. You know the kind.

After the Department tried to stonewall the VP by explaining that it was the students who were to blame, or maybe the students’ parents, or the high schools, or the use of calculators, or the lack of time or maybe El Niño, or who knows what, the VP thought of applying some pressure on the Department by threatening to create a separate Developmental Mathematics Department.

The pressure was not all the VP could have wished because, if the VP could probably have forced the Chairperson of this Developmental Mathematics Department to be an “Educologist”, since there is a Collective Bargaining Agreement most of the current mathematics faculty, including yours truly, would have ended up teaching in the Developmental Mathematics Department.

Still, the fear in the Department was such that it bestirred itself to the point that . . . it created a new Committee of Elementary Education and its Effects on the Curriculum (CEMEC) which, sure enough, eventually came up with a Proposal.

The Proposal invoked all the appropriate “authorities”, the “Crossroad in Mathematics Standards”, etc, and said that it had

examined our curriculum and the way it is taught and concluded that:

It is quite often presented in a very repetitive manner. A teacher presents a problem and the same type of problem, where the variation does not explore the complexities inherent in the recursive scheme of the algorithm, is solved by students several times, with no additional gain in insight beyond the trivialities of the variation.

The textbooks that are commonly used (and, in some way, adopted by the Department) reinforce that type of teaching.

The conclusion was lofty enough but the implementation was essentially to be fewer topics—which was good but one might perhaps be forgiven for wondering what took the Department so long to come to that conclusion—and increasing the number of instruction hours—which was debatable and most certainly a tactical mistake. And, last but not least, texts were of course to be written.

The “area of attention” was confined to: “Teaching issues, Faculty preparation and responsibiliites, Students’ issues, Structure of remedial mathematics program.” In other words, rather ironically, exactly what you would have expected from . . . Educologists!

What was glaringly absent was any”attention” to what was to be learned by the students. The 116 page long document did “specify” the courses with detailed lists of topics, each illustrated with sample questions but the emphasis throughout remained on mathematical “factoïds” and the issue of what bound them and why they should be investigated was not even raised. Typical was the use of phrases such as “understanding of concepts”—even though they were to remain in isolation—and “proper use of mathematical language and symbols”—as if in a “finishing school”.

For example, in the exercise

Write using exponential notation whenever possible {(-z)(-z)(-z)} / {z+z+z}

students must not only recognize the operations that are performed on the variable z but also recall that exponential notiation applies only to multiplication. Each time a student decides whether exponential notation can be used, he reinforces his understanding of this concept.

And, in fact, the texts that were eventually to be written were entirely prescriptive.

The VP agreed to a “pilot testing” but, a couple of years later, the results were rather unsurprising:

It is clear that pass rates for students in CEMEC sections were no better and in almost all cases worse than for students in the non-experimental sections.

But of course this was because

the CEMEC materials are more demanding and since the students in CEMEC sections were consistently held to higher standards than in most other sections, it is not surprising that pass rates in CEMEC sections are lower than in other sections.

Some attention was given to the outcome in further mathematical courses:

The success of students in subsequent mathematics courses was always considered by CEMEC to be a prime indicator of the success of this approach. The results are mixed. The success of Arithmetic students in passing Basic Algebra is roughly the same for students in CEMEC and non-CEMEC sections. The success of Basic Algebra students in passing Intermediate Algebra is clearly better for students in CEMEC sections compared to students in non-CEMEC sections.

However,

The positive reports of faculty in the Pilot provide qualitative evidence for the sustainability and possible expansion of the CEMEC plan. The materials and methods of CEMEC have been embraced by a diverse group of mathematics instructors with different histories, practices and assumptions.

And therefore

The CEMEC Pilot Project has shown promise in accomplishing its primary goal: to have students who successfully complete Arithmetic and Basic Algebra obtain a better understanding of Arithmetic and Algebra. Where it has fallen short is in getting a larger percentage of students to successfully complete these courses.

The VP was not happy, rejected CEMEC’s conclusions, demanded Exit Criteria and independently administered Final Exams with, again, the threat of a separate Developmental Mathematics Department lurking in the background.

The Department caved-in re. Math Ed degrees and approved “Revised” Course Descriptions for Arithmetic and Basic Algebra together with mandatory Common Final Exams.

Regarding the latter, I wrote the following:

I have long been (40+ years) an advocate of common exams for the simple reason that most courses at CCP are part of sequences—whether at CCP or continuing elsewhere—so that the instructor in course n has to know what s/he can count on the students having learned in courses i < n. Moreover, since most courses at CCP are multi-sections, instructors need to tune their violins.

I have long been (50+ years) an anarcho-syndicalist. For those to whom the term is less than familiar, it designates the school of anarchism whose main concern is to prevent the accretion of power in the hand of a minority but which realizes that, while small may be beautiful, certain systems need to be large, e.g. airlines, railroads, manufacture, Basic Algebra, etc. However, anarcho-syndicalism holds that the solution is not management by a small minority but by “all involved” through syndicalism—as partially opposed to unionism in its current meaning.

As such, here is the kind of system that I advocated back then. Let me take an exam for Basic Algebra as an example. Say the exam is to consist of 25 questions.

PART ONE: Specifying the exam. (Or maybe the course?)

1. Let everybody—everybody who wants (?)—submit, say, 10 different questions
2. This should add up to at least a couple of hundred questions.
3. Partition these questions according to roughly what they intend to check, e.g. addition in Z, division of polynomials, etc so that each part corresponds to what I will call, for lack of a better term, a “checkable item”.
4. If necessary, subdivide the above parts. For instance, division of polynomials might be divided into: with all coefficients in N, or in Z, or in Q^+ or in Q. Will division in ascending exponents be included? Etc.
5. Work on this until an acceptable list of, say, 30 “checkable items” has been arrived at. This list of checkable items is just an embodiment of a list of descriptions of “intermediate performance objectives” which may or may not actually be written down eventually—although I don’t see the point. Hopefully, the fact that the length of the list is bounded (here 30)—although not by the number of questions on the exam (here 25)—should help a consensus to be arrived at. The fact that there are more checkable items than items actually checked on any exam actually given out ought to prevent, at least to some extent, “teaching to the exam”.

PART TWO: Implementing the exam.

1. For each one of the 30 “checkable items”, let everybody—who wants (?)—submit, say, one “instantiation”, that is an actual question.
2. Let everybody see all the instantiations proposed for all the checkable items and let everybody have the right to reject any instantiation(s) but only with an explanation as to why. For example, if I submit an instantiation of “division of polynomials” in which a polynomial with seven terms has to be divided by a polynomial with four terms, one may object that such length is not necessary to check whether the student can really divide a polynomial by another. Else, I would have to explain exactly why such long polynomials are needed. On the other hand, were you to submit 2+3 as an instantiation of addition in N, I would reject it on the basis of various objections.
3. Note: The distinction between “checkable item” and “instantiation” of a checkable item is thus paramount: One might agree with a “checkable item” but not with a “instantiation” proposed for that checkable item.

Given that there are very many sections of Basic Algebra, such a system would ensure that all the students would be taking essentially the same exam while no two sections would have exactly the same version and an instructor might even use two different version in her/his class. This would call for, I think, about a dozen instantiation of each checkable item for the system to be reasonably reliable.
It would also require a computerized system to produce the actual exams. Fortunately, there are several LaTeX packages that can do this. The one I am using is probsoln. I used it to write my own implementation of the system described above. For each instantiation, this implementation allows both a multiple-choice format and an open format. For instance, given (+3) + (-2) + (-3) + (+5) = ?, this implementation can, for instance, propose the choices (a) +3, (b) –3 . . . (e) None of the preceding. It can also give a “response space”as well as other options. Since I was able to write this implementation, anyone should be able to but, in any case, it will soon be available for free download under a GNU Free Documentation License.
Beyond this issue of common exams, we should also think about the specific “goals” of each sequence and then how to achieve these goals over the length of the sequence as opposed to listing a laundry list of “skills” as nowadays offered in place of “intermediate performance objectives”.

Of course, the Department found it easier to delegate the whole thing to a committee and I started seriously to worry about how I would protect my Basic Algebra students—who are using Reasonable Basic Algebra—from a final exam likely to focus on “understanding of concepts” and “proper use of mathematical language and symbols”.

This was when, somehow, something stirred along the lines of the Memorandum For A Reasonable Sequence.

It started with an accidental but long conversation with the “presumably sick and tired administrator” mentioned at the outset. The subject had swiftly turned to the lack of continuity between Basic Algebra and Intermediate Algebra and I came up with . . .

[To be continued]

Summary of Past Episodes: We left me rather amazed at the fact that, somehow, someone, somewhere in academia, had actually noticed that the great masses were being served rather substandard mathematics, didn’t in fact seem to think that this was something pre-ordained and therefore that one might at least think about it and, possibly even do something about it.

So, … having been enraptured by such political overture in a call for proposals to discuss issues of “quality versus equity”, but also for other reasons to be discussed later, I responded with the following:

Please find below my “expression of interest“:

An often overlooked aspect of “developmental mathematics education”, at least in the USA, is that commercially available textbooks are memory based so that, to facilitate memorization, the subject matter is atomized into “topics” presented independently during a couple of short semesters so that nothing can make any sense anymore. All connective tissues have been removed and no build-up can take place. Indeed, typically, instructors deplore that their students cannot remember the simplest things past the test. This has a number of dire consequences:

1. Students who wish eventually to learn, say, Differential Calculus, the “mathematics of change”, face an inordinate number of courses: Arithmetic, Elementary Algebra (8/9th grade Algebra I), Intermediate Algebra (10/11th grade Algebra II), College Algebra, College Trigonometry, Calculus I.

2. Success is defined internally rather than by success in later courses with the result that it is rarely noticed that the chances of overall success are extremely low—in the above example, usually no more than one percent.

3. Developmental students are ghettoized into identifying learning with having recourse to experts and into belittling the power of personal logical thinking while, to quote Colin McGinn, “One of the central aims of education, as a preparation for political democracy, should be to enable people to get on better terms with reason—to learn to live with the truth.”

Yet, there is nothing inevitable about this situation and the object of the proposal is the description of a three course sequence in which the intention is to get the students to change, in John Holt’s words in “How Children Fail”, from being “answer oriented”, the inevitable result of “show and tell, drill and test”, to being “question oriented” and thus, rather than to try to remember things, the students can reconstruct them if and when needed.

Some of the characteristics of this sequence are:

—An expositional approach based on what is known in mathematics as Model Theory which carefully distinguishes “real-world” situations from their “paper-world” representations.

—Contents carefully structured into an architecture designed to create systematic reinforcement and thus foster an exponential learning curve based, in Liping Ma’s terms, on a “Coherent View of Mathematics” and thus help students acquire a “Profound Understanding of Fundamental Mathematics”.

—Systematic attention given to linguistic issues that often prevent students from being able to focus on the mathematical concepts themselves.

—Continuing insistence on convincing the students that the reason the things they are dealing with are the way they are is not because “experts say so” but because common sense says they cannot be otherwise. But, while the standard way of establishing truth in mathematics is by way of proof, Edward Thorndike showed a century ago that proofs do not transfer into convincing arguments. So, the sequence uses a mode of arguing somewhat like that used by lawyers in front of a court. See Toulmin’s The Uses of Argument.

Such a “fast track” as described above would of course seem utterly improbable and the reason it works is that the investigation of functions is based on (Laurent) polynomial approximations as an extension of decimal approximations. i.e. Lagrange’s approach.

Part Two, Algebraic Functions, and Part Three, Transcendental Functions, were indeed originally specified, designed and experimented with under the terms of a 1988 NSF calculus grant as an alternative to the conventional sequence, Precalculus I, Precalculus II and Calculus I (Differential). The 1992 report of my school’s Office of Institutional Research said in part:

“Of those attempting the first course in each sequence, 12.5% finished the [conventional three semester 10 hour] sequence while 48.3% finished the [integrated two semester 8-hour] sequence, revealing a definite association between the [integrated two semester 8 hour ] sequence and completion (χ2 (1) = 82.14, p < .001).”

The report also said that the passing rates in Calculus II (Integral) for the students coming from the above two sequences were almost identical but that this was not significant because, in both sequences, most students did not continue into Calculus II (Integral).

Because of the work done for Differential Calculus I and Differential Calculus II, the contents of the Algebraic Functions course and of the Transcendental Functions course are well specified and therefore the contents of Part One, Decimal Arithmetic and Basic Algebra are fairly well specified as being exactly, no more, no less, what is needed for the Algebraic Functions and Transcendental Functions courses. They are currently under development.

Then, upon a little bit of reflection in which I thought that the above sounded a bit too much like self-promotion, I emailed the following:

Being at the end of my career, I am not in need of publications nor do I even particularly wish to publish as the situation described in the first half of the rational I submitted concerns me enough that I got myself involved in the rather large project alluded to in the second half of the rational. See freemathtexts.org for more detail.

On the other hand, the alternative given in the call for proposals, i.e. “[q]uality mathematics […] seen as a reflection of its rigor, formality and generalisability” versus “utilitarian importance” struck me as rather restrictive and I would argue that there is a middle ground in which the situation mentioned in the first half of my proposal is dealt on the basis of an appeal to “reason” as opposed to a reliance on memory and which addresses utilitarian needs much better than the passive ingurgitation of “recipes” as only sketched in the second half of my proposal.

As it happens, I have just begun reading Tao’s “Why are solitons stable” in the current Bulletin and I was struck by the way Tao was dealing with the subject and the way he was talking about it: Neither the usual, “Let f …” nor the degrading style found in today’s textbooks which take their readers for utter idiots.

So, it occurred to me that I should have perhaps proposed a different light than I did and put the emphasis on the “third road”, that is one in which the mathematical exposition is neither “rigourous” in a pseudo Bourbaki manner—and, essentially, unreadable—nor down-graded into utilitarian cookbooks—with built-in obsolescence—and is accessible to the “great unwashed masses”. So, I thought that I should argue that a logical mind is an asset, not only in many practical situations (I still do general construction work in the summer) but also in societal matters as noted above in the quote from Colin McGinn and what is of paramount importance is that mathematics can be the simplest environment in which to develop one’s mind. It is not “problem solving” that is important, it is that a logical mind is what allows us to deal with many a priori different problems, recognize that, after all, they have similarities and learn from the one to deal with the other. Of course, after he got the Field medal, that this is the way that Tao works has been criticized, for not being specialized.

So, rather than to present an actual “solution” (mine), I should perhaps discuss the necessary parameters of any solution (using my own, easily accessible, solution for examples), in particular the time necessary for the learning curve to become exponential … but also for the conditions necessary for other, not a priori convinced people to give it a try:

“Early in my career, I naively thought that if you give a good idea to competent mathematicians or physicists, they will work out its implications for themselves. I have learned since that most of them need the implications spelled out in utter detail.”

–Hestenes, Oersted Lecture, page 38

Now, while I have very little respect for, and patience with, referees and all because they always want you to have written the piece that they would have wanted you to write, and because I think that editors, by contrast, at least know who the readership will be, I would like to ask you which proposal would be more in line with the goals of the volume.

In any case, I hope you will forgive me for this overlong afterthought.

Thereupon, on the response that “[the thoughts] are definitely worth expanding”, I sent in the following:

Mathematics education has been confronting the problem of how to bring mathematics to the “great unwashed masses” for at least thirty years but with no discernible success or even progress. In fact, the only conspicuous thing is that mathematics textbooks during that time have devolved to exposition by way of “template examples” and that the subject matter has been atomized into “topics” presented independently to facilitate memorization while, typically, instructors deplore that their students cannot remember the simplest things past the test.

Of course, it is not difficult to show how the stress generated by memorization on the scale required by, say, a year of mathematics must necessarily have that result. However, the operating, if tacit, assumption is that “developmental” students are incapable of learning on the basis of logic, the only alternative to memorization. And by an unfortunate, even if possibly unavoidable, coincidence, not only has research in mathematical learning also largely dealt with isolated topics but, even more unfortunately, it too has essentially equated learning with memorizing.

I would argue that the alternative in the call for proposals, i.e. “[q]uality mathematics […] seen as a reflection of its rigor, formality and generalisability” versus “utilitarian importance” would seem to be rather beside the point given that there is a third approach in which, i. learning is done on the basis of an appeal to “reason” as opposed to a reliance on memory and, ii. utilitarian needs are much better addressed than with the passive ingurgitation of “recipes” whose obsolescence is built-in in that a logical mind is very much an asset in practical situations not to mention that “One of the central aims of education, as a preparation for political democracy, should be to enable people to get on better terms with reason—to learn to live with the truth.” (Colin McGinn).

The first thing that this third approach requires is of course what Liping Ma called a “coherent view of mathematics” which, in turn requires a carefully designed contents architecture in which students cope with progressively more complicated situations and, even more importantly, in which one thing leads naturally to another and which gives students time to reflect.

While the recourse to “concrete applications” has finally been shown to be rather counter-productive, this is not to say that mathematics should not derive from the real world. In fact, what is quite natural is an expositional approach based on what is known in mathematics as Model Theory which carefully distinguishes “real-world” situations from their “paper-world” representations.

The third thing that has to be carefully dealt with is the meta-language, that is the language used to present and discuss the object language used by mathematics to represent the real-world. Systematic attention has to be given to linguistic issues that often prevent students from being able to focus on the mathematical concepts themselves.

Last, and most importantly, the students must become convinced that the reason the things they are dealing with are the way they are is not because “experts say so” but because common sense says they cannot be otherwise. But, while the standard way of establishing truth in mathematics is by way of proof, Edward Thorndike showed a century ago that proofs do not transfer into convincing arguments. So, a mode of arguing more like that used by lawyers in front of a court is necessary. See Toulmin’s The Uses of Argument.

Quite obviously, this cannot be done in the course of a couple of semesters but, perhaps surprisingly, there is strong evidence that it can be done in three four-hour semesters.

I would propose to discuss the above in some depth and to offer, as proof of concept, work done under the terms of a 1988 NSF calculus grant as an alternative to the conventional sequence, Precalculus I, Precalculus II and Calculus I (Differential). The 1992 report of my school’s Office of Institutional Research read in part:

   “Of those attempting the first course in each sequence, 12.5% finished the [conventional three semester 10 hour] sequence while 48.3% finished the [integrated two semester 8-hour] sequence, revealing a definite association between the [integrated two semester 8 hour] sequence and completion (?2 (1) = 82.14, p < .001).”

The report also said that the passing rates in Calculus II (Integral) for the students coming from the above two sequences were almost identical but that this was not significant because, in both sequences, most students did not continue into Calculus II (Integral).

What is directly relevant to “developmental” mathematics is that what made the above sequence work is the systematic use of (Laurent) polynomial approximations (Lagrange’s approach) and that these are of course nothing but an extension of decimal approximations so that a “profound understanding of fundamental mathematics”, in this case functions, decimal approximations, equations and inequations, and (Laurent) polynomials, is all that is necessary and is likely achievable in one four-hour semester. Given the pass rate mentioned above, an overall success rate from Arithmetic to Differential Calculus ought, no matter what, to be considerably higher that the current one, usually no more than one percent. See freemathtexts.org

But then common sense, you might say, prevailed and I forgot all about it.

[To be continued, though.]

Once upon a time, at the start of my career, the Monthly (of the MAA) turned down a paper of mine on the basis of two referees: one had said that it was highly controversial and the other that it was completely trivial. Obviously, I was very sore but I had been vaccinated: After that, I never “submitted” anything and just about everything else I published was “by invitation”.

And then, forty years later, I came across

A Call for Chapter Proposals for a Forthcoming Book on Quality and Equity Agendas in Mathematics Education. Editors: Bill Atweh, Mellony Graven, Walter Secada and Paola Valero.

Specifically,

Proposals for chapter authorship [are] requested from mathematics teachers, educators, researchers and policy makers for an edited collection dealing with two important agendas for mathematics education in societies around the world, namely quality and equity.

I will now quote in full:

Concerns about quality mathematics education are often posed in terms of the types of mathematics that are worthwhile and valuable for both the student and society in general, and about how to best support students so that they can develop this mathematics. Quality mathematics is sometimes measured from within the discipline of mathematics itself and is seen as a reflection of its rigor, formality and generalisability. Alternatively, the value of mathematics is often argued based on perceptions of its utilitarian importance such as individual mathematical literacy, the economic and technological well being of a society, the participation of an informed citizenry in the challenges of a democratic society, and/or for opening up future opportunities for students in terms of their career goals and access to higher education. Trends gleaned from international comparisons have ignited debates within many countries about the low level of achievement of their students internationally regardless whether mathematics is valued for its academic rigor or utilitarian literacy. Less often do international comparisons result in local media and public discourse vis à vis equity issues that they might raise.

Concerns about equity are about who is excluded from the opportunity to develop quality mathematics within our current practices and systems, and about how to remove social barriers that systematically disadvantage those students. Equity concerns in mathematics education are no longer seen at the margins of mathematics education policy, research and practice. Issues relating to ability, gender, language, multiculturalism, ethnomathematics, the effects of ethnicity, indigeneity, and the significance of socio-economic and cultural backgrounds of students on their participation and performance in mathematics are regularly discussed in the literature. This is not to say,however, that the problem of equity is exclusive of students who are positioned as disadvantaged due to their association to any of the categories above; nor that the growing focus on the issue has resolved it between countries and, indeed, within any society. Rather, insofar as access to quality mathematics is thought to confer benefits on individuals and the larger society, concerns for equity and access revolve around the impacts on an individual’s life and social participation and on the larger society’s continued well being when that access and its benefits are systematically restricted from and/or systematically provided to people on the basis of their or their parents’ social placements.

This collection of chapters attempts to summarise our learning about the achievement of both equity and quality agendas in mathematics education and to move forward the debate on their importance for the field. Some educators may take the stance that a focus on one may necessarily lead to a sacrifice in the other. Others may see the two agendas as necessary for each other and that a focus on one without the other is problematic. Finally, some educators may position their work within one or another agenda but in opposition to how that agenda has been historically construed In this collection we are interested in a variety of conceptualizations and mappings of the terrain on how quality and equity agendas relate. Following are examples of issues that authors might want to address:

Theoretical Issues

1. What discourses are useful for understanding equity and quality?

2. How are concerns for equity and quality contradictory and/or synergistic?

3. How do the themes of equity and quality in mathematics education play out within an international context and/or in different local contexts?

Research Findings

1. What does research in general say about the achievement of quality and equity agendas?

2. What can we learn from specific research studies or programs relevant to either/both agendas?

3. What kind of research is needed to deal with the achievement of both quality and equity?

Reform in Mathematics Education

1. What can we learn from teachers’ experiences in classrooms or schools about balancing, increasing and/or tradeoffs between quality and equity in our programs?

2. What lessons can we learn about collaboration between different stakeholders about achieving the equity/quality agendas?

3. What policies exist/are needed in different international contexts for the achievement of equity/quality?

Authors are invited to submit papers in one or at the intersection of some of the topics above. They are also encouraged to produce different kinds of papers such as:

1. Theoretical papers

2. Papers reporting empirical research

3. Research-based essays and debate papers

Theoretical Issues

1. What discourses are useful for understanding equity and quality

2. How are concerns for equity and quality contradictory and/or synergistic?

3. How do the themes of equity and quality in mathematics education play out within an international context and/or in different local contexts?

Research Findings

1. What does research in general say about the achievement of quality and equity agendas?

2. What can we learn from specific research studies or programs relevant toeither/both agendas?

3. What kind of research is needed to deal with the achievement of both quality and equity?

Reform in Mathematics Education

1. What can we learn from teachers’ experiences in classrooms or schools about balancing, increasing and/or tradeoffs between quality and equity in our programs?

2. What lessons can we learn about collaboration between different stakeholders about achieving the equity/quality agendas?

3. What policies exist/are needed in different international contexts for the achievement of equity/quality?

Authors are invited to submit papers in one or at the intersection of some of the topics above. They are also encouraged to produce different kinds of papers such as:

1. Theoretical papers

2. Papers reporting empirical research

3. Research-based essays and debate papers

Process of contribution

We seek expressions of interest from authors who wish to contribute a chapter on any issues relevant to the focus of the collection. In particular we are interested in teams consisting of more experienced and novice writers, and in teams that involve authorship from and across) different societies and across different types of participants (teachers, researchers, students, policy makers, parents; etc.)

This was truly amazing: people actually interested in mathematics for the great unwashed masses? There are of course a lot of people teaching the great unwashed masses but there had always seemed to me to be some sort of quiet resignation about not being able to do much in that regards. And this seemed to be something a bit different.

So …

[To be continued]

At the end of 2008, the self-imposed but already postponed target date for uploading Reasonable Algebraic Functions, came and went for all the usual reasons and then some. One reason was that I kept being interrupted in my mighty effort to meet the deadline by all sorts of considerations about “Develomental Mathematics” in particular and about teaching/learning in general. While this didn’t help with meeting the deadline, I didn’t feel I was wasting any time. All things pertaining to teaching are, indeed, rather enthralling if perplexing.

1. Consider a child, say, 5 or 6 years old. Say he expresses interest in the violin; you will likely get in search of a violin teacher to give him lessons, say an hour once or even twice a week. The child, though, will be expected to do a lot of practicing at home and the lesson will probably consist of the teacher listening to the child’s playing his exercises and “critiquing”. Then, the teacher will give the next assignment, annotating the partition with fingering, giving such and such indications on how to play it, maybe demonstrating how to hold one’s hand. And that will be it. Back home, the child will keep on practicing, perhaps with some prodding on your part. And, things will take their course. Maybe, the child will give up, maybe he will decide that he has other, more pressing interests, maybe the child will continue to become a “gifted amateur”, or maybe even a professional musician or perhaps only … a violin teacher.
   Now consider the child’s twin sister. Say she is “good at figuring numbers and puzzles”; you will likely congratulate her on her good grades at school and that will likely be the end of that and not only because you are a male chauvinist: even if you thought of getting in search of a mathematics teacher, it is rather unlikely that you would find one. But suppose, somehow, that you were able to find someone—say your sister is a research mathematician and she is willing to help out with her niece. The question now is: what is your sister going to do? Help her niece with her math homework? Her niece is not likely to need her. Start her niece on, say, a precalculus textbook. The goddesses help her: See SimWobpa. So, then, what?
2. The teaching-learning couple is really mostly a societal matter. First, there is the influence of the parents. That is not to say that I concur with, say:
   

   I see the problem as a parental and student one and NOT a school problem. K-8 math generally is very successful. The students learn all the basic math they need to continue into the algebra sequence and some even start algebra in 7th or 8th grade. National testing and local testing here in Massachusetts verify this. Then something happens when kids hit high school. Students start to become disinterested in school in general and math in particular and they are not encouraged at home by their parents to continue in math. I think we all realize that if one does not keep up with math we tend to forget it. This is true of anything we learn but more pertinent to the study of math. I have student who continue in my elementary to intermediate algebra sequence over a summer and show signs of forgetting what they just learned. When parents cannot get their kids to do well in school they criticize the school system rather than their own children and themselves. If they can’t get their kids to pay attention in school how are the schools going to get them to?”
   Ted Panitz, Cape Cod Community College, [mathspin] Teaching arithmetic to college students, August 8, 2008 10:46:06 AM EDT

   because a lot could be said about the causes of the students’ loss of interest. For one, it may well be that it is only when they come of age that they see for what it is what their teachers are trying to feed them.
   But what you might call the atmosphere at home does matter.
   But what the teachers are is what society wants them to be, as implemented by the Schools of Education. So, if anyone is to blame, it should be the distinguished “educologists” who train the teachers. But then how did said educologists reach the position they are in. Why were they permitted to get there and stay there?

3. And then there is the interference of, to put it as politely as possible, people for whom “making” money is a glorious end in itself: Given the shallowness and disconnectedness of today’s (high-priced) mathematics textbooks, students in general, and developmental students in particular, have no idea that one can learn from reading.
   But the fact that today’s textbooks are atomized has two very specific reasons: a) short easy pieces can be memorized and retained until the exam—which proves the efficiency of the textbook … and of the instructor who is thereby complicit and, b) once any cross reference, explicit or implicit, has been expurgated, the textbook can be “cafeteria-ed” to “suit the particular needs of your institution”: You want this topic before that topic? No problem! It is not for us to judge if it makes sense from any point of view. We are here only to satisfy the customer and you are the customer since you are the one to choose the textbooks. As for those who actually pay for the books …
   One may still wonder, though, why teachers, at least at the college level, are getting along with it.  
4. And what of “machine learning”?
   

   In 1922 Thomas Edison predicted that “the motion picture is destined to revolutionize our educational system and … in a few years it will supplant largely, if not entirely, the use of textbooks. ” Twenty-three years later, in 1945, William Levenson, the director of the Cleveland public school’s radio station, claimed that “the time may come when a portable radio receiver will be as common in the classroom as is the blackboard.” Forty years after that, the noted psychologist B. F. Skinner, referring to the first days of his “teaching machines,” in the late 1950s and early 1960s, wrote, “I was soon saying that, with the help of teaching machines and programmed instruction, students could learn twice as much in the same time and with the same effort as in a standard classroom.

   Oppenheimer, T. (1997, July). The Computer Delusion. The Atlantic Monthly. 45-62.
   

   Now it is web homework, etc When will we the teachers learn?

5. The latter question is actually rather an interesting one, especially in mathematics. The fact is that there has been a dumbing down of America. The question is by whom and how it was done. The instigators are not necessarily those pointed at by Charlotte Thomson Iserbyt’s the deliberate dumbing down of america or by Allan Bloom’s The Closing of the American Mind but dumbing down has occurred. And those who went about implementing it or, at least, were on the front lines, i.e. the teachers at all levels, may not have realized what the consequences would be. And, even if the dumbing down may not have been deliberately engineered, it certainly was facilitated by us, whichever way it happened and for whatever reasons.
   So the question is how we the mathematics teachers were not able to see where all that was happening with, say “Why Johnny can’t add”, was going to get us and, a lot more importantly, the students as a whole and now the republic almost as a whole: As the Daily Mirror of Thursday, November 4, 2004 headline screamed:

   How can 59,017,382 people be so DUMB?

   Along these lines, by the way, one may note that the election of Obama seemed to be the result of some deep down reaction of the people. At the very least, it seemed to be the result of a popular desire for a more reasonable organization of society. One way or the other, consciously or not, many people didn’t seem to believe that

6. (~Z) [The market] is not a zero sum game.

   It even seems that there was some questioning along the lines of “Where did the money go”. Just a little bit and just for a short while, though, as it was quickly made to seem … unseemly. But the future has yet to speak.

7. Mathematics education has been confronting the problem of how to bring mathematics to the “great unwashed masses” for at least thirty years but with no discernible success or even progress. In fact, the only conspicuous thing is that mathematics textbooks during that time have devolved to exposition by way of “template examples” and that the subject matter has been atomized into “topics” presented independently to facilitate memorization while, typically, instructors deplore that their students cannot remember the simplest things past the test.
   Of course, it is not difficult to show how the stress generated by memorization on the scale required by, say, a year of mathematics must necessarily have that result. However, the operating, if tacit, assumption is that “developmental” students are incapable of learning on the basis of logic, the only alternative to memorization. And by an unfortunate, even if possibly unavoidable, coincidence, not only has research in mathematical learning also largely dealt with isolated topics but, even more unfortunately, it too has essentially equated learning with memorizing.
8. And how much “research” has been done, if any? The short of it is that while there is of course a lot of research on the educational front, there isn’t much that is really operational. For instance, do you know of any study that compares various “teaching techniques”? Do you know of any study that tries to estimate the “exponential effect” in learning. For that matter, most research, even if perforce, deals with the learning of some, at most few, items. So, essentially, learning is identified with learning by heart. At best, learning is identified with familiarizing oneself with the stuff.
   I know of no research that, given a body of knowledge, tries to identify which of the various Hamiltonian paths, or approximations thereof, is optimal. Or even to compare them. Or even to specify the Hamiltonian paths in the form, say, of tables of contents. To give just an example: Why do we have to take for granted—and, please don’t argue—that fractions must be dealt with before integers? Isn’t that dumb?
   Of course, I would appreciate being proven wrong by being sent any reference to even only a discussion of the pros and cons of the matter. Or by being responded to here.

And now, back to Reasonable Algebraic Functions:

First I have to get enough stuff ready for the first third of the semester in the class in which I am using the”bundle” (In order not to confuse LaTeX people, I gave up on the term “package” which is already used there.) But I am almost done.

Second, the site which was built around Reasonable Basis Algebra has to be rebuilt. Unfortunately, this is not something I can do alone. But help is on the way.

Once the site has been redone, though, I will upload right away the first third of the RAF bundle.

By the way, one of the things that threw me off schedule was designing a system that will handle both the Homeworks and the Review/Exams from a single QuestionBase so as to ensure consistency and to permit a much wider and easier choice among the “checkable items” that can be used in either a Homework or an Exam. As a matter of fact, writing this is a break from the crashing bore that transferring the stuff from the two old questions bases to the new single QuestionBase is.

Having ended the previous entry in these NOTES with an announcement that I would “discuss developmental mathematics as embodied in the Arithmetic-Basic Algebra-Differential Calculus sequence”, I found myself somewhat at a loss as to exactly how to resume matters.

Fortunately, being known at my school as a militant, radical fault-finder of Developmental Mathematics courses as currently practiced, and therefore a royal pain, I was recently asked by a presumably sick and tired administrator how I would specify a program intended for the kind of students currently enrolling into Developmental Algebra. Obliging as ever in such cases, I wrote a few pages to sketch a program that would bring these students, in three four-credit semesters and with an acceptable success rate, to the level achieved in Calculus I (Differential) and to explain what it is that would make such a program work.

What will ensue at my school is anyone’s guess but it seemed to me that I might as well publish here (a slightly edited version of) that sketch as it might be of some small use to someone out there.

NOTE: So far, I have not been able to convert the formulas written below in LaTeX into HTML but I am still trying. In the meantime, I apologize.

Description Of The Sequence

The most important part of such a program is of course the contents of the three courses in the sequence and, most especially, their architecture. One possibility would be for the sequence to consist of:

1. An Arithmetic-Algebra course, to be discussed at some length below, specially and specifically designed to serve as an introduction to:
2. An Algebraic Functions course dealing with their introduction, algebraic discussion and differential calculus,
3. A Transcendental Functions course dealing with their introduction, algebraic discussion and differential calculus.

Of course, one reason for this architecture is that it is that of the two courses, Differential Calculus I and Differential Calculus II, that were originally specified, designed and experimented with under the terms of a 1988 NSF grant as an alternative to the conventional sequence, Precalculus I, Precalculus II and Calculus I (Differential). The 1992 report of the school’s Office of Institutional Research said in part:

“Of those attempting the first course in each sequence, 12.5% finished the [conventional three semester 10 hour] sequence while 48.3% finished the [integrated two semester 8-hour] sequence, revealing a definite association between the [integrated two semester 8 hour ] sequence and completion (χ2 (1) = 82.14, p < .001).”

The report also said that the passing rates in Calculus II (Integral) for the students coming from the above two sequences were almost identical but that this was not significant because most students did not continue into Calculus II (Integral).

However, for reasons that may or may not be obscure, the integrated sequence all but died out. (This is of course what happened to most, if not all, the courses and textbooks developed under the NSF grant program. In particular, this was the eventual fate of the book developed at Harvard under, if memory serves, a one and a quarter million dollars NSF grant.)

The approach used in Differential Calculus I and Differential Calculus II to bypass a lot of the usual stumbling blocks on the and to provide the students with a continuous and sustained conceptual development is due to Lagrange, one of the preeminent nineteenth century mathematicians, who wanted to avoid the use of limits in the development of calculus.
While the standard objection to Lagrange’s treatment is that it cannot deal with all functions, it certainly does handle all functions likely to be encountered by students in Calculus I (Differential) and can serve as a basis from which to develop the Bolzano-Cauchy-Weirstrass treatment of limits that is nowadays de rigueur in, or at least underlies, “elementary” calculus texts. What it cannot deal with is only the kind of esoteric functions encountered by research mathematicians.
The details of how Lagrange’s approach achieves this is beyond the scope of this paper but an important fact to keep in mind in terms of the sequence sketched here is that it is based solely on the use of something called polynomial approximations. (More about this below.)

Description Of The Arithmetic-Algebra Course

Because of the work done for Differential Calculus I and Differential Calculus II, the contents of the Algebraic Functions course and of the Transcendental Functions course are already well specified and therefore the contents of the Arithmetic-Algebra course are fairly well specified as being exactly, no more, no less, what is needed for the Algebraic Functions and Transcendental Functions courses.
In fact, the package Reasonable Basic Algebra already contains a large part of what is needed, namely

• Elementary equations and “inequations”,
• Laurent polynomials i.e. polynomials that can include negative powers.

What is not clear is exactly how much to do with these contents to ensure a “profound understanding”—in the sense of Liping Ma’s “profound understanding of fundamental mathematics“. So, here, some experimentation would certainly be necessary.

In any case, what is missing from the Reasonable Basic Algebra package, yet is absolutely necessary for a profound understanding of Algebraic Functionsand Transcendental Functions, is a profound understanding of three more concepts:

• Decimal Numbers
• Functions
• Approximations

While the first part of the Reasonable Basic Algebra package does deal with numbers, it does so mostly with counting numbers, plain and signed. But, even though decimal numbers are constantly used in Part 2 and 3, for lack of time, the Reasonable Basic Algebra package could not discuss decimal numbers per se and had to take for granted their profound understanding. This of course precluded any discussion of Approximations.

Similarly, for lack of time, functions are not really introduced in the Reasonable Basic Algebra package even though, without saying it, operations are introduced as functions, that is, for instance, as adding 2 to 3 which can be visualized as

$latex 3 \xrightarrow{\hspace{2mm} +2\hspace{2mm}}\;5$

as opposed to adding 3 and 2 which can be visualized as

$latex 3, 2 \xrightarrow{\hspace{2mm} +\hspace{2mm}}\;5$

The distinction may appear overly subtle but, in fact, it facilitates considerably the understanding of how we work with numbers. For instance,

• We can then say that subtracting 2 “undoes” adding 2; visually:
  

  $LATEX 3 \xrightarrow{\hspace{2mm} +2\hspace{2mm}}5 \xrightarrow{\hspace{2mm} -2\hspace{2mm}}3$

• Similarly, the equation x + 2 = 5 can be understood as the question “to what number should we add 2 to get 5?”. Visually:
  

  $latex x \xrightarrow{\hspace{2mm} +2\hspace{2mm}}5$

Had functions been actually introduced, the above question would then have been recast as:

For which input(s), if any, does the function whose input-output rule is

$latex x \xrightarrow{\hspace{2mm} JILL\hspace{2mm}}JILL(x) = x+2$

return the output 5?

The latter is of course cumbersome but it does familiarize the students with a way to look at things that is crucial in Algebraic Functions where we might ask, for instance,

For which input(s), if any, does the function whose input-output rule is

$latex x\xrightarrow{\hspace{2mm}QUAD\hspace{2mm}}QUAD(x) = -3x^{2}+5x-7$

return the output −23?

which leads naturally to the quadratic equation

which can then be solved simply by counting inputs from the vertex.

Thus, the Arithmetic-Algebra course should be designed something as follows:

1. Introduce functions as the mathematical version of real-world input-output devices.
2. Introduce counting numbers, plain and signed, essentially along the lines of the Reasonable Basic Algebra package but now in the context, and with the full aid, of functions. In particular, plotting functions when the data set consists of signed counting numbers would go a long way towards the students’ familiarization with these.
3. Introduce decimal numbers with, immediately, the then necessary concept of approximation.

   For instance, while the question

   For which input(s), if any, does the function whose input-output rule is

   $latex x \xrightarrow{\hspace{2mm} DIL\hspace{2mm}}DIL(x) = 3x$

   return the output 12?

   has the solution 4, the question

   For which input(s), if any, does the function whose input-output rule is

   $latex x \xrightarrow{\hspace{2mm} DIL\hspace{2mm}}DIL(x) = 3x$

   return the output 12?

   does not admit of any exact decimal solution. However, we may say that any of the following

   4 + […]

   4.3 + […]

   4.33 + […]

   4.333 + […]

   where […] is to be read as “something too small to matter in the current situation” is a solution.

   Which approximation we will choose will depend of course on the actual situation and so, in the meantime, we will say that the solution is

   $latex \dfrac{13}{4}$

   which is then read as “code” for setting up the division of 4 into 13 with the decision as to where to stop the division left to when we know what the actual situation requires.

4. Very little in the investigation of equations and inequations in Part 2 of the Reasonable Basic Algebra package would have to be modified to take full advantage of the above.
5. In fact, and more generally, once the sequence is a given, there would be no need to keep the contents in the particular courses described above and, presumably, one could vastly improve the learning curve by bringing in the contents on an “as needed” basis.It is for instance quite probable that the study of affine functions currently in the Algebraic Functions course could occur in the Arithmetic-Algebra course where affine equations are already dealt while leaving most of the study of approximations for the investigation of quadratic functions and cubic functions in the Algebraic Functions course where it is really needed. (Of course, the courses would then have to be renamed accordingly.)

Rationale

Some explanation as to why the contents have to be something like what was described above is probably in order.

1. The students entering Developmental Mathematics courses are “damaged” in the sense that they want only to be shown “how to do” the problems understood to appear on the exam. When reminded that “show and tell, drill and test” is precisely what got them into Developmental Mathematics courses, many even rise to the defense of their school/teachers and say that this is only because “they were never good in math”.
2. Compounding the problem is the fact that all commercially available textbooks are memory based and that, to facilitate memorization, the subject matter is atomized into “topics” that are then presented independently of each other. All connective tissues have been removed and, as a result, nothing can make sense anymore and no build-up can take place. Indeed, typically, instructors deplore that the students cannot remember the simplest things past the test.
3. Given these circumstances, such a “fast track” as described above would seem utterly improbable. The point, though, is that the content architecture sketched above is extremely efficient in that everything serves to support all that follows and thus fosters an environment in which the students are able and begin to see that things mathematical are the way they are, not because someone or some book says so, but because of the way they connect to each other so that it makes sense that they should be the way they are.
   For example, consider the number 2345.67. In the Arithmetic-Algebra course, it initially stands as a shorthand for

   $latex 2\cdot10^{+3} + 3\cdot10^{+2} + 4\cdot10^{+1} + 5\cdot10^{0} + 6\cdot10^{-1} + 7\cdot10^{-2}$

   to represent

   $latex 2 Clevelands \& 3 Franklins \& 4 Hamiltons \& 5 Wahingtons \& 6 Dimes \& 7 Cents$

   The exponents are “code” for the number of 0s to be placed after the 1 when the exponent is + or to be placed before the 1 when the exponent is -:

   $latex \begin{align*}
   2345.67
   &=2\cdot1000. + 3\cdot100. + 4\cdot10. + 5\cdot1. + 6\cdot0.1 + 7\cdot0.01
   \\
   &= 2000. + 300. + 40. + 5. + 0.6 + 0.07
   \end{align*} $

   With the details that are quite necessary but omitted here, the above makes complete sense to the students. Most important is that the concept of multiplication is not required at that point and that this is all that is needed to start the development of arithmetic (In the above, the symbol + should be read as “and”).
   For instance, the above is enough to discuss why, depending on the situation, we can write

   $latex \begin{align*}
   2345.67
   &= 2000 + […]
   \\
   &= 2300 + […]
   \\
   &= 2340 + […]
   \\
   &= 2345 + […]
   \\
   &= 2345.6 + […]
   \end{align*}
   $

   But then, once we have introduced multiplication, we realize that, say,

   
   $latex 2 · 10^{+3}$

   can be read as

   
   2 multiplied by 3 copies of 10

   and

   
   $latex 2 · 10^{-3}$

   can be read as

   
   2 divided by 3 copies of 10

   From there, it is an easy transition to reading

   $latex 2 · x^{+3}$

   as

   2 multiplied by 3 copies of x

   and reading

   $latex 2 · x^{-3}$

   as

   2 divided by 3 copies of x

   And then, we can see why, when x stands for a large number, we can write, depending on the situation,

   
   \begin{align*}
   \hspace{-7mm}2x^{+3}+3x^{+2}+4x^{+1}+5x^{0}+6x^{-1}+7x^{-2}
   &=2x^{3} + […]
   \\
   &=2x^{+3}+3x^{+2} + […]
   \\
   &=2x^{+3}+3x^{+2} +4x^{+1} +[…]
   \\
   &=2x^{+3}+3x^{+2} +4x^{+1}+5x^{0} +[…]
   \\
   &=2x^{+3}+3x^{+2} +4x^{+1}+5x^{0}+6x^{-1} +[…]
   \end{align*}

   A profound understanding of decimal numbers and of how they are approximated is absolutely crucial to the profound understanding of functions afforded by Lagrange’s viewpoint because not only are polynomial functions approximated essentially in the same manner as decimal numbers but polynomial functions in fact serve to approximate all the functions normally encountered in calculus in exactly the same manner that decimal numbers serve to approximate all numbers, for instance √2, π, e, the golden ratio, etc.

4. Eventually, as the students’s understanding deepens, the attitude mentioned above begins to change and many students begin to be more willing to take the time to consider a question, what it means and how to cope with it, one way or the other, and then to take whatever more time it takes to get a result and/or to make a case for whatever result they have come to.
   Incidentally, the amount of time students are willing to stay on a given question is an extremely good indicator of their progress in the direction of thinking for themselves.
   Significantly, the questions to the instructor get to be less and less about whether they “got it wrong” and more and more about where they made the wrong turn.
5. However, the problem is that this change in attitude starts being noticeable only about two thirds of the way down the first course. And of course, if the second course is “standard”, everything goes back to square one: “teach me, show me”. In other words, the convalescence cannot be expected to take place within a single semester.
6. Some students, though, have already reached a “denial” stage where, against all evidence, continue to say that they will just memorize and pass the course.

Practical Considerations

While a content architecture such as the one sketched above is necessary, it is unfortunately not sufficient to ensure acceptable retention and success rates.

1. Students need to be able to read (and write) mathematical explanations and this is far from being initially the case. It is thus necessary to link the sections of the Arithmetic-Algebra course with sections of a Developmental English Reading course in which the texts assigned for reading are those used in the Arithmetic- Algebra course: Textbook and Review DISCUSSIONS.
   The link was tried once in my school and the instructor who had taught the Remedial English Reading course later wrote that “The students that stayed to the end also appreciated [the approach] whether they passed or not. If we pursue another link, the English teacher should definitely read the math text with the students. Unfortunately, because I had my own reading to do, we did not read the math in English class as we should have done.“
2. Students need to have the time necessary to discover and experience what is for them a completely new modus operandi, namely “thinking” as opposed to memorizing.
   There should therefore be a “study period”, after each class, in a manner of a lab, to ensure that the students will have the necessary time to do the homeworks provided in the package while reading the Textbook and the Review DISCUSSION. (See for example the Reasonable Basic Algebra package). Just as with physics or chemistry labs, an instructor should be present as a “resource”.
3. There should be as much continuity as possible:
   1. The Arithmetic-Algebra course should be offered only in the Fall with the Algebraic Functions course offered in the Spring and the Transcendental Functions course offered in the following Fall (or in the Summer but only in a fourteen week course.)
   2. The instructor ought to be able to follow her/his students from one course to the next throughout the entire sequence—except of course when students need to repeat a class.
   3. Some “contract” should be passed between entering students and the school to ensure:
      • That the students will attend classes and study periods,
      • That the school will offer the subsequent courses on schedule,
      • That the school will give the students a specific number of credits upon completion of the sequence,
      • That the school will have seen to it that the sequence transfers appropriately in four-year schools.
   4. Some screening would seem to be necessary but should absolutely not be done on the basis of “knowledge” and only to determine the likelihood that the student’s commitment is realistic and seriously understood.
      There is a subtle difference between students testing into Developmental Arithmetic and students testing into Developmental Algebra, although not entirely in favor of the Developmental Algebra students. So, although it seems a priori safer to start such a program with students testing into Developmental Algebra, once a screening has been found to successfully predict the students’ level of commitment, there does not remain much reason for not accepting in the program students testing into Developmental Arithmetic so that, eventually, the distinction could be dispensed with entirely.

All I have ever written has been, one way or another, concerned with “developmental” mathematics. I have written and still write mathematics from a point of view that I like to think is the real developmental one. I have presented and discussed developmental issues with colleagues on various venues, in my own department, at joint AMS-MAA meetings, at meetings of the American Mathematical Association of Two Year Colleges (AMATYC), in the AMATYC Review, on Mathedcc and Mathspin, etc. But, for all that and for just as long, even though I have kept musing about my own position in relation to developmental mathematics and while I have never had any doubt about said position, I have always been uneasy about formulating and stating it.

I. On this site, the closest I have come to mentioning my motivation was in the preface to Reasonable Basic Algebra in which I quoted from one of my Notes From the Mathematical Underground in the AMATYC Review, Spring 1996 issue:

Also directly relevant to the issue is an article by Colin McGinn, Homage to Education, in the August 16, 1990 issue of the London Review of Books [. . . ]. The article is a review of a book of, and of a book about, R. G. Collingwood. The relevant part is where McGinn “spell[s] in [his] own way what [he] thinks Collingwood is getting at here.” “Democratic States are constitutively committed to ensuring and furthering the intellectual health of the citizens who compose them: indeed, they are only possible at all if people reach a certain cognitive level . . . . [. . . ]. Democracy and education (in the widest sense) are thus as conceptually inseparable as individual rational action and knowledge of the world.” [. . . ]. But what is education? “Plainly, it involves the transmission of knowledge from teacher to taught. But what exactly is knowledge? ” […]. [It] is true justified belief that has been arrived at by rational means.” […]. Thus the norms governing political action incorporate or embed norms appropriate to rational belief formation. […]. The educational system of schools and universities is one central element in this cognitive health service […].

[…]

The quasi-mathematical language in which this is stated should have a special resonance for mathematicians. “It would be a mistake to suppose that the educational duties of democratic state extended only to political education, leaving other kinds to their own devices. […]. How do we bring about the cognitive health required by democratic government? A basic requirement is to cultivate in the populace a respect for intellectual values, an intolerance of intellectual vices or shortcomings. […]. The forces of cretinisation are, and have always been, the biggest threat to the success of democracy as a way of allocating political power: this is the fundamental conceptual truth, as well as a lamentable fact of history.

[…]

However, “people do not really like the truth; they feel coerced by reason, bullied by fact. In a certain sense, this is not irrational, since a commitment to believe only what is true implies a willingness to detach your beliefs from your desires. […]. Truth limits your freedom, in a way, because it reduces your belief-options; it is quite capable of forcing your mind to go against its natural inclination. This, I suspect, is the root psychological cause of the relativistic view of truth, for that view gives me license to believe whatever it pleases me to believe. […]. One of the central aims of education, as a preparation for political democracy, should be to enable people to get on better terms with reason—to learn to live with the truth.”

Indeed,

A political act “to enable people to get on better terms with reason—to learn to live with the truth.

is the phrase I used when trying to explain what this site and its contents are all about. And, in a way, that does say it all. However it certainly doesn’t spell it out and I have long felt the need to discuss this a bit further but, somehow, have always had trouble with it.

II. It was a recent article, again in the London Review of Books, 6 March 2008, Is It Glamorous? by David Simpson, a review of Absent Minds: Intellectuals in Britain by Stefan Collini, Oxford 2007, that, although it had of course nothing to do with developmental mathematics, finally gave me, for whatever reason, what I needed to discuss how I see developmental mathematics and why.

The part of the article that resonated with my own feelings and motivations is where Simpson discusses Collini’s attitude regarding Edward Said in general and Said’s Representations of the Intellectual [The 1993 Reith Lectures] in particular. Following are a few excerpts of Simpson’s article directly relevant to what I will say below about developmental mathematics. However, I should say immediately that the whole article is well worth reading independently of the issues that concern me here.

[I]t seems symptomatic that the figure [Collini] finds most wanting is Edward Said.

Collini finds [Representations of the Intellectual] a ‘poor book’ marked by ‘simplistic binary alternatives’, a ‘kind of political free association’ […]. Above all it succumbs to an ‘insidious kind of glamour, that of being the champion of the wretched of the earth’.

[Said was] throughout his career […] a defender of those who couldn’t speak for themselves or get a fair hearing when they did.

[Intellectuals] should call our attention to ‘all those […] issues that are routinely forgotten or swept under the rug’ and should universalise every crisis so as to bring it into line with as many other crises as possible, to the point of being ‘embarrassing . . . even unpleasant’.

Collini does not like Said’s ‘existential drama’ and its ‘inescapable logic of choice’

and, last but not least,

If Collini is right that with a few variations and exceptions the view of intellectuals has been much the same across the West in the 20th century, that there is a ‘larger international pattern’ at work, what is the common influence or structure that would explain it? He says at the end of the book that there is such a structure, but somehow the matter of ‘structural rather than merely local explanations’ dwindles down to a matter of ‘alarmist cultural pessimism’ which he has ‘taken issue with on other occasions’. There might be interesting reasons why capitalist economies in tandem with representative democracies are felt to have the power to impose despair or desperation on their intellectuals. But do they do so on all of them? Are there not some who maintain an optimism of the will, and must it be ‘culpably romantic’ to do so?

Well, of course, I am not Said and since I have long admired Said I feel that directly invoking the above would be at least presumptuous, in fact impertinent, and certainly quite ridiculous. So, I will now proceed with my stance regarding developmental mathematics and leave any connection with the above entirely to the reader.

III. The first issue is why would anyone at all want to “learn” mathematics. I can see three possible answers which however involve three different meanings of the word “learn”.

1. One can see mathematics as a chore, as something necessary to be able to do other, specific things such as being able to register for another course or being able to cut rafters for a roof. But the apprentice carpenter neither needs nor wants to go through Euclid’s books in order to be able to cut rafters and the English major neither needs nor wants to factor quadratics in order to be able to discourse on Shakespeare. Developmental mathematics is therefore the prerequisite for Precalculus which, as the name indicates, is the prerequisite for Calculus which is the prerequisite for Physics and other “advanced” courses, etc.

   Of course, the dual of the question is why would most curriculums require mathematics in the first place. The answer is usually that curriculum designers include some mathematics so as to give weight to, or simply pad, their curriculum … and, by a fortunate coincidence, give jobs to mathematicians who are then expected to, and duly do, return the favor.

   The carpentry curriculum thus claims “geometry” to be a necessary background for learning how to use a carpenter’s square and the English departments thus claims that set theory and abstract algebra are essential to understand the linguistics of Harris or Chomsky, the structures of Lévi-Strauss, the Borromean knots of Lacan, as well as Catastrophe Theory and now Chaos Theory for whatever literary theory is currently fashionable. But of course, none of these claims ever held any water. Harris didn’t know any mathematics beyond the definitions of equivalence relation and semi-group and never did anything with either. Being honest, Chomsky never claimed to do anything mathematical in the first place, Lévi-Strauss didn’t know what a structure was—he just liked the word, Lacan never exhibited the least interest in rational discourse, etc.

   When all else fails, as it invariably does, recourse is then made to needs like having to be able to compute unit prices, discounts and markups at the store, etc. When you point out that very few people feel these supposed needs and, anyhow, that most everybody nowadays has a calculating cell phone, your opponent, very likely to be into teaching with calculators, gets vaguer and vaguer until s/he declares her/himself outraged by something or the other and walks away in a huff and with a snort of disgust.

   What I find amazing under these conditions is how we are able to convince most everybody that they need to know a certain amount of arithmetic and a certain amount of algebra and, when we catch them in time, how we succeed in corralling them into developmental mathematics programs. The bitter irony here is that success in these programs is largely determined by those running these very programs. There might be a good reason here as when, occasionally, the success of a developmental program is measured by the success of its graduates in ulterior courses, the results turn out to be horrendous. See, for instance, LongitudinalStudy

2. Another view is that mathematics, in some way, is somehow formative: being able to factor a few quadratics is good for you. Period. A variant is the belief that a—small—amount of mathematics is a necessary ingredient of “general knowledge”, the panoply of the well-rounded, cultivated gentleman: Gauss as well as Michelangelo, Shakespeare and Einstein. See whatever “liberal arts mathematics” book you happen to have at hand.

   Yet, there is something to that view but it is impossible to delineate and, in any case, most people don’t have the leisure or the financial means and that type of course is very much on the way out.

3. My own view is that mathematics is the simplest universe in which to learn how to make a case for one’s conjecture, in which to distinguish what we can show is true from what we can show is false and from what we don’t know to be true or false, etc. In short, mathematics is the simplest place in which to learn how to operate rationally. In that, mathematics is a lot closer to law than to what is currently peddled as mathematical proof in geometry textbooks. See The Uses of Argument by S. Toulmin.

IV. The second issue is what mathematics ought to be learned when. As pointed out above, most students are not really free to choose what mathematics they are to learn. And when it us who decide for them, as we usually do, we invariably choose developmental mathematics and/or precalculus mathematics and for the—very few—survivors, calculus. All of which according to the gospel of texts carefully packaged by an industry driven by greed bordering on the pathologically insane which, though, give us great opportunities to deploy our teaching skills, that is essentially our ability to sugarcoat the pill and grease the plank. The students be damned.

Many alternatives would of course be possible.

• One could be the arithmetic and algebra of collections-of-items and unit-prices with co-multiplication because this can quickly be generalized to “baskets” of collections-of-items and “lists” of unit-prices, that is, in other words, LinearAlgebra.
• Another could be Discrete Mathematics but it seems to lack any story line and I have nothing to suggest. All I can say is that the texts I have seen appeared to be collections of topics: a bit of sentential logic here, a bit of set theory there, some graph theory possibly somewhere in-between, etc.
• Another could be Geometry and/or Group Theory starting, say from the notion of tiling. But, while I have played a bit with the idea, I am not sure how to let it go beyond the obvious. It is a bit as with Incidence Geometry which is initially enticing since there are so few axioms but which quickly degenerates into counting arguments.
• The alternative which I have chosen to develop materials for is a strongly integrated Arithmetic-Basic Algebra-Differential Calculus three-semester sequence and this for a variety of reasons. The main one is that the mathematics of change is a well defined goal well within reach of a lot more students than learn it in the traditional sequence. Another one is that there is a simple, very strong story line which I will discuss later. Yet another is that this alternative can be fitted without too much upheaval in the current college framework. Last but not least is that I happen to like the subject of polynomial approximations and that I see this sequence as the ideal developmental mathematics. (Of course, how well it will work in practice will have to be ascertained by others than myself.)

In the next installment of this blog, I will thus discuss developmental mathematics as embodied in the Arithmetic-Basic Algebra-Differential Calculus sequence and from the point of view that “mathematics is the simplest universe in which to learn how to make a case for one’s conjecture, …”

As ever, any criticism, critique, feedback, etc is of course welcome, the more detailed, the more welcome.

A. Schremmer

This Note is more of a “progress report” than about issues concerning learning mathematics although, as it happens, the two are deeply linked.

The estimated time of arrival for Reasonable Algebraic Functions (RAF) is now sometimes in Fall 2008 and that for Reasonable Decimal Arithmetic (RDA) sometimes in Spring 2009.

The reason or, if you prefer, the excuse, for the delay. I spent an inordinate amount of time, even for me, trying again and again to redesign the first of the three parts that comprise RAF. The issue was that while Polynomial Functions (Part Two of RAF) and Rational Functions (Part Three of RAF) had long seemed stable enough in terms of their mathematical treatment, there had always been two problems: (1) How to present Power Functions and (2) several issues associated with the methodology used in the treatment of Polynomial Functions and Rational Functions.

1. Essentially, the mathematical approach, due to Lagrange, is to use local polynomial approximation and I discussed in some detail how I used it in a Precalculus-Calculus One sequence. By itself, its implementation does not raise any problem, at least certainly nothing like what is involved when using limits:
   • For Affine Functions, all that is needed is replacing x by x₀+h.
   • For Quadratic Functions, all that is needed is an addition formula for the expansion of (x₀+h)².
   • For Cubic Functions, all that is needed is an addition formula for the expansion of (x₀+h)³.
   • For Quartic Functions, etc …
   • For Polynomial Functions of degree n, we need the Binomial Theorem.
   • For Rational Functions, all that is needed is polynomial division albeit both in ascending powers to localize near bounded inputs including the poles, if any, and in descending powers to localize near infinity.
   • For Radical Functions, we need to solve a—fortunately sparse—system of equations to find the coefficients of the expansion.

   While it might not entirely satisfy mathematicians, this approach is, I think, very much in the spirit of mathematics in that cases can be made—and readily turned into proofs—for most of the theorems being used. At the very least and if nothing else, there is none of that “After plotting these points, you can see that they appear to lie on a line, as shown on Figure 1.1. The graph of the equation is the line that passes through the five plotted points.” (Larson-Hostetler, Precalculus 6th edition. For the effect this kind of “philosophy” is likely to have on serious students, see Sim Wobpa which was written à propos the 4th edition of the very same opus.) In the rare occasions when, for whatever reason, a case is not made, this is duly noted and explained.

2. There is, however, an important difference between RAF and Lagrange DC which is that: a) in order to be usable in Precalculus One, the concept of derivative had to go and, b) in order to still make sense as a “mathematical theory”, the treatment in RAF had to be, in some sense, “minimum and closed”. As it turned out, though, these two requirements were easy to satisfy. Concerning a), the analysis in RAF could be done without mentioning derivatives and by just using the coefficient of hⁿ. Concerning b), first RAF does not include Radical Functions on the grounds that the way the localization is obtained brings them closer to Transcendental Functions defined as solutions of differential equations and then, second, since Affine Functions have no turning point and Quadratic Functions have no inflection point with Cubic Functions being able to exhibit both, RAF did not have to deal with any other Polynomial Functions to illustrate these behaviors and the list of contents for RAF is:
   • Power Functions
   • Constant Functions
   • Affine Functions
   • Quadratic Functions
   • Cubic Functions
   • Rational Functions

3. A very important unifying fact is that, in some ways, both Polynomial Functions and Rational Functions are “essentially” little more than Power Functions with some number of fluctuations, that is of minimum-maximum pairs, thrown in. To be a bit more precise, though, a big difference is that the fluctuations are bounded in the case of Polynomial Functions whereas they can be infinite in the case of Rational Functions. Specifically, the essential bounded graph is defined as the simplest bounded graph compatible with the local graph near infinity and the local graph near the pole(s). Beyond that, of course, the essence of their nature is that Polynomial Functions are locally approximately polynomial while Rational Functions are locally approximately Laurent-polynomial.

   Still, an important issue is that of locating these fluctuations. Fortunately, in the above list of contents, things are naturally taken care of: after expanding near x₀, we locate the turning point by killing the coefficient of h in the localization which, in the case of a quadratic function, entails solving an affine equation and in the case of a cubic function entails solving a quadratic equation. Similarly, we locate the inflection point of a cubic function by killing the coefficient of h² which entails solving an affine equation. So, in a way, there is a “closure” of sorts.

4. However, in the case of students who need to learn that mathematics has a logical development, as opposed to “math” being a set of “skills”, an extremely important issue is that of the order in which to introduce the necessary language and the necessary concepts so as to create an impression of logical flow. The “progression” from Polynomial Functions to Rational Functions thus seemed quite “logical”, at least to me. But I realized eventually that it still resulted in a certain “ad hoc” feeling with the students. And another place where the flow was far from being “obvious” occurred when moving from Part Two - Polynomial Functions to Part Three - Rational Functions: there was this sudden, new concern as to whether there might be bounded inputs with infinite outputs. Again, at first, this didn’t seem to be a big deal as, after all that is why they were dealt with in two different Parts, but, still, it sure didn’t contribute to continuity in the story line. And, of course, there was the issue that the flow in the progression
   • Positive-power Functions
   • Negative-power Functions
   • Polynomial Functions (of degree ≤ 3)
   • Rational Functions

   which, if it certainly works, was somewhat less than felicitous. But then, the flow in the progression

   • Positive-power Functions (Needed to localize Polynomial
     Functions.)
   • Polynomial Functions (of degree ≤3)
   • Negative-power Functions (Needed to localize Rational
     Functions.)
   • Rational Functions

   while it was on an “as needed basis” and thus seemed a bit more logical, didn’t deal with the concern about the sudden concern about whether there might be bounded inputs with infinite output.

5. There was also another problem in that I was also introducing the terminology on an “as needed” basis. For instance, the term “turning point” was introduced in the context of Quadratic Functions and the term “inflection point” in the context of Cubic Functions. While it felt right not to start with a massive set of definitions at the beginning of the course, later on, it created problems of reference and ended up exacting a heavier price than I had thought.

   And then, given that RAF is a standalone, there was the issue of where to introduce the various necessary concepts, those usually thrown-in into a catch all “review chapter. What took me four months to arrive at was the idea of introducing all that was needed on functions defined by graphs. In particular, given the local graph near infinity of a function, a natural question is under what conditions does joining smoothly the local graph near –∞ to the local graph near +∞ result in the bounded graph. The condition of course was the answer to the Essential Question—do all bounded inputs have bounded outputs or is there a bounded input with an infinite output?—which until now had come up only with Rational Functions—and Chapter 3 was entirely devoted to the issue of smooth interpolation. But then, finally, I came to realize that it was the notion of outlying graph, that is of the local graph near infinity together with the local graph near the poles, if any, that provided the means to make the contents consistent in that for “all” functions it was the outlying graph which controlled the (essential) bounded graph.

All of this to say that, as of this Bastille Day, the list of contents for the first five chapters of RAF is:

1 - Introduction
1.1 Relations
1.2 Functions
1.3 Functions Specified by an Input-Outpur Rule
1.4 Signed-numbers Graphically
1.5 Signed-numbers Qualitatively
1.6 Large and Small Numbers
1.7 Qualitative Rulers
1.8 Screens
1.9 Functions Defined By A Graph
1.10 The Fundamental Problem*

2 - Graphic Local Analysis
2.1 Local Graphs
2.2 Local Language
2.3 Place of a Local Graph
2.4 ∞-Height Inputs and 0-Height Inputs
2.5 Shape of a Local Graph
2.6 Local Behavior
2.7 Feature-Sign Change Inputs
2.8 0-Slope and 0-Concavity Inputs
2.9 Extremum Inputs

3 - From I-O Rule To Global Graph
3.1 Smoothness
3.2 Interpolation
3.3 The Essential Question
3.4 The Essential Bounded Graph
3.5 Essential Notable Inputs

Part I - Power Functions

4 - Positive-Power Functions
4.1 Input-Output Pairs
4.2 Normalized Input-Output Rule
4.3 Local Graph Near ∞
4.4 Types of Local Graphs Near ∞
4.5 The Essential Question
4.6 Essential Bounded Graph
4.7 Notable Inputs
4.8 Local Graph near 0
4.9 Types of Local Graphs Near 0
4.10 Essential Global Graph
4.11 Types of Global Graphs

5 - Negative-Power Functions
5.1 Input-Output Pairs
5.2 Normalized Input-Output Rule
5.3 Local Graph Near ∞
5.4 Types of Local Graphs Near ∞
5.5 The Essential Question
5.6 Types of Local Graphs Near 0
5.7 Essential Bounded Graph
5.8 Local Graph near 0
5.9 Notable Inputs
5.10 Essential Global Graph
5.11 Types of Global Graphs

*The Fundamental Problem is the problem of deriving a graph from an input-output rule. It provides the story line of RFA. Larson-Hostetler and all the others notwithstanding, this cannot be done by “joining five points smoothly”.

Hopefully, this architecture will support the rest of RFA in a manner satisfactory to Students interested In mathematics With only a background in polynomial algebra.

As ever, any criticism, critique, feedback, etc is of course welcome, the more detailed, the more welcome.

A. Schremmer

P.S. While the original entry was written on Bastille Day, it was slightly modified on July 23.

P.P.S. The navigation in FreeMathTexts is of course perfectly atrocious and, as soon as I understand css better, I will use some open source code I found on the web to remedy this rather unfortunate situation. In other, unrelated news, I am also considering learning about how to set up a forum/listserv.