Category Archives: MATH

Math

Statistics and the Scientific Method • Bayesian vs Frequentist Analysis of Results

Posted on by

INDEPENDENT LEARNING PROJECT:

  1. Read Darwin “Seeds & Salt Water” Experiment
  2. Read the full article (link below) and research Bayesian vs Frequentist Statistics
    1. Explain why: “Scientists who have learned Bayesian statistics often marvel that it propels them through a different kind of scientific reasoning”
    2. Explain what the author means when he says, “Bayesian calculations go straight for the probability of the hypothesis, factoring in not just the data from the coin-toss experiment but any other relevant information”
    3. Explain what counter-intuitive advice Bayesian statistics would recommend to a “Let’s Make a Deal” game show contestant.
  3. Analyze Results using BOTH Methods

When people think of statistics, they may imagine lists of numbers — batting averages or life-insurance tables.

  • But the current debate is about how scientists turn data into knowledge, evidence and predictions.
  • Concern has been growing in recent years that some fields are not doing a very good job at this sort of inference.

In 2012, for example, a team at the biotech company Amgen announced that they’d analyzed 53 cancer studies and found it could not replicate 47 of them.

Similar follow-up analyses have cast doubt on so many findings in fields such as neuroscience and social science that researchers talk about a “replication crisis”

Some statisticians and scientists are

  • optimistic that Bayesian methods can improve the reliability of research by
  • allowing scientists to crosscheck work done with the more traditional or “classical” approach, known as frequentist statistics.

The two methods approach the same problems from different angles.

The essence of the frequentist technique is to apply probability to data. If you suspect your friend has a weighted coin, for example, and you observe that it came up heads nine times out of 10, a frequentist would calculate the probability of getting such a result with an unweighted coin. The answer (about 1 percent) is not a direct measure of the probability that the coin is weighted; it’s a measure of how improbable the nine-in-10 result is — a piece of information that can be useful in investigating your suspicion.

  • By contrast, Bayesian calculations go straight for the probability of the hypothesis, factoring in not just the data from the coin-toss experiment but any other relevant information — including whether you’ve previously seen your friend use a weighted coin.
  • Scientists who have learned Bayesian statistics often marvel that it propels them through a different kind of scientific reasoning than they’d experienced using classical methods.

“Statistics sounds like this dry, technical subject, but it draws on deep philosophical debates about the nature of reality,” said the Princeton University astrophysicist Edwin Turner, who has witnessed a widespread conversion to Bayesian thinking in his field over the last 15 years.

via The Odds, Continually Updated – NYTimes.com.

Some Universities Crack Code in Drawing Women to Computer Science – NYTimes.com

Posted on by

Claire Cain Miller

One of the reasons so few women work in tech is that few choose to study computer science or engineering. Only 18 percent of computer science graduates in the United States are women, down from 37 percent in 1985.

At a few top college programs, though, that appears to be changing.

At Carnegie Mellon University, 40 percent of incoming freshmen to the School of Computer Science are women, the largest group ever. At the University of Washington, another technology powerhouse, women earned 30 percent of computer science degrees this year. At Harvey Mudd College, 40 percent of computer science majors are women, and this year, women represented more than half of the engineering graduates for the first time.

These examples provide a road map for how colleges can help produce a more diverse group of computer science graduates. They also help answer a controversial question: Does the substance of computer science instruction need to be adjusted to attract women, or does recruitment and mentorship? It’s an important question because tech companies have so many jobs to fill, and because computer science skills have become necessary in almost every other industry, too.

via Some Universities Crack Code in Drawing Women to Computer Science – NYTimes.com.

If the Robots Kill Us, It’s Because It’s Their Job – Patrick Tucker – The Atlantic

Posted on by

The math that explains why that is Omohundro calls the formula for optimal rational decision making. It speaks to the way that any rational being will make decisions in order to maximize rewards and lowest possible cost. It looks like this:

In the above model, A is an action and S is a stimulus that results from that action. In the case of utility function, action and stimulus form a sort of feedback loop. Actions that produce stimuli consistent with fulfilling the program’s primary goal will result in more of that sort of behavior. That will include gaining more resources to do it.

For a sufficiently complex or empowered system, that decision-making would include not allowing itself to be turned off, take, for example, a robot with the primary goal of playing chess.

“When roboticists are asked by nervous onlookers about safety, a common answer is ‘We can always unplug it!’

But imagine this outcome from the chess robot’s point of view,” writes Omohundro. “A future in which it is unplugged is a future in which it cannot play or win any games of chess. This has very low utility and so expected utility maximisation will cause the creation of the instrumental subgoal of preventing itself from being unplugged. If the system believes the roboticist will persist in trying to unplug it, it will be motivated to develop the subgoal of permanently stopping the roboticist,” he writes.

In other words, the more logical the robot, the more likely it is to fight you to the death.

via If the Robots Kill Us, It’s Because It’s Their Job – Patrick Tucker – The Atlantic.

American 15-Year-Olds Lag, Mainly in Math, on International Standardized Tests – NYTimes.com

Posted on by

In the United States, just 9 percent of 15-year-olds scored in the top two levels of proficiency in math, compared with an average of 13 percent among industrialized nations and as high as 55 percent in Shanghai, 40 percent in Singapore, and 17 percent in Germany and Poland.

Jack Buckley, the commissioner of the National Center for Education Statistics, noted that American students from families with incomes in the highest quartile did not perform as well as students with similar backgrounds in other countries.

via American 15-Year-Olds Lag, Mainly in Math, on International Standardized Tests – NYTimes.com.

A Better Way to Teach Math – No Step Too Small to Ignore

Posted on by

April 19, 2011 by questmiddleschool

Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps. Breaking things down this finely allows a teacher to identify the specific point at which a student may need help. “No step is too small to ignore,” Mighton says. “Math is like a ladder. If you miss a step, sometimes you can’t go on. And then you start losing your confidence and then the hierarchies develop. It’s all interconnected.”

Mighton saw that if he approached teaching this way, he could virtually guarantee that every student would experience success. In turn, the children’s math anxiety diminished. As they grew more confident, they grew excited, and they began requesting harder challenges. “More than anything, kids love success,” he says, “and they love getting to higher levels, like in a video game.”

As the children experienced repeated success, it seemed to Mighton that their brains actually began to work more efficiently. Sometimes adding one more drop of knowledge led to a leap in understanding. One day, a child would be struggling; the next day she would solve a problem that was harder than anything she’d previously handled. Mighton saw that if you provided painstaking guidance, children would make their own discoveries. That’s why he calls his approach “guided discovery.”

via A Better Way to Teach Math – NYTimes.com.

OTA Roboticists Kick Bot!!

Posted on by

10 OTA students participating in 3 teams qualified for entry to the next level of robotics competition at the Southern California FLL Championship Tournament on Sunday Dec. 2nd. Representing team Ospreys were  Ethan, Brian, RT & Matthew. Representing the first all-girls robotics team for OTA was team Major Pain consisting of Kiki, Maddy, Lupe & Zoe. Xavier & Asia competed on the last team comprised of students from their local neighborhood. In order to advance Continue reading →