This post, like part of yesterday’s, brings up an educational dilemma:
- On the one hand, we want students to work hard. That means that we need to provide incentives as rewards for working hard. Grades are pretty much the only currency we have in high school, so students expect to get good grades if they put in a lot of effort. “I worked really hard, so I deserve a B” is a common refrain.
- On the other hand, we want students to succeed at learning math. That means that we shouldn’t reward someone who tries but doesn’t succeed. If I give a B to students who have earned a D, merely because they’ve been trying, it sends the wrong message to those students, to their parents, and to their future teachers.
There’s the dilemma. How do we get students to work hard, if they don’t connect it with success? Many — perhaps most — high-school students see the process like this:
What they’re missing, of course, is the step in the middle of the link. In reality the process goes like this:
Teenagers are notoriously bad at connecting distant consequences to current actions, so it’s no surprise that so many don’t see the step in the middle. As educators we can’t promise that hard work will necessarily lead to high grades, but we can help our students see how the connection works. Of course they may point to counterexamples: either classmates who do well without working hard or those who work hard and still get low grades. Those counterexamples do make our argument more difficult. Let’s create a sort of Punnett square of four types of students:
|
1
works hard,
is successful
|
2
works hard,
is unsuccessful
|
|
3
works very little,
is successful
|
4
works very little,
is unsuccessful
|
It’s easy to agree about students 1 and 4. Every teacher admires and is pleased with student 1; every teacher is frustrated and unhappy about student 4. But students 2 and 3 are the difficult cases. We admire and praise student 2, but we (and the student) are still frustrated. Working hard is supposed to lead to success, so how do we reward it when it doesn’t? But we can’t in good conscience give such a student a good grade.
Student 3 may be annoying but is less of an issue. We definitely do not want to praise such students, either publicly or privately. But we have to admit that they are successful despite the fact that they don’t work hard. Oh well, that’s life. We try to provide extra challenges for them, but it’s hard to push difficult challenges on those who don’t want them.
Finally, what about the argument that success is what counts in “real life”? Frankly, I’m unimpressed with the view that school needs to be like work. I hear some people say that we shouldn’t allow retakes, as “there are no second chances in real life.” Putting aside the fact that that claim simply isn’t true, as many a politician and business executive can tell you, we still don’t have to accept the implication for school situations. There are plenty of reasons to refuse to yield to the desire to reward hard work that doesn’t happen to lead to success, but the “real life” claim isn’t one of them.
City council candidates and math education? Those are two utterly unrelated topics, aren’t they? But there turns out to be a connection.
First of all, this afternoon I had already been intending to comment on an op-ed piece from this morning’s New York Times, titled “How to Fix Our Math Education.” And I was going to relate it to a comment by Frank Baker, candidate for Boston City Council from the Third District (where I live). But I didn’t actually finish the post before dinner, and then I had to leave immediately in order to go hear all seven of the candidates participate in a candidates’ forum. So let’s discuss all three topics: the op-ed piece, the candidates’ forum, and the connecting link: Frank Baker.
The op-ed piece, by the distinguished Sol Garfunkel and David Mumford, argues for a major change in the emphasis of high-school mathematics. Their views are basically correct, so go read their entire essay. Here is a brief excerpt so you can see what their claim is:
This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life…. Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.
The reason I say that their views are only “basically” correct is that such a curriculum would not be sufficient for students preparing for any career that requires a traditional mathematical background. Even though only a small minority of college-students study science, engineering, and pre-med courses, they need to be well served, and it would be folly to think that we can identify exactly who those students will be when they’re in tenth grade (although many countries do exactly that). We could just offer Garfunkel and Mumford’s solution to non-honors students, but who wants to increase the difference between honors and non-honors courses? That wouldn’t be good either. Some kind of a combination is needed, where students could move more in one direction or another in their last two years of high school, and where the ideas presented in the op-ed piece are incorporated into the traditional program as well. The first of these solutions is what we do at Weston High School, where we offer three semesters of “Applied Discrete Math Concepts” to those who want something different from the traditional pre-calculus and calculus courses. While many of the students who elect this course tend to be our weaker math students, that is far from universally true, and some excellent math students even take Applied Discrete Math in addition to precalculus.
Now let’s move to the apparently unrelated candidates’ forum. We have a surprisingly large field this year, as our long-term councilor, the hard-working Maureen Feeney, is retiring after serving many years. Of the seven candidates who are vying to succeed her, two are clearly Republicans (even though they might not admit it) and I won’t comment on their ideas or their presentations at the forum. The Boston City Council is officially non-partisan, but the other five candidates are clearly Democrats. Three of them gave pretty weak presentations tonight, leaving only the remaining two, whose supporters are coincidentally the only ones whose supporters you will see in this picture:
It may be a little difficult to tell from the picture, but this crowd on both sides of the street was almost entirely white, a bad sign in racially mixed Dorchester. Inside the hall I counted close to 300 people in the audience, of whom 98% were white. So much for “racially mixed.”
Finally, how are these two topics related? The connection is a paragraph last week in the Dorchester Reporter:
Baker suggested the city bring back trade schools and attempt to replicate popular schools like the Richard Murphy K-8 School. “Why aren’t we looking at that and trying to apply it to other schools?” he said.
But here’s the dilemma. On the one hand, college isn’t for everyone, and vocational programs can do a lot to keep kids in school and teach them useful skills. On the other hand, it seems unbearably classist to say that Weston should have a college-preparatory program and Boston should be oriented toward life skills. I don’t know how to resolve this dilemma. Weston is so college-oriented that “college-prep” is the name for our lowest level of courses! Everything that isn’t honors or AP is college-prep. Students who want vocational training can go to Minuteman Career and Technical High School, which offers an excellent program — but it is socially deprecated in Weston and kids don’t want to be separated from their friends. So basically everyone in Weston is assumed to be college-bound. Boston, of course, is far more diverse. Even if it doesn’t bring back trade schools as an option, the kind of math curriculum proposed by Garfunkel and Mumford would at least be a start.
A recent article in Salon opens with the conventional view of “kids today”:
They live in a state of perpetual, endless distraction, and, for many parents and educators, it’s a source of real concern. Will future generations be able to finish a whole book? Will they be able to sit through an entire movie without checking their phones? Are we raising a generation of impatient brats?
Been there, done that, you yawn. But in actuality the article isn’t rehashing these old ideas but is rebutting them in a refreshing new way. Duke professor Cathy Davidson (no relation) has written a book called Now You See It: How Brain Science of Attention Will Transform the Way We Live, Work, and Learn, in which she argues that “much of the panic about children’s shortened attention spans isn’t just misguided, it’s harmful.” That’s not to say that Davidson believes in multitasking:
[W]hen we pay attention to one thing, it means we’re not paying attention to something else. When we’re multitasking, what we’re actually really doing is what Linda Stone calls “continuous partial attention.” We’re not actually simultaneously paying equal attention to two things: One of the things that we’re doing is probably being done automatically, and we’re sort of cruising through that, and we’re paying more attention to the other thing. Or we’re moving back and forth between them.
I particularly like her remarks on the common view that our brains develop more neurons as we get older:
We used to think that as we get older we develop more neural pathways, but the opposite is actually the case. You and I have about 40 percent less neurons than a newborn infant does. A baby pays attention to everything. You’ve probably witnessed this — if there are shadows in the ceiling or sand blades are making peculiar patterns, we adults don’t recognize that, but it can be utterly mesmerizing to a child. They learn what not to pay attention to over and over and over again, and learn what to pay attention to, and that makes for neural pathways that are very efficient. They’re what we tend to call reflexes or automatic behaviors, because we’ve done them so many times we don’t pay attention to them anymore. As an adult, you feel distracted when you learn something new and you can’t depend on those automatic responses or automatic reflexes that have been streamlined neurally over a lifetime of use.
One reason why I liked that passage was that it resonated with one of my favorite quotations: “Civilization advances by extending the number of important operations which we can perform without thinking about them.” This observation, written by Alfred North Whitehead one hundred years ago, seems to fly in the face of what we teachers always say — students should think before they speak or write — but that’s what makes it so appealing. In fact, what we want is for students to think about the big things but to be automatic about the little things. If you have to think what 2x + 3x is whenever you do an algebra problem, you’ll never be able to pay attention to the Big Ideas.
Here are two more quotations, which I’ll give without comment, since you really should just read the entire article:
[on multiple-choice tests] How do you teach a kid to be able to make a sound judgment about what is and what isn’t reliable information? How do you synthesize that into a coherent position that allows you to make informed decisions about your life? In other words, all of those things we think of as school were shaped for a vision of work and productivity and adulthood that was very much an industrial age of work, productivity and adulthood. We now have a pretty different idea of work, productivity and adulthood, but we’re still teaching people using the same institutionalized forms of education.
[on points and grading]
There are all these really stunning computer scientists that are just frustrated as heck about how badly we’re training scientists. And many of them feel that A,B,C,D and numeric grades are disincentives to exactly the kind of inductive thinking, creative thinking that is the scientific method. Top Coder is the world’s most important certification system for people who are doing open Web development around the world, and they’ve come up with an incredibly complex badging system, where if I’m working with you on code and I see you’re doing a great job, it’s part of my job as a member of the Top Coding community to give somebody points. So if I think you’re doing a great job solving some problem in C++ that I can’t see a solution to, I might give you 20 points. If I’m a third developer, and I say I really need somebody who can help me with some really complicated stuff on C++ , and I see you have a badge with 1,000 points on it on your website, I can click on your badge and it will give me in minute and excruciating detail how you earned every one of those points. There are now a group of computer scientists who are working together to see if we can’t come up with ways that textbooks — particularly online and interactive textbooks; there’ve been some wonderful ones for algebra, for example — could be based on testing that works in some similar way, where a teacher would give you points for succeeding at a problem, where you would automatically get points for getting the correct answer. You wouldn’t even worry about giving negative points because it doesn’t matter; all you do is get points when you do something well. Even saying that is a conceptual breakthrough. When I told my students that we don’t have to worry about trolls and criticism, all we have to do is make really sound, conscientious, articulate judgments about positive things, it was as if a cloud opened.
Several different threads have recently been coming together under the heading of “inverting the classroom.” The basic idea is that modern technology has let some of us come to the conclusion that the traditional model of the classroom has it all backwards:
- Students currently spend a lot of class time in a group of 25 (or 35, or even more, depending on the school) listening to a teacher lecture to them. They could just as well watch a lecture at home — on YouTube, say — where they could pause whenever necessary and watch difficult material many times.
- Students currently get most of their practicing done at home — it’s called homework, after all — where there is no teacher there to help them.
My department head has been fighting this model for several years now, mostly by devoting 90% of his class time to helping kids as they work individually or in groups. Homework becomes classwork.
But that won’t work for most teachers and most classes. I have no idea what percent of class time is typically devoted to lecturing; in my case I would guess 30%, but I fear that the national average is more like 80%. In any case, all of that time could be better spent. The most well-known example of the getting-the-lectures-at-home-through-technology point-of-view is Khan Academy, which offers over 2000 free videos giving short lectures on topics ranging from simple arithmetic to quantum mechanics. I’ve only watched four or five of these, but the ones I’ve seen look pretty good. I think I’ll recommend some of them to my classes this fall. They have their use.
So what’s the downside? The first issue is that “lecture” is usually a misnomer. When I lecture in class, I try to pause to let students ask questions, I constantly look at them to see whether they seem to be understanding, I give them quick exercises to work on, I vary my pace according to my audience. Sometimes we even have a discussion as part of a lecture, or as a follow-up to it. None of that is possible with a pre-recorded video. That worries me.
The second issue is that the public in the current political climate will use this model as an excuse to fire teachers, decrease class time, and increase class size. If that happens, it will be exactly backwards in a different way: after all, if class time is to be used primarily for getting individual help from the teacher, then we’ll need smaller classes and more teachers.
The third issue is that inverting the classroom will make it more difficult for a teacher to create discovery learning opportunities. If I have a carefully staged series of questions that are all designed to let my geometry students figure out a certain theorem, I don’t want them to be watching a video on that theorem ahead of time. This difficulty can be overcome by using a significant portion of classtime for such a purpose, but teachers who think that homework is primarily for practice will have trouble implementing that idea.
Finally, the fourth issue is that the whole approach atomizes a course into bite-size chunks of facts, all taught in a way that can’t possibly integrate into the story-line of a course. If I’m teaching trigonometry, for example, I may want a particular lecture to use radians rather than degrees, arcsin rather than sin-1, and so forth. I may want a linear equation to be y = a + bx, not y = mx + b. Sometimes I even make words up, for well-thought-out pedagogical purposes. How do I control all this if my students are watching Salman Khan give his pre-recorded lecture? The philosophy behind this sort of lecture seems to be that teachers are fungible, but we aren’t.
So…I have many doubts. It’s not that these issues can’t be resolved, but they won’t be unless we put a lot of thought into finding solutions. Inverting the classroom is a great idea, with wonderful potential, and it’s definitely worth pursuing — but only if it’s done right.
Be sure to watch Weston vs. Hamilton-Wenham in the quarterfinals of High School Quiz show, to be broadcast tomorrow night, 4/24, at 7:00 PM on Channel 2!
In last month’s post about our Fractal Fair, I made the following promise:
Stay tuned for a post on one project in particular, a spectacular children’s book on fractals.
So here’s the follow-up, or at least a preliminary follow-up. I still haven’t figured out the ideal way to take pictures of the eighteen 11″-by-14″ hand-drawn and hand-lettered pages of The Fractal Adventure, written and drawn by my students Anna, Ali, and Eye. The pages are too big for my scanner, and I suspect that I need stronger ambient light in order to take high-quality digital photos of them. For the moment, though, here are a couple of less-than-ideal images of pages 4 and 5, so you can at least get the idea:
Everyone who has any connection with education — teacher, student, parent, administrator — needs to read Todd Farley’s Making the Grades: My Misadventures in the Standardized Testing Industry. Yes, the book is a bit repetitive, and of course it reflects only one person’s views, and it doesn’t match my colleagues’ experiences scoring AP exams…but you still should read it. Not a statistical study, it is an easy-to-read narrative of Todd Farley’s work with Pearson (“the world’s leading education company”) and ETS (which “conducts assessment and policy research and develops assessments and related services to advance quality and equity in learning worldwide”). Farley started out scoring open-response questions, progressed to being a table leader and a trainer, and eventually wrote rubrics and test questions, all over a period of 15 years. Note, please, that none of this has anything to do with multiple-choice tests, which may have their own issues but at least are scored objectively and consistently.
Quoting from the book will be more effective than merely commenting on it. We’ll start with a portion of a rubric describing how to score an eighth-grade descriptive writing task on a statewide test (not MCAS, but similar), quoted verbatim including layout, punctuation, and capitalization:
A good response (3) includes
Good organization, including appropriate use of the five-paragraph format.
Good focus and development.
Good style and sentence fluency.
Good grammar, usage, and mechanics.
The excellent (4), inconsistent (2), and poor (1) portions of the rubric are identical to this one, as long as you do a find-and-replace accordingly. Now you have to understand that the typical abysmally paid and undereducated scorer somehow has to decide whether an essay is “good” in all four categories, with no more guidance than “good” = “good.” Of course the scorers and table leaders have to be trained, which means they have to pass a qualifying test. Here’s how the trainer, Maria, ensured that all the table leaders (Caitlin, Ricky, Harlan, etc.) would pass the qualifying test, on which they had to score seven out of ten essays “correctly”:
Maria held up her hand to tell Caitlin not to move. After Maria checked the scores, she handed the score sheet back to Caitlin, whispered something to her, and sent her back to her desk, where Caitlin started to rescore the ten essays. Then Maria whispered something to Ricky at her side, a something Ricky turned and whispered into the ear of the table leader closest to him. The whispering continued through the room. Harlan, on my left, passed on to me the useful nugget: “The same score is never given to successive essays. Pass it on.”
…
Then Maria passed to Ricky who passed to the table leaders information that essay 2 “was absolutely considered appropriate five-paragraph format,” meaning it would earn at least a 3.
And so forth.
Other examples of cheating pervade the first half of the book. For example:
I rarely, of course, actually looked at the essays in question, because I simply didn’t have the time. If I was looking at the score sheets of two scorers I didn’t trust (Louise and Harry, for example), eventually I compromised and erased bubbles from the score sheets of each, changing the scores until their agreement went from an unacceptable 50 or 60 percent up to an acceptable 70 percent.
The horrifying thing about examples like this one is that statistics are determining data, not the other way around. These are real students’ lives that they’re playing with. Even if, say, 70% of the scores are in some sense “correct,” that is of no comfort to you if you are in the remaining 30%.
Not that you would ever find out.
With all the pressures from politicians and the public these days, so-called “standardized tests” are going to become more and more prevalent. But the scorers won’t become better educated or better paid. And the more pressure there is for favorable statistics, the more likely it is that adults will cheat, as we’ve seen recently in Washington, New York, and Springfield. Replacing open-response questions with multiple-choice questions will solve several of these problems (though it would introduce other problems, of course).
We might get a rebuttal from one of my colleagues. But in the meantime…read this book!
I’m sure you’re familiar with all the controversy surrounding Amy Chua’s memoir, Battle Hymn of the Tiger Mother. Some of the controversy is well-deserved, but much is not.
This book came to the world’s attention through an excerpt published in the Wall Street Journal and splashed all over the Web. Chua seemed to be claiming that Americans raised their kids all wrong, that Chinese (and other Asian) parents knew how to do it right, that it’s OK (and even desirable) to emotionally abuse one’s children in public, and that the important thing was to emphasize doing huge amounts of homework and spending even more time on practicing piano and/or violin. No other instruments would do. Theater was certainly out of the question. Grades of A– were also out of the question; only straight A’s sufficed.
Needless to say, this seemed exaggerated and full of stereotypes. But Chua is a successful law professor at Yale, the second-best law school in the country (well…some people do call it the best…), so she has to be taken seriously. I decided to get the whole story by reading the entire book, not just the brief excerpt that had caused such extreme reactions.
The book is worth reading. It’s written in a breezy style that makes it quick to read, and no deep thinking is required. That’s a mixed blessing, of course. Clearly there were some logical inconsistencies, such as the claimed insistence of every Chinese family in a class that their child had to be #1. More interestingly, it became apparent that the Journal had unfairly selected the most sensationalistic portions of the book, no matter how unrepresentative they might be. The full realization doesn’t appear until close to the end of the book, where the reader discovers that Chua recants half of what she had said earlier. It turns out that her child-rearing methods worked for one daughter and not for the other. (It also turns out that this memoir is only partly about child-rearing; like any memoir it covers an entire range of events.) It also turns out that Chua admits to unfair generalizations by contrasting Chinese child-rearing with American, since she admits that it isn’t ethnically based after all. So don’t believe everything you read in the papers, even the Wall Street Journal.
In closing, however, I need to point out that there are, of course, several grains of truth hidden in the excerpts and in the earlier parts of the book. It’s not coincidence that over half of the students at the state math meet are always Asian. We know that it’s not genetic; it’s because of parental expectations. That much is believable.
Congratulations to the Weston High School Math Team for coming in fifth in the state at the Massachusetts State Math Playoffs in Shrewsbury on Monday! We have just learned that those results have qualified us to enter the New England playoffs in Canton on 4/29, so stay tuned…
A terrific turnout last night at the Driscoll School in Brookline. More than half (!) of the fourth- and fifth-graders (and their parents) showed up for an evening event revolving around Penny Noyce’s Lost in Lexicon. My role was to be the Pi Man, representing the Village of Irrationality. Kids (and often their parents) would measure various circular bowls, dividing the circumference by the diameter in each case. This being Brookline, most of them already knew about pi and expected to get the “correct” value, so the activity tended to turn into the surprise they experienced when the average of their ratios for three different bowls turned out to be less precise than they had expected. One boy decided to measure the entire round table to get a better result. Everyone had a great time, being totally engaged in a variety of activities relating to math and language. What better combination could there be?
Continuing yesterday’s theme, I offer you a Latin test from 1964, with no comment about how well I would do on it today:
Deep in my attic, in an old file folder, I discovered an AP US History test that I took back in October of 1964:
I wonder how many of those questions I could answer correctly today, especially within a 50-minute time limit. You may wonder why it says “United States History” at the top, instead of “Advanced Placement United States History” or the like.
The answer to the first question is…well…maybe 10? or at most 12? (Is it OK to answer a question with a question?)
The answer to the second is that we were all required to take AP US History, so they didn’t have to call the course anything but “United States History.”
We’re all being pressed to use rubrics. For those of you not in the ed biz, a rubric is described pretty well in Wikipedia:
A rubric is a scoring tool for subjective assessments. It is a set of criteria and standards linked to learning objectives that is used to assess a student’s performance on papers, projects, essays, and other assignments. Rubrics allow for standardised evaluation according to specified criteria, making grading simpler and more transparent….
Rubrics are generally thought to promote more consistent grading and to develop self-evaluation skills in students as they monitor their performance relative to the rubric.
Clearly rubrics are useful. Those two paragraphs are full of Good Things: criteria, standards, assessment, standardization, simplicity, transparency, consistency, self-evaluation skills. Everyone is in favor of these, so everyone is in favor of rubrics.
We generally use rubrics to determine partial credit on math tests. Instead of subjectively deciding that one student’s work is worth 10 points out of 16 and another student’s work is worth 12 points, a teacher just consults the rubric (usually developed in conjunction with other teachers on the team):
4 pts for stating appropriate law, 4 for finding one angle R, 4 for finding one PR, 4 for other; –1 pt for less precision than requested, no penalty for extra precision.
There are still unstated but implied decisions to be made. For instance, if the student is to get 4 points for finding one angle but makes a mistake along the way, is it a 2-point mistake? Nevertheless, the rubric is extremely useful and helps to promote fairness. More importantly, it promotes the perception of fairness.
In math, however, we can’t usually share the rubric with students in advance (except perhaps on projects), since the rubric itself gives away the solution method. Having the rubric in advance could reduce the problem to mere rule-following rather than thinking. As a result, we can’t “develop self-evaluation skills in students as they monitor their performance relative to the rubric.”
Rubrics are also used effectively by the College Board in scoring answers to open-response questions on Advanced Placement exams. Experience shows a very high degree in reliability among the numbers given by different well-trained scorers to the same response.
But there are some downsides to rubrics. My upcoming post on Making the Grades will explore some of the them. An additional problem not covered in that book is that the current push for rubrics includes some preposterous goals. For example, here’s what the New England Association of Schools and Colleges lists as #1 among the 2011 accreditation standards for “Assessment of and for Student Learning” :
The professional staff continuously employs a formal process, based on school-wide rubrics, to assess whole-school and individual student progress in achieving the school’s 21st century learning expectations.
“School-wide rubrics”? There’s the rub. If a rubric is general enough to be used school-wide, from math to art, from history to science, then it’s going to be so general as to be meaningless. If a rubric is going to be useful to a math student in assessing his or her work, it has to be specific to math.
A colleague who does not teach in our Math Department was tutoring one of my students. Not being familiar with our mildly unusual Honors Geometry course, she found that she herself did not know how to do the last problem on his test. “But I figured it out,” she then reported. “Looking back at the earlier problems, I could see that they told a story, so I understood what the last problem must be all about.”
Naturally I loved the resolution of this difficulty. She was 100% correct, though that certainly wouldn’t have happened all the time. A test should tell a story. It should have a theme; a beginning, a middle, and an end; a conflict and resolution; and a plot that exhibits a well-defined arc. A test should tell a story, but all too often it doesn’t, whether it’s one that I wrote or one that someone else wrote. Occasionally I write a test that literally tells a story, one in which the student has to fill in some blanks and solve some problems along the way, but that distresses too many kids who are unaccustomed to that form for a math test. Most of the time the story can be discerned only by reading between the lines, but I hope it’s still there, at least on most tests.
Not only should a test tell a story, but a course should as well. A couple of years ago a colleague commented on an Algebra II course at another school by saying something like this: “It isn’t a course; it’s a collection of topics.” Unfortunately all too many Algebra II courses suffer from this failing, which is one of the many reasons why I tend to prefer precalculus over Algebra II. Precalculus, at least as I teach it at Weston, definitely tells a story, with all of the parts I outlined above: theme, beginning, middle, end, conflict, resolution, plot, and arc. Algebra II gets only halfway, though one of the attractions behind the decision to spend the fourth quarter on cryptography is that it truly helps to complete the story of Algebra II, with lots of attention to functions, inverses, matrices, exponents, representations, and of course real-world applications.
When I was talking with a Weston English teacher the other day, I realized that my own high-school experience with literature as assigned by English teachers was badly skewed. “This is an English department, not an American department” was one teacher’s lame explanation for why we read almost nothing written by Americans. One of the many reasons why it was unconvincing was that our readings were not actually limited to English authors, though admittedly they did form the bulk of the curriculum. It was true that Hardy, Dickens, Milton, Donne, Herrick, Herbert, Tennyson, Shaw, Yeats (OK, those two were Irish, not English), T.S. Eliot (technically American, but English in spirit), Orwell, Jonson, Marlowe, and of course Shakespeare tended to predominate, supporting the “English department” explanation. (Naturally there was nothing by Austen, the Brontés, or George Eliot, but I’m sure you can figure out why they didn’t count any more than the Americans did.) But the explanation collapses because we read plenty of authors who were neither English nor American, not just Joseph Conrad but also lots of Ibsen in translation and even more ancient Greek literature in translation, ranging from Homer to Sophocles to Aristophanes. All this amounted to a rather peculiar collection of authors, though I admit to enjoying most of it (all but Hardy, of whom we had read one novel a year). It’s odd that F. Scott Fitzgerald and Eugene O’Neill were considered “foreign” when Ibsen and Sophocles were not, but that’s what happens when an American school considers itself to be English. This was Phillips Academy in the early ’60s; I’m sure it’s different today.
In all fairness, I have to admit that my wonderful AP English class taught by Dudley Fitts included not only a huge amount of Greek literature, which I loved, but also an entire collection of poetry by a contemporary American author — a woman, no less. Unfortunately almost nobody has heard of her today. But do check out Jean Valentine’s website, from which I re-learned something that I had forgotten: her collection Dream Barker, which is what we read in 1965, was almost literally hot off the press, having been published mere weeks before we read it. That was truly an unusual opportunity in those pre-Internet days.
Finally, as you probably know, National Poetry Month is coming up in April. We were each asked to select our favorite poem to put in the school library’s display case. I chose Alfred Lord Tennyson’s Ulysses, apparently for the second year in a row, though I didn’t remember that. “Well, it’s still my favorite poem,” I explained.
The great Art Benjamin, whom we’ve had the pleasure of listening to twice at Weston High School, made the following remarks in his TED talk:
If I had an extra minute, I’d also talk about how we shouldn’t only show the mathematics that’s useful — and statistics is useful for being an educated consumer and citizen. We could replace a lot of the drudgerous mathematics that’s being taught with math that’s purely fun, with no real promise of “you’re going to use this,” but just “this is beautiful stuff.”
You can go ape over patterns in Pascal’s triangle, in the Fibonacci numbers, in chaos, in fractals. These things that are just positively inspirational. We don’t make — I mean, I’m listening to this music. It’s inspirational. But I didn’t have to be drilled with how to draw my notes properly and learn all this music theory before I got exposed to that kind of music. I think the same sort of thing could happen in mathematics.
Why not give them a taste of beautiful mathematics in addition to the useful stuff?
He’s right, of course. But what’s most interesting is the interplay among the three different ideas of usefulness, fun, and beauty. Too often we end up with none of the above. Benjamin advocates more statistics and less calculus (and preparation for calculus). That path certainly wins on the usefulness score, though many would question it on grounds of fun and beauty. He cites wonderful examples for those, and we do find that a great many students enjoy studying chaos and fractals, finding both fun and beauty in them. Pascal’s Triangle and Fibonacci numbers are in our curriculum, but we could do more with them, especially if we want students to see their beauty and enjoy studying them.
Today we participated in an intense professional development (PD) program and worked on our preparation for NEASC accreditation. NEASC work is often frustrating but often useful as well (more on that later). Today’s PD was quite interesting. The main part of the day’s program opened with “Where Good Ideas Come From,” a great video by Steven Johnson. Drop whatever you’re doing right now, click on that link, and watch the entire four-minute video…
…So now you know why “chance favors the connected mind.” Keep that in mind, no matter what organization(s) you’re a part of. I think I’ll show it to my students.
After that video, as well as other more-or-less related introductory material, we split up into small groups to participate in parallel sessions in which various “professional learning communities” presented progress reports. (You have to keep up with the current jargon, you know.) I chose to attend sessions on Metco, mathematical discussions, and iPads. All were well worth it:
- The Metco presentation was the first time I had heard a report on the entire program, K–12. The central question, how to improve Metco students’ success in math, was of course relevant and important.
- The “mathematical discussions” presentation concerned a project in fourth and fifth grades revolving around Suzanne Chapin’s work, which was reported to result in “increase in civility and logical thinking.” Because the presentation consisted almost entirely of video shot in real classrooms, the findings were powerful and convincing; “they’re articulate and they try to use precise language” was the observation of one teacher.
- Finally, the iPad presentation concerned a pilot project in which an entire class of seventh-graders have been given iPads for second semester to use in science, social studies, and English/language arts. While I teach none of those subjects, the relevance to me is that I was recently approved for participating in an iPad project of my own for one month in Algebra II. So stay tuned for my results and my comparison with what I saw in the report from the seventh-grade teachers.
In the afternoon we watched the film Race to Nowhere and discussed it in small groups. This “documentary,” in the style of Michael Moore, is definitely worth seeing despite its obvious biases. It makes a strong case that our students are being stressed out and pressured to focus entirely on a race to attend the most prestigious colleges. That is certainly true for many students in Weston. But I have several problems with the film: math teachers are always the bad guys (too much math homework drives a girl to suicide???); scenes of black students in Oakland were obviously edited in, probably in response to the otherwise white-suburban bias; scenes from The Blue School in New York City made unfounded generalizations about all other schools; the film claims that 95% of American high-school students cheat, which seems unbelievably high; they also claim that American schools aren’t preparing independent thinkers, which strikes me as a gross over-generalization. Almost everything in the movie is worth thinking and talking about, but I wish it had been more balanced.
Strange but true (like many of the other news reports heard on Wait Wait…Don’t Tell Me):
A Manhattan mom is suing a $19,000-a-year preschool, claiming it jeopardized her daughter’s chances of getting into an elite private school….
and the elite private school would be “the first step to the Ivy League.” According to the Huffington Post (and confirmed elsewhere), Nicole Imprescia, mother of four-year-old Lucia, claims that her daughter’s preschool “proved not to be a school at all, but just one big playroom.”
Oh my. What can I say? Sue your preschool because you’re not going to get into Harvard? Oh my.
This is a few days late, but…
We held our annual observation of Pi Day on Monday in two of my classes and on Tuesday in the other two (since they didn’t meet on Monday). But one of my students pointed me to a couple of posts claiming that pi is wrong — not wrong in the sense that the ratio of the circumference of a circle to its diameter isn’t actually π, nor wrong in the sense that π doesn’t have the value we think it does (of course it’s the right ratio and does have that value) but wrong in the sense that it would be much more useful and pedagogically better to use the ratio of the circumference to the radius. This ratio, called tau (τ) is explored by Vi Hart in her usual inimitable manner. Do watch her video!
Professor Robert Devaney of Boston University gave two excellent talks to our precalculus classes (consisting mostly of juniors, with a sprinkling of advanced sophomores and freshmen) on Tuesday. His talk to the college-prep classes (”Precalculus Part One”) focused on the use of geometric transformations to create fractals which in turn could become artificial but convincing landscapes in movies. This combination of pure and applied math was a stunning example of real-life applications of what appears to be a highly theoretical piece of pure mathematics.
I only wish the audience had been more consistently respectful. As Bob’s introducer, I happened to be sitting in front, where I got to see a non-representative sample of the audience. The kids near me were not only respectful but were also attentive and engaged. They were duly appreciative when the apparently random activity of the Chaos Game turned into the highly regular Sierpinski’s Triangle, and when Barnsley’s Fern emerged out of chaos. But teachers in the back of the room reported a different cohort there: kids using cell phones, sleeping, talking, etc. Since students sat where they pleased, the distribution was certainly not coincidental. But the question to me is why this audience was so extremely different from the honors math students (see next paragraph). Of course it’s easy to claim that students in honors classes are almost always better behaved than those in non-honors classes, as those who don’t want to take a subject seriously are unlikely to sign up for an honors class. And there is indeed a certain measure of truth in that observation. But it’s clearly not the whole truth. For instance, my non-honors Algebra II class is far more respectful, polite, and better behaved than my D Block Honors Geometry class. I wonder what accounts for these differences; is it merely the chance distribution of students?
The talk to the honors classes was almost entirely about the Mandelbrot Set, although it had to involve some necessary preliminaries about Julia Sets. The students were attentive and learned a lot from this presentation, including some surprising interpretations of “how to count” and “how to add.” Although I had heard almost all of this many times before, there was one important nugget that was brand new to me: how to insert sliders into Excel spreadsheets. The resulting dynamic graph became a wonderful tool for visualizing (and therefore understanding) the chaotic effect of varying a single factor when looking at the orbit as a function is iterated. I will have to try using that myself some time.
Also, as a follow-up, yesterday’s Fractal Fair was extremely successful. Almost all the projects were solid, many were excellent, and we got a lot of visitors of all ages. Stay tuned for a post on one project in particular, a spectacular children’s book on fractals. Here are a few photos, taken by the school librarian:
  
If you’re in or around Weston on Wednesday, come to our Ninth Annual Fractal Fair! It’s from 10:00 to 12:15 in the Weston High School Library. The exhibits and presentations, by 50 Honors Precalculus students (mostly juniors), will focus on ideas of iteration, recursion, fractals, and chaos. Although these are primarily mathematical in nature, many of them will also have tie-ins to other subjects, especially science and art. We’re all used to seeing athletic, musical, artistic, and drama performances — but how often do you get to see exciting math exhibits from high school students???
Be sure to watch High School Quiz Show tonight: Channel 2 at 7:00! The match is between Weston and Woburn, starring Mir Bokhari, Grace Huckins, Jon Birjiniuk, and Matthew Chernick, as seen left-to-right in this brief promotional video.
Weston High School is a great place! With a student body of only 748 students, we had about 150 show up for a chess match of all things! What a delightfully geeky experience. The context was a fund-raiser for the chess club, in which chess player extraordinaire Alex Kaye challenged students to beat him in chess. He played 27 opponents simultaneously, moving rapidly from board to board around in a circle as the onlookers cheered. While it certainly wasn’t silent as a chess match is supposed to be, there was great excitement and school spirit.
After 35 minutes, a fellow senior finally won. Everyone had a great time. Here are a couple of pictures showing what is was like:
Congratulations to the Weston High School Math Team for moving on to post-season! We will be competing in the State Meet on April 1 because of our current standing, which is #3 in the state among medium-sized high schools: at this point Worcester Academy has 810 points, Winchester has 797, and Weston has 657. It will be a challenge to become state champions, but it’s not out of reach!
On the whole it was refreshing to read Why Don’t Students Like School? A Cognitive Scientist Answers Questions about How the Mind Works and What it Means for the Classroom. Despite the misleading title and overly long subtitle, Daniel Willingham’s new book offers an interesting mixture of the obvious and the unconventional.
First, let’s get that title out of the way. The publisher obviously pulled it out of a small portion of Chapter One, the only part of the book to which it actually applies. Presumably the reason they did this was that it would sell more copies than the subtitle alone would do. (Speaking of disservices committed by the publisher, Jossey-Bass, I also have to observe that the tiny font size is very difficult for readers over the age of 40.)
Willingham limits himself to nine principles, chosen on the basis of three criteria (quoted verbatim here, except for punctuation):
- Using versus ignoring a principle had to have a big impact on student learning.
- There had to be an enormous amount of data, not just a few studies, to support the principle.
- The principle had to suggest classroom applications that teachers might not already know.
These criteria lead to principles that Willingham casts in the form of questions, a form that I like in this context:
- Why don’t students like school?
- How can I teach students the skills they need when standardized tests require only facts?
- Why do students remember everything that’s on television and forget everything I say?
- Why is it so hard for students to understand abstract ideas?
- Is drilling worth it?
- What’s the secret to getting students to think like real scientists, mathematicians, and historians?
- How should I adjust my teaching for different types of learners?
- How can I help slow learners?
- What about my mind?
Some of these are yawn-inducing, but some are truly intriguing. Willingham certainly goes against the conventional wisdom in #5–7. He makes a compelling case for devoting more attention to practice (“drilling”) than is currently in vogue; he makes an even more compelling case for not “getting students to think like real scientists, mathematicians, and historians”; and he debunks the current orthodoxy that there are different types of learners, such as auditory, visual, and kinesthetic. In this last context he observes that Howard Gardner is usually credited with this distinction even though it comes much more from Gardner’s followers than from Gardner himself. I’ve never been convinced that there’s a good reason to refer to Gardner’s list as intelligences rather than talents, and Willingham also claims that “most psychologists think Gardner didn’t really get it right.” Most interestingly, he observes that not only Willingham but also Gardner himself disagrees with the following idea that is usually attributed to him:
Many or even all of the intelligences should be used as conduits when presenting new material. That way each student will experience the material via his or her best intelligence, and thus each student’s understanding will be maximized.
Finally, it’s worth mentioning Willingham’s observation that much of psychology can be learned from your grandmother.
And the award for coolest math video ever goes to…Vi Hart, for her Doodling in Math Class series.
These videos are “subversive,” as one of my colleagues (approvingly) labels them. The common theme appears to be that math classes in high school are boring, because they tend to focus on minute details rather than the big ideas of interesting mathematics. So how does the bored student react? By doodling, of course. But…as you watch the frenetically paced video, you realize that Hart is actually teaching the very mathematical concept that she pretends to be avoiding by doodling.
I’m not sure which one is my favorite. Perhaps it’s “Infinity Elephants,” which I’ll show to my precalculus class. Or perhaps “Binary Trees,” which would work both for that class and for Algebra II. Do check out all four, actually. And while you’re doing that, I’ll explore Hart’s three online publications, all of which look intriguing:
I’ll let you know about these after I’ve had a chance to explore them.
Quotation from one of my students: “I hate it when people don’t like negative attitudes.”
(Context: He had been solving a problem at the board and made a negative remark, causing one of his classmates to say that he was damaging the learning environment in the class. The above quotation was his response. He meant it seriously.)
A course ought to tell a story. If it doesn’t, it’s just a collection of topics, not a course.
Honors Precalculus at Weston definitely does tell a story. I was thinking about the themes of that story today, and I realized that a big one is the idea of expanding a domain to go more broadly and more deeply into a topic. We begin the year with a review of right-triangle trigonometry, where the domain of the sine and cosine functions is the interval from 0° to 90° (exclusive). We then expand it to the inclusive interval, then to obtuse angles, and then to all angles. Through the use of the unit circle and the switch to radians we have a domain that consists of all real numbers. At the end of the year we’ll expand the domain to complex numbers, through the use of infinite series.
In the meantime, we are turning to complex numbers. Over the years our notion of “number” has expanded from whole numbers to rational numbers to non-negative rational and irrational numbers to real numbers and now to complex numbers. Eventually we’ll break out of the idea of “number” altogether and will explore different infinities.
In the area of polynomials we’ve moved from linears to quadratics, and later this year we’ll explore cubics and beyond.
I’m sure fractals can fit this theme also, but that’s for another day.
Why do so many of my students use incorrect names for various polygons? They claim that they are merely recalling what they have been taught; maybe this is so, maybe not. I suppose there are two major possibilities:
- They are remembering incorrectly.
- They really were taught incorrectly.
Since this is Weston, I would prefer to believe it’s #1…but I have to admit that it might be #2, even in Weston.
Of course we shouldn’t just throw around the claim that certain names are incorrect without producing an argument for what the correct names are. Some of my students want to look in Wikipedia or count Google hits, but those methods lead to popularity contests, not truths. As I said in an earlier post, you can usually trust Wikipedia for mathematical information, but names occupy a middle ground between math and English, so Wikipedia is less reliable in this case than with pure math. As a better starting point, here is Wolfram Mathworld’s reasonably authoritative list of names for polygons with n sides:
| n |
polygon |
| 2 |
digon |
| 3 |
triangle (trigon) |
| 4 |
quadrilateral (tetragon) |
| 5 |
pentagon |
| 6 |
hexagon |
| 7 |
heptagon |
| 8 |
octagon |
| 9 |
nonagon (enneagon) |
| 10 |
decagon |
| 11 |
hendecagon (undecagon) |
| 12 |
dodecagon |
| 13 |
tridecagon (triskaidecagon) |
| 14 |
tetradecagon (tetrakaidecagon) |
| 15 |
pentadecagon (pentakaidecagon) |
| 16 |
hexadecagon (hexakaidecagon) |
| 17 |
heptadecagon (heptakaidecagon) |
| 18 |
octadecagon (octakaidecagon) |
| 19 |
enneadecagon (enneakaidecagon) |
| 20 |
icosagon |
| 30 |
triacontagon |
| 40 |
tetracontagon |
| 50 |
pentacontagon |
| 60 |
hexacontagon |
| 70 |
heptacontagon |
| 80 |
octacontagon |
| 90 |
enneacontagon |
| 100 |
hectogon |
| 10000 |
myriagon |
Let’s see what we can do with this list. I make the following observations:
- The very existence of a two-sided polygon sounds doubtful to most people. We’ll discuss this one below.
- Three- and four-sided polygons, being the most common ones, commonly have Latin names (triangle and quadrilateral), even though there are also alternative Greek names, which are very rarely used.
- All other polygons have Greek names. Therefore nobody ever calls a six-sided polygons sexagon or sextagon, and nobody calls a seven-sided polygon septagon, no matter what my students claim.
- For some mysterious reason, the 11-sided polygon is listed here not only as hendecagon (the correct name, from the Greek hendeca, meaning 11), but also with an incorrect alternative Latin-Greek name, undecagon. I see no reason to do this. In fact, another Wolfram Mathworld page makes this observation:
A hendecagon is an 11-sided polygon, also variously known as the undecagon or unidecagon. The term “hendecagon” is preferable to the other two since it uses the Greek prefix and suffix instead of mixing a Roman prefix and Greek suffix.
- Somewhat similarly, but worse, the 9-sided polygon is listed in both the Greek form, enneagon, and the hybrid, nonagon — but here Mathworld oddly prefers the Latin-Greek hybrid to the pure Greek. On their other page, however, they make this observation:
The nonagon, also known as an enneagon, is a 9-sided polygon. Although the term “enneagon” is perhaps preferable (since it uses the Greek prefix and suffix instead of the mixed Roman/Greek nonagon), the term “nonagon,” which is simpler to spell and pronounce, is used in this work.
Even though counting Google hits is a useless way to decide these issues, let’s check them out just for fun:
- 14,900 hits for “hendecagon”; 12,400 for “undecagon.” Hooray!
- 18,400 hits for “enneagon”; 69,500 for “nonagon.” Boo, hiss!
Oh — I also promised a discussion of two-sided polygons, didn’t I? Most people think they don’t exist, so they don’t need to be named. (Unicorns don’t exist, but they still have a name. Hmm….) Actually, however, they do exist: for example, start at the North Pole, draw a line segment along the prime meridian until it reaches the South Pole, and then draw another line segment from the North Pole along the 90° longitude line, also stopping at the South Pole. Voilà: a two-sided polygon! You may think I’ve cheated, since this polygon exists on the surface of a sphere, not on a plane, but it might be worth imagining that you lived on the surface of a sphere, not on a plane… Anyway, I’ve never heard the term digon before; I’ve seen biangle and bigon, however. Be sure to pronounce bigon with a long i, and think of the famous saying, “Let bigons be bigons.” Again we can check Google hits, useless though it may be: 14,600 hits for “digon,” 487 for “biangle,” and 7,330 for “bigon.” Even though “biangle” loses the popularity contest, I suspect that it’s the best choice, since it’s consistent with the general principle: use Latin names for polygons with four sides or fewer, Greek names for those with more than four sides, and hybrid names for none.
The Weston High School Library recently posted a slide show from Rutgers University explaining why students shouldn’t use Wikipedia. This carefully produced polemic deserves a thoughtful rebuttal; I have endeavored to write one here. Be sure to watch the slide show before reading the rest of this essay.
Those of us of a certain age will remember Hamilton Burger’s frequent cry of “Incompetent, irrelevant, and immaterial!” from the old Perry Mason TV shows. The Rutgers slide show on Wikipedia immediately prompted me to raise the same objection. First, however, I do need to acknowledge the considerable amount of truth in the Rutgers argument. Yes, of course there are many biased statements, inaccuracies, and downright lies on Wikipedia. Yes, it should not be cited as an authoritative source in a formal research paper. Yes, Wikipedia sometimes falls short when we’re looking for accuracy, authority, objectivity, and currency.
But that’s just one side of the picture. The authors of this tendentious slide show fail to meet their own criteria by ignoring the other side! Here, as Paul Harvey used to say, is the rest of the story:
- Every single example in the slide show comes from a narrow range of subjects in which the perceptive reader should immediately be aware of bias. Indeed Wikipedia should not be relied upon for information about history, politics, or biography — subjects where opinion is likely to substitute for facts, whether intentionally or inadvertently. But there are other subjects where Wikipedia is exceptionally reliable. In two fields that I know a lot about, mathematics and linguistics, it is unquestionably the first place to look for accurate information. Go to Wikipedia to find out about vowel harmony in Turkish, but not about political harmony in Turkey. Go there to find out about number theory, but not about the theory of evolution.
This is not to say that Wikipedia is 100% accurate — of course it isn’t. But so-called “authoritative” sources aren’t 100% accurate either. I recently gave an assignment in which I asked my freshmen to comment on some statements about geometry from presumably authoritative sources; these contained errors that I was unable to find on Wikipedia but quickly found elsewhere. Even textbooks are far from immune. I am reminded of the late Richard Feynman’s famous critique of a middle-school science textbook that contained questions like, “John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?” Of course professors like textbooks; they write them, after all. The rest of us should be skeptical of all sources, even textbooks.
- Regardless of the subject, Wikipedia is a fine location for beginning one’s research. It should be your first stop, not your last. I’m reminded of those advertisements in which a dealership or carpet store says, “Shop us last!” Aside from the newfangled use of “shop” as a transitive verb, which I can’t help noticing, I also observe that the slogan makes more sense than the “Shop us first!” that one sometimes sees. Wikipedia is indeed not the place to cite in your footnotes, but shop there first.
- Finally, the authors of the slide show make misleading use of their own sources. Why on earth do they cite Stephen Colbert of all people — not just once, but twice — as an authoritative source? Their whole argument is undermined by quoting a comedian in this role. Furthermore, although it is cute for them to cite the founders of Wikipedia in support of their argument, they do so in a highly misleading way. They quote one statement, “Wikipedia acknowledges that it should not be used as a primary source for serious research,” without emphasizing the word “primary”; of course it’s not a primary source, but it’s a great place to start in order to continue on to those primary sources. Then they quote Larry Sanger as a “co-founder of Wikipedia” as if he were still on board, whereas in reality he has every reason in the world to be bitter and biased; he is hardly an authoritative or neutral source. Finally, they quote Jimmy Wales (who is still very much on board) in a manner that is clearly incompetent, irrelevant, and immaterial: “For God sake, you’re in college; don’t use the encyclopedia.” Go read the original source, and it becomes clear that Wales is talking about encyclopedias in general, not just Wikipedia. He does say that Wikipedia is a good place to start, though you would never know it from reading this quotation that was taken out of context.
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