You are here: Home / Archives for problem solving

Alfred North Whitehead on “inert ideas”

May 22, 2011 by

One of the most remarkable essays I have read on education is “The Aims of Education” by Alfred North Whitehead. First published in 1917, it calls some of our current “wars” into question, particularly the apparent battles between progressives and traditionalists. When Whitehead argues against the danger of “inert ideas,” he seems both progressive and traditional at once.

Whitehead (1861-1947) was a mathematician and philosopher. He co-authored the Principia Mathematica with Bertrand Russell. He is the founder (to some degree) of “process philosophy,” which he explains in Process and Reality: An Essay in Cosmology.

Already, I am bristling, because the very idea of “process philosophy” sounds like so much nonsense. But when Whitehead says something, he makes you think–in a way that differs from what you might expect. His points don’t fall in the usual classifications.

The second paragraph of “The Aims of Education” reads:

In training a child to activity of thought, above all things we must beware of what I will call “inert ideas”–that is to say, ideas that are merely received into the mind without being utilised, or tested, or thrown into fresh combinations.

Now, this is interesting, because such “inert ideas” could consist of disjointed facts and big, vague concepts. In other words, schools that emphasize isolated bits of information and schools that emphasize ungrounded “critical thinking and problem-solving” are committing a similar error. They are giving students material out of context. As commenters on Michael’s most recent post have suggested, it is the motion of a topic that makes it interesting and memorable. Daniel T. Willingham has made similar points in his book, Why Don’t Students Like School?

But am I reading things into Whitehead? Not at all; here’s more:

Furthermore, we should not endeavour to use propositions in isolation. Emphatically I do not mean, a neat little set of experiments to illustrate Proposition I and then the proof of Proposition I, a neat little set of experiments to illustrate Proposition II and then the proof of Proposition II, and so on to the end of the book. Nothing could be more boring. Interrelated truths are utilised en bloc, and the various propositions are employed in any order, and with any reiteration. Choose some important applications of your theoretical subject; and study them concurrently with the systematic theoretical exposition. … Also the theory should not be muddled up with the practice. The child should have no doubt when it is proving and when it is utilising. My point is that what is proved should be utilised, and that what is utilised should–so far, as is practicable–be proved. I am far from asserting that proof and utilisation are the same thing.

Very interesting. So there should be “theoretical exposition,” short and thorough, alongside (and clearly distinct from) practical application. The theory should be presented in a systematic manner, but “interrelated truths” should be utilized “en bloc.”

In none of this can the details of the subject or the hard work of practice be avoided:

All practical teachers know that education is a patient process of the mastery of details, minute by minute, hour by hour, day by day. There is no royal road to learning through an airy path of brilliant generalisations. There is a proverb about the difficulty of seeing the wood because of the trees. That difficulty is exactly the point which I am enforcing. The problem of education is to make the pupil see the wood by means of the trees.

But what of the aims of education? What are they? Whitehead writes:

What education has to impart is an intimate sense for the power of ideas, for the beauty of ideas, and for the structure of ideas, together with a particular body of knowledge which has peculiar reference to the life of the being possessing it.

Here’s where things get a little shaky for me. What does he mean by “peculiar reference”? Does he mean that studies should be of personal relevance to each student? Or does he mean that a subject taught in motion is a subject made relevant–that the very motion, the procession from one idea to another, consitutes the relevance, as it helps us see where a particular idea comes from and where it is going? I believe he means the latter. He continues:

The appreciation of the structure of ideas is that side of a cultured mind which can only grow under the influence of a special study. I mean that eye for the whole chess-board, for the bearing of one set of ideas on another. Nothing but a special study can give any appreciation for the exact formulation of general ideas, for their relations when formulated, for their service in the comprehension of life. A mind so disciplined should be
both more abstract and more concrete. It has been trained in the comprehension of abstract thought and in the analysis of facts.

There is much more to the essay than I am conveying here. What’s tantalizing is that some of his ideas are so good and can be misunderstood so easily. They resemble, at first glance, some of the education jargon out there (regarding the “joy of discovery,” for instance) but mean something quite different. One need not agree with all of his points, but they raise the possibility that there is something beyond the oppositions familiar to us today.

I bring up Whitehead in my forthcoming book, Republic of Noise: The Loss of Solitude in Schools and Culture. I am grateful to the mathematician who brought Whitehead’s essay to my attention.

Filed Under: Education Tagged With: , , , ,

Put argument at the core

April 24, 2011 by

Common Core Standards need More Argument, Fewer Standards, argue Mike Schmoker and Gerald Graff in Ed Week.

Argument . . .  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.

Many educators don’t realize the importance of argument or the research showing that students learn more — and earn higher test scores — when they have “in-school opportunities to argue and debate about current issues, literary characters, and the pros and cons of a math solution.”

Argument not only makes subject matter more interesting; it also dramatically increases our ability to retain, retrieve, apply, and synthesize knowledge. It works for all students—from lowest- to highest-achieving.

The new standards affirm the importance of argument, but ask teachers to do too many other things too, they argue.  “For all their merits, these standards are still overlong, redundant, and often confusing.”

All standards are not created equal. We believe it is far more critical for teachers to help students to analyze, evaluate, and support their conclusions with evidence than it is for them to spend precious time on puzzling standards like these:

“Compare and contrast the structure of two or more texts and analyze how the differing structure of each text contributes to its meaning and style”; or

“Analyze different points of view of the characters and the audience or reader (e.g., created through the use of dramatic irony) creating such effects as suspense or humor.”

In addition, there are too many “foundations” standards with “long lists of mechanical skills well into the later grades,” they argue.

When I was in high school, we didn’t write journals, much less design posters. It was all expository writing all the time. Make an argument. Support it.  The dread 3-3-3 paragraph consisted of a thesis statement supported by three topic sentences, each supported by three subtopic sentences, each supported by three “concrete and specific” details. I never used the 3-3-3 in college. I didn’t need to.

Filed Under: Education Tagged With: , , ,

Real math

April 12, 2011 by

Students learn by solving real math problems, argues Dan Meyer on a wildly popular TED video, Math Class Needs a Makeover.  An algebra teacher and dy/dan blogger, Meyer is now working on a doctorate in curriculum design.

Rejected as a film student, Meyer tells Ed Week about the “narrative arc” of a real-world math problem. Intead of “shark terrorizes seaside town,” it might be “how long will it take me to get to Los Angeles?”

During what he calls the “second act” of a film, the characters encounter obstacles and find out what they need to do. In a math problem, the second act involves measuring, determining a formula, or finding out what information is missing.

The third act brings the exciting conclusion — with potential for a sequel.

Textbooks label the variables, present the measurements  and ask leading questions in an attempt to help students, Meyer says.  That can overwhelm students.

He starts with the hook: The final question.

For example, when teaching high schoolers, Meyer uses the digital projector to display a photo of himself shooting a basketball. Meyer has doctored the photo so that it shows the ball at several different points along the trajectory, stopping at the apex. “When I put that up on the board, the premise of that problem is obvious to every student. I don’t even have to say it. ‘Will the ball go in?’ That’s what we’re all wondering,” he says.

Then Meyer asks the students to figure out what information they need to determine whether his shot will go in. The students discover they have to measure the arc and need a protractor to do so—in a way writing their own second act. A textbook would have provided this information, Meyer says. But in the real world, “When on earth do you get all the information you need before you know you need it?”

The students can then solve the problem on their own.

Then they watch the video to see if they’re right.

Meyer believes in “delegating the sense-making of math to students.”

In my day, people were always rowing against the current, which seemed like a waste of energy. Or they were trying to calculate when trains going opposite directions would pass, instead of reading the train schedule.

 Get the Math, an educational reality TV show produced by WNET in New York, shows the real-world applicatons of algebra, reports Ed Week.

The single-episode program, as well as the companion website, features three short video segments designed to provide an introduction to teen-favored industries—music recording, fashion design, and video game development. . . . the professionals featured in each video offer examples of how they use mathematical knowledge as part of their creative processes.

Then comes the “challenge.” At the end of each segment, the pro gives a pair of two-student teams a specific industry-related algebraic problem to solve. The videos show the teams working through the problems and then presenting their solutions. The idea, of course, is that other students can play along in their classrooms.

The program, lesson plans and classroom activites are available at no charge at www.getthemath.org.

Filed Under: Education Tagged With: , , , ,

Teaching creativity

July 14, 2010 by

While IQ scores rise over time, creativity scores are declining in the U.S., write Po Bronson and Ashley Merryman in Newsweek. It’s not clear why, though Bronson and Merryman think passive TV watching and video game playing may be crowding out creative play.

Other nations are trying to encourage students to think creatively and solve problems, while U.S. schools often concentrate on teaching basic skills.  Creativity is seen as something that happens in art class. Here’s where the article got interesting for me:

The age-old belief that the arts have a special claim to creativity is unfounded. When scholars gave creativity tasks to both engineering majors and music majors, their scores laid down on an identical spectrum, with the same high averages and standard deviations. Inside their brains, the same thing was happening — ideas were being generated and evaluated on the fly.

. . . Creativity isn’t about freedom from concrete facts. Rather, fact-finding and deep research are vital stages in the creative process. Scholars argue that current curriculum standards can still be met, if taught in a different way.

Problem solving requires using both sides of the brain, switching rapidly between convergent to divergent thinking, Bronson and Merryman write. The solver considers known facts and strategies, then scans “remote memories that could be vaguely relevant,” searching for  “unseen patterns, alternative meanings, and high-level abstractions.”  The brain locks on to a possible answer — aha! — then evaluates whether it’s worth pursing.

(Yesterday, my husband, an electrical engineer who holds many patents, told me his advice to a friend who’s working for an inventor with a divergent idea. “Try to impress the investors with your competence so they’ll recommend you for a job when this fails.”)

Creativity training helps students learn to solve problems, say researchers at the University of Oklahoma, the University of Georgia, and Taiwan’s National Chengchi University.

The National Inventors Hall of Fame School, a new public middle school in Akron that admits students by lottery, teaches problem solving as part of its STEM mission. Fifth graders were given four weeks to design proposals for reducing noise in the library, which has windows looking out on a public space.

Working in small teams, the fifth graders first engaged in what creativity theorist Donald Treffinger describes as fact-finding. How does sound travel through materials? What materials reduce noise the most? Then, problem-finding — anticipating all potential pitfalls so their designs are more likely to work. Next, idea-finding: generate as many ideas as possible. Drapes, plants, or large kites hung from the ceiling would all baffle sound. Or, instead of reducing the sound, maybe mask it by playing the sound of a gentle waterfall? A proposal for double-paned glass evolved into an idea to fill the space between panes with water. Next, solution-finding: which ideas were the most effective, cheapest, and aesthetically pleasing? Fiberglass absorbed sound the best but wouldn’t be safe. Would an aquarium with fish be easier than water-filled panes?

Then teams developed a plan of action. They built scale models and chose fabric samples. They realized they’d need to persuade a janitor to care for the plants and fish during vacation. Teams persuaded others to support them — sometimes so well, teams decided to combine projects. Finally, they presented designs to teachers, parents, and Jim West, inventor of the electric microphone.

Teachers had designed the project to meet Ohio’s curriculum standards. That was reflected in the school’s first-year test scores, which placed it third among Akron schools.

Sixth-grader Brandon Smith’s Hamster Cleaner 3000 made the finals of a local TV stations’ Coolest Creations contest, after competing at the Invention Convention at the Cleveland Great Lakes Science Center.

Filed Under: Children, Education Tagged With: , , , ,

Learning to speak data

May 3, 2010 by

Statistics is the new grammar, writes Clive Thompson in the May issue of Wired. The statistically illiterate can’t understand public policy debates, which increasingly come down to what the data mean. Is the economy improving? Do childhood vaccines increase the risk of autism? Is global warming for real?  Is the latest political poll reliable?

Statistics should now be a core part of general education. You shouldn’t finish high school without understanding it reasonably well — as well, say, as you can compose an essay.

Schools teach probability — red, blue and yellow marbles in a bag — and “a bit of basic data analysis,” responds Mr. D of I Want to Teach Forever. But math teachers often gloss over “problem solving, finding reasonable answers and determining what data is needed to solve a problem.”

Aside from problem solving skills, we don’t spend enough time on proportional thinking (everything from using percents to measurement and scale) and just plain number sense that everyone could use on a daily basis. What we’re left with is a nation of people who fear math, who run to a calculator for the most rudimentary problems.

Some people live the data-driven life, writes Gary Wolf in the New York Times Magazine.

Filed Under: Education Tagged With: , , , , ,
Newer Posts»