Geometry without proofs

These days, high school geometry is light on proofs, writes Barry Garelick on Education News. Students may know the sum of the measures of angles in a triangle equals 180 degrees, but most can’t prove the proposition.

If done right, the study of geometry offers students a first-rate and very accessible introduction to the nature and techniques of logical argument and proof which is central to the spirit of mathematics itself.  As such, a proof-based geometry course offers to students—for the first time—an idea of what mathematics means to mathematicians, and how it is used.  Also, unlike algebra and pre-calculus, since geometry deals with shapes, it is easier for students to visualize what it is that must be proven, as opposed to more abstract concepts in algebra.

Most geometry textbooks give students “one or two proofs that are not very challenging in a set of problems devoted to the application of theorems rather than the proving of propositions,” he writes. Many problems indicate missing angles or segments as algebraic expressions. It is, to quote Mr. Spock, “illogical.”

Teach algebra via programming?

Schools can teach mathematical reasoning through software programming rather than conventional algebra classes,writes Julia Steiny on Education News.

In the 1980′s, when Providence, Rhode Island tried College Board’s Equity 2000, she served on the school board. “Business” and “consumer” math were eliminated in favor of algebra for all. The goal was to get everyone through geometry and advanced algebra. Providence assigned all sixth graders to pre-algebra.

The smart kids zipped through quickly, doing algebra in seventh, geometry in eighth and advanced algebra in ninth grade. Teachers created many levels of slower-paced classes for weaker students.

“In time, Equity 2000 got many more urban kids into college,” but it only helped “kids for whom low expectations were the only real problem,” Steiny writes. It will take “new approaches to lure students into the puzzles of mathematical reasoning.”

My now-grown sons, two of whom became software developers, have been arguing since high school that learning computer software programming is essentially learning algebra, only infinitely more fun, interesting, and useful.

Seymour Papert, author of Mindstorms, created Logo to enable young children to explore mathematical ideas.

Why students can’t write

From trivium to triviality: Students can’t write because they don’t learn grammar or logic, an instructor argues.

Also, all students need a liberal education, even those training for careers, writes a community college president on Community College Spotlight.

The failure of U.S. higher education

Higher education is failing almost as badly as K-12, writes Robert D. Atkinson, president of the Information Technology and Innovation Foundation.

Over the years, he’s interviewed many recent college graduates for jobs at this think tank.

Because the quality of so many of the graduates was so poor, ITIF has taken to giving the small share of the most promising applicants (based on their resumes and cover letters) a short test that we email them to complete at home in one hour. The questions are pretty simple: “Go to this person’s bio online and write a three- or four-sentence version of their bio for us to include in a conference packet,” or, “Enter these eight items in a spreadsheet and tell us the average for the ones that end in an odd number.”

. . . In our current hiring process (for an office manager/research assistant) we have so far given the test to approximately 20 college grads. Only one did well enough to merit an interview.

Most of the 19 were graduated from  top-ranked institutions. A recent Princeton graduate  “submitted a test that was full of spelling and grammar mistakes.”

“Why can’t colleges turn out graduates who can write basic sentences and do basic math?” Atkinson asks. He blames professors who want to teach their favorite subject, but don’t teach logic, debate, writing, research or other workplace skills.

One of the best college grads I ever hired (a graduate of Dartmouth) majored in history. In his job . . .  he didn’t need to know history. What he needed to know was how to think, how to write, how to speak intelligently, how to find information and make sense out of it, how to argue coherently, and how to do basic math. Fortunately, he had acquired these skills. But other graduates of colleges such as Kenyan, Bowdoin, Bates, or the University of Pennsylvania, whom I have hired over the years, clearly had not, or at least not nearly as well.

Science and engineering graduates need to know their subjects, he writes. Liberal arts and social sciences majors need practical skills, which they may or may not pick up by accident while studying French literature or the history of the comic book.

Atkinson wants a national test for college graduates of logic, reasoning, basic writing and math skills.

Next, he calls for a national employer survey to determine “what are the specific skills employers are looking for in recent graduates.” The survey also should ask which colleges and universities have provided the best employees.

Finally, we need radical experimentation in college design. It’s time for a foundation or wealthy individual to endow an entirely new college founded on teaching 21st century skills, not 20th century subjects.

I can’t imagine teaching these skills without teaching subject matter as well. Of course, I can’t imagine a Princeton grad who hasn’t mastered grammar and spelling — or, at least, spellchecker.

Many colleges do try to teach writing to students; most professors require research papers. K-12 students do lots of oral presentations. Are college graduates really that hopeless?