More Kakaes Followup

In his blog today, Dan Meyers skewers (rightly) Kakaes's basketball metaphor.  Kakaes writes

Math and science can be hard to learn—and that’s OK. The proper job of a teacher is not to make it easy, but to guide students through the difficulty by getting them to practice and persevere. “Some of the best basketball players on Earth will stand at that foul line and shoot foul shots for hours and be bored out of their minds,” says Williams. Math students, too, need to practice foul shots: adding fractions, factoring polynomials.And whether or not the students are bright, “once they buy into the idea that hard work leads to cool results,” Williams says, you can work with them.

and, as Dan points out,

1. Drills aren't a basketball player's first, only, or most prominent experience with basketball.
 Drills come after a student has been sufficiently enticed by the game of basketball — either by watching it or playing it on the playground — to sign up for a more dedicated commitment. If a player's first, only, or most prominent experience with basketball is hours of free-throw and perimeter drills, she'll quit the first day — even if she's six foot two with a twenty-eight inch vertical and enormous potential to excel at and love the game.
2. Basketball players aren't bored shooting foul shots.
 Long before "math teacher" was on my resume, I was a lanky high school basketball player trying to get his foul shooting above 50%. I'd shoot for hours but I wouldn't get bored, as Williams suggests I must have been. That's because I knew my practice had a purpose. I knew where that practice would eventually be situated. I knew it would pay off in a game where I'd be called to the line for a shot that had consequences.

Dan's right on target on both points, but I don't think he goes far enough.

1. Our (national) approach to teaching math is to avoid doing anything requiring actual thought or creativity until we've convinced as many students as possible that there's nothing worth thinking about in math;  eventually, the few "survivors" get to do actual mathematics.  If we taught English that way, it would be all grammar and spelling until senior year, when a lucky few would get to read actual poetry.  Right now, the problem isn't that the U.S. curriculum doesn't have enough skill practice; it's that it doesn't consist of much besides skill practice.
2. As Dan suggests, what *makes* skills important is their placement within the big picture of doing actual mathematics.  Being able to multiply accurately isn't worth a darn--especially in the age of calculators--if you don't have good ideas about when and what to multiply (and when and what not to).  We can be excited when kids know their times tables, the way we might be excited about a kid being able to spell really well, or lift something really heavy, but by itself, multiplying not a really useful skill except in the context of multiplication tests.

If we taught athletes the way we taught mathematics, there would be no Kobe Bryant, although there would be handful of strikingly eccentric bodybuilders who would get together to run around, lift heavy things, and engage in odd activities that make no sense to the rest of us couch potatoes.

Another idiotic "calculators = bad" article

The Slate article, "Why Johnny Can't Add Without a Calculator" is so poorly argued that I even hesitate to cite it here, but it's getting so much play that somebody has to rebut it.

Konstantin Kakaes's argument essentially boils down to three elements:

1. Many math teachers who use technology do so ineffectively.
2. Many students who are taught mathematics with calculators don't have a good grasp of basic arithmetic, or other "traditional" mathematics.
3. A few teachers teach math successfully without calculators to some students, where "successfully" is defined as "according to traditional criteria."
   
   Therefore, teaching math with calculators results in students who can't do math.

Although I would agree with #1, I would take it further:  for as long as there have been math teachers, there have been many ineffective math teachers, with or without technology.  As Kakaes himself acknowledges late in the article (when he claims that software won't be able to teach children "any time soon"), teaching (anything) is a complex process.  Teaching math requires actually understanding math, and people who understand math have always been in short supply, in and outside of the teaching profession.  So a different, simpler explanation for the failure of students to learn math is that there aren't a lot of excellent (or even mediocre--see my previous post) teachers out there teaching math.

A bigger problem with Kakaes's argument is that, if it were true, we would expect to see declining math achievement in the U.S.  In fact, the opposite is true.  TIMSS and NAEP scores have been rising steadily for the last twenty or so years--the exact time period in which calculators became standard equipment in high school mathematics classrooms.  To mention one statistic, the number of students in the U.S. passing the AP Calculus BC exam each year--half of which is no-calculator--is now more than five times the number who even took the exam in any year in the 1980's.

In fact, I would argue that calculators have made possible one of the great sea changes in mathematics education in the western world.  In 1960, there was a dropout rate of 27%, and of the 73% of U.S. students who graduated from high school, very few took any math beyond geometry or trigonometry, which was still a course offered at many colleges.  In 2009, there was a dropout rate of 8.1%, and of the 91.9% of U.S. students who graduated from high school, something like 50% (77% in 2004) had trigonometry or higher.  Put differently: we are now in a world in which about half of U.S. students are expected to learn substantial amounts of advanced algebra and trigonometry before graduating from high school.  Technology makes it possible, as great teachers like my friends John Benson and Natalie Jakucyn, to name two have shown, to increase students' access to higher mathematics. With technology, it's possible for a student who doesn't know how to add fractions to learn what a derivative is, what it means, and what you can do with it--and how to let a computer do the computations that he needs to use the derivative in an actual application.

Finally, Kakaes never engages what is, to me, the central question technology poses to the mathematics teacher, namely, what of the traditional pencil-and-paper mathematics is worth teaching?  Kakaes writes:

If you learn how to multiply 37 by 41 using a calculator, you only understand the black box. You’ll never learn how to build a better calculator that way.

Besides the inaccurate alarmism of his example--even calculator-active elementary school curricula like Trailblazers and Connected Math expect that students will be able to multiply two-digit numbers by hand (and explain their computations, a higher cognitive skill than was demanded in my day)--he proves too much.  If it were necessary to teach everyone a skill to ensure the supply of programmers able to create machines in the future, we would presumably also teach the following:

• Computation of decimal approximations of square roots, using the "two digits at a time" method found in old textbooks, or continued fractions, or the Babylonian method.
• Approximations of transcendental functions using Taylor series.
• Approximations of trignonometric functions using matrix multiplication (faster and better for most angles, actually).
• Approximations of transcendental functions using tables and linear interpolation.

But while there's a pedagogical value to each of these (my advanced students think that continued fractions are pretty cool, as any number theorist will attest), we just don't teach them anymore.  Why would we?  It's inconceivable that anyone would need to know these values accurately without a calculator, and while Kakaes is correct that many university math departments are stuffed full of old-fashioned mathematicians, even they use calculators (actually, Mathematica or Maple) to do these problems--and expect their students to do the same.  My point is just that we all agree that there's a line to be drawn between what math students should be able to do by hand and what they can (and should) use a calculator for--we're only arguing about where that line is.  

That line is porous at best.  Zalman Usiskin has pointed out (in his NCTM Yearbook article on technology), even paper-and-pencil algorithms are technology, every bit as much as computer software.  Old-fashioned long multiplication, as I've pictured it at right, is one:

This "killer app" version is fast and correct if you actually do it right--but many students find it hard to understand, hard to apply consistently, and--in practice--extremely inaccurate, because the most common errors (not shifting the second row over, for example) actually have huge effects on the results.  Other algorithms (partial products, estimation with corrections, etc.) are not as fast, and (sometimes) only produce approximate answers.  But students can actually understand and explain them, and apply them correctly.  And if you really need a completely accurate answer quickly, in a grocery store or at a worksite, wouldn't you do what I do -- that is, pull out the calculator on your phone?


Kakaes does raise some valid points.  Technology by itself isn't the sole indicator of high-quality math instruction:  there's lots of low-quality math instruction with technology (just as there's lots of low-quality math instruction without technology).  Promethean boards do not raise outcomes by themselves.  And (as Sugatra Mitra says, in his argument for why technology can be transformative for the poorest children), for kids in affluent districts (which, by his standards, is much of the U.S.), the marginal impact of any given new technology might be quite low.  But as Zalman argues, the question is never "should we teach students to use technology?", but "which technologies should we teach students to use?"  That question--not this fake "can Johnny learn math with a calculator?" question--is where the discussion should start.


Update:  A version of this article is now a posted response on Slate!  

We Don't Need More Great Teachers

First, hats off to my friend Peter DeCraene, currently in D.C. for the Presidential Awards Recognition Program.  This post is kind of in his honor.

I've been doing a lot of thinking about what would improve outcomes, on a large scale, for the truly underserved kids from around the U.S.  I've had the good fortune to know many truly great teachers, and to have been part of a department that developed several of them (with more in the pipeline!).  One of my students once compared being in one of my classes to watching a perfect game in baseball, and--except for the "watching" part--I have to say I've had few nicer compliments.  But I've come to think that the emphasis on great teaching in our nation's current dialogue about the importance of education is at best unhelpful and at worst counterproductive.  We don't need more great teachers.

Visiting China, I was struck by how rigorous the mathematics is at the nation's best schools, but also--and I was really surprised by this--how decidedly vanilla the teaching is.  A typical classroom--at a top school--looks like this:

Notice:  fifty or so students listening to a lecture by one teacher at the front, on a chalk board (no technology), and--this doesn't come through on the still photo--little to no actual student input.  On this day, students were going over questions on a practice test for their regional end-of-year exams (already an activity I'm not sure I'd spend time on, certainly not with me doing the presenting), and for forty or so minutes, literally all that happened was that students were told how to solve problems they had gotten incorrect, and marked the correct methods in their test booklets.  But the problems were something else--deep, challenging, multidimensional.  For example, the problem below (from the same class's papers) asks about what happens when a trapezoid is folded into a solid:

It's almost inconceivable that even an honors geometry class in the U.S. would ask a question this complicated.

The disjunction here poses a real question: how is it that Chinese students get to the level where they can do and appreciate such challenging problems, without getting excellent instruction?  And after a lot of asking and thinking and soul-searching, I think I have the answer:  there are very few bad teachers.  The typical middle-to-strong Chinese student gets, so far as I can tell, much more consistent instruction than a similar student in the U.S.  Ask any student at my school--selective, very high-level--and you'll get the story of fifth grade, when they didn't really do any math in math class.  (Or sixth grade, or third grade, or whatever.)  So far as I can tell, this almost never happens in China.  In China, there are some good teachers, a lot of mediocre teachers, but almost no bad teachers.  Go into any Chinese math class on a given day, and my guess is, you'll see kids getting fair-to-middling teacher-led instruction in mathematics that is reasonably clear and factually correct.  I'm claiming that Chinese students--unlike my students--don't get told that "zero isn't even or odd, it's special" or that you can't subtract 7 from 3, or that "there's a formula for solving polynomials of degree five or higher, we just haven't found it yet."  And then this instruction is supported by a consistent experience of solving rich problems on homework and on tests.  Eleven consecutive years of this kind of solid, not particularly imaginative teaching produces literally tens thousands of students who can tackle very challenging math problems--which is the point.

I've observed a lot of teachers and tried my hardest to help the teachers in my department improve their own practice, as I'm always trying to improve mine.  But I don't know what to tell a teacher to make them into another Peter, or John, or Ray, or Natalie, or even me.  I'm not sure it's possible to communicate to one person how they can become a great teacher, because one of the things about truly great teaching is that it's idiosyncratic: what John does has influenced me, but I can no more do John's teaching than I can do Groucho Marx's repartee.

On the other hand, I think it's possible, and probably not even that difficult, to delineate what it takes to be a reasonable, middle-of-the-road math teacher who produces a solid year of growth in the vast majority of his or her students.  If we could have more of those, we wouldn't have to play catch-up--which is hard even for terrific teachers, not to mention the mediocre ones.  Almost all of our students would, like the Chinese, finish eighth grade with some working knowledge of algebra and geometry--not just a collection of area formulas jumbled together--and the ability to tackle multi-step problems.  In high school, kids finishing trig would actually know enough trigonometry to apply it in precalculus and calculus, because they weren't spending trig relearning facts about functions, equations, and geometry that they should have learned a year or more previously.

I'll finish this rant post with a brief list of items I'm looking for in the next generation of mediocre teachers.  The expectations may not strike you as very high: but imagine what would happen if we could really expect them every day, every year from kindergarten through 12th grade.

1. Except for testing days, each day's class has an objective: something students are to know, understand, or be able to do that they didn't know or understand, or weren't able to do nearly as well the previous day.  Content is not simply repeated from year to year or even day to day.
2. The day's objective is clearly related to overall course goals, to local and national standards, and to what the students already know.
3. Assessment is frequent and individual:  at least a couple of times per week, students' work is collected (or assessed in class) individually to find out what they know, to give them feedback on what they need to improve, and to adjust instruction.  Assessment tasks are nontrivial, especially on formative assessments.
4. The mathematics presented each day is correct.
5. The mathematics is presented each day in a reasonably logical order.  When asked, a teacher can explain the motivation for each step, not just what the step is.
6. The time allocated to mathematics is spent actually doing mathematics, not graduation practice, watching a non-math movie, or taking a break.  (I don't make these up, but please don't ask me to name names.)
7. The time allocated to mathematics is spent with the students either (a) doing mathematics, (b) listening to brief explanations about how to do mathematics, or (c) asking each other or the teacher questions about mathematics.  ["Will this be on the test?" is not a question about mathematics.]

I'm sure there are more ... leave them in the comments.

What if we held professional development workshops to the same standards as our classes?

Every so often, a kind parent says to me "My child really felt that every minute in your class was valuable."  Of course, I don't think that's literally true, but I'm glad that this family understood my most important goal:  to make every minute valuable, in fact totally crucial.  I believe that anything else is disrespectful. Think about it:  by law, students are not just asked, but compelled to be in my classroom for (46 minutes, 90 minutes, whatever) each day.  How can you justify forcing someone to be someplace where you then waste their time?

So I hold my classes to high standards:

1. If everyone already knows it, we don't cover it.  If most but not everyone knows it, we don't cover it as a class;  I provide an opportunity to review or relearn the idea either as a pull-out, or as part of a larger task, or as one option among many activities.  If a few people know it, I give them something else to do while the rest of the class learns.
2. I figure out ahead of time and at the time how many people can already do what I want them to do, and how well, so I can do item #1.
3. I help students connect each day's lesson to course themes and to material from other courses (and also to real life).  I make sure they know why that day's lesson is important. 
4. Class time is for work that can't be done at home: because it involves high-level problem-solving, demands that they share ideas, requires higher-level thinking that they can't do independently, or because they need guided practice or reinforcement that isn't available online or with a worksheet with answers.
5. Class time is not for watching movies, reading, lecture, or even whole-class discussion, unless I expect ideas to build on each other, students to critique each others' ideas, etc.  In particular, we don't "report out" results unless there's something to do or discuss from the reports.  Time I spend talking is, as far as I can tell, mostly time wasted.
6. When the assigned work is done, I always have more math for students to work on, so that the ones who get done early don't sit around getting bored.  This strategy also decreases the incentive for students to rush through the material without thinking carefully.

Items 4-6 can be summarized simply: class time is for doing mathematics, not for watching other people do mathematics.

Now let's turn to the typical professional development session:

1. "Who here knows about Gardner's Multiple Intelligences? [or Bloom's Taxonomy, or the Common Core Standards for Mathematical Practice, or ... ]"  The teachers are all over the place, but it's hard to tell exactly what each teacher knows, because "Who knows about ____ ?" is not exactly a fine-grained assessment.
2. "Everyone do this worksheet reviewing the different Intelligences [levels of Bloom's//Standards for Mathematical Practice//etc."  Now there's no opportunity for choice or differentiation.  When you're done, you just wait around until everyone else is finished.  There's no immediate followup task.
3. "Let's watch this TED talk about ___ ".  Or:  "Read this article about ___ " I could have done this at home.  In fact, I love watching TED talks at home, so I'd be happier watching it at home and using the class time productively.  Also, what am I supposed to get out of the TED talk or reading?  Why not tell me up front?  Occasionally, the TED talk actually shows a process or strategy that would be hard to summarize, like this one by Dan Meyer.
4. "Let me tell you about ... " What is my take-away?  What do I need to get out of this?  Could I just read what you're planning to say?  and then spend group time doing some task related to the take-away?
5. "Well, we can wind up at many different places with this ... " Obviously, we're all professionals, and so it's hard to tell someone they're flat-out wrong.  But it is important to have standards and to communicate them clearly.  If the point of the activity is to rewrite a textbook activity to achieve a certain aim, and the proposed rewrite doesn't achieve it, then whom does it help to let the activity slide by?

In this area, I think we teachers are our own worst enemies.  In my classes, one norm is that everyone is wrong at least sometimes, and that correcting an error or misconception is an important job for everyone.  But how often do we sit in PD and watch someone say something that is clearly incorrect without challenging it?  Maybe one reason why in-school or departmental PD is more effective (at least for me) than inter-school PD is that we're only willing to challenge people we know well and trust.

Tony Wagner's article Rigor on Trial lists seven questions he poses to students during a lesson to assess the level of rigor; note #6 and #7.

1. What is the purpose of this lesson?
2. Why is this important to learn?
3. In what ways am I challenged to think in this lesson?
4. How will I apply, assess, or communicate what I've learned?
5. How will I know how good my work is and how I can improve it?
6. Do I feel respected by other students in this class?
7. Do I feel respected by the teacher in this class?

He asks whether these questions could "be used as a set of standards for planning and assessing both adult and student learning across a district?"  It's hard to imagine how much things would change if they--and the other standards to which we hold our own classes--were implemented as basic principles of PD.

Update:  In this morning's PD, taking my own maxim to heart, I challenged a teacher who said that you have to go over every homework problem and every answer to in-class tasks.  I said that what I see is that when the teacher "goes over" problems and answers, the energy level and engagement drop dramatically, and that time spent going over homework is mostly wasted.  Immediately another teacher said "Where you do you teach?  Oh, Payton." I stuck to my guns, pointing out--as we've discussed on this blog--that no matter what high school we're at, if more than 20-30% of the studentscan't do a particular homework problem, then that problem probably wasn't appropriate for independent work, and that if that's the situation for many problems on the assignment, then the assignment itself was too hard.  But they'd already stopped listening....

ISEF Question: Do math contests decrease math research?

I'm way behind in my postings, so this is a quick one to get back in the groove ...

I'm at Intel ISEF, the world's largest HS math, science, and engineering fair, as part of a team from Chicago trying to increase the number of students doing math research in high school.  On Monday, I walked down the math aisle and talked to the five or six kids I found setting up their projects--cool ideas, like using fractal dimension to quantify the distinction between cancerous and noncancerous cells, or linking quadratic residues to the number of digits in the base b expansion of 1/p.  And I asked them three questions:  Do you do a lot of math contests?  Have you ever done a summer math program?  Are you part of a math circle?  All eighteen answers:  no.

Now I like to think that doing these extracurricular math activities makes kids more interested in math and more likely to investigate mathematical ideas on their own, but this makes me wonder.  Some hypotheses in search of more data...let me know what you think and I'll report back after more extensive conversations on Thursday.

1. Maybe the kids who do math research are doing it because they don't have any other outlets for their math interest, as a sort of last resort.
2. Maybe the kids who do lots of other math stuff simply don't have the time or energy to do math research, because the other math stuff they do consumes all that time and energy.
3. Maybe the kids who do lots of other math stuff are also the kids driven (literally) to lots of other "Race to Nowhere" activities, so that they don't have time or energy to explore and play, not because of the math they do, but because of everything they do.
4. Maybe the kids who are driven to do high-quality research are exactly the kinds of curious loners unlikely to be attracted to math contests and summer math programs (ugh! other people!) in the first place.

Hypotheses #3 and #4 are most benign, because #1 and #2 are suggesting--to my chagrin--that part of the reason more kids aren't doing the most authentic math--the "I'm wondering about..." kind of math--is that many of them are doing math contests instead.  And that seems oddly backward.

Thoughts?

== pjk

Target group

To whom do you aim the problems you are using to teach the lesson?
This question assumes that you teach by asking interesting questions and allowing students to figure out the math. If you do not teach in this mode, I am not sure this question applies. I am also not sure I have much to offer, because I think good teaching starts by recognizing that our job is to ask interesting questions and to help students figure out the math behind the questions. I think a teacher helps by watching, listening and letting the students do the work. Telling a student how it works does not work.
But I have talked about that before. This idea is new, I think. P.J. reminded me of this yesterday so I thought I would write about it before he did.

The problem should almost always be aimed at the top group of kids, say the top 25%. There is a myth that says teach to the middle. Do that, and over half of your students learn nothing. The ones above the middle already know it so you are wasting their time.

The problem needs to be accessible to everyone, but difficult enough, challenging enough, that no one can just solve it. That means it needs to be designed so the middle kids and below always have to stretch a lot, while the best kids are still challenged.

One thing to consider is that the top 25% one day will be a different group of students another day. The problems should not be aimed at a specific student but rather at a specific level of challenge. Problem solving has to happen for every student every day, or students will not learn how to solve problems.

"Oh, but students will give up, because it will be too hard for them," you say. I say, you are with them, you are walking around watching them work, and listening. You can push them in the right direction if they have a good thought, and redirect them if they don't. If everyone is about to give up, you will know it, and you can immediately fix it by intervention.

Differentiation does not mean make it really easy. It means teach students how to think. Good teaching means helping students learn what to do when they don't know what to do. That can only happen face to face in your classroom.
It is our job.

A True Story

Back in the year 2000, there was a lot of fuss about "Reform Calculus." The Advanced Placement committee had announced changes in the AP test, and there were several calculus books being offered that were quite different from the Thomas book that most high schools had been using for years and years. After much deliberation, Evanston decided to make the change, and we adopted the Ostebee Zorn calculus book for BC Calculus. There were two of us teaching the class that year: Ron Selke and myself. We approached the year with excitement and fear.

As we worked through chapter one with the students, we both learned a lot about how to make calculus meaningful and understandable to our students. We had decided to collaboratively write tests, and so we did. The first test covered the ideas in chapter 1 rather well we thought, and we were eager to see how students performed.

To say the first test was a disaster would be an understatement. There was not one student who even tried to work all of the problems. Many students left three or four blank. Ron and I looked at the test, and it measured what we thought was important, but because of the difficulty it measured little or nothing and created considerable discontent among our students. And these are the best students we had. We adjusted the grading scale on the test, admitting that we had totally failed to create a fair test, amd promised that we would do better for the next chapter.

In considering how to fix the problem, we had several ideas. One was to break the chapters into two tests. We rejected that because it would mean giving up too much instructional time for formal assessment; we would spend the year writing and grading tests. Another idea was to only test the easy stuff. We rejected that approach as not being in the best interests of our students. We were committed to making the class a rich mathematical experience that matched the wonderful way Ostebee and Zorn were allowing the course to unfold. Then one morning Ron came to me with one of the best ideas I have encountered. And I resisted at first. I offered reasons why it was a bad idea. After all of that, I agreed to try it. I have never looked back.

Ron's idea was to make a collaborative problem part of the test. The original plan said the day before the test, we would give each student a collaborative problem. This collaborative problem would consist of several parts and would encompass the main ideas of the chapter. The problem would allow us to address some of the subtle concepts or more complicated aspects of the material covered in the chapter. Each student would be required to work with at least two other people, and each person would turn in one copy of the team's perfect, well-organized, well-written solution when the student came in to take the in-class part. It would count as about one seventh of the test grade.

After a couple of tries, we modified the conditions a bit. In particular: we gave students the collaborative problem several days before the test, and always so it would be in their hands over a weekend. We posted the collaborative problem on our websites to allow absent students to access it. We helped shy students find collaborators. We encouraged collaboration with students in the other sections of BC Calculus.

The collaborative problem turned in to one of the best educational experiences in my career. Most of the work was correct, making them easy to grade. The in-class part was now manageable, but we were assessing all of the material. More importantly, students were learning mathematics while taking a test. What an amazing experience! Ron and I listened to their conversations as they worked the problems in the math lab, we heard them talking at the beginning of class, and we flat-out asked them about their experiences with the collaborative problem. All of what we heard was exciting. There was an outcry when we did not offer a collaborative problem for a test we gave on a half chapter. We had found a way to help them consolidate their ideas before they took the in-class part, so the in-class tests were also done better.

We began to notice that the collaborative groups became entities in themselves. The students started to get together just to study. Some of them met during a common free period every day in the math lab or the cafeteria and went over homework questions. Parents praised us for the learning they saw taking place in their homes as students gathered. Collaborative groups compared reaults with other collaborative groups. Students made friends and learned that learning is not an isolated activity.

I also taught a class in Multivariable Calculus and another in Linear Algebra for those students who had finished BC Calculus and had not graduated. There was a clamor for a collaborative problem in that class, so I happily agreed to their demands.

The second year, we realized that we did not have to rewrite the collaborative problem except to fix questiosn that didn't quite go where we thought they would. By now these questions are establsihed. The students understand their value as learning tools, and so there is little evidence that they are looking at old tests.

One moral to the story: when you try something new, it rarely works the way you
intended it. But the thing to do is not to throw it out--then you wind up
doing the same things you were dissatisfied with before that led you to make
the change. Rather, you need to try and identify what's not working and
fix that piece, By iterating several times, you come up with a new
strategy that does accomplish your goals--and even gets you places you hadn't
realized you wanted to be!


By the way, our students performed better than ever on the AP test and have ever since. It is nice when one's observations are valideted by an outside source.