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### Panos Ipeirotis

Shared publicly -I don't know how to do get bibtex from the browser but this works on the command line:

curl -LH "Accept: application/x-bibtex" http://dx.doi.org/10.1007/s11083-012-9252-6

Here is the magic output:

@article{Dorais_2012,

doi = {10.1007/s11083-012-9252-6},

url = {http://dx.doi.org/10.1007/s11083-012-9252-6},

year = 2012,

month = {mar},

publisher = {Springer Science $\mathplus$ Business Media},

volume = {30},

number = {2},

pages = {415--426},

author = {Fran{\c{c}}ois Gilbert Dorais and Steven Gubkin and Daniel McDonald and Manuel Rivera},

title = {Automorphism Groups of Countably Categorical Linear Orders are Extremely Amenable},

journal = {Order}

}

http://www.crosscite.org/cn/

### Panos Ipeirotis

Shared publicly -**Demographics of Mechanical Turk: Now Live! (April 2015 edition)**

One of the most common question that I receive is whether I have new data about the demographics of Mechanical Turk workers. The latest data that I had collected were back in 2010, and it was not clear how things have changed since then. The key problem was...

### Panos Ipeirotis

Shared publicly -### Panos Ipeirotis

Shared publicly -**An API for MTurk Demographics**

A few months back, I launched demographics.mturk-tracker.com , a tool that runs continuously surveys of the Mechanical Turk worker population and displays live statistics about gender, age, income, country of origin, etc. Of course, there are many other rep...

### Panos Ipeirotis

Shared publicly -**Postdoc Position for Quality Control in Crowdsourcing**

The Center for Data Science at NYU invites applications for a post-doctoral fellowship in statistical methodology relating to evaluating rater quality for a new research program in the application of crowdsourcing ratings of human speech production. Duties ...

### Panos Ipeirotis

Shared publicly -**The World Bank Report on Online Labor**

I am often asked about statistics and data about the global population of "crowdsourcing" workers, going beyond Mechanical Turk. I am happy to say that from now on I will be able to point everyone to a study from The World Bank, which I was fortunate to par...

### Panos Ipeirotis

Shared publicly -### Panos Ipeirotis

Shared publicly -### Panos Ipeirotis

Shared publicly -I began by asking her something like "If f(x) = x^2 and g(y) = y+1, then what is g(f(x))?" I thought I was making things clearer by saying that g was a function of y, so that one could substitute y=x^2 rather than substituting x=x^2. This, however, was a mistake and led to her making statements like, "Well, y must be x^2 -1," which I couldn't really do much about given that I couldn't talk about quantifiers.

Actually, I tried to talk about quantifiers without explicitly mentioning them, by saying things like "f takes any number and squares it, while g adds 1." But it didn't really help. When I gave up and said "f(x) = x^2 and g(x) = x+1," she was no longer confused, even though in some sense she ought to have been confused.

Well, I say she wasn't, but then a new problem emerged, which was that she consistently composed functions the wrong way round. So I'd ask her what g(f(x)) was when g(x)=x+1 and f(x)=x^2 and she would say (x+1)^2. I tried hard to think what could possibly be going on in her mind, which was difficult when I find the notation g(f(x)) utterly transparent:

*obviously*you rewrite f(x) as x^2 to get g(x^2), and then since g(x) is x+1, g(x^2) must be x^2+1. But somehow she wasn't seeing it like that.

Writing this, I now think that perhaps she read the g and thought "OK, that gives me x+1," then read the f and thought "That's x^2, so I must square the x+1," ending up with (x+1)^2. In other words, she was simply doing the functions in the order they were written. So she wasn't reading g(f(x)) as "Do g to f(x)". Rather, she was reading it as "Do g and then f to x".

At some point in the conversation I discovered something that suddenly shed light on the situation. When I was her age, if I had been told that f(x) = sin(x+30) and had then been asked to work out f(10) on a calculator, I would have had to type in 1 0 + 3 0 = SIN. Similarly, if I had had to work out exp(sqrt(log 20)) I would have had to type in 2 0 LN SQRT EXP. But she had been issued with a calculator where you simply type in the expression as it is written on the page. So for those examples, she would have typed SIN ( 1 0 + 3 0 ) = and EXP ( SQRT ( LN 2 0 ) ) =. The result: without being conscious of it, I was internalizing the way functions worked, every time I used my calculator, while she could simply switch off her brain and copy expressions directly from the page, with no need to consider what they meant. This calculator, by the way, is the standard one that everyone in the country taking the exam is supposed to use.

The end of the story is that she did in the end get the idea and did her functions questions without any problem. So this post is not about her but about the way she, and presumably hundreds of thousands of others, have been taught mathematics.

- New York UniversityAssociate Professor, 2004 - present
- Computer Technology Institute, Patras, Greece
- Columbia University, New York, NY, USA1999 - 2004
- Microsoft Research, Cambridge, UK

Phone | +1-212-998-0803 |

- panos@stern.nyu.edu
| |

Address | 44 West 4th Street, Ste 8-84New York, NY 10012 |

- Columbia University, New York, NY, USA
- University of Patras, Greece

- New York: Old and New (current)

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