Science Is Not About Facts, But About Reasoning

Researchers at Ohio State University appear astounded to discover that teaching students "science facts" does not appear to help students with scientific reasoning.

(PhysOrg.com) -- A study of college freshmen in the United States and in China found that Chinese students know more science facts than their American counterparts -- but both groups are nearly identical when it comes to their ability to do scientific reasoning.

Neither group is especially skilled at reasoning, however, and the study suggests that educators must go beyond teaching science facts if they hope to boost students' reasoning ability.

Researchers tested nearly 6,000 students majoring in science and engineering at seven universities -- four in the United States and three in China. Chinese students greatly outperformed American students on factual knowledge of physics -- averaging 90 percent on one test, versus the American students' 50 percent, for example.

But in a test of science reasoning, both groups averaged around 75 percent -- not a very high score, especially for students hoping to major in science or engineering.

The research appears in the January 30, 2009 issue of the journal Science.

Lei Bao, associate professor of physics at Ohio State University and lead author of the study, said that the finding defies conventional wisdom, which holds that teaching science facts will improve students' reasoning ability. _PO

Reasoning without facts is fanciful, and facts without reasoning are dead. Education seems to flit from one approach to the other, without understanding that children need exposure to both -- without political indoctrination (eg "climate catastrophe"), and respecting the critical developmental periods of the brain.

Educational methods pass through fads, lacking a deep understanding of the nature of human learning. One current fad is the use of computers in place of books or dialogue. There may be problems with this fad.

"By using more visual media, students will process information better," she said. "However, most visual media are real-time media that do not allow time for reflection, analysis or imagination — those do not get developed by real-time media such as television or video games. Technology is not a panacea in education, because of the skills that are being lost.

"Studies show that reading develops imagination, induction, reflection and critical thinking, as well as vocabulary," Greenfield said. "Reading for pleasure is the key to developing these skills. Students today have more visual literacy and less print literacy. Many students do not read for pleasure and have not for decades." _PO

Many teachers want students to use the computer during school time, and to have parents make the children read at home. Parents may have difficulty separating the children from their video games, cell phones, texting, messaging, MTV, videos, etc. long enough to get them to read, however. With both parents working, there may be little actual "home time" at all.

Reading, thinking, and thoughtful, informed dialogue help to teach children reasoning. Computer games can teach multi-tasking and eye-hand coordination. But the deeper reasoning that higher level modern life requires seems to fall between the cracks.

It was once thought that the $100 "laptop for every child" was the answer to educating the entire world's children. In India, they even have the $10 laptop for every child. Indian officials have high hopes for the device, naturally.

Learning to reason with objective facts -- forming hypotheses, then finding ingenious and elegant ways to test them -- is at the heart of scientific reasoning, which is what science is about.

Modern education seems to be about something entirely different. Call it a cross between indoctrination (academic lobotomy), expensive baby-sitting, and peer group socialisation into psychological neoteny. Not the best way to prepare new generations of problem solvers. More like the programming of brain-dead consumers and lifelong helpless adolescents.

Even the National Science Foundation Couldn't Care Less About Real Science

American culture and American schools are not preparing American students for a world of science and technology. Even the National Science Foundation is more concerned about what is between the legs of scientists than what their heads are producing. The 21st century will not be kind to cultures with such misplaced priorities.

In the United States, students are insulated from the commercial market's demand for their knowledge and skills. That market lies a long way off — often too far to see. But they are not insulated one bit from the worldview promoted by their teachers, textbooks, and entertainment. From those sources, students pick up attitudes, motivations, and a lively sense of what life is about. School has always been as much about learning the ropes as it is about learning the rotes. We do, however, have some new ropes, and they aren't very science-friendly. Rather, they lead students who look upon the difficulties of pursuing science to ask, "Why bother?"

Success in the sciences unquestionably takes a lot of hard work, sustained over many years. Students usually have to catch the science bug in grade school and stick with it to develop the competencies in math and the mastery of complex theories they need to progress up the ladder. Those who succeed at the level where they can eventually pursue graduate degrees must have not only abundant intellectual talent but also a powerful interest in sticking to a long course of cumulative study. A century ago, Max Weber wrote of "Science as a Vocation," and, indeed, students need to feel something like a calling for science to surmount the numerous obstacles on the way to an advanced degree.

At least on the emotional level, contemporary American education sides with the obstacles. It begins by treating children as psychologically fragile beings who will fail to learn — and worse, fail to develop as "whole persons" — if not constantly praised. The self-esteem movement may have its merits, but preparing students for arduous intellectual ascents aren't among them. What the movement most commonly yields is a surfeit of college freshmen who "feel good" about themselves for no discernible reason and who grossly overrate their meager attainments.

The intellectual lassitude we breed in students, their unearned and inflated self-confidence, undercuts both the self-discipline and the intellectual modesty that is needed for the apprentice years in the sciences. Modesty? Yes, for while talented scientists are often proud of their talent and accomplishments, they universally subscribe to the humbling need to prove themselves against the most-unyielding standards of inquiry. That willingness to play by nature's rules runs in contrast to the make-it-up-as-you-go-along insouciance that characterizes so many variants of postmodernism and that flatters itself as being a higher form of pragmatism.

The aversion to long-term and deeply committed study of science among American students also stems from other cultural imperatives. We rank the manufacture of "self-esteem" above hard-won achievement, but we also have immersed a generation in wall-to-wall promotion of diversity and multiculturalism as being the worthiest form of educational endeavor; we have foregrounded the redistributional dreams of "social justice" over heroic aspirations to discover, invent, and thereby create new wealth; and we have endlessly extolled the virtue of "sustainability" against the ravages of "progress." Do all that, and you create an educational system that is essentially hostile to advanced achievement in the sciences and technology. Moreover, those threads have a certainty and unity that make them not just a collection of educational conceits but also part of a compelling worldview.

The antiscience agenda is visible as early as kindergarten, with its infantile versions of the diversity agenda and its early budding of self-esteem lessons. But it complicates and propagates all the way up through grade school and high school. In college it often drops the mask of diffuse benevolence and hardens into a fascination with "identity."

That could be a good thing if the introspections were enriched by professors who could show students where Plato or Shakespeare had touched such depths, or who could startle them by showing where Hobbes or Tocqueville had seen them coming. But in a curriculum dissolved in the sea of minutiae and professorial enthusiasms, the opportunity to pass through moody introspection and back into the sturdy world of real people grows rare.

The science "problems" we now ask students to think about aren't really science problems at all. Instead we have the National Science Foundation vexed about the need for more women and minorities in the sciences. President Lawrence H. Summers was pushed out of Harvard University for speculating (in league with a great deal of neurological evidence) that innate difference might have something to do with the disparity in numbers of men and women at the highest levels of those fields. In 2006 the National Academy of Sciences issued a report, "Beyond Bias and Barriers: Fulfilling the Potential of Women in Academic Science and Engineering." Officials of the National Science Foundation and the Department of Education are looking to use Title IX to force science graduate programs to admit more women. The big problem? As of 2001, 80 percent of engineering degrees and 72 percent of computer-science degrees have gone to men.

A society that worries itself about which chromosomes scientists have isn't a society that takes science education seriously. In 1900 the mathematician David Hilbert famously drew up a list of 23 unsolved problems in mathematics; 18 have now been solved. Hilbert has also bequeathed us a way of thinking about mathematics and the sciences as a to-do list of intellectual challenges. Notably, Hilbert didn't write down problem No. 24: "Make sure half the preceding 23 problems are solved by female mathematicians."

Obsession with the sex and race of scientists is just one more indication of how American higher education has swung into orbit around the neutron star of identity politics. _Commentary

It should no longer be such a mystery how science can be so easily corrupted and perverted to pimp political movements such as climate catastrophe. The deeper spirit of science as the pursuit of insight into the deep and dynamic state of the universe has been bred out of students along the way. Some of these new breeds are now tenured professors, and will be training grad students to see science as social construct like themselves.

Basic Concepts in Science: Lists

Evolving Thoughts blog is helping to maintain a list of postings that explain basic concepts in science. It is an ambitious and worthy project, and I congratulate Science Blogs and Evolving Thoughts on the ongoing achievement.

Here is a similar list of posts and applets explaining basic concepts in modern physics.

This list of basics science concepts is geared toward lesson plans for adolescent students and younger.

This list from Cocktail Party Physics explains some basic physics concepts.

If any readers know of similar lists, feel free to post links in comments, and I will add them here.

Update: Sensible Energy suggested HyperPhysics, a great collection of short physics tutorials on basic physics concepts.

Education in Finland: Can High Finnish Test Scores be Replicated in Other Countries?

What Makes Finnish Kids So Smart?, reads a Wall Street Journal Headline. It seems that test scores for Finnish children are among the highest of all advanced countries, and a lot of educators from North America and elsewhere want to know "why?"

...by one international measure, Finnish teenagers are among the smartest in the world. They earned some of the top scores by 15-year-old students who were tested in 57 countries. American teens finished among the world's C students even as U.S. educators piled on more homework, standards and rules....In the most recent test, which focused on science, Finland's students placed first in science and near the top in math and reading, according to results released late last year. An unofficial tally of Finland's combined scores puts it in first place overall, says Andreas Schleicher, who directs the OECD's test, known as the Programme for International Student Assessment, or PISA.

...The academic prowess of Finland's students has lured educators from more than 50 countries in recent years to learn the country's secret, including an official from the U.S. Department of Education. What they find is simple but not easy: well-trained teachers and responsible children. Early on, kids do a lot without adults hovering. And teachers create lessons to fit their students.

...The Norssi School is run like a teaching hospital, with about 800 teacher trainees each year. Graduate students work with kids while instructors evaluate from the sidelines. Teachers must hold master's degrees, and the profession is highly competitive: More than 40 people may apply for a single job. Their salaries are similar to those of U.S. teachers, but they generally have more freedom.

Finnish teachers pick books and customize lessons as they shape students to national standards. "In most countries, education feels like a car factory. In Finland, the teachers are the entrepreneurs," says Mr. Schleicher, of the Paris-based OECD, which began the international student test in 2000.

...Finland separates students for the last three years of high school based on grades; 53% go to high school and the rest enter vocational school. (All 15-year-old students took the PISA test.) Finland has a high-school dropout rate of about 4% -- or 10% at vocational schools -- compared with roughly 25% in the U.S., according to their respective education departments.

...Once school starts, the Finns are more self-reliant. While some U.S. parents fuss over accompanying their children to and from school, and arrange every play date and outing, young Finns do much more on their own. At the Ymmersta School in a nearby Helsinki suburb, some first-grade students trudge to school through a stand of evergreens in near darkness. At lunch, they pick out their own meals, which all schools give free, and carry the trays to lunch tables. There is no Internet filter in the school library. They can walk in their socks during class, but at home even the very young are expected to lace up their own skates or put on their own skis.___WSJ

It seems that teachers in Finland are much better trained and are given more freedom to teach to their students' needs and interests. Teaching is seen as a desirable job in Sweden, and the enthusiasm of teachers helps keep students interested.

Although Finnish people do not have higher IQs than other Europeans--and it is not likely the children are born with higher Executive Function or short-term memories--the nature of child up-bringing, and early education likely contributes to a more independent mind-set, and a tendency to understand the need to follow the guidance of parents and teachers in preparing for the future. Life in Finland can be hard, due to extremes of climate and insolation. All of these things are likely to help Finnish children learn more self-reliance than more coddled children of lower Europe and North America.

Psychological neoteny is a very real problem for North American children and children in many other first world countries. Children are too often treated as fragile, pampered parental trophies, rather than as human beings who have to learn to deal for themselves with a real and often dangerous world.

H/T to Dennis Mangan and Steve Sailer , who have provided interesting comments on the WSJ article, and its topic. Check out their take on the article.

Europe, UK: Brain Drain in Science, Medicine

Europe and the UK are suffering from a growing stampede of educated professionals, scientists, and technical specialists out of the region.

Record numbers of Britons are leaving - many of them doctors, teachers and engineers - in the biggest exodus for almost 50 years.

Skilled professionals, including doctors, are leaving the UK in record numbers Over a quarter of qualified professionals who have moved abroad had health or education qualifications

There are now 3.247 million British-born people living abroad, of whom more than 1.1 million are highly-skilled university graduates, say the researchers.

More than three quarters of these professionals have settled abroad for more than 10 years, according to the study by the Organisation of Economic Co-operation and Development (OECD).

No other nation is losing so many qualified people, it points out. Britain has now lost more than one in 10 of its most skilled citizens, while overall only Mexico has had more people emigrate.

The figures, based on official records from more than 220 countries, will alarm Gordon Brown as tens of thousands of pounds of taxpayers' money is spent on educating graduates. The cost of training a junior doctor, for example, is £250,000.

The most popular destinations are English-speaking countries such as Australia, America, Canada and New Zealand and holiday areas including France and Spain.

Almost 60 per cent of those leaving take jobs, although hundreds of thousands of retired people live abroad.___Telegraph

Feedback on this issue from Telegraph readers can be found here. Many informed blog comments on the same article can be found here. North Americans may find it difficult to understand why so many Europeans are so eager to leave. Particularly leftist North Americans, who so badly want to reproduce Western Europe within the borders of North America.

Unfortunately, there are many good reasons for Europeans and Britons to leave their home countries, as the comments linked to above will demonstrate. I see two overwhelming themes dominating the reasons behind this brain drain.

• 1. Out of control immigration into Europe by uneducated, undisciplined, intolerant newcomers, leading to higher violent crime and property crime rates.
• 2. Excessive taxation and regulation of the productive classes to benefit the non-productive, quasi-parasitic classes (including many of the new immigrants).

The trend will only accelerate with time and continued immigration of violent and religiously intolerant, unassimilable newcomers. When you lose residents at the high end of education and income, and gain residents at the lowest end of education and income--in fact gaining criminals and welfare dependents--then the long-term prospects look grim. It is no wonder that European women are reproducing at only half the replacement levels.Bonus image from Noodle Food demonstrates the flow of emigrant MDs between the four countries: the UK, the US, Canada, and Australia. It is important to realise that although the flow of MDs goes both ways between each pair of countries, there is a distinct lopsidedness there. The desirability rankings, based upon the emigration:immigration ratio, go like this:

1. US
2. Canada
3. Australia
4. UK

The MDs are mostly leaving the UK, and eventually mostly ending up in the US. It would be helpful to have similar information regarding other medical professionals, scientists, teachers, engineers, IT professionals, etc. who are emigrating from the UK and Europe.

Arab World Plans Large Science Infrastructure: What Are The Odds?

the Arab world has stagnated. Per capita income in Arab countries grew at an annual rate of just 0.5% during the last quarter century - less than half the global average. Despite being blessed with massive quantities of "black gold," Arabs have seen their average standard of living decline relative to the rest of the world. The combined GDP of all Arab countries ($531.2 billion) is today less than that of Spain (a country that Arabs once ruled).

SourceAccording to the best studies by intelligence researchers, the mean IQ for the arab world is near 85--exactly that of the african-american population in the US. While the Arab world reaps many billions of dollars yearly from oil and gas revenues, the scientific and educational levels of the Arab world are dismally low. Is it theoretically likely that a larger monetary investment in a Science Education/Research infrastructure could raise scientific achievement in the arab world up to western levels?

Earlier this year, the 22 nations of the Arab League approved a 10-year plan to boost scientific research. It calls for member states to raise their allocation to science twelvefold to 2.5 percent of GDP—more than the average 2.3 percent spent by developed nations.

...Sheikh Mohammed bin Rashid Al-Maktoum of the United Arab Emirates recently launched a new pan-Arab foundation with a monumental endowment of $10 billion—one of the largest charitable donations in history. The foundation's stated mission is to "develop world-class knowledge" in the Arab region, and many are hoping it will foster broad-based scientific research.

...With a $1.5 billion annual allocation to science in a country with a population of less than a million, Qatar is intent on reform. Education City is Qatar's new university system—a 2,500-acre campus that is home to branches of five of the world's top universities, including Cornell and Carnegie Mellon. The Qatar Science and Technology Park (QSTP) has enticed foreign labs and international companies by offering top-notch research facilities. The country is bringing in foreign expertise to achieve a long-term vision—to make Qatar a knowledge-based society. "QSTP is a 20-year program," says director Eulian Roberts, "but we're working hard now so that we can achieve a change in culture, a change in mentality."

Source

Oman and Saudi Arabia plan to join Qatar and the Emirates in their aggressive thrust to build large new scientific infrastructures for education and research. We know that the Arab world is proficient at "bringing in foreign expertise." That is how the oil and gas fields were developed, how the modern urban infrastructures were constructed and maintained, how the entire Arab civilisation keeps from falling apart. But the key question that any knowledgeable person is forced to ask in connection with this new putsch for Arab science is: Where will they get all the promising young math, physics, chemistry, biology students?

Good science, math, and engineering students at the university level do not spring up from thin air. They come from good programs at lower levels of education. They come from families that typically encourage curious young minds to explore. Where will they find this type of family, this type of K-12 education and top notch undergrad training? In the muslim world--particularly the arab muslim world--curiosity is too often beaten out of young minds, and too many questions are forbidden to children and youth. Women are seen as second-rate minds and third-rate citizens, which eliminates half of youth intelligent enough to pursue a scientific career. Much of science conflicts with rigid Islamic teaching. Where will the religious police be during all of this buildup?

The chart above comparing a population with mean IQ of 85 with a population with a mean IQ of 100 (SD 15) indicates the relative portions of the two populations with enough intelligence for the different careers. While this type of chart has its limitations, it is useful as a broad guide.

Is this type of promotional thrust yet another example of "cargo cult science and education?" I suspect so. "If you build it, they will come . . ." If you build the huge and expensive universities and science/engineering labs, the students and professors will come, the researchers will come, the international regard for homegrown science and technology will come . . . or will it?

While the faculty and researchers for these new institutions can be imported from abroad, the students will have to come from home turf--if the program is to have any meaning at all. And once you do train world class Arab youth in science and technology, how do you keep them from emigrating to Europe and the Anglosphere? That is always a perennial problem for the third world.

An Interesting Approach to Teaching Science to Children

North American children are lagging in science behind children in other parts of the developed world. A lot of the blame for that belongs with the standard government school approach to teaching science--memorizing facts and formulas.

Imagine a school where children learned scientific concepts hierarchically, in a logical progression much as they were originally discovered? What if science classes consisted almost entirely of experiments?

One day, I taught my students the principle that water pressure increases only with depth. I gave them a powerful demonstration by poking holes at the same depth in two vessels of dramatically different diameters, and observing identical jets of water coming out of the holes. They were shocked and fascinated, and when one student’s mother came to pick him up, he immediately went to the board and started drawing diagrams and testing her about this principle to see if she understood it as well as he did. Such enthusiasm springs from a first-hand grasp of relevant principles, which can be achieved only by means of a hierarchy-driven curriculum.

The principle of hierarchy is just as crucial in teaching more abstract scientific knowledge to older children as it is in teaching the simplest scientific knowledge to younger children. Consider the subject of physics.

Most science teachers present the highly abstract laws of physics as if they are self-contained truths, unrelated to the long history of scientific development. For example, Newton’s discovery of universal gravitation, one of the most extraordinary discoveries in the history of thought, is usually presented as an out-of-context commandment to be memorized—as knowledge that, along with Newton’s apple, fell from the sky.

A proper science teacher, by contrast, recognizes what the students must know for this law to be intelligible. He explains the steps in Newton’s reasoning, and ensures that the students have already learned the discoveries leading up to Newton’s theory, the principles they must know if they are to follow his reasoning.

In the famous incident with the apple, Newton asked himself if the same attractive force from the Earth caused both the apple’s descent and the moon’s orbit. In order to check the idea, Newton needed to know the acceleration of the apple (which he learned from Galileo), the size of the Earth (which had been measured by Eratosthenes), and the distance to the moon (which was calculated by Aristarchus). If the students are to grasp the law at hand, they must first grasp these facts—as did Newton.

Further, in arriving at this hypothesis, Newton was relying on Galileo’s principle of inertia, Kepler’s laws of planetary motion, and the law of circular acceleration (which Newton himself had discovered a few months earlier). Without this knowledge, Newton could not even have raised the question. Therefore, without this knowledge, the students cannot grasp the question and they certainly cannot understand Newton’s final answer.

Having been taught physics as it progressed historically, the students at VanDamme Academy know the discoveries of Aristarchus, Eratosthenes, Kepler, and Galileo. When guided through the ingenious process by which Newton integrated this knowledge and built upon it, the students thoroughly grasp the principle of universal gravitation: They see that it is true and why it must be true. The law of gravitation is, in their own minds, connected to reality. It is real knowledge.

Does the hierarchical approach to teaching science require that students be taught the entire history of science, including every detail of every experiment ever performed? No. A crucial part of teaching in accordance with the principle of hierarchy is to select only the essentials. This is in contrast to the common view, expressed in a local newspaper by a high school biology teacher, that the hardest part of his job is keeping up with all the latest discoveries in his field. The latest developments in biology are properly the concern of Ph.D. biologists who have the context to understand them and the need to apply them.

High school students should be taught a carefully selected list of the most essential discoveries in the field, and should be taught them in hierarchical order. Only if they are taught by this method will they emerge with a sound understanding of the fundamental concepts of science and a genuine ability to think. Anything else deprives them of independently grasped, real knowledge, in favor of passively accepted pseudo-knowledge.

Source.

For more about the VanDamme Academy--a modified Montessori approach--go here.

Children need to learn science and technology experimentally. Educational methods need to exploit the strengths of the child's mind and motivation, but too often education is designed to undermine a child's confidence and prevent the acquisition of competence.

Featured Online Portals to Learning

Continuing with the online learning theme from a few days ago, today's posting features two of the best collections of links to learning materials on the web.

First is Merlot.org. Merlot links to Arts(507 sites), Business(2417), Education(2012), Humanities(2282), Mathematics and Statistics(1151), Science and Technology(5807), and Social Sciences(987) websites. Some of the sites linked to might qualify as portals in themselves, others are java animations suitable for teaching small children. The site has to be explored in order to understand the wide variety of approaches to learning, from pre-school to post-graduate level. Highly recommended.

Then there is 101 Science, a well stuffed and eclectic website with a large set of links to many large sets of links to . . . . and so on. The theme is science and technology. Check out these meta-portals to biology and mathematics. There are many more. While you stand a good chance of finding what you are looking for at 101 Science, you will almost certainly find much more.

Both of these featured sites are well worth visiting. I also recommend the many reference links in the sidebar of the Al Fin main page. One of the main purposes of this website is to encourage readers to prepare themselves for more demanding times. Being well trained in only one field is not sufficient. The underpinnings of once stable societies are beginning to loosen and the fabric starting to unravel. Once trustworthy institutions have become worthless. Many things have to happen for the future to arrive, some of them a bit frightening. It would be better for readers and their loved ones not to be caught unaware.

Budgeting more time to learning, and learning how to learn, has gone from a luxury to a necessity.

How People Learn: Can Schools Be Made Safe and Effective?

The National Academies of Sciences Press has made available online free of charge two books relevant to this topic. How People Learn: Brain, Mind, Experience, and School: Expanded Edition (2000) from the Commission on Behavioral and Social Sciences and Education, and How Students Learn: History, Mathematics, and Science in the Classroom (2005) from the Board on Behavioral, Cognitive, and Sensory Sciences and Education. Both books can be read online or printed for non-electronic reading.

If you have read the recent essays from Derek Bok on the state of university education, you will not be comforted to learn that K-12 education is worse--much worse. Education is resistant to reform, is almost impossible to improve, because education is an industry--a heavily politicised industry. A lot of money is at stake for vested interests, who strongly resist change out of political and financial motivations. The children and students are the helpless pawns whose lives are injured by the largely indifferent industry of education. Colleges of education at universities are in the pockets of this industry, serving the industry's interests rather than that of the population of students.

The most trivial allergy pill goes through far more scrupulous testing than the teaching methods used on helpless children in schools. There is a lot of new knowledge from cognitive science on how children learn. This knowledge is not being implemented in schools, from K-12 through universities. The knowledge is being ignored by the education industry in hopes it will go away, and leave the status quo undisturbed.

One of the reasons this is so important is the phenomenon of massive unassimilated immigration. The dropout rates for these masses of unassimilated immigrants is huge--approaching 60% and more in many areas. Their mothers and fathers are happy to be making more in their new country, but the failing, dropout children will more likely be radicalised by class envy, given a popular culture that promotes the idea of conspicuous affluence as the measure of personal worth.

If ever there was a time for adopting education methods that work, this is the time. There is no time to waste. No more dumbing down can be tolerated.

Math: More Getting What You Need

Most of you will probably agree that government schools--and schools in general--are shortchanging our children, particularly in the areas of math, science, and in world view perspectives. For now, we will concentrate more on math. As a follow-on to my previous post, I wanted to explain that reading wikipedia-math for 15-30 minutes daily is not just for adults who want to brush up and round off their math knowledge. It might be particularly helpful for those young brains that are in the middle of the pruning process. A good parent will lead his children to such resources, and help the child to utilise them, while the children are still young enough to have faith in the parents' judgment.

Here is a good essay on "how to read mathematics." It is meant to aid in reading math journal articles and texts, but it can also be adapted to the wikipedia and mathworld approach.

A reading protocol is a set of strategies that a reader must use in order to benefit fully from reading the text. Poetry calls for a different set of strategies than fiction, and fiction a different set than non-fiction. It would be ridiculous to read fiction and ask oneself what is the author's source for the assertion that the hero is blond and tanned; it would be wrong to read non-fiction and not ask such a question. This reading protocol extends to a viewing or listening protocol in art and music. Indeed, much of the introductory course material in literature, music and art is spent teaching these protocols.

Mathematics has a reading protocol all its own, and just as we teach students to read literature, we should teach them to read mathematics.This article categorizes some of the strategies for a mathematics reading protocol. I am sure my readers will think of many strategies that I missed. The point is that there *is* such a protocol, that we all know and use it, and that we should make an attempt to share the secret with our students.

....."Reading Mathematics is not at all a linear experience ...Understanding the text requires cross references, scanning, pausing and revisiting" (ibid page 16).

Don't assume that understanding each phrase, will enable you to understand the whole idea. This is like trying to see a painting by staring at each square inch of it from the distance of your nose. You will get the detail, texture and style but miss the picture completely. A math article has a story! Try to see what the story is before you delve into the details. You can go in for a closer look once you have a framework to fill with details, just as you might reread a novel.

....Mathematics says a lot with a little. The reader must participate! At every stage, he must decide whether or not the idea presented was clear. Why is it true? Do I really believe it? Could I convince someone else that it is true? Why didn't the author use a different argument? Do I have a better argument or method of explaining the idea? Why didn't the author explain it the way that I understand it? Is my way wrong? Do I really get the idea? Am I missing some subtlety? Did this author miss a subtlety? If I still can't understand the point, perhaps I can understand a similar but simpler idea? Which simpler idea? Is it really necessary to understand the idea? Perhaps I will just accept this point without understanding the details? Perhaps, my understanding of the whole story will not suffer from this?

Putting too little effort into this participation, is like reading a novel without concentrating. After half an hour, you wake up to realize the pages have turned, but you have been day dreaming and don't remember a thing you read.

....Reading mathematics too quickly, results in frustration. A half hour of concentration in a novel buys you 20-60 pages with full comprehension (depending on how experienced you are at reading novels). The same half hour in a math article buys you 0-3 lines (depending on how experienced you are at reading mathematics). There is no substitute for work and time. You can speed up your math reading skill by practicing, but be careful. Like any skill, trying too much too fast can set you back and kill your motivation. Imagine trying to do an hour of high energy aerobics if you have not worked out in two years. You may make it through the first class, but you are not likely to come back. The frustration from seeing the experienced class members effortlessly do twice as much as you, while you moan the whole next day from soreness, is too much to take.

And so on. Scan through the essay as you can, then apply the approach to your next 15 minute math session. Persistence plus good technique. It takes time to learn the right technique for you--the best approach--but persistence is there for everyone, for just a little willpower.

You might also want to check out something called Visual Math. Math can be very beautiful, visually and esthetically. Here are some links to visual geometry.

Finally, here is a math reference website, to add to all the others in my previous posting, and the ones on the sidebar.

Combining the information in this posting and the previous one, you may be in a better postion to compensate for any deficits in your own math education, and to prevent large gaps from forming in your children's math educations. Math is not all they will need to know, but it will be a key part.

Do It Yourself and Self Teaching

Here is a useful set of links to do it yourself webpages of several types. Enjoy thesedo it yourself science links for kids from a number of different sites, courtesy of The Science Club.org.

Here is some information on self teaching resources. These are several links to homeschooling organisations in the US and around the world, including Canada. This is an excellent source for Canadian homeschooling information.

Online Great Books, freely available for the price of a mouse click.

As a special added bonus, a one-time only link to a never-ending list of eclectic science links. Keep scrolling. Just when you think there is nothing more, another big group of surprising links appears.