No, Weston teachers do not have the highest salaries in the state. According to today’s Boston Globe, Weston ranks only 16th in the state in average teacher salaries! At $73,338, we can be compared to a high of $79,444 (Old Colony), though we’re still well above the state median of $61,800. If you look at the alphabetical district-by-district listings, you can compare us to five of our immediately neighboring communities, and we’re higher than any of them:
| Weston |
$73,338 |
| Wayland |
$73,015 |
| Wellesley |
$71,128 |
| Newton |
$70,961 |
| Lincoln |
$69,778 |
| Waltham |
$65,017 |
The real problem is that statistics can be so misleading. The really relevant criterion is the salary scale, since these figures probably reflect more about the average age and teaching experience of the faculty than they tell you about the minimum or maximum salaries in any particular district.
“I make order out of chaos.” This is how an old friend whom I hadn’t seen in years explains her transition from linguistics to statistics, when people think it’s a complete change of field. It’s how she explains it to non-linguists, of course — as I already knew the connection. But the phrasing really resonates with me. I’ve described elsewhere how the search for patterns and abstract generalizations is what unites linguistics and math teaching in my mind, but I rather like the step up the ladder of abstraction implied by “I make order out of chaos.”
It also got me thinking about why I like the mystery and science fiction genres in popular fiction. My liking for science fiction is no mystery, so to speak: anyone with a mathematical bent is likely to enjoy the conventions of that genre. But what about mysteries? I’d been thinking about that lately, and it occurred to me that mystery writers also make order out of chaos: the unsolved crime is cognitively chaotic, and the solution creates order out of it. Furthermore, the puzzle that’s often involved bears definite kinship to the kinds of puzzles we solve in both math and linguistics. Just a thought…
“Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.”
So says Paul Lockhart, and I couldn’t agree more. It’s great having cooperative students who will correctly follow directions in solving problems — or should I say exercises — but following directions is a cheap virtue. As Lockhart observes, you don’t develop your mental faculties that way. On March 28, 2008, I wrote a brief laudatory piece about Lockhart’s fascinating essay, which he has now turned into an irritating book, also called Mathematician’s Lament. That’s too bad, as he has a lot of valid things to say. But most readers will be unable to see what’s good because it’s surrounding by so much that’s annoying. In particular, Lockhart seems to take an extreme view in favor of throwing out all curriculum and all direct instruction, replacing everything with student-directed problem solving. I say “seems to take” because in fact that’s not actually his position; it’s just that he gets so carried away with his radical POV that everything else gets lost. So, if you read this book, you need to star the following sentence in particular:
If I object to a pendulum being too far to one side, it doesn’t mean I want it to be all the way on the other side.
Keep that in mind. It’s just that everything Lockhart discusses is in fact all the way on the other side. Consider, for example, this provocative chapter title: “High School Geometry: Instrument of the Devil.” Certainly some students do like geometry, though Lockhart claims that those students would like it even more if it were taught his way. And surely most adults remember their high school geometry class with something less than fondness. The big complaint about high school geometry — and here I agree with Lockhart — is that the central themes of proof and definition are presented so woodenly. Writing proofs about claims that are obvious feels arbitrary and useless, and yet that’s what most of the early months of geometry are filled with. And the two-column format is arbitrary and restrictive, a peculiar American custom that no real mathematician would ever use. As Lockhart observes, “A proof should be an epiphany from the gods, not a coded message from the Pentagon.” But it’s rare experience in high school geometry for students to spend a long time struggling with a non-obvious problem, then to come up with a non-obvious conjecture, and finally to write a convincing proof that shows how the conjecture connects with other knowledge. That’s how it should be done.
A similar issue occurs with definitions:
Definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem-generated. To make a definition is to highlight and call attention to a feature or a structural property. Historically this comes out of working on a problem, not as a prelude to it.
Hear, hear!
All of this, of course, is driven by one’s concept of what math really is. Lockhart is a pure mathematician, viewing problem-solving and puzzle-solving as rewarding for their own sake, and I agree with him there. But his contempt for applied mathematics will do nothing but turn off most of his readers. It’s important for students to understand that applications come after the math is developed and hardly ever motivate the discovery of new mathematics, but it’s also important for them to work with those applications. Some students will be motivated by that, and everyone will learn something that their future teachers will expect. Nevertheless, Lockhart’s characterization of what math really is is spot on:
Math is not about a collection of “truths” (however useful or interesting they may be). Math is about reason and understanding.
Unfortunately this characterization flies in the face of so much of what is expected of math teachers and math students. MCAS and SATs and science teachers inadvertently encourage the “collection of truths” misconception, even though they of course also want reason and understanding.
Finally, I need to mention the subtitle of Dan Meyer’s blog, dy/dan. The subtitle is simply less helpful. This characterization may seem like an odd one, especially when it’s the subtitle of a blog that’s well worth reading. But Meyer’s resolution to be less helpful is an important one. Like most math teachers, my unthinking inclination is usually to try to be helpful, to answer questions, to point students in the right direction. But Lockhart’s response to a certain question from a student is to observe that “the right thing for me to do as your math teacher would be nothing.” In other words, to be less helpful. That, in the long run, is what will actually be helpful to the student. I just wish that Lockhart had limited himself to a more tempered criticism and had been clearer about taking a balanced approach; he will turn off too many readers who would have a lot to gain from his wisdom if they could only pay attention to what’s good rather than what’s irritating in this provocative book.
Ten years ago, the highly respected mathematician Lynn Arthur Steen wrote an article entitled, “Algebra for All in Eighth Grade: What’s the Rush?” Well, now we know what the rush is…or do we? Steen sets up the issue with a couple of rhetorical questions:
How can a subject that for many adults serves as a metaphor for frustration suddenly be the top priority for soccer moms and internet dads? And why do so many parents suddenly demand of their schools and their children something they themselves neither mastered nor loved?
He then proceeds to give several arguments in favor of algebra: it provides access to higher education and jobs, it is the language of the information age, it is the mark of a rigorous education… in short, it is “the key to access in our technological society.”
But then come the counterarguments:
- Relatively few students finish seventh grade prepared to study algebra. At this age students’ readiness for algebra — their maturity, motivation, and preparation — is as varied as their height, weight, and sexual maturity. Premature immersion in the abstraction of algebra is a leading source of math anxiety among adults.
- Even fewer eighth grade teachers are prepared to teach algebra. Most eighth grade teachers, having migrated upwards from an elementary license, are barely qualified to teach the mix of advanced arithmetic and pre-algebra topics found in traditional eighth grade mathematics. Practically nothing is worse for students’ mathematical growth than instruction by a teacher who is uncomfortable with algebra and insecure about mathematics.
- Few algebra courses or textbooks offer sufficient immersion in the kind of concrete, authentic problems that many students require as a bridge from numbers to variables and from arithmetic to algebra. Indeed, despite revolutionary changes in technology and in the practice of mathematics, most algebra courses are still filled with mindless exercises in symbol manipulation that require extraordinary motivation to master.
- Most teachers don’t believe that all students can learn algebra in eighth grade. Many studies show that teachers’ beliefs about children and about mathematics significantly influence student learning. Algebra in eighth grade cannot succeed unless teachers believe that all their students can learn it.
So, where does this leave us? Steen’s conclusion is a sensible one: everyone should take algebra, but not necessarily in eighth grade. As the title of his article asks, What’s the rush?
The rush is that many states, including California and Massachusetts, are now mandating algebra in eighth grade, which moves the argument from whether we should implement this to how we should implement it; Steen’s four bullet points are real, and passing laws won’t wash them away. This is not to say that all eighth-graders really do study algebra, but Weston is surely not the only system in which Algebra I is simply not even offered at the high school: we expect all incoming ninth-graders to enter with an Algebra I background. I don’t know about other school systems, but Weston has attempted to address all four of Steen’s points, though the frst three are of course easier to remedy than the fourth. Nationwide about one third of eighth-graders study algebra, for better or for worse. Weston, of course, is Lake Wobegon, so all of our students are capable of learning algebra in eighth grade.
For much more depth of this question, read Tom Loveless’s article, “The Misplaced Math Student: Lost in Eighth-Grade Algebra,” or his full report from the Brown Center of the Brookings Institution. Here are a few interesting excerpts from this 16-page document:
At first glance, this appears to be good news… Research also suggests that students who take algebra earlier rather than later subsequently have higher math skills. These findings, however, are clouded by selection effects — by the presence of unmeasured factors influencing who takes algebra early and who takes it late…
The push for universal eighth-grade algebra is based on an argument for equity, not on empirical evidence. By completing algebra in eighth grade… students are able to take calculus in the senior year of high school… From this point of view, expanding eighth-grade algebra to include all students opens up opportunities for advancement to students who previously had not been afforded them, in particular students of color and from poor families. Democratizing eighth-grade algebra promotes social justice.
…
One catch. Course-taking is a means to an end, not an end in itself. Students take math courses to learn mathematics. Will policies mandating algebra for all eighth graders mean that the nation’s students learn more math? Not necessarily…
Loveless then goes on to cite statistics that show that “the typical eighth grader in an advanced math course knows less today than in 2000.” Hmmm…
…Any teacher who stops to teach misplaced students fractions shortchanges the well-prepared students who sit in that algebra class… There will be advocates, despite the data presented here, who will continue to argue for placing low-performing eighth graders in algebra classes. They believe that a more rigorous course is always preferable to a less rigorous one. Many do not believe that students must learn basic mathematics in order to successfully tackle higher-level mathematics… Algebra teachers already feel the strain of such unrealistic expectations.
Anyway, do read the entire article.
Here are some excerpts from Loveless’s conclusion:
One hundred twenty thousand students are misplaced in their eighth-grade math classes. They have not been prepared to learn the mathematics that they are expected to learn… Two groups of students pay a price. The misplaced eighth-graders waste a year of mathematics, lost in a curriculum of advanced math when they have not yet learned elementary arithmetic… Their clasmates also lose — students who are good at math and ready for algebra. These well-prepared but ill-served students also tend to be black and Hispanic and to come from low socioeconomic backgrounds. Teachers report that classes of students with widely diverse mathematics preparation impede effective teaching, that too many students arrive in algebra classes unmotivated to learn… Universal eighth-grade algebra is creating more problems than it solves, with 120,000 students not learning the mathematics that they need to know and hundreds of thousands of their classmates paying an educational price along with them.
Fortunately Weston is different. But read the whole article, as I said above.
On a slightly different but closely related matter, I need to mention a comment I overheard at the next table at Tavolo: “I don’t understand why kids have so much trouble with algebra. It’s nothing but finding the value of x.” No, that’s not what algebra is about. Sigh.
Some of us can barely remember anything from third grade, but last night at a restaurant in Dorchester I met someone my age who was truly traumatized for life by a single experience way back in third grade. We’ll call her Laura. When she found out that I’m a math teacher, she had to tell me her story. It went something like this:
A couple of weeks into second grade, Laura’s teacher determined that she was so bright that she should skip a grade, and so Laura was instantly promoted to third grade — with the approval of her mother, but the frowning disapproval of the third-grade teacher, whose plans and groups were all messed up by this unexpected child. The third-graders had been adding two-digit numbers with carrying, but of course Laura didn’t know how to do this, since she had missed all but the first two weeks of second grade, not to mention the beginning of third. When she was unable to do the assigned problems, the teacher called her up in front of the room and said to the class, “Laura thinks she’s so smart because she skipped a grade, but in fact she’s stupid. She can’t even add 28 and 47.”
And to this day — despite success in future math courses, and eventually getting into med school — Laura has a phobia about math.
This is the cue for my students to roll their eyes… Yesterday I got into a heated discussion with another math teacher about an important issue: how to define a trapezoid. He was arguing in favor of the position that a trapezoid has exactly one pair of parallel sides; I was arguing in favor of the position that a trapezoid has at least one pair of parallel sides. We both agree that it’s a quadrilateral.
My opponent made several good points:
- Our current textbook defines the word his way.
- So do some other textbooks.
- The common image of a trapezoid has two non-parallel sides.
- We don’t expect someone to look at a parallelogram and exclaim, “That’s a trapezoid!”
But I made, IMHO, several better points:
- Nowhere else do we define a geometric object in this exclusionary way. We don’t say that a rectangle cannot have four congruent sides. We all agree that squares are rectangles, rhombuses are parallelograms, circles are ellipses, etc.
- Many textbooks, including Moise’s and UCSMP, do define it my way.
- In the software we use, the Geometer’s Sketchpad, it’s straightforward to construct a trapezoid with my definition but not with his.
- Most importantly, the quadrilateral hierarchy should show parallelograms as a subset of trapezoids because theorems about trapezoids also apply to parallelograms.
Of course it all leads to a teachable moment — or more than a moment, actually. In my honors geometry class we devoted more than an hour of class and homework time to exploring the ramifications of the two definitions. Then the issue emerged in a question on the next quiz:
Always/Sometimes/Never: Under Mr. Davidson’s preferred definition of “trapezoid,” the diagonals of a trapezoid are congruent.
And then in a four-part question on the next test:
- Define “trapezoid” as the textbook does.
- Define “trapezoid” in the way that Mr. Davidson prefers.
- Former Harvard professor Edwin Moise has the following theorem (not definition) in his book: “A trapezoid is a parallelogram if its diagonals bisect each other.” Can you tell which of the two definitions of “trapezoid” must have preceded this theorem? Explain convincingly.
- Prove Moise’s theorem (using whichever definition you identified in part c).
Bonus: little did I realize that the embedded “if” in Moise’s theorem would confuse some students. Apparently they had never seen a theorem where the consequent preceded the antecedent. So that led to a worksheet for another assignment.
Thinking about what you’re learning is a good thing. And, as Humpty Dumpty said in Through the Looking Glass, “When I use a word, it means just what I choose it to mean — neither more nor less.”
This pie chart from Fox News speaks for itself:
At this week’s Math Department meeting, we spent the first 15 minutes or so discussing what we do to help “struggling students” succeed in our courses — particularly what resources we provide. Something was bothering me about the whole discussion, so I waited a few minutes before I said anything. Then I realized what was bothering me: the participle “struggling” was apparently being used as a synonym for “unsuccessful.”
This usage has long seemed completely wrong to me. To my mind, I have some students who struggle and do well. I also have some students who are unsuccessful — precisely because they don’t struggle.
It all comes down, of course, to the meaning of the verb “struggle.” Let’s see what a couple of reputable dictionaries say about the matter. In each case I’ve selected the appropriate sense of the word:
- to make strenuous…efforts in the face of difficulties… <struggling with the problem>
- to proceed with difficulty or with great effort <struggled through the high grass> <struggling to make a living>
—Merriam-Webster
- to be strenuously engaged with a problem, a task, or an undertaking
- to progress with difficulty <struggled with calculus>
—American Heritage Dictionary
Linguists, of course, always insist on being descriptive rather than prescriptive, and yet they usually rely on introspection or on the use of a small number of informants. I suppose a more accurate technique in this context would be to survey a large number of people and find out how they use the word “struggle”; I have no idea what we would find, but at least the dictionary definitions make it absolutely clear to me that we should stop using this verb as a synonym for “be unsucessful.”
On another front, we spent the next 25 minutes of the department meeting discussing how to solve the equation x2 = 2x. I told my Algebra II class about this, since we’re currently transitioning from quadratic functions to exponential functions, and one of their homework problems called for a comparison between y = x2 and y = 2x. They found it an unlikely topic for a meeting — and they were especially surprised that we were so geeky that the meeting ran ten minutes over before anybody looked at the clock and noticed that we had gone past the announced end of the meeting.
By the way, there are three solutions to this equation. One solution, 2, is immediately obvious; a second solution, 4, is not at all obvious until you give it some considerable thought, at which point it “becomes obvious.” The third solution can be estimated by looking at a graph. Finding this solution is left as an exercise for the reader.
A major topic of high-school math is the study of transformations. My colleague, Jim McLaughlin, wants you to know that his desk has somehow undergone a miraculous transformation:
 |
 |
| Before |
After |
I was recently asked whether a Boston voter should always vote for the full allotment of four at-large City Council candidates, or whether bullet voting made sense. I unhelpfully replied, “It depends.”
It occurred to me that I had already dealt with this issue four years ago. So read that link if you want to read the mathematical arguments (actually, not too much math!) for or against bullet voting, depending on the situation.
I don’t know why I never knew about the Cornell Math Explorers Club before now. Its website is a terrific enrichment resource for high-school math students and their teachers, with a wonderful assortment of slightly offbeat topics that are right up my alley for all the programs in which I teach: Weston High School, Crimson Summer Academy, and Saturday Course. Their modules include cryptography, graph theory, probability, set theory, topology, game theory, and the mathematics of voting and elections. I wish I had known about this site earlier. Check it out!
On the other hand, the season opener of Numb3rs — Season Six, which is hard to believe! — was pretty good, even it was skimpy on the math and a bit long on tensions between Charlie and Amita. But this is television, after all.
Math content included Fibonacci spirals in nature, and the Unexpected Hanging paradox in its original form. In order to make the latter less gruesome and more relevant to students, it is usually changed to an unexpected quiz, as Jim Loy points out in the linked article. It goes something like this:
I’m going to give you a pop quiz next week, and I guarantee that it will come as a surprise to you. Of course I can’t give it on Friday, since if you walk into class on Friday and haven’t had the quiz yet, it will no longer be a surprise. So you rule out Friday.
Could I give it Thursday? Since you’ve already ruled out Friday, you will expect the pop quiz when you walk into class on Thursday. Then it won’t be a surprise. So I can’t give it Thursday.
By the same reasoning, I can’t give it Wednesday. Similarly, I can’t give it Tuesday. Or even Monday!
So, I guess I’ll just have to give the quiz right now.
And I hand out the quiz.
Where’s the flaw?
No, not football — too late for that.
And not basketball — although it’s the right season for that.
I’m talking, of course, about the New England Association of Math Leagues Playoffs, which took place today at Canton High School. I’m not sure yet just how Weston High School did, but we were in sixth place after the individual rounds and just before the final round (the team round, and I don’t have its results yet). In fact, we were close to fifth, so maybe we finished fifth. Or maybe not. At any rate, congratulations to our team for an excellent job: Alex Bruce ’09, Katie Hsia ’09, Ernest Zeidman ’10, Jonathan Birjiniuk ’11, Grace Huckins ’12, and Stephanie Palocz ’12!
A few years ago, one of my former students from Honors Precalculus informed me that my course had been “unnecessarily difficult.” An interesting phrase. “What does that mean?” was my puzzled response. Let’s call her Rachel (not her real name).
It turned out that Rachel interpreted the name of the course rather too literally; she saw no reason for us to do anything more than the bare minimum to prepare her for calculus. This was AB Calculus, of course; she wasn’t interested in all the parts of my course that would have prepared her for BC Calculus had she chosen to take it. So I explained that to Rachel, but I also pointed out that every math class has many goals, only one of which is the need to prepare students for the next course in sequence. What else did she get out of precalculus? (I thought she might say “a B minus”…but in fact she just shrugged and repeated her criticism.)
This all made me wonder yet again about the value of feedback from students. I learned something about Rachel, but I learned almost nothing that I could use in rethinking the course. (Actually, I did learn something useful: I realized that I should be more explicit about the goals of the course when talking to students.) Nevertheless, when it came time recently to give questionnaires to students, I of course complied with this requirement as did all Weston teachers. (Question to my current students: They all did, didn’t they?) We were allowed to develop our own questionnaires, and we could share as much or as little of the results as we felt appropriate. I chose to share all the results, both with my students and with my department head, and I’ll present a few interesting excerpts here.
First, as I have already mentioned Honors Precalculus, let’s see what I learned from the 43 students who responded (that’s a 100% return from both sections — one of 23, the other of 20). It’s hard to tally open-ended responses, since a prompt like “The best things about this course so far have been…” can elicit anything from a topic to pedagogy to teaching style. I would surely have gotten different results had I prompted for responses to a particular topic or a particular type of lesson. Nevertheless, it intrigued me that 5 students listed groupwork among their “best things about this course” and 6 listed it among their “worst things about this course.” Hard to know what to do with that. So let’s abandon the four open-ended questions and move on to Part Two, which presented students with a couple of dozen statements, each of which they could agree with or not. Here are a few thought-provoking numbers, showing how many agreed with each statement:
| 28 |
I feel I have worked as hard as a reasonable person could expect of me. |
| 8 |
I feel I have worked harder than a reasonable person should expect of me. |
| 7 |
I feel I have not worked as hard as a reasonable person would expect of me. |
| 6 |
More class time should have been devoted to working on assignments/worksheets individually. |
| 12 |
More class time should have been devoted to working on assignments/worksheets in groups. |
| 23 |
More class time should have been devoted to lectures/discussions. |
| 30 |
More class time should have been devoted to going over homework at the board. |
| 6 |
More class time should have been devoted to going over homework in groups. |
| 8 |
More class time should have been devoted to tests and quizzes. |
One student agreed with all six of this last set of statements! Where would we find the time? Anyway, the rest of these numbers may be thought-provoking, but it’s not necessarily clear what to do in response.
Now let’s turn to my other course, Algebra II, which is a “college-prep” course at Weston. (As discussed elsewhere, all non-honors courses at Weston are “college-prep” and tend to attract an enormous range of students: by definition all those who choose not to take honors or are unable to do so. Since Algebra II is a graduation requirement, no one escapes it, even those who are weakest at math as well as those few who are not intending to go to college.) I have two sections, one of 21 and one of 24; 41 of the 45 responded. Here are some numbers from those 41 in response to the same statements quoted above:
| 24 |
I feel I have worked as hard as a reasonable person could expect of me. |
| 6 |
I feel I have worked harder than a reasonable person should expect of me. |
| 11 |
I feel I have not worked as hard as a reasonable person would expect of me. |
| 11 |
More class time should have been devoted to working on assignments/worksheets individually. |
| 20 |
More class time should have been devoted to working on assignments/worksheets in groups. |
| 7 |
More class time should have been devoted to lectures/discussions. |
| 12 |
More class time should have been devoted to going over homework at the board. |
| 12 |
More class time should have been devoted to going over homework in groups. |
| 8 |
More class time should have been devoted to tests and quizzes. |
What do these results tell you (if anything)?
I was just thinking about some of the difficulties that many high-school students have when attempting to learn math. Aside from those who face external obstacles — such as brain damage, severe emotional problems, or extremely inadequate teaching — we have those who don’t work hard enough and those who do. It’s easy to dismiss those who don’t work hard enough (“just work harder,” we advise, even though that might or not might help and they might or might have time), so let’s focus on those who have difficulty even though they have no visible external difficulties and they do enough work in math. “Enough” differs from student to student, of course, so we won’t try to quantify it. We’ll just say that if you do enough work, your problem is not that you need more practice.
What I’ve observed is two diametrically opposite subgroups of people who work hard but still have trouble with math: those who miss the trees and those who miss the forest.
- The first subgroup, which is overwhelmingly male (but by no means exclusively so), contains those who are apparently following Tom Lehrer’s satirical advice: “The important thing is to understand what you’re doing rather than to get the right answer.” These students get the big picture, or at least claim to do so, but they haven’t learned the details. They often say that they’ve made “dumb mistakes” and confidentally plan to do better on the retake with no additional studying. They correctly view math as a field with Big Ideas, but they incorrectly believe that that’s all there is. Because they can’t reliably execute the Big Ideas in the form of solving problems, they can’t actually apply what they know (or think they know). Usually they are deceiving themselves about their level of understanding; often they are not merely making “dumb mistakes” but actually are facing gaps in their knowledge. But it’s hard for them to improve, since they feel assured that they understand what’s important, and the rest is just details.
- The second subgroup, which has a slight preponderance of females (but not significantly so), contains those who focus on rote learning of algorithms. They view math as a bag of tricks, a collection of skills that can be mastered without context. They don’t see the big picture and often don’t believe there even is a big picture. If they have unimaginative teachers who always test them on problems that are exactly like the ones they’ve already seen with only the numbers changed, they are misled into thinking that they are successful learners of mathematics. And then they have a rude awakening when they eventually find that skills aren’t sufficient: understanding is also necessary. As John Holt put it, “The true test of intelligence is not how much we know how to do, but how we behave when we don’t know what to do.”
Just about everyone can speak, so we all have an opinion about language. Just about everyone can count, so we all have an opinion about math. Everyone’s an expert. After reading uninformed opinions about both, I decided to compare and contrast. Off the top of my head I can think of three points of comparison and four points of contrast; if I were writing a thesis on this subject I would undoubtedly find many more.
First let’s look at some similarities that go beyond what I hinted at in the first paragraph:
- For some reason, people who state their uninformed opinions in public almost always take the conservative side of issues in both of these areas. I suppose that shouldn’t surprise me, given the general tenor of Rush Limbaugh, Fox News, and other instances of talk radio and its television allies. But it’s still striking that your typical layperson wants to preserve or resurrect obsolete grammar “rules,” wants dictionaries to be prescriptive rather than descriptive, thinks that some languages are inferior to others, etc. And it’s equally striking that the same typical layperson advocates “back to basics” approaches in mathematics and thinks that math is all about memorization of facts, rote algorithms, and the like.
- Members of the general public are confused equally about what linguists and mathematicians do. They think that a linguist is someone who speaks several languages fluently, so why should they ask a linguist for an expert scientific opinion about language in general or English in particular? Similarly, they think that a mathematician is someone who calculates quickly and accurately, so why should they ask a mathematician for an expert scientific opinion on learning mathematical concepts?
- Because the general public understands neither subject, they are confused about the connections between the two. The fact that I moved from studying linguistics to teaching math sounds like a complete change of direction, not a natural evolution. It’s the rare person who sees cryptology as a connection or bridge between the two, or who understands that in some sense math is a language (only in some sense, I hasten to add), or who can imagine that one can apply mathematics in any way to the study of language.
So much for the similarities. Are there ways in which we can contrast the general public’s opinions about language and math? Sure. Here are a few:
- People freely profess ignorance about math. When someone at a party asks what you do, and you say that you’re a math teacher, the most common response is, “I was never any good at math.” Rarely are they ashamed. Often they even seem proud of it. But no one admits to be ignorant about reading, writing, and speaking.
- On what might be the same lines, people at least have some idea that there is such a thing as a mathematician, even if they don’t know what mathematicians do. (“A mathematician is a device for turning coffee into theorems,” according to Alfred Renyi.) But when I tell people that I majored in linguistics, the usual response is either a blank stare or “What’s that?”
- Perhaps as a consequence of the previous observation, a great many myths about language float around in the popular culture, but there are not nearly so many myths about math.
- Everyone thinks they can teach English — they can’t, of course — but very few people think they can teach math.
Perhaps in a later post I’ll give some citations to exemplify these seven points, but this will do for now.
Congratulations to the Weston High School math team for their excellent showing in the state playoffs on Friday in Shrewsbury! We finished fourth in the state among medium-sized high schools (the schools with which we compete) and will therefore be going on to the New England finals in Canton in early May. The state playoffs are particularly tough compared to the earlier season since all six individual rounds are totally calculator-free (usually there’s just one such), though the team round allows calculators.
We send our top six mathletes to both the state playoffs and the New England. Please recognize the fine work of Jonathan Birjiniuk ’11, Alex Bruce ’09, Katie Hsia ’09, Grace Huckins ’12, Stephanie Palocz ’12, and Ernest Zeidman ’10 — and wish them the best in Canton!
Seven is a lucky number, so no one was surprised that the seventh annual Fractal Fair at Weston High School turned out to be the best one so far. Of course there were many great exhibits in each of the previous fairs, but never was there such a consistently high level of quality and enthusiasm; there was not a mediocre or poor project among them! And, in contrast to last year, there were no tears.
OK, let’s agree that it wasn’t actually a matter of luck. Here in the reality-based community we look for genuine explanations, not fanciful ones. The fair got rave reviews from many visitors, especially colleagues and parents of students. While my co-teacher and I would love to take credit for that, we can’t: we don’t deserve more than a tiny fraction of the credit. Yes, we did tweak the requirements a bit — for example, each group had to come up with a question that would let them assess whether visitors and listeners got the “Big Idea” of their project, each student had a structured opportunity to see their classmates’ projects, and we scheduled the fair at a better time — but the lion’s share of the credit has to go to the students. They consistently produced projects that were interesting, accurate, attractive, and thorough. No group was excessively ambitious (in the past there were always some who bit off more than they could chew); no group settled for making sloppy posters that were marred by typos or spelling errors. Every group was able to explain their work, and visitors noted their enthusiasm in doing so. Congratulations to all!
My students sometimes ask me whether the mathematics in the television show Numb3rs is real. This question, among others, is explored in a fascinating book, The Numbers behind Numb3rs: Solving Crime with Mathematics, by mathematicians Keith Devlin and Gary Lorden. Most of the book consists of excursions into specific mathematical topics that arise in various episodes of the show; these excursions are really spinoffs, as they take an idea (that might be mentioned in passing or might be the entire basis of an episode) and discuss it in an accessible manner that goes far beyond anything on the show. If you’re interested in real-world applications of math, these chapters are well worth reading for their own sake, even if you’ve never watched Numb3rs.
But I specifically want to comment on my students’ question with regard to what’s in this book. Not surprisingly, the authors get asked the same question that I do. Here are some excerpts from their answer:
Is the math in Numb3rs real?
Both of us are asked this question a lot. The simplest answer is “yes.” The producers and writers go to considerable lengths to make sure that any math on the show is correct…
A more difficult question to answer is whether the mathematics shown could really be used to solve a crime in the way depicted. In some cases the answer is a definite “yes.” Some episodes are based on real cases where mathematics actually was used to solve crimes…. But even when an episode is not based ona real case, the use of mathematics depicted is generally, though not always, believable — it could happen… The skepticism critics express after viewing an episode is sometimes based on their lack of awareness of the power of mathematics and the extent to which it can be applied.
In many ways, the most accurate way to think of the series is to compare it to good science fiction: In many cases the depiction in Numb3rs of a particular use of mathematics to solve a crime is something that could, and maybe even may, happen someday in the future.
So there! Read the book for more details.
But the views of Devlin and Lorden may be out of date. A more recent and contrarian view comes from Mark Bridger, a mathematician at Northeastern University who maintains a blog about Numb3rs, from which these excerpts are taken:
January 3, 2009:
Last Friday’s Numb3rs was a repeat of the exciting episode “The Chinese Box” — aired December 14, 2007. This was yet another show where either the math consultants made a bunch of mistakes or the writers garbled the technicalities…
Since the Numb3rs folks eliminated independent script reviewers — mathematicians such as yours truly — the show’s math has gotten very sloppy, to put it politely. As far as I know, the math these days is injected exclusively by the Wolfram people. They seem prone to making mistakes, but Big companies such as CBS-Paramount like to deal with other Big companies such as Wolfram, not with individuals whom they have no control over. (And Wolfram gets to advertise its product Mathematica on the CBS website.) So what else is new?
December 15, 2007:
…Charlie whines that people are dissing him, and that he sees exactly what’s going on but can’t put it into words. This is an aspect of Charlie’s personality we have not seen before. The whole point of mathematics is to elucidate the structure of things. To say “I see things but can’t explain them” is pre-mathematical; Charlie can hardly expect people to recognize an expertise that he can’t communicate…
Now we come to some actual mathematical topics. Charlie describes a game called “Chomp” in which players take turns removing cookies from a grid… Exactly how this is relevant to the situation in the elevator is unclear, but at least there is mathematics here. Charlie identifies Sinclair with the “first player,” who makes his first move by getting into the elevator. Of course, we already know that it is not known what a winning first move is in Chomp, so I don’t see the analogy. Then Charlie throws in a real clinker: “Chaos Theory holds that outcome is sensitive to initial conditions. We must restore the decision making process to the man who started it.” This is a total non-sequitur. Yes, it’s true that a chaotic process is very sensitive to initial conditions: a small change in the beginning set-up can result in a tremendous change in the outcome. But how do we know that the Chomp game — or the elevator hostage situation — is chaotic? It would seem just the opposite: we simply don’t know what effect the first player’s first move will have: we just know that, as the game progresses, the first player can force a win. Furthermore, Sinclair stepping or not stepping into the elevator can hardly be described as a small change in initial conditions. On the other hand, Charlie’s conclusion turns out to be exactly correct: return the decision-making process to Sinclair. That’s exactly what the FBI doesn’t do, nearly resulting in Sinclair’s death (only his bullet-proof vest saves him).
All this reflects a disturbing trend in the show. Instead of using mathematics to solve the kind of physical or logical problems that are its natural setting, Charlie is trying to apply it to human behavior in complex situations. This is over-reaching, and the results simply do not ring true. In the early days of Numb3rs (season I, May 6, 2005) there was an episode called “Sacrifice” in which a young computer scientist kills his boss because the senior scientist is developing a program that uses mathematics to profile neighborhoods — this in order to determine where federal education money would be best spent. Charlie is admonished to look at the nature of his own research to see if he is not misusing mathematics to make social projections. It is interesting that he is, in recent shows, routinely using game theory, profiling, and data-mining to do just that: predict how humans will behave. We see once again, as in the “Chinese Room,” that the nature of human thought, behavior and language is very complicated and difficult to pin down. It can be very dangerous to exaggerate what we know and (think) we can predict.
It’s been a stressful week here in Lake Wobegon, but February break is finally almost upon us; now I have a chance to catch up on things from the past few days. Of course it’s always nice to see the occasional reference to mathematics in the Boston Globe, and there was an interesting article on Monday that was primarily about math, although you might not know that from the article. That’s excusable, since its topic — theoretical computer science — isn’t necessarily considered part of mathematics. Note the emphasis that reporter Carolyn Johnson places on science and scientists rather than mathematics and mathematicians:
…for some scientists, knowing what we can’t figure out is just as important.
A little-known discipline of science called computational intractability studies the boundaries of our understanding…
Johnson has interviewed several theoretical computer scientists, including Scott Aaronson, whose blog I have included in my blogroll for some years now (under Mathematics, not Science, by the way). But it would be unfair to accuse Johnson of ignoring mathematicians altogether. For example:
The Clay Mathematics Institute in Cambridge is offering a $1 million prize as part of its “millennium awards” to the person who can prove that problems that seem unsolveable given today’s computational tools are — in fact — intractable.
Sometimes what is intractable looks deceptively straightforward. Take the challenge of making housing assignments. Imagine that there are 100 spots in dorms, and a pool of 400 students to choose from. The dean has a list of pairs of students who are incompatible and cannot appear on the final list.
“The total number of ways of choosing students 100 from the 400 applicants is greater than the number of atoms in the known universe!” the Clay Mathematics Institute wrote in its $1 million challenge. “Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force.”
Well, yes, but I think that Johnson partly misses the point. Computers can’t and won’t solve it by brute force; no one has yet figured out how to solve it by other means, so at the moment it remains intractable, but that doesn’t make it impossible. No one knows yet whether problems that are currently intractable will remain so forever — but if they become reasonable to solve by non-brute-force means they will no longer be intractable.
Despite all that, it was good to see an article about mathematics. Perhaps the million-dollar prize will attract some people; I can think of certain students of mine…
In all six sections of college-prep Algebra II (taught by three teachers, with two sections apiece), we have just completed a project in which each student has to understand a scenario (written by one of my colleagues), complete some mathematics with exponential and other functions, and use the mathematics to support a political argument:
Protect the Earth, an environmental advocacy group, has just offered you a short-term contract. Coincidentally, an energy corporation, Mr. Burns Inc., has made a similar offer. Each employer is willing to pay you $2000 for two weeks of part-time work! You will have to decide which to accept.
The money’s the same, so you want to know what the responsibilities are. Surprisingly, they seem to be the same for both jobs! Each employer wants you to get an interview with Senator Kerry to discuss a proposal by Mr. Burns Inc., which wants to get permission to dump 400 pounds of Strontium-90 into a vacant field in Weston. Strontium-90 has a half life of 28 years. The corporation is willing to provide a number of incentives in exchange for permission to dump the waste. These include paying Weston $5,000,000 for care of the dump site; a payment for each year thereafter of $10,000 less than the previous year’s amount; and free glow-in-the-dark bumper stickers with funny sayings such as “I’d Rather be Fission”, “Nuke-u-lar energy lights up my life”, and “I went to Three Mile Island and all I got was this lousy third eye.”
If you accept the Protect the Earth job, you will be arguing against Mr. Burns. If you accept his job, you will of course be arguing in favor of his offer.
Whichever job you take, it has two parts. First, make an advertisement to sway public opinion to your side. It could be a PowerPoint presentation, a video, a pamphlet, a newspaper ad, or a letter to the editor. (Any other ideas must have prior approval from your teacher.) Second, hold an interview with Senator Kerry. (Since he may be unavailable, a Weston teacher might take his place.) The senator or the teacher will have some specific questions for you to answer, so you need to be prepared!
Part One must include the following information, showing all computations and any other work. (Think carefully about a useful time frame and scale.)
- An appropriate graph displaying the amount of Strontium-90 with respect to time.
- An equation that will let you compute the amount of Strontium-90 after n years.
- An appropriate table showing the amount of Strontium-90 with respect to time.
- An equation and table that represent the payments from Mr. Burns Inc.
- The exact amount of Strontium-90 when the payments from Mr. Burns Inc. run out.
- The exact number of days when the Strontium-90 reaches 10 pounds in weight.
- The percent decrease between the years 2009 and 2029.
- The percent decrease between the years 2209 and 2229.
- A clear explanation of how any or all of these support your argument.
After handing in Part One, each student was interviewed by a Senator Kerry surrogate: a Weston High School math teacher other than the student’s own teacher. What was particularly interesting was that we managed to get most of the department participating in the interviewing. It’s all too rare in the culture of schools for teachers to have anything to do with their colleagues’ students, since we all tend to close the classroom door and teach within our own castles. So it was a great experience for my students to be interviewed by teachers other than myself, and it was a great experience for those teachers to see what we’re doing in Algebra II and to see how well these sophomores and juniors can (or cannot, as the case may be) explain their conclusions in their projects.
After I finish reading and grading the projects, I will let you know how they turned out.
This post is a follow-up to my post of November 30, where I brought up two points that can illuminate one’s views on the big ideas of algebra:
…we discussed the assignment of partial credit for work in solving a problem — more on this later, but it definitely reflects one’s views on what the big ideas are — and whether the study of algebra is distinct from (and prior to) the study of functions…
Partial credit doesn’t sound like a deep issue, but it really is. All you have to do is gather a group of math teachers, give them a student’s solution to a problem, and ask how many points should be assigned. Regardless of whether it’s out of four (where there are only three partial-credit possibilities) or out of ten (where there are nine), there will be significant disagreements; I reached this conclusion from having participated in such activities many, many times, with various groups of colleagues. And I don’t just mean that one teacher will give two out of four and another will give three, or that one will give seven out of ten and one will give five. No, I mean that one teacher will give nine points and one will give zero! And this is in mathematics, which is supposed to be an objective discipline — unlike English, where such disparities might not be surprising.
So, what does it mean when major disagreements surface in this area?
It usually means one (or both) of the following types of differences:
- differences of opinion about what the big ideas are
- differences in what one values
For instance, consider these three examples of student work that we discussed in the seminar in which I participated in November:
- The problem read, “The sum of three consecutive odd integers is 81. Find the integers.” One student’s solution was like this:
Let x = 1st odd integer
x+1 = 2nd odd integer
x+2 = 3rd odd integer
x + x + 1 + x + 2 = 81
3x + 3 = 81
3x = 78
x = 26
The integers are 26, 27, 28
How many points (out of ten) is this worth? If a big idea is that odd numbers differ by 2, not by 1, then the setup at the beginning of the solution represents a significant error — especially since the student wrote the word “odd” each time, thus showing that s/he didn’t merely skip over the word “odd” in the problem statement. On the other hand, the solution is otherwise correct, the work is clearly shown, and the answers will even check, being three consecutive integers adding up to 81 (ignoring the word “odd” again). If you highly value all the skills of combining like terms, backtracking to solve an equation, and recording a solution, then the solution is worth a lot of points. But if the idea of consecutive odd integers is important, it may be worth very few points. My colleagues rated the solution all the way from one to nine; I gave it a six.
- Next we have a different student’s solution to the same problem:
The grades on this one ranged all the way from zero to ten! Some teachers gave it only a few points — or even zero points — because no algebra was used. But I was one of those who gave it a ten, because not only was the solution correct but it also showed a thorough understanding of what the problem was asking for, and of course the answer was checked. If an algebraic solution was meant to be required, that requirement should have been specified.
- Finally, here’s a different problem, along with one student’s solution: “Solve 2(x – 10) – (12x – 4) = 20.”
The issue here is that the student couldn’t read his own handwriting (possibly her own handwriting, but the odds are against that): “20” got transcribed sloppily and then read as 26. I gave it a nine out of ten, since that error struck me as a very minor one. But other teachers’ scores ranged all the way down to zero, on the theory that the student had nobviously never checked his answer. My own values are that checking one’s answer is a big idea of algebra, if we mean that it’s important to understand that what we mean by a solution is a number that will satisfy the equation. But failure to check, especially in a time-sensitive situation where there are no instructions to check, strikes me as a very minor offense.
Your mileage may vary.
The second point that I was intending to discuss — whether algebra is distinct from functions — will have to wait for another post. This one is already too long.
“What do I need to do to get an A?” asks one of my students in an honors math course.
I wish I had a magic recipe. I can say with reasonable confidence that it’s possible to get a B by studying hard, by studying smart, by working hard to understand concepts, by getting enough practice in math skills. But an A? Every student sees some classmates getting A’s by some mysterious method, in some cases working very hard and in other cases magically earning the A with seemingly little effort. Surely there must be a secret recipe that the teacher isn’t revealing.
Of course there isn’t such a recipe. Think of some non-academic endeavors. What can you do to gain a place on the varsity basketball team or the Greater Boston Youth Symphony Orchestra? Sure, working hard is important and necessary (think of the old joke about how to get to Carnegie Hall: “practice, practice, practice”), but unfortunately it’s not sufficient. There’s no way that I’m going to be a top-ranked musician or a top-ranked athlete, no matter how hard I try. And yet, as a teacher, I definitely don’t want to be the bearer of discouraging news. I want my students to work hard, to do their best. But their best might or might not match their hopes. There’s nothing wrong with a B in an honors math course at Weston High School — in fact, I have a couple of students who work very hard and are delighted when they achieve B’s. But, in Lake Wobegon and similar communities, nothing less than an A will do.
So here’s the dilemma: how do I motivate students to achieve their personal best — which, after all, is the aim in music and athletics and other endeavors — without telling them that their personal best might not be good enough? I know how to help students get B’s, but after 34 years of teaching I don’t know how to help them get A’s, at least in honors courses. Some do, most don’t, but it’s not clear what effect I can have on the outcome. I can help a willing, able, and motivated student get an A in a college-prep course, but such a student might well try his or her best and only get a B in an honors course. As I say, there’s nothing wrong with that, but in today’s world of competitive parents and even more competitive college admissions, my point of view won’t be compelling. This is discouraging. The last thing I would do is say ahead of time that any given student is incapable of earning an A. And yet, at some point, one is forced to admit that a particular student is trying as hard as any reasonable person could expect and is earning a B. I don’t know what to say, except to repeat that there’s nothing wrong with that.
Strip Search, by William Bernhardt, is an irritating novel.
Why do I say that? Well, it’s not just because Bernhardt portrays math teachers as weird and psychotic, though that’s certainly a major part of it. And it’s not just because the plot is so implausible, though that too is part of it. And it’s not just that the book is riddled with mathematical errors, though of course that definitely bothered me. And it’s not just that the amount of violence is excessive and unnecessarily explicit, though that would certainly put off many readers. No, the most irritating characteristic of Strip Search is that it reads like a “good idea” that someone had. My impression is that someone said to the author, “Here’s a proposal for a novel. Go write it.” Not surprisingly, a coherent novel was not the result.
You may wonder why I started reading this book. In the past I’ve found Bernhardt to be a competent and engaging writer, even if not a memorable one. And I had heard that Strip Search featured a combination of mathematics (equations left as clues at each crime scene) and a major character (Darcy) who’s an autistic savant. No math teacher could resist that enticing combination. Some readers (in customer reviews on Amazon, for instance) were annoyed by the characters and found none of them likeable. Personally I didn’t have that problem, although I can see why others might. But anyone who has taught students who have Asperger’s or autism will find Darcy likeable enough, to coin a phrase. And the detective is no more unlikeable than many a highly flawed protagonist.
You may also wonder why I bothered finishing Strip Search if I was so irritated by it; I’m not one of those people who feel compelled to finish a book once they’ve started it. But I kept irrationally hoping that things would get better, that there would be a good reason for all the flaws. Unfortunately I was wrong, so here is your warning. Don’t read this post any further if you’re intending to read Strip Search, as I can’t write what I need to write without introducing spoilers.
*** SPOILER ALERT*** SPOILER ALERT *** SPOILER ALERT ***
OK, so we have a detective who’s actually a police psychologist (the protagonist) and fits into the genre stereotypes of being insubordinate and an alcoholic. Later she turns to pills. She is a psychologist without a doctorate, and she reaches most of her conclusions by intuition and guesswork. Since she’s also the first-person narrator, I’ve forgotten her name. Oh, that’s right, it’s Susan.
But don’t think that Bernhardt extends genre stereotypes to gender stereotypes. No, we also have Esther Goldstein, a female mathematician who not only teaches math but also has apparently solved the Riemann Hypothesis (misspelled “Reimann” throughout the book). For reasons that apparently stem in some undefined way from an unhappy childhood, she is also reviving the ancient Pythagorean religion, the Brethren of Purity. Unfortunately she also turns out to be a psychotic mass murderer. But then again she is a math teacher, so you can’t expect her to be normal, can you? “Math has been riddled with positively brilliant madmen,” as she explains at one point.
- The sympathetic characters, such as police lab technician Amelia, say things like, “I gave up on math after my second semester of algebra.”
- Susan, even though she presumably has at least a master’s degree in psychology, says, “I hadn’t taken a math class since junior high school.”
- The puzzle expert says, “I’m a word boy. Left brain. Math freaks are a whole different breed. And this doesn’t look like a real puzzle anyway. How can you solve an equation if you don’t have any of the numbers?
Bernhardt’s mathematical errors include confusing variables with unknowns and referring to expressions as equations. For example, on page 161, we have this excerpt:
It was another equation:
Bernhardt’s account of the Pythagoreans’ attitude toward the irrationality of the square root of 2 is also muddled. For instance, Esther, the professional mathematician who’s an expert on the Pythagoreans, says, “The square root of two was a problem with no solution.”
OK. That’s enough. Don’t bother reading the book.
Continuing yesterday’s theme… There has been renewed interest in Larry Summers’s supposed sexist remarks. When Senator Obama (I almost said “President Obama”) announced that he would appoint Summers to be his senior White House economic advisor, bloggers and others revived the old canard that Summers believed that women were deficient in their math and science abilities. For instance, Wendy Hansen in the LA Times wrote as follows:
The notion that boys are better than girls at math simply doesn’t add up, according to a study being published Friday in the journal Science. An analysis of standardized test scores from more than 7.2 million students in grades 2 through 11 found no difference in math scores for girls and boys, contradicting the pervasive belief that most women aren’t hard-wired for careers in science and technology.it
The study also undermined the assumption — infamously espoused by former Harvard University President Lawrence Summers in 2005 — that boys are more likely than girls to be math geniuses. Girls scored in the top 5% almost as often as boys, the data showed.
The trouble, of course, is that Summers did not espouse that position. Summers did observe that there is a gender disparity among the very top mathematicians and scientists (as no one could deny) and proposed that it would be helpful to investigate why: to what extent is it genetic, and to what extent is it societal?
Although this question is precisely what a scientist in a research university should ask, it created great controversy. The very act of asking the question suggested to many people that Summers was assuming that women are less capable than men in math and science.
Summers had great strengths as President of Harvard, especially in his sponsorship of the Crimson Summer Academy (where I teach in the summers, so I can’t claim objectivity) and in his insistence on huge scholarships for low- and moderate-income students. Unfortunately his lack of social skills caused him to lose support in the Faculty of Arts & Sciences (though not in Harvard’s ten other schools), and he was forced to resign. Some say he has Asperger’s Syndrome; he might well, but who knows? Anyway, he has returned to being a professor of Economics at Harvard, and now he is a top advisor to Obama. I don’t believe that he’s sexist, but he clearly has some problems communicating his ideas; nevertheless, he is a distinguished economist with a lot to contribute, and he is an excellent pick for the Obama administration.
So why is it that the top two mathletes on Weston High School’s Math Team are freshmen girls? And a year young for their grade, at that?
Check out the situation from ten months ago. But it’s only two data points.
I recently installed an unusual application on my iPhone: Ocarina. This program turns your iPhone into a four-hole ocarina, with the holes outlined on the iPhone’s touch-sensitive screen. But the really cool thing is that you actually blow into your iPhone to simulate blowing into the ocarina! Try it: it really works!
Many of my students agree that this is really cool, although some adults think that it’s a waste of time. I don’t really understand their point of view, since they are likely to spend their time on useless things like watching football games, but anyway….
The reason that I had to demo this product for my precalculus class is that we have just finished studying the use of trigonometric and exponential functions to model musical sounds, and one of the issues that arose is what the dependent variable represents when graphing an oscilloscopic rendering of a tone. Sure, if Middle C is 262 Hz, we notice that the frequency is 262 cycles per second since the period of the independent variable is 1/262 of a second. But what does the y-axis represent? We say pressure, and we may measure it in pascals or mV, but what does this have to do with the loudness of a sign? The direct analog construction of the iPhone ocarina application — with no intermediate abstractions of digital software — provides a clear understanding of this phenomenon, since the user’s breath blowing into the iPhone moves the membrane of the microphone, illustrating pressure in a literal way.
Should high-school math classes be teaching Excel? Or, more generally, should we be teaching spreadsheet use — and Excel just happens to dominate the market? We have been exploring these issues at Weston High School.
Certainly the right point of view is to teach spreadsheets rather than Excel, although the dominance of Excel means that it will inevitably be hard to distinguish it from spreadsheets in general. Anyway, I think the real questions are the following:
- What can spreadsheets add to high-school math?
- When and in what order should we teach various spreadsheet techniques?
- Should we teach non-mathematical skills such as formatting cells, creating headers, etc.?
A side point is that knowledge of Excel will be very helpful to our students in college courses and in a great many job situations, so somebody should teach it in high school. I suppose the responsibility falls to the math department by default, even if it isn’t really math, though that conclusion makes me uncomfortable. The only other likely places are the science department — since science courses also do a fair amount with Excel — and the business department, though that wouldn’t touch all students by any means.
There are a lot of issues, large and small, in using Excel. The notion of a variable is quite different from that of a variable in mathematics. Order of operations is important, and that’s almost identical to what we do in math and can therefore reinforce it. Sorting becomes important, and that’s a mathematical concept that comes up a lot in computer science courses but rarely in pure math. Graphing and regression are possible in Excel, but both are clumsy. Sometimes the syntax can be confusing, such as beginning a formula with an equals sign, but getting used to different syntactic conventions is a useful mathematical skill. A great many mathematical functions are built into Excel and can therefore be reinforced when we use spreadsheets. Perhaps most important is the level of abstraction involved in creating formulas that can be dragged vertically or horizontally independent of pre-existing data.
We’ve just completed an Excel activity in my college-prep Algebra II class: Saving for College. My hope is that this activity will not only give some experience with spreadsheets (that’s the secondary goal) but will also reinforce some of the important concepts in exponential functions (that’s the primary goal).
Incidentally, it turns out that many of those who teach Excel only have a very narrow view of this powerful piece of software. (As much as I don’t like Microsoft, I do have to admit that Excel has a great many excellent features and has an enormous number of useful options, many of which I barely know myself.) So I wonder if Weston needs to have a workshop in which we try to plumb the depths of Excel and figure out which aspects of this software will be most useful in teaching high-school math.
Earlier this month I participated in a fascinating two-day seminar on The Big Ideas of Algebra, taught by Deborah Hughes-Hallett and sponsored by Teachers as Scholars. Although I undoubtedly talked too much, I figure that that was because I had a lot to contribute. Nevertheless, I learned a number of valuable things from the seminar, and they will be as useful both at Weston High School and at the Crimson Summer Academy. Mostly this was a matter of focusing my attention on things I already knew and believed but wasn’t thinking much about. In particular, I am now convinced than ever that we need to pay a lot more attention to symbolic literacy (understanding the symbols and combinations of symbols used in algebraic expressions) and to the distinctions among expressions, equations, and functions. Because of my linguistics background I have always believed that one important way to look at mathematics is that it is a language — in fact, that’s probably what got me to make a smooth transition from linguistics to math — but I hadn’t thought enough about the implications of that point of view when teaching math.
In addition, among the other matters we talked about in the seminar, we discussed the assignment of partial credit for work in solving a problem — more on this later, but it definitely reflects one’s views on what the big ideas are — and whether the study of algebra is distinct from (and prior to) the study of functions. More on that later as well.
By this point I’ve taught simplified versions of the RSA algorithm to ten different cohorts of teens: four years’ worth of Honors Algebra II students at Weston High School, juniors for four summers at Crimson Summer Academy, and two years’ worth of college-prep Algebra II students at Weston. Tweaking the details as I’ve gone along, and benefitting from changes in technology, I’ve learned a lot from these experiences.
There are several types of benefits for the students. Some benefits are conceptual, involving understanding ideas about public-key cryptography, ranging from technical questions like how a cryptosystem can use a public key, why that’s necessary, and why it’s secure, to public-interest issues like whether we can trust so-called “secure” financial transactions on the Internet. I could have predicted these benefits; the unit was designed to try to achieve them, after all. And I could have predicted the success of some of the more concrete mathematical benefits as well, since RSA involves exponentiation, prime numbers, modular arithmetic, factoring, representation of characters as integers, and other operations with numbers. But a third type of benefit is more of a surprise: because the technical details are complicated, and even a single mistake can doom the effort to failure, most of my students have been doggedly persistent in paying attention to details and getting them right. Too often we can fall into the trap parodied by Tom Lehrer: “The important thing is to understand what you’re doing rather than to get the right answer.” We give so much partial credit that a student can get a B without ever producing a correct result. Part of the way that my colleagues and I have avoided this trap with RSA is that our sequence of activities and assignments concludes with a two-way exchange of messages: each student sends me a message using my public key, and I reply with a message using the student’s public key. This gives everyone practice in figuring out their private and public keys, enciphering, and deciphering. But fewer than half the kids get it right the first time, since there are so many opportunities to make mistakes. Unlike the usual math problem, they can’t settle for having a couple of points taken off; the message simply won’t work. So they try over and over again — sometimes four or five times — in order to get it right. If they don’t, I can’t read their message, or they can’t read mine.
If you’re interested in checking out how I’ve simplified RSA so it can be studied at the level of Algebra II, take a look at my worksheets, starting at RSA Phase One. (If you keep incrementing the “1” in that URL, you’ll find the next three worksheets at the expected URLs.)
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