Pre-Calculus Mathematics Courses (open thread)
As I mentioned in the comments yesterday, I may well have to be teaching math courses below the calculus level this year. The closest I’ve come to it before is “tutoring” some of my parents’ friends’ kids, which amounted to reminding them to do their homework. Amazingly, when they actually did their homework (and extra similar problems from the book before exams), their grades improved immediately. So I’m a bit nervous as to how this will play out. But I’m not saying it’ll go badly, so please don’t fire me quite yet, Dr. Hamburger.
Anyhow, I’ve got the impression that there’s a subtle, yet distinct difference between calculus and pre-calculus (including “college algebra”) math teaching. There’s a reason that a lot of schools don’t teach anything below calculus, and I’ve been able to feel my classes teetering on the edge of a conceptual continental divide. My intuition is that the calculus is the leading edge of a sea change from “how” to “why” in mathematics teaching — though I think that elementary mathematics could do with a lot more “why”, especially abstract reasoning. But I could be wrong about this, having little hands-on experience.
But when it comes down to it, I understand the position of the pre-calculus college math course. Most students taking it are scraping through whatever minimal mathematics requirement the institution sets, and will stop once it’s completed. And I understand that this will hold for most of them no matter how inspiring a teacher I am. The University of Maryland had mathematics courses below calculus, but I never knew a mathematics or physics major who started there. There was the occasional computer science major, but that was the heyday of the internet bubble, so you couldn’t spit on campus without hitting one.
The upshot is that such courses exist with certain “The Student Will Be Able To” line-items, and it’s my job to do the best I can to help them meet those. What I don’t know is the current thoughts on their pedagogy. What is the role (and the balance) of “how” vs. “why” in these courses? Are they really that different from calculus courses? If so, in what ways?
Since I’m now here in Bowling Green and taking a bit of a break while I look for an apartment, I thought I’d open this thread up to public discussion. JVP and JackieB have already started the ball rolling a bit. Jackie’s a high school math teacher, which provides her bona fides here, and I’ve yet to find the subject JVP doesn’t have some insight on. But there’s a lot of people out there from a lot of backgrounds. What do you think? And call up your friends teaching (or who have taught) below the calculus level in high schools and colleges, or those who might end up doing it some day, so they can weigh in as well.
29 Comments »
Leave a comment
Recent Posts- Upper and Lower Integrals and Riemann’s Condition
- Higher-Dimensional Riemann Integrals
- Sunday Samples 149 (one day late)
- Extrema with Constraints II
- Extrema with Constraints I
- Classifying Critical Points
- Local Extrema in Multiple Variables
- Sunday Samples 148
- The Implicit Function Theorem II
- The Implicit Function Theorem I
- Recent Comments
- Comment on Riemann’s Condition by Upper and Lower Integrals and Riemann’s Condition « The Unapologetic Mathematician
- Comment on Upper and Lower Integrals by Upper and Lower Integrals and Riemann’s Condition « The Unapologetic Mathematician
- Comment on Darboux Integration by Upper and Lower Integrals and Riemann’s Condition « The Unapologetic Mathematician
- Comment on Higher-Dimensional Riemann Integrals by Upper and Lower Integrals and Riemann’s Condition « The Unapologetic Mathematician
- Comment on Parallelepipeds and Volumes III by Higher-Dimensional Riemann Integrals « The Unapologetic Mathematician
- Comment on Nets, Part I by Higher-Dimensional Riemann Integrals « The Unapologetic Mathematician
- Comment on The Mean Value Theorem by Higher-Dimensional Riemann Integrals « The Unapologetic Mathematician
- Comment on Riemann Integration by Higher-Dimensional Riemann Integrals « The Unapologetic Mathematician
- Comment on Limits of Topological Spaces by Higher-Dimensional Riemann Integrals « The Unapologetic Mathematician
- Comment on What was that about my head? by Weltfremd » Idiocracy, you’re welcome!?!
I TAed for a Precalc course at Rutgers one semester, and from what I saw there, you need to go pretty heavy on the How. I focused on the Why, and it didn’t work very well…pretty much the students who’d seen the material before aced the class, and the ones that hadn’t failed, and not much in the middle. I’m not sure how much of that was me and how much was the lecturer, but it’s the extent of my experience in the subject of Precalc teaching.
Charles, I’m curious, was it only the why without any of the how? If it was both, which came first?
I can speak only from having tutored many college students in calculus, as well as several high school students in precalculus. You’re absolutely right that precalculus focuses much more on the “how” than the “why”, although the calculus courses at Union also deal very little with the “why”. The other thing I noticed, though, was that the calculus students I encountered almost all struggled very much with the mechanics of algebra, and this was their biggest problem. (They also had no intuition for what the derivatives, integrals, etc. actually mean, but I believe this was largely due to the algebra problems.) On the other hand, most of the high school precalculus students actually didn’t struggle too much with their algebra, and in general were much better at solving the problems they were given than the college calculus students.
I made a conscious effort to treat the precalculus sessions exactly like the calculus sessions, in that I focused as much as I could on why a certain technique worked as opposed to simply drilling the method. I tried to ask questions along the lines of “What kind of pattern do you see here?” and “What do you predict will happen if we change the problem in this way?” It seemed to work pretty well, and I like to think the precalculus students benefited from being made to think more carefully about why things worked.
I was trying to mix the two together. Do an example problem and try to talk the students through why the method works, but that didn’t work so well for me. I’m kind of hoping that most of it was that the students showed up already thinking “I’m going to pass” or “I’m going to fail” and wasn’t just that I suck at teaching, but it is a possibility.
Charles,
While I’ve never been in your class, I doubt it is that you suck at teaching. Students do not think of math as a subject that involves thinking about the why. It takes a long time to change this mindset. From early on they are “drilled” to just answer the question. Those preconceived notions of what math is and what it means to “do” math are hard to change.
As musesusan said, asking the right questions helps. Then the issue becomes learning to wait for their answers. I try not to answer any of my own questions anymore.
I like to give a lesson that takes a step back from Mathematics itself, and philosophize about its relationship to Art, the Humanities, politics, and religion. That gets us away from the question and answer drill.
My challenge is to establish and maintain the proper interconnection
of Instruction, Management, and Assessment, in any course that I teach, regardless of Content Area. This quotation is in the core of my educational philosophy.
“Beauty is truth, truth beauty,” – that is all
Ye know on earth, and all ye need to know.
“Ode on a Grecian Urn”, lines 49-50, John Keats [1795–1821].
Shakespeare, other poets and other literary figures were grappling in their own ways with the Big Questions. Science has developed into an alternative approach. William Blake made his own etchings, by his own invented technology, to illustrate his own quirky take on cosmology and other weighty issues. Blake considered himself radically opposed
to Isaac Newton [4 January 1643 – 31 March 1727], even though both Blake and Newton were influenced by a common metaphysical thinker, Jakob Boehm [1575 — 21 Nov 1624] mystic and theosophist who founded modern theosophy; and influenced others such as George Fox[1575-1624].
“In his “Ode on a Grecian Urn” Keats will say exactly the same thing, more elegantly but more cryptically also: “Beauty is truth, truth beauty”—which some English professors have called “surely the most famous equation in English literature and precisely correct in suggesting the Newtonian origin of the unstated ‘proof.’”
“The urn, in other words, begins by quoting Sir Joshua (for Keats and his readers, the world’s greatest authority on art of all kinds), implicitly affirms the sufficiency of human intellect, explicitly affirms the equation of beauty and truth, and pronounces this knowledge entirely sufficient to create the elegant geometry of such superb art as the urn. Because of the uniformity of human minds and passions, moreover, the figures inscribed on the urn (which puzzle the observer at first glance) become intelligible as we relate them to our own experience. The first stanza of the poem is filled with questions;
the last, with none. Being art, the urn retains its ability to ’speak’ to all who observe it, reminding us of our paradoxical dilemma as mortals who exist in finite time.”
['Some Quotations in Keats's Poetry' by Dennis R. Dean. From the Philological Quarterly. Volume: 76. Issue: 1, 1997.]
Oh yes, we do indeed exist in finite time, yet my life as
Mathematician and artist is deeply connected to Infinity.
So here’s what these lines mean to me as a professional Mathematician, Scientist, Poet, and Teacher.
Everything that I tell my students is the Truth. I tell them so. I want them to respond as complete human beings, aware of beauty and ugliness in the world and within themselves. That gives me a chance, (again, regardless of the content area) to discuss with them what “Beauty” and “Truth” are, to me, and to them.
How do we reach the highest level in Bloom’s Taxonomy, synthesizing and judging the major players in what C. P. Snow famously (and I think incorrectly) identified as “The Two Cultures?” How can I teach both Newton and Keats, and make them part of the same tapestry, to my students?
My mentor’s mentor’s mentor Albert Einstein, when pressed on the subject, would say that he believed in the God of Spinoza, that is, that all matter, energy, time, space existed “in the mind of God.” The phrase “the mind of God” was used by Hawking and others since, to indicate what some Physicists think that they are trying, by mathematico-scientific means, to read from what Galileo called “the Book of Nature.” But in the secular world (including my classroom), how can we find a Humanist framework in which to address and appreciate both art and science?
I have come to believe, from many sources that there are are least 5 kinds of “truth” that each have their own notions of “proof”, of deduction, of evidence, of social protocol.
(1) Axiomatic Truth, the beating heart of pure Mathematics, from Euclid on. Given a set of axioms, and rules of deduction, and two people can sit down together or apart and prove the same truths or disprove the same falsehoods, up to the limits described by Godel, Turing, Church, Post [Emil Post, famous Logician, not a relative], et al. — but that is not the Physical world.
(2) Empirical Truth, from the Scientific methods, and, more recently, from Experimental Mathematics a la Borwein et al. That is, an evolved articulation of trial and error, with open publication and peer review, with a standard of independent verifiability in diverse laboratories.
(3) Legal-political Truth. If a jury declares O.J. Simpson “not guilty” of murder, then he is, by law, not guilty of that criminal charge. Another jury may find him guilty of a civil charge of wrongful death, as did happen. If a politician is elected by a plurality, he or she may claim a mandate from the people, and that is a political truth, regardless of circumstance.
(4) Aesthetic Truth. A song is beautiful or ugly to you regardless of what the composer, singer, or critic says. Same for a painting, a sculpture, a building, or (to a Mathematician), an equation or a sequence of integers. Except that one grows and changes over time, with education and with acculturation and with maturity. What first seems discord can become beautiful. People stormed out of Beethoven symphony premiers, or stormed out of art museums, outraged, and we now wonder why.
(5) Revealed or religious or mystical Truth. My mention of Einstein is about as close as I can come in the classroom to discussing this, though I can reply to quotes from the scripture of several major religions with my quotes from the same sources.
No two of these forms of truth are the same, and much agony comes from the philosophical category error of confusing one with another. Legislating the value of pi to be 3 or 22/7 (as was alleged for the Tennessee Legislature). Outlawing an art form. China enforcing laws about Tibetan reincarnation. Seeking beauty in a test tube, or equations in prayer (unless you’re Ramaujan).
My students in the urban classroom, as I have seen hundreds of times in the past year alone, within essentially all the middle schools and high schools of Pasadena Unified School District, have a cramped, unhappy, inconsistent, and ignorant conception of “Truth” in all its complexity.
(1) Axiomatic Truth, and the rest of Mathematics, is something that see as if underwater and though a cloud of squid ink. Almost all of them hate Math. Yet I have been able to open the eyes and hearts and minds of many students in secondary and post-secondary education, and make Math intelligible to them, often for the first time in their
lives.
(2) Empirical Truth, from the Scientific method, is known poorly to my students, who have had second-rate Science Teachers (who themselves do not, by my standards, know Science). Again, I have been able to give inspiration from glimpses of the splendor of the scientific world, and
elicted child-like delight in my students in subjects with which they have inherent interest, such as dinosaurs, earthquakes, sunlight, explosions, microbes, bird flight, and flower colors.
(3) Legal-political Truth is familiar, again in a degraded form, to many of my urban students. Many have been arrested, many have been jailed, many are on probation, many have lives blighted by gangs, drugs, divorce, crime, violence, and death. It is important in the classroom that these students here my mantra: “I am not a cop; I am not a snitch; I am not a rat; I am not your boss.” It is important that I do not act as a Judge; they tend to have a negative view of judges and the judicial system. Nor do they, not yet of voting age, have a record of participation in the democratic process, and know little more than sound bytes on TV about local, state, national, or international politics. They often do not see this arena as one dominated by truth, but, rather, a cesspool of lies.
(4) Aesthetic Truth is real to all my students, and very inadequately addressed by schooling. The are very aware of color and style in clothing, tattoos, cars, make-up, hairdo, and find the black-and-white word of the printed page and the Xeroxed handout to be as drab and inhumane as the colors of the floor and wall of the disintegrating
classroom. Teachers castigate the music that they listen to, confiscate their iPods and radios, and leave them in angry silence. They are usually stunned that I can defend the lyrics of Eminem’s rap songs, and that I have performed Rock music onstage, and wrote lyrics that were heard on MTV. I fight to bring more beauty into their lies, and validate their Aesthetic Truth, while leading them to a more sophisticated context.
(5) Revealed or religious or mystical Truth. We are required by law to respect the diversity of individual beliefs in this domain, when teaching (as I do now, in public schools), and to respect the “separation of church and state” in specified ways.
In conclusion, of this meandering preface, the classroom is, to me and my students, a human microcosm in which Truth and Beauty are our shared source of value.
Dude, you are an intelligent man with an Ivy League PhD. No reason to take a one-year position at the University of Western Kentucky and teach high school math. Gotta be more assertive, man!
Be that as it may, it was this or nothing. In fact, I just landed this gig at the last minute. I was about to take a non-academic job, in fact.
I like your blog,
I’m basically a math groupie. Congratulations on your new gig, I’ll think you will be a fine instructor and your students will be lucky to have you for their teacher!
Conversely from Norbert, no matter how qualified and educated you are, you are lucky to have a job in the field you were trained in. Now go make history!
Testing for algebra
As California debates college-prep math testing, educators must remember to put learning first.
July 9, 2008
Los Angeles Times, Editorial page
http://www.latimes.com/news/opinion/la-ed-algebra9-2008jul09,0,3505640.story
California has made great strides toward teaching algebra in the eighth grade. Six years ago, fewer than a third of eighth-graders took the course that’s considered the linchpin to college-prep mathematics. Now, more than half do.
But the state is caught between the rigid mandates of the No Child Left Behind Act and its own lofty academic ambitions. The federal act requires the state’s proficiency exams to test whatever the standard is, and the state’sstandard is for all eight-graders to take algebra. What to do about the 48% who take lower-level math?
The answer to be considered by the state Board of Education this week is to give those students a new, tougher test with a sprinkling of algebra questions, while algebra students would continue to take the full algebra test. That makes the feds happy. That makes the state Department of Education happy. The only problem is that this is unsound education.
Though we support standardized testing as a way to measure the progress of schools, testing students on material they haven’t learned is the educational tail wagging the dog. Teachers will throw a few simple algebra concepts into a curriculum in which they make no sense in the hopes of a better score; it’s the proverbial “teaching to the test” in its worst form.
The state already has inducements in place to prod schools toward eighth-grade algebra, which has led to the progress so far. But the eighth-grade algebra standard is not a requirement, and though there’s a movement afoot to make it one, that would be a mistake. Math demands progressively sophisticated skills. Students have to master Step 1 before they can successfully attempt Step 2, and the public schools have long allowed students to move on to the next step while they’re still shaky on the previous one. That’s why algebra remains the single biggest obstacle to high school graduation.
The problem begins long before middle school; in fact, one of the major factors is failure to master multiplication tables. The state needs to think out its curriculum before it starts testing students on it.
California has adopted materials for an algebra-readiness course for middle schools, but it will be years before the curriculum is fully in place. Let’s not lose sight of the real goal here, which is to ensure that students learn math — not just take it, or pass it, but actually learn it. Better for lagging students to be prepared properly for algebra in ninth grade than for them to take it early, only to fail, and fail again.
Do you know any specifics about your students? I’ve taught classes like these, and there was a dramatic difference between classes where people were seeing the material for the first time, and people who were taking the class for the nth time. For the first group, it’s like a normal class — if you were effective at teaching Calculus I, you can teach this the same way (just going slower). For the second group, you have to overcome their built-up hostility towards the subject.
Walt, I don’t even know what I’ll be teaching, and I just talked to my chair on Tuesday!
Be that as it may, it was this or nothing. In fact, I just landed this gig at the last minute. I was about to take a non-academic job, in fact.
I thought you were an assistant professor at Tulane!??
Past tense. “Was”. They didn’t pick me up again for another year, preferring to keep the two visiting assistant professors who hadn’t applied for more than a handful of positions.
Math plan doesn’t add up
Article Launched: 07/10/2008 07:31:34 PM PDT
http://www.pasadenastarnews.com/ci_9844739
GOV. Arnold Schwarzenegger is no doubt right on the money, as he so often is, by this week calling algebra “the key that unlocks the world of science, innovation, engineering and technology.”
Its fundamental arithmetical methods, first unlocked by ancient Arab scholars – algebra means “the mathematics” in Arabic – form the building blocks for understanding everything in higher math beyond mere addition and subtraction, multiplication and division – and by introducing concepts such as place-holding, allows us to understand those simpler concepts in much greater depth.
But if 2 is the average student, and it plus x equals 4 – well, for some very young teens still struggling to add large groups of numbers, or to fully understand long division, sometimes the seemingly fairly clear abstraction that is x is just a little bit hard to grasp.
Still, California public schools, with all their educational problems, should be proud to have the largest percentage of eighth-grade students in the nation who are already taking algebra.
But taking such a class doesn’t always end up in full comprehension – right now, just 52 percent of those middle-schoolers taking algebra are able to pass a standardized algebra test indicating a high level of algebraic understanding. But even having just over half of the students passing indicates we’re doing something right.
It was just a couple of generations ago that very few California students took algebra
Advertisement
until the ninth grade, long considered the standard age for average students being ready to comprehend the formulas that form the subject’s basis.
Truth be told, statistically 48 percent of those eighth graders are not quite ready today. But it’s nice to push students a bit, and keeping at that current push is likely to pay off over the years.
Still, when you’re in a bit of a hole already, the first thing you do is to stop digging. Instead, for almost incomprehensible reasons, the ordinarily quite sensible governor this week proposed at the last minute, and with the support of his hand-picked state Board of Education, that all California eighth graders be required to take algebra and be subjected to a standardized test on it within three years – all! And then proposed that they all be mandated to take that standardized test to prove full comprehension.
If the program were to go through, just imagine how that 52 percent passage rate is going to plunge. Why, a 30 percent rate would be a miracle.
The governor acknowledges that it would be tough, and that it would take investment of significant educational monies – “billions,” he says – to raise the passage rate.
But one of the reasons he and his board are using in justifying such loony new standards is that if we don’t set them, California could lose a paltry $4.1 million in federal funding on Aug. 1. Spread out over the largest state, that means virtually nothing.
So where are the billions he admits would have to be invested to achieve algebraic success supposed to come from?
There is already a significant shortfall in the number of qualified math and science teachers available to teach the subjects in California schools. This artificial timetable for all eighth-grade students to toe the mark simply sets an unachievable goal. Realistically, such a lofty program – certainly worthwhile over time – is the equivalent of committing the country to landing on the moon. When wise leaders do that, they give a reasonable time frame for their ambitions – “within the decade” sounds about right.
The governor and the board worry that not setting this standard for all would set up a two-tiered system in which some students speed ahead into the wild blue yonder of calculus and trigonometry while others are left behind in the Earth-bound realms of general math.
We say, what precisely is the problem with sometimes having two-tiered systems? It’s exactly akin to the state’s for the most part having dropped vocational training in its schools. Not everyone is going on to graduate school in string theory – and that’s as it should be. To pretend otherwise is a pipe dream.
Superintendent of Instruction Jack O’Connell said of the plan: “It is quite distressing that the governor would forward a proposal that would have significant impact on thousands and thousands of children with literally less than 24 hours notice so as to guarantee those affected most – teachers, students, and parents – would have virtually no opportunity to engage in the discussion. I strongly disagree with the governor’s proposal to require all eighth graders to take algebra within three years without also offering any of the support for our school districts and schools to successfully make this major change.
“Our system simply has more work to do to put in place the necessary tools to ensure every child is ready to participate and succeed in algebra.”
That’s the kind of straight talk that adds up. We strongly encourage the governor to back off from this rash plan, one that could disastrously backfire by significantly adding to California’s already high drop-out rate.
Okay, JVP. I understand that you care deeply about public school mathematics education in California. However, that’s drifting further and further from the actual subject of the thread, which is pedagogical theory for college mathematics courses below the calculus level.
You’re right, John. Here’s the nugget of pedagogy.
(1) Before Calculus comes, Algebra (often divided into chunks such as Pre-Algebra, Algebra 1, Algebra 2), Geometry, Trigonometry, and “Pre-Calculus.” Probability, Statistics, and related Data-Analysis is squeezed in there someplace. There may be a Computer class, more about practical use of application software than about theory. There may be some Linear Algebra stuffed into one or more of the Algebra classes, on the use of matrices and determinants for solving systems of simultaneous linear equations. But there is no real coherence in the courses, whatever Standards there are at the State level that defines the goal on Instruction.
(2) Algebra may well be “the key that unlocks the world of science, innovation, engineering and technology.” However, as currently taught in many states, little or none of that unlocking is done in the classroom. Unless the student is lucky enough to have a Physics class to use the Algebra and Trigonometry (there are fewer High School Physics teachers than there are High Schools), or a Quantitative Chemistry class or a Biotechnology class where data analysis is done seriously, the connection with real world application is little more than lip-service
(3) Hence the college teacher of Math below Calculus must make up for the incoherence, lack of motivation, and disconnect from the professions of Science, Engineering, and Technology.
(4) If the student is on a B.B.A. to M.B.A. business sequence, they have some statistics and data analysis, but lacking theory. They may have had Economics, but it was qualitative, and surely without Calculus. They know about Google as a corporation, but not what the algorithmic basis of web search is.
(5) If they are majoring in Chemistry or Biology, they will know that there are atomic orbitals for the electrons, and that has something to do with molecules and spectra, but they will not have had that mathematically. They will not know the Quantum Mechanics that won Pauling his first Nobel Prize. They may have seen the exponential in radioactive decay, but not the basis of magic numbers in nuclear physics, or the diffusion equation for neutrons in a fission device. They may know about the 64 codons in RNA or DNA, but not understand the basis of codes and coding, let alone the basis of error-detecting and error-correcting codes.
(6) They may have had computer classes to promote “computer literacy” but not the basics of Numerical Analysis. They will never have seen a slide rule, and so are vague about logarithms. They do not have a Theory of Algorithms, know who Godel or Turing was. Hence they do not — even if they took a “Finite Math” class — know what Computing is at the abstract level.
(7) They know nothing of the History of Mathematics, except for some little sidebars in a textbook with the bare bones of of data under a photo of of a famous woman mathematicians, or a dead white European whose name they don’t pronounce. They have never had any Philosophy of Mathematics. They may have seen some Propositional Logic in a Philosophy or class, or on a Debate team, but no clue that Predicate Calculus exists, or Fuzzy Logic.
So what are you to do, John Armstrong? How can you “teach to the standards” when those standards them selves are disconnected, incoherent. ahistorical, non-biographical, and have not used color animations, online search, or actual programming? Will you have them read some of the excellent fiction about Mathematics? See movies such as “Flatland” or “A Beautiful Mind”? Will you be invited to meet with Astronomy, Biology, Chemistry, Engineering, and Economics faculty, Chairpersons, Deans, to bring coherence and mathematical rigor? Will you coerce, seduce, and elbow your way into such discussions? Or will you be floundering in the vacuum at the heart of a failed education system, without the access or ability to improve the context?
You inherit the whirlwind. I admire you in many ways, and await the insights that shall come from grappling with your attempt to illuminate students who have been stumbling in darkness. This is the most important job in the world. Whatever you do, there will be students whom you shall awaken from slumber, who shall succeed in their professions and, more importantly, become part of the Enlightenment dialogues on the 3 big questions:
* What is the universe, and how does it work?
* What is a Human being?
* What is the role of a Human Being in that Universe.
I believe that Mathematics is at the heart of that discourse, but not in isolation.
Is it really a pre-calculus class, or is it just an algebra class?
I remember when I took pre-calculus (in my first semester, back when I was a mere computer science major), we actually got to limits (the eps-delta definition, too) and derivatives by the last quarter of the semester. Granted, I was at UMCP, and they do have an excellent math program.
Jon, I’m not sure how to refer to all those courses. I decided to write “pre-calculus” for everything that comes before the calculus level, and “precalculus” for the mixed-bag course that tries to give all the tools that will then be used in the calculus.
Will you be blogging about the new courses? I would like to read about your experiences (and insights) in teaching these “pre-calculus” and/or “precalculus” classes.
I might every so often, but I don’t tend to talk much about my teaching. I usually stick to the exposition.
I’m still trying to puzzle out how Pre-Calculus fits into the grand scheme myself, but from my two years of teaching it I have found the main thing that needs to be added is a rigorous sense of functions. In prior classes they can get away with an ad hoc understanding, but in Pre-Calculus they need to get it solid.
As far as the “why” and “how” distinction goes, unfortunately it isn’t as simple as picking one and running with it. There are *specific* things you can go into detail on “why”, and certain things you must still hold back. The only approach I know is to get one’s hands messy and analyze each chapter individually for what the students are capable of.
What textbook are you using?
Jason, I still don’t know exactly what I’m teaching. It might be something even below “precalculus”.
One other thing I’d add from personal experience. (I understand better now that you don’t know what courses you’ll actually be teaching, but I’ll assume for the time being that we’re talking about a precalculus course that comes right before calculus, which usually covers things like functions, exponents and logarithms, conic sections, limits, and maybe some basic derivatives.)
Many of the students I work with in calculus classes are genuinely afraid and baffled in a way that they aren’t in any earlier class (except maybe algebra), because they’re suddenly being asked to use and understand an entirely new set of tools and a new way of thinking (the way of thinking boils down to “Hey, let’s break something into small pieces and then add them up!” but they don’t really understand that). Part of the problem is that they don’t understand how to use the tools, but the other aspect is that they don’t even understand that these ARE tools, and very flexible ones.
I think precalculus is an excellent opportunity to expose students to all these ideas, and ease the transition. The sooner you can get them to make up their own method for solving a problem, the better they will understand that mathematics is almost never about plugging the numbers into a pre-made formula. If you emphasize figuring out how to solve new problems with the techniques at hand, rather than simply practicing the algorithm for solving problems of each type, I think the students will come out of the class much better equipped to tackle the kind of open-ended problems they may encounter in calculus, and will certainly encounter in the real world.
The deep purpose of Mathematics is enlightenment, including self-enlightenment. Your course can — as suggested clearly by musesusan — teach students how rigorously to break ideas into smaller ideas (analysis) and build new ideas from old ones (synthesis) and from themselves (recursion).
Mathematics is a way of handling mental or social constructs whether or not they correspond to physical reality, for interesting and infinite sets of constructs and operations on constructs, which are consistent and persistent.
They COULD have been taught Algebra and Geometry that way, but almost always were not.
This will benefit them in every discipline. And, once the light goes on in each and every head, it will give them legitimate self-esteem, for having broken through the wall of abstraction at last.
[...] taught this way instead of the older one. And again the proponents of this style (rehashing the why vs. how discussion) come off as arrogant. Their concerns aren’t for the parents’ ability to [...]
Pingback by New Math on its way back? « The Unapologetic Mathematician | July 19, 2008 |
Here’s a very clear and readable article which summarizes the role of the Mathematicians as (poorly) seen be his Dean, professors in other departments, to resentful people at cocktail parties. He explains the role of the mathematician in a university as incomprehensible to most members of the community, except as a teacher, and then clarifies the rewards of that teaching. Too many quotable lines for this posting. I note that one of the people acknowledged is Jonathan Borwein, the biggest name in “Experimental Mathematics.” Good summary of how to tell nonmathematicians — including students and the Press — what the field is about, by presentations with Music or Art.
New submissions for Fri, 18 Jul 08
http://arxiv.org/pdf/0807.2656
Title: Through a Glass Darkly
Authors: Steven G. Krantz
Subjects: History and Overview (math.HO)
We consider the question of how mathematicians view themselves and how non-mathematicians view us. What is our role in society? Is it effective? Is it rewarding? How could it be improved? This paper will be part of a forthcoming volume on this circle of questions.
During my time at MIT, I did a fair amount of note-comparing with other students about the strengths and weaknesses of our high-school educations. Mostly, we talked about weaknesses, because that was more fun (the amount of total BS we collectively snuck past our English teachers far exceeds Alan Sokal’s boundary transgression output). Judging from these conversations, the class known as “pre-calculus” in Alabama and Florida is just a repeat of the second year of algebra (“Algebra II/Trigonometry”) with a couple extra worksheets on limits thrown in. One could skip it and jump straight into calculus without disadvantage.
The curriculum track for high-achievers ran as follows:
Algebra I (eighth grade)
Geometry (ninth grade, taken in eighth by those who pass some kind of placement test and skip ahead)
Algebra II/Trigonometry
Pre-Calculus (Algebra II/Trig redux)
AP Calculus
Those who skipped ahead and took geometry early and consequently ran out of math classes would enroll in AP Computer Science their senior year, or in less frequent cases, take differential equations at the local university.
Hey! Thanx for this beautiful place of the Inet!!