Should You Opt Out of PARCC?

Today’s post is a discussion of education reform, Common Core, standardized testing, and PARCC with my friend Kristin Wald, who has been extremely kind to this blog. Kristin taught high school English in the NYC public schools for many years. Today her kids and mine go to school together in Montclair. She has her own blog that gets orders of magnitude more readers than I do.

We’re cross-posting this on Kristin’s blog and also on Mathbabe (thank you, Cathy O’Neil!)

ES: PARCC testing is beginning in New Jersey this month. There’s been lots of anxiety and confusion in Montclair and elsewhere as parents debate whether to have their kids take the test or opt out. How do you think about it, both as a teacher and as a parent?

KW: My simple answer is that my kids will sit for PARCC. However, and this is where is gets grainy, that doesn’t mean I consider myself a cheerleader for the exam or for the Common Core curriculum in general.

In fact, my initial reaction, a few years ago, was to distance my children from both the Common Core and PARCC. So much so that I wrote to my child’s principal and teacher requesting that no practice tests be administered to him. At that point I had only peripherally heard about the issues and was extending my distaste for No Child Left Behind and, later, Race to the Top. However, despite reading about and discussing the myriad issues, I still believe in change from within and trying the system out to see kinks and wrinkles up-close rather than condemning it full force.

Standards

ES: Why did you dislike NCLB and Race to the Top? What was your experience with them as a teacher?

KW: Back when I taught in NYC, there was wiggle room if students and schools didn’t meet standards. Part of my survival as a teacher was to shut my door and do what I wanted. By the time I left the classroom in 2007 we were being asked to post the standards codes for the New York State Regents Exams around our rooms, similar to posting Common Core standards all around. That made no sense to me. Who was this supposed to be for? Not the students – if they’re gazing around the room they’re not looking at CC RL.9-10 next to an essay hanging on a bulletin board. I also found NCLB naïve in its “every child can learn it all” attitude. I mean, yes, sure, any child can learn. But kids aren’t starting out at the same place or with the same support. And anyone who has experience with children who have not had the proper support up through 11th grade knows they’re not going to do well, or even half-way to well, just because they have a kickass teacher that year.

Regarding my initial aversion to Common Core, especially as a high school English Language Arts teacher, the minimal appearance of fiction and poetry was disheartening. We’d already seen the slant in the NYS Regents Exam since the late 90’s.

However, a couple of years ago, a friend asked me to explain the reason The Bluest Eye, with its abuse and rape scenes, was included in Common Core selections, so I took a closer look. Basically, a right-wing blogger had excerpted lines and scenes from the novel to paint it as “smut” and child pornography, thus condemning the entire Common Core curriculum. My response to my friend ended up as “In Defense of The Bluest Eye.”

That’s when I started looking more closely at the Common Core curriculum. Learning about some of the challenges facing public schools around the country, I had to admit that having a required curriculum didn’t seem like a terrible idea. In fact, in a few cases, the Common Core felt less confining than what they’d had before. And you know, even in NYC, there were English departments that rarely taught women or minority writers. Without a strong leader in a department, there’s such a thing as too much autonomy. Just like a unit in a class, a school and a department should have a focus, a balance.

But your expertise is Mathematics, Eugene. What are your thoughts on the Common Core from that perspective?

ES: They’re a mix. There are aspects of the reforms that I agree with, aspects that I strongly disagree with, and then a bunch of stuff in between.

The main thing I agree with is that learning math should be centered on learning concepts rather than procedures. You should still learn procedures, but with a conceptual underpinning, so you understand what you’re doing. That’s not a new idea: it’s been in the air, and frustrating some parents, for 50 years or more. In the 1960’s, they called it New Math.

Back then, the reforms didn’t go so well because the concepts they were trying to teach were too abstract – too much set theory, in a nutshell, at least in the younger grades. So then there was a retrenchment, back to learning procedures. But these things seem to go in cycles, and now we’re trying to teach concepts better again. This time more flexibly, less abstractly, with more examples. At least that’s the hope, and I share that hope.

I also agree with your point about needing some common standards defining what gets taught at each grade level. You don’t want to be super-prescriptive, but you need to ensure some kind of consistency between schools. Otherwise, what happens when a kid switches schools? Math, especially, is such a cumulative subject that you really need to have some big picture consistency in how you teach it.

Assessment

ES: What I disagree with is the increased emphasis on standardized testing, especially the raised stakes of those tests. I want to see better, more consistent standards and curriculum, but I think that can and should happen without putting this very heavy and punitive assessment mechanism on top of it.

KW: Yes, claiming to want to assess ability (which is a good thing), but then connecting the results to a teacher’s effectiveness in that moment is insincere evaluation. And using a standardized test not created by the teacher with material not covered in class as a hard percentage of a teacher’s evaluation makes little sense. I understand that much of the exam is testing critical thinking, ability to reason and use logic, and so on. It’s not about specific content, and that’s fine. (I really do think that’s fine!) Linking teacher evaluations to it is not.

Students cannot be taught to think critically in six months. As you mentioned about the spiraling back to concepts, those skills need to be revisited again and again in different contexts. And I agree, tests needn’t be the main driver for raising standards and developing curriculum. But they can give a good read on overall strengths and weaknesses. And if PARCC is supposed to be about assessing student strengths and weaknesses, it should be informing adjustments in curriculum.

On a smaller scale, strong teachers and staffs are supposed to work as a team to influence the entire school and district with adjusted curriculum as well. With a wide reach like the Common Core, a worrying issue is that different parts of the USA will have varying needs to meet. Making adjustments for all based on such a wide collection of assessments is counterintuitive. Local districts (and the principals and teachers in them) need to have leeway with applying them to best suit their own students.

Even so, I do like some things about data driven curricula. Teachers and school administrators are some of the most empathetic and caring people there are, but they are still human, and biases exist. Teachers, guidance counselors, administrators can’t help but be affected by personal sympathies and peeves. Having a consistent assessment of skills can be very helpful for those students who sometimes fall through the cracks. Basically, standards: yes. Linking scores to teacher evaluation: no.

ES: Yes, I just don’t get the conventional wisdom that we can only tell that the reforms are working, at both the individual and group level, through standardized test results. It gives us some information, but it’s still just a proxy. A highly imperfect proxy at that, and we need to have lots of others.

I also really like your point that, as you’re rolling out national standards, you need some local assessment to help you see how those national standards are meeting local needs. It’s a safeguard against getting too cookie-cutter.

I think it’s incredibly important that, as you and I talk, we can separate changes we like from changes we don’t. One reason there’s so much noise and confusion now is that everything – standards, curriculum, testing – gets lumped together under “Common Core.” It becomes this giant kitchen sink that’s very hard to talk about in a rational way. Testing especially should be separated out because it’s fundamentally an issue of process, whereas standards and curriculum are really about content.

You take a guy like Cuomo in New York. He’s trying to increase the reliance on standardized tests in teacher evaluations, so that value added models based on test scores count for half of a teacher’s total evaluation. And he says stuff like this: “Everyone will tell you, nationwide, the key to education reform is a teacher evaluation system.” That’s from his State of the State address in January. He doesn’t care about making the content better at all. “Everyone” will tell you! I know for a fact that the people spending all their time figuring out at what grade level kids should start to learn about fractions aren’t going tell you that!

I couldn’t disagree with that guy more, but I’m not going to argue with him based on whether or not I like the problems my kids are getting in math class. I’m going to point out examples, which he should be well aware of by now, of how badly the models work. That’s a totally different discussion, about what we can model accurately and fairly and what we can’t.

So let’s have that discussion. Starting point: if you want to use test scores to evaluate teachers, you need a model because – I think everyone agrees on this – how kids do on a test depends on much more than how good their teacher was. There’s the talent of the kid, what preparation they got outside their teacher’s classroom, whether they got a good night’s sleep the night before, and a good breakfast, and lots of other things. As well as natural randomness: maybe the reading comprehension section was about DNA, and the kid just read a book about DNA last month. So you need a model to break out the impact of the teacher. And the models we have today, even the most state-of-the-art ones, can give you useful aggregate information, but they just don’t work at that level of detail. I’m saying this as a math person, and the American Statistical Association agrees. I’ve written about this here and here and here and here.

Having student test results impact teacher evaluations is my biggest objection to PARCC, by far.

KW: Yep. Can I just cut and paste what you’ve said? However, for me, another distasteful aspect is how technology is tangled up in the PARCC exam.

Technology

ES: Let me tell you the saddest thing I’ve heard all week. There’s a guy named Dan Meyer, who writes very interesting things about math education, both in his blog and on Twitter. He put out a tweet about a bunch of kids coming into a classroom and collectively groaning when they saw laptops on every desk. And the reason was that they just instinctively assumed they were either about to take a test or do test prep.

That feels like such a collective failure to me. Look, I work in technology, and I’m still optimistic that it’s going to have a positive impact on math education. You can use computers to do experiments, visualize relationships, reinforce concepts by having kids code them up, you name it. The new standards emphasize data analysis and statistics much more than any earlier standards did, and I think that’s a great thing. But using computers primarily as a testing tool is an enormous missed opportunity. It’s like, here’s the most amazing tool human beings have ever invented, and we’re going to use it primarily as a paperweight. And we’re going to waste class time teaching kids exactly how to use it as a paperweight. That’s just so dispiriting.

KW: That’s something that hardly occurred to me. My main objection to hosting the PARCC exam on computers – and giving preparation homework and assignments that MUST be done on a computer – is the unfairness inherent in accessibility. It’s one more way to widen the achievement gap that we are supposed to be minimizing. I wrote about it from one perspective here.

I’m sure there are some students who test better on a computer, but the playing field has to be evenly designed and aggressively offered. Otherwise, a major part of what the PARCC is testing is how accurately and quickly children use a keyboard. And in the aggregate, the group that will have scores negatively impacted will be children with less access to the technology used on the PARCC. That’s not an assessment we need to test to know. When I took the practice tests, I found some questions quite clear, but others were difficult not for content but in maneuvering to create a fraction or other concept. Part of that can be solved through practice and comfort with the technology, but then we return to what we’re actually testing.

ES: Those are both great points. The last thing you want to do is force kids to write math on a computer, because it’s really hard! Math has lots of specialized notation that’s much easier to write with pencil and paper, and learning how to write math and use that notation is a big part of learning the subject. It’s not easy, and you don’t want to put artificial obstacles in kids’ way. I want kids thinking about fractions and exponents and what they mean, and how to write them in a mathematical expression, but not worrying about how to put a numerator above a denominator or do a superscript or make a font smaller on a computer. Plus, why in the world would you limit what kids can express on a test to what they can input on a keyboard? A test is a proxy already, and this limits what it can capture even more.

I believe in using technology in education, but we’ve got the order totally backwards. Don’t introduce the computer as a device to administer tests, introduce it as a tool to help in the classroom. Use it for demos and experiments and illustrating concepts.

As far as access and fairness go, I think that’s another argument for using the computer as a teaching tool rather than a testing tool. If a school is using computers in class, then at least everyone has access in the classroom setting, which is a start. Now you might branch out from there to assignments that require a computer. But if that’s done right, and those assignments grow in an organic way out of what’s happening in the classroom, and they have clear learning value, then the school and the community are also morally obligated to make sure that everyone has access. If you don’t have a computer at home, and you need to do computer-based homework, then we have to get you computer access, after school hours, or at the library, or what have you. And that might actually level the playing field a bit. Whereas now, many computer exercises feel like they’re primarily there to get kids used to the testing medium. There isn’t the same moral imperative to give everybody access to that.

I really want to hear more about your experience with the PARCC practice tests, though. I’ve seen many social media threads about unclear questions, both in a testing context and more generally with the Common Core. It sounds like you didn’t think it was so bad?

KW: Well, “not so bad” in that I am a 45 year old who was really trying to take the practice exam honestly, but didn’t feel stressed about the results. However, I found the questions with fractions confusing in execution on the computer (I almost gave up), and some of the questions really had to be read more than once. Now, granted, I haven’t been exposed to the language and technique of the exam. That matters a lot. In the SAT, for example, if you don’t know the testing language and format it will adversely affect your performance. This is similar to any format of an exam or task, even putting together an IKEA nightstand.

There are mainly two approaches to preparation, and out of fear of failing, some school districts are doing hardcore test preparation – much like SAT preparation classes – to the detriment of content and skill-based learning. Others are not altering their classroom approaches radically; in fact, some teachers and parents have told me they hardly notice a difference. My unscientific observations point to a separation between the two that is lined in Socio-Economic Status. If districts feel like they are on the edge or have a lot to lose (autonomy, funding, jobs), if makes sense that they would be reactionary in dealing with the PARCC exam. Ironically, schools that treat the PARCC like a high-stakes test are the ones losing the most.

Opting Out

KW: Despite my misgivings, I’m not in favor of “opting out” of the test. I understand the frustration that has prompted the push some districts are experiencing, but there have been some compromises in New Jersey. I was glad to see that the NJ Assembly voted to put off using the PARCC results for student placement and teacher evaluations for three years. And I was relieved, though not thrilled, that the percentage of PARCC results to be used in teacher evaluations was lowered to 10% (and now put off). I still think it should not be a part of teacher evaluations, but 10% is an improvement.

Rather than refusing the exam, I’d prefer to see the PARCC in action and compare honest data to school and teacher-generated assessments in order to improve the assessment overall. I believe an objective state or national model is worth having; relying only on teacher-based assessment has consistency and subjective problems in many areas. And that goes double for areas with deeply disadvantaged students.

ES: Yes, NJ seems to be stepping back from the brink as far as model-driven teacher evaluation goes. I think I feel the same way you do, but if I lived in NY, where Cuomo is trying to bump up the weight of value added models in evaluations to 50%, I might very well be opting out.

Let me illustrate the contrast – NY vs. NJ, more test prep vs. less — with an example. My family is good friends with a family that lived in NYC for many years, and just moved to Montclair a couple months ago. Their older kid is in third grade, which is the grade level where all this testing starts. In their NYC gifted and talented public school, the test was this big, stressful thing, and it was giving the kid all kinds of test anxiety. So the mom was planning to opt out. But when they got to Montclair, the kid’s teacher was much more low key, and telling the kids not to worry. And once it became lower stakes, the kid wanted to take the test! The mom was still ambivalent, but she decided that here was an opportunity for her kid to get used to tests without anxiety, and that was the most important factor for her.

I’m trying to make two points here. One: whether or not you opt out depends on lots of factors, and people’s situations and priorities can be very different. We need to respect that, regardless of which way people end up going. Two: shame on us, as grown ups, for polluting our kids’ education with our anxieties! We need to stop that, and that extends both to the education policies we put in place and how we collectively debate those policies. I guess what I’m saying is: less noise, folks, please.

KW: Does this very long blog post count as noise, Eugene? I wonder how this will be assessed? There are so many other issues – private profits from public education, teacher autonomy in high performing schools, a lack of educational supplies and family support, and so on. But we have to start somewhere with civil and productive discourse, right? So, thank you for having the conversation.

ES: Kristin, I won’t try to predict anyone else’s assessment, but I will keep mine low stakes and say this has been a pleasure!

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Why Your Kids Should Help in the Kitchen

We were emptying the dishwasher in the morning, and my younger son’s job was putting away the silverware. He brought the silverware basket over to the silverware drawer, and said:

You know what I’m going to do? First I’m going to collect together all the spoons, and put them in the spoon bin. Then I’ll take all the knives and put them in the knife bin. Then I’ll take the forks…

Any idea that a five year old can come up with must be really simple, right? But simple ideas can still be deep and powerful. Among other things, this one is at the heart of an important mathematical technique called Lebesgue integration, which one of my favorite math teachers once explained to me like this:

Say you’re trying to count a really big pile of money. You can stack it really high and count it a bill at a time. Or you could separate it into piles of ones, fives, tens, and twenties, count how many bills are in each pile, multiply the count in each pile by the denomination, and add the results. Lebesgue integration is when you break the bills into separate piles first.

What my son figured out was that if you focus on one kind of utensil at a time, you can work faster because everything you pick up goes in the same place, so you don’t need to think about switching from one bin to another all the time. From a computer science perspective, you’re doing fewer operations. From a math perspective, you’re representing a single function that appears complex (because it jumps around all the time, from knife to fork to spoon to knife or from $1 to $10 to $5 to $1) in terms of a few simple (constant) functions defined on different domains. I’ve written before about how math is about finding, creating, and making use of order, and this is a great example.

You can apply this idea to the problem of finding the area under a really jumpy curve. Henri Lebesgue is famous, with an integration technique named after him, because he worked out the details, about 100 years ago. But the underlying idea truly is accessible to a five year old. At least, as long as that five year old pays attention to his chores.

Once in a Lifetime: Happy Pi Day!

Today is 3/14/15, which might be an appealing date if you like math and circles, because 3.1415 are the first few digits in the decimal expansion of pi. (Some people celebrate 3/14 as Pi Day every year, and for them, having the year be 2015 is just icing on the cake, or, um, the pie.) While I am not big on numerology, everybody’s talking about pi today, so here are a few choice tastes of it. Toward the end, there will be monkeys.

First off, let’s agree on what pi is. It’s not 3.1415, because that’s not exactly pi, and it’s not 3.1415…, because it’s not so clear what’s supposed to come after that “…” . Also, please don’t say it’s “some important mathematical constant,” because heaven knows there are plenty of those. Pi is the circumference of a circle of diameter 1. Which is both a definition and a math problem: if a circle is a mile across, how far is it around?

Since today is 3/14/15 and we’re talking about pi, you already know that the answer is “about 3.1415 miles.” Let’s try to picture that. Have a look at the diagram below:

pi There’s a circle, a hexagon inside the circle, and a square outside the circle, and they are all the same distance across — a mile, say. Let’s figure out how far each one is around.

  • The square is easy: since it’s a mile across in the center, each side is a mile too, so it’s 4 miles around.
  • The hexagon is a little harder. The dashed lines divide the hexagon up into six equilateral triangles. Each triangle has two inner sides, which are 1/2 mile long each, and an outer side, which must be 1/2 mile long too. The outside of the hexagon is made up of 6 of those outer sides, so it’s 3 miles across.

The circle fits between them, longer around than the hexagon, not as long around as the square. So pi, the distance around the circle, is between 3 and 4. The circle is a lot closer to the hexagon than it is to the square, so pi must be a lot closer to 3 than it is to 4. The picture may not make the number 3.1415 pop into your head, but it certainly makes that number look plausible.

Here’s another picture you might like:

circle_wedges

This one is an illustration of perhaps the most famous fact about pi. The idea is to rearrange the circle into something that looks like a rectangle. (The more wedges you cut the circle into, the more rectangular the rearrangement; you get a true rectangle “in the limit.”) The height of the rectangle is r, the length of the circle. The length of the base is half the circumference of the circle (blue wedges only). We can express the length of the circumference in terms of pi: if a circle that’s a mile across is pi miles around, then our circle here, which is 2r miles across, must be 2πr miles around. So the base of the rectangle is πr miles long. And if you know the base and the height of a rectangle, you can compute the area: πr times r, or πr². Which must be the area of the circle as well.

Oh yeah, I promised you monkeys. A hundred monkeys typing for a hundred years might not produce the works of William Shakespeare, but they can get you a pretty good estimate of pi. Here’s how:

  1. Generate a bunch of pairs (x, y) of random numbers between –1 and 1. If you are doing this by the monkey method, you need to take all the letters your monkeys have typed and figure out some way of turning them into numbers. For example, you can break up their typescript into four-letter segments, read each segment as a 4-digit decimal in base 26, and convert it to base 10. Then add a minus sign in front if the last base 26 digit was odd. These days, using a computer is probably faster, and less smelly.
  2. For each pair, decide if it lies inside the unit circle ( + < 1) or outside. Circle_area_Monte_Carlo_integration.svg
  3. Find the fraction of pairs that landed inside the unit circle (709/900 in the picture above) and multiply it by 4. The idea is that the circle has area pi (r = 1), and the square has area 4 (because each side has length 2), so each pair you generated had a π/4 chance of landing inside the circle. That means that pi should be about 4 times the probability that you actually observed. In the example above, 709/900 × 4 is about 3.151, which is pretty close, although it would mean that you’d have to wait for your pi till tomorrow.

So that’s pi for you, folks. A a smidgen more than 3, a lot less than 4, and monkeys people have been trying to say exactly how much it is for over 2000 years. Happy Pi Day!

Fractions, Vodka, Pizza, Fibonacci, and the Golden Mean

I’m teaching a weekend math workshop for school kids (grades 3-6) in Montclair again this spring. This week we started in on comparing fractions. I always like to work with real life problems, and my inspiration for this past Sunday’s class was a remark on teaching by the great Russian mathematician I.M. Gelfand, who once said:

You can explain fractions even to heavy drinkers. If you ask them, ‘Which is larger, 2/3 or 3/5?’ it is likely they will not know. But if you ask, ‘Which is better, two bottles of vodka for three people, or three bottles of vodka for five people?’ they will answer you immediately. They will say two for three, of course.

Well, it wasn’t quite as easy as that. Figuring that my kids might have less intuition for vodka than Gelfand might be counting on, I asked them if they’d rather have two bottles of soda for three people or three bottles for five. Sadly, I got back only random guesses and bored stares. Perhaps kids don’t drink enough soda anymore. So I decided to frame the question in terms of pizza instead.

The advantage of working with pizza is that you can divide it up into slices. Your basic large pizza has 8 slices, so two pizzas means 16 slices. If you have 3 people and two pizzas to divide up between them, then you can give each person 5 slices (3 × 5 = 15), with a single slice left over. Similarly, three pizzas means 24 slices. If you’re dividing three pizzas among five people, four of them get 5 slices (4 × 5 = 20), but then there are only 4 left for the fifth person. In other words: with two pizzas for three people, you have a bit more than five slices a person, whereas with three pizzas for five you have a bit less. So two for three is indeed better, just as Gelfand and the vodka drinkers promised. Coming back to fractions: just by counting pizza slices, we’ve shown that 2/3 is larger than 3/5. Easy-peasy, right? It worked very nicely as an activity in my class.

But there is some very famous and lovely math hiding behind our humble slice-counting. To find it, let’s take our one group of three people with two pizzas, and our other group of five people with three pizzas, and combine them — eight people, five pizzas. Thinking in terms of slices again, five pizzas means 40 slices, which is nice and even: each person gets exactly five. So the group with a little more than five slices a person exactly balanced out the other group with a little less. In terms of fractions, 3/5 < 5/8 < 2/3.

Perhaps the numbers that appear in these fractions — 2, 3, 5, 8 — look familiar to you. What you’re recognizing are some of the first numbers in the Fibonacci sequence, in which you add pairs of adjacent elements to get the next element. So we have 5 = 2 + 3, and 8 = 3 + 5. The next element after that is 5 + 8, which is 13. The next ones after that are 21, 34, 55, 89, and so on.

The pizza eater’s analog of adding the last two numbers is combining the last two groups. If we take eight people with five pizzas and five people with three pizzas and combine them, we get thirteen people with eight pizzas. Because 3/5 < 5/8 from the above, there was more pizza per person in the first group than in the second. In the combined group, the amount of pizza per person has to be somewhere in between. That average amount in the combined group is 8/13 of a pizza, which has to squeeze somewhere between the pizza per person ratios of the two smaller groups. Without any more slice counting, we’ve shown that 3/5 < 8/13 < 5/8.

And you can keep right on going. Combining our new, 13 (people)-with-8 (pizzas) group with the previous 8-with-5 group, we get a group of 21-with-13. In this group, there is more pizza per person than in the 13-with-8 group, but less than in the 8-with-5 group. Since everyone in the 21-with-13 group gets 13/21 of a pizza, 13/21 has to squeeze in between 8/13 and 5/8. In other words: 8/13 < 13/21 < 5/8. Continuing on, we get:

  • 8/13 < 21/34 < 13/21,
  • 21/34 < 34/55 < 13/21,
  • 21/34 < 55/84 < 34/55,

and on and on.

Let’s notice a few things that happen as we iterate and the numbers get bigger:

  1. Each fraction is the ratio of two successive Fibonacci numbers.
  2. Up steps and down steps alternate. We went down from 2/3 to 3/5, then up from 3/5 to 5/8, then down from 5/8 to 8/13, then up to 13/21, down to 21/34, up to 34/55, and so on.
  3. As the numbers get bigger, successive ratios get squeezed into a tighter and tighter range. Each step is smaller in magnitude, and opposite in direction, to the one that preceded it.

In the same way that each Fibonacci number is the sum of the two numbers that came before it, each Fibonacci ratio has to squeeze in between the two ratios that came before it. A calculus fact known as the alternating series test guarantees that numbers that oscillate up and down forever in a tighter and tighter range like that have to be homing on something. A limit, if you will. But what is that limit here?

If we write Fi for the i-th Fibonacci number, the i-th ratio is Fi / Fi+1, and the ratio after that is Fi+1 / Fi+2, which you can rewrite as Fi+1 / (Fi+1 + Fi). Here’s a picture, with Fi = 21, Fi+1 = 34, and Fi+2Fi+1Fi = 55:

Golden-Rectangles

Fi / Fi+1 is the ratio of the side lengths of the small rectangle, and Fi+1 / Fi+2 is the ratio of the side lengths of the large rectangle. These ratios get closer and closer to each other as we iterate Fibonacci numbers and ratios. In the limit, they become equal, and the two rectangles become similar:

220px-SimilarGoldenRectangles.svgIf we write b/a for the ratio in the limit, and equate the ratios of the side lengths of the small and large rectangles, we get the equation b/a = a/(a+b). To find a formula for the ratio, we can set a = 1 and solve the resulting quadratic equation for b.

The ancient Greeks thought that the picture above indicated perfect proportionality, and called the ratio (really its reciprocal) the Golden Ratio, or Golden Mean. So as you iterate the Fibonacci numbers, the ratios between them become more and more perfectly proportioned. And it all started with sharing some pizza.

So try not to despair the next time your kid insists on pizza for dinner yet again. Perhaps you can explain fractions to heavy drinkers, but look how much tasty math you can explain to pizza eaters.