It seems he was asked to read a book, The Bee Tree, and prepare a presentation on it for his son's third-grade class.
So he dug into his graduate-school calculus:
Bees, as it turns out, create a completely optimal space for their honey, raising their young, and storing food. The hexagonal shape of honeycomb cells actually maximizes interior volume while minimizing surface area. Since bees must eat about 8 times the volume of honey for a given volume of wax that they need to create their hives, wax is a precious commodity, so optimization is key. Obviously I wouldn’t expect the third-graders to be able to handle calculus-based optimization problems (in fact, the so-called Honeycomb Conjecture, [the] idea carried forward from the ancient Greeks that honeycombs were optimally arranged, was only proved about 10 years ago).
However, if the third-graders were meeting the math standards outlined for their grade here in Massachusetts, they should know enough geometry to be able to have a pretty cool discussion about hexagons and other shapes that might fit together with the smallest perimeter and largest surface area.
The upshot (and read his post for details) was that it worked very well indeed, and -- despite dire predictions from his wife and eldest son -- without dumbing it down. The kids latched onto it, ran with it, and learned from it.
His point is that if we raise the bar, kids will come up to it. At least until such time as they have been hammered down into the habit of mediocrity.
Here's the thirsty:
What do you do in your classes to raise the bar, to challenge your students? Does it work? Can and do you get away with it? Does it rouse students from their torpor, or do you just see a lot of thousand-mile stares?
Inquiring minds want to know.
I used to make my students read readings from the 'normal' readers--Back to the Lake and other such drivel. Now I ask them to read primary sources--Machiavelli, Rawls, Aristotle, Locke, Hobbs---and we're all having a great time. They feel like college students (and not 13th graders), and I feel like a professor. They're managing the readings, the writing, and the thinking, and we're all completely engaged in the class. So, yes--add some rigor. Students can manage it, they appreciate it, and it transforms an otherwise boring class into a gathering of scholars.
ReplyDeleteBees don't have little protractors that they use measure each angle. Technically, bees make round chambers. Look at the top of a honeycomb. The most circles you can place around a circle (of the same size) that will touch the middle circle is six. As more chambers (circles) are constructed the pressure from the weight of the added chambers forces the wax chamber walls to flatten out. Bam: Hexagon.
ReplyDeleteDid I just dumb it down? Sorry.
I think the issue at the compulsory education level is the lowest common denomiator factor. Teachers have to teach to the slowest kid in the class. In college that shouldn't happen. I am a huge proponent of challenging students, especially at the college level. I've had numerous discussions with faculty about setting standards and holding students to them. My attitude is, "Fail 'em if they can't hit your minimum standard."
I love this. If we taught children to paint the way we teach math, we would start off with the theory of optics and light, and only when they got into college could the students begin to actually do some fingerpainting.
ReplyDeleteReal math is interesting at the conceptual level, and this makes some Numerophobes realize that there is no such thing as a "math person" and a "non math person," at least not to the degree many students believe.
Bravo!
I showed my televen-year-old nephew how to calculate the area under a curve with summation, and he got it right away. I think pausing from time to time to show the students where this really leads can transform the experience in really positive ways.
Of course, it's time to get back to the serious business of learning the language of math, so fun's over for now: do your problems!
The caps of the honeycomb appears spherical but the hexagonal prisms themselves don't maximize volume subject to a certain surface area. A sphere does that.
ReplyDeleteA regular hexagonal prism is just the shape that maximizes Volume subject to a certain surface area if we are requiring a tiling of the plane the comb is on. There are only 3 regular (i.e. all sides have same length and all interior angles are the same) polygons which tile the plane: a triange, a square, and hexagon. The hexagon is "closest" to a circle so it fits the most stuff inside. Thus the bees build regular hexagonal prisms to (1) leave no gaps, (2) create the "biggest" area, and (3) make the most symmetrical comb possible subject to desires (1) and (2).
I like to give concept questions more than procedure based question since that's a good way determine how well a student understands what something means. But as far as I know the research doesn't support the idea that procedural knowledge and conceptual knowledge depend on one another in any significant way.
Back to the question at hand....
I like to query my students before we start a topic. For instance, if we are going to talk about derivatives I will start with chatting about slope. I will ask them what slope is and what it means/represents. Then I will ask if "curvy" functions might have slope. Basically I'm trying to get them to poke at the meaning of a derivative without all the fancy notation and words. It works in some classes but not all of them. My mileage varies.
I don't want to get into too many details here, since this touches very closely on current discussions at my university and in my department, but my university has recently launched a major initiative to teach students to do something I've been having my students do, in a small way, for years. They don't do it perfectly (the class falls about halfway through their college careers, and this is something we hope they'll be able to do with some competence by the time they graduate), but they do show decent comprehension of the basic principles involved, and an average of about 75% success in applying those principles to their own research and writing, albeit sometimes in rather tangled syntax (which is a pretty common result of writers trying to wrestle with ideas at a new level of complexity).
ReplyDeleteI'm in the process of grading a set of papers that are supposed to exhibit the skills involved, and, while I've given grades from a low A to a C, I've only given one F (and I'm pretty sure that resulted from laziness and not reading directions rather than lack of ability; there might also be some sort of undiagnosed learning or behavioral disability involved; the student acknowledges some difficulties, which he attributes to English being his second language, but I'm not seeing any traces of ESL in his writing; instead, I think there might be a reading comprehension problem. If, as I have requested, he gets in touch with me, I'll try to help him figure out what's going on, and point him in the direction of possibly-appropriate resources).
So, yes, they're rising to the challenge -- not perfectly, but adequately, as part of a learning process that is meant to stretch over several years. I'm quite satisfied with their work; I wish I could find a better way of communicating its worth both to the students themselves and to those who evaluate my teaching.
I teach freshman comp, and one thing I tell students at the beginning of the semester is that in college, unlike in high school, we write about things we don't know yet--we write to explore and to push our minds where they, in Star Trek terms, have never gone before. I'm very upfront about the challenge and promise them that I will make their heads hurt. The assignments are not research papers but analytical writing prompts that require in-depth critical thinking. Many students are so used to writing prompts like "write about your best friend" or "explain three causes of headaches" that they tend to be dumbstruck by analytical writing assignments. With these "new" prompts, the "writing process" that some of them have learned--i.e. forming the thesis after five minutes of prewriting and then keeping that same thesis until the paper is turned in--utterly fails them. More than likely, they cannot use their beloved 5-paragraph format either. Some quit or fail. Many moan. A few cry (though rarely in front of the class). There are very not many Bs and fewer As, though the students who earn them are afterward in awe of what they were able to do.
ReplyDeleteI'm lucky to have support not only from my department chair, but also from the dean--they're okay with my drop rates because they support this idea of raising the bar.
Thanks for this. My university is always dumbing the base class requirements down... and when I receive the course, I raise the bar right back up. I frequently receive notes that my class was the "hardest I've ever taken" but my kids adjust and learn the material. The dears are even able to pull together a large research project at the end of the course -- when I'm supposed to simply ask for a thought piece on how they "feel" about basket-weaving.
ReplyDeleteWithout any parameters, they always flail their arms wildly on the "feeling" thoughtpiece. The highly structured research project is hard work but able to do it in the end.
@Thin Woman: I'm going to borrow that talk, I think. I do this anyway, but I've been avoiding actually telling them what they're learning (because they are learning it, some better than others) because I was afraid to scare them. Maybe if I make it look scary and then they realize they can do it, it will work better.
ReplyDelete@Eric Gates:
ReplyDeleteI have news for you. We don't teach art. Nor penmanship, nor spelling, nor geography, nor grammar, not history, nor science. All of this was abandoned years ago.
So, what do K-12 kids spend the time doing? Partly being told over and over and OVER again how special every last one of them is, and partly being told how to use a condom. I learned this in about ten minutes from reading the instructions on the back of the box, but I suppose this can be much harder, if one can't (or just plain doesn't) read.
I recently saw some 4th graders working with the Fibonacci sequence.
ReplyDeleteOnce, in a 19th century American literature class, I had my 70 undergrads do primary archival research in special collections, digitize their findings, collect them in an online archive, annotate them, and use them to analyze a work of fiction. It was remarkably successful, though a huge amount of labor for me.
@Marcia -- I want to do that! (and have heard several conference presentations on similar projects, maybe even one from you). But in a 70-student class? Wow, you're brave.
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