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A rational number is any number that can be written as a ratio of two integers (hence the name!). In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers. The term "rational" comes from the word "ratio," because the rational numbers are the ones that can be written in the ratio form p/q where p and q are integers. Irrational, then, just means all the numbers that aren't rational. Every integer is a rational number, since each integer n can be written in the form n/1. For example 5 = 5/1 and thus 5 is a rational number. However, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers. So the set of all rational numbers will contain the numbers 4/5, -8, 1.75 (which is 7/4), -97/3, and so on. Is .999 repeating a rational number? Well, a number is rational if it can be written as A/B (A over B): .3 = 3/10 and .55555..... = 5/9, so these are both rational numbers. Now look at .99999999..... which is equal to 9/9 = 1. We have just written down 1 and .9999999 in the form A/B where A and B are both 9, so 1 and .9999999 are both rational numbers. In fact all repeating decimals like .575757575757... , all integers like 46, and all finite decimals like .472 are rational.

If our ideas seem smaller nowadays, it’s not because we are dumber than our forebears but because we just don’t care as much about ideas as they did. In effect, we are living in an increasingly post-idea world — a world in which big, thought-provoking ideas that can’t instantly be monetized are of so little intrinsic value that fewer people are generating them and fewer outlets are disseminating them, the Internet notwithstanding. Bold ideas are almost passé.

we live in a post-Enlightenment age in which rationality, science, evidence, logical argument and debate have lost the battle in many sectors, and perhaps even in society generally, to superstition, faith, opinion and orthodoxy. While we continue to make giant technological advances, we may be the first generation to have turned back the epochal clock — to have gone backward intellectually from advanced modes of thinking into old modes of belief.

Post-Enlightenment refers to a style of thinking that no longer deploys the techniques of rational thought. Post-idea refers to thinking that is no longer done, regardless of the style.

If you doubt that we live in a winner-take-all economy and that education is the trump card, consider the vast amounts the affluent spend to teach their offspring.

This power spending on the children of the economic elite is usually — and rightly — cited as further evidence of the dangers of rising income inequality.

But it may be that the less lavishly educated children lower down the income distribution aren’t the only losers. Being groomed for the winner-take-all economy starting in nursery school turns out to exact a toll on the children at the top, too.

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patchwork collection of online courses that's difficult for students to navigate.

they can improve their online-education efforts to help students find streamlined, affordable pathways to a degree.

One effort I started in June was to announce a seed grant program to support individual faculty and small teams that wanted to try different ways of teaching their course. So the internal funding helps support students or assistants or web developers or other people to help faculty recraft all or part of their course in order to see if new approaches really help.

here isn't an actual fixed fund. We got about 40 applicants. Maybe 20 of those things we funded. Each one was up to $25,000. I think I'd like to continue that on a quarterly basis. And really, we're new at all this. So the scale of this effort will depend on the faculty input and the outcome of how effective we find this to be.

We now have the ability for individual faculty or programs on campuses to produce appealing online content with relatively low effort and distribute that wide

teenagers would love to socialize face-to-face with their friends. But adult society won’t let them. “Teens aren’t addicted to social media. They’re addicted to each other,” Boyd says. “They’re not allowed to hang out the way you and I did, so they’ve moved it online.”

today’s teens have neither the time nor the freedom to hang out. So their avid migration to social media is a rational response to a crazy situation. They’d rather socialize F2F, so long as it’s unstructured and away from grown-ups.

If you want your kids to learn valuable face-to-face skills, conquer your own irrational fears and give them more freedom. They want the same face-to-face intimacy you grew up with.

First, students tend to lose interest in whatever they’re learning. As motivation to get good grades goes up, motivation to explore ideas tends to go down. Second, students try to avoid challenging tasks whenever possible. More difficult assignments, after all, would be seen as an impediment to getting a top grade. Finally, the quality of students’ thinking is less impressive. One study after another shows that creativity and even long-term recall of facts are adversely affected by the use of traditional grades.

Unhappily, assessment is sometimes driven by entirely different objectives--for example, to motivate students (with grades used as carrots and sticks to coerce them into working harder) or to sort students (the point being not to help everyone learn but to figure out who is better than whom)

Standardized tests often have the additional disadvantages of being (a) produced and scored far away from the classroom, (b) multiple choice in design (so students can’t generate answers or explain their thinking), (c) timed (so speed matters more than thoughtfulness) and (d) administered on a one-shot, high-anxiety basis.

Leaders who lose their way are not bad people; rather, they lose their moral bearings

we all have the capacity for actions we deeply regret unless we stay grounded.

Self-reflection: a path to leadership development

So what we're seeing is less of a notion of a big idea carried through and much more little bursts and snippets. And we see that across media, across film, across, in Web sites, this idea of just do a little bit and then you can run away.

anytime you switch from one task to another, there's something called the "task switch cost," which basically, imagine, is I've got to turn off this part of the brain and turn on this part of the brain. And it's not free; it takes time.

One of the biggest delusions we hear from students is, "I do five things at once because I don't have time to do them one at a time." And that turns out to be false. That is to say, they would actually be quicker if they did one thing, then the next thing, then the next. It may not be as fun, but they'd be more efficient.

most of us would like to ascribe authority to sites and authors who support our conclusions and deny authority to publications that disagree with our worldview

Wikipedia’s guidelines for determining the reliability of publications. These guidelines were developed to help people with diametrically opposed positions argue in rational ways about the reliability of sources using common criteria.

defined by process, aim, and expertise.