Mathematical Illiteracy in the NYT

I know I’m late to the game here, but I can’t resist taking a moment to dive in to the furor surrounding yesterday’s appalling NY Times op-ed, “Is Algebra Necessary?”. (Yesterday was my birthday, and I just couldn’t face reading something that I knew would make me so angry.)

In case you haven’t seen it yet, let’s start with a quick look at the argument:

A typical American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.

My question extends beyond algebra and applies more broadly to the usual mathematics sequence, from geometry through calculus. State regents and legislators — and much of the public — take it as self-evident that every young person should be made to master polynomial functions and parametric equations.

There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong — unsupported by research or evidence, or based on wishful logic. (I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame.)

Already, this is a total disgrace. The number of cheap fallacies in this little excerpt is just amazing. To point out a couple:

  1. Blame the victim. We do a lousy job teaching math. Therefore, math is bad, and we should stop teaching it.
  2. Obfuscation. The author really wants to make it look like math is really terribly difficult. So he chooses a couple of silly phrases that take simple mathematical concepts, and express them in ways that make them sound much more complicated and difficult. It’s not just algebra: It’s polynomial functions and parametric equations. What do those two terms really mean? Basically, “simple algebra”. “Parametric equations” means “equations with variables”. “Polynomial equations” means equations that include variables with exponents. Which are, of course, really immensely terrifying and complex things that absolutely never come up in the real world. (Except, of course, in compound interest, investment, taxes, mortgages….)
  3. Qualification. The last paragraph essentially says “There are no valid arguments to support the teaching of math, except for the valid ones, but I’m going to exclude those.”

and from there, it just keeps getting worse. The bulk of the argument can be reduced to the first point above: lots of students fail high-school level math, and therefore, we should give up and stop teaching it. It repeats the same thing over and over again: algebra is so terribly hard, students keep failing it, and it’s just not useful (except for all of the places where it is)

One way of addressing the stupidity of this is to just take what the moron says, and try applying it to any other subject:

A typical American school day finds some six million high school students and two million college freshmen struggling with grammatical writing. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.

My question extends beyond just simple sentence construction and applies more broadly to the usual english sequence – from basic sentence structure and grammar through writing full-length papers and essays. State regents and legislators — and much of the public — take it as self-evident that every young person should be made to master rhetoric, thesis construction, and logical synthesis.

Would any newspaper in the US, much less one as obsessed with its own status as the New York Times, ever consider publishing an article like that claiming that we shouldn’t bother to teach students to write? It’s an utter disgrace, but in America, land of the mathematically illiterate, this is an acceptable, respectable argument when applied to mathematics.

Is algebra really so difficult? No.

But more substantial, is it really useful? Yes. Here’s a typical example, from real life. My wife and I bought our first house back in 1997. Two years later, the interest rate had gone down by a substantial amount, so we wanted to refinance. We had a choice between two refinance plans: one had an interest rate 1/4% lower, but required pre-paying of 2% of the principal in a special lump interest payment. Which mortgage should we have taken?

The answer is, it depends on how long we planned to own the house. The idea is that we needed to figure out when the amount of money saved by the lower interest rate would exceed the 2% pre-payment.

How do you figure that out?

Well, the amortization equation describing the mortgage is:

m = p frac{i(i+1)^2}{(i+1)^n - 1}

Where:

  • m is the monthly payment on the mortgage.
  • p is the amount of money being borrowed on the loan.
  • i is the interest rate per payment period.
  • n is the number of payments.

Using that equation, we can see the monthly payment. If we calculate that for both mortgages, we get two values, m_1 and m_2. Now, how many months before it pays off? If D is the amount of the pre-payment to get the lower interest rate, then D = k(m_2 - m_1), where k is the number of months – so it would take k=frac{D}{m_2 - m_1} months. It happened that for us, k worked out to around 60 – that is, about 5 years. We did make the pre-payment. (And that was a mistake; we didn’t stay in that house as long as we’d planned.)

But whoa…. Quadratic equations!

According to our idiotic author, expecting the average american to be capable of figuring out the right choice in that situation is completely unreasonable. We’re screwing over students all over the country by expecting them to be capable of dealing with parametric polynomial equations like this.

Of course, the jackass goes on to talk about how we should offer courses in things like statistics instead of algebra. How on earth are you going to explain bell curves, standard deviations, margins of error, etc., without using any algebra? The guy is so totally clueless that he doesn’t even understand when he’s using it.

Total crap. I’m going to leave it with that, because writing about this is just making me so damned angry. Mathematical illiteracy is just as bad

Weekend Recipe: Catfish Banh Mi!

I’m a huge fan of Banh Mi, the amazing sandwich that came from the fusion of Vietnamese and French food during the 20th century. Banh Mi is, basically, a sandwich made on a Vietnamese baguette (which is like a french baguette, except that it’s got some rice flour mixed with the wheat, giving it a crisper texture), topped with a spicy mayo (or some other sauce), some kind of protein, pickled vegetables, and cilantro.

I only learned about these glorious things recently. When I started working out foursquare last september, our office was in the East Village in Manhattan, just a couple of blocks away from a terrific banh mi restaurant, called Baoguette. Baoguette was one of our standard lunch joints. It’s a tacky little run-down hole-in-the-wall place – which is exactly the kind of place you want for what is, basically, street food.

In January, we moved to a new office space down in Soho. It’s a great space – I love our new office. But I miss my banh mi. So since the big move, I’ve been trying to figure out how to replicate my favorite sandwich. And today, I finally did it: I made a home-made catfish banh mi that is, I think, good enough to call an unqualified success. And I’m going to share how I did it with you!

The first thing you need is really good bread. In an ideal world, you want a baguette from a Vietnamese bakery. But since those are hard to find, go french. A really good french baguette is the closest thing – it’s got the right basic texture: a crisp but not crunchy crust.

Next, the veggies. You need to make these at least a couple of hours ahead, ideally at least a day.

You take a daikon radish (also called a chinese turnip), and julienne it, into strips about 1/4 inch by 1/8th of an inch by 3 inches or so. Do the same thing to a couple of carrots. Sprinkle with about a tablespoon of salt. (Don’t worry; it’s not going to turn out super-salty – we’re trying to extract water from the turnip.) Set it aside for about 1/2 an hour, and then drain off the liquid that will have accumulated.

Now, mix together about 1/2 cup of rice vinegar, and about 3/8ths of a cup of sugar. When you’ve got the sugar nicely dissolved, dump this over your veggies, and add cold water until the veggies are just covered. Add in a couple of fronds of cilantro, and a sliced chili pepper, and put the whole shebang into the fridge. Give it a stir once in a while. It’ll be ready to eat in about 4 hours, and it’ll be perfect in a day. After that, drain off the liquid, and put it into a tightly covered container to keep for about a week.

Now, finally, we’re up to the catfish. You really do want catfish for this. Catfish has a great, most, tender texture, and a strong enough flavor to stand up to what we’re going to do to it. Take a good size catfish filet, and cut it in half down the rib line, and then in half again perpendicular to the ridge line. Depending on how big your bread is, you’ll want between 2 and 4 of these catfish strips per sandwich.

Finely mince a clove of garlic, and a couple of slices of fresh ginger. Crush them up with just a bit of salt, to get them really pureed. Then mix them with about one teaspoon each of fish sauce and soy sauce, and about 1/4 teaspoon of korean chili pepper or cayenne (or more, if you like to spicier. I actually did more like 1/2 a teaspoon, but that’s a lot!). Toss the fish into this mixture, to get it nice and coated, and let it marinade for about 10 minutes.

Now, you want to prepare your bread. Then cut a section of the bread about 8 inches long, and then slice it open about 2/3rds of the way, and rip out some of the interior to make room for fillings!

Now, sauce: The sauce is simple: half and half mayo and sriracha. Whip some of that up. Give your bread a good layer of the sriracha mayo all around. On top of that, lay a sprig of fresh cilantro. (In the past, I’ve tried using a homemade aioli, but this is one of the rare cases where the premade is actually just as good.)

Now you’re almost ready to cook the fish. Mix together equal parts of general purpose flour and cornstarch, and give the fish a light coat. Heat up a frying pan on medium heat, and add just enough oil to cover the bottom of the pan. (We’re frying here, but not deep-frying!). When the oil is hot, put in the fish, and cook it for about 1 1/2 minutes on each side. You want it to be just barely cooked through.

Put some fish inside the bread, then add a generous helping of the pickled vegetables, and top it all of with more sriracha. And then prepare to eat one of the best sandwiches you’ve ever tasted!

(I’d show you a picture, but we totally devoured these. There’s nothing left to photograph.)

The American Heat Wave and Global Warming

Global warming is a big issue. If we’re honest and we look carefully at the data, it’s beyond question that the atmosphere of our planet is warming. It’s also beyond any honest question that the preponderance of the evidence is that human behavior is the primary cause. It’s not impossible that we’re wrong – but when we look at the real evidence, it’s overwhelming.

Of course, this doesn’t stop people from being idiots.

But what I’m going to focus on here isn’t exactly the usual idiots. See, here in the US, we’re in the middle of a dramatic heat wave. All over the country, we’ve been breaking heat daily temperature records. As I write this, it’s 98 degrees outside here in NY, and we’re expecting another couple of degrees. Out in the west, there are gigantic wildfires, cause by poor snowfall last winter, poor rainfall this spring, and record heat to dry everything out. So: is this global warming?

We’re seeing lots and lots of people saying yes. Or worse, saying that it is, because of the heat wave, while pretending that they’re not really saying that it is. For one, among all-too-many examples, you can look at Bad Astronomy here. Not to rag too much on Phil though, because hes just one among about two dozen different example of this that I’ve seen in the last 3 days.

Weather 10 or twenty degree above normal isn’t global warming. A heat wave, even a massive epic heat wave, isn’t proof that global warming is real, any more than an epic cold wave or blizzard is evidence that global warming is fake.

I’m sure you’ve heard many people say weather is not climate. But for human beings, it’s really hard to understand just what that really means. Climate is a world-wide long-term average; weather is instantaneous and local. This isn’t just a nitpick: it’s a huge distinction. When we talk about global warming, what we’re talking about is the year-round average temperature changing by one or two degrees. A ten degree variation in local weather doesn’t tell us anything about the worldwide trend.

Global warming is about climate. And part of what that means is that in some places, global warming will probably make the weather colder. Cold weather isn’t evidence against global warming. Most people realize that – which is why we all laugh when gasbags like Rush Limbaugh talk about how a snowstorm “proves” that global warming is a fraud. But at the same time, we look at weather like what we have in the US, and conclude that “Yes, global warming is real”. But we’re making the same mistake.

Global warming is about a surprisingly small change. Over the last hundred years, global warming is a change of about 1 degree celsius in the global average temperature. That’s about 1 1/2 degrees fahrenheit, for us Americans. It seems miniscule, and it’s a tiny fraction of the temperature difference that we’re seeing this summer in the US.

But that tiny difference in climate can cause huge differences in weather. As I mentioned before, it can make local weather either warmer or colder – not just by directly warming the air, but by altering wind and water currents in ways that create dramatic changes.

For example, global warming could, likely, make Europe significantly colder. How? The weather in western Europe is greatly affected by an ocean water current called the atlantic conveyor. The conveyor is a cyclic ocean current, where (driven in part by the jet stream), warm water flows north from the equator in a surface current, cooling as it goes, until it finally sinks and starts to cycle back south in a deep underwater current. This acts as a heat pump, moving energy from the equator north and east to western Europe. This is why Western Europe is significantly warmer than places at the same latitude in Eastern North America.

Global warming could alter the flow of the atlantic conveyor. (We don’t know if it will – but it’s one possibility, which makes a good example of something counter-intuitive.) If the conveyor is slowed, so that it transfers less energy, Europe will get colder. How could the conveyor be slowed? By ice-melt. The conveyor works as a cycle because of the differences in density between warm and cold water: cold water is denser than warm water, so the cold water sinks as it cools. It warms in the tropics, gets pushed north by the jet stream, cools along the way and gradually sinks.

But global warming is melting a lot of artic and glacier ice, which produces freshwater. Freshwater is less dense than saltwater. So the freshwater, when it dilutes the cold water at the northern end of the conveyor, it reduces its density relative to the pure salt-water – and that reduces the tendency of the cold water to sink, which could slow the conveyor.

There are numerous similar phenomena that involve changes in ocean currents and wind due to relatively small temperature variations. El Nino and La Nina, conveyor changes, changes in the moisture-carrying capacity of wind currents to carry – they’re all caused by relatively small changes – changes well with the couple of degrees of variatio that we see occuring.

But we need to be honest and careful. This summer may be incredibly hot, and we had an unsually warm winter before it – but we really shouldn’t try to use that as evidence of global warming. Because if you do, when some colder-than-normal weather occurs somewhere, the cranks and liars that want to convince people that global warming is an elaborate fraud will use that the muddle things – and when they do, it’ll be our fault when people fall for it, because we’ll be the ones who primed them for that argument. As nice, as convenient, as convincing as it might seem to draw a correlation between a specific instance of extreme weather and global warming, we really need to stop doing it.

The Meaning of the Higgs

This isn’t exactly my area of expertise, but I’ve gotten requests by both email and twitter to try to explain yesterday’s news about the Higgs’ boson.

The questions.

  • What is this Higgs’ boson thing?
  • How did they find it?

  • What does the five sigma stuff mean?
  • Why do they talk about it as a “Higgs’-like particle”?

So, first things first. What is a Higgs’ boson?

When things in the universe interact, they usually don’t actually interacts by touching each other directly. They interact through forces and fields. What that means is a bit tricky. I can define it mathematically, but it won’t do a bit of good for intuition. But the basic idea is that space itself has some properties. A point in space, even when it’s completely empty, it has some properties.

Outside of empty space, we have particles of various types. Those particles interact with each other, and with space itself. Those interactions are what end up producing the universe we see and live in.

Fields are, essentially, a property of space. A field is, at its simplest, a kind of property of space that is defined at every point in space.

When particles interact with fields, they can end up exchanging energy. They do that through a particular kind of particle, called an exchange particle. For example, think about an electromagnetic field. An electron orbits an atomic nucleus, due to forces created by the electromagnetic fields of the electrons and protons. When an electron moves to a lower-energy orbital, it produces a photon; when it absorbs a photon, it can jump to a higher orbital. The photon is the exchange particle for the electromagnetic field. Exchange particles are instances of a kind of particle called a boson.

So.. one of the really big mysteries of physics is: why do some particles have mass, and other particles don’t? That is, some particles, like protons, have masses. Others, like photons, don’t. Why is that?

It’s quite a big mystery. Based on our best model – called the standard model – we can predict all of the basic kinds of particles, and what their masses should be. But we didn’t have a clue about why there’s mass at all!

So, following the usual pattern in particle physics, we predict that there’s a field. Particles moving through that field, if they interact with the field, experience a sort of drag. That drag is mass. So – just like particles like neutrinos aren’t affected by electromagnetic fields, some particles like photons won’t have mass because they don’t interact with the field that produces mass. We call that field the Higgs’ field.

(The previous paragraph formerly contained an error. The higgs field produces mass, not gravity. Just a stupid typo; my fingers got ahead of my brain.)

So physicists proposed the existence of the Higgs’ field. But how could they test it?

It’s a field. Fields have exchange particles. What would the exchange particles of the Higgs’ field be? Exchange particles are bosons, so this one is, naturally, called a Higgs’ boson. So if the Higgs’ field exists, then it will have an exchange particle. If the standard model of physics is right, then we can use it to predict the mass that that boson must have.

So – if we can find a particle whose mass matches what we predict, and it has the correct properties for a mass-field exchange particle, then we can infer that the Higgs’ field is real, and is the cause of mass.

How did they find the Higgs’ boson?

We have a pretty good idea of what the mass of the Higgs’ boson must be. We can describe that mass in terms of a quantity of energy. (See the infamous Einstein equation!) If we can take particles that we can easily see and manipulate, and we can accelerate them up to super-super high speed, and collide them together. If the energy of a collision matches the mass of a particle, it can create that kind of particle. So we slam together, say, two protons at high enough energy, we’ll get a Higgs’ boson.

But things are never quite that easy. There are a bunch of problems. First, the kind of collision that can produce a Higgs’ doesn’t always produce one. It can produce a variety of results, depending on the specifics of the collision as well as purely random factors. Second, it produces a lot more than just a Higgs’. We’re talking about an extremely complex, extremely high energy collision, with a ton of complex results. And third, the Higgs’ boson isn’t particularly stable. It doesn’t really like to exist. So like many unstable things in particle physics, it decays, producing other particles. And many of those particles are themselves unstable, and decay into other particles. What we can observe is the last products of the collision, several steps back from the Higgs’. But we know what kind of things the Higgs’ can decay into, and what they can decay into, etc.

So, we slam these things together a couple of thousand, or a couple of million times. And we look at the results. We look at all of the results of all of the collisions. And we specifically look for a bump: if there’s really a specific collision energy level at which Higgs’ bosons are produced, then we’ll see a bump in the number of Higgs’ decay products that are produced by collisions at that energy. And what the announcement yesterday showed is that that’s exactly what they saw: a bump in the observations inside the expected range of values of the mass of a Higgs’ boson.

The bump

What does five sigmas mean?

Whenever we’re making observations of a complex phenomenon, there are all sorts of things that can confound our observations. There are measurement errors, calculation errors, random noise, among many other things. So we can’t just look at one, or two, or ten data points. We need to look at a lot of data. And when you’ve got a lot of data, there’s always a chance that you’ll see what appears to be a pattern in the data, which is really just the product of random noise

For example, there are some people who’ve won the lottery multiple times. That seems crazy – it’s so unlikely to win once! To win multiple times seems crazy. But probabilistically, if you keep observing lotteries, you’ll find repeat winners. Or you’ll find apparent patterns in the winning numbers, even though they’re being drawn randomly.

We don’t want to be fooled by statistics. So we create standards. We can compute how unlikely a given pattern would be, if it were occuring do to pure randomness. We can’t even absolutely rule out randomness, but for any degree of certainty, we can determine just how unlikely a given observation is to be due to randomness.

We describe that in terms of standard deviations. An observation of a phenomenon has a roughly 68% chance of being measured within one standard deviation (one sigma) of the actual value, or a roughly 32% chance of being observed outside of one sigma. At two sigmas, there’s only a roughly 5% chance of being outside. At three sigmas out, you’re down to a roughly 0.3% chance of randomly observing an event outside. The odds continue that way.

So, the Higgs’ hunters computed probabilities of observing the data that they found if they assumed that there was no Higgs’. The amount of data that they found exceeded 5 sigmas away from what you would expect by random chance if there was no Higgs’. That translates as damned unlikely. The ultimate choice of 5 sigmas is arbitrary, but it’s accepted as a threshold for certainty in particle physics. At five sigmas, we realistically rule out random chance.

Why do they keep saying Higgs’-like particle?

Remember up above, I said: “So – if we can find a particle whose mass matches what we predict, and it has the correct properties for a mass-field exchange particle, then we can infer that the Higgs’ field is real, and is the cause of mass”? There are two thing we need to show to conclude that we’ve found the mediator of the Higgs’ field. There needs to be a particle with the right mass, and it needs to have the properties of a mass-mediator. What we’ve got right now is an observation that yes, there is a particle at the mass we’d expect for a Higgs’. But we don’t have observations yet of any properties of the particle other than its mass. Assuming the standard model is right, the odds of finding another particle with that mass is damned unlikely, but the standard model could be wrong. It’s not likely at this point, but people like to be careful. So at this point, to be precise, we’ve observed a Higgs’-like particle – a particle that according to all of the observations we’ve made so far appears to be a Higgs’; but until we observe some properties other than mass, we can’t be absolutely certain that it’s a Higgs’.