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Love StackOverflow and MathOverflow? Now there’s StemForums!

August 14, 2014 Comments off

Everyone I know who codes uses stackoverflow.com for absolutely everything.

Just yesterday I met a cool coding chick who was learning python and pandas (of course!) with the assistance of stackoverflow. It is exactly what you need to get stuff working, and it’s better than having a friend to ask, even a highly knowledgable friend, because your friend might be busy or might not know the answer, or even if your friend knew the answer her answer isn’t cut-and-paste-able.

If you are someone who has never used stackoverflow for help, then let me explain how it works. Say you want to know how to load a JSON file into python but you don’t want to write a script for that because you’re pretty sure someone already has. You just search for “import json into python” and you get results with vote counts:

Screen Shot 2014-08-14 at 7.32.52 AM

Also, every math nerd I know uses and contributes to mathoverflow.net. It’s not just for math facts and questions, either, there are interesting discussions going on there all the time. Here’s an example of a comment in response to understanding the philosophy behind the claimed proof of the ABC Conjecture:

Screen Shot 2014-08-14 at 7.37.27 AM

OK well hold on tight because now there’s a new online forum, but not about coding and not about math. It’s about all the other STEM subjects, which since we’ve removed math might need to be called STE subjects, which is not catchy. 

It’s called stemforums.com, and it is being created by a team led by Gary Cornell, mathematician, publisher at Apress, and beloved Black Oak bookstore owner.

So far only statistics is open, but other stuff is coming very soon. Specifically it covers, or soon will cover, the following fields:

  1. Statistics
  2. Biology 
  3. Chemistry
  4. Cognitive Sciences
  5. Computer Sciences
  6. Earth and Planetary Sciences
  7. Economics
  8. Science & Math Education
  9. Engineering
  10. History of Science and Mathematics
  11. Applied Mathematics, and
  12. Physics

I’m super excited for this site, it has serious potential to make peoples’ lives better. I wish it had a category for Data Sciences, and for Data Journalism, because I’d probably be more involved in those categories than most of the above, but then again most data science-y questions could be inserted into one of the above. I’ll try to be patient on this one.

Here’s a screen shot of an existing Stats question on the site:

Screen Shot 2014-08-14 at 7.45.00 AMThe site doesn’t have many questions, and even fewer answers, but as I understand it the first few people to get involved are eligible for Springer books, so go check it out.

Nerding out: RSA on an iPython Notebook

Yesterday was a day filled with secrets and codes. In the morning, at The Platform, we had guest speaker Columbia history professor Matthew Connelly, who came and talked to us about his work with declassified documents. Two big and slightly depressing take-aways for me were the following:

  • As records have become digitized, it has gotten easy for people to get rid of archival records in large quantities. Just press delete.
  • As records have become digitized, it has become easy to trace the access of records, and in particular the leaks. Connelly explained that, to some extent, Obama’s harsh approach to leakers and whistleblowers might be explained as simply “letting the system work.” Yet another way that technology informs the way we approach human interactions.

After class we had section, in which we discussed the Computer Science classes some of the students are taking next semester (there’s a list here) and then I talked to them about prime numbers and the RSA crypto system.

I got really into it and wrote up an iPython Notebook which could be better but is pretty good, I think, and works out one example completely, encoding and decoding the message “hello”.

The underlying file is here but if you want to view it on the web just go here.

The platonic solids

July 18, 2014 Comments off

I managed to record this week’s Slate Money podcast early so I could drive up to HCSSiM for July 17th, and the Yellow Pig Day celebration. I missed the 17 talk but made it in time for yellow pig carols and cake.

This morning my buddy Aaron decided to let me talk to the kids in the last day of his workshop. First Amber is working out the formula for the Euler Characteristic of a planar graph with the kids and after that I’ll help them count the platonic solids using stereographic projection. If we have time we’ll talk about duals (update: we had time!).

I can never remember which one is the icosahedron.

I can never remember which one is the icosahedron.

Tonight at Prime Time I’ll play a game or two of Nim with them.

Categories: math, math education

Guest post: What is the goal of a college calculus course?

This is a guest post by Nathan, who recently finished graduate school in math, and will begin a post-doc in the fall. He loves teaching young kids, but is still figuring out how to motivate undergraduates.

The question

Like most mathematicians in academia, I’m teaching calculus in the fall. I taught in grad school, but the syllabus and assignments were already set. This time I’ll be in charge, so I need to make some design decisions, like the following:

  1. Are calculators/computers/notes allowed on the exams?
  2. Which purely technical skills must students master (by a technical skill I mean something like expanding rational functions into partial fractions: a task which is deterministic but possibly intricate)?
  3. Will students need to write explanations and/or proofs?

I have some angst about decisions like these, because it seems like each one can go in very different directions depending on what I hope the students are supposed to get from the course. If I’m listing the pros and cons of permitting calculators, I need some yardstick to measure these pros and cons.

My question is: what is the goal of a college calculus course?

I’d love to have an answer that is specific enough that I can use it to make concrete decisions like the ones above. Part of my angst is that I’ve asked many people this question, including people I respect enormously for their teaching, but often end up with a muddled answer. And there are a couple stock answers that come to mind, but each one doesn’t satisfy me for one reason or another. Here’s what I have so far.

The contenders.

To teach specific tasks that are necessary for other subjects.

These tasks would include computing integrals and derivatives, converting functions to power series or Fourier series, and so forth.

Intuitive understanding of functions and their behavior.

This is vague, so here’s an example: a couple years ago, a friend in medical school showed me a page from his textbook. The page concerned whether a certain drug would affect heart function in one way or in the opposite way (it caused two opposite effects), and it showed a curve relating two involved parameters. It turned out that the essential feature was that this curve was concave down. The book did not use the phrase “concave down,” though, and had a rather wordy explanation of the behavior. In this situation, a student who has a good grasp of what concavity is and what its implications are is better equipped to understand the effect described in the book. So if a student has really learned how to think about concavity of functions and its implications, then she can more quickly grasp the essential parts of this medical situation.

To practice communicating with precision.

I’m taking “communication” in a very wide sense here: carefully showing the steps in an integral calculation would count.

Not Satisfied

I have issues with each of these as written. I don’t buy number 1, because the bread and butter of calculus class, like computing integrals, isn’t something most doctors or scientists will ever do again. Number 2 is a noble goal, but it’s overly idealistic; if this is the goal, then our success rate is less than 10%. Number 3 also seems like a great goal, relevant for most of the students, but I think we’d have to write very different sorts of assignments than we currently do if we really want to aim for it.

I would love to have a clear and realistic answer to this question. What do you think?

Categories: education, math education

Getting rid of teacher tenure does not solve the problem

There’s been a movement to make primary and secondary education run more like a business. Just this week in California, a lawsuit funded by Silicon Valley entrepreneur David Welch led to a judge finding that student’s constitutional rights were being compromised by the tenure system for teachers in California.

The thinking is that tenure removes the possibility of getting rid of bad teachers, and that bad teachers are what is causing the achievement gap between poor kids and well-off kids. So if we get rid of bad teachers, which is easier after removing tenure, then no child will be “left behind.”

The problem is, there’s little evidence for this very real achievement gap problem as being caused by tenure, or even by teachers. So this is a huge waste of time.

As a thought experiment, let’s say we did away with tenure. This basically means that teachers could be fired at will, say through a bad teacher evaluation score.

An immediate consequence of this would be that many of the best teachers would get other jobs. You see, one of the appeals of teaching is getting a comfortable pension at retirement, but if you have no idea when you’re being dismissed, then it makes no sense to put in the 25 or 30 years to get that pension. Plus, what with all the crazy and random value-added teacher models out there, there’s no telling when your score will look accidentally bad one year and you’ll be summarily dismissed.

People with options and skills will seek other opportunities. After all, we wanted to make it more like a business, and that’s what happens when you remove incentives in business!

The problem is you’d still need teachers. So one possibility is to have teachers with middling salaries and no job security. That means lots of turnover among the better teachers as they get better offers. Another option is to pay teachers way more to offset the lack of security. Remember, the only reason teacher salaries have been low historically is that uber competent women like Laura Ingalls Wilder had no other options than being a teacher. I’m pretty sure I’d have been a teacher if I’d been born 150 years ago.

So we either have worse teachers or education doubles in price, both bad options. And, sadly, either way we aren’t actually addressing the underlying issue, which is that pesky achievement gap.

People who want to make schools more like businesses also enjoy measuring things, and one way they like measuring things is through standardized tests like achievement scores. They blame teachers for bad scores and they claim they’re being data-driven.

Here’s the thing though, if we want to be data-driven, let’s start to maybe blame poverty for bad scores instead:

dc-public-schools-poverty-versus-reaching-ach-2010

 

I’m tempted to conclude that we should just go ahead and get rid of teacher tenure so we can wait a few years and still see no movement in the achievement gap. The problem with that approach is that we’ll see great teachers leave the profession and no progress on the actual root cause, which is very likely to be poverty and inequality, hopelessness and despair. Not sure we want to sacrifice a generation of students just to prove a point about causation.

On the other hand, given that David Welch has a lot of money and seems to be really excited by this fight, it looks like we might have no choice but to blame the teachers, get rid of their tenure, see a bunch of them leave, have a surprise teacher shortage, respond either by paying way more or reinstating tenure, and then only then finally gather the data that none of this has helped and very possibly made things worse.

Categories: education, math education, news

How Not To Be Wrong by Jordan Ellenberg

You guys are in for a treat. In fact I’m jealous of you.

I had a little secret about my survival in grad school, and that secret has a name, and that name is Jordan Ellenberg. We used to meet every Tuesday and Thursday to study schemes at the CallaLily Cafe a few blocks from the Science Center on Kirkland Street, and even though that sounds kind of dull, it was a blast. It was what kept me sane at Harvard.

You see, Jordan has an infectious positivity about him, which balances my rather intense suspicions, and moreover he’s hilariously funny. He’s really somewhere between a mathematician and a stand-up comedian, and to be honest I don’t know which one he’s better at, although he is a deeply talented mathematician.

Screen Shot 2014-05-29 at 7.21.14 AMThe reason I’m telling you this is that he’s written a book, called How Not To Be Wrong, and available for purchase starting today, which is a delight to read and which will make you understand why I survived graduate school. In fact nobody will ever let me complain again once they’ve read this book, because it reads just like Jordan talks. In reading it, I felt like I was right back at CallaLily, singing Prince’s “Sexy MF” and watching Jordan flirt with the cashier lady again. Aaaah memories.

So what’s in the book? Well, he talks a lot about math, and about mathematicians, and the lottery, and in fact he has this long riff which starts out with lottery math, then goes to error-correcting codes and then to made-up languages and then to sphere packing and then arrives again at lotteries. And it’s brilliant and true and beautiful and also funny.

I have a theory about this book that you could essentially open it up to any page and begin to enjoy it, since it is thoroughly enjoyable and the math is cumulative but everywhere so well explained that it wouldn’t take long to follow along, and pretty soon you’d be giggling along with Jordan at every ridiculous footnote he’s inserted into his narrative.

In other words, every page is a standalone positive and ontological examination of the beauty and surprise of mathematical discovery. And so, if you are someone who shares with Jordan a love for mathematics, you will have a consistently great time with this book. In fact I’m imagining that you have an uncle or a mom who loves math or science, in which case this would be a seriously perfect gift to them, but of course you could also give that gift to yourself. I mean, this is a guy who can make nazi jokes funny, and he does.

Having said that, the magic of the book is that it’s not just a collection of wonderful mathy tidbits. Jordan also has a point about the act of scrutinizing something in a logical and mathematical fashion. That act itself is courageous and should be appreciated, and he explains why, and he tells us how much we’ve already benefited from people in the past who have had the bravery to do so. He appreciates them and we should too.

And yet, he also sends the important message that it’s not an elitist crew of the usual genius suspects, that in fact we can all do this in our own capacity. It’s a great message and, if it ends up allowing people to re-examine their need for certainty in an uncertain world, then Jordan will really end up doing good. Fingers crossed.

That’s not to say it’s a perfect book, and I wanted to argue with points on basically every other page, but mostly in a good, friendly, over-drinks kind of way, which is provocative but not annoying. One exception I might make came on page 256: no, Jordan, municipal bonds do not always get paid back, and no, stocks do not always go up, not even in expectation. In fact to the extent that both of those statements seem true to many people is the result of many cynical political acts and is damaging, mostly to people like retired civil servants. Don’t go there!

Another quibble: Jordan talks about how public policy makers make proclamations in the face of uncertainty, and he has a lot of sympathy and seems to think the should keep doing this. I’m on the other side on this one. Telling people to avoid certain foods and then changing stances seems more damaging than helpful and it happens constantly. And it’s often tied to industry and money, which also doesn’t impress.

Even so, even when I strongly disagree with Jordan, I always want to have the conversation. He forces that on the reader because he’s so darn positive and open-minded.

A few more goodies that I wanted to adore without giving too much away. Jordan does a great job with something he calls “The Great Square of Men” and Berkson’s Fallacy: it will explain to many many women why they are not finding the man they’re looking for. He also throws out a bone to nerds like me when he almost proves that every pig is yellow, and he absolutely kills it, stand-up comedian style, when comparing Ross Perot to a small dark pile of oats. Holy crap he was on a roll there.

So here’s one thing I’ve started doing since reading the book. When I give my 5-year-old son his dessert, it’s in the form of Hershey Drops, which are kind of like fat M&M’s. I give him 15 and I ask him to count them to make sure I got it right. Sometimes I give him 14 to make sure he’s paying attention. But that’s not the new part. The new part is something I stole from Jordan’s book.

The new part is that some days I ask him, “do you want me to give you 3 rows of 5 drops?” And I wait for him to figure out that’s enough and say “yes!” And the other days I ask him “do you want me to give you 5 rows of 3 drops?” and I again wait. And in either case I put the drops out in a rectangle.

And last night, for the first time, he explained to me in a slightly patronizing voice that it doesn’t matter which way I do it because it ends up being the same, because of the rectangle formation and how you look at it. And just to check I asked him which would be more, 10 rows of 7 drops or 7 rows of 10 drops, and he told me, “duh, it would be the same because it couldn’t be any different.”

And that, my friends, is how not to be wrong.

Categories: math, math education

Interview with a middle school math teacher on the Common Core

Today’s post is an email interview with Fawn Nguyen, who teaches math at Mesa Union Junior High in southern California. Fawn is on the leadership team for UCSB Mathematics Project that provides professional development for teachers in the Tri-County area. She is a co-founder of the Thousand Oaks Math Teachers’ Circle. In an effort to share and learn from other math teachers, Fawn blogs at Finding Ways to Nguyen Students Over. She also started VisualPatterns.org to help students develop algebraic thinking, and more recently, she shares her students’ daily math talks to promote number sense. When Fawn is not teaching or writing, she is reading posts on mathblogging.org as one of the editors. She sleeps occasionally and dreams of becoming an architect when all this is done.

Importantly for the below interview, Fawn is not being measured via a value-added model. My questions are italicized.

——

I’ve been studying the rhetoric around the mathematics Common Core State Standard (CCSS). So far I’ve listened to Diane Ravitch stuff, I’ve interviewed Bill McCallum, the lead writer of the math CCSS, and I’ve also interviewed Kiri Soares, a New York City high school principal. They have very different views. Interestingly, McCallum distinguished three things: standards, curriculum, and testing. 

What do you think? Do teachers see those as three different things? Or is it a package deal, where all three things rolled into one in terms of how they’re presented?

I can’t speak for other teachers. I understand that the standards are not meant to be the curriculum, but the two are not mutually exclusive either. They can’t be. Standards inform the curriculum. This might be a terrible analogy, but I love food and cooking, so maybe the standards are the major ingredients, and the curriculum is the entrée that contains those ingredients. In the show Chopped on Food Network, the competing chefs must use all 4 ingredients to make a dish – and the prepared foods that end up on the plates differ widely in taste and presentation. We can’t blame the ingredients when the dish is blandly prepared any more than we can blame the standards when the curriculum is poorly written.

Similary, the standards inform testing. Test items for a certain grade level cover the standards of that grade level. I’m not against testing. I’m against bad tests and a lot of it. By bad, I mean multiple-choice items that require more memorization than actual problem solving. But I’m confident we can create good multiple-choice tests because realistically a portion of the test needs to be of this type due to costs.

The three – standards, curriculum, and testing – are not a “package deal” in the sense that the same people are not delivering them to us. But they go together, otherwise what is school mathematics? Funny thing is we have always had the three operating in schools, but somehow the Common Core State Standands (CCSS) seem to get the all the blame for the anxieties and costs connected to testing and curriculum development.

As a teacher, what’s good and bad about the CCSS?

I see a lot of good in the CCSS. This set of standards is not perfect, but it’s much better than our state standards. We can examine the standards and see for ourselves that the integrity of the standards holds up to their claims of being embedded with mathematical focus, rigor, and coherence.

Implementation of CCSS means that students and teachers can expect consistency in what is being in taught at each grade level across state boundaries. This is a nontrivial effort in addressing equity. This consistency also helps teachers collaborate nationwide, and professional development for teachers will improve and be more relevant and effective.

I can only hope that textbooks will be much better because of the inherent focus and coherence in CCSS. A kid can move from Maine to California and not have to see different state outlines on their textbooks as if he’d taken on a new kind of mathematics in his new school. I went to a textbook publishers fair recently at our district, and I remain optimistic that better products are already on their way.

We had every state create its own assessment, now we have two consortia, PARCC and Smarter Balanced. I’ve gone through the sample assessments from the latter, and they are far better than the old multiple-choice items of the CST. Kids will have to process the question at a deeper level to show understanding. This is a good thing.

What is potentially bad about the CCSS is the improper or lack of implementation. So, this boils down to the most important element of the Common Core equation – the teacher. There is no doubt that many teachers, myself included, need sustained professional development to do the job right. And I don’t mean just PD in making math more relevant and engaging, and in how many ways we can use technology, I mean more importantly, we need PD in content knowledge.

It is a perverse notion to think that anyone with a college education can teach elementary mathematics. Teaching mathematics requires knowing mathematics. To know a concept is to understand it backward and forward, inside and outside, to recognize it in different forms and structures, to put it into context, to ask questions about it that leads to more questions, to know the mathematics beyond this concept. That reminds me just recently a 6th grader said to me as we were working on our unit of dividing by a fraction. She said, “My elementary teacher lied to me! She said we always get a smaller number when we divide two numbers.”

Just because one can make tuna casserole does not make one a chef. (Sorry, I’m hungry.)

What are the good and bad things for kids about testing?

Testing is only good for kids when it helps them learn and become more successful – that the feedback from testing should inform the teacher of next moves. Testing has become such a dirty word because we over test our kids. I’m still in the classroom after 23 years, yet I don’t have the answers. I struggle with telling my kids that I value them and their learning, yet at the end of each quarter, the narrative sum of their learning is a letter grade.

Then, in the absence of helping kids learn, testing is bad.

What are the good/bad things for the teachers with all these tests?

Ideally, a good test that measures what it’s supposed to measure should help the teacher and his students. Testing must be done in moderation. Do we really need to test kids at the start of the school year? Don’t we have the results from a few months ago, right before they left for summer vacation? Every test takes time away from learning.

I’m not sure I understand why testing is bad for teachers aside from lost instructional minutes. Again, I can’t speak for other teachers. But I do sense heightened anxiety among some teachers because CCSS is new – and newness causes us to squirm in our seats and doubt our abilities. I don’t necessarily see this as a bad thing. I see it as an opportunity to learn content at a deeper conceptual level and to implement better teaching strategies.

If we look at anything long and hard enough, we are bound to find the good and the bad. I choose to focus on the positives because I can’t make the day any longer and I can’t have fewer than 4 hours of sleep a night. I want to spend my energies working with my administrators, my colleagues, my parents to bring the best I can bring into my classroom.

Is there anything else you’d like to add?

The best things about CCSS for me are not even the standards – they are the 8 Mathematical Practices. These are life-long habits that will serve students well, in all disciplines. They’re equivalent to the essential cooking techniques, like making roux and roasting garlic and braising kale and shucking oysters. Okay, maybe not that last one, but I just got back from New Orleans, and raw oysters are awesome.

I’m excited to continue to share and collaborate with my colleagues locally and online because we now have a common language! We teachers do this very hard work – day in and day out, late into the nights and into the weekends – because we love our kids and we love teaching. But we need to be mathematically competent first and foremost to teach mathematics. I want the focus to always be about the kids and their learning. We start with them; we end with them.

Categories: math, math education

Interview with a high school principal on the math Common Core

In my third effort to understand the Common Core State Standards (CC) for math, I interviewed an old college friend Kiri Soares, who is the principal and co-founder of the Urban Assembly Institute of Math and Science for Young Women. Here’s a transcript of the interview which took place earlier this month. My words are in italics below.

——

How are high school math teachers in New York City currently evaluated?

Teachers are now evaluated on 2 things:

  1. First, measures of teacher practice, which are based on observations, in turn based on some rubric. Right now it’s the Danielson Rubric. This is a qualitative measure. In fact it is essentially an old method with a new name.
  2. Second, measures of student learning, that is supposed to be “objective”. Overall it is worth 40% of the teacher’s score but it is separated into two 20% parts, where teachers choose the methodology of one part and principals choose the other. Some stuff is chosen for principals by the city. Any time there is a state test we have to choose it. In terms of the teachers’ choices, there are two ways to get evaluated: goals or growth. Goals are based on a given kid, and the teachers can guess they will get a certain slightly lower score or higher score for whatever reason. Otherwise, it’s a growth-based score. Teachers can also choose from an array of assessments (state tests, performance tests, and third party exams). They can also choose the cohort (their own kids/ the grade/the school). The city also chose performance tasks in some instances.

Can you give me a concrete example of what a teacher would choose as a goal?

At the beginning of year you give diagnostic tests to students in your subject. Based on what a given kid scored in September, you extrapolate a guess for their performance in the June test. So if a kid has a disrupted homelife you might guess lower. Teacher’s goal setting is based on these teachers’ guesses.

So in other words, this is really just a measurement of how well teachers guess?

Well they are given a baseline and teachers set goals relative to that, but yes. And they are expected to make those guesses in November, possibly well before homelife is disrupted. It definitely makes things more complicated. And things are pretty complicated. Let me say a bit more.

The first three weeks of school are all testing. We test math, social studies, science, and English in every grade, and overall it depending on teacher/principal selections it can take up to 6 weeks, although not in a given subject. Foreign language and gym teachers also getting measured, by the way, based on those other tests. These early tests are diagnostic tests.

Moreover, they are new types of tests, which are called performance-based assessments, and they are based on writing samples with prompts. They are theoretically better quality because they go deeper, the aren’t just bubble standardized tests, but of course they had no pre-existing baseline (like the state tests) and thus had to be administered as diagnostic. Even so, we are still trying to predict growth based on them, which is confusing since we don’t know how to predict performance on new tests. Also don’t even know how we can consistently grade such essay-based tests- despite “norming protocols”, which is yet another source of uncertainty.

How many weeks per year is there testing of students?

The last half of June is gone, a week in January, and 2-3 weeks in the high school in the beginning per subject. That’s a minimum of 5 weeks per subject per year, out of a total of 40 weeks. So one eighth of teacher time is spent administering tests. But if you think about it, for the teachers, it’s even more. They have to grade these tests too.

I’ve been studying the rhetoric around the CC. So far I’ve listened to Diane Ravitch stuff, and to Bill McCallum, the lead writer of the math CC. They have very different views. McCallum distinguished three things, which when they are separated like that, Ravitch doesn’t make sense.

Namely, he separates standards, curriculum, and testing. People complain about testing and say that CC standards make testing easier, and we already have too much testing, so CC is a bad thing. But McCallum makes this point: good standards also make good testing easier.

What do you think? Do teachers see those as three different things? Or is it a package deal, where all three things rolled into one in terms of how they’re presented?

It’s much easier to think of those three things as vertices of a triangle. We cannot make them completely isolated, because they are interrelated.

So, we cannot make the CC good without curriculum and assessment, since there’s a feedback loop. Similarly, we cannot have aligned curriculum without good standards and assessment, and we cannot have good tests without good standards and curriculum. The standards have existed forever. The common core is an attempt to create a set of nationwide standards. For example, without a coherent national curriculum it might seem OK to teach creationism in place of evolution in some states. Should that be OK?

CC is attempting to address this, in our global economy, but it hasn’t even approached science for clear political reasons. Math and English are the least political subjects so they started with those. This is a long time coming, and people often think CC refers to everything but so far it’s really only 40% of a kid’s day. Social studies CC standards are actually out right now, but they are very new.

Next, the massive machine of curriculum starts getting into play, as does the testing. I have CC standards and the CC-aligned test, but not curriculum.

Next, you’re throwing into the picture teacher evaluation aligned to CC tests. Teachers are freaking out now – they’re thinking, my curriculum hasn’t been CC-aligned for many years, what do I do now? By the way, importantly, none of the high school curriculum in NY State is actually CC-aligned now. DOE recommendations for the middle school happened last year, and DOE people will probably recommend this year for high school, since they went into talks with publication houses last year to negotiate CC curriculum materials.

The real problem is this: we’ve created these new standards to make things more difficult and more challenging without recognizing where kids are in the present moment. If I’m a former 5th grader, and the old standards were expecting something from me that I got used to, and it wasn’t very much, and now I’m in 6th grade, and there are all these raised expectations, and there’s no gap attention.

Bottomline, everybody is freaking out – teachers, students, and parents.

Last year was the first CC-aligned ELA and math tests. Everybody failed. They rolled out the test before any CC curriculum.

From the point of view of NYC teachers, this seems like a terrorizing regime, doesn’t it?

Yes, because the CC roll-out is rigidly tied to the tests, which are in turn rigidly tied to evaluations of teachers. So the teachers are worried they are automatically going to get a “failure” on that vector.

Another way of saying this is that, if teacher evaluations were taken out of the mix, we’d have a very different roll-out environment. But as it is, teachers are hugely anxious about the possibility that their kids might fail both the city and state tests, and that would give the teacher an automatic “failure” no matter how good their teacher observations are.

So if I’m a special ed teacher of a bunch of kids reading at 4th and 5th grade level even through they’re in 7th grade, I’m particularly worried with the introduction of the new and unknown CC-aligned tests.

So is that really what will happen? Will all these teachers get failing evaluation scores?

That’s the big question mark. I doubt it there will be massive failure though. I think given that the scores were so clustered in the middle/low muddle last year, they are going to add a curve and not allow so many students to fail.

So what you’re pointing out is that they can just redefine failure?

Exactly. It doesn’t actually make sense to fail everyone. Probably 75% of the kids got 2’s or 1’s out of a 4 point scale. What does failure mean when everyone fails? It just means the test was too hard, or that what the kids were being taught was not relevant to the test.

Let’s dig down to the the three topics. As far as you’ve heard from the teachers, what’s good and bad about CC?

My teachers are used to the CC. We’ve rolled out standards-based grading three years ago, so our math and ELA teachers were well adjusted, and our other subject teachers were familiar. The biggest change is what used to be 9th grade math is now expected of the 8th grade. And the biggest complaint I’ve heard is that it’s too much stuff – nobody can teach all that. But that’s always been true about every set of standards.

Did they get rid of anything?

Not sure, because I don’t know what the elementary level CC standards did. There was lots of shuffling in the middle school, and lots of emphasis on algebra and algebraic thinking. Maybe they moved data and stats to earlier grades.

So I believe that my teachers in particular were more prepared. In other schools, where teachers weren’t explicitly being asked to align themselves to standards, it was a huge shock. For them, it used to be solely about Regents, and also Regents exams are very predictable and consistent, so it was pretty smooth sailing.

Let’s move on to curriculum. You mentioned there is no CC-aligned curriculum in NY. I also heard NY state has recently come out against the CC, did you hear that?

Well what I heard is that they previously said they this year’s 9th graders (class of 2017) would be held accountable but now the class of 2022 will be. So they’ve shifted accountability to the future.

What does accountability mean in this context?

It means graduation requirements. You need to pass 5 Regents exams to graduate, and right now there are two versions of some of those exams: one CC-aligned, one old-school. The question is who has to pass the CC-aligned versions to graduate. Now the current 9th grade could take either the CC-aligned or “regular” Regents in math.

I’m going to ask my 9th grade students to take both so we can gather information, even though it means giving them 3 extra hours of tests. Most of my kids pass 2 Regents in 9th grade, 2 in 10th, and 3 in 11th, and then they’re supposed to be done. They only take those Regents tests in senior year that they didn’t pass earlier.

What are the good and bad things about testing?

What’s bad is how much time is lost, as we’ve already said. And also, it’s incredibly stressful. You and I went to school and we had one big college test that was stressful, namely the SAT. In terms of us finishing high school, that was it. For these kids it’s test, test, test, test. I don’t think it’s actually improved the quality of college students across the country. 20 years ago NY was the only one that had extra tests except California achievement tests, which I guess we sometimes took as well.

Another way to say it is that we did take some tests but it didn’t take 5 weeks.

And it wasn’t high stakes for the teacher!

Let’s go straight there: what are the good/bad things for the teachers with all these tests?

Well it definitely makes the teachers more accountable. Even teachers think this: there is a cadre of protected teachers in the city, and the principals didn’t want to take the time to get rid of them, so they’d excess them out of the schools, and they would stay in the system.

Now with testing it has become much more the principal’s responsibility to get rid of bad teachers. The number of floating teachers is going down.

How did they get rid of the floaters?

A lot of different ways. They made them go into the schools, take interviews, they made their quality of life not great, and a lot if them left or retired or found jobs. Principals took up the mantle as well, and they started to do due diligence.

Sounds like the incentive system for over-worked principals was wrong.

Yes, although the reason it became easier for the principals is because now we have data. So if you’re coming in as ineffective and I also have attendance data and observation data, I can add my observational data (subjective albeit rubric based) and do something.

If I may be more skeptical, it sounds like this data gathering was used as a weapon against teachers. There were probably lots of good teachers that have bad numbers attached to them that could get fired if someone wanted them to be fired.

Correct, except those good teachers generally have principals who protect them.

You could give everyone a bad number and then fire the people you want, right?

Correct.

Is that the goal?

Under Bloomberg it was.

Is there anything else you want to mention? 

I think testing needs to be dialed down but not disappear. Education is a bi-polar pendulum and it never stops in the middle. We’re on an extreme but let’s not get rid of everything. There is a place for testing.

Let’s get our CC standards, curriculum, and testing reasonable and college-aligned and let’s keep it reasonable. Let’s do it with standards across states and let’s make sure it makes sense.

Also, there are some new tests coming out, called PARCC assessments, that are adaptive tests aligned to the CC. They are supposed to replace Regents down the line and be national.

Here’s what bothers me about that. It’s even harder to investigate the experience of the student with adaptive tests.

I’m not sure there’s enough technology to actually do this anyway very soon. For example, we were given $10,000 for 500 student. That’s not going to go far unless it takes 2 weeks to administer the test. But we are investing in our technology this year. For example, I’m looking forward  to buying textbooks and get my updates pushed instead of having to buy new books every year.

Last question. They are redoing the SAT because rich kids are doing so much better. Are they just trying to get in on the test prep game? Because, here’s the thing, there’s no test that can’t be gamed that’s also easy to grade. It’s gotta depend on the letters and grades. We keep trying to shortcut that.

Listen, this is what I tell the kids. What’s going to matter to you is the letter of recommendation, so don’t be an jerk to your fellow students or to the teachers. Next, are you going to be able to meet the minimum requirements? That’s what the SAT is good for. It defines a lower bound.

Is it a good lower bound though?

Well, I define the lower bound as 1000 in total. My kids can target that. It’s a reasonable low bar.

To what extent do your students – mostly inner-city, black girls interested in math and science – suffer under the wholly gamed SAT system?

It serves to give them a point of self-reference with the rest of the country. You have to understand, they, like most kids in the nation, don’t have a conception of themselves outside of their own experience. The SAT serves that purpose. My kids, like many others, have the dream of Ivy League minus the understanding of where they actually stand.

So you’re saying their estimates of their chances are too high?

Yes, oftentimes. They are the big fish in a well-defined pond. At the very least, The SAT helps give them perspective.

Thanks so much for your time Kiri.

Billionaire money and academic freedom

If you haven’t seen this recent New York Times article by William Broad, entitled Billionaires With Big Ideas Are Privatizing American Sciencethen go take a look. It generalizes to all of scientific research my recent post entitled Billionaire Money in Mathematics.

My favorite part of Broad’s article is the caption of the video at the top, which sums it up nicely:

Funding the Future: As government financing of basic science research has plunged, private donors have filled the void, raising questions about the future of research for the public good.

In his article Broad makes a bunch of great points.

First, the fact that rich people generally ask for research into topics they care about (“personal setting of priorities”) to the detriment of basic research. They want flashy stuff, bang for their buck.

Second, academics interested in getting funding from these rich people have to learn to market themselves. From the article:

The availability of so much well-financed ambition has created a new kind of dating game. In what is becoming a common narrative, researchers like to describe how they begged the federal science establishment for funds, were brushed aside and turned instead to the welcoming arms of philanthropists. To help scientists bond quickly with potential benefactors, a cottage industry has emerged, offering workshops, personal coaching, role-playing exercises and the production of video appeals.

If you think about it, the two issues above are kind of wrapped up together. Flashy academic content goes hand in hand with flashy marketing. Let’s say goodbye to the true nerd who doesn’t button up their cardigan correctly. And I don’t know about you but I like those nerds. My mom is one of them.

This morning I thought of another way to express this issue, from the point of view of the individual scientist or mathematician, that might have profound resonance where the above just sounds annoying.

Namely, I believe that academic freedom itself is at stake. Let me explain.

I’m the last person who would defend our current tenure system. It’s awful for women, especially those who want kids, and it often breeds a kind of arrogant laziness post-tenure. Even so, there are good things about it, and one of them is academic freedom.

And although theoretically you can have academic freedom without tenure, it is certainly easier with it (example from this piece: “In Oklahoma, a number of state legislators attempted to have Anita Hill fired from her university position because of her testimony before the U.S. Senate. If not for tenure, professors could be attacked every time there’s a change in the wind.”).

But as we’ve seen recently, tenure-track positions are quickly declining in number, even as the number of teaching positions is growing. This is the academic analog of how we’ve seen job growth in the US but it’s majority shitty jobs. And as I’ve predicted already, this trend is surely going to continue as we scale education through MOOCs.

The dwindling tenured positions means there are increasing number of people trying to do research dependent upon outside grants and funding, and without the safety net of tenure. These people often lose their jobs when their funding flags, as we’ve recently seen at Columbia.

Now let’s put these two trends together. We’ve got fewer and fewer tenure jobs, which are precariously dependent on outside funding, and we’ve got rich people funding their own tastes and proclivities.

Where does academic freedom shake out in that picture? I’m going to say nowhere.

Categories: education, math, math education

Report from an MSRI MOOC conversation

I am back from Berkeley where I attended a couple of hours of conversations about MOOCs last Friday up at MSRI.

It was a panel discussion given mostly by math and stats people who themselves run MOOCs, and I was wondering if the people who are involved have a better sense of the side effects and feedback loops involved in the process. After all, I’m claiming that the MOOC Revolution will lead to the end of math research, and I wanted to be proven wrong.

Unfortunately, I left feeling like I have even more evidence that my fears will be realized.

I think the critical moment came when Ani Adhikari spoke. Professor Adhikari is in the second semester of giving her basic stats MOOC, and from how she described it, she is incredibly good at it, and there’s a social network aspect of the class which seems like it’s going really well – she says she spends 30 minutes to an hour a day on it herself, interacting with students. I think she said 28,000 students took it her first semester in addition to her in-class students at Berkeley. I know and respect Professory Adhikari personally, as I taught for her at the Berkeley Mills summer program for women many years ago. I know how devoted she is to good teaching.

Even so, she lost me late in the discussion when she explained that EdX, the platform which hosts her stats MOOC, wanted to offer her class three times a year without her participation. She said something to the effect that MOOC professors had to be “extra vigilant” about this outrageous idea and guard against it at all costs.

After all, she said, at the end of the day the MOOC videos are something like a fancy textbook, and we don’t hand out textbooks and claim they are courses, so we by the same token cannot hand out MOOC videos (and presumably the social networks associated with them) and claim they are courses.

When I pressed her in the Q&A session as to how exactly she was going to remain vigilant against this threat, she said she has a legal contract with EdX that prevented them from offering the course without her approval.

And I’m happy for her and her great contract, but here are two questions for her and for the community.

First, how long until someone in math or stats makes a kick-ass MOOC and doesn’t remember to have that air-tight legal contract? Or has an actual legal battle with EdX and realized their lawyers are not as expensive? Or believes that “information should be free” and does it with the express intention of letting the MOOC be replayed forever?

Second, how much sense does it make to claim that you and your presence are super critical to the success of a MOOC if 28,000 people took this class and you interacted at most one hour a day? Can you possibly claim that the average student benefitted from your presence? It seems to me that the value proposition for the average MOOC student is very similar whether you are there or not.

Overall the impression I got from the speakers, who were mostly MOOC evangelists and involved with MOOCs themselves, was that they loved MOOCs because MOOCs were working for them. They weren’t looking much beyond that point to side effects.

There was one exception, namely Susan Holmes, who listed some side effects of MOOCs including a decreased need for math Ph.D.’s. Unfortunately the conversation didn’t dwell on this, though, and it happened at the very end of the day.

Here’s what I’d like to see: a conversation at MSRI about the future of math research funding in the context of MOOCs and a reduced NSF, where hopefully we come up with something besides “Jim Simons”. It’s extra ironic that the conversation, if it happens, would be held in the Simons Theater.

Categories: math education

Interview with Bill McCallum, lead writer of Math Common Core

This is an interview I had on 2/4/2014 with Bill McCallum, who is a University Distinguished Professor of Mathematics and member of the Department of Mathematics at the University of Arizona. Bill also led the Work Team which recently wrote the Mathematics Common Core State Standards, and was graciously willing to let me interview him on that.

Q: Tell me about how the Common Core State Standards Mathematics Work Team got formed and how you got onto it.

A: There were actually two separate documents and two separate processes, and people often get confused between them.

The first part happened in the summer of 2009 and produced a document called “College and Career Readiness Standards“. It didn’t go grade by grade but rather described what a high school student leaving and ready for college and career looks like. The team that wrote that pre-cursor document consisted of people from the College Board, ACT, and Achieve and was organized and managed by CCSSO (which represents state education commissioners and the like) and NGA (which represents Governors). Gene Wilhoutthen executive director of CCSSO, led the charge.

I was on that first panel representing Achieve. Achieve does not write assessments, but College Board and ACT do, and I think that’s where the charge that the standards were developed by testing companies comes from. It’s worth noting in the context of that charge that both ACT and College Board are non-profits.

The second part of the process, called the Work Team, took that document and worked backwards to create the actual Common Core State Standards for mathematics. I was the chair of the Work Team and was one of the 3 lead writers, the other two being Jason Zimba and Phil Daro. But the other members of the Work Team represent many educators, mathematicians, and math education folks, as well as DOE folks, and importantly there are no testing or textbook companies represented. The full list is here.

Q: Explain what Achieve is and is not.

A: Achieve is not a testing company, so let’s put that to rest.

Without going into too much historical detail (some of which is available here), Achieve was launched as an initiative of a combination of business and education leaders with the goal to improve education. It’s a non-profit think tank, which came out with benchmarks for mathematics education and tried to get states to align standards to them.

I started working with Achieve around 2005 and pretty soon I found myself chairing a committee to revise the benchmarks, which is how I got involved in the drafting of the first document I talked about above.

One more word about getting states on board with the Common Core State Standards (CCSS). There were 48 states that had committed to being involved in the writing, but not necessarily to adopt the standards. The states were involved in the review process as the CCSS were being written in 2009-2010. And here by “states” we usually mean teams from the various Department of Education, but different states had different team makeups.

For example, Arizona heavily involved teachers and some other states had their mathematics specialists at the DOE look things over and make comments.The American Federation of Teachers took the review quite seriously, and Jason and I met twice over the weekend to talk to teams of teachers assembled by AFT, listening to comments and making revisions.The National Council of Teachers of Mathematics was also quite involved in reviews and meetings.

Q: What are the goals of CCSS?

A: The goal is educational: to describe the mathematical achievements, skills, and understandings that we want students to have by the time they leave high school, broken down by grades.

It’s important to note at this point that this is not a new idea. Indeed states have had standards since the early 1990’s. But those standards were pretty unfocused and incoherent in many cases. What’s new is that we have common standards, and that they are focused and coherent.

Q: So what’s the difference between standards and tests?

A: Standards are descriptions of what you want students to learn. What you do with them is up to you. Testing is something you do if you want to know if they’ve learned what you wanted them to learn. It’s assessment.

Q: What’s your view on tests?

A: I would say that it doesn’t make sense to have no tests, no assessments. It doesn’t make sense to spend the money we spend on education ($12000 per student each year) and then not bother to see if it has had an effect. But nobody tests as much as the United States, and it seems quite overdone. This is a legacy of the No Child Left Behind bill, which had punitive measures for schools based on assessments embedded into law.

From my perspective, education in this country goes between extremes, and right now we are undeniably on the extreme with respect to testing. But I’d like to be clear that standards don’t cause testing.

Q: OK, but it’s undeniable that CC makes testing easier, do you agree?

A: Yes, and isn’t that a good thing? Having common standards also makes good testing easier. I’d also argue that they make it possible to spend less money on testing, and to make testing more centered on what you actually want. It puts more, not less, power into the hands of the consumers of the test. And that’s a good thing.

A word about testing companies. There’s no question that testing companies are trying to grab their share of money for tests. But before they could get paid for 50 different tests based on 50 different standards. What’s better?

There are two new assessment consortia, groups of states which are developing common assessments based on the standards. The consortia will have more power in the marketplace than individual states had.

I believe that people are conflating two separate issues which I’d like to separate. First, do we do a good job of choosing tests? Second, do the CCSS make that worse?

I believe that the CCSS have the power to make things better, although it’s possible that nobody will take advantage of the “commonness” in CCSS. And I’m not saying I’m not worried – the assessment consortia might do a good job but they might fail or get caught up in politics. The campaign for teacher accountability is causing fear and anger. I think you are right to be suspicious of VAM, for example. But that’s not caused by CCSS. Having common standards gives us power if we use it.

I’d also like to make the point that having common standards helps gives power to small players in curriculum publishing. When 50 different states had 50 different standards, the big publishing players with huge sales forces were able to send people to every state and adapt books to different standards. But now we will have smallish companies able to make something work and prove their worth in Tennessee and then sell it in California or wherever.

Q: What would you say to the people who might say that we don’t need more tests, we need to address poverty?

A: I’d say that having good standards can help.

Look, we need a good education system and to eliminate poverty. And having good common standards helps that second goal as well as the first. Why do I say that? A lot of what is good about the CCSS is that they are pretty focused, whereas many of the complaints about the old state standards were that they had tended to be “mile wide inch deep,” meaning having laundry lists of skills which were overall unfocused and incoherent.

We wanted to make something focused, which translated into having fewer things per grade level and doing them right, and making the overall standards a progression which tells a story that makes sense. Good standards, as I believe the CCSS represent, help everybody by providing clear guidance, which particularly helps struggling students and poor schools with less than ideal conditions.

Q: Do CC standards make teachers passive? Are they sufficiently flexible?

A: I don’t even get that.

Here’s the thing, standards are not curriculum.

Curriculum is what teachers actually follow in the classroom. We’ve always had standards, so what changed? Why are we suddenly worried about this new concept which isn’t new at all?

Here’s a legitimate fear: regimented, overly-prescriptive curricula that tell you what to do every day, like in France. Fair enough. But standards don’t say you need to have that. They just say what we want students to learn. It’s true that an overzealous implementation of standards could make teachers passive.

Maybe what’s new is that previously most people ignored their state standards and now people are actually paying attention. But that still doesn’t imply boring or rigid curricula.

Q: Are the CCSS “alive”?

A: How living do we want CC standards to be? Countries like Singapore revise standards on a 10-year cycle. After all we don’t want it to move too quickly, since we need to have time to implement stuff. In fact I’d argue that instability has been a big problem: there’s always a new fad, a new thing, and people never get a chance to figure out what we’re doing with what we’ve got. We should study what works and what doesn’t. And of course there should be revisions when that makes sense.

Q: Is there anything else you’d like to say?

A: Two things. First, I’d like to stress that people are conflating CCSS and testing, and they’re also conflating CCSS and curriculum. It’d be nice for people to separate their issues.

And one last thing. We as a country don’t understand common anything. We don’t see advantages of standards. Think about how useful it is to have standards, though. My recent project is a website called Illustrative Mathematics, which could not exist without standards. it’s a national community of teachers figuring out what they need to know – across state lines. That’s neat, and it’s only one of many benefits of having shared standards.

Categories: math education

Researching the Common Core

I’m in the middle of researching the Common Core standards for math. So far I’ve watched a Diane Ravitch talk, which I blogged about here, which was interesting but raised more questions than it answered, at least for me.

I’ve also interviewed Bill McCallum this week, who was a lead writer and chair of the Work Team that wrote the Common Core standards for mathematics. I’m still writing up that interview but I should have it done soon.

Next up I plan to interview a long-time teacher and current principal of a Brooklyn-based girls school for math and science, Kiri Soares, on her perspective on the Common Core standards and standardized tests in general.

One thing I can say already for sure: people who are not insiders here conflate a bunch of different issues. I’m hoping to at least separate them and understand where people stand on each issue, and if I at the very least get to the point of agreeing to disagree on well-defined points then I will have done my job.

Tell me if you think I need to go further to fully understand the issues at hand. Of course one thing I’m not doing is delving directly into the content of the standards, and that may very well be essential to understanding them. I’d love your thoughts.

Categories: math education

Guest Post: Beauty, even in the teaching of mathematics

This is a guest post by Manya Raman-Sundström.

Mathematical Beauty

If you talk to a mathematician about what she or he does, pretty soon it will surface that one reason for working those long hours on those difficult problems has to do with beauty.

Whatever we mean by that term, whether it is the way things hang together, or the sheer simplicity of a result found in a jungle of complexity, beauty – or aesthetics more generally—is often cited as one of the main rewards for the work, and in some cases the main motivating factor for doing this work. Indeed, the fact that a proof of known theorem can be published just because it is more elegant is one evidence of this fact.

Mathematics is beautiful. Any mathematician will tell you that. Then why is it that when we teach mathematics we tend not to bring out the beauty? We would consider it odd to teach music via scales and theory without ever giving children a chance to listen to a symphony. So why do we teach mathematics in bits and pieces without exposing students to the real thing, the full aesthetic experience?

Of course there are marvelous teachers out there who do manage to bring out the beauty and excitement and maybe even the depth of mathematics, but aesthetics is not something we tend to value at a curricular level. The new Common Core Standards that most US states have adopted as their curricular blueprint do not mention beauty as a goal. Neither do the curriculum guidelines of most countries, western or eastern (one exception is Korea).

Mathematics teaching is about achievement, not about aesthetic appreciation, a fact that test-makers are probably grateful for – can you imagine the makeover needed for the SAT if we started to try to measure aesthetic appreciation?

Why Does Beauty Matter?

First, it should be a bit troubling that our mathematics classrooms do not mirror practice. How can young people make wise decisions about whether they should continue to study mathematics if they have never really seen mathematics?

Second, to overlook the aesthetic components of mathematical thought might be to preventing our children from developing their intellectual capacities.

In the 1970s Seymour Papert , a well-known mathematician and educator, claimed that scientific thought consisted of three components: cognitive, affective, and aesthetic (for some discussion on aesthetics, see here).

At the time, research in education was almost entirely cognitive. In the last couple decades, the role of affect in thinking has become better understood, and now appears visibly in national curriculum documents. Enjoying mathematics, it turns out, is important for learning it. However, aesthetics is still largely overlooked.

Recently Nathalie Sinclair, of Simon Frasier University, has shown that children can develop aesthetic appreciation, even at a young age, somewhat analogously to mathematicians. But this kind of research is very far, currently, from making an impact on teaching on a broad scale.

Once one starts to take seriously the aesthetic nature of mathematics, one quickly meets some very tough (but quite interesting!) questions. What do we mean by beauty? How do we characterise it? Is beauty subjective, or objective (or neither? or both?) Is beauty something that can be taught, or does it just come to be experienced over time?

These questions, despite their allure, have not been fully explored. Several mathematicians (Hardy, Poincare, Rota) have speculated, but there is no definite answer even on the question of what characterizes beauty.

Example

To see why these questions might be of interest to anyone but hard-core philosophers, let’s look at an example. Consider the famous question, answered supposedly by Gauss, of the sum of the first n integers. Think about your favorite proof of this. Probably the proof that did NOT come to your mind first was a proof by induction:

Prove that S(n) = 1 + 2 + 3 … + n = n (n+1) /2

S(k + 1) = S(k) + (k + 1)

= k(k + 1)/2 + 2(k + 1)/2

= k(k + 1)/2 + 2(k + 1)/2

= (k + 1)(k + 2)/2.

Now compare this proof to another well known one. I will give the picture and leave the details to you:

Screen Shot 2014-02-04 at 6.53.05 AM

Does one of these strike you as nicer, or more explanatory, or perhaps even more beautiful than the other? My guess is that you will find the second one more appealing once you see that it is two sequences put together, giving an area of n (n+1), so S(n) = n (n+1)/2.

Note: another nice proof of this theorem, of course, is the one where S(n) is written both forwards and backwards and added. That proof also involves a visual component, as well as an algebraic one. See here for this and a few other proofs.

Beauty vs. Explanation

How often do we, as teachers, stop and think about the aesthetic merits of a proof? What is it, exactly, that makes the explanatory proof more attractive? In what way does the presentation of the proof make the key ideas accessible, and does this accessibility affect our sense of understanding, and what underpins the feeling that one has found exactly the right proof or exactly the right picture or exactly the right argument?

Beauty and explanation, while not obvious related (see here), might at least be bed-fellows. It may be the case that what lies at the bottom of explanation — a feeling of understanding, or a sense that one can ”see” what is going on — is also related to the aesthetic rewards we get when we find a particularly good solution.

Perhaps our minds are drawn to what is easiest to grasp, which brings us back to central questions of teaching and learning: how do we best present mathematics in a way that makes it understandable, clear, and perhaps even beautiful? These questions might all be related.

Workshop on Math Beauty

This March 10-12, 2014 in Umeå, Sweden, a group will gather to discuss this topic. Specifically, we will look at the question of whether mathematical beauty has anything to do with mathematical explanation. And if so, whether the two might have anything to do with visualization.

If this discussion peaks your interest at all, you are welcome to check out my blog on math beauty. There you will find a link to the workshop, with a fantastic lineup of philosophers, mathematicians, and mathematics educators who will come together to try to make some progress on these hard questions.

Thanks to Cathy, the always fabulous mathbabe, for letting me take up her space to share the news of this workshop (and perhaps get someone out there excited about this research area). Perhaps she, or you if you have read this far, would be willing to share your own favorite examples of beautiful mathematics. Some examples have already been collected here, please add yours.

Diane Ravitch speaks in Westchester

One thing I learned on the “Public Facing Math” panel at the JMM was that I needed to know more about the Common Core, since so much of the audience was very interested in discussing it and since it was actually a huge factor in the public’s perception of math, both in the sense of high school math curriculum and in the context of the associated mathematical models related to assessments. In fact at that panel I promised to learned more about the Common Core and I urged other mathematicians in the room to do the same.

As part of my research I listened to a recent lecture that Diane Ravitch gave in Westchester which centered on the Common Core. The video of the lecture is available here.

Diane Ravitch

If you don’t know anything about Diane Ravitch, you should. She’s got a super interesting history in education – she’s an education historian – and in particular has worked high up, as the U.S. Assistant Secretary of Education and on the National Assessment Governing Board, which supervises the National Assessment of Educational Progress.

What’s most interesting about her is that, as a high ranking person in education, she originally supported the Bush “No Child Left Behind” policy but now is an outspoken opponent of it as well as Obama’s “Race to the Top“, which she claims in an extension of the same bad idea.

Ravitch writes an incredibly interesting blog on education issues and, what’s most interesting to me, assessment issues.

Ravitch in Westchester

Let me summarize her remarks in a free-form and incomplete way. If you want to know exactly what she said and how she said it, watch the video, and feel free to skip the first 16 minutes of introductions.

She doesn’t like the Common Core initiative and mentions that Gates Foundation people, mostly not experienced educators, and many of them associated to the testing industry, developed the Common Core standards. So there’s a suspicion right off the bat that the material is overly academic and unrealistic for actual teachers in actual classrooms.

She also objects to the idea of any fixed and untested set of standards. No standard is perfect, and this one is rigid. At the very least, if we need a “one solution for all” kind of standard, it needs to be under constant review and testing and open to revisions – a living document to change with the times and with the needs and limits of classrooms.

So now we have an unrealistic and rigid set of standards, written by outsiders with vested interests, and it’s all for the sake of being able to test everyone to death. She also made some remarks about the crappiness of the Value-Added Model similar to stuff I’ve mentioned in the past.

The Common Core initiative, she explains, exposes an underlying and incorrect mindset, which is that testing makes kids learn, and more testing makes kids learn faster. That setting a high bar makes kids suddenly be able to jump higher. The Common Core, she says, is that higher bar. But just because you raise standards doesn’t mean people suddenly know more.

In fact, she got a leaked copy of last year’s Common Core test and saw that it’s 5th grade version is similar to a current 8th grade standardized test. So it’s very much this “raise the bar” setup. And it points to the fact that standardized testing is used as punishment rather than diagnostic.

In other words, if we were interested in finding out who needs help and giving them help, we wouldn’t need harder and harder tests, we’d just look at who is struggling with the current tests and go help them. But because it’s all about punishment, we need to add causality and blame to the environment.

She claims that poverty causes kids to underperform in schools, and blaming the teachers on poverty is a huge distraction and meaningless for those kids. In fact, she asks, what are going to happen to all of those kids who fail the Common Core standards? What is going to become of them if we don’t allow them to graduate? And how do we think we are helping them? Why do we spend so much time with developing these fancy tests and on assessments instead of figuring out how to help them graduate?

She also points out that the blame game going on in this country is fueled by bad facts.

For example, there is no actual educational emergency in this country. In fact, test scores and graduation rates have never been higher for each racial group. And, although we are alway made to be afraid vis a vis our “international competition” (great recent example of this here) we actually historically never scored at the top of international rankings. But we didn’t think that meant we weren’t competitive 50 years ago, so why do we suddenly care now?

She provides the answer. Namely, if people are convinced there is an emergency in education, then the private companies – test prep and testing companies as well as companies that run charter school – stand to make big money from our response and from straight up privatization.

The statistical argument that poverty causes educational delays is ready to be made. If we want to “fix our educational system” then we need to address poverty, not scapegoat teachers.

Categories: math education, modeling

How do you define success for a calculus MOOC?

I’m going to strike now, while the conversational iron is hot, and ask people to define success for a calculus MOOC.

Why?

I’ve already mostly explained why in this recent post, but just in case you missed it, I think mathematics is being threatened by calculus MOOCs, and although maybe in some possibly futures this wouldn’t be a bad thing, in others it definitely would.

One way it could be a really truly bad thing is if the metric of success were as perverted as we’ve seen happen in high school teaching, where Value-Added Models have no defined metric of success and are tearing up a generation of students and teachers, creating the kind of opaque, confusing, and threatening environment where code errors lead to people getting fired.

And yes, it’s kind of weird to define success in a systematic way given that calculus has been taught in a lot of places for a long time without such a rigid concept. And it’s quite possible that flexibility should be built in to the definition, so as to acknowledge that different contexts need different outcomes.

Let’s keep things as complicated as they need to be to get things right!

The problem with large-scale models is that they are easier to build if you have some fixed definition of success against which to optimize. And if we mathematicians don’t get busy thinking this through, my fear is that administrations will do it for us, and will come up with things based strictly on money and not so much on pedagogy.

So what should we try?

Here’s what I consider to be a critical idea to get started:

  • Calculus teachers should start experimenting with teaching calculus in different ways. Do randomized experiments with different calculus sections that meet at comparable times (I say comparable because I’ve noticed that people who show up for 8am sections are typically more motivated students, so don’t pit them against 4pm sections).
  • Try out a bunch of different possible definitions of success, including the experience and attitude of the students and the teacher.
  • So for example, it could be how students perform on the final, which should be consistent for both sections (although to do that fairly you need to make sure the MOOC you’re using covers the critical material to do the final).
  • Or it could be partly an oral exam or long-form written exam, whether students have learned to discuss the concepts (keeping in mind that we have to compare the “MOOC” students to the standardly taught students).
  • Design the questions you will ask your students and yourself before the semester begins so as to practice good model design – we don’t want to decide on our metric after the fact. A great way to do this is to keep a blog with your plan carefully described – that will timestamp the plan and allow others to comment.
  • Of course there’s more than one way to incorporate MOOCs in the curriculum, so I’d suggest more than one experiment.
  • And of course the success of the experiment will also depend on the teaching style of the calc prof.
  • Finally, share your results with the world so we can all start thinking in terms of what works and for whom.

One last comment. One might complain that, if we do this, we’re actually speeding on our own deaths by accelerating the MOOCs in the classroom. But I think it’s important we take control before someone else does.

Categories: math education, modeling

If it’s hocus pocus then it’s not math

A few days ago there was a kerfuffle over this “numberphile” video, which was blogged about in Slate here by Phil Plait in his “Bad Astronomy” column, with a followup post here with an apology and a great quote from my friend Jordan Ellenberg.

The original video is hideous and should never have gotten attention in the first place. I say that not because the subject couldn’t have been done well – it could have, for sure – but because it was done so poorly that it ends up being destructive to the public’s most basic understanding of math and in particular positive versus negative numbers. My least favorite line from the crappy video:

I was trying to come up with an intuitive reason for this I and I just couldn’t. You have to do the mathematical hocus pocus to see it.

What??

Anything that is hocus pocus isn’t actually math. And people who don’t understand that shouldn’t be making math videos for public consumption, especially ones that have MSRI’s logo on them and get written up in Slate. Yuck!

I’m not going to just vent about the cultural context, though, I’m going to mention what the actual mathematical object of study was in this video. Namely, it’s an argument that “prove” that we have the following identity:

1 + 2 + 3 + 4 + \dots = - \frac{1}{12}.

Wait, how can that be? Isn’t the left hand side positive and the right hand side negative?!

This mathematical argument is familiar to me – in fact it is very much along the lines of stuff we sometimes cover at the math summer program HCSSiM I teach at sometimes (see my notes from 2012 here). But in the case of HCSSiM, we do it quite differently. Specifically, we use it as a demonstration of flawed mathematical thinking. Then we take note and make sure we’re more careful in the future.

If you watch the video, you will see the flaw almost immediately. Namely, it starts with the question of what the value is of the infinite sum

1 -1 + 1 -1 + \dots.

But here’s the thing, that doesn’t actually have a value. That is, it doesn’t have a value until you assign it a value, which you can do but then you might want to absolutely positively must explain how you’ve done so. Instead of that explanation, the guy in the video just acts like it’s obvious and uses that “fact,” along with a bunch of super careless moving around of terms in infinite sums, to infer the above outrageous identity.

To be clear, sometimes infinite sums do have pretty intuitive and reasonable values (even though you should be careful to acknowledge that they too are assigned rather than “true”). For example, any geometric series where each successive term gets smaller has an actual “converging sum”. The most canonical example of this is the following:

1/2 + 1/4 + 1/8 + \dots + 1/2^k + \dots = 1.

What’s nice about this sum is that it is naively plausible. Our intuition from elementary school is corroborated when we think about eating half a cake, then another quarter, and then half of what’s left, and so on, and it makes sense to us that, if we did that forever (or if we did that increasingly quickly) we’d end up eating the whole cake.

This concept has a name, and it’s convergence, and it jibes with our sense of what would happen “if we kept doing stuff forever (again at possibly increasing speed).” The amounts we’ve measured on the way to forever are called partial sums, and we make sure they converge to the answer. In the example above the partial sums are 1/2, 3/4, 7/8, and so on, and they definitely converge to 1.

There’s a mathematical way of defining convergence of series like this that the geometric series follows but that the 1-1+1-1 \dots series does not. Namely, you guess the answer, and to make sure you’ve got the right one, you make sure that all of the partial sums are very very close to that answer if you go far enough, for any definition of “very very close.”

So if you want it to get within 0.00001, there’s a number N so that, after the Nth partial sum, all partial sums are within 0.00001 of the answer. And so on.

Notice that if you take the partial sums of the 1-1+1-1 \dots series you get the sequence 1, 0, 1, 0,1,0,1, \dots, which doesn’t get closer and closer to anything. That’s another way of saying that there is no naively plausible value for this infinite sum.

As for the first infinite sum we came across, the 1 +2 + 3 + 4 +\dots, that does have a naively plausible value, which we call “infinity.” Totally cool and satisfying to your intuition that you worked so hard to achieve in high school.

But here’s the thing. Mathematicians are pretty clever, so they haven’t stopped there, and they’ve assigned a value to the infinite sum 1-1+1-1 \dots in spite of these pesky intuition issues, namely \frac{1}{2}, and in a weird mathematical universe of their construction, which is wildly useful in some contexts, that value is internally consistent with other crazy-ass things. One of those other crazy-ass things is the original identity 1 + 2 + 3 + 4 + \dots = - \frac{1}{12}.

[Note: what would be really cool is if a mathematician made a video explaining the crazy-ass universe and why it's useful and in what contexts. This might be hard and it's not my expertise but I for one would love to watch that video.]

That doesn’t mean the identity is “true” in any intuitively plausible sense of the word. It means that mathematicians are scrappy.

Now here’s my last point, and it’s the only place I disagree somewhat (I think) with Jordan in his tweets. Namely, I really do think that the intuitive definition is qualitatively different from what I’ve termed the “crazy-ass” definition. Maybe not in a context where you’re talking to other mathematicians, and everyone is sufficiently sophisticated to know what’s going on, but definitely in the context of explaining math to the public where you can rely on number sense and (hopefully!) a strong intuition that positive numbers can’t suddenly become negative numbers.

Specifically, if you can’t make any sense of it, intuitive or otherwise, and if you have to ascribe it to “mathematical hocus pocus,” then you’re definitely doing something wrong. Please stop.

Categories: math, math education, rant

The coming Calculus MOOC Revolution and the end of math research

I don’t usually like to sound like a doomsayer but today I’m going to make an exception. I’m going to describe an effect that I believe will be present, even if it’s not as strong as I am suggesting it might be. There are three points to my post today.

1) Math research is a byproduct of calculus teaching

I’ve said it before, calculus (and pre-calculus, and linear algebra) might be a thorn in many math teachers’ side, and boring to teach over and over again, but it’s the bread and butter of math departments. I’ve heard statistics that 85% of students who take any class in math at a given college take only calculus.

Math research is essentially funded through these teaching jobs. This is less true for the super elite institutions which might have their own army of calculus adjuncts and have separate sources of funding both from NSF-like entities and private entities, but if you take the group of people I just saw at JMM you have a bunch of people who essentially depend on their take-home salary to do research, and their take-home salary depends on lots of students at their school taking service courses.

I wish I had a graph comparing the number of student enrolled in calculus each year versus the number of papers published in math journals each year. That would be a great graphic to have, and I think it would make my point.

2) Calculus MOOCs and other web tools are going to start replacing calculus teaching very soon and at a large scale

It’s already happening at Penn through Coursera. Word on the street is it is about to happen at MIT through EdX.

If this isn’t feasible right now it will be soon. Right now the average calculus class might be better than the best MOOC, especially if you consider asking questions and getting a human response. But as the calculus version of math overflow springs into existence with a record of every question and every answer provided, it will become less and less important to have a Ph.D. mathematician present.

Which isn’t to say we won’t need a person at all – we might well need someone. But chances are they won’t be tenured, and chances are they could be overseas in a call center.

This is not really a bad thing in theory, at least for the students, as long as they actually learn the stuff (as compared to now). Once the appropriate tools have been written and deployed and populated, the students may be better off and happier. They will very likely be more adept at finding correct answers for their calculus questions online, which may be a way of evaluating success (although not mine).

It’s called progress, and machines have been doing it for more than a hundred years, replacing skilled craftspeople. It hurts at first but then the world adjusts. And after all, lots of people complain now about teaching boring classes, and they will get relief. But then again many of them will have to find other jobs.

Colleges might take a hit from parents about how expensive they are and how they’re just getting the kids to learn via computer. And maybe they will actually lower tuition, but my guess is they’ll come up with something else they are offering that makes up for it which will have nothing to do with the math department.

3) Math researchers will be severely reduced if nothing is done

Let’s put those two things together, and what we see is that math research, which we’ve basically been getting for free all this time, as a byproduct of calculus, will be severely curtailed. Not at the small elite institutions that don’t mind paying for it, but at the rest of the country. That’s a lot of research. In terms of scale, my guess is that the average faculty will be reduced by more than 50%, and some faculties will be closed altogether.

Why isn’t anything being done? Why do mathematicians seem so asleep at this wheel? Why aren’t they making the case that math research is vital to a long-term functioning society?

My theory is that mathematicians haven’t been promoting their work for the simple reason that they haven’t had to, because they had this cash cow called calculus which many of them aren’t even aware of as a great thing (because close up it’s often a pain).

It’s possible that mathematicians don’t even know how to promote math to the general public, at least right now. But I’m thinking that’s going to change. We’re going to think about it pretty hard and learn how to promote math research very soon, or else we’re going back to 1850 levels of math research, where everyone knew each other and stuff was done by letter.

How worried am I about this?

For my friends with tenure, not so worried, except if their entire department is at risk. But for my younger friends who are interested in going to grad school now, I’m not writing them letters of recommendation before having this talk, because they’ll be looking around for tenured positions in about 10 years, and that’s the time scale at which I think math departments will be shrinking instead of expanding.

In terms of math PR, I’m also pretty worried, but not hopeless. I think one can really make the case that basic math research should be supported and expanded, but it’s going to take a lot of things going right and a lot of people willing to put time and organizing skills into the effort for it to work. And hopefully it will be a community effort and not controlled by a few billionaires.

Categories: math, math education, musing

JMM

It occurs to me, as I prepare to join my panel this afternoon on Public Facing Math, that I’ve been to more Joint Math Meetings in the 7 years since I left academic math (3) than I did in the 17 years I was actually in math (2). I include my undergraduate years in that count because when I was a junior in college I went to Vancouver for the JMM and I met Cora Sadosky, which was probably my favorite conference ever.

Anyhoo I’m on my way to one of the highlights of any JMM, the HCSSiM breakfast, where we hang out with students and teachers from summers long ago and where I do my best to convince the director Kelly and myself that I should come back next summer to teach again. Then after that I spend 4 months at home convincing my family that it’s a great plan. Woohoo!

yp

Besides the above plan, I plan to meet people in the hallways and gossip. That’s all I have ever accomplished here. I hope it is the official mission of the conference, but I’m not sure.

Categories: math, math education

How do I know if I’m good enough to go into math?

Hi Cathy,

I met you this past summer, you may not remember me. I have a question.

I know a lot of people who know much more math than I do and who figure out solutions to problems more quickly than me. Whenever I come up with a solution to a problem that I’m really proud of and that I worked really hard on, they talk about how they’ve seen that problem before and all the stuff they know about it. How do I know if I’m good enough to go into math?

Thanks,

High School Kid

Dear High School Kid,

Great question, and I’m glad I can answer it, because I had almost the same experience when I was in high school and I didn’t have anyone to ask. And if you don’t mind, I’m going to answer it to anyone who reads my blog, just in case there are other young people wondering this, and especially girls, but of course not only girls.

Here’s the thing. There’s always someone faster than you. And it feels bad, especially when you feel slow, and especially when that person cares about being fast, because all of a sudden, in your confusion about all sort of things, speed seems important. But it’s not a race. Mathematics is patient and doesn’t mind. Think of it, your slowness, or lack of quickness, as a style thing but not as a shortcoming.

Why style? Over the years I’ve found that slow mathematicians have a different thing to offer than fast mathematicians, although there are exceptions (Bjorn Poonen comes to mind, who is fast but thinks things through like a slow mathematician. Love that guy). I totally didn’t define this but I think it’s true, and other mathematicians, weigh in please.

One thing that’s incredibly annoying about this concept of “fastness” when it comes to solving math problems is that, as a high school kid, you’re surrounded by math competitions, which all kind of suck. They make it seem like, to be “good” at math, you have to be fast. That’s really just not true once you grow up and start doing grownup math.

In reality, mostly of being good at math is really about how much you want to spend your time doing math. And I guess it’s true that if you’re slower you have to want to spend more time doing math, but if you love doing math then that’s totally fine. Plus, thinking about things overnight always helps me. So sleeping about math counts as time spent doing math.

[As an aside, I have figured things out so often in my sleep that it's become my preferred way of working on problems. I often wonder if there's a "math part" of my brain which I don't have normal access to but which furiously works on questions during the night. That is, if I've spent the requisite time during the day trying to figure it out. In any case, when it works, I wake up the next morning just simply knowing the proof and it actually seems obvious. It's just like magic.]

So here’s my advice to you, high school kid. Ignore your surroundings, ignore the math competitions, and especially ignore the annoying kids who care about doing fast math. They will slowly recede as you go to college and as high school algebra gives way to college algebra and then Galois Theory. As the math gets awesomer, the speed gets slower.

And in terms of your identity, let yourself fancy yourself a mathematician, or an astronaut, or an engineer, or whatever, because you don’t have to know exactly what it’ll be yet. But promise me you’ll take some math major courses, some real ones like Galois Theory (take Galois Theory!) and for goodness sakes don’t close off any options because of some false definition of “good at math” or because some dude (or possibly dudette) needs to care about knowing everything quickly. Believe me, as you know more you will realize more and more how little you know.

One last thing. Math is not a competitive sport. It’s one of the only existing truly crowd-sourced projects of society, and that makes it highly collaborative and community-oriented, even if the awards and prizes and media narratives  about “precocious geniuses” would have you believing the opposite. And once again, it’s been around a long time and is patient to be added to by you when you have the love and time and will to do so.

Love,
Cathy

Categories: math, math education

The case against algebra II

There’s an interesting debate described in this essay, Wrong Answer: the case against Algebra II, by Nicholson Baker (hat tip Nicholas Evangelos) around the requirement of algebra II to go to college. I’ll do my best to summarize the positions briefly. I’m making some of the pro-side up since it wasn’t well-articulated in the article.

On the pro-algebra side, we have the argument that learning algebra II promotes abstract thinking. It’s the first time you go from thinking about ratios of integers to ratios of polynomial functions, and where you consider the geometric properties of these generalized fractions. It is a convenient litmus test for even more abstraction: sure, it’s kind of abstract, but on the other hand you can also for the most part draw pictures of what’s going on, to keep things concrete. In that sense you might see it as a launching pad for the world of truly abstract geometric concepts.

Plus, doing well in algebra II is a signal for doing well in college and in later life. Plus, if we remove it as a requirement we might as well admit we’re dumbing down college: we’re giving the message that you can be a college graduate even if you can’t do math beyond adding fractions. And if that’s what college means, why have college? What happened to standards? And how is this preparing our young people to be competitive on a national or international scale?

On the anti-algebra side, we see a lot of empathy for struggling and suffering students. We see that raising so-called standards only gives them more suffering but no more understanding or clarity. And although we’re not sure if that’s because the subject is taught badly or because the subject is inherently unappealing or unattainable, it’s clear that wishful thinking won’t close this gap.

Plus, of course doing well in algebra II is a signal for doing well in college, it’s a freaking prerequisite for going to college. We might as well have embroidery as a prerequisite and then be impressed by all the beautiful piano stool covers that result. Finally, the standards aren’t going up just because we’re training a new generation in how to game a standardized test in an abstract rote-memorization skill of formulas and rules. It’s more like learning student’s capacity for drudgery.

OK, so now I’m going to make comments.

While it’s certainly true that, in the best of situations, the content of algebra II promotes abstract and logical thinking, it’s easy for me to believe, based on my very small experience in the matter that, it’s much more often taught poorly, and the students are expected to memorize formulas and rules. This makes it easier to test but doesn’t add to anyone’s love for math, including people who actually love math.

Speaking of my experience, it’s an important issue. Keep in mind that asking the population of mathematicians what they think of removing a high school class is asking for trouble. This is a group of people who pretty much across the board didn’t have any problems whatsoever with the class in question and sailed through it, possibly with a teacher dedicated to teaching honors students. They likely can’t remember much about their experience, and if they can it probably wasn’t bad.

Plus, removing a math requirement, any math requirement, will seem to a mathematician like an indictment of their field as not as important as it used to be to the world, which is always a bad thing. In other words, even if someone’s job isn’t directly on the line with this issue of algebra II, which it undoubtedly is for thousands of math teachers and college teachers, then even so it’s got a slippery slope feel, and pretty soon we’re going to have math departments shrinking over this.

In other words, it shouldn’t surprised anyone that we have defensive and unsympathetic mathematicians on one side who cannot understand the arguments of the empathizers on the other hand.

Of course, it’s always a difficult decision to remove a requirement. It’s much easier to make the case for a new one than to take one away, except of course for the students who have to work for the ensuing credentials.

And another thing, not so long ago we’d hear people say that women don’t need education at all, or that peasants don’t need to know how to read. Saying that a basic math course should become and elective kind of smells like that too if you want to get histrionic about things.

For myself, I’m willing to get rid of all of it, all the math classes ever taught, at least as a thought experiment, and then put shit back that we think actually adds value. So I still think we all need to know our multiplication tables and basic arithmetic, and even basic algebra so we can deal with an unknown or two. But from then on it’s all up in the air. Abstract reasoning is great, but it can be done in context just as well as in geometry class.

And, coming as I now do from data science, I don’t see why statistics is never taught in high school (at least in mine it wasn’t, please correct me if I’m wrong). It seems pretty clear we can chuck trigonometry out the window, and focus on getting the average high school student up to the point of scientific literacy that she can read a paper in a medical journal and understand what the experiment was and what the results mean. Or at the very least be able to read media reports of the studies and have some sense of statistical significance. That’d be a pretty cool goal, to get people to be able to read the newspaper.

So sure, get rid of algebra II, but don’t stop there. Think about what is actually useful and interesting and mathematical and see if we can’t improve things beyond just removing one crappy class.

Categories: math education, statistics
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