Archive for the ‘what does this have to do w/ org theory?’ Category
A long, long time ago, I used to teach math. One of the central questions in mathematical education at the college level is how to teach mathematical proofs. Sometimes, you had pessimistic conversations. People simply had “mathematical maturity” and there wasn’t much you could do about it. There is truth to this – some people simply can’t grasp what a proof would entail.
Beyond this simple observation, there was remarkably little thinking about how to teach proofs. Of course, there are occasional books that try to break down the process of creating and writing proofs, such as How to Prove It. Still, I felt there was something missing in the conversation about proof teaching. This blog post is my modest contribution to the topic.
My hypothesis: An important barrier to teaching math proofs is that they combine two very, very hard skills and that most math teachers only focus on one of the those skills. Specifically, proofs entail (a) symbolic manipulation and (b) recipes that get you from A to B. Math teachers and books are actually pretty good at (a). For example, almost every text will teach you about the symbols – set theory; formal logic; deltas and epsilons; etc. What is almost completely overlooked is that students find it hard to glimmer the “recipes” that make up proofs and there is no theory, or set of instructional strategies, for helping students intuitively understand recipes. In practice, you simply take courses on various topics (numerical analysis or matrix theory) and you mimic the proofs that people give you. Not great, but better than nothing.
The old Dolciani high school text books had an interesting response to this issue. In the geometry text, the proofs would always have two parts: “analysis” (outline of the idea) and “proof” (traditional proof with all details). You also see this in advanced texts and journal articles. When a long, hard proof is coming up, the author will present an outline.
Here is my modest suggestion: When teaching proofs, always outline the proof as a flow chart. In other words, take the old notion of the proof outline (or “analysis” in Dolcian’s terms), make it visual, and then put it in front of all proofs that require more than a few sentences. By repeatedly visualizing proofs as chains, teachers will be forced to extract the recipe from the text in a way that more students can understand. They will also more easily identify common themes that appear in multiple visualizations of proofs. Also, pictures are easier to remember than dense, equation filled masses of text.
The official history of Japanese food in the United States says that Americans didn’t get a taste of raw fish and vinegared rice until the late 1960s, when groovy Hollywood stars and trendy Buddhist humbugs began turning the squares onto the best thing since sliced bologna: sushi.
But that’s wrong. The truth is that two generation earlier, in the first two decades of the 20th century, Americans knew all about Japanese food and enjoyed it so much that labor unions and American restaurant owners conspired to run the Japanese out of business and out of the country. Worse, these angry agents of change were mostly successful in that effort, launching a thirty-year-long campaign of hysteria, intimidation and misinformation, one that ended in 1924 with the passage of the Japanese Exclusion and Labor Act.
And, xenophobia rears its ugly head:
On May 20, 1907, however, things blew up. A group of union men caught four of their fellow unionists eating at the ten-cent Horseshoe Restaurant at 1213 Folsom Street. Beatings were handed out to the two men who were foolish enough to exit the restaurant through the front door. A crowd of Barbary Coast hoodlums, who had gathered to watch, turned it into a genuine donnybrook, rushing into the restaurant armed with crowbars, bats, fist and feet, bashing everyone and everything inside.
When the police declined an invitation to become involved, the fun spread to a Japanese bathhouse across the street. The demonstrations resumed again the next night, with less vigor, and four more nights after that, until the police, led by the Japanese counsel, Kazuo Matsubara, stood watch at the edge of Japantown.
Thus, began the decline of Japanese food in America. The story of the first sushi in America is worth reading as well.