Back to school: this year, geometry and AP Statistics

This year I’m back in the math department, teaching geometry and AP Statistics. This is the first year ever that I’ve had a year teaching subjects I’ve taught before. One of the things I realized at the end of the last academic year–the end of my tenth year of teaching middle- and high-school students–that I’d taught 24 different subjects in those years, or taught out of textbooks that were so dramatically different from others I’d used that it might as well have been teaching a new course. It’s nice to return to something familiar and concentrate on teaching and classroom management instead of reinventing a new curriculum.

I have 5 geometry classes and 1 AP Statistics class. With 31 students in a room that comfortably holds 24, it’s a bit tight, but it’s great fun to be teaching juniors and seniors after an absence of 5 years.

The lead-in picture was our first data gathering, a classification of people’s birthdays. The top diagram shows what the composite looked like for the entire class; the students then separated the birthdays by gender to see if there were any patterns. With only 31 data points, it’s hard to draw any real conclusions, but it started the students thinking about gathering statistics about themselves and analyzing them.

The next class I had a harder but more insightful exercise. The students measured their heights (in inches; we are an Imperial-system country, alas, unlike the rest of the world) and then posted it. This exercise was my reintroduction to high-school students after 4 years in teaching middle school. They follow succinct instructions well, figure things out, and were off measuring without any additional help from me, probably taking about one-third the time middle-schoolers would have taken.

The major problem is that I now have 3 whiteboards in my class, and there’s not a lot of clear wall to measure. They adapted creatively. Here as some of them measuring against the Greek alphabet on a concrete column:

Others measured by the closed classroom door:

And others found whatever space they could:

At the end, they placed their measurements using Post-its to create a histogram. I’d never tried using Post-its for the purpose before this year, but they work remarkably well. Even without segregating by gender, the height differences between male and female were obvious (pink is female; blue is male, and yes, I know it’s a cliche):

You’ll be seeing more of their work as the year progresses!

N.B.: I will be finishing the various journeys we’ve taken at some point, but everything has been preparation for the beginning of school since August 1. Stay tuned (or “watch this space,” which I gather has replaced “stay tuned”)

Sea depth and sea height: teaching and measuring

The Maury Project is about teaching teachers to teach other teachers. While that may be a mouthful, it also means that we do presentations for each other. I haven’t documented one of these yet, and today’s was a good one.

The purpose of this lesson is to teach student how to measure the depth of an ocean or lake without being able to see the bottom. Here, Alicia Fisher (on the left, who teaches eat Seat Pleasant Elementary in Capitol Heights, Maryland) and Candice Autry (on the right, who teaches middle-school science at the Sheridan School in Washington, D.C.) are placing a 3-D contour of the ocean floor inside a styrofoam cooler.  They then cover the cooler with a filter from a heating system, which allows the students to pass a rod through the fiberglass insulation of the filter:

They can’t see what’s down below, but they can push their rods through until they stop on the bottom:

Tthe styrofoam cooler has masking tape on the sides with co-ordinates, letters running along the length and numbers across the width, rather like a spreadsheet. The students then record their data and then graph the data and cut out the resulting curve. If they take enough measurements, they can get a contour map of the ocean floor. It’s a really good exercise, and one that can work at almost any level, from elementary school to high school.

Don McManus, one of the Maury Project instructors, then showed us how you can do something similar and build a 3-D contour using Post-its. Through some complicated gravitational calculations, it can be shown that if there’s a trench in the ocean, the sea level is lower there. Similarly, if there’s a high point, such as a seamount or mid-ocean ridge, the water will be higher. This is a good thing, since we can determine sea-level height from satellites bouncing radar waves off the ocean surface. Here’s Don demonstrating the weird and counter-intuitive vector mechanics of how this works:

We then did the Post-it exercise. You graph the sea level reported by the satellite on a Post-it, and cut it out. Assemble a couple of these, and you have an approximation of the contour of the ocean floor, like this:

Claire, as always, does it better than I do. She’s justifiably proud of what she did.

What works for mountains also works for ocean trenches. Here’s one modeling the Java (or Sunda) trench off of Indonesia, whose tectonic movement generated the Boxing Day earthquake and tsunami of 2004:

Both of these exercises are simple and elegant ways of mapping 3-dimensional data that are within the grasp of most students. The more I teach math, the more I’m convinced that lots of exercises exactly like this are what students need to hone their mathematics skills.

Common Core panacea…the drumbeat continues

Friends and family have again directed me to a several new articles/editorials about Common Core standards. The first was a very short—and factually empty—editorial from the New York Times on August 14, 2015 bemoaning the fact that 20% of New York families withdrew their students from standardized testing this year. The Center for American Progress, a Washington-based think tank founded by former members of the Bill Clinton administration, has also published an opinion piece arguing in favor of Common Core math standards. A somewhat contrary opinion is that of Carol Burris, a retired and highly-regarded regarded superintendant from New York state, who demonstrated convincingly in the Washington Post Common Core tests in math in New York state had barely moved the needle at all toward proficiency.

The New York Times editorial is breathtakingly vapid. Here is what now passes for analysis at the New York Times:

This ill-conceived boycott could damage educational reform — desperately needed in poor and rural communities — and undermine the Common Core standards adopted by New York and many other states. The standards offer the best hope for holding school districts accountable for educating all students, regardless of race or income.

How the mighty have fallen. What’s more interesting to me is an issue it raises but doesn’t state, which is why parents are in open revolt in New York state against Common Core standards and tests. The revolt seems to startle the editorial board of the Times. Why they should be startled or dismayed is mystifying, other than that they are totally out of touch with the reality in the schools. As the editorial points out, “For the most part, those opting out were white and in wealthy or middle-class communities.”

Wow. What a surprise. Wealthier, politically-active people who are used to democratically controlling their children’s education through elected school boards are forced to implement tests they think are harmful to their children, and when the bureaucrat who forces the issue (John King), comes to “listen” to their concerns, he in fact does nothing but hector the parents about their inadequacies and the brilliance of the imposed plan. If the elites like the New York Times and the state educational bureaucracies are this arrogant and politically inept, I don’t have much sympathy that they get bad results. What do you expect when you spit in people’s eyes?

What the editorial doesn’t address is whether Common Core is all that it’s touted to be by the Gates Foundation and other proponents. (The Times assumes as given that it is.) Now, I actually have a lot of sympathy for the standards. I know some of the folks who drafted them, and they are serious and concerned people. Some are even educators. As aspirational goals, the Common Core standards are fine and, with some exceptions, are well-articulated and make sense. Certainly the idea of teaching conceptual mathematics rather than procedural mathematics (where you learn a method but have no idea of why it works) is the route we ought to be following. This is what the opinion piece from the Center for American Progress focuses on, making a decent argument that conceptual understanding gives superior results to simply learning procedures without understanding why they work.

But that’s not enough to make the standards work, which is what the Times and the Center for American Progress just don’t understand. I have no argument with the idea that conceptual mathematical knowledge is superior to teaching mere procedures. That’s the way I teach because I think it’s the only truly effective method. Where I part company with is the way in which the Common Core standards have been implemented. We went from being an entirely procedurally-driven instruction (with occasional dissenters and trouble-makers like me) overnight to doing conceptually-based instruction. Does anybody see any problem with this immediate 180° shift?

Making a radical change like this overnight virtually guarantees failure. Students who have been raised do attack problems by rote cannot suddenly turn around and explain why their procedures work. It requires years of training and preparation, slow and often painful. Throw them in suddenly to a new world and what happens? They start failing. I’m sure everyone is shocked—shocked!—to hear that.

This is precisely the point that Carol Burris makes in the Washington Post. Why would any rational person expect scores to go up? They haven’t, and they won’t unless we get serious about phasing in Common Core gradually. Sudden immersion results in shock and system failure. If you throw a lot of students into unfamiliar and choppy waters, I expect a lot of them to drown. And that’s what’s happening. Scores are appalling everywhere.

If Common Core is going to be adopted, we need to have a discussion we haven’t had, which is how so complete a change in educational pedagogy is going to be accomplished. Simply imposing it from above will result in failure in 100% of the cases. Where the conversation needs to be is about not about the should but about the how: how long will it take students to master the basic skills? How do we retrain teachers to teach conceptually rather than procedurally? (It’s a lot easier to teacher procedure than concepts, which is why it persists as a teaching method to overworked teachers.) How do we deal with the students who are so far along that we’re realistically not going to be able to retrain them?

You will hear a lot from educational bureaucrats and textbooks manufacturers that they’ve taken care of all these problems. Anyone saying such things is unworthy of further trust. The textbook manufacturers have, for the most part, plucked ideas like Singapore Math (a very sophisticated and profoundly different way of approaching mathematics that’s shown great success in Singapore), plopped them down in a textbook with the context and structure that Singapore gives them, and claims they’ve met Common Core standards. They’re not even close. This is a time for real conversation among serious adults, not among ideologues promoting some reform agenda or centralizing bureaucrats thinking that imposition from above or textbook manufacturers looking to score a quick buck off the lastest fad. We need to start talking seriously about what’s feasible in the how. Let the real discussion begin.

NB: for those unfamiliar with the words or the sentiment behind the phrase “I’m shocked—shocked!—” enjoy the following clip from “Casablanca” where Claude Raines, the corrupt French policeman in World-War-II-Casablanca, caves to political pressure from the Germans. Priceless, and a necessary part of anyone’s cultural understanding:

https://www.youtube.com/watch?v=nM_A4Skusro

Rethinking the high-school science curriculum: keeping the current structure but changing the focus

Look at this picture. See everybody smiling? How can you NOT smile when you’re in the field, dressed up in waders, and stomping through the water like a 4-year-old? (This may be truer of males than of females, though I’ve found that most teachers and students love doing science outdoors.)

How does this compare with what science is actually like in our schools? The comparison is not favorable. Most science classes are lecture-and-calculation. As education budgets get cut, science budgets for supplies get cut, so there are fewer demonstrations or hands-on experiments. In addition, many charter schools are in facilities that make no allowance for a real science classroom with running water, hoods, ventilators, and disposal of toxic chemicals, so there’s not a lot of student work. Instead, it’s mainly lecture, note-taking, some on-line research, and worksheets. In short, boring!

Why do we do this when we could kindle an undying passion for finding out stuff about nature? A big answer, I think, is that the science standards that we’ve been using for the past 20 years are impossibly broad. The idea seems to be that, since most students aren’t going to take another course in the field, we need to give them the broadest exposure possible. The result is much like the Advanced Placement courses: knowledge that’s a mile-wide and an inch deep, and frankly, not very interesting knowledge at that. But, hey, we hit all the standards before the year-end exams, right?

This alone is good reason to junk most of the Every-Child-Left-Behind testing regime (yes, yes, I know it’s actually “No Child Left Behind,” but since the law is a complete failure, I prefer to give it an honest name). It’s foolish and silly to let a test, and a bad one at that, drive the curriculum. So what’s an alternative that might keep student interested?

Well, another reform to the science curriculum might be to narrow it, while at the same time actually teaching the students something that they had to use in a real-life situation. As it is right now, most students get exposed to pollution or acidification of water sources by looking at a bunch of equations that many don’t understand in the slightest, like this:

Imagine, instead, of what the students might do if they had to go out and monitor local water supplies and test them for pollutants or imbalanced pH. Would you rather be doing what you see up above, or actually doing something in the field, like this?

If you chose the former, good for you, though most students would definitely opt for the latter field situation. It’s more immediate, it’s real, and they have to use what they know to solve the problems. It would engage and inspire them, and I’m convinced would produce much better results. Back in the 60’s, they called this “relevance.” Many sneered at its application to science, but why on earth would you want to study something you can’t apply to the world around you? To dismiss relevance is to deny human curiosity, which is the driving force of science. This is what I want my students to look like when doing science:

Thus, another possible reform is to keep the existing sequence of biology, then chemistry, then physics, but narrow the curriculum and have the students work on a bunch of real, practical problems in some depth. Instead of a lot of rarified, boring, and forgettable factoids, they’d have a real understanding of the world and the process of doing science.

Sound great. Hard to do, because there’s no textbook that does this. We have no existing curriculum that’s widely used that has a reasonable scope combined with depth, which means teachers will have to make some hard decisions about what they teach so that they have enough time to do develop some really good, interesting hands-on experiences for their students instead of worrying about covering a curriculum that’s been broaden to the point of triviality. Further, if the silliness about using end-of-year test scores to evaluate teachers, what incentive do teachers have to do anything other than help the students cram and memorize enough miscellaneous facts to do well on a shallow multiple-choice exam? To implement a project-based learning program like this would require a sea-change in the way we approach science and our willingness to fund it adequately. I’m not remotely convinced that we’re there, yet, either as teachers or as a public.

Knowing very little about a lot is not rigor. It’s rote memorization with no analytical skills being imparted. It’s unworthy of being called education. We frankly ought to be ashamed that we’re forcing our children to eat so much thin gruel when they could be dining on really meaty and difficult problems that would capture their interest.

In all honesty, I don’t think this proposal—narrowing the curriculum and developing of series of real-world problems with which to teach the subject—will work in the current industrial structure of American education. However, no analysis would be complete with at least mentioning it.

This proposal has the further problem that it presumes that students have adequate math skills for the course they are taking. I had not originally planned on interweaving the math issue so closely with the science issues, but as I’ve developed my thoughts, the two are clearly inseparable. You can’t solve one without having a viable solution to the other.

Next: integrated science and the NGSS (Next Generation Science Standards)

Rethinking the high school mathematics curriculum

Most California high schools require 3 years of high-school mathematics to graduate. Last time I checked the Legislature only requires two year (algebra 1 and geometry), but most school districts—and more importantly, the University of California—requires three years high school math to graduate or to get admitted to college.

As a practical matter, this means that most high school students are mandated to take algebra 2, which in my opinion is the hardest math course in the curriculum Moreover, Algebra 2 is a disaster for most students, particularly in urban high schools, though the middle-class, suburban high schools aren’t running that far ahead. To most, it’s a meaningless jumble of words and concepts

What led to this disaster?

Yes, disaster. This is a failure of unprecedented magnitude. There is serious support for the argument that the dropout rate in California high schools is driven by the algebra 2 requirement (pass or don’t graduate). Yet none of the so-called adults in the conversation is addressing the question of why we’re at this point.

Somebody somewhere observed that students who took algebra 2 were more likely to take college math courses. Aha! A simple and cheap solution: make everybody take algebra 2, and they’ll automatically improve in college math, right? I’ll be writing more about this kind of “thinking” (sic) in an upcoming series of blog posts. For now, suffice it to say that it confused causation with correlation. Taking algebra 2 didn’t magically produce students who were ready for college mathematics; rather, students who were prepared to take college math courses took algebra 2 and did well in it because they were prepared for the subject.

As any math teacher will tell you, students fail algebra because they don’t understand basic arithmetic. (Algebra is nothing more than a more abstract form of arithmetic.) To succeed in algebra, students must have mastered the four basic arithmetic operations (additional subtraction, multiplication and division), fractions, decimals, percentages, and ratios (including conversion problems). Typically I find that a majority of urban students understand multiplication, though many don’t. Practically none understand or can do division, and from fractions onward…..forget about it. These students have no chance of every understanding algebra 1 enough to pass it, if it’s being taught rigorously. Without algebra 1, algebra 2 is a lost cause.

What, then, should we do?

I have two suggestions:

• First, let’s evaluate what students really need to become intelligent voters. That, after all, ought to be our goal. In brief, if we’re going to require a year of math beyond algebra 1 and geometry, let’s drop algebra 2 and substitute statistics and probability.
• Second, let’s address honestly the real problem in American math education. That means the subjects I just discussed, namely, that we are letting students graduate from high school without having any understanding or computation abilities in multiplication, division, fractions, decimals, percentages, and ratios. Let’s train our kids/students to master these subjects before we worry about algebra 2, pre-calculus, calculus, and linear algebra (or even differential equations).

The first topic is the easiest to deal with. When I was in high school—you know, when I walked 10 miles through the snow each morning to haul water back to the family homestead while fighting off the wolves in sub-zero cold—there were only two math subjects required for graduation: algebra 1 and geometry. Few students took more.

Do they need more? No. Most people don’t need algebra 2, or conic sections, or polar coordinates, or even trigonometry in their daily lives. They’re useful to know, but most people won’t ever use them outside of the classroom.

What people DO need is a solid understanding of statistics and probability, and not just to figure gambling odds. As voters, our students are going to be asked to evaluate different proposals to make political decisions. Good knowledge of statistics is invaluable in such cases.

Most of my classmates from high school never took anything beyond geometry. To the best of my knowledge, that never prevented them from getting through college, going on to graduate school, and making their way successfully in the world. Some even became accountants. I’m not worried that the world we collapse if we go back to where we were.

The only drawback for this proposal is that some students will self-select out of areas where math is needed. Any student who is going to college should complete the standard math curriculum through pre-calculus. Even sociologists these days are becoming highly quantitative, and business has more number crunching that I like to think about. Opting out of “hard” math courses limits choice down the road. That’s just a fact. It doesn’t mean, however, that we assume that every students needs to be ready for calculus, because the reality is that most don’t.

If we cut back on the algebra 2, we really must address the issue of widespread innumeracy in our schools. (Innumeracy is to math with illiteracy is to writing and literature.) One of the reasons that algebra 2 got foisted on the high schools is that too many high schools were routing students—particularly students from modest socioeconomic backgrounds and students of color—into “dummy” courses because it was thought that they couldn’t do math. That’s silly and obscene. I have yet to meet a student who couldn’t do math and wouldn’t do math if they were operating at a level where they could be successful. Sometimes that level is the 2^nd grade. I’m perfectly OK with that and will gladly take the student wherever they are. Further, if I’m given the students and permission to fix what needs fixing (read below: many school districts don’t permit that), I can get most students ready for algebra 1 in a year. I can’t do it AND get them through algebra 1; nobody can.

As part of the backlash against meaningless math courses, well-intentioned reformers have done a serious disservice to students. For example, no course lower than algebra 1 can be offered to high school students in the Los Angeles Unified School District. That’s absurd when you have students who can’t multiply or divide or do fractions, and is a guarantee of failure for the suffering students. The last thing we need to do is to add to the student’s burden of shame and embarrassment at not succeeding in math.

Not only do we need reform on the district level, we need an adjustment of attitude in many math teachers. Many of my colleagues are frustrated when middle-school or high-school students are at an elementary level. A lot of what we teach as remediation is hard stuff. For example, it took me several months to figure out how to teach the long-division algorithm we all learned in grade school to struggling students. Adminstrators and math teachers themselves must place value and esteem on teachers who are willing and able to take on the challenge of teaching a remedial course in mathematics instead of delegating such a course to the least experienced and least senior members of the faculty.

I’m not saying that these changes will produce lots more scientists and engineers. For that to happen, we’ll need to have a change in the culture so that math and science knowledge are admired and valued instead of being scary subjects. But unless we make the changes, math is going to be a miserable experience for most and a humiliating time for many who are behind. Surely we’re better people than this and don’t need to inflict this kind of damage on our children.