For some people, and not only professional mathematicians, the essence of mathematics lies in its beauty and its intellectual challenge.
For others, including many scientists and engineers, the chief value of mathematics is how it applies to their own work. Because mathematics plays such a central role in modern culture, some basic understanding of the nature of mathematics is requisite for scientific literacy. To achieve this, students need to perceive mathematics as part of the scientific endeavor, comprehend the nature of mathematical thinking, and become familiar with key mathematical ideas and skills.
This chapter focuses on mathematics as part of the scientific endeavor and then on mathematics as a process, or way of thinking. Recommendations related to mathematical ideas are presented in Chapter 9, The Mathematical World, and those on mathematical skills are included in Chapter 12, Habits of Mind. Mathematics is the science of patterns and relationships. As a theoretical discipline, mathematics explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world.
The abstractions can be anything from strings of numbers to geometric figures to sets of equations. In addressing, say, "Does the interval between prime numbers form a pattern? In deriving, for instance, an expression for the change in the surface area of any regular solid as its volume approaches zero, mathematicians have no interest in any correspondence between geometric solids and physical objects in the real world.
A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced.
Mathematicians, like other scientists, are particularly pleased when previously unrelated parts of mathematics are found to be derivable from one another, or from some more general theory. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest elaborateness or complexity but on the contrary, in finding the greatest economy and simplicity of representation and proof. These cross-connections enable insights to be developed into the various parts; together, they strengthen belief in the correctness and underlying unity of the whole structure.
Mathematics is also an applied science. Many mathematicians focus their attention on solving problems that originate in the world of experience. They too search for patterns and relationships, and in the process they use techniques that are similar to those used in doing purely theoretical mathematics. The results were intriguing…. Much like several surveys that had gone before, the scientists found that the average number of sexual partners was actually lower than you might think: around seven for heterosexual women and around 13 for heterosexual men.
But before we start reinforcing any old-fashioned theories about promiscuous men and chaste women, the eagle-eyed among you might question this discrepancy.
By virtue of the fact that there are roughly the same number of heterosexual men and women in the world and that sex has to occur between two people, the average number of partners for both men and women should be the same.
And yet, the difference in male and female averages comes up time and time again in surveys of this kind. There are a few possible explanations for this difference.
Perhaps men are more likely to exaggerate. Perhaps men and women have different definitions of what has to take place to add a partner to their total. But perhaps more significantly, it appears that the way men and women arrive at their number is different. Women tend to count upwards, listing their partners by name.
This does tend to give quite accurate results, but if you forget anyone while counting, you are prone to underestimating your true number of partners. This theory is strengthened when you realise that a surprising number of male answers happen to be divisible by five. Beyond looking at averages, though, the Swedish study also provided the data for a revolutionary finding. In Fredrik Liljeros and a team of mathematicians plotted all of the responses from the Swedish survey on a graph and found a startlingly simple underlying pattern.
The list of 2, responses all lie on a near-perfect curve like that here, showing a clear pattern in the number of partners each participant had admitted to.
Most people had had relatively few sexual partners — which is why the left-hand side of the curve is so high. But there were some responses from people with an extraordinary number of conquests, which is why the right-hand side of the line on the graph never quite reaches zero. If the Swedish survey is representative of the population at large, the curve suggests that there will always be some chance of finding someone with any number of sexual partners, however large.
If you pick a person in the world at random, the chances that they will have had more than x sexual partners is just x -a. The value of a comes directly from the data. If this number were representative of all of us, the chances of someone having more than partners would be 0. The probability drops the higher the numbers go, but the chances of finding someone with more than 1, partners would then be 0. Carl 15 months old looked at the shape-sorter—a plastic drum with 3 holes in the top.
The holes were in the shape of a triangle, a circle and a square. Carl looked at the chunky shapes on the floor. He picked up a triangle. He put it in his month, then banged it on the floor. He touched the edges with his fingers. Then he tried to stuff it in each of the holes of the new toy. It fell inside the triangle hole! Carl reached for another block, a circular one this time….
Math skills are just one part of a larger web of skills that children are developing in the early years—including language skills, physical skills, and social skills.
Each of these skill areas is dependent on and influences the others. Trina 18 months old was stacking blocks. She had put two square blocks on top of one another, then a triangle block on top of that. She discovered that no more blocks would balance on top of the triangle-shaped block. Her physical ability allows her to manipulate the blocks and use her thinking skills to execute her plan to make a tower. She uses her language and social skills as she asks her father for help.
Her effective communication allows Dad to respond and provide the helps she needs further enhancing her social skills as she sees herself as important and a good communicator. This then further builds her thinking skills as she learns how to solve the problem of making the tower taller. The tips below highlight ways that you can help your child learn early math skills by building on their natural curiosity and having fun together.
Note: Most of these tips are designed for older children—ages 2—3. Younger children can be exposed to stories and songs using repetition, rhymes and numbers. Play with shape-sorters.
Talk with your child about each shape—count the sides, describe the colors. Make your own shapes by cutting large shapes out of colored construction paper. Gather together a basket of small toys, shells, pebbles or buttons. Count them with your child. Sort them based on size, color, or what they do i.
With your 3-year-old, begin teaching her the address and phone number of your home. Talk with your child about how each house has a number, and how their house or apartment is one of a series, each with its own number. Notice the sizes of objects in the world around you: That pink pocketbook is the biggest.
The blue pocketbook is the smallest. Under the chair? Even young children can help fill, stir, and pour. While the computer chugged, he clicked open a second window to check his OkCupid inbox.
McKinlay, a lanky year-old with tousled hair, was one of about 40 million Americans looking for romance through websites like Match. He'd sent dozens of cutesy introductory messages to women touted as potential matches by OkCupid's algorithms. Most were ignored; he'd gone on a total of six first dates. On that early morning in June , his compiler crunching out machine code in one window, his forlorn dating profile sitting idle in the other, it dawned on him that he was doing it wrong.
He'd been approaching online matchmaking like any other user. Instead, he realized, he should be dating like a mathematician. OkCupid was founded by Harvard math majors in , and it first caught daters' attention because of its computational approach to matchmaking. Members answer droves of multiple-choice survey questions on everything from politics, religion, and family to love, sex, and smartphones. The closer to percent—mathematical soul mate—the better. But mathematically, McKinlay's compatibility with women in Los Angeles was abysmal.
OkCupid's algorithms use only the questions that both potential matches decide to answer, and the match questions McKinlay had chosen—more or less at random—had proven unpopular. When he scrolled through his matches, fewer than women would appear above the 90 percent compatibility mark. And that was in a city containing some 2 million women approximately 80, of them on OkCupid.
On a site where compatibility equals visibility, he was practically a ghost. He realized he'd have to boost that number. If, through statistical sampling, McKinlay could ascertain which questions mattered to the kind of women he liked, he could construct a new profile that honestly answered those questions and ignored the rest. He could match every woman in LA who might be right for him, and none that weren't.
He then sorted female daters into seven clusters, like "Diverse" and "Mindful," each with distinct characteristics. Maurico Alejo. Even for a mathematician, McKinlay is unusual. Raised in a Boston suburb, he graduated from Middlebury College in with a degree in Chinese.
In August of that year he took a part-time job in New York translating Chinese into English for a company on the 91st floor of the north tower of the World Trade Center. The towers fell five weeks later. McKinlay wasn't due at the office until 2 o'clock that day. He was asleep when the first plane hit the north tower at am. The experience kindled his interest in applied math, ultimately inspiring him to earn a master's and then a PhD in the field.
Now he'd do the same for love. First he'd need data. While his dissertation work continued to run on the side, he set up 12 fake OkCupid accounts and wrote a Python script to manage them. To find the survey answers, he had to do a bit of extra sleuthing. OkCupid lets users see the responses of others, but only to questions they've answered themselves. McKinlay watched with satisfaction as his bots purred along. Then, after about a thousand profiles were collected, he hit his first roadblock.
OkCupid has a system in place to prevent exactly this kind of data harvesting: It can spot rapid-fire use easily. One by one, his bots started getting banned. He turned to his friend Sam Torrisi, a neuroscientist who'd recently taught McKinlay music theory in exchange for advanced math lessons.
Torrisi was also on OkCupid, and he agreed to install spyware on his computer to monitor his use of the site. With the data in hand, McKinlay programmed his bots to simulate Torrisi's click-rates and typing speed.
He brought in a second computer from home and plugged it into the math department's broadband line so it could run uninterrupted 24 hours a day. After three weeks he'd harvested 6 million questions and answers from 20, women all over the country. McKinlay's dissertation was relegated to a side project as he dove into the data.
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