There were differences in income between the regions of the world; but as you can see from Figure 1. Nobody thinks the world is flat today, when it comes to income. Countries are arranged according to GDP per capita from the poorest on the left of the diagram Liberiato the richest on the right Singapore.
There is some overlap among these topics, so I recommend reading the whole page. Ultimately, what are the sources of errors and of misunderstanding? What kinds of biases and erroneous preconceptions do we have?
Two of my favorite historic discoveries are Einstein's discovery of relativity and Cantor's discoveries of some of the most basic rules of infinities. These discoveries are remarkable in that neither involved long, involved, complicated computations. Both are fairly simple, in retrospect, to anyone who has studied them.
But both involved "thinking outside the box" par excellence -- i. As philosopher John Culkin said, "We don't know who discovered water, but we are certain it wasn't a fish. Errors in Communication Some teachers are hostile to questions. That is an error made by teachers.
Teachers, you will be more comfortable in your job if you try to do it well, and don't think of your students as the enemy. This means listening to your students and encouraging their questions.
A teacher who only lectures, and does not encourage questions, might as well be replaced by a book or a movie. To teach effectively, you have to know when your students have understood something and when they haven't; the most efficient way to discover that is to listen to them and to watch their faces.
Perhaps you identify with your brightest students, because they are most able to appreciate the beauty of the ideas you are teaching -- but the other students have greater need of your help, and they have a right to it.
A variant of teacher hostility is teacher arrogance. Actually, most of the errors listed below can be made by teachers, not just by students. However, most teachers are right far more often than their students, so students should exercise great caution when considering whether their teachers could be in error.
If you're a student with a hostile teacher, then I'm afraid I don't know what advice to give you; transfer to a different section or drop the course altogether if that is feasible.
The remarks on communication in the next few paragraphs are for students whose teachers are receptive to questions. For such students, a common error is that of not asking questions.
When your teacher says something that you don't understand, don't be shy about asking; that's why you're in class! If you've been listening but not understanding, then your question is not a "stupid question.
Think of yourself as their spokesperson; you'll be doing them all a favor if you ask your question. You'll also be doing your teacher a favor -- your teacher doesn't always know which points have been explained clearly enough and which points have not; your questions provide the feedback that your teacher needs.
If you think your teacher may have made a mistake on the chalkboard, you'd be doing the whole class a favor by asking about it.
To save face, just in case the error is your own, formulate it as a question rather than a statement. For instance, instead of saying "that 5 should be a 7", you can ask "should that 5 be a 7?
Don't wait until the very end of the example, or until the end of class. As a teacher, I hate it when class has ended and students are leaving the room and some student comes up to me and says "shouldn't that 5 have been a 7?
Now all your classmates have an error in the notes that they took in class, and they may have trouble deciphering their notes later.
In the early s, I managed a computer retail store. Several of my employees were college students. One bright your man was having difficulty with his Freshman college algebra class.
I tutored him and he did very well, but invariably, he would say, "the professor worked through this problem on the board, and it was nothing like this. I sure hope we got the correct answer. The professor was from the music department, and didn't normally teach college algebra he had been pressed into duty when over enrollment forced the class to be split.
During the class, he picked a problem from the assignment to work out on the board. Very early in the problem, he made an error. I don't recall the specifics, but I'm sure it was one of the many typical algebra errors you list.
Because of the error, he eventually reached a point from which he could no longer proceed. Rather than admitting an error and going to work to find it, he paused staring at the board for several seconds, then turned to the class and said, " The English language was not designed for mathematical clarity.Education rules concerning Texas Essential Knowledge and Skills (curriculum standards) for middle school mathematics.
Feb 22, · Learn how to write the inequality given a graph on a number line. To do this we will need to know the difference between open and closed points as .
What is Bash? Bash is the shell, or command language interpreter, for the GNU operating system. The name is an acronym for the ‘Bourne-Again SHell’, a pun on Stephen Bourne, the author of the direct ancestor of the current Unix shell sh, which appeared in the Seventh Edition Bell Labs Research version of Unix.
Bash is largely compatible with sh and incorporates useful features from the. Transforming media into collaborative spaces with video, voice, and text commenting. Follow us: Share this page: This section covers: Basic Linear Inequalities; Compound Inequalities; Absolute Value and Inequalities; Linear Inequalities with Two Variables.
Write a compound inequality for each graph. 62/87,21 This graph represents an intersection. Both endpoints are closed circles which include the endpoints.