Archive for the ‘math education’ Category

Educational feedback loops in China and the U.S.

Today I want to discuss a recent review in New York Review of Books, on a new book entitled Who’s Afraid of the Big Bad Dragon? Why China Has the Best (and Worst) Education System in the World by Yong Zhao (hat tip Alex). The review was written by Diane Ravitch, an outspoken critic of No Child Left Behind, Race To The Top, and the Common Core.

You should read the review, it’s well written and convincing, at least to me. I’ve been studying these issues and devoted a large chunk of my book to the feedback loops described as they’ve played out in this country. Here are the steps I see, which are largely reflected in Ravitch’s review:

  1. Politicians get outraged about a growing “achievement gap” (whereby richer or whiter students get better test scores than poorer or browner students) and/or a “lack of international competitiveness” (whereby students in countries like China get higher international standardized test scores than U.S. students).
  2. The current president decides to “get tough on education,” which translates into new technology and way more standardized tests.
  3. The underlying message is that teachers and students and possibly parents are lazy and need to be “held accountable” to improve test scores. The even deeper assumption is that test scores are the way to measure quality of learning.
  4. Once there’s lots of attention being given to test scores, lots of things start happening in response (the “feedback loop”).
  5. For example, widespread cheating by students and teachers and principals, especially when teachers and principals get paid based on test performance.
  6. Also, well-off students get more and better test prep, so the achievement gap gets wider.
  7. Even just the test scores themselves lead to segregation by class: parents who can afford it move to towns with “better schools,” measured by test scores.
  8. International competitiveness doesn’t improve. But we’ve actually never been highly ranked since we started measuring this.

What Zhao’s book adds to this is how much worse it all is in China. Especially the cheating. My favorite excerpt from the book:

Teachers guess possible [test] items, companies sell answers and wireless cheating devices to students, and students engage in all sorts of elaborate cheating. In 2013, a riot broke out because a group of students in Hubei Province were stopped from executing the cheating scheme their parents purchased to ease their college entrance exam.

Ravitch adds after that that ‘an angry mob of two thousand people smashed cars and chanted, “We want fairness. There is no fairness if you do not let us cheat.”’

To be sure, the stakes in China are way higher. Test scores are incredibly important and allow people to have certain careers. But according to Zhao, this selection process, which is quite old, has stifled creativity in the Chinese educational system (so, in other words, test scores are the wrong way to measure learning, in part because of the feedback loop). He blames the obsession with test scores on the fact that no Chinese native has received a Nobel Prize since 1949, for example: the winners of that selection process are not naturally creative.

Furthermore, Zhao claims, the Chinese educational system stifles individuality and forces conformity. It is an authoritarian tool.

In that light, I guess we should be proud that our international scores are lower than China’s; maybe it is evidence that we’re doing something right.

I know that, as a parent, I am sensitive to these issues. I want my kids to have discipline in some ways, but I don’t want them to learn to submit themselves to an arbitrary system for no good reason. I like the fact that they question why they should do things like go to bed on time, and exercise regularly, and keep their rooms cleanish, and I encourage their questions, even while I know I’m kind of ruining their chances at happily working in a giant corporation and being a conformist drone.

This parenting style of mine, which I believe is pretty widespread, seems reasonable to me because, at least in my experience, I’ve gotten further by being smart and clever than by being exactly what other people have wanted me to be. And I’m glad I live in a society that rewards quirkiness and individuality.

Student evaluations: very noisy data

I’ve been sent this recent New York Times article by a few people (thanks!). It’s called Grading Teachers, With Data From Class, and it’s about how standardized tests are showing themselves to be inadequate to evaluate teachers, so a Silicon Valley-backed education startup called Panorama is stepping into the mix with a data collection process focused on student evaluations.

Putting aside for now how much this is a play for collecting information about the students themselves, I have a few words to say about the signal which one gets from student evaluations. It’s noisy.

So, for example, I was a calculus teacher at Barnard, teaching students from all over the Columbia University community (so, not just women). I taught the same class two semesters in a row: first in Fall, then in Spring.

Here’s something I noticed. The students in the Fall were young (mostly first semester frosh), eager, smart, and hard-working. They loved me and gave me high marks on all categories, except of course for the few students who just hated math, who would typically give themselves away by saying “I hate math and this class is no different.”

The students in the Spring were older, less eager, probably just as smart, but less hard-working. They didn’t like me or the class. In particular, they didn’t like how I expected them to work hard and challenge themselves. The evaluations came back consistently less excited, with many more people who hated math.

I figured out that many of the students had avoided this class and were taking it for a requirement, didn’t want to be there, and it showed. And the result was that, although my teaching didn’t change remarkably between the two semesters, my evaluations changed considerably.

Was there some way I could have gotten better evaluations from that second group? Absolutely. I could have made the class easier. That class wanted calculus to be cookie-cutter, and didn’t particularly care about the underlying concepts and didn’t want to challenge themselves. The first class, by contrast, had loved those things.

My conclusion is that, once we add “get good student evaluations” to the mix of requirements for our country’s teachers, we are asking for them to conform to their students’ wishes, which aren’t always good. Many of the students in this country don’t like doing homework (in fact most!). Only some of them like to be challenged to think outside their comfort zone. We think teachers should do those things, but by asking them to get good student evaluations we might be preventing them from doing those things. A bad feedback loop would result.

I’m not saying teachers shouldn’t look at student evaluations; far from it, I always did and I found them useful and illuminating, but the data was very noisy. I’d love to see teachers be allowed to see these evaluations without there being punitive consequences.

Guest Post: Bring Back The Slide Rule!

This is a guest post by Gary Cornell, a mathematician, writer, publisher, and recent founder of StemForums.

I was was having a wonderful ramen lunch with the mathbabe and, as is all too common when two broad minded Ph.D.’s in math get together, we started talking about the horrible state math education is in for both advanced high school students and undergraduates.

One amusing thing we discovered pretty quickly is that we had independently come up with the same (radical) solution to at least part of the problem: throw out the traditional sequence which goes through first and second year calculus and replace it with a unified probability, statistics, calculus course where the calculus component was only for the smoothest of functions and moreover the applications of calculus are only to statistics and probability. Not only is everything much more practical and easier to motivate in such a course, students would hopefully learn a skill that is essential nowadays: how to separate out statistically good information from the large amount of statistical crap that is out there.

Of course, the downside is that the (interesting) subtleties that come from the proofs, the study of non-smooth functions and for that matter all the other stuff interesting to prospective physicists like DiffEQ’s would have to be reserved for different courses. (We also were in agreement that Gonick’s beyond wonderful“Cartoon Guide To Statistics” should be required reading for all the students in these courses, but I digress…)

The real point of this blog post is based on what happened next: but first you have to know I’m more or less one generation older than the mathbabe. This meant I was both able and willing to preface my next point with the words: “You know when I was young, in one way students were much better off because…” Now it is well known that using this phrase to preface a discussion often poisons the discussion but occasionally, as I hope in this case, some practices from days gone by ago can if brought back, help solve some of today’s educational problems.

By the way, and apropos of nothing, there is a cure for people prone to too frequent use of this phrase: go quickly to YouTube and repeatedly make them watch Monty Python’s Four Yorkshireman until cured:

Anyway, the point I made was that I am a member of the last generation of students who had to use slide rules. Another good reference is: here. Both these references are great and I recommend them. (The latter being more technical.) For those who have never heard of them, in a nutshell, a slide rule is an analog device that uses logarithms under the hood to do (sufficiently accurate in most cases) approximate multiplication, division, roots etc.

The key point is that using a slide rule requires the user to keep track of the “order of magnitude” of the answers— because slide rules only give you four or so significant digits. This meant students of my generation when taking science and math courses were continuously exposed to order of magnitude calculations and you just couldn’t escape from having to make order of magnitude calculations all the time—students nowadays, not so much. Calculators have made skill at doing order of magnitude calculations (or Fermi calculations as they are often lovingly called) an add-on rather than a base line skill and that is a really bad thing. (Actually my belief that bringing back slide rules would be a good thing goes back a ways: when that when I was a Program Director at the NSF in the 90’s, I actually tried to get someone to submit a proposal which would have been called “On the use of a hand held analog device to improve science and math education!” Didn’t have much luck.)

Anyway, if you want to try a slide rule out, alas, good vintage slide rules have become collectible and so expensive— because baby boomers like me are buying the ones we couldn’t afford when we were in high school – but the nice thing is there are lots of sites like this one which show you how to make your own.

Finally, while I don’t think they will ever be as much fun as using a slide rule, you could still allow calculators in classrooms.

Why? Because it would be trivial to have a mode in the TI calculator or the Casio calculator that all high school students seem to use, called “significant digits only.” With the right kind of problems this mode would require students to do order of magnitude calculations because they would never be able to enter trailing or leading zeroes and we could easily stick them with problems having a lot of them!

But calculators really bug me in classrooms and, so I can’t resist pointing out one last flaw in their omnipresence: it makes students believe in the possibility of ridiculously high precision results in the real world. After all, nothing they are likely to encounter in their work (and certainly not in their lives) will ever need (or even have) 14 digits of accuracy and, more to the point, when you see a high precision result in the real world, it is likely to be totally bogus when examined under the hood.

Love StackOverflow and MathOverflow? Now there’s StemForums!

Everyone I know who codes uses 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 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, 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

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

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