Because a large segment of the material in IPI is presented in programmed form, the questions often require filling in blanks or selecting a correct answer. Therefore, this mode of instruction places and emphasis on answers rather than on the mathematical processes involved.

In other words, students who used IPI may have identified that 1 + 1 = 2 on a multiple choice test, even without understanding the concept of addition. As we might expect, Erlwanger concluded that IPI did not help students learn math in any meaningful way, but instead

Through using IPI, learning mathematics has become a “wild good chase” in which [the student] is chasing particular answers.

Erlwanger’s “wild goose chase” finding should come as no surprise. As anyone who’s ever taken traffic school knows, there’s a big difference between passing a test and learning the material.
 
To recap: rote procedure; multiple choice and fill-in-the-blank questions; lack of conceptual understanding; ineffective. Though this was written in the 1970s, it still applies today:
 

 
Of course, fans of Khan Academy — which, to be fair, includes many teachers, parents and administrators — say that their students are engaged and performing better than ever. Still, this may be a false sense of security. The Khan Academy computer may conclude that students have learned math — the students, teachers and parents may, too — but all they will have really learned is how to game the system. Indeed, that’s all it asks them to do!
 
In the end, Erlwanger concluded what numerous studies have since: that the best way for a student to learn math is to work closely with a teacher trained in pedagogy and supported by effective curriculum. There are no shortcuts.
 
Yet even with all this evidence, many people continue to believe…
 
 
THIS TIME IT’S DIFFERENT
 
I attended a conference recently where an influential proponent of Khan Academy began his keynote by promising, “You know, every year we say, ‘This time it’s different,’ but we always end up right back where we started. Well, this time it really is.”
 
The reasons, he explained, were twofold. First, technology has finally gotten to a point on the exponential growth curve where we can create “adaptive learning” systems that analyze student behavior and tailor personalized learning opportunities just for them. Second, schools and districts are finally under such budgetary pressure that they’ll have no choice but to adopt resources like Khan Academy.
 
There are a number of problems with this argument. For one, proponents of educational technologies — laptops, tablets, interactive whiteboards, etc. — have been making similar claims for years, yet student outcomes are as bad as ever (if not worse). Not only that, the budget cuts he seemed so giddy about invariably mean fewer teachers, and to argue that this is somehow beneficial to learning is to argue against years of research and practice.
 
When we say that Khan Academy will revolutionize education, what we’re really saying is that the rules no longer apply; that everything we’ve learned about how humans learn is just one microprocessor away from obsolescence.
 
Nowhere is this philosophy more widespread than in Silicon Valley, which in recent years has become the epicenter for educational innovation and investment. On one hand this is great. Education is a mess — it’s bureaucratic, resistant to change — and could use some of the ingenuity that characterizes the Bay Area.
 
On the other hand, Silicon Valley has a very particular mindset: a secular faith that technology is the solution to everything. Have a bunch of stuff that you want to sell? eBay. Tired of waiting in line at the bank? PayPal. This works well when you want a new way to distribute coupons, but it’s problematic when we try to impose technological solutions on problems that aren’t technological in nature.
 
And at its heart, education remains a fundamentally human endeavor. Yes, there are places technology can help — Edmodo helps teachers connect with one another, while Better Lesson provides a platform for them to share their lessons — but these only work if there are good teachers and better lessons in the first place. (Of course, these companies already know this: both were founded by teachers!)
 
In the end, the “exponential growth + budget cuts = Khan Academy” argument is a non sequitur. After all, industrial farmers have made incredible leaps in bioengineering, while thousands of families across the country struggle to make ends meet. Still, it doesn’t follow that McDonald’s is therefore the solution to the hunger crisis.
 

 
Ultimately, the biggest problem with the “this time it’s different” mentality is what it says about our collective ability to tackle large, complicated challenges. By appointing what is objectively a website for homework help as the “next big thing” in education, we’re simply holding our breath for a simple solution. Instead of getting our hands dirty, we’re putting all of our chips on a shiny new toy.
 
And that’s the danger. Today it’s Khan Academy. But when that doesn’t pan out — and it won’t — we’ll move on to something else tomorrow. Meanwhile, the longer we keep telling ourselves that this time it’s different, the longer things will stay the same.
 
Still, this may not matter so long as …
 
 
GOOD ENOUGH IS GOOD ENOUGH
 
There’s a telling scene in the movie Pirates of Silicon Valley in which a young Steve Jobs and Bill Gates are debating the merits of Macintosh vs. Windows:
 

 
To a large extent, Gates’s best doesn’t matter mentality has come to define what we expect from public education in the United States. For all the talk about turning students into creative problem solvers and 21st century critical thinkers, the truth is that our current system still very much revolves around the institutionalized mediocrity of standardized testing.
 
In his recent State of the Union address, President Obama urged:
 

 
Yet presidential pardons notwithstanding, as long as we tell kids that the state test is the raison d’etre of the school year — as long as we evaluate (and even pay) teachers based on test scores, and shutter schools that fail to meet adequate yearly progress — this is just rhetoric. We can talk all we want about moving beyond testing, but it’s still the only thing that people are immediately incentivized to care about.
 
In this environment, Khan Academy is very much like Windows itself: good enough. It helps students do their homework. It may even help them pass the end-of-year state tests. Indeed, the website is a wonderful resource for people looking to bone up on basic skills, and certainly makes good on its promise to “help you learn what you want, when you want.”
 
But that’s the problem. What we want isn’t necessarily what we need.
 
Of course, because so many of us experienced math education in the same rote way, we may not even know what we need; we may have trouble envisioning it. It’s like living in a world of black & white; someone can tell you about color, but it doesn’t mean much until you actually experience it!
 
By all measures, Khan Academy offers substandard math instruction. It focuses exclusively on basic skills. It has no pedagogical underpinnings, and we have decades of research that demonstrate that it doesn’t work. Yet we love it, and think it has the potential to revolutionize education. It’s nobody’s fault. We just don’t know any better. It’s the inevitable product of optimism and ignorance.
 
Still, even if we woke up and were suddenly able to see in color, it might be too late. And this is one of the most overlooked — and ironic — consequences of Khan Academy:
 
 
KHAN ACADEMY MAKES IT DIFFICULT FOR SOMETHING BETTER TO COME ALONG
 
One of Khan Academy’s biggest selling points is that it’s free. For schools and districts across the country faced with budget cuts, Khan Academy sounds like a godsend.
 
Except it’s not really free. Just as corn subsidies allow the price of a Big Mac to remain artificially low, the millions of dollars that well-meaning philanthropists have donated to Khan Academy have distorted the market for educational resources. In the name of leveling the playing field for students, they’ve inadvertently created barriers for other organizations whose products may be better, but also more expensive.
 
At first glance this may not seem like a big deal. And who can begrudge Sal’s commitment to providing, as the website claims, a “free world-class education to anyone anywhere?”
 
The problem, of course, is it’s not world-class. It’s just free. Yet for an administrator in a cash-strapped district — and particularly one desperate to raise test scores — this may be all that matters. They save money in the short term, but end up losing out in the long.
 
It’s a conundrum.
 
On one level, it’s great that Khan Academy doesn’t charge. I’ve heard Sal speak at various conferences, and I truly admire his commitment to sharing his knowledge with the world at no cost. (As cheesy as that sounds, it’s true. I’m sure he could have sold Khan Academy for millions, but he hasn’t.)
 
On another level, though, Khan Academy and its donors may preclude better products from coming along: products built by experts that actually can improve how students learn. And ultimately, this may be the most dangerous thing about Khan Academy: not that it exists, but that it’s the only thing that does.
 
For as long as we remain hypnotized by the Siren song of free, then we risk postponing the conversation about what better might look like. As long as students and parents are convinced that earning a Black Hole badge is the same as learning — as long as schools and districts giddily delude themselves that it’s possible to solve the problem with the problem — then we’ll remain shipwrecked on the rocks.
 
And if this happens, “free” will end up costing a lot.
 
 
THAT WAS A BUMMER. SO NOW WHAT?
 
I know this seems like a downer. I’m sure it sounds like I’m a Khan Academy hater. I’m not.
 
I actually really like Khan Academy. (Seriously.) I’ve used it, enjoyed it, and think the world is better for it. Thousands of videos on everything from biology to credit default swaps and millions of “lessons delivered?” This is incredible, and I’d be a fool to argue against that.
 
But we’d be equally foolish to hold it up as the panacea for all that ails education. Khan Academy is great for what it is — a supplemental resource; homework help — but we’ve turned it into something it’s not. Indeed, something it was never intended to be.
 
It’s worth remembering the origins of Khan Academy. Sal started the site to help tutor his cousins in math, not to become the so-called “Moses of math education.” And yet that’s exactly how we’ve coronated him. In a fit of irrational exuberance mixed with an obsession with celebrity, we’ve taken someone who never taught a day in his life and appointed him America’s Professor. It’s silly. And also very dangerous.
 
But it’s not Sal’s fault. It’s ours. We have allowed desire to trump reason, and a rose-colored media narrative to overshadow everything we’ve learned about how students learn (not to mention our own common sense). We’ve begun to beat the drums, and that rarely turns out well.
 
There’s no question that we have a crisis in education, but fixing it requires more than a Wacom tablet. To address our challenges, we need to do what every other successful country has done: invest in professional development; give teachers more time to collaborate; and provide them with resources that help them not only meet the learning standards, but exceed them.
 
This isn’t rocket science. It’s not new. We’ve always needed to do these things, and deluding ourselves that “this time it’s different” won’t just perpetuate the problem. It’ll make it worse.
 

 
I heard Sal speak at a conference once, and he explained why people like his videos so much. “They sound so casual,” he said. “Sometimes I’ll sit down and start recording without even knowing what I’m going to say.”
 
It’s fine for Sal to be casual; he gives homework help. But teaching is different. It’s intentional. Socrates didn’t just wing it.
 
And this is the biggest danger of all: that we ignore this. Because the president was right: teachers do matter. Steve Jobs was right: quality does matter. Because in all things — but education especially — Good Enough isn’t.

In his recent book Born to be Good, author Dacher Keltner describes a “jen ratio” in which positive messages/observations are in the numerator, while negative ones are in the denominator. For instance, imagine we’re at a park and witness the following:

• A brother shares his ice-cream with his sister (+)
• A mother hugs her child (+)
• Two teenagers tease another (-)
• A man hits his dog with a newspaper and yells, “No!” (-)
• A roller-blader falls, and a stranger helps her up (+)

Here, the ratio of positive:negative observations is 3:2, so our jen ratio is 1.5. The lower the ratio, the worse we feel. The higher, the better, and the more attuned we are to the better angels of our nature.

Of course, these examples are hypothetical, so let’s look at something real. If aliens visited this morning’s CNN homepage, what would they conclude about the state of human jen? (Click image to enlarge.)

In the Mathalicious lesson Jen Playlist, students explore this jen ratio in more depth. They begin Act Two by watching a mashup of news clips & TV shows which are predominantly negative, including the 1999 “breaking news” about Columbine, and clips from the X-Factor in which Simon Cowell insults everyone on stage. Told that many consider a good jen ratio to be 5:1, students calculate the number of positive messages they’d need to make up for the negativity, and then watch a series of positive videos in attempt to get the jen ratio back on track. The lesson ends on a high note, and hopefully students will carry that positive energy with them for the rest of the day, and even share it with others.

What was so interesting to me when writing the lesson, though, was how much easier it was to find negative examples than positive ones. Newspaper headlines. Music videos. I considered including Ice-T’s It Was a Good Day until I remembered why it was such a good day in the first place (“I didn’t even have to use my AK”).

I was excited, then, to read Nicholas Kristoff’s editorial in this morning’s New York Times. His article, Are We Getting Nicer?, begins,

It’s pretty easy to conclude that the world is spinning down the toilet. So let me be contrary and offer a reason to be grateful this Thanksgiving. Despite the gloomy mood, the historical backdrop is stunning progress in human decency over recent centuries. War is declining, and humanity is becoming less violent, less racist and less sexist — and this moral progress has accelerated in recent decades. To put it bluntly, we humans seem to be getting nicer.

I appreciate Kristoff’s sentiment, and would love to agree that humans are “getting nicer.” But notice how he defines “progress.” It’s not that we’re necessarily becoming more peaceful and more tolerant, but simply less violent and less racist. We no longer expect women to stay at home with the kids, but we’re not exactly rallying for equal pay, either.

Mathematically, the absence of negative is not the same as positive. It’s one thing to refrain from kicking the roller-blader when she’s down, but quite another to help her up. And so I’m left to wonder: is it that we humans are getting nicer, or simply less mean?

Ultimately, there’s no way to know. Newspapers, political debates: they all feed off of some kind of division, and using them as a barometer for humanity’s jen ratio sets us up for disappointment at worst, ambivalence at best. Ultimately, I suppose it’s an act of faith of sorts: do you believe that humans are inherently negative, or do you believe that we’re positive? Put another way, do you agree with Dacher Keltner that we’re “Born to Be Good?”

I do.

Since writing the Jen Playlist lesson, I’ve found myself passing over the negative headlines and focusing more on the positive ones. It’s not conscious. It’s not that I’m intentionally trying to increase my jen ratio as part of some mathematical experiment, but simply that the positive messages feel more authentic. They resonate more deeply. The negative messages don’t feel false, but they don’t feel true, either.

Fortunately, there are examples out there that do feel true. Examples that vibrate with such pure energy that I can’t help but be moved, that I can’t help but feel more optimistic. That I can’t help but become…nicer.

Here’s one. It’s great and I love it. I hope you enjoy it. Happy Thanksgiving. Happy jen ratio’ing.

As anyone who’s taught a Mathalicious lesson knows, we structure our lessons in acts (see Dan Meyer, This American Life, playwrights, etc.). The goal is to create a narrative that flows so naturally that it seems almost effortless, as though the conversation existed already and we just wrote it down. In some cases it feels like this–like we’re not so much writing math lessons as conducting them (as in “conduit”)–but in other cases lessons go through a lot of massaging and work behind-the-scenes.

We’ll be releasing a lesson on cryptography soon in which students use modular mathematics (think “clock” arithmetic, where 17:00 = 5:00 MOD 12) to encrypt and decode secret messages.  To help give a peak under the Mathalicious hood, we hope you enjoy the following.

A huge shout-out to Professor Janet Beissinger @ the University of Illinois at Chicago for her generous explanation.  It’s nice to get schooled every now and then!

Good morning, Professors Pless and Beissinger:

I’m writing a lesson on cryptography and had a quick question for you.  I found your interaction on the UIC site and would be grateful for your help. My question has to do with how to decrypt multiplicative ciphers.

For instance, imagine someone uses the multiplicative cipher x3.  If A = 1…

1. L = 12     x3 = 36 = 10 mod 26 = J
2. Y = 25     x3 = 75 = 23 mod 26 = W

All good so far.  When students decrypt this:

1. J = 10     +26 = 36          36/3 = 12 = L (correct)
2. W = 23   +26 = 49          49/3 = 16.33 = not possible
3. W = 23   +26(2) = 75     75/3 = 25 = Y (correct)

I assume your computer algorithm is set to something like, “If N/(int N) > 1, N + 26 –> N” (i.e. that it checks that the result is an integer and, if not, “rewinds” the wheel another 26 spaces), but I’m concerned that this may throw students for a loop given the grade-level target.  With a multiplicative cipher, is there any way around this second step?

Thanks for your help.

Karim

#############

Hello,

We find the topic of decrypting multiplicative ciphers very interesting because it brings up the notion of a multiplicative inverse in modular arithmetic. You can decrypt multiplicative ciphers with inverses. In regular arithmetic, if you multiply by 3, then you can undo that multiplication by multiplying by the inverse, which is 1/3. This is something students know or at least we want them to know in middle school.

As you point out, 1/3 does not exist when we work mod 26. But it turns out other numbers act as inverses mod 26. In the example you gave, if you multiply by 3, then multiplying by 9 and reducing mod 26 will give you back what you started with: Try Y = 25 from your example: 25 x 3 = 75 = 23 mod 26.

Now to decrypt, multiply by 9: 23 x 9 = 207 = 25 mod 26, which is what you started with.

In regular arithmetic, 3 and 1/3 are inverses because their product is 1. The same holds for modular inverses: 3 and 9 are inverses because their product is 3 x 9 = 27 = 1 mod 26.

I don’t know how deeply you want to go into this in your curriculum, but in our Cryptoclub curriculum, we have middle grade students find the inverses of the numbers that have them (the numbers that are relatively prime* to 26) and then use these to decrypt.

Your method would work too, but what I like about exploring inverses is that it builds on what they are learning in their work with Real Numbers: that to undo multiplication by a number, you can multiply by the inverse of the number.

We are currently working on a new website (cryptoclub.org) which will have tutorials about this. But for now, the only thing we have published is in our book The Cryptoclub: Using Mathematics to Make and Break Secret Codes–you can find details on the old Cryptoclub website: http://cryptoclub.math.uic.edu/about/aboutbook.htm.

I am glad to hear you are putting some cryptography into your material. It is a great application of the math students learn in middle school and they find the topic of secret codes very engaging.

Good luck,
Janet

It also makes sense that Radiolab only has 29,925 followers on Twitter. Compare this to Ashton Kutcher’s 7,667,517 or Kim Kardashian’s 9,982,835. With over 1 million followers, each of The Situation’s abs has over six times as many as does Radiolab in total. But again, this is to be expected. Most players are within a few yards of the line of scrimmage, yet Radiolab is throwing the ball out of the stadium. The money is in the mean, while “+3 standard deviations” is by definition a bad investment.

Which is why it was so heartening to wake up this morning and read that the MacArthur Foundation had awarded Jad (and by extension, everyone at Radiolab) a rarified “genius award.” Other recipients include a poet, a cellist, a silversmith; an athletic trainer, chemist and architect. They stand lonely at the periphery, generations too early, and reveal tomorrow’s truths. While the rest of us sleep, Herschel discovers Uranus. And though it doesn’t seek recognition (indeed, that would be a paradox), and even if we can’t understand it yet, it’s good that we at least remember that Genius exists.

It’s also good that we remember that Jad, Robert, the cellist & the trainer were students once, too. That they needed teachers, too. We spend all of our time talking about test scores, all of our money trying to get the majority of students to meet minimum standards. And that’s fine. But what about the non-standard deviants? What about the 8th grader in the back row who’s bored out of his skull but can’t take Geometry because there’s nobody in the building certified to teach it? The kid whom the principal sees in the hallway and thinks, “It’s a bummer, but not a priority.”

A teacher told me recently about his experience teaching the lesson, “Roots of Squares.” It explores how the Pythagoreans used pebbles to construct perfect squares and find square roots, and ends with a discussion of irrationality and whether a square with a side length of root-10 can possibly exist (and if so, whether it’s infinitely wide). It’s a hard lesson. It’s meant to be. Yet the teacher, originally so enthusiastic about engaging his students in this conversation, says to me, totally deflated, “We never got that far. Some of my ninth graders couldn’t even build the squares.”

Meanwhile there’s a kid who not only constructed the square, not only constructed the cube, but is now daydreaming about other dimensions.

No Child Left Behind. It’s a nice idea. But “behind” is relative, and the so-called “gifted” (or Indigo, whatever) kids are nowhere near where they could be. And that’s a shame. Because one day Jad’s voice will fail, and we’ll need someone to take the mic and tell us what it looks like on the other side of the stadium walls.

Duh.

It’s no surprise that we as a society have a sort-of blind faith that technology is able to solve all of our problems. Yet while the iPad can and should replace textbooks, it can’t replace common sense. Unfortunately that’s exactly what’s happening in education reform. We’re focused so much on the device that we’re ignoring what’s on it.

Take math. Students dislike it and perform badly in it. Each year they ask, “What does this mean?” and “When will I use this?” And what’s our answer? A new platform. This is like reading a novel, hating it, and concluding it would be better on the Kindle. Students find the book disengaging and irrelevant, but instead of rewriting it, we simply reformat it.

So what can explain this? I’d argue there are a few factors:

1. Evaluating quality content is harder than evaluating quality technology. Try this. Which is better: Connected Math or Everyday Math? How about: the iPhone/iOS or Android?
2. We often confuse the platform for the content itself. Houghton Mifflin* made news when it announced that it was creating iPad versions of its textbooks, and a host of websites now promise students a “revolutionary” new way to access education. Yet in each of these cases the material–the thing that’s actually being taught–is exactly the same as its always been. The media may herald these as dramatic steps forward, but crtl-v is by definition not innovation. Hormel can design all the cans it wants but it’s still SPAM.
3. Much of the funding for education reform comes from large foundations, many of whom view their role as to push the envelope in public education. Organizations such as NewSchools Venture Fund and the Gates Foundation tend to support initiatives like alternative teacher preparation programs, technology platforms and charter schools. Because their entrepreneurial emphasis is to reshape the future rather than build upon the present, there’s often an unavoidable disconnect between what teachers want today and what foundations want them to want tomorrow. Ask a teacher what they’d rather have: a dynamic learning management system that tracks students by individual skill, or an engaging lesson on percents. Then ask what a foundation would rather fund. (Incidentally, we were recently contacted by a school district which had been awarded a $30,000 grant to buy iPads but had no money leftover for content. It’s not the district’s fault: surely the grant was only for the tablets themselves. But if a funder is going to spend that much money on devices, wouldn’t it make sense to also ensure that the schools can put something good on them?  There’s a reason Apple advertises apps: without the App Store the iPad is useless.  Just ask HP.)
4. Just as there’s a disconnect between foundations and teachers, there’s often a disconnect between administrators and teachers as well. Teachers answer to principals who answer to the superintendent who answers to the school board, many of whom have never taught. When they say they want schools to look different, the easiest way to do that is to dress the schools up with projectors, interactive white boards, laptops, tablets, etc. School boards have elections and there’s no easier sound bite than “technology.”
5. As a country, we seem to care more about style than substance. Want proof?  Two words: Jersey Shore.

Perhaps the most important factor, though, is the sixth one: we humans are very good at seeing only what we want to see, finding only what we’re looking for. You believe the world is flat? You’ll find evidence for that. You don’t believe in global warming? There’s a scientists somewhere who will back you up. You think technology will fix education? The high school in my town is failing despite its laptop-for-every-student program, but that’s okay: try the next town over. I’m sure the New York Times will be happy to reprint the same article next year…and the next…and the next.

Technology is great. I love my iPhone. It can do all sorts of things, but making me a better dancer isn’t one of them. Every day parents ask their kids, “What did you learn today?” It’s never “How did you learn it?” or “On what device did you learn it?” but always, “What?” Yet so long as the answer to that doesn’t change, neither will educational outcomes.

We need to stop pretending that technology can fix problems that aren’t technological in nature. Kids are bored. They don’t know why they’re learning what they’re learning. The solution isn’t asking the question better. The solution is asking a better question.

—-

Correction: the original version of this post said that it was Pearson who had released an iPad version of its textbook. It was actually Houghton Mifflin. Sorry about that.

In the coming weeks millions of students around the country will head back to school. They’ll buy spiral notebooks and number two pencils. They’ll get locker combinations and class schedules. And they’ll sit in math class, open their textbooks and encounter something like this:

8 is to 3 as x is to 12. What’s x?, or
24 is 76% of what number?

According to a recent Raytheon survey, 61 percent of middle school students say they’d rather take out the garbage than do their math homework. And can you blame them?

Too many students learn math as a bunch of random skills, and don’t have a chance to apply them to the real world. They don’t know what math means or when they’ll ever use it. Imagine a woodworking class where you learn how to use a hammer but never build anything, an English class where you diagram sentences but never have a conversation. That would be ridiculous, yet it’s exactly what happens in math classes every day.

This type of teaching is ineffective. It bores students and sets teachers up to fail. It’s also dishonest, because math is not just a bunch of skills that exist in a textbook. In fact, skills are a small part of what math really is: a way of thinking about the world.

• What are the odds of finding life on other planets? To answer that, we need to multiply fractions.
• Do people with small feet pay more for shoes, and should Nike charge by weight? We can use unit rates & proportions for that.
• How far would you have to run to burn off an Extra Value Meal? That’s order of operations.

Among math educators, there’s often a debate between skills and applications. That’s a false choice. We don’t have to choose between making math rigorous and making it interesting. When we contextualize math through real-world examples, students are engaged and perform better. Meanwhile, teachers feel more confident and are more effective.

And there’s another benefit. When students use math to explore game shows and marketing, healthy eating and exercise, they learn how the world works. They become curious and more discerning. They don’t just become better mathematicians. They become more informed citizens.

Under ‘No Child Left Behind,’ 62 percent of public schools in Virginia have fallen short of testing goals. In D.C., 87 percent have. Much of this is because of math: students see no reason to learn it, so they don’t. We can fix that. We can fundamentally transform what it means to teach and learn math in this country, and accomplish so much more in the process. All we have to do is keep it real.

A 2006 New York Times article described a 4-year-old boy who presented with a persistent fever and brown spots on his skin.  Doctors concluded it was leukemia and ordered a painful round of chemotherapy.  It turned out that the boy did have leukemia, but a rare form that chemotherapy could not cure.  In fact, each round would have weakened the boy further, increasing his risk of death.

Medical misdiagnoses are an inevitable part of medicine.  “Once you start down one of these clinical pathways,” his oncologist explained, “it’s very hard to step off.”

Today, are we witnessing a similar inertia in the conversation surrounding education reform?  Schools are failing.  Students are failing.  There’s no question about that.  But why?  What’s the diagnosis, and is it correct?

One group believes that the problem is teachers.  They think the solution is in removing ineffective teachers from the classroom and hiring better ones to take their place.  They may bemoan unions and support organizations like Teach for America and the New Teacher Project, as well as DC Public Schools’ contract that rewards teachers for performance.  For them, the emphasis is on personnel: the who.

Another group points to the underlying structure of school itself: the where.  They argue that traditional public schools lack incentives to improve, and that the answer lies in increased competition through charter schools like KIPP.  Organizations such as the Walton Family Foundation have generously given millions to promote school choice, which makes sense given Wal-Mart’s own experience in the free market.  For them, fixing education is a question of business planning and requires an MBA-type solution.

Finally, a third group believes that the problem is technological and views education more as an engineering challenge.  Each week the excellent newsletter EdSurge features “edupreneurs” from Silicon Valley who are developing online learning platforms like Rocketship, Dreambox and Khan Academy.  This group wants to transform the way students access education: the how.

The who.  The where.  The how.  Collectively these represent the three main diagnoses for what ails education in America.  They’re the narrative.  But are they really the whole problem?

A 2009 Raytheon study found that 20% of eighth graders say they “hate” math, while 61% of middle school students say they’d rather take out the garbage than do their homework.  When you ask why, the answer is simple: they don’t know why they’re learning what they’re learning or how it applies to their lives.  Each morning they open a textbook and calculate a dozen square roots or summarize the causes of the Mexican-American War.  After lunch they conjugate “aburrir” or memorize the differences between igneous, sedimentary and metamorphic rock.

For students, the problem is not that teachers are ineffective, that schools aren’t accountable or that the textbook is an inefficient technology for delivering content.  Their problem is the content itself.  Students are disengaged because they’re bored, and they’re bored because the material is often irrelevant and meaningless.  For them, the issue is not the who, the where or the how.  It’s the what.

A textbook asks, “You want to find the product of 5 x 215, but the ‘2’ button on your calculator is broken.  How can you use the distributive property and your calculator to find the product?”

In the past decade philanthropic organizations such as the Gates and Annenberg foundations have contributed upwards of $1 billion to improving public education, yet almost all of this has gone towards training teachers, expanding charters and promoting technology: fixing the calculator, but not the question.  Earlier this year a major textbook publisher made news when it announced a pilot program to teach Algebra on the iPad.  Many heralded it as a giant leap for mankind, though you have to wonder: if the content is the same, why do we expect students to react differently?

The who, the where and the how.  These are important.  But for them to be truly effective, we also need to fix the what.

So why haven’t we?  In an era of change, why is content the constant?

Since 2000 the Carnegie Corporation of New York has given almost $100 million to support education reform, and the generosity of this cannot be overstated.  Yet  according to their website, of their 37 staff members, only one has taught in a K12 classroom in the United States.

The reason this matters is that foundations often represent the main source of funding for new educational organizations: the philanthropic equivalent of venture capitalists.  And because they naturally focus on their “core competencies,” their understanding of education effectively determines what gets innovated or, in the case of curriculum, doesn’t.

Of course the obvious question is, “If we train better teachers, won’t they naturally develop better content on their own?”

This makes sense, but it’s not the way teaching works.  First-year teachers are preoccupied dealing with classroom management, calling parents, making seating charts, and figuring out whether they’re supposed to be a friend to their students or an authoritarian.  Even if they have the time, these teachers may not yet have the experience to develop effective content, and therefore default to textbooks that not only fail to support their development but may in fact undermine it.

This is not limited to new teachers.  A 2009 study from New York City highlights the difficulty in staffing classrooms with teachers who are highly qualified in their content areas.  And when teachers aren’t confident in the subject they’re teaching, they have little choice but to rely on textbooks and their silly questions about broken calculator buttons.

Clearly this is a problem, and it’s about to get worse.  For over the next five years more teachers will retire than at any other time in American history.  Meanwhile the new Common Core standards in math have pushed what were once high school topics into middle school.  What this means is that the need to support teachers with effective and engaging content has never been more urgent.  Yet so long as funding organizations continue to overlook the centrality of curriculum, we risk being blinded by the inertia of an incomplete diagnosis.  Not a wrong one, but an incomplete one.

The who.  The where.  The how.  We have to address these pieces of the puzzle.  But we have to address content, too.  For we can hire a generation of enthusiastic teachers, build cathedrals of education and give every student an iPad.  And then what?

And that’s just it.  And then, what.

I’ve not heard anyone argue that applied math and pure math are mutually exclusive–indeed, I don’t recall ever hearing the expression “fake math” until I read the post–but Dan does bring up a very good question: how do we as educators balance the need to make math concrete on one hand, and abstract on the other.  We want students to recognize how math relates to “real” life (as they experience it), but also to be able to extend it in hitherto unknown directions: known-knowns, and also unknown-unknowns.

In this sense, perhaps it’s a bit like parenting.  You provide a structure and framework to help children make sense of the world.  You make the act of growing up familiar and comfortable, gradually introducing elements of adulthood.  What you’re really doing, though, is preparing them for the day you show them the door.  It’s an imperfect metaphor, of course, but not too far off.  For the real question isn’t whether we should teach pure vs. applied math, but when.  And I would argue that there are a number of factors which compel us towards a concrete, then abstract approach:

1. Real-world (again, as experienced by the student) scenarios helps students get their footing, and act as a sort-of trail map should they get lost later.  In the iCost lesson, students use iPad pricing to come up with the equation of the line between (16GB, $499) and (32GB, $599).  First, they calculate the cost per additional GB of memory, i.e. the slope.  At $6.25/GB, they know that for the 16GB model they’re spending $100 on memory; for the 32GB model, $200 on memory.  In either case, the final price is $399 more, which must be the base price of the iPad, i.e. the y-intercept.  And so assuming iPad pricing is linear (more on that later), Apple appears to be charging a $399 base price, plus $6.25 for each additional GB: y = 6.25x + 399.  Of course, we may not really care about iPad pricing per se, and certainly we want students to be able to extend beyond this limited example.  But what’s useful about this upfront contextualization is that, when students confront something like (18, 200) and (25, 235) later, they can say to themselves, I haven’t been here exactly, but I’ve been somewhere like here, and the first thing I did was…

 

2. Real-world scenarios allow us to ask more questions–to cover more math–in a way that makes sense to students.  If we want them to solve 6.25x + 399 = 1000, how do we explain “subtract 399 from both sides” without it sounding totally random, like some trick we just made up?  But ask, “What size iPad could you get for $1000?”, and a student may instinctively say, The first thing I need to do is get rid of the $399 base price, leaving the $601 I’m spending on memory.  At $6.25/GB… Other questions we can ask include: If you double the size of the iPad, does the price also double? (proportions).  Would it make sense for the 16GB version to cost $499 AND $529? (vertical line test).  For the same size, will the Wi-Fi model ever cost the same as the 3G model? (parallel lines).  Assuming linear pricing, how much should the 64GB model cost? (evaluating equations). According to website, the 64GB model costs $699, $100 less than we’d expect.  What does this mean? (linear vs. non-linear functions…and marketing to boot).  The list goes on, and a small amount of contextualization allows us to do more math.  Indeed, the iCost lesson addresses approximately 30% of the Common Core standards for Grade 8.  From the teacher’s perspective: a third of the year in 90 minutes.  From the student’s: I get it.

 

3. In addition to being more accesible to students, real-world math can also be more accessible to teachers.  This may not matter to everyone, but it’s worth remembering that in many states 6th grade teachers can teach on an elementary license, while many middle school teachers teach on a Grades 6-8 license (which doesn’t necessarily include Algebra).  To the extent that many teachers themselves feel a certain math-phobia, contextualizing skills can assuage their angst and make them more comfortable, confident and effective.  Again, this doesn’t concern everyone, but it absolutely must concern curriculum developers whose content can act as both a tool for instruction as well as professional development.

 

4. Not only is the concrete, then abstract approach more effective–especially for students yet to fully develop the capacity for abstract thinking–it’s also more reflective of the K12 trajectory.  In elementary school, kids use coins to understand place value, marbles to learn addition and make groupings, etc.  Topics become a bit more abstract in middle school–we begin to substitute one thing for another, variables for values, etc.–though it’s mostly pretty concrete.  It’s really only in high school that math starts to transition towards the pure, until eventually it’s almost exclusively abstract.  Of course, even before this happens there’s some serious number theory going on beneath the surface, and we would do well to develop this type of thinking as early as possible (which I imagine is even earlier than we tend to think possible!).  Still, it would be strange to teach modular arithmetic before addressing “Tell and write time…to the nearest five minutes” (2.MD.7), which is why even in college it’s called “clock.”

 

5. Not only does the concrete, then abstract parallel the K12 trajectory, it more importantly parallels the evolution of mathematics itself.  Just as humans developed a screwdriver in order to build something, humans developed mathematics as a tool to construct a more comprehensive understanding of how the world works.  Put more simply: we’ve always had a practical/utilitarian relationship to math.  The Egyptians wanted to calculate field dimensions and army formations.  The Arabs, Indians and Italians: to improve commerce.  This is not to minimize the contributions of pure mathematicians such as Pythagoras, Cantor & Euler, but simply to highlight that whoever notched the Ishango bone was probably motivated less by some fascination with prime numbers, and more by the immediate need to count his sheep and make sure they all came home at night.  We went to the moon because it was there, and there’s something profoundly beautiful about exploration for exploration’s sake.  Still, aviation began when some dude strapped feathers to his arms, which is to say: it was pretty concrete (and flew like it, too).

 

6. By using real-world topics to teach how math works, we end up teaching something else, too: how the world works.  A student goes home and her mother says, “Hi, honey, what’d you learn today?”  And the girl doesn’t say “how to write the equation of the line between two points” (in all of human history, has that conversation ever happened?), but rather “I learned how Apple prices its products.”  Or, I learned that…video game consoles haven’t followed Moore’s Law, but still may be changing exponentially; …even though people with small feet pay more for shoes, Nike probably still shouldn’t charge by weight; …Wheel of Fortune may be rigged, but I have to watch more episodes to find out for sure. And isn’t that what’s so unique about math: it’s ability to shed light on…everything?  And isn’t that what makes being a math teacher so cool: our ability to ask…anything?

 

Which gets us right back to the question-at-hand: real vs. fake.  Pure vs. applied.  Embedded in Dan’s post are really two questions: what’s the point of math; and how do students learn it?  There’s no easy answer to the first–certainly it’s a tool and an object of inquiry in its own right–and how you address it depends on any number of things, from your own experience as a student to the grade level you teach (or are writing for).  In terms of “how students learn it,” though, I think the answer is a bit clearer cut: ”The way we [humans] developed it in the first place.”  There seems to be this rate of change phenomenon that keeps coming up.  Let’s call it something.  How’s “slope?”

How do we learn math?  By using it.  And indeed, I think Dan’s WCYDWT materials are wonderful examples of this.  Pure or applied? Yes, and a scenario like the basketball parabola allows students to access quadratics in a real-world setting, while also providing them an opportunity for “pure” extension later (e.g. prove why the parabola opens down iff a is negative).  Mathalicious lessons do this, too, albeit in their own way and with their own focus.

Again, the question isn’t “real” or fake, concrete or abstract.  It’s not drill & kill or open-ended exploration, but rather: Why do so many kids hate math, where does this drop-off happen and and how do we fix this? And before we can answer How do we get where we’re going, where must first ask, Where do we start?

“Real world” may be an overused expression, but it’s definitely an underused launching point in most math classrooms.  Yet just as a swim coach doesn’t teach his kids to dive from the raft, we’d do well to begin on solid ground, get our footing first and go from there.