# Numbers: Rational and Irrational

From Numbers: Rational and Irrational by Ivan Niven

# The Calculus Bottleneck

What if someone told you that learners in high-school don’t actually need calculus as a compulsory subject for a career in STEM? Surely I would disagree. After all, without calculus how will they understand many of the topics in the STEM. For example basic Newtonian mechanics? Another line of thought that might be put forth is that calculus allows learners to develop an interest in mathematics and pursue it as a career. But swell, nothing could be farther from truth. From what I have experienced there are two major categories of students who take calculus in high school. The first category would be students who are just out of wits about calculus, its purpose and meaning. They just see it as another infliction upon them without any significance. They struggle with remembering the formulae and will just barely pass the course (and many times don’t). These students hate mathematics, calculus makes it worse. Integration is opposite of differentiation: but why teach it to us?

The other major category of students is the one who take on calculus but with a caveat. They are the ones who will score in the 80s and 90s in the examination, but they have cracked the exam system per se. And might not have any foundational knowledge of calculus. But someone might ask how can one score 95/100 and still not have foundational knowledge of the subject matter? This is the way to beat the system. These learners are usually drilled in solving problems of a particular type. It is no different than chug and slug. They see a particular problem – they apply a rote learned method to solve it and bingo there is a solution. I have seen students labour “problem sets” — typically hundreds of problems of a given type — to score in the 90s in the papers. This just gives them the ability to solve typical problems which are usually asked in the examinations. Since the examination does not ask for questions based on conceptual knowledge – it never gets tested. Perhaps even their teachers if asked conceptual questions will not be able to handle them — it will be treated like a radioactive waste and thrown out — since it will be out of syllabus.

There is a third minority (a real minority, and may not be real!, this might just be wishful thinking) who will actually understand the meaning and significance of the conceptual knowledge, and they might not score in the 90s. They might take a fancy for the subject due to calculus but the way syllabus is structured it is astonishing that any students have any fascination left for mathematics. Like someone had said: the fascination for mathematics cannot be taught it must be caught. And this is exactly what MAA and NCTM have said in their statement about dropping calculus from high-school.

What the members of the mathematical community—especially those in the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM)—have known for a long time is that the pump that is pushing more students into more advanced mathematics ever earlier is not just ineffective: It is counter-productive. Too many students are moving too fast through preliminary courses so that they can get calculus onto their high school transcripts. The result is that even if they are able to pass high school calculus, they have established an inadequate foundation on which to build the mathematical knowledge required for a STEM career. (emphasis added)

The problem stems from the fact that the foundational topics which are prerequisites for calculus are on shaky grounds. No wonder anything build on top of them is not solid. I remember having very rudimentary calculus in college chemistry, when it was not needed and high-flying into physical meaning of derivatives in physics which was not covered enough earlier. There is a certain mismatch between the expectations from the students and their actual knowledge of the discipline as they come to college from high-school.

Too many students are being accelerated, short-changing their preparation in and knowledge of algebra, geometry, trigonometry, and other precalculus topics. Too many students experience a secondary school calculus course that drills on the techniques and procedures that will enable them to successfully answer standard problems, but are never challenged to encounter and understand the conceptual foundations of calculus. Too many students arrive at college Calculus I and see a course that looks like a review of what they learned the year before. By the time they realize that the expectations of this course are very different from what they had previously experienced, it is often too late to get up to speed.

Though they conclude that with enough solid conceptual background in these prerequisites it might be beneficial for the students to have a calculus course in the highschool.

# On mathematics

Mathematics is regarded as a demonstrative science. Yet this is only one of its aspects. Finished mathematics presented in a finished form appears as purely demonstrative, consisting of proofs only. Yet mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies; you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.

George Polya (Induction and Analogy – Mathematics of Plausible Reasoning – Vol. 1, 1954)

# Unreal and Useless Problems

We had previously talked about problem with contexts given in mathematics problems. This is not new, Thorndike in 1926 made similar observations.

Unreal and Useless Problems

In a previous chapter it was shown that about half of the verbal problems given in standard courses were not genuine, since in real life the answer would not be needed. Obviously we should not, except for reasons of weight, thus connect algebraic work with futility. Similarly we should not teach the pupil to solve by algebra problems which in reality are better solved otherwise, for example, by actual counting or measuring. Similarly we should not set him to solve problems which are silly or trivial, connecting algebra in his mind with pettiness and folly, unless there is some clear, counterbalancing gain.
This may seem beside the point to some teachers, ”A problem is just a problem to the children,” they will say,

“The children don’t know or care whether it is about men or fairies, ball games or consecutive numbers.” This may be largely true in some classes, but it strengthens our criticism. For, if pupils^do not know what the problem is about, they are forming the extremely bad habit of solving problems by considering only the numbers, conjunctions, etc., regardless of the situation described. If they do not care what it is about, it is probably because the problems encountered have not on the average been worth caring about save as corpora vilia for practice in thinking.

Another objection to our criticism may be that great mathematicians have been interested in problems which are admittedly silly or trivial. So Bhaskara addresses a young woman as follows: ”The square root of half the number of a swarm of bees is gone to a shrub of jasmine; and so are eight-ninths of the swarm: a female is buzzing to one remaining male that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.” Euclid is the reputed author of: ”A mule and a donkey were going to market laden with wheat. The mule said,’If you gave me one measure I should carry twice as much as you, but if I gave you one we should bear equal burdens.’ Tell me, learned geometrician, what were their burdens.” Diophantus is said to have included in his preparations for death the composition of this for his epitaph : ” Diophantus passed one-sixth of his life in childhood one-twelfth in youth, and one-seventh more as a bachelor. Five years after his marriage was born a son, who died four years before his father at half his father’s age.”

My answer to this is that pupils of great mathematical interest and ability to whom the mathematical aspects of these problems outweigh all else about them will also be interested in such problems, but the rank and file of pupils will react primarily to the silliness and triviality. If all they experience of algebra is that it solves such problems they will think it a folly; if all they know of Euclid or Diophantus is that he put such problems, they will think him a fool. Such enjoyment of these problems as they do have is indeed compounded in part of a feeling of superiority.

# Dialectic vs Algorithmic Mathematics

Dialectic mathematics is a rigorously logical science, where statements are either true or false, and where objects with specified properties either do or do not exist. Algorithmic mathematics is a tool for solving problems. Here we are concerned not only with the existence of a mathematical object, but also with the credentials of its existence. Dialectic mathematics is an intellectual game played according to titles about which there is a high degree of consensus. The rules ol the game of algorithmic mathematics vary according to the urgency of the problem on hand. We never could have put a man on the moon if we had insisted that the trajectories should be computed with dialectic rigor. The rules may also vary according to the computing equipment available. Dialectic mathematics invites contemplation. Algorithmic mathematics invites action. Dialectic mathematics generates insight. Algorithmic mathematics generates results.

# On not learning or con in the context

We will, we will, fail you by testing what you do not know…

We live in a rather strange world. Or is it that we assume the world
to be non-strange in a normative way, but the descriptive world has
always been strange? Anyways, why I say this is to start a rant to
about some obviously missed points in the area of my work. Namely,
educational research, particularly science and mathematics education
research.

In many cases the zeal to show that the students have
‘misunderstandings’ or are simply wrong, and then do a hair-splitting
(micro-genetic) exercise on the test the students were inflicted
with. Using terse jargon and unconsequential statistics, making the
study reports as impossible to read as possible, seem to be the norm.

But I have seen another pattern in many of the studies, particularly
in mathematics education. The so-called researchers spent countless
nights in order to dream up situations as abstract as possible (the
further far away from real-life scenarios the better), then devise
problems around them. Now, these problems are put in research studies,
which aim to reveal (almost in evangelical sense) the problems that
plague our education. Unsuspecting students are rounded, with
appropriate backgrounds. As a general rule, the weaker socio-economic
background your students come from, the more exotic is your study. So
choose wisely. Then these problems are inflicted upon these poor,
mathematically challenged students. The problems will be in situations
that the students were never in or never will be. The unreal nature of
these problems (for example, 6 packets of milk in a cup of coffee! I
mean who in real life does that? The milk will just spill over, the
problem isn’t there. This is just a pseudo-problem created for satisfying the research question of the researcher. There is no context, but only con.

Or finding out a real-life example for some weird fractions) puts many off. The fewer students perform correctly happier the researcher is. It just adds to the data statistic that so many % students cannot perform even this elementary task well. Elementary for
that age group, so to speak. The situation is hopeless. We need a
remedy, they say. And remedy they have. Using some revised strategy,
which they will now inflict on students. Then either they will observe
a few students as if they are some exotic specimens from an
uncontacted tribe as they go on explaining what they are doing or why
they are doing it. Or the researcher will inflict a test (or is it
taste) in wholesale on the lot. This gives another data
statistic. This is then analysed within a ‘framework’, (of course it
needs support) of theoretical constructs!

Then the researcher armed with this data will do a hair-splitting
analysis on why, why on Earth student did what they did (or didn’t
do). In this analysis, they will use the work of other researchers before
them who did almost the same thing. Unwieldy, exotic and esoteric
jargons will be used profusely, to persuade any untrained person to
giveup on reading it immediately. (The mundane, exoteric and
understandable and humane is out of the box if you write in that
style it is not considered ‘academic’.) Of course writing this way,
supported by the statistics that are there will get it published in
the leading journals in the field. Getting a statistically significant
result is like getting a license to assert truthfulness of the
result. What is not clear in these mostly concocted and highly
artificial studies is that what does one make of this significance
outside of the experimental setup? As anyone in education research
would agree two setups cannot be the same, then what is t

Testing students in this way is akin to learners who are learning a
new language being subjected to and exotic and terse vocabulary
test. Of course, we are going to perform badly on such a test. The
point of a test should be to know what students know, not what they
don’t know. And if at all, they don’t know something, it is treated as
if is the fault of the individual student. After all, there would be
/some/ students in each study (with a sufficiently large sample) that
would perform as expected. In case the student does not perform as
expected we can have many possible causes. It might be the case that
the student is not able to cognitively process and solve the problem,
that is inspite of having sufficient background knowledge to solve the
problem at hand the student is unable to perform as expected. It might
be the case that the student is capable, but was never told about the
ways in which to solve the given problem (ZPD anyone?). In this case, it might be that the curricular materials that the student has access
to are simply not dealing with concepts in an amenable way. Or it
might be that the test itself is missing out on some crucial aspects
and is flawed, as we have seen in the example above. The problem is
systemic, yet we tend to focus on the individual. This is perhaps
because we have a normative structure to follow an ideal student at
that age group. This normative, ideal student is given by the so-called /standards of learning/. These standards decide, that at xx age
a student should be able to do multiplication of three digit
numbers. The entire curricula are based on these standards. Who and
what decides this? Most of the times, the standards are wayyy above
the actual level of the students. This apparent chasm between the
descriptive and the normative could not be more. We set unreal
expectations from the students, in the most de-contextualised and
uninteresting manner, and when they do not fulfil we lament the lack
of educational practices, resources and infrastructure.

# What is a mathematical proof?

A dialogue in The Mathematical Experience by Davis and Hersh on what is mathematical proof and who decides what a proof is?

Let’s see how our ideal mathematician (IM) made out with a student who came to him with a strange question.

Student: Sir, what is a mathematical proof?

I.M.: You don’t know that? What year are you in?

I.M.: Incredible! A proof is what you’ve been watching me do at the board three times a week for three years! That’s what a proof is.

Student: Sorry, sir, I should have explained. I’m in philosophy, not math. I’ve never taken your course.

I.M.: Oh! Well, in that case – you have taken some math, haven’t you? You know the proof of the fundamental theorem of calculus – or the fundamental theorem of algebra?

Student: I’ve seen arguments in geometry and algebra and calculus that were called proofs. What I’m asking you for isn’t examples of proof, it’s a definition of proof. Otherwise, how can I tell what examples are correct?

I.M.: Well, this whole thing was cleared up by the logician Tarski, I guess, and some others, maybe Russell or Peano. Anyhow, what you do is, you write down the axioms of your theory in a formal language with a given list of symbols or alphabet. Then you write down the hypothesis of your theorem in the same symbolism. Then you show that you can transform the hypothesis step by step, using the rules of logic, till you get the conclusion. That’s a proof.

Student: Really? That’s amazing! I’ve taken elementary and advanced calculus, basic algebra, and topology, and I’ve never seen that done.

I.M.: Oh, of course, no one ever really does it. It would take forever! You just show that you could do
it, that’s sufficient.

Student: But even that doesn’t sound like what was done in my courses and textbooks. So mathematicians don’t really do proofs, after all.

I.M.: Of course we do! If a theorem isn’t proved, it’s nothing.

Student: Then what is a proof? If it’s this thing with a formal language and transforming formulas, nobody ever proves anything. Do you have to know all about formal languages and formal logic before you can do a mathematical proof?

I.M.: Of course not! The less you know, the better. That stuff is all abstract nonsense anyway.

Student: Then really what is a proof?

I.M.: Well, it’s an argument that convinces someone who knows the subject.

Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons.Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?

I.M.: No, no. There’s nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you’ll catch on.

Student: Are you sure?

I.M.: Well – it is possible that you won’t, if you don’t have any aptitude for it. That can happen, too.

Student: Then you decide what a proof is, and if I don’t learn to decide in the same way, you decide I don’t have any aptitude.

I.M.: If not me, then who?

# Implicit cognition in the visual mode

Images become iconified, with the image representing an object or
phenomena, but this happens by enculturation rather by training. An
example to elaborate this notion is the painting Treachery of
Images by Belgian surrealist artist René Magritte. The painting is
also sometimes called This is not a pipe. The picture shows a
pipe, and below it, Magritte painted, “Ceci n’est pas une pipe.”,
French for “This is not a pipe.”

When one looks at the painting, one
exclaims “Of course, it is a pipe! What is the painter trying to say
here? We can all see that it is indeed a pipe, only a fool will claim
otherwise!” But then this is what Magritte has to say:

The famous pipe. How people reproached me for it! And yet, could you
stuff my pipe? No, it’s just a representation, is it not? So if I had
written on my picture `This is a pipe’, I’d have been lying!

Aha! Yess! Of course!! you say. “Of course it is not a pipe! Of
course it is a representation of the pipe. We all know that! Is this
all the painter was trying to say? Its a sort of let down, we were
expecting more abstract thing from the surrealist.” We see that the
idea or concept that the painting is a \emph{representation} is so
deeply embedded in our mental conceptual construct that we take it for
granted all the time. It has become so basic to our everyday social
discourse and intercourse that by default we assume it to be so. Hence
the confusion about the image of the pipe. Magritte exposes this
simple assumption, that we so often ignore. This is true for all the
graphics that we see around us. The assumption is implicit in all the
things we experience in the society. The representation becomes the
thing itself, for it is implicit in the way we talk and communicate.

Big B and D

When you look at a photo of something or someone, you recognize
it. “This is Big B!” you say looking at the painting! But then you
have already implicitly assumed that the representation of Big B is Big B. This implicit assumption comes from years of implicit training from being submerged in  the sea of the visual artefacts that surround and drown us. This association between the visual representation and the reality it represents had become the central theme of the visual culture that we live in. The training that we need for such an association comes from the peers and mentors that surround us from the childhood. The meaning and the association of the images is taught/caught over the years, so much so that we assume the abstract association is the normal way things are. In this way it becomes the implicit truth, though when one is pressed, the explicit connections are brought out.

Yet when it comes to understanding images in science and mathematics, the same thing doesn’t happen. There is no enculturation of children into understand the implicit meaning in these images. Hardly there are no peers or mentors whose actions and practices can be imitated by the young impressible learners. The practice which comes so naturally in other domains (identifying actor with a picture of the actor, or identifying a physical space with a photo) doesn’t happen in science and mathematics classrooms. The notion of practice is dissociated from the what is done to imbibe this understanding in the children. A practice based approach where the images become synonymous with their implied meaning is used in vocabulary might one very positive way out, this is after all practitioners of science and mathematics learn their trade.

# Mathematical Literacy Goals for Students

National Council of Teachers for Mathematics NCTM proposed these five goals to cover the idea of mathematical literacy for students:

1. Learning to value mathematics: Understanding its evolution and its role in society and the sciences.
2. Becoming confident of one’s own ability: Coming to trust one’s own mathematical thinking, and having the ability to make sense of situations and solve problems.
3. Becoming a mathematical problem solver: Essential to becoming a productive citizen, which requires experience in a variety of extended and non-routine problems.
4. Learning to communicate mathematically:  Learning the signs, symbols, and terms of mathematics.
5. Learning to reason mathematically: Making conjectures, gathering evidence, and building mathematical arguments.
National Council of Teachers of Mathematics. Commission on Standards for School Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Natl Council of Teachers of.