The setting was a summer course, which the department routinely gave to graduate students as a way of keeping them in the luxurious lifestyle to which grad students become accustomed. For five weeks and a couple days I’d spend several hours explaining the elements of vector calculus to students who either didn’t get it the first time around or who wanted to not have to deal with it during the normal term. (It’s the expansion of calculus to deal with integrals and differentials along curves, and across surfaces, and through solid bodies, and remarkably is not as impossibly complicated as this sounds. It’s probably easier to learn, once you know normal calculus, than it is to learn calculus to start. It’s essential, among other things, for working out physics problems in space, since it gives you the mathematical background to handle things like electric fields or the flow of fluids.)

What I thought was: the goal of the class is to get students to be proficient in a variety of techniques — that they could recognize what they were supposed to do, set up a problem to use whatever technique was needed, and could carry out the technique successfully. So why not divide the course up into all the things that I thought were different techniques, and challenge students to demonstrate proficiency in each of them? With experience behind me I understand at least one major objection to this, but if the forthcoming objection were to be dealt with, I’d still have blown it in the implementation.

Dividing a course into its atoms is a fine enough idea and probably useful for people who work to a much more detailed syllabus than I do. But here’s how I graded proficiency: students would get homework problems, and exam problems, and if they did great work on either homework or exam, or decent work on both together, that counted as proving their skills on that topic. Another objection should come here, one which proved pretty important to the failure.

To get a scheme this loose to fit to the A/B/C/D/F selection the Registrar wanted, I made the course grade to be the number of topics with demonstrated proficiency divided by the number of topics in the course. I think it worked out to something like 40 topics, so, people who proved they were able to do problems in 36 topics or more got an A. 32 and up got a B, and so on down to miserable failure.

Here’s how complete my failure was: nobody failed. Or got a D. Or a C, for that matter. As I remember it — I think accurately, but surely I would? — that of the 13 students in the summer course, four got B’s and nine got A’s, the most inflated grade curve I’ve ever given out. Mercifully, the department didn’t ask me to explain this, possibly trusting that between the small class size and the abnormal population there would be weird grading artifacts, possibly trusting that however I screwed up the grading the students would recover.

So how did it get so screwed up? I think the core problem was the demonstration-of-proficiency standard. As it worked out, this was binary: the student had shown an ability to do it, or hadn’t. But proficiency is a more fluid concept than that. It requires more subtle gradations, which is one of the benefits of an exam (or homework) being out of 100 (or whatever) points: you can recognize the distinction between A-level work and C-level work.

One of the side effects of this mistake was that I had to divide the course into many, many little sections — the 40 I mentioned above, and I remember trying to think whether I could get it up to 50. After all, a true/false test can measure with pretty fine gradation how well someone knows a subject, if there are enough questions, covering the subject broadly enough. So the binary grading of each topic implied having many topics.

Another consequence was that since I made the course depend on so many topics, I couldn’t ask too many questions about each of the topics. It would just produce a terrible workload, for myself as grader and for my students as people trying to figure out what the heck a contour integral is, to do otherwise. I couldn’t say with certainty that a student who proved proficiency actually was any good at it — the way working out a bunch of problems covering similar themes would — or just got lucky. Or did homework with someone who was good at it.

This brings me to another way that this scheme failed: students could pretty easily game the system. Students could skip, for example, exam problems which covered topics they’d already proved proficiency in, and if they were managing their time wisely certainly should. The exams could be second chances for poorly done homeworks instead, and I believe that is how the students treated them. This was great for getting exams graded more rapidly — students only worked on a couple problems and the most blessed thing to encounter on an exam is a blank page (edging out a perfectly correct answer neatly written) — but I don’t think that was a good trade-off to make.

Another of the core problems of this approach is that dividing the course into so many topics meant I had to divide up homework and exam problems into each topic, and — to be fair to the students — identify which problems covered which topics. I couldn’t do a multiple-part problem that tested different topics, lest I force students to redo topics that, by the rules, they didn’t have to.

A smaller problem, although one I probably could have managed, was that I couldn’t write exams the way I normally like. I like to have an exam be several long-form questions, a page or two of short-answer questions, another page of true/false or multiple choice answers. These are good ways to check tiny bits of knowledge, certainly, and I admit they’re easy grading, but they also mean that points which I have to be sure a student knows but which aren’t worth a full question can be asked. But this was my limited thinking. I could’ve overcome that if, for example, I’d required answering a set of multiple-choice questions on a single topic for the proof of proficiency.

But now I think even that wouldn’t have fixed the deepest flaw in the way it constrained my exams. I implicitly promised to ask questions about every topic since the past exam. At least some of the value of exams has to be that students should consider what topics are likely to be on the test, and what aren’t, and how important each topic is, and evaluate what they should have learned and what they have to get better on, swiftly.

If I wanted to salvage this scheme — and I’m not sure it’s worth it — I’d have to fix the definition of “proficiency” so that it had a broader range than just demonstrated/not demonstrated, first of all. I think this would require dividing the course into fewer topics, as well, so that “proficiency” can be a more graduated measure, and so that I can have some of the demonstration be in homework and some in exams without having either be pointless or a last-chance makeup.

If someone else wants to salvage this scheme, you’re welcome to try. I’d be interested to know of ways to make it functional.

I have  a post on this under my category Romance and Love called Logic plus Logic Equals Win.

My karma, not me, reap the fruits of tilled soil,
Mine’s not to yearn or intend, mine’s just to hush and toil. (I., Sl.47, Ch.2, Bhagwad Gita)

- May 2013

Differential Calculus. Applications of Differentiation. Stationary Points. Mathematical Methods (CAS) Units 3 and 4 as defined by the Victorian Curriculum and Assessment Authority in Victoria, Australia

Seriously, how did this even happen? I’m down to 30 days today.

I have my first AP test for the year tomorrow… Calculus. I obviously hate myself.

I just finished my senior project for my Journalism class.

My final choir concert is in one week.

What the heck is happening? How did this day come so fast? How on earth did I make it this far?

I need sleep or I’m going to get a 0 on my calc test.

Love you!

Abby

Về ý nghĩa, local normalization có thể xem là một kĩ thuật regularization cho neural network. Tuy nhiên thay vì tập trung vào thuật toán Backpropagation như nhiều kĩ thuật khác, phương pháp này trực tiếp thay đổi kiến trúc mạng. Cụ thể phương pháp này tỏ ra có hiệu quả khi sử dụng với các non-linearity không bị chặn như rectified linear unit (ReLU) vì nó ngăn không cho activation của các neuron có giá trị quá lớn so với các neuron chung quanh.

Tuy vậy, như sẽ thấy ngay sau đây, local normalization thêm vào 4 hyper-parameter cho mạng neural, làm tăng thêm gánh nặng hyper-parameter tuning vốn đã là “ác mộng” trong việc huấn luyện mạng neural. Cứ mỗi lớp normalization là có 4 hyper-parameter kèm theo.

Bài này ghi lại công thức và đạo hàm tương ứng của các thể loại local normalization. Chi tiết sẽ được viết sau, khi có dịp.

1. Local response normalization

Across maps:

Same map:

Đạo hàm:
Do local normalization không có tham số nào, nên ta chỉ cần tính đạo hàm của output đối với input. Tuy nhiên việc này hơi tricky vì ta phải tính hai thành phần:

•  đạo hàm của đối với
• đạo hàm của đối với , với là output ở trong phạm vi neighborhood của , và dĩ nhiên

Ta có:

Vậy nên đạo hàm của output đối với là tổng các đạo hàm riêng tại tất cả các vị trí trong local neighborhood của nó:

Trong đó là kí hiệu tắt cho mẫu số trong biểu thức ban đầu:

Trong cài đặt của thuật toán Backpropagation, ta sẽ quan tâm đến đạo hàm của hàm lỗi (tạm gọi là ) đối với input . Sử dụng chain rule, ta có:

Đây là công thức cuối cùng. Công thức này viết cho trường hợp across maps, trường hợp same map hoàn toàn tương tự.

2. Local contrast normalization

Trong khi local response normalization tính correlation trong vùng neighborhood thì local contrast normalization tính variance bằng cách tính thêm mean của vùng neighborhood. Chi tiết này làm tăng đáng kể sự phức tạp của đạo hàm cũng như khi cài đặt trên máy tính

Across maps:

Same map:

Đạo hàm:

Tương tự như local response normalization, đạo hàm cũng gồm 2 thành phần:

Và công thức đạo hàm của hàm lỗi theo input là:

More generally:

Given k independent random variables which are uniformly distributed amongst the integers 1 to n, how does the probability that some pair of random variables takes the same value behave as a function of n and k?

Clearly, if , then there will be at least one pair of random variables which share the same value. (Notice that this is just the pigeonhole principle.) So from now on we assume that .

Let be the probability that all random variables take on distinct values. Then

Indeed, if the first values are already all distinct, the probability that is not equal to any of the preceding values is . The above expression is all well and good, but how can we get an intuitive handle on how it behaves as a function of ? In particular, we might be interested in knowing the order of magnitude of relative to when . Of course, we don’t necessarily expect there to actually be an integral value of for which is exactly 1/2, but we are only interested in the order of magnitude of when is approximately 1/2.

As a matter of fact, we will be even lazier and content ourselves with an upper bound. We use the inequality , which is true for all . Then we have

so will be less than or equal to

When is this less than a half? Taking logarithms, we see

So, we see that when is about , we can expect to be about 1/2. In other words, when has order of magnitude about , there will be a `non-trivial’ (on the order of 1/2) probability that will not all be distinct.

This makes precise the idea that we need only sample a small (order , as opposed to , say) number of before expecting to find an identical pair. In subsequent posts we will explore some applications of the birthday problem.

Exercises.

• Prove the inequality . (Hint: calculus is helpful.)
• Write a program which actually computes the smallest value of for which . How close are these values of to the estimate derived earlier?

]]> http://thatstephanieclark.wordpress.com/2013/05/06/06-05-2013-the-beginning-of-the-worst-week-of-my-life/ Tue, 07 May 2013 03:24:25 +0000 thenerdfighter http://thatstephanieclark.wordpress.com/2013/05/06/06-05-2013-the-beginning-of-the-worst-week-of-my-life/ I know I’m supposed to be doing a poetry response, but I have just embarked on what I truly believe will be the worst week of my life thus far. I’m going to use this space to shamelessly complain in hopes that it will be counted as a completion grade. This probably won’t be read until after I graduate, anyway.

Anyway, this week is packed even on the topmost layer. I have rehearsal from 6:30 until whenever they let us out every night, one of which I’ll be missing for the band concert on Thursday, and two AP tests. Friday and Saturday are the show dates; 7:00 on Friday and Saturday and 2:30 on Saturday.

At the next level, I’m really stressed out because I just found out about a series of break-ins in my neighborhood. When I got home after rehearsal tonight, one of my neighbors flagged me down and told me someone’s house had been broken into last night. Apparently some people in a black truck with a broken muffler and tinted windows had been cruising, as well. I’ve been alone since Saturday evening (for reasons I will discuss later), and, naturally, I was kind of afraid. I called my parents in Hickory. My father told me he’d been planning on coming home, anyway.

He just got home and the first thing he asked was,  “Where’s my Tahoe?”. I had been under the impression that he and my mother had driven separately to Hickory. I didn’t think it was strange that his car wasn’t in the garage. I was wrong. He didn’t drive his car to Hickory. Someone stole his car three days ago and I only just noticed it.

Well, obviously, that makes me very comfortable about my own powers of observation and the sense of security I feel at any given moment in my house.

The tertiary level of suckiness comes from the fact that my grandmother is, quite frankly, dying. Hospice in involved. I’m terrified that it’s going to happen before I get the chance to see her again. I’m visiting after my AP test on Wednesday, but I’m still really scared. If /when it does happen, I’m probably not going to be able to come immediately. I’d have to finish out the week, or the month, and how am I supposed to live with that? I don’t know what to do at this point. Everything is happening at once and I can’t explain it to my friends because I know it’s a bunch of bullshit. This kind of stuff happens all the time and I have no right to complain about them because it could be much worse.

Somehow, that doesn’t help at all.

The Washington State Board for Community and Technical Colleges (SBCTC) has updated the Open Course Library, which is a collection of expertly developed educational materials for 81 of the state’s highest-enrolled college courses. The materials are freely available online under an open license for use by the state’s 34 public community and technical colleges, four-year colleges and universities, and anyone else worldwide. The subjects included in this update  are: Business, Communications, English, Fine Arts, Humanities, Mathematics,  and Science.

This resource series contains the IMS Common Cartridge files of the Open Course Library catalog. These files are compatible with many learning management systems (LMS) and contain all of the course content including syllabi, course activities, readings, and assessments. Some courses have content available via Google Docs which enables users to download the course resources directly to their Google account.

All of the course titles included in this series are:

American Government, American Literature I, American Sign Language I, American Sign Language II, American Sign Language III, Art Appreciation, Biology I, Biology II, Biology III, Business Calculus, Business Law, Calculus I, Calculus II, Calculus III, College Success Course, Cultural Anthropology, Elementary Algebra, Engineering Physics I, English Composition I, English Composition II, French I, French II, French III, General Biology with Lab, General Chemistry with Lab I, General Chemistry with Lab II, General Chemistry with Lab III, General Psychology, Health for Adult Living, Human Anatomy and Physiology 1&2, Intermediate Algebra, Interpersonal Communication, Intro to Communication, Intro to Drama, Intro to Humanities, Intro to Mass Media, Intro to Political Science, Intro to Sociology, Introduction To Astronomy, Introduction To Business, Introduction to Chemistry (Inorganic), Introduction To Literature I, Introduction To Logic, Introduction To Oceanography, Introduction To Philosophy, Introduction to Physical Geology, Introduction to Statistics, Lifespan Psychology, Macroeconomics, Math in Society, Microbiology, Microeconomics, Music Appreciation, Nutrition, Pacific NW History, Physical Anthropology, Physics: Non Science Majors, Precalculus I, Precalculus II, Pre-College English, Principles of Accounting I, Principles of Accounting II, Principles of Accounting III, Public Speaking, Research for the 21st Century, Small Group Communication, Social Problems, Spanish I, Spanish II, Spanish III, Survey of Anthropology, Survey of Biology, Survey of Environmental Science, Symbolic Logic, Technical Writing, US History I, US History II, US History III, Western Civilization, Women in US History, and World Civilizations I.

Creative Commons License
This content is licensed under the Creative Commons Attribution 3.0 Unported License which means that you are free to reuse the course in its entirety, edit it and use a your own modified version, or pick out only pieces which can be incorporated into your own course, as long as you credit the original author for their work.

The NCCCS courses that could be enhanced by using these resources are:

ACC111, ACC115, ACC118, ACC119, ACC120, ACC221, ACC3107, ANT210, ANT220, ANT230, ART111, ASL111, ASL112, ASL211, ASL221, ASL222, AST151, BIO100, BIO106, BIO110, BIO111, BIO112, BIO155, BIO160, BIO163, BUS110, BUS115, BUS116, BUS121, CHM115A, CHM121A, CHM131, CHM131A, CHM151, COM110, COM120, COM150, COM160, COM231, COM3709, ECO251, ECO252, EGR150, EGR220, ENG090, ENG110, ENG111, ENG116, ENG131, ENG132, ENG138, ENG231, ENV110, FLI3714, FLI3717, FRE111, FRE112, FRE211, FRE212, FRE221, GEL120, HEA110, HIS111, HIS121, HIS122, HIS131, HIS132, HIS162, HIS165, HIS232, HUM211, LIB210, MAT070, MAT080, MAT101, MAT151, MAT171, MAT172, MAT210, MAT223, MAT271, MAT272, MAT273, MSC160, MUS110, PHI210, PHI216, PHI230, PHY080, PHY090, POL110, POL120, PSY110, PSY150, SOC210, SOC220, SPA110, SPA111, SPA211, SPA212, SPA241,and SPA242.

To access this series, you can do any of the following:

• Log into the NCLOR and click on “Browse by Resource Series” and then the Open Course Library series.
• If you are NCCCS faculty, you can add the links to the resources directly to your LMS based course. (learn how)
• Simply, click here to view the series as a guest.

English

Science

Calculus

Class

Just

An excuse.

Love is the subject

–

About the poem:

I know now that teaching is just an excuse to hang out with cool people. It’s a precious opportunity to love them and be loved. Realizing that love is a teacher’s top priority makes things richer, clearer. It’s similar to how shooting hoops with friends is not really about winning the game of H-O-R-S-E; it’s about spending fun, important, bonding time with friends.

When a student enrolls in class, the end goal is to have a good life …one filled with love and purpose. But, if our focus is love, appreciation and encouragement, we can deliver that moment, now!

About the author:

Joan just really, really hopes you figure out how to enjoy your life before it’s over. She writes books and blogs about this very important, fun topic!

– Bud Norman

This is about limits in mathematics: both the technical notion that arises in calculus, and the barriers to comprehension that one might reach in one’s own studies. I am going to say a few technical things about the technical notion, but there is no reason why this should be a barrier to your reading: you can just skip the paragraphs that have special symbols in them.

Looking up something else in the online magazine called Slate, I noted a reprint of an article called “What It Feels Like to Be Bad at Math” from a blog called Math With Bad Drawings by Ben Orlin. Now teaching high-school mathematics, Mr Orlin recalls his difficulties in an undergraduate topology course. His memories help him understand the difficulties of his own students. When students do not study, why is this? It is because studying makes them conscious of how much they do not understand. They feel stupid, and they do not like this feeling.

Such is Mr Orlin’s conclusion, as I understand it. I hope it is the right conclusion. Another possible reason for students’ not studying is mere lack of interest. Thinking about mathematics is less fun than hanging out at the local shopping mall. But if a student avoids studying because it causes her self-doubt, this is a more hopeful situation. It means the student might actually appreciate the feeling of being smart. The student may learn to achieve this feeling through actually coming to understand some piece of mathematics.

“Not understanding Topology”, writes Mr Orlin, “doesn’t make me stupid. It makes me bad at Topology.” This is imprecise. Not understanding topology means he is bad, right now, at what is being presented as topology. This momentary badness may change with time, and the presentation of the subject may be changed for the better. Here is where a teacher might help.

We all vary. When, in a graduate course, I first encountered the axioms for a topological space, I was so fascinated by them that I wrote them down in a letter to a non-mathematical friend from college. He was not amused.

But like Mr Orlin, I have had difficulties in mathematics. The hardest time of my own mathematical life was in a calculus course, when I had to understand limits and write epsilon-delta proofs. I copy the following from my high school notes.

The entire structure of the calculus rests upon the foundation known as the Theory of Limits…

The rigorous definition of limit is a simple translation of the following statement. The function f approaches the limit L near a if given any preassigned tolerance ε > 0 we can find a control δ > 0, so that when x is within δ of a, and unequal to a, f(x) is within ε of L. The following definition took 2500 years and is attributed to Cauchy and Weierstraß. It is the definition on which all of the calculus rests.

It is fine to point out the importance of limits. Their difficulty might also be acknowledged. Indeed, the definition is anything but simple, at least when one has to use it. Epsilon-delta proofs are a common stumbling-block. A friend of mine transferred to another calculus class, thought to be easier, but apparently still rigorous; he later reported that, after the switch, he finally understood epsilon-delta proofs. When I teach calculus now, I recall to the students my own difficulties with limits. Students should be aware that the concept is hard for just about everybody. [See comments.]

There are schemes for making the learning of calculus easier. Donald Knuth’s proposal to use the big-O notation is reprinted in Alexandre Borovik’s blog, Mathematics Under the Microscope. Another possibility is to use the so-called non-standard approach of Abraham Robinson.

Non-standard analysis is a wonderful subject, and everybody who teaches calculus ought to know something about it. I have thrice taught a week-long course of non-standard analysis for undergraduates at a summer math camp. (In the course, I go back to the origins of calculus in Archimedes.) The non-standard approach to calculus makes limits easier in retrospect; but this is the retrospect of somebody who has already struggled with the epsilon-delta definition.

The definition is difficult, because it involves two alternations of logical quantifiers:

lim_x→af(x) = L means ∀ε ∃δ ∀x (ε > 0 ⇒ δ > 0 & (0 < |x – a| < δ ⇒ |f(x) – L| < ε)).

This is the “simple translation” referred to in the notes quoted above. The non-standard definition involves no alternations of quantifiers:

lim_x→af(x) = L means ∀x (x ≈ a & x ≠ a ⇒ f(x) ≈ L).

But the reduction in quantifiers is only an illusion. The quantifiers are hidden in the new symbol ≈. The formula x ≈ a means the difference |x – a| is infinitesimal, so that

x ≈ a & x ≠ a means ∀δ (δ ∈ ℝ & δ > 0 ⇒ 0 < |x – a| < δ).

Here δ must be restricted to the field ℝ of real numbers, because otherwise the formula would have no solution. As it is, if a ∈ ℝ, then the formula has no solution in ℝ. It has solutions in a larger field, *ℝ, consisting of hyper-real numbers.

One can just declare, by fiat, that *ℝ exists as desired. It is a proper elementary extension of ℝ, when the latter is considered as a structure in a perfectly enormous signature. One wants this signature to contain a symbol for every subset of every finite power ℝⁿ. Ideally, one also has a sort for each power set ℘(ℝⁿ), so that one can quantify over elements of this (as for example when defining the Riemann integral). One still does calculus in ℝ as usual; the non-standard aspect is that one can use the help of elements of *ℝ, as in the second definition of limits above, where x ranges over *ℝ, even though a and L are in ℝ. This all needs explicit discussion of symbolic logic.

If one knows “abstract” algebra, and in particular rings, then one can let *ℝ be a quotient ℝ^ω⁄p, where p is a non-principal maximal ideal of the Cartesian power ℝ^ω. One might also write this power as the product ∏_ωℝ. A proper ideal of the power is non-principal if and only if the ideal includes the ideal ∑_ωℝ. The field ℝ embeds in ℝ^ω⁄p under the diagonal map x → (x, x, x, …) + p; the embedding is proper because the ideal p is non-principal. Actually choosing such an ideal does indeed require a special case of the Axiom of Choice. So one does not and cannot make the choice explicitly; one just assumes it has been done.

I mention all of these details, just to make the point that understanding limits rigorously is bound to be hard, no matter how you go about it. The non-standard approach adds its own difficulties, and there is good reason why this approach has not caught on. There are calculus textbooks that take the non-standard approach. I am aware of the examples by Keisler and by Henle & Kleinberg. The latter authors write in their preface,

A most natural place for Robinson’s insight is as a next (and possibly final) point in the evolution of the teaching of calculus. We can now develop calculus using infinitesimals and enjoy all of their simplicity and intuitive power, yet at the same time work in a mathematically precise and rigorous atmosphere. This approach, although quite new, has been used at a number of universities with remarkable success.

This success has not been so great that Robinson’s non-standard approach has become standard. Perhaps in time it will become standard. It is however foolish to suggest that any approach represents the ultimate stage of evolution. It is dangerous to suggest to students that anything in mathematics is simple. If the mathematics really is simple, then we need not waste any time telling this to the students; we need only show them.

I am familiar with a younger contemporary of mine who struggled with mathematics. In a college course, she asked an instructor to explain the manipulations that he had performed on the board. He told her, “It’s easy!” Perhaps he also repeated the manipulations, by way of showing how easy they were.

No Sir, the mathematics is not easy; this is why you are being asked to explain it. Instead of explaining, you cause your student to be ashamed of her own confusion. Obviously she must be really stupid, if she cannot see how easy your mathematics is. This is what you are telling her.

I suppose it is just possible that the shame of feeling stupid may cause a student to work harder. But I think it is our job as teachers to find a better power of motivation than this. If our subject is not intrinsically interesting, beautiful, captivating, fascinating, then why are we teaching it?

Moreover, if mathematics is easy, why need we bother to teach it? Χαλεπὰ τὰ καλά as the saying goes (Plato’s Republic 435c): Fine things are difficult. We cannot make the pain of learning go away. To deny this only makes the pain worse.

References

1. Henle & Kleinberg, Infinitesimal Calculus, MIT Press, 1979; republished by Dover, 2003.

2. H. Jerome Keisler, Elementary Calculus: An Infinitesimal Approach, second edition, Prindle, Weber & Schmidt, 1986; available from the author’s website.

This is the first of several blogs that resulted from my re-evaluation of how I assess my students’ performance. I will make my calculus class last year as a case study, and perhaps relate this with other classes that I had in other subjects in the other years, with some input from observations from other colleagues’ classes.

Last school year, I taught Math 5, that is Elementary Analysis, for the first time. It was actually the first time that I handled a calculus class for high school. I have mixed feelings about meeting the results of the assessments I have given my students. For one, I doubt that the grades that I gave really reflected what my students learned and didn’t learn. I feel that I have cheated myself thinking that I have done well in teaching my students calculus since all of them passed anyway.

I met a lot of problems during the last school year. For one, I saw some students who made it a habit to get sick during long tests and periodical examinations. And that was not just in my class but in almost all other classes. It was also disheartening to see some students who looked and acted really bored during lectures. This should not be the case. I am a math guy and I really passionate about mathematics–teaching mathematics and learning mathematics–but, my students did not seem to get into the passion
thing that I am in. There was a disconnect between what I feel about mathematics and the response that I was getting about mathematics. (I will write more about this in another blog.)

It seemed that my calculus class was a game that my students and I played, a game that most of my students did not find enjoyable at all–much like how I felt during basketball lessons in P.E. when I was in high school. I’d rather run in the oval or play chess back then. I knew that if I learned how to perform the basic sequences and scored a point or two, I would pass, since I have met the minimum learning competencies. But I could forget about it all later, and I did. Most of my students in the calculus class, aparently, were doing the same thing.

I don’t know if I am right so please correct me if I am wrong. But what I would like to talk about, and perhaps the first thing that should be addressed is the calculus curriculum itself.

The Current PSHS Calculus Syllabus

The syllabus is divided into four quarters. The first half of the first quarter is alotted to conic sections: parabola, ellipse, hyperbola, and unified treatment of the conics. The student should be able to identify and analyze the properties of all conic sections based on a given equation, a graph, or a situation in which the concept of a conic section can be applied. Of course, a long test is given at the end of the unit.

The lesson on limits and continuity starts on the next half of the first quarter, beginning with a review of functions, definition of limits, building on this through continuity, Intermediate Value Theorem, Squeeze Theorem, and Limits of trigonometric functions.

The succeeding topics are then built upon the last half of the first quarter until chain rule, implicit differentiation, and differentials.

The Jump Discontinuity

So what is wrong? Maybe nothing. I am not so sure. Perhaps I am not looking at it more objectively. The curriculum seems to have been borrowed from the late Louis Leithold’s sixth edition of the The Calculus with Analytic Geometry (or more popularly known among Isko‘s as TCWAG). When TCWAG became TC7, the PSHS curriculum seemed not to be able to catch up with the change.

One of my hypotheses regarding the syllabus is that most of my students could not see the continuity of concepts from the lesson on conics. The development of concepts from conic sections is abruptly halted when the syllabus inserted a review of functions to brush up on perhaps forgotten pre-calculus skills. Nowhere in the review lessons is there a use for the skills acquired during the lessons on conics aside from possibly mentioning examples of quadratic functions. But the applications of the quadratic functions in this context is far from the very technical and detailed discussions and graphing exercises conducted in lessons on conics. If one believes, as I do, that in mathematics, a skill not often used is a skill lost, then students taking calculus in Philippine Science High School will easily forget the conics. As my students did, as can be seen in the results of their first quarter examination.

Of course a background on analytic geometry is essential in learning calculus. But the idea is not even touched until parametric equations, which is not covered in the PSHS calculus syllabus! I think the lesson on conic sections is a waste of time and energy both on the part of the teacher and the student. It is a source of confusion among students, who are left asking what the use of the lessons on conic sections is. It is a gaping discontinuity in the syllabus. I am planning to remove it altogether the next time that I teach calculus.

Perhaps you will disagree with me. There is a school of thought that says, "Oh, it is not important now, but as the student digs deeper into mathematics, he or she will find the relevance of the topic." But there are a lot of loopholes into this kind of thinking. Firstly, not all of the students plan to take a mathematics-intensive degree in college. When can they find the time to dig more deeply into mathematics if they are not going to use it in the future? Secondly, for the mathematically-inclined students, don’t they get more out of calculus if the time they are going to spend on a topic whose applications are not immediate is used for mastering topics whose applications are?

I think the PSHS system math teachers need to consider these questions, specially in light of the overhaul the entire curriculum is undergoing due to the shift to the K-12 curriculum. One consideration that the system has in these major changes in the entire curriculum is decongesting the topics. Perhaps we should scrap the conic sections part from the calculus syllabus and place it in a pre-calculus course where it can be taught in a more thematic way. This also means looking into the present pre-calculus course (Math 4, Advanced Algebra and Trigonometry), and removing topics that seem to be not that immediately useful in calculus. For example, that small part on combinatorics can be placed in a specialized course like a problem-solving elective, or perhaps probability and statistics.

Again, I am not so sure. One thing about a scientific way of thinking is that you can never be too sure. Later evidence or a more objective evaluation of the syllabus might suggest a better way of going about the calculus syllabus. But my point is I could have saved my students from a lot of yak, yak, yak, and yadadidada that they did not understand because I am trying to stick with a syllabus, unable to prevent myself from hurried discussions and expositions in order to be able to finish the whole thing in the specified number of meetings.

I did not plan for this post to be very long. So I will end it here and leave for future posts the other matters that I want to talk about. Let me hear from you what you think about the things I mentioned here.

Mathematical Methods (CAS) Units 3 and 4.
Differential Calculus. Applications of Differentiation.
Linear Approximation. This course is a fully prescribed course for Year 12 students as specified by the Victorian Curriculum and Assessment Authority, Victoria, Australia

Professor Lewin derives the displacement current term:

Here’s another, by Dr. Phys:

We had our second Calculus exam yesterday. It had to answer 3 items worth 25 points and for the first half hour, all I did was stare at it. It’s not that it’s really hard. It’s just that maybe I lacked practice and so I was taken aback by the fact that I didn’t know where to begin at each item. I felt ashamed not because I lacked the capability to answer the exam but because my professor is a really close friend, and I didn’t want him saying that maybe he lacked the capability to teach me because I can’t seem to pass most of his exams.

I feel bad.

After my Calculus exam, a friend of mine from my hometown asked me a huge favor. It was something about his shifting papers in the university. Since I was not really in a happy mood and would do anything just to get the Calculus exam out of my head, I happily agreed to do hid bidding. I went to the college that he was applying in only to find out that he filled out the wrong application form. I had to fill up another application form for him again and had to photocopy his previous application form. I also had to wait to know if he was qualified to take the written examination because the woman in charge still had to calculate his grades.

Fortunately though, after almost 3 hours, I found that he was indeed qualified and he’d have to take the exam on the 6th of May. 

I directly went back to the dormitory after that and change clothes for my PE class. While walking to the gym, I realized that yesterday wasn’t so bad after all. I realized that my world doesn’t revolve around that Calculus exam. Of course, it doesn’t mean I’m happy that I flunked it. It’s just that I don’t think my purpose in this world revolves around perfecting that exam. What makes me happy is helping other people. When they think it’s game over already, I gladly extend a hand and say, “No, not on my watch”. 

You may say that maybe I’m just looking for ways to comfort myself because that exam really got me down, but whatever. I think that helping my classmate was a step closer to reaching his goal, and knowing that I was part of reaching that goal? That made me feel better.

Mathematical Methods (CAS) Units 3 and 4. Differential Calculus. Applications of Differentiation. Tangents and Normals. This video is prepared for Year 12 students studying this topic of the subject as defined by the Victorian Curriculum and Assessment Authority, Victoria, Australia.

1. Likely it’s full of errors that some won’t catch.
2. They really ought to be doing this sort of thing themselves.
3. Some will try to memorize without understanding – a tactic of little value.

Anyway, here it is.