In our department meeting today, we had a guest speaker from the Wellness Department* who talked with us about building closer connections with students, an atmosphere of trust, and greater engagement by our students. All good things, certainly. Part of the pitch was that the few minutes spent on such tasks in each lesson would more than pay off in increased learning, and I do believe that.
So why do I feel uncomfortable about the whole idea? I suppose it’s because it just doesn’t feel like me. Even the initial idea, shaking hands with each student on entering the classroom, feels unnatural; I’m not at all convinced that I could carry it off. The speaker says that his students not only welcome this practice but explicitly ask for it if he forgets. Maybe so, but I can’t see myself doing it.
On the other hand, I don’t want to rule out the idea, either in detail or in the big picture, and I’m willing to try. I’m even willing to sign up for a proposed summer workshop on the subject, if the calendar permits.
* Yes, I know…
Bored math students often ask, “When am I ever going to use this stuff?” Even math students who are not bored often articulate the same question, albeit in a more polite phrasing, such as “Can you give us an example of an application of the theoretical math we’ve been learning?” The question deserves a respectful answer.
I discussed this issue two years ago, but I have a slightly different point of view now. Here’s what I said then:
I know, I’ve written about this topic before, but it bears further consideration. Too often I hear the question, “When will I ever use this stuff?” This is a common question in Weston, and surely elsewhere as wel
Sometimes the question comes from a bored student who is really asking a deeper question, something like, “I don’t like this, I don’t understand this, why should I have to learn it?” In that case it’s hard to know whether to answer the explicit question or the implicit question.
But sometimes the question comes from an otherwise engaged student who actually wants an answer. And it’s hard to give a satisfactory answer. There are at least two reasons for this — probably more. First of all, no high-school student really knows what he or she is going to be doing in life. It’s important to keep the doors open, in case the unanticipated economics course in college or statistical analysis in a job turns out to require something from a high-school math course. But that’s pretty vague and abstract, and of course it isn’t a very satisfactory answer for most students, even though it’s a true answer.
The other reason why the question is hard to answer is that the hidden but more important curriculum in high school has nothing to do with the specifics of logarithms, cosines, etc. When a student takes Algebra II or Precalculus or whatever, the important things that s/he is learning have to do with problem solving, approaches to mathematics, and learning itself. Sure, you might never see logs again (although the odds are that you actually will); but the analytic techniques and reasoning methods that you learn will stand you in good stead.
The only trouble is that most Weston students don’t want to hear this, or it doesn’t make sense to them. They want to know how they are going to use the precise content in the job that they imagine that they will have, even though the probability is that they will be doing something else entirely. How do we give them an answer that they will consider satisfactory?
I finally have an answer that I can believe in. I still believe everything I wrote two years ago, but I hadn’t figured out the answer to my concluding question. Maybe I still don’t have one, but I truly like what the Math Curmudgeon has to say on the matter. Even though I often disagree with the Curmudgeon (whose real name I don’t even know), I have taken to heart his words on this matter. He says the right things in his speech to his students (complete with a fractal tree). Here’s the first half; read the rest of it yourself:
“When are We EVER going to have to use this?”
“Beats me,” I usually answer. “You can’t even tell me definitively what you’ll be doing next month, forget about four years from now. How can I definitively say when or whether you’ll use THIS? All I can say is that it is useful in certain situations (the word problems in this section are limited versions of the same problems some people face daily), useful as mathematical development for later work (which may be a prerequisite for the course or job you really wanted) or is mental development to expand your brain beyond the limited understanding and very limited world-view you currently have. I’m not being critical here — you really have no experience at life. How could you possibly know the utility of everything you’re learning?”
 “You have millions of possibilities ahead of you, thousands of doors along this hallway you call life. Writing ability will unlock many of them, artistic ability others, mathematical ability many more. Some may require that you speak English well — certainly 95% of the jobs in this country do. Some will require a little of everything. Each of these doors is along a different stretch of hallway, sort of like that fractal tree over to the right. Each educational decision you make takes you down one branch or another, closing off some possibilities and making others available. To switch from one branch of the tree to another may require a little backtracking to pick up things that you could be learning now. I have no idea which doors will interest you so I have to lay a very broad groundwork and push you in directions you may not immediately see any need for. You have to trust that, over the course of many years and many students, I have a good sense of what you might need and of what you may find interesting after we’re done.”
“How do I know this? I talk to my students after they graduate. They tell me what they found useful or pointless. I get all kinds of stories about topics that we covered here that directly applied to something they were working on, stories about being the only one who really understood something the professor was trying to say. There aren’t many complaints that we spent too much time on a topic they never saw agai
“If there is ever a commonality in the comments of returning graduates, it’s this: ‘I never imagined that THAT would be useful. I was surprised when it showed up. So was the professor — he was grateful SOMEONE knew about it.’”
What are grades supposed to represent? What is the appropriate connection between assessments (whether formative or summative) and grades?
I’ve recently been reading some interesting discussions about these questions in several math teachers’ blogs (including those of Matt Townsley, Karl Fisch, David Cox, and Dan Meyer), all sparked by Shawn Cornally’s fascinating blog, Think Thank Thunk. Cornally poses the problem like this:
The Message Grades Send:
Problem: Kids want to play games to get points in order to get an ‘A’. This is a problem because it puts emphasis on accumulat ing points and not on what the points are supposed to represent: learning. You must migrate your system of grading away from grading every single assignment summatively (that is assigning a static grade for everything a kid does), and towards grades that are indexed by content.
Students could not care less about their score on “Quiz 5″ from last month; they don’t even know what was on that quiz. Don’t put that in your grade book. Put the individual ideas that that quiz assessed in your grade book, so that the students know what it is you care about. I do this, and my grade book has ballooned to about three times its previous size. Oh well.
Reporting Should Be Dynamic:
Let’s say you really care about a certain bit of knowledge, so much so that you’re going to put in on a test. In other words, you want students to know it really badly. Like, say, the Pythagorean Theorem, and you consider your class worthless if the student hasn’t learned that piece of knowledge, then your grading system should be set up to help students remediate their misunderstandings, not screw them over for not getting it the first time.
It’s hard to disagree with that, and I’m not going to try; I merely want to consider three of the reasons why it won’t work, according to us skeptics (but of course we could be wrong). Do read the entire article in order to get the full context, since I’m focusing only on a few specific issues. If you have time, also read the many articles on the subject in Cornally’s blog.
Here are my objections:
- The first reason why it won’t work is that kids and their parents won’t buy into it. Cornally does address this point: “I’ve had kids cry over this, but I have to hold my ground. Parent emails be damned; Johnny didn’t improve from a 8/10 to a 9/10. He just didn’t, sorry.” It’s hard for me to see this point-of-view succeeding in Weston, but of course I could be wrong.
- The second reason is that it’s too time-consuming. We’re already too busy with all the demands on our time, so how can we keep track of all these finely itemized skills and concepts? But of course I could be wrong.
- The third reason is that I fear that it would inevitably lead to a de-emphasis on concepts and big ideas, because atomized skills are so much easier to measure. But of course I could be wrong.
I would like to be proved wrong. I would like to see if this concept of standards-based grading could fly in a place like Weston. Maybe it can.
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.
Apparently this is becoming more and more common in Massachusetts. Our students got a four-day weekend in January, but the faculty only had a three-day weekend, in order to schedule a day of professional development. Not that that’s a bad thing. Just saying.
Our focus today (as in all of Weston’s professional development this year) was Professional Learning Communities, usually known as PLCs. You may be wondering what this latest piece of jargon means. Google produces thousands of hits (172,000 as of this moment, or so it claims), but let’s look at allthingsplc.info, which sounds like (and is) a good source. Although I labeled this as the “latest piece of jargon,” in fact it‘s not particularly knew. For instance, the site includes an article from six years ago by Richard DuFour. Here is an excerpt from it:
The idea of improving schools by developing professional learning communities is currently in vogue. People use this term to describe every imaginable combination of individuals with an interest in education… The term has been used so ubiquitously that it is in danger of losing all meaning.
…
The professional learning community model flows from the assumption that the core mission of formal education is not simply to ensure that students are taught but to ensure that they learn. This simple shift — from a focus on teaching to a focus on learning — has profound implications for schools… Every professional in the building must engage with colleagues in the ongoing exploration of three crucial questions that drive the work of those within a professional learning community:
- What do we want each student to learn?
- How will we know when each student has learned it?
- How will we respond when a student experiences difficulty in learning?
This emphasis, of course, fits in perfectly with the current national obsession with No Child Left Behind, statewide standardized testing, and other related initiatives. But it’s much more reasonable than No Child Left Behind or statewide standardized testing, since the three questions are clearly important and don’t suggest a dumbed-down or cookie-cutter approach, leading to teaching-to-the-test.
To my mind, however, the issue isn’t the three questions. The real issue is buried within the quotation above — a simple three-word phrase: engage with colleagues. Throughout my entire teaching career of 36 years I have enjoyed working with colleagues and have considered it to be an essential ingredient of our success. This observation flies in the face of the common (mis)perception that the teacher closes the classroom door and is a solo practitioner in charge of a group of students, but I’ve never believed in that model. Everywhere I’ve taught, the math department shares a common office and works together on curriculum, instruction, and assessment. For whatever reason, not all departments have operated like this, but it is always a major plus for math teachers. What hasn’t always happened is an emphasis on any of the three questions above. I look forward to further work in this area.
In recent days I’ve talked with several colleagues and a couple of students concerning overrides into honors-level math classes. All high schools have to face the question of what to do when a student and his or her math teacher disagree about the appropriate next course. There seem to be three different models:
- The teacher’s recommendation is law, and the student cannot override it.
- The teacher’s recommendation is the default, but the student and/or parent can jump through a number of virtual hoops in order to override it.
- The student can sign up for whatever s/he wants, regardless of the teacher’s recommendation.
Weston follows model #2. The hoops consist of talking with the current teacher, a prospective teacher of the desired course, the counselor, and the department head, as well as filling out a form. This deliberately cumbersome procedure is designed to ensure that the override is not being done casually, and that everyone is going through this with open eyes. After all the steps are carried out, the student is admitted to the desired course (assuming that there is space available). The usual context for this process is when a student is recommended for a college-prep (CP) course but wants to be in honors.
The Weston parent community would never stand for model #1, but I have sometimes heard people advocate model #3. There is a certain logic to that: a student picks a course (with parental signature) and lives with the consequences. If s/he picks an honors course inappropriately, the results are either a bad grade or a decision to drop down to CP. What’s the harm?
Well, I’ll tell you what the harm is. Suppose we follow model #3, and as a result we schedule six sections of honors geometry and only four of CP. And then it turns out that 42 honors geometry students find out that they are over their heads and try to drop down to CP. But since there are only four sections of CP, it turns out that only 8 slots are available, so 34 extra kids are stuck in honors, being understandably discouraged and making it nearly impossible for the teachers to maintain a positive honors-level atmosphere.
“Why not just switch one or two honors sections to CP?” you may ask. You can’t do that becuase the 34 kids in question are spread out among six sections meeting in six different blocks, and it would be horribly disruptive to the entire schedule to try to switch everyone around. More than disruptive, it would probably be impossible.
In order to maintain an appropriate level of classroom discourse, it’s essential to have a critical mass of appropriately placed students. It’s OK to have two or three who are over their heads, but if half the class doesn’t belong there it will poison the well for everyone. That’s not being elitist, it’s just reality.
I’ve just sent off my final college recommendations — for a couple of schools that have surprisingly late deadlines of January 10 or January 15. My spreadsheet shows that the students who asked me to write recommendations for them this year altogether applied to an average of 7.5 colleges apiece (ranging from 2 to 14). Last year’s seniors applied to an average of 10.4 colleges apiece (ranging from 5 to 16).
What accounts for this dramatic drop? Of course we’re talking about only a small subset of the senior class at Weston, so it might not be statistically significant. But I do wonder whether there is finally a reaction to the excessive number of applications that the Common App has encouraged, or whether this year’s senior class includes a lot more students admitted early (under Early Decision or Early Action), so they are less likely to apply to a lot of schools in round two, or whether there’s some other explanation I haven’t thought of.
“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.”
Paper clips are up to 25 now:
Getting closer to donuts.
“Do you know about the Xerox Alto and Xerox Star computers from back in the ’70s?” asked one of my fifth-graders in The Saturday Course.
“Yes,” I replied, “but I’ve never before met a fifth-grader who knows about them!” This mut have been ancient history to him.
He went on to tell me that the Alto was the first computer to use windows, menu, and a mouse, but that Xerox’s marketing department wasn’t able to sell it to home users.
Actually, I wrote once before about a Saturday Course student who surprised me with her wisdom and knowledge — and that was a fourth-grader that time — so I guess I shouldn’t have been surprised. But this blast from the past was definitely unexpected.
Junior Lauren Avery, one of the editors of Weston High School’s student newspaper, Wildcat Tracks, asked if she could interview me. Of course I said yes, and the result was a half-page article that focused on my transition from linguistics to teaching math. I was pleased with the depth and breadth of the writing, as well as by its unusually high degree of accuracy. “It’s much more accurate than Fox News,” I said to one of my colleagues.
“That’s not a very high bar,” she replied. She’s right, of course. This article was probably 99% accurate, which is as much as anyone could ask for — and I was just kidding about Fox News.
Here are a few excerpts from Lauren’s article:
Davidson’s smooth switch between two seemingly incompatible fields often surprises his students. Despite this, Davidson sees a great deal of similarities between linguistics and mathematics, and to this day he continues to pursue both subjects.
…
A linguist is a person who studies the origins and usage of ancient and modern languages…. By studying multiple languages instead of focusing on a single language, Davidson was able to begin to identify trends and patterns between languages, a concept that played a major role in his interest in mathematics later on.
…
To his current and former students, Davidson’s ability to switch between two fields has given them a new perspective about choosing a career in the future. “It lets me think that it’s not really too late to change what you are passionate about,” said junior Mir Bokhari.
…
Davidson’s switch between two fields has affected him both as a teacher and as a person, and it reflects some valuable lessons concerning education. “You never know if sometimes something you’re interested in can come back. My jobs make use of all the linguistics I had done 20 years earlier in new contexts. My linguistics training helped in math,” Davidson said. “There are surprising connections. Nothing you learn is ever wasted.”
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:
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| Before |
After |
It doesn’t seem possible for a student to lose his quiz while taking it. If it’s possible at all, it should surely be a once-in-a-lifetime event.
But no! For the second time in my years at Weston, this unlikely event has occurred. About eight years ago a sophomore managed to lose his quiz while taking an open-note quiz (it disappeared into a black hole in his notebook). And now, what do you know, a freshman has repeated the same feat, again causing his quiz to disappear into his notebook while he was taking it.
What are the odds?
Three different freshmen approached me at various times today with concerns about whether they should stay in Honors Geometry or drop down to College Prep. That’s OK. But for two of them the reason was that their current average is B+. Not D+, not even C+, but B+!
What’s wrong with a B+? B+ has always been above the average grade for Honors Geometry. It’s something for anyone to be proud of. So what’s the problem? Is it merely that they’ve always gotten an A in math? Are their parents pressuring them to get an A? Are we teachers inadvertently pressuring them this much? Are they already so worried about getting into college that they think a grade in first quarter of freshman year has to be an A? A B+ in Honors Geometry looks great — and term grades don’t go on the transcript anyway.
We all know about grade inflation. But we don’t know what to do about it. In an honors course I can live with the idea that everyone should get some sort of A or B (although I’ve known a number of students who learned a huge amount of math despite getting a C in honors math); what I can’t live with is the idea that everyone should get an A.
Check out the blogs for all of my classes! We rotate each day that a class meets, so that students take turns posting class notes. So far this has led to a number of positive effects:
- Students who miss class for any reason have a resource for finding class notes.
- Students who were present in class but have inadequate notes (yes, this has been known to happen, even in Weston) have a backup.
- Students get the experience of taking and writing up detailed notes for a real audience to read.
- Misconceptions are revealed to me way in advance of the next quiz or test, as I can sometimes see that a student misunderstood what was done in class.
The first three effects were expected, but I hadn’t anticipated the fourth.
Currently up to 18 paper clips…
This year I’ve been trying something new, and I already love the effects. Back in August I had read this wonderful idea from a math teacher whom I don’t know except from her post:
I start a chain of paper clips at the top of the whiteboard. When students catch a mistake I add another paper clip, then when it reaches the floor they receive an award. I use one chain for all classes and it usually reaches the floor 2–3 times a year. It’s highly motivating and a visual reminder each day.
So I shamelessly stole the idea — actually, I credited the otherwise anonymous blogger who calls herself Jade — and promised donuts to all classes when the chain reaches the floor. (I award a paper clip only the first time that a mistake is caught; it has to be a mistake in either math or English, not a judgment call about an explanation or a lesson plan; and it has to be caught by a student, not by me.)
The benefits so far are quite visible:
- Students pay extra-close attention in order to catch me in a mistake.
- They get the message that everyone makes mistakes, that doing so doesn’t damage one’s self-esteem, and that the important thing is to take risks and to make quick corrections as needed without embarrassment.
- Class morale is boosted because students enjoy catching my mistakes.
So far we’re up to twelve paper clips. I haven’t measured, but eyeballing the chain suggests that we’re about one-third of the way down to the floor.
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!
I always start off each class in September with a seating chart where the students are seated in alphabetical order. This arrangement is the quickest way for me to learn the names of the 75–95 names in all my classes, and it’s also a reasonably objective way of seeing how the students behave as a class. If they are quiet and behave appropriately, I may allow them to choose their own seats after a few weeks — or I may simply create a new seating chart.
This year, in two of my classes, I decided to try out another teacher’s method of assigning seats. After each test he has everyone stand up, and then he returns the tests in random order. As each student receives a test back, s/he gets to pick a seat. Clearly this method won’t work for a class where careful assignment of seats is necessary, but it should work well in a group of serious, well-behaved students. So I tried it in two sections of the same course. In one of them the large majority of the students were satisfied, and many of them enjoyed the change of pace provided by the activity. Of course you can’t satisfy everybody, so we just left things as they turned out as a result of the randomly ordered free choice. But in the other section (of the same course, remember) there was massive dissatisfaction. Earlier — when I had assigned seats alphabetically — nobody claimed at all. But as soon as they had the illusion of choice, almost everyone who didn’t get his or her first choice was unhappy. Basically that meant all but the first seven or eight who got to choose. Somehow they preferred no choice to some choice.
It turned out that practically everyone wanted to sit in the front row. Yes, that’s a pleasant and surprising dilemma for a teacher, and I pointed out that it was obviously impossible for more than five to sit there — or six if I slightly rearranged the furniture. So, I asked the students to answer three questions:
- Where do you want to sit?
- Why?
- On a scale of 1 to 10, how strongly do you feel about it?
I then created a new seating chart that would maximize everyone’s wishes, giving preference to those who had compelling reasons for those wishes. (The result worked pretty well, by the way.) What amazed me was the detail that some of these freshmen put into their answers, including diagrams of the room! Here are four examples (with a few details changed to protect the students’ privacy):
| On a scale of 1–10 I want to be in the seat three rows back against the wall. I give it an 8! I feel like I can focus in the mid-side but by the window I get distracted. |
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| Anywhere inside the shaded area would be nice because my vision isn’t amazing but sometimes it’s hard from the way back. It’s not the end of the world but it will make a difference. |
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| I would like to sit in the middle seat the 2nd row. I would also like to sit near Griselda and Matilda. 10: I cannot see very well, especially if tall people sit in front of me. That seat is the perfect distance. |
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| I either want to sit in the 1st or 2nd row — I feel really strongly to sit in the front b/c I need to be able to see the board, and I stay more foxued when I sit closer to you and the board. I feel a 10 (really strongly). The circled places are where I wish to sit. |
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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?
Go see the musical version of Spring Awakening at the Colonial Theater if you’re a parent or a teacher or a teen, or if you’ve ever been one of those. This disturbing German play from 1891 is not exactly typical raw material for a musical, but it survives the transition admirably. As you probably know by now, Frank Wedekind’s original play was banned because of its themes and how they are presented. Sexuality, sex between young teens, teen suicide, abortion, unethical teaching and parenting, radical politics — even one of these would get a worked banned in late Victorian times, whether in England, America, or Germany — and the combination of all of them was surely so far over the top that there wouldn’t have been any doubt.
Today, of course, these topics aren’t shocking. I suppose that’s why the producers made no attempt to translate the play to modern times. The audience is always conscious of the time and the place, despite the presence of rock music — which somehow doesn’t seem out of place. One oddity is that they’ve cast a single actor to play all of the adult male roles, and another to play all of the adult female roles. I assume that this is trying to convey a message — something to the effect that the adults are all interchangeable, and only the kids have individual personalities. Fortunately it’s not as confusing as it might be, since the contexts are clear and the characters are often addressed by name.
Some bits of trivia: How often do you hear quadratic equations and lines from Vergil’s Aeneid mentioned in a musical? They probably didn’t register on most of the audience, but the math references were appropriate and the Latin class convincing (though terrifying). Also, I can’t find anyone else who was aware of this play in the ’60s, but I first became acquainted with it in 1966, when my roommate was reading it (in the original) for a college freshman German course. So I knew it as Frühlingserwachen, and it clearly made an impression on my impressionable roommate, who had a lot of related issues. For similar reasons, I thoroughly recommend it to my high-school students, most especially if they can later have an in-depth discussion with their parents and/or teachers. I was glad to hear that a couple of dozen Weston students will be going to see it next week on a field trip.
I can’t keep up with Andrew Sullivan, since he posts about 42 entries a day. (I’m not exaggerating!) But I just read the following email from one of his readers and I have to pass it on:
First they tortured in ticking time bomb cases but I didn't mind because it was a clear and imminent danger.
Second they tortured "slow-fuse" high value detainees and I didn't mind, because you never know what might happen.
Third they tortured Iraqi and Afghan prisoners who weren't high value, but who might have had useful information, and I didn't mind, because they were acting in good faith.
Fourth they tortured prisoners to establish a link between Al Qaeda and Saddam, and I didn't mind, because surely there must have been such a connection.
Finally, they came to torture me, and nobody cared, because if I was being tortured, I obviously deserved to be tortured, and, as Peggy Noonan says, some things are just mysterious and it's best to just keep on walking.
Once a teacher, always a teacher, so…in case you didn’t catch the references, here are some pointers.
- The entire quotation is an homage to Martin Niemöller’s famous text:
In Germany, they came first for the Communists, And I didn’t speak up because I wasn’t a Communist;
And then they came for the trade unionists, And I didn’t speak up because I wasn’t a trade unionist;
And then they came for the Jews, And I didn’t speak up because I wasn’t a Jew;
And then . . . they came for me . . . And by that time there was no one left to speak up.
- Charles Krauthammer is a right-wing pundit who consistently supports the use of torture. (OK, Krauthammer would object to my word “consistently,” but follow the link and judge for yourself.)
- Peggy Noonan is a commentator and former Reagan speechwriter from New Jersey and Boston, like many of us (I mean the New Jersey and Boston part, not the Reagan speechwriter part). The last sentence of the passage quoted above from the Andrew Sullivan blog refers to her remark about the torture memos: “Sometimes in life you just want to keep walking. Some of life has to be mysterious.”
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!
I don’t usually read books just on the basis of advertising, so I first checked with my colleague the music teacher before I decided to read the new book with such intriguing ads: Daniel J. Levitin’s This is Your Brain on Music. My colleague’s enthusiastic recommendation confirmed me in my inclination to go ahead. After all, combining neuroscience with music? How much more interesting could you get? Or was I just setting myself up for disappointment?
The quick answer is no, I wasn’t. Despite being a bestseller on the New York Times non-fiction list, This is Your Brain on Music is well worth reading. As a neuroscientist with a background as a musician, sound engineer, and producer, Levitin is the perfect person to write a popularization of the intersection between music and brain science. My only complaint is that I would have welcomed greater depth, but then it wouldn’t be a popularization, would it? You’ll learn something about the anatomy and physiology of the brain, something about music theory, a smattering about the psychology of sound, and a lot about the connections among all of them. And if you’re hungering for greater depth, the 29 pages of annotated bibliography will satisfy your hunger. So read it!
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)?
Kathryn Cramer writes about the new book, Free-Range Kids: Giving Our Children the Freedon We Had Without Going Nuts with Worry, by Leonore Skenazy. I’ve reserved a copy through the Minuteman Library Network; maybe I’ll write a review in this blog after I’ve read it. But at least I can respond to Cramer’s observations even before reading the book:
Skenazy was dubbed “America’s Worst Mom” after she wrote about letting her 9-year-old ride the New York City subways by himself… America is now gripped with terrible anxiety about what will happen to kids if they are not constantly under the watchful eye of a parent or some paid professional. And, as Lenore Skenazy points out, the crime statistics do not bear out the claim that this is a more dangerous era. It is not. We only behave as though it is. Skenazy discusses the issue of balancing children’s freedom and safety and aims to empower parents to give their children the kind of freedom they themselves enjoyed as children.
These remarks resonated with me for several reasons, not the least of which was that I rode the Newark subway by myself when I was ten (OK, not New York, and not when I was nine, but close enough). I don’t remember how old I was when I first went into New York by myself, but it certainly was before I was a teenager. I felt trusted, not abandoned. I felt safe.
Cramer asks, “Why the de-liberation of both mother and child?”
Whose interest does it serve? Certainly not the children. It serves the interests of towns that don’t want to pay for sidewalks. It serves the interests of rating-hungry media like CNN (known in this household as Child-abuse News Network). It serves the interests of cultural conservatives. It serves the interests of car makers if our kids have to be driven everywhere. It serves the interests of lawyers, especially divorce lawyers. It serves the interests of insurance companies. In short, there are many conflicting social forces at work.
I don’t know. Is Cramer being too cynical? Or just realistic? Certainly her observations are on the mark. As a teacher since 1969, I unquestionably notice that parents hover around their kids much more than they used to. And we’re not just talking about pre-teens: even colleges suffer from the attention of helicopter parents. Teachers and parents are doing kids a disservice by curtailing their freedoms so much.
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