Learning Strategies

Our distinguished governor, Mitt Romney, advocates merit pay for teachers, based on the standardized test scores of their students. Vivian Troen and Katherine C. Boles wrote an op-ed piece about this idea in the Boston Globe on 9/28. I was distracted by the fact that they managed to commit two logical errors in a single ten-word phrase, accusing Romney of “proving the axiom that no bad idea stays dead forever.” Aside from this flaw, their essay makes many important points, such as the bad track record that merit pay has achieved:

Teaching became more mechanical as teachers found that drill and rote repetition produced the “best” results. Both teachers and administrators were tempted to falsify results, and many did.

But they don’t bring up the three most important flaws in Romney’s proposal:

• Merit pay destroys collaboration and collegiality. If teachers are in competition with each other, there is little incentive for experienced teachers to share good ideas and improve their colleagues’ success levels. (In some schools, like Weston, such helpfulness is noticed and could well be a component of what’s rewarded in merit pay, but Romney is talking about measuring merit by test scores, not by success as a team player.)


• If standardized tests such as MCAS measure anything, they measure learning that has taken place over a period of years. It makes no sense to reward or punish this year’s teacher for the work of many teachers and parents over a period of years.


• Most important is this related point: Schools aren’t businesses; if the “raw materials” aren’t up to our standards, we can’t reject them and ask for better students. We have to teach whoever shows up. Objective tests aren’t going to measure how well we do this. Who will want to teach the weakest students if we are measured by their test scores?

Labels: teaching and learning, Weston

Yesterday evening we held our annual Back-to-School Night at Weston High School. You know the drill: the parents come to school, arrive late at their first-period class because they can’t find a parking space, go to one ten-minute class after another in a frantic parody of a school day — and everyone goes home exhausted at 9 PM. And yet the experience is useful (for all) and even exhilarating (for some). Even someone who talks as fast as I do can’t say a whole lot in ten minutes, but that isn’t really the point. In fact, in our modern world of the Internet, the purpose of Back-to-School Night isn’t to communicate information; parents can use the World Wide Web to find out everything they need to know about their kids’ courses, at least if we’re talking about Weston High School math courses. The real purpose is to establish some brief human contact between parent and teacher, so the parents can make us feel appreciated by letting us know how much their sons and daughters are enjoying our courses, and the teachers can make the parents feel positive by showing them that we’re enthusiastic and competent.

You can probably tell that I’m in the camp that finds Back-to-School Night exhilarating.

Labels: teaching and learning, Weston

One of my students — let’s call her Artemis — muses in her blog:

Giving partial credit may be helpful to a student’s grade in school but in real life, people don’t want to know how you did something, they just want the result. The how part gets shuffled into a stack of papers that some poor soul will have to look over later. The answer seems to be more important than how you got it.

But of course this is highschool, so partial credit is better for getting into college.

Artemis is more right than wrong here, so I would give her substantial partial credit. (No, wait — this wasn’t something for me to grade! It was merely a post in a student’s blog.) It’s true that if I go to a doctor or a lawyer or a computer programmer, what I want is a correct diagnosis, correct legal advice, or a working program. I will cerainly not be happy if the doctor notices five of my six symptoms and therefore prescribes the wrong meds, or if the lawyer gets most of the law right and offers the wrong advice, or if the programmer sells me a program that contains a fatal bug.

So in that sense Artemis is right. I’m not happy about giving “partial credit” to the adult professional who did most of the work correctly but got the wrong result. But...no adult professional is perfect. Programmers make mistakes — all the time! Lawyers make mistakes and lose cases they should have won. Even doctors make mistakes, and I’m not just referring to malpractice. If we can’t get perfection — and we know we can’t — good work is still better than bad. Otherwise we fall into the well-known trap where the perfect is the enemy of the good. If we refuse to give partial credit — in high school or in real life — we are equating all qualities of work that fall short of perfect.

Anyway, why isn’t high school “real life”? But that’s a question for another day.

Labels: teaching and learning

Key Curriculum Press publishes our new Algebra II textbook as well as two software products that have had a significant impact on many high-school math teachers, Geometer’s Sketchpad and Fathom. Their recent catalog includes a moderately long article entitled “The seven habits of highly effective math teachers.” Now what do you expect these teachers to be doing? Since the article was in a Key Curriculum Press catalog, I had figured that they "use Geometer’s Sketchpad" and "incorporate lessons from Discovering Algebra” — but I was surprised to see that the seven habits turned out to be much subtler than mere product placement. In fact, it’s not a bad list, even though a little bit of blatant advertising certainly does find its way in. The list, of course, consists of seven sections:

1. Use hands-on activities and real data. [In this section, Discovering Algebra is mentioned briefly, but Fathom isn’t, even though it’s chock-full of real data.]


2. Try cooperative learning groups.


3. Incorporate technology. [Aha! Sketchpad is mentioned twice in this section.]


4. Create a curriculum that works for your students.


5. Be a little silly. [a surprise]


6. Connect math to other subjects.


7. Keep learning.

Each section includes relevant quotations and observations from real teachers. All in all, the catalog shows refreshing self-restraint.

Labels: teaching and learning

On Tuesday, Boston voters will go to the polls in the “preliminary election” for City Council. Something like a primary, the preliminary election narrows each race down to a number of candidates equal to twice the number who will be elected. Thus we will narrow the 15 candidates for at-large City Councilor down to 8, since 4 will be elected in the final election; and where there are more than two candidates for a district City Councilor we will narrow them down to two, since one is chosen in each district. (When there are only two candidates, as there are for Mayor this year, no preliminary election is needed and therefore none is held.) I say that the preliminary is something like a primary, but it’s non-partisan since practically every candidate in the city of Boston is a Democrat.

Anyway, the important mathematical issue is this: in both the preliminary and final elections, each voter is entitled to cast four votes for at-large City Councilor, but is it wise to exercise that right or should one cast only one vote, which is known as a bullet vote? (Intermediate positions — casting two or three votes — are also possible of course.) I’ll present both sides with an example from the final election, though the argument holds equally well for the preliminary except that the cut-off point is between the eighth and ninth positions rather than between the fourth and fifth:

• Suppose you and 20 others agree that your first choice is Sam Yoon and your second choice is Matt O’Malley. And suppose, before your 21 ballots are counted, Yoon is in a tight race for fourth place, just 13 votes behind O’Malley. If the 21 of you bullet-vote for Yoon, your votes put him 8 votes ahead of O’Malley for fourth place, letting Yoon squeak in. But if the 21 of you dilute your support by voting for both of these fine candidates, Yoon remains 13 votes behind O’Malley, who then pushes Yoon out of fourth place. Net result: your second choice would be elected and your first choice defeated. Bullet voting is clearly the correct strategy.


• On the other hand, there is also an argument against bullet voting. To understand this argument we have to consider a candidate of medium popularity whom you definitely do not want elected. Let’s call him Steve Murphy, to pick a name at random. Now suppose Murphy is 13 votes ahead of O’Malley for fourth place before the 21 ballots from you and your friends are counted. And suppose that Yoon is way ahead (or way behind, it doesn’t matter which), so your 21 votes for Yoon won’t make any difference in his election. If you bullet-vote for Yoon, what happens? Murphy gets elected by finishing in fourth place! But if you each cast two votes — one for Yoon and one for O’Malley — then O’Malley finishes 8 votes ahead of Murphy and squeaks in for a fourth-place victory! Net result of refusing to bullet-vote: your second choice would be elected and your last choice would be defeated. Bullet voting is clearly not the correct strategy!

So, to bullet-vote or not? The choice depends entirely on which scenario you think is more probable. If the race for fourth place is between your two favorite candidates, you should bullet-vote. If the race for fourth place is between your second choice and someone you strongly dislike, you should not bullet-vote. Maybe it’s not how they would handle it on Numb3rs, but math can’t give you a definite answer here. Actually, come to think of it, it is rather like last week’s Numb3rs, since it’s all a matter of Bayesian probabilities.

Labels: Dorchester, life, math

Standards-based education encourages us to give untimed tests. This idea makes a lot of sense: if I want to tell whether a student can solve a quadratic equation, I shouldn’t be testing how fast s/he can solve the equation. The important thing is knowledge of skills and concepts and the ability to apply that knowledge — not speed.

And yet...

Consider two students. Student A has learned all the material thoroughly and can efficiently solve problems, even when they are unfamiliar and require creative thinking. Student B has done no studying or other preparation and has to reconstruct how to solve every problem, either by guess-and-check or by elaborate work-arounds. What purports to be a 45-minute test is completed by A in 30 minutes and by B in 90. Both students eventually get 88% of the problems right. Do they deserve the same grade?

Perhaps B deserves a higher grade, for having done such a good job of thinking. Or perhaps A does, for having learned the material.

B has demonstrated an important skill, but is that skill the one that’s being tested? B still doesn’t know how to solve quadratic equations.

And yet...

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.” This is indeed the true test of intelligence, but we aren’t doing high-school students any service if we leave them unprepared for SATs and for college exams because we let them take as much time as they want in order to make up for what they haven’t learned. There must be a place for timed work and a place for untimed work.

(A separate issue is the one of limited extra time. Certain students, who have demonstrated learning disabilities, are legally allowed 50% or 100% extra time on tests. Their tests are still timed, but an adjustment is made in the amount of time.)

Labels: teaching and learning

If you look at almost any set of modern standards for mathematics teaching — such as the NCTM’s or the Massachusetts curriculum frameworks or Weston’s own standards — you will see a prominent role for applications of mathematics. This is a welcome change from the ’60s, when I attended high school (together with George W. Bush, but that’s another story and will wait for another day).

But it’s important for us to us to understand why it’s a welcome change. The reason is almost entirely motivational. Some students love math for its own sake and never need to worry about applications. At the other extreme, some students hate math and ask “When will I ever need to use this?” without ever waiting to listen to an answer, because the question is really a complaint rather than a request for information. But the vast majority of students, who are in between these two groups, are more motivated to learn math when they can see that it has authentic applications than when it’s totally abstract (though even they can still be hooked by intriguing puzzles).

The trouble comes when students reverse cause and effect in trying to understand mathematicians’ motivations. Mathematicians practically never start with an application and then develop the theory to support it. Instead, they develop the theory and then eventually it finds an application. This has even happened to prime numbers, long considered one of the most abstract and least applicable ideas in math, being part of number theory, which for millennia was safe from applications. Now number theory in general and prime numbers in particular have become the cornerstone of encrypting messages over the Internet, a practical application if there ever was one.

Eugene Wigner’s famous 1960 article, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” discusses fundamental philosophical justifications for the surprising applicability of math, at least in the area of physics. If you haven’t read the article, do so.

One side-comment on this idea: although science teachers and math teachers have similar sensibilities and overlapping interests, a source of continual dispute between the two groups is the use of units. Science teachers and science textbooks rightly insist on the careful use of units, as their principles and their answers can’t make sense without the correct units. But math teachers and math textbooks usually refer to things like “3-4-5” triangles, with no units attached. Kids are often confused by the mixed messages that this conflict sends, especially since math teachers sometimes do insist on units. Students want to know who’s right, the math teacher or the science teacher. After I say that of course the math teacher is right (at least if the student has a sense of humor), I point out that both of the teachers are right, becasuse they’re engaged in different endeavors. Science almost always requires correct units in order for statements to make sense, but math is more abstract, and part of its unreasonable effectiveness is that mathematical theorems are general and do not depend on particular units. When we’re doing an applied problem, we do use units and make sure that they are correct.

Labels: math, teaching and learning, Weston

One key tenet of standards-based education is the observation that some students take more time than others to master any given skill or concept. No one can disagree with the observation, but the conclusions to be drawn from it are another matter.

Let’s call our topic X. We’re supposed to believe in the following conclusions:

1. All students can learn X.


2. Since some students will learn X faster than others, we need to grade them on their final achievements, not on their progress as of a specific date.


3. In order to be able to schedule a summative assessment for X (final test or whatever) on a particular date, we have to provide extra hours of learning time for any student who needs them.


4. If X is deemed important, it’s important for all students, since we don’t want to close the doors on any particular future for any student.

There is definitely something appealing about these four conclusions, but all four are fundamentally wrong-headed unless they are significantly modified. Let’s look at them one by one:

1. In fact, not all students can learn X. Some are severely brain-damaged. Some don’t have the prerequisite knowledge. Some have such severe emotional problems or disrupted home lives that they are not prepared to learn. Even if for some reason you exclude all those students from consideration, the reality is that June 21 is going to come around right on schedule — right after June 20 — and not everyone can learn everything by that date! The only way to believe that all students can learn the material of every course by the end of that course is to dumb down the demands and thereby penalize the kids who are most capable and most interested in learning. That’s one of the effects of the No Child Too Far Ahead Act, or whatever it’s called.


2. In an ideal world, it would be great to grade students on their final achievements, not on their progress as of a specific date. But in the real world, any teacher who does that is inviting students to procrastinate and procrastinate. I would love to think that grades aren’t important, but if I don’t grade tomorrow’s quiz, there will be too many kids who will just slough it off.


3. Sure, we should provide extra hours of learning time for any student who needs them. We should also have class sizes limited to 18 (the average in Switzerland). But the number of hours in the school week and the realities of school budgets make both of these dreams impossible. High school class sizes average 23 in the U.S. (29 in California!), and they’re not likely to drop any time soon. Weston does provide 50% extra classtime for Algebra 2 students who have been identified as needing it, but even in Weston you can’t get that for geometry or precalculus. (And if we had the budget to offer it, how many students could fit it into their schedule? And what if they also need extra time in history and science and Latin?)


4. Item 4 is particularly well-intentioned. Indeed we don’t want to close the doors on any particular future for any student. That’s why Weston requires Algebra II for all students. But that’s a baseline, not a general conclusion about all topics. There’s simply too much material that’s important, and what’s essential for the student who’s going to take AP Physics simply isn’t the same as what’s essential for the student who’s going to take AP European History, no matter how much we might like to pretend that everything’s that one student needs is also needed by another student.

Labels: teaching and learning, Weston

In this week’s New York Times Magazine, language expert William Safire observes that there are no bad puns:

Remember, there are no “bad” puns — all plays on words are good, and the louder the groans they elicit, the better. And never forget, do not insult your audience by calling attention to the coming wordplay.

Labels: life, linguistics

Math teachers who are looking for short, narrowly focused math activities should check out the Shodor Foundation. Although they tend to focus mostly on middle-school math, they have plenty of interactive activities that are suitable for high-school students of all levels, ranging from factoring integers to Pascal’s triangle to recursion to affine ciphers to normal distributions. Be sure to pay attention to the what?, how?, and why? at the top of each activity.

Labels: teaching and learning

Some lawyers, including one of my colleagues, point out the irony that the new law requiring all schools and colleges to observe yesterday’s Constitution Day may be unconstitutional. (Techically, it’s not “all schools and colleges” — just those receiving federal dollars. But effectively that’s all of them.) Since yesterday was a Saturday, some schools observed it on Friday and others will do so on Monday. Perhaps some of them will pick the most appropriate way to recognize the Constitution: encouraging students and librarians to exercise their First Amendment rights.

There is an interesting article on the subject in Inside Higher Ed:

Edward Rubin, the law dean at Vanderbilt [University], is anything but muted on the topic. “I’m surprised that the Congress and the president would choose to honor the Constitution by violating it,” Rubin said. “Nothing could be further from the meaning of the Constitution than compelling speech about a particular topic at a particular time.”

Labels: teaching and learning

This is a follow-up to my post of September 14 concerning my student’s iPod with the picture of Jesus on its screen.

First, Keith got himself interviewed by Fox News the other day and showed the iPod on camera; I didn’t get to see that, since for some reason I don’t watch Fox News. Then he was interviewed yesterday by Channel 7, and I did get to see the segment on their evening news show. (As soon as my wife caught the promo, she woke me up out of a sound sleep at 11:15 last night so I could see the broadcast.) I have it on good authority that the segment also aired several times this morning during Channel 7’s Today in New England show.

What will Keith do for an encore? Can he use this as the basis for the computer game he’ll be writing later this semester as his final project in Computer Science?

Labels: teaching and learning, technology

This is a follow-up to my post of September 13. Today’s topic is also testing, but from a different POV: the use of “formative assessment”.

We all know that testing has four purposes:

1. To provide feedback, both to the teacher and to the student, giving a snapshot of how the student is doing and how much s/he has learned. (For those who like euphemisms, we say that a test provides an opportunity for the student to demonstrate competence.)


2. To create a learning experience that supplements classwork and homework.


3. To furnish data that the teacher will use to calculate a grade.


4. To satisfy requirements of colleges, the state of Massachusetts, and the No Child Left Behind act.

It’s not true, of course, that we all know this. I have always insisted on the importance of #2 and try to provide something new on each test so that students can learn something from taking it — but many people (students and teachers alike) reject this notion. Some colleagues have vehemently argued with me about it. In any case, the first two purposes are the ones we admire, and the last two are the ones we endure.

But in our Department meeting this week we all read an article by Paul Black and Dylan Wiliam, who argued for a fascinating variation on purpose #2:

...formative assessment produces significant student learning gains and...helps to narrow the achievement gap between low and high achievers.

Although a good editor could reducing the length of this article by about 50% and thereby greatly improve it, its research offers a potentially critical conclusion. Perhaps we finally have an answer here to the vexing question of how to narrow the achievement gap. (See my post of July 1.)

Labels: teaching and learning

One of my ninth-graders accidentally dropped his iPod, and the screen shattered into an image of Jesus! I told him he could probably sell it to the National Enquirer for tens of thousands of dollars, but he decided to sell it on eBay instead, as you can see in the link. (Check out the second and larger image on the eBay page.) Don’t read his description if you’re easily offended.

Labels: teaching and learning, technology

Several years ago the entire faculty of the Weston Public Schools participated in a series of workshops on so-called Standards-based Education (SBE). There were actually a lot of good ideas in these workshops. In fact, I estimated that I had already been doing about 60% of what we learned there, so those must have been good ideas. Being standards-based is a good thing.

But approximately another 30% consisted of ideas that I disagreed with and considered wrongheaded. I will discuss these in later posts.

So that left 10% that was actually useful to me.

This is the first of a series of posts reflecting on SBE with the perspective of six more years of teaching since the original workshops.

Today’s topic is the idea of writing the unit test (“summative assessment” in educationese jargon) ahead of time. The theory is that you should “begin with the end in mind” rather than go day by day and see where you end up. That’s a fine theory, reminding me of a poster I had in my college dorm room back in the sixties: “If you don’t know where you’re going, you’ll probably end up somewhere else.”

So of course I want to begin by thinking through what I’m going to want my students to know and to be able to do at the end of the current unit. I have no problem with that.

But I have a very big problem — two big problems, in fact — with the notion that all the teachers who teach sections of a course should actually agree on and write the final unit test ahead of time:

1. Teaching to the test. In this time of federally mandated standardized testing and so-called “no child left behind,” it’s supposed to be OK to teach to the test. But let’s look at what really happens if the teacher knows the final test when beginning the unit. We always run short of time. We can’t possibly teach everything we want to teach on a topic, and a 60- or 75-minute test can’t possibly test everything that should be tested on a topic. Both teaching and assessment are necessarily selective. If I know the final test ahead of time, I will inevitably (even if only subconsciously) make decisions based on what’s on the test rather than on what’s most important for students to know. Worse yet, many teachers will become competitive and want their students to do better than other teachers’ students, so they may inadvertently become corrupt and actually teach problems nearly identical to ones on the test, perhaps only changing the numbers. There’s a reason why standardized tests are never revealed to teachers.


2. Cheating. Unless all the sections of a course meet at the same time, some students will find out the test questions from others. That’s the main reason why Weston High School’s Math Department gives all final exams at the same time. But that’s impossible with a regular unit test. In fact, there’s never even a single day when all sections meet. It’s just inviting kids to cheat when the same test is given over a period of two days. (I know, you’re shocked, shocked to hear that there might be cheating in Weston, and I hate to disillusion you...)

Labels: teaching and learning, Weston

Now that Michael Brown has resigned, maybe we can learn about everything else that’s wrong with FEMA. One thing that’s wrong is that FEMA requires everyone to use Internet Explorer:

Hurricane Katrina victims seeking to file claims with the Federal Emergency Management Agency (FEMA) may only do so with Internet Explorer 6.0 or above.

...the use of Windows PCs and Explorer have also caused some consternation among relief workers who have reportedly complained of the time it takes to set up and secure a Windows machine versus other operating systems, which rely on other browsers.

...many of the IT professionals who have rushed into the relief effort carry Apple PowerBooks...

Labels: technology

How do you display a calculator to a large group of people, such as a class? Simple if it’s a TI graphing calculator: just download the free Virtual TI, which displays not only the calculator screen but also the entire keyboard. It’s a perfect emulator of the TI and can be controlled entirely by the mouse as well as partially by the keyboard. Better yet, it’s free and completely legal! The only downside is that you need to use Virtual PC if you want to run it on a Mac.

Labels: teaching and learning, technology

Bruce Schneier describes the Case of the Kutztown 13:

...a group of high schoolers charged with felonies for bypassing security with school-issued laptops, downloading forbidden internet goodies and using monitoring software to spy on district administrators.

The students, their families, and outraged supporters say authorities are overreacting, punishing the kids not for any heinous behavior — no malicious acts are alleged — but rather because they outsmarted the district’s technology workers....

The...district issued some 600 Apple iBook laptops to every student at the high school about 50 miles northwest of Philadelphia. The computers were loaded with a filtering program that limited Internet access...

But those barriers proved easily surmountable: The administrative password that allowed students to reconfigure computers and obtain unrestricted Internet access was easy to obtain: ...the password was taped to the backs of the computers.

That’s high security for you.

Labels: teaching and learning

According to an article in this morning’s Boston Globe, the principal of Hopkinton High School has raised the grades assigned by a math teacher with 25 years of experience:

Hopkinton High School teacher Rachel Bartlett appeared before the School Committee this week to complain about being asked to bump up grades for a mediocre group of 11th-grade students. She refused, but the grades were raised anyway, lifting the overall class average from 70 to 77...

Bartlett, a 25-year veteran of the Hopkinton schools, told the School Committee Thursday that she tries to maintain high standards for all her students and wasn’t pleased with what this year’s group achieved. But, she said, “This past year I had two sections of students with particularly weak skills and poor work habits. The grades reflected that level of skill and effort.” She added that few students took advantage of extra help when it was offered...

Bartlett told the committee that Gould asked her to change the grades and “meet the students where they were.”...

Yes, of course we have to meet the students where they are. That’s in September. But then we take them somewhere else, and we should grade them according to what they know and what they’ve learned along the way. It doesn’t do students a service to give them high grades when they don’t know anything.

The problem is exacerbated when students don’t take advantage of opportunities for extra help. But even when they do, effort can only take you so far. If you haven’t demonstrated competence in what you’re supposed to be learning, what message is being sent when you are given a high grade?

Labels: teaching and learning

Our westonmath.org website looked great in Safari on Mac OS X. But then we discovered that Microsoft misinterprets much of the CSS code, so the site looked terrible in Internet Explorer on both Macintosh and Windows. Worse yet, it looked terrible in two different ways!

It’s extremely frustrating trying to make a single stylesheet work in all three browsers. Why do we have so much trouble with padding and width?

Labels: technology