Month: June 2010

A major study of recent international data on school mathematics performance casts doubt on some common assumptions about gender and math achievement — in particular, the idea that girls and women have less ability due to a difference in biology.

“We tested some recently proposed hypotheses that try to explain a supposed gender gap in math performance and found the data did not support them,” says Janet Mertz, senior author of the study and a professor of oncology at the University of Wisconsin-Madison.

Instead, the Wisconsin researchers linked differences in math performance to social and cultural factors.

The new study, by Mertz and Jonathan Kane, a professor of mathematical and computer sciences at the University of Wisconsin-Whitewater, was published in Dec 2011 in Notices of the American Mathematical Society. The study looked at data from 86 countries, which the authors used to test the “greater male variability hypothesis” famously expounded in 2005 by Lawrence Summers, then president of Harvard, as the primary reason for the scarcity of outstanding women mathematicians.

That hypothesis holds that males diverge more from the mean at both ends of the spectrum and, hence, are more represented in the highest-performing sector. But, using the international data, the Wisconsin authors observed that greater male variation in math achievement is not present in some countries, and is mostly due to boys with low scores in some other countries, indicating that it relates much more to culture than to biology.

The new study relied on data from the 2007 Trends in International Mathematics and Science Study and the 2009 Programme in International Student Assessment.

“People have looked at international data sets for many years”, Mertz says. “What has changed is that many more non-Western countries are now participating in these studies, enabling much better cross-cultural analysis.”

The Wisconsin study also debunked the idea proposed by Steven Levitt of “Freakonomics” fame that gender inequity does not hamper girls’ math performance in Muslim countries, where most students attend single-sex schools. Levitt claimed to have disproved a prior conclusion of others that gender inequity limits girls’ mathematics performance. He suggested, instead, that Muslim culture or single-sex classrooms benefit girls’ ability to learn mathematics.

By examining the data in detail, the Wisconsin authors noted other factors at work. “The girls living in some Middle Eastern countries, such as Bahrain and Oman, had, in fact, not scored very well, but their boys had scored even worse, a result found to be unrelated to either Muslim culture or schooling in single-gender classrooms,” says Kane.

He suggests that Bahraini boys may have low average math scores because some attend religious schools whose curricula include little mathematics. Also, some low-performing girls drop out of school, making the tested sample of eighth graders unrepresentative of the whole population.

“For these reasons, we believe it is much more reasonable to attribute differences in math performance primarily to country-specific social factors,” Kane says.

To measure the status of females relative to males within each country, the authors relied on a gender-gap index, which compares the genders in terms of income, education, health and political participation. Relating these indices to math scores, they concluded that math achievement at the low, average and high end for both boys and girls tends to be higher in countries where gender equity is better. In addition, in wealthier countries, women’s participation and salary in the paid labor force was the main factor linked to higher math scores for both genders.

“We found that boys — as well as girls — tend to do better in math when raised in countries where females have better equality, and that’s new and important,” says Kane. “It makes sense that when women are well-educated and earn a good income, the math scores of their children of both genders benefit.”

Mertz adds, “Many folks believe gender equity is a win-lose zero-sum game: If females are given more, males end up with less. Our results indicate that, at least for math achievement, gender equity is a win-win situation.”

U.S. students ranked only 31st on the 2009 Programme in International Student Assessment, below most Western and East-Asian countries. One proposed solution, creating single-sex classrooms, is not supported by the data. Instead, Mertz and Kane recommend increasing the number of math-certified teachers in middle and high schools, decreasing the number of children living in poverty and ensuring gender equality.

“These changes would help give all children an optimal chance to succeed,” says Mertz. “This is not a matter of biology: None of our findings suggest that an innate biological difference between the sexes is the primary reason for a gender gap in math performance at any level. Rather, these major international studies strongly suggest that the math-gender gap, where it occurs, is due to sociocultural factors that differ among countries, and that these factors can be changed.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to University of Wisconsin-Madison

Mathematicians at The University of Manchester have answered the question: How many lottery tickets do you need to buy to guarantee wining something on the U.K. National Lottery?

Focusing on the National Lottery’s flagship game “Lotto,” which draws six random numbersfrom 1 to 59, Dr. David Stewart and Dr. David Cushing found that 27 is the lowest possible number of tickets needed to guarantee a win—although, importantly, with no guarantee of a profit.

They describe the solution using a mathematical system called finite geometry, which centers around a triangle-like structure called a Fano plane. Each point of the structure is plotted with pairs of numbers and connected with lines—each line generates a set of six numbers, which equates to one ticket.

It takes three Fano planes and two triangles to cover all 59 numbers and generate 27 sets of tickets.

Choosing tickets in this way guarantees that no matter which of the 45,057,474 possible draws occurs, at least one of the tickets will have at least two numbers in common. From any draw of six, two numbers must appear on one of the five geometric structures, which ensures they appear on at least one ticket.

But Dr. Stewart and Dr. Cushing say that the hard work is actually showing that achieving the same outcome with 26 tickets is not possible.

Dr. David Stewart, a Reader in Pure Mathematics at The University of Manchester, said, “Fundamentally there is a tension which comes from the fact that there are only 156 entries on 26 tickets. This means a lot of numbers can’t appear a lot of times. Eventually you see that you’ll be able to find six numbers that don’t appear on any ticket together. In graph theory terms, we end up proving the existence of an independent set of size six.”

Although guaranteed a win, the researchers say that the chances of making a profit are very unlikely and shouldn’t be used as a reason to gamble.

The 27 lottery tickets would set you back £54. And Peter Rowlett, a mathematician from The Aperiodical website, has shown that in almost 99% of cases, you wouldn’t make that money back.

When putting the theory to the test in the lottery draw on 1 July 2023; the researchers matched just two balls on three of the tickets, the reward being three lucky dip tries on a subsequent lottery, each of which came to nothing.

The researchers say that the finding is interesting from a computational point of view. They use a fifty-year-old programming language called Prolog, which they say makes it one of the oldest examples of real artificial intelligence.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to University of Manchester

My intention with this article is to give an intuitive and non-technical introduction to the field of evolutionary algorithms, particularly with regards to optimisation.

If I get you interested, I think you’re ready to go down the rabbit hole and simulate evolution on your own computer. If not … well, I’m sure we can still be friends.

Survival of the fittest

According to Charles Darwin, the great evolutionary biologist, the human race owes its existence to the phenomenon of survival of the fittest. And being the fittest doesn’t necessarily mean the biggest physical presence.

Once in high school, my lunchbox was targeted by swooping eagles, and I was reduced to a hapless onlooker. The eagle, though smaller in form, was fitter than me because it could take my lunch and fly away – it knew I couldn’t chase it.

As harsh as it sounds, look around you and you will see many examples of the rule of the jungle – the fitter survive while the rest gradually vanish.

The research area, now broadly referred to as Evolutionary Algorithms, simulates this behaviour on a computer to find the fittest solutions to a number of different classes of problems in science, engineering and economics.

The area in which this area is perhaps most widely used is known as “optimisation”.

Optimisation is everywhere

Your high school maths teacher probably told you the shortest way to go from point A to point B was along the straight line joining A and B. Your mum told you that you should always get the right amount of sleep.

And, if you have lived on your own for any length of time, you’ll be familiar with the ever-increasing cost of living versus the constant income – you always strive to minimise the expenditures, while ensuring you are not malnourished.

Whenever you undertake an activity that seeks to minimise or maximise a well-defined quantity such as distance or the vague notion of the right amount of sleep, you are optimising.

Look around you right now and you’ll see optimisation in play – your Coke can is shaped like that for a reason, a water droplet is spherical for a reason, you wash all your dishes together in the dishwasher for a reason.

Each of these strives to save on something: volume of material of the Coke can, and energy and water, respectively, in the above cases.

So we can safely say optimisation is the act of minimising or maximising a quantity. But that definition misses an important detail: there is always a notion of subject to, or satisfying some conditions.

You must get the right amount of sleep, but you also must do your studies and go for your music lessons. Such conditions, which you also have to adhere to, are known as “constraints”. Optimisation with constraints is then collectively termed “constrained optimisation”.

After constraints comes the notion of “multi-objective optimisation”. You’ll usually have more than one thing to worry about (you must keep your supervisor happy with your work and keep yourself happy and also ensure that you are working on your other projects). In many cases these multiple objectives can be in conflict.

Evolutionary algorithms and optimisation

Imagine your local walking group has arranged a weekend trip for its members and one of the activities is a hill climbing exercise. The problem assigned to your group leader is to identify who among you will reach the hill in the shortest time.

There are two approaches he or she could take to complete this task: ask only one of you to climb up the hill at a time and measure the time needed, or ask all of you to run all at once and see who reaches first.

That second method is known as the “population approach” of solving optimisation problems – and that’s how evolutionary algorithms work. The “population” of solutions are evolved over a number of iterations, with only the fittest solutions making it to the next.

This is analogous to the champion girl from your school making to the next round which was contested among champions from other schools in your state, then your country, and finally winning among all the countries.

Or, in our above scenario, finding who in the walking group reaches the hill top fastest, who would then be denoted as the fittest.

In engineering, optimisation needs are faced at almost every step, so it’s not surprising evolutionary algorithms have been successful in that domain.

Design optimisation of scramjets

At the Multi-disciplinary Design Optimisation Group at the University of New South Wales, my colleagues and I are involved in the design optimisation of scramjets, as part of the SCRAMSPACE program. In this, we’re working with colleagues from the University of Queensland.

Our evolutionary algorithms-based optimisation procedures have been successfully used to obtain the optimal configuration of various components of a scramjet.

Some of these have quite technical names, that in themselves would require quite a bit of explanation but, if you want, you can get a feel for the kind of work we do, and its applications for scramjets, by clicking here.

There are, at the risk of sounding over-zealous, no limits to the application of evolutionary algorithms.

Has this whetted your appetite? Have you learnt something new today?

If so, I’m glad. May the force be with you!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Amit Saha*