” ‘Use logic in math’ is my motto.” — Sophia S (October 31, 2010)
That, and persistent, hard work, are why she gets mostly As. I love that quote.
July 25, 2011
July 12, 2011
The Importance of Mathematics: An Example
The Lake Peigneur salt dome drilling disaster, which you can read about on Wikipedia and see on YouTube (and on EducatedEarth.com), demonstrates the importance and truth of the dictum “measure twice before you cut once.”
June 20, 2011
Advice For Writing — and Math. And All Other Subjects.
In “Can You Drop It on Your Foot?” (National Association of Scholars, June 02, 2011), John Maguire says:
Student papers are often unreadable not only because their grammar is bad, and sentences incomplete, but also because they are way, way too abstract. Abstractions really trap students. Assigned to write about some idea, students get caught in the sphere of abstract words and stay there. Abstract words multiply on the page in unpleasant clusters. If you ask freshmen to write about, say, The relationship between wealth and productivity in a market society, watch out. Few will notice that the four abstract terms relationship, wealth, productivity and market society need definition or examples. They will just move those abstract terms around like checkers on a board, repeating them, and hoping through repetition that something will be said. The resulting paper will be mush.
…
An alternate approach might be to start the course with physical objects, training students to write with objects, and to understand that every abstract idea summarizes a set of physical facts. I do in fact take that approach. “If you are writing about markets, recognize that market is an abstract idea, and find a bunch of objects that relate to it,” I say. “Give me concrete nouns. Show me a wooden roadside stand with corn and green peppers on it, if you want. Show me a supermarket displaying six kinds of oranges under halogen lights. Show me a stock exchange floor where bids are shouted and answered.”
…
Students led into writing this way at the start of a course…find it strange at first, but…they will learn after six or eight weeks of practice that it does work; about that time they start to smile because their thinking on paper is clearer, they can see what they are talking about, and their writing has become vivid.© 2011 National Association of Scholars. All rights reserved.
All subjects, from math to physics to chemistry to history to philosophy, should be taught in the same way. Concepts are classifications of real, individual things, and if a student does not know what real things the concepts refer to, he does not know what he is thinking, writing, or talking about; the students mind is then being destroyed and divorced from reality. Not healthy!! Don’t do it, please! In education, as in all else, please be objective: follow a human method (logic) in knowing the facts.
Mr. Maguire’s advice is good advice for the SAT and ACT, too.
June 17, 2011
Khan Academy
Some people seem to be sold on the videos of Mr. Khan of Khan Academy. However, in “tutoring” mathematics, Mr. Khan does nothing but deliver the exact same methodology as the texts and as current public education. He is a video version of paper texts, showing, in essentials and fundamentals, nothing original whatsoever. He basically reads from textbooks.
If Mr. Khan comes to grasp his errors, states them publicly, and starts promoting a rational, objective pedagogy, curriculum, and philosophy, then I will give him kudos. Until then, he is part of the problem, not part of the solution, and I recommend against him and his work.
In “The Dangerous Mr. Khan” (National Association of Scholars, June 07, 2011), David Clemens attacks also Mr. Khan’s teaching of history. Mr. Clemens says at the beginning of his article:
So is the Khan video approach a “disruptive technology” which undermines the existing deathbed educational model by doing it faster, better, and cheaper? Mr. Gates thinks so. “It’s a revolution,” he enthuses. “Everyone should check it out.” (www.khanacademy.org) Wearing his education reformer hat, Mr. Gates declares himself “superhappy.”
© 2011 National Association of Scholars. All rights reserved.
Mr. Gates is fundamentally wrong; he certainly knows computer software and business, but he does not know education or philosophy. Khan might do it faster and cheaper, but he does it in the same way, not better. He is in no way different in fundamentals or philosophy from, as Mr. Clemens puts it, “the existing deathbed educational model.” Hence there is no “undermining” or “revolution,” and certainly nothing to be “superhappy” about here; there is merely conformity and agreement, and reason to worry.
Mr. Clemens goes on to say:
Mr. Khan observes that “from FDR’s point of view, Hitler definitely was in the wrong here.” This observation is so odd, that I have to hit the pause button and take a moment to think about it. In Mr. Khan’s History, whether Hitler should have invaded Poland or not is just a matter of viewpoint, wrong in FDR’s (and probably Poland’s) but okey-dokey in Hitler’s. Everything is a matter of viewpoint, perspective, and cultural positioning, therefore nothing is essentially right or wrong, to be applauded or condemned. Here Mr. Khan stands exposed as possessing a historical perspective steeped in academia’s standard issue, postmodern, left-leaning narrative of cultural relativism, multiculturalism, and moral equivalence.
© 2011 National Association of Scholars. All rights reserved.
This is a major criticism. If true (I have not seen Mr. Khan’s history videos), then Mr. Khan is teaching skepticism: the denial of certainty, principle, and timeless truth. No thanks. Skepticism is a major philosophic error, as are moral and cultural relativism, all of which lead to personal and cultural frustration and disaster. And if true, this pretty much trashes Mr. Khan’s videos; they would be of no objective value whatsoever.
Mr. Clemens also wrote:
May 6, 2011
Not All That Glitters Is Gold
In “More Argument, Fewer Standards” (Education Week, April 19, 2011), Mike Schmoker and Gerald Graff start off seemingly hot, saying:
If we want record numbers of students to succeed in postsecondary studies and careers, an ancient, accessible concept needs to be restored to its rightful place at the center of schooling: argument. In its various forms, it includes the ability to analyze and assess our facts and evidence, support our solutions, and defend our interpretations and recommendations with clarity and precision in every subject area. Argument is the primary skill essential to our success as citizens, students, and workers.
Sounds good, right? They call for teaching students to use “evidence” and “clarity” and “precision.” Can’t argue with that! Sounds perfect!
But let’s see what else they say. What is their context? What is their underlying epistemology? What is their view of logic and of concepts?
In the meantime, let’s immediately begin, as the new standards urge us, to give students hundreds of opportunities, every year, to dismantle and defend arguments about increasingly rich, complex texts. From the earliest grades, let’s have them argue about the pros and cons of almost anything: literary characters and interpretations, global warming, capitalism vs. socialism, Sarah Palin, or the comparative quality of life in the United States and Canada (based on statistical analysis). Let’s ask students to explain their reasoning for which alternative-energy source we should invest in as they read, talk, and write about what they are learning in novels, textbooks, newspapers, and magazines.
© 2011 Editorial Projects in Education
And there we have what they are really after: divorcing concepts and minds from reality. Students, the authors say, should be encouraged to argue about things about which they know nothing or about which they are cognitively not ready for. The hierarchy of knowledge should be ignored and violated — but, like a building, if you take away the lower floors, the building cannot stand; like a tree, if you take away the roots and grounds, the tree cannot survive, live, and flourish.
Divorce argument from objectivity, hierarchy, context, and the evidence of the senses, and you have a Platonic hash.
April 12, 2011
How Not To Teach Math
No wonder Dan Meyer is a heralded educator: he’s a pragmatist who would have our kids’ minds be crammed into and restricted to the immediate moment. Would he have us look at the universe, look at broad abstractions and large contexts, a la the Greeks and Galileo? No! Heck no! He says to look at the local basketball court — where you can find an application of math no one cares about and no one uses. Such examples are easily dismissed as artificial, since no one ever uses algebra and parabolas to shoot a basket — first-hand experience in the class room tells me so. And such examples are as narrow as the book examples Mr. Meyer is trying to oppose. And such examples do not naturally lead to further thought and investigation.
He responds to Platonism by giving us Pragmatism — but they are two sides of the same false coin. We need a rational approach, a la Aristotle, Archimedes, Newton, Galileo and Gauss.
December 22, 2009
A Great Book on Philosophy
A History of Philosophy by Wilhelm Windelband, available for viewing or download on Google books.
HT: Andy C.
October 28, 2009
Theoretical Statistics Is Practical and Life-Giving
In “The Median Isn’t the Message,” Stephen Jay Gould (evolutionary biologist who taught at Harvard University) wrote:
My life has recently intersected, in a most personal way, two of Mark Twain’s famous quips. One I shall defer to the end of this essay. The other (sometimes attributed to Disraeli), identifies three species of mendacity, each worse than the one before – lies, damned lies, and statistics.
Many people make an unfortunate and invalid separation between heart and mind, or feeling and intellect. In some contemporary traditions, abetted by attitudes stereotypically centered on Southern California, feelings are exalted as more “real” and the only proper basis for action – if it feels good, do it – while intellect gets short shrift as a hang-up of outmoded elitism. Statistics, in this absurd dichotomy, often become the symbol of the enemy. As Hilaire Belloc wrote, “Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death.”
This is a personal story of statistics, properly interpreted, as profoundly nurturant and life-giving. It declares holy war on the downgrading of intellect by telling a small story about the utility of dry, academic knowledge about science. Heart and head are focal points of one body, one personality.
Mr. Gould also goes on to discuss how the Platonic view that the type or kind is (most) real is false; what is true is the Aristotelian view that the individual (“variation”) is real. He says:
We still carry the historical baggage of a Platonic heritage that seeks sharp essences and definite boundaries. (Thus we hope to find an unambiguous “beginning of life” or “definition of death,” although nature often comes to us as irreducible continua.) This Platonic heritage, with its emphasis in clear distinctions and separated immutable entities, leads us to view statistical measures of central tendency wrongly, indeed opposite to the appropriate interpretation in our actual world of variation, shadings, and continua. In short, we view means and medians as the hard “realities,” and the variation that permits their calculation as a set of transient and imperfect measurements of this hidden essence. If the median is the reality and variation around the median just a device for its calculation, the “I will probably be dead in eight months” may pass as a reasonable interpretation.
But all evolutionary biologists know that variation itself is nature’s only irreducible essence. Variation is the hard reality, not a set of imperfect measures for a central tendency. Means and medians are the abstractions.
Notice how Mr. Gould is talking about kinds of things as being separate from variation, shadings, and continua. I don’t know if he’d say everything was like that, even individuals, but if so, I’d have to disagree: individuals are distinct and separate; this is given clearly (by real, immutable cause-effect relationships) in perception. Kinds of things, conceptual categories, come about only by recognizing things in their reality- and perceptually-given background of variation: tables grasped as related to but contrasted with furniture and other items in a house; trees grasped as related to but contrasted with grass and bushes; people grasped as related to but contrasted with other animals; engineers grasped as related to but contrasted with other human professions.
Concepts are only ways of categorizing individuals based on cause-effect and explanatory relationships. Individuals are most real; types or kinds are real, but have a “secondary, dependent existence” to individuals.
September 30, 2009
Poetry of Parmenides
The Stanford Encyclopedia of Philosophy has some fragments of the (philosophic) poetry of Parmenides, with philosophic commentary. They say about Parmenides:
Parmenides of Elea, active in the earlier part of the 5th c. BCE., authored a difficult metaphysical poem that has earned him a reputation as early Greek philosophy’s most profound and challenging thinker. His philosophical stance has typically been understood as at once extremely paradoxical and yet crucial for the broader development of Greek natural philosophy and metaphysics. He has been seen as a metaphysical monist (of one stripe or another) who so challenged the naïve cosmological theories of his predecessors that his major successors among the Presocratics were all driven to develop more sophisticated physical theories in response to his arguments.
Copyright © 2008 by John Palmer
© Metaphysics Research Lab, CSLI, Stanford University
Here are some lines of his poetry:
And the goddess received me kindly, and in her hand she took/ my right hand, and she spoke and addressed me thus:/ “O young man, accompanied by immortal charioteers/ [25] and mares who bear you as you arrive at our abode,/ welcome, since a fate by no means ill sent you ahead to travel/ this way (for surely it is far from the track of humans),/ but Right and Justice did.” (Fr. 1.1-28a)
You must needs learn all things,/ both the unshaken heart of well-rounded reality/ [30] and the notions of mortals, in which there is no genuine conviction./ Nonetheless these things too will you learn, how what they resolved/ had actually to be, all through all pervading. (Fr. 1.28b-32)
Come now, I shall tell—and convey home the tale once you have heard—/just which ways of inquiry alone there are for thinking:/ the one, that [it] is and that [it] is not not to be,/ is the path of conviction, for it attends upon true reality,/ [5] but the other, that [it] is not and that [it] must not be,/ this, I tell you, is a path wholly without report:/ for neither could you apprehend what is not, for it is not to be accomplished,/ nor could you indicate it. (Fr. 2)
It is necessary to say and to think that What Is is; for it is to be,/ but nothing it is not. These things I bid you ponder./ For I shall begin for you from this first way of inquiry,/ then yet again from that along which mortals who know nothing/ [5] wander two-headed: for haplessness in their/ breasts directs wandering thought. They are borne along/ deaf and blind at once, bedazzled, undiscriminating hordes,/ who have supposed that it is and is not the same/ and not the same; but the path of all these turns back on itself. (Fr. 6, supplementing the lacuna at the end of fr. 6.3 with arxô and taking s’ earlier in the line as an elision of soi, as per Nehamas 1981, 103-5; cf. the similar proposal at Cordero 1984, ch. 3, expanding parts of Cordero 1979.)
August 26, 2009
Our Educational System is Broken: An Anecdote
In “Arne Duncan, Among the Scientists” (Curriculum Matters Blog on Education Week; August 25, 2009 12:21 PM) Sean Cavanagh writes:
[Secretary of Education Arne Duncan] also stressed the potential for the money to help states and schools recruit new teachers in math and science. Today, too many students are taught by educators who don’t know the content in those subjects, the secretary noted. He made another pitch for differential pay for math and science teachers, as well as for teachers in other high-need subjects, possibly spec-ed and foreign languages. He also mentioned the importance of increasing access to AP programs, and singled out a teacher-training program, the University of Texas’ “UTeach,” for helping produce the “next generation of great leadership” in schools. Interestingly, those remarks came on the same day that the Dallas-based organization that’s seeking to replicate the UTeach model and expand AP access said that its participating schools have seen a major increase in AP passing scores.
…
Overall, American schools need to churn out students with better math and science skills, said Duncan, who, as he has previously, cited mediocre U.S. scores on international tests as a source of worry.“We’ve become complacent,” he told the audience. “We’ve sort of lost our way. This is huge challenge for us.”
© 2009 Editorial Projects in Education
And they are looking to be complacent in the future. Sean Cavanagh, in “Grounded in Content” (Published Online: December 4, 2007, Published in Print: December 5, 2007) gives us an illustration of the fact that UTeach seems to be the same old John Dewey-influenced mess:
