Happy Friday! Hope everyone has their kids registered for the AMC next week. If you haven’t already subscribed to the AMC problem-a-day from the MAA, you should! It’ll keep you sharp :-).
Here are a few nice things seen ’round the web recently:
- The Scrambler, by Dan Meyer & co. Here, Dan challenges us to analyze a classic carnival ride, and asks us to predict where you end up at the end of the ride. And by Dan & “co”, I mean “comment” folks who have generated lots of fun solutions and applets. Dan made a great interactive version here, too.
- And finally, this lengthy article “Reflections on mathematics and Democracy” by Lynn Arthur Steen is well worth the time [ht: Gene Chase]. He thoughtfully discusses the need for math education among the citizenship. Is “usefulness” to the democracy the highest goal of secondary math education? Do we aim to create quantitatively literate citizens? Or do we put them on the Calculus track and prepare them for college-level STEM careers? Does teaching “quantitative literacy” even count as Mathematics with a capital M? This is obviously something I’ve been thinking a lot about recently. Here are a few of my favorite excerpts:
Ten years ago I addressed the first question posed to this panel in Mathematics and Democracy—a collection of essays from a variety of professionals both inside and outside mathematics.4 (These essays are available for free downloading on the MAA website.) The chief message of this volume is that the mathematics taught in school bears little relationship to the mathematics needed for active citizenship. That mathematics we called quantitative literacy (QL) to contrast it with traditional school mathematics which, historically, is the mathematics students needed to prepare for calculus.
Mathematics and quantitative literacy are distinct but overlapping domains. Whereas mathematics’ power derives from its generality and abstraction, QL is anchored in specific contexts and real world data. An alternative framing of the challenge for this panel is to ask whether perhaps QL might be a more effective approach to high school mathematics for all.
What we forget, however, is that when NCTM initiated its standards work, most mathematics teachers did not actually believe in the goal of teaching mathematics to all high school students. Whereas now we argue about how much and what kind of mathematics to teach in high school, three decades ago debate centered on who should learn high school mathematics. At that time, the curriculum was designed to efficiently sort students into those who were capable of learning high school mathematics and those who were not. So between grades 7 and 9, somewhere between one-third and one-half of the students were placed in a course called General Math—an enervating, pointless review of arithmetic.
Another decade has passed, and our ambitions are now much higher: a common core for all, with everyone emerging from high school ready for college. In one generation, the political view of high school mathematics has progressed from something only some need (or can) learn to a core subject in which all students can and must become proficient. That’s quite a rapid change in ends, which has been matched by a major change in means. The very idea of a common curriculum enforced with common assessments was all but unthinkable back in the 1980s.
Here’s a “stellar” application of the Fundamental Theorem of Calculus, created by one of my students. It is a stellated icosahedron made up of 30 individual pieces of paper, all of which have the FTC printed on them. Doesn’t it just make you smile? :-)
Check out this awesome blog post by Richard Clark on the Alpha Blog.
Follow the link to see lots of great pictures made with equations. These pictures are so complicated it makes you wonder, is there any picture we can’t make with equations? My first answer is NO.
Think about vector-based graphics. Vector graphics, for those who’ve never heard the term, are pictures/graphics that are stored as a set of instructions for redrawing the picture rather than as a large array of pixels. You’ve used vector graphics if you have ever used clip-art or used the drawing tools in Microsoft Office, or if you’ve ever used Adobe Illustrator, or Inkscape. The advantages of vector graphics include very small files and infinite loss-less resizeability. How can vector graphics achieve this? Well, like I said, vector graphics are stored as rules not pixels. And by rules, we could just as easily say equations.
So the answer is certainly YES we can make any picture using equations. I think the harder question is can we make any picture using ONE equation? Or one set of parametric equations? Or one implicit equation?
What constraints do we want to impose? Do fractals/iterative/recursive rules count?
I am curious to find out how the creators of these picture-equations came up with them. It seems infeasible to do this by trial and error, given the massive size of these equations.
Oh, and if you haven’t yet seen the Batman Curve, you better go check that out too.
Have you heard of the “The Incompatible Food Triad” problem? This was first introduced formally here by George Hart (father of the now famous Vi Hart). Here’s the statement, taken straight from his website:
Can you find three foods such that all three do not go together (by any reasonable definition of foods “going together”) but every pair of them does go together?
Hart’s page is full of various suggested solutions, most of which are shot down in one way or another. It’s obviously not a rigorous mathematical question, since much of the success of a solution depends on culturally-defined taste. But it still doesn’t stop us from trying solutions.
pictured above: chocolate and strawberries on a waffle, one suggested pairing. just don’t you dare also add peanut butter!!
Here’s mine. My wife and I sat down to have waffles one weekend and we think we discovered an Incompatible Food Triad: chocolate, strawberries, and peanut butter.
- Chocolate & strawberries obviously taste good together on a waffle.
- Chocolate & peanut butter also obviously taste good together on a waffle.
- Strawberries & peanut butter is okay if you think of it like a peanut butter & jelly sandwich (mash the strawberries!)
But we found that all three did not taste good together on a waffle. A quick web search reveals that people do sometimes eat these three things together, but not on a waffle.Let me make a slightly more rigorous statement of my suggestion then: I claim (chocolate AND waffle), (strawberries AND waffles), and (peanut butter AND waffles) are an incompatible food triad.
Can you see how this is not a very mathematical/scientific question? :-)
Would you agree with our incompatible food triad? Do you have any other suggested solutions?
I think I’ll show this to my Algebra 2 class this week.
[HT: Fred Connington]
Check out this pretty music! [ht: Prisca Chase]