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.
One of the components that I use in my class is student presentations. While I like having students present, I had always a hard time grading the presentations. Plus, many students seemed to target the presentation to me, trying to sound too technical and ...
"...The stray object in the path that might once have immobilised a robot won't immobilise it if the robot can ring up tech support and informed that it's just a stray object, go around dummy...[...].... It's very hard to design a machine that can improvise when confronted by the unfamiliar or reason its way through most difficulties—just as it's rare to find a human who can seamlessly navigate his way across all of America's public roads, large and small, without some sort of guide. But just as any regular joe with access to Wikipedia can do a passable impression of someone with enormous intellectual powers, the extended mind of the cloud could lead to impressive improvements in robot capabilities."
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...
A school wanted to ask students if they have ever taken drugs (yes or no). The survey was anonymous, but school was still worried that students will not report drug use.
So they introduced randomness to help. Each student was asked to flip a coin (privately) before answering. Heads, the student would have to say "yes". Tails, they would have to say the truth.
So, students could safely select “yes, I have taken drugs” and even if personally identified, the answer could be justified as the coin telling them to do so.
If no one had been taking drugs, 50% of the final result would be positive for drug use (those who got heads), and 50% would be negative (those who got tails).
In practice, it was something closer to 60%-40%, which meant about 20% of students had been taking drugs.
Did you know you can import tables available online directly into ? That can be done using the ImportHTML function on Google Spreadsheets and will save you a lot of time. The image below shows how to do it (source goo.gl/19mojE).
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- Computer Technology Institute, Patras, Greece
- Columbia University, New York, NY, USA1999 - 2004
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- University of Patras, Greece
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Big Data, Stupid Decisions / Strata Jumpstart 2011 / Panos Ipeiroti...
Strata Jumpstart, New York, September 19 2011 Big Data , Stupid Decisions: The Importance Of Measuring What We Should Be Measuring Video
Humans Plus Computers Equals Better Crowdsourcing - BusinessWeek
Greek-born computer scientist Panagiotis Ipeirotis is developing technology that gets computers to help people work smarter, and vice versa