“Mathematics and Learning”

What subject should be learned first?  The question rightly troubles anyone who’s interest is in education.  Of course, young children often must learn in a very basic and rote fashion, applying their apt memorization skills to simple tasks that will serve them very well in later years when they go one to apply such knowledge to more complex topics.  However, when the time comes to designing curricula, an important question must be answered for older students, namely “What is most important first topic in these students’s education?”

An argument can be made for the use of mathematics as a tool for teaching students how to reason more clearly.  This is not because mathematics is the basis of all knowledge.  Indeed not.  There are many important subjects including not only the humanities like poetry and history but sciences like biology and physiology too.   These topics are not strictly speaking mathematical in nature, even though mathematics can be used in it in many ways.

Our minds are best geared for learning things that we can sense, things that are visible and tangible.  Although mathematics is abstract, it can begin with this kind of sense derived experience.  Beginning with simple everyday examples, children can be taught the more abstract and difficult skills that must be learned for the sake of the development of mathematical skills.  In the process of learning these topics, the children will begin to learn important rules about reasoning.  He or she will learn how several propositions can serve as the basis for conclusions.  They will learn how certain properties are related to various geometric figures and arithmetical rules.  Although much of this will be memorized at first, with time, they will have the opportunity to see that human reasoning in mathematical subjects is orderly and logical.  On the basis of such “logical experience,” young learners can then begin to be taught the rules of logic that they have been using all along.  As the medievals used to say, they could go from logica utens, logic used in other subjects, to logica docens logic taught, as a unique, and separate subject.

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“Intellectual Virtues”

Whenever someone talks about being “virtuous,” we immediately think of someone whose very moral.  Perhaps we even think of people who are a bit boring for virtuous people can appear to have no fun at least in the popular imagination.  Whatever the case might be, almost any reader would be surprised to see the expression “intellectual virtues.”  What could this expression mean to designate!  At best, most people would say, “Such virtues must describe people for who knowledge is combined with devotion and rigorous discipline.”  That is; they would seem to describe the person who has a disciplined character in addition to being intelligent.

However, in ancient and medieval philosophy, certain intellectual capacities were considered virtues.  These character traits were not quite the same as moral character traits or virtues.  To understand this idea, it can be helpful to consider two example people, one whose skills are the fruit of a so-called intellectual virtue and the other whose skills are not.

It is easier to start with the person who does not have a given intellectual virtue.  We all know someone who is not very good at math, that is, someone for who math is difficult even though he or she might be quite skilled at many other tasks   It makes sense to say that this person doesn’t have an intellectual virtue.  Likewise, think of the person who is only able to memorize formulas.  Such a person is often very good at working through many problems with deft skill.  This person seems to be a “wiz” at geometry and algebra, quickly solving equations and proofs. 

However, this latter person might suddenly be presented with a difficult, new problem.  When we notice that he or she does not have the creative skill and insight to solve the problem, we realize that he or she does’nt have a so-called “intellectual virtue.”  This person merely has a habit—a particular skill that is helpful but does not indicate true and complete mathematical knowledge.  The person who is able to understand the mathematics and creatively apply this knowledge to solve new problems.  This person has a true intellectual virtue.  They have a particular ability for intellectual insight, able to probe the difficult domain of this topic.  This is much more noble as the mere habit of being able to balance equations and repeat facts about geometric figures!

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"Lincoln as a Child" by Caleb Zimmerman (2013)

 Abraham Lincoln's forefathers were pioneers. People that left their homes to open up the wilderness and make the way clear for others to follow them. For one hundred and seventy years, ever since the first Lincoln came from England to Massachusetts in 1638, he had been moving slowly westward as new settlements were made in the forest. They faced solitude, privation, and all the dangers and hardships that beset those who take up their homes where only beasts and wild men have had homes before; but they continued to press steadily forward, though they lost fortune and sometimes even life itself in their westward progress.

Back in Pennsylvania and New Jersey, some of the Lincolns had been people of wealth and influence. In Kentucky, where the future President was born on February 12, 1809, his parents live in deep poverty. Their home was a small log cabin of the rudest kind, and nothing seemed more unlikely than that their child, coming into the world in such humble surroundings, was destined to be the greatest man of his time and true to his heritage, he also was to be a pioneer—not into new woods and unexplored fields like his ancestors, but a pioneer of a nobler and grander sort, directing the thoughts of people ever toward the right, and leading the American people, through difficulties and dangers and a mighty war, to peace and freedom.

Choose the answer that best corrects the bolded portion of the passage. If the bolded portion is correct as written, choose "NO CHANGE."

“What is Leisure?”

It would likely surprise modern readers if he or she were told that the meaning of life is leisure. This seems to be the same thing as saying that the meaning of life is nothing more than relaxing by the side of the pool. One can imagine almost anyone thinking to himself, “What a preposterous idea.” This idea is not as foreign as it might appear at first glance. Indeed, it could be considered the classic Western position about the very meaning of life. 

Of course, we need to understand what is meant by the word leisure if we are going to understand this assertion. The best way to understand this is to consider a contrast between two different kinds of activity. On the one hand, there are many activities that are for the sake of something else. On the other hand, there are those activities that are done for their own sake. These latter kinds of activities are those that are properly understood as being leisurely. For example, when someone cuts onions for a meal, the cutting of the onions is not done for their own sake. The easiest way to figure out if something is being done for the sake of another end is to ask, “Why are you doing that.” The onion cutter will answer, “I am doing it so that I can make dinner.” This helps us to see that the cutting of the onions is for the sake of something else. Indeed, even the eating of dinner is for the sake of something else, namely attaining adequate nutrition.

In contrast to these examples, leisurely activities are those that are done for they’re own sake. For example, ask someone who enjoys organ music why he is listening to an organ concert. The answer will almost certainly be, “For the sake of listening to organ music. It is that important and beautiful!” Many things can be considered leisurely, including forms of games that are played merely for their own enjoyment; however, it is important to bear in mind that there is a hierarchy of goods. Some types of leisure are better than others and likely are more preferable.

How should the underlined selection be changed?

“What is Leisure?”

It would likely surprise modern readers if he or she were told that the meaning of life is leisure. This seems to be the same thing as saying that the meaning of life is nothing more than relaxing by the side of the pool. One can imagine almost anyone thinking to himself, “What a preposterous idea.” This idea is not as foreign as it might appear at first glance. Indeed, it could be considered the classic Western position about the very meaning of life. 

Of course, we need to understand what is meant by the word leisure if we are going to understand this assertion. The best way to understand this is to consider a contrast between two different kinds of activity. On the one hand, there are many activities that are for the sake of something else. On the other hand, there are those activities that are done for their own sake. These latter kinds of activities are those that are properly understood as being leisurely. For example, when someone cuts onions for a meal, the cutting of the onions is not done for their own sake. The easiest way to figure out if something is being done for the sake of another end is to ask, “Why are you doing that.” The onion cutter will answer, “I am doing it so that I can make dinner.” This helps us to see that the cutting of the onions is for the sake of something else. Indeed, even the eating of dinner is for the sake of something else, namely attaining adequate nutrition.

In contrast to these examples, leisurely activities are those that are done for they’re own sake. For example, ask someone who enjoys organ music why he is listening to an organ concert. The answer will almost certainly be, “For the sake of listening to organ music. It is that important and beautiful!” Many things can be considered leisurely, including forms of games that are played merely for their own enjoyment; however, it is important to bear in mind that there is a hierarchy of goods. Some types of leisure are better than others and likely are more preferable.

How should the underlined selection be changed?

Choose the answer that best corrects the underlined portion of the sentence. If the underlined portion is correct as written, choose "NO CHANGE."

"Well, I wouldn't necessarily go so far as to say that!" he calmly replied.

Josh suffered from a fear of dogs ever since he is a little boy. He would shiver in his boots thinking of a dog barking at him, biting him, or that could chase him. He had never met a single dog he liked. The scarier dog he had ever seen was Mr. Green’s dog, whom he had to pass every day. On his way to school. This dog barked so loudly it made Josh’s hairs stand on end.

She said that there was no need to be afraid. Josh’s mom told him that dogs only barked because they were excited to see him. She told him that he just needed to be confident and not act afraid of this dogs’ barking, and speak in a reassuring tone. Then the dog would quiet down and act more friendly.

Josh felt that neither confidence or a reassuring tone would make any difference. He didn’t think he would ever get over his fear. However, one day he had enough. Barking for hours, he realized the dog was bored and just needed a friend. He wasn’t sure if his mother’s advice would prove true? Hiding his fear, he approached the dog and spoke to it in a reassuring tone. To his surprise, the dog wagged it’s tail and quieted down right away. 

Choose the answer that best corrects the underlined portion of the sentence. If the underlined portion is correct as written, choose "NO CHANGE."

Choose the answer that best corrects the underlined portion of the sentence. If the underlined portion is correct as written, select “NO CHANGE.”

"What do you think you are doing!" Vanessa screamed at the top of her lungs.

 Are you trying to stick to a budget? Using coupons for purchases, also known as “couponing” is a great way to save money on groceries. Coupons are a little piece of paper that can give you a discount on what you buy. You will be amazed at the great bargains and amazing savings you can get!

It’s easy to get started. When you open up your daily newspaper, one might find a glossy insert full of coupons. Some of the coupons will be for things you don’t buy, some will be for things you buy all the time. Go through the coupons and chop out the ones you can use.

The key to successful couponing is getting multiple copies of coupon circulars. Ask your friends, your neighbors, and family if they have any extras. Some coupon users even go through the recycling at their office to find more coupons! Have you ever heard of such a thing in your life! Completely devoted, these circulars help coupon users to get even more savings.

Couponing might sound like hard work, but for many people, it’s also a hobby. Not only does it help them save hundreds of dollars per year, but also it gives them a fun challenge every time they do their shopping. 

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Choose the answer that best corrects the underlined portion of the text. If the underlined portion is correct as written, select “NO CHANGE.”

I wonder when I will reach the next level in my video game? I wouldn't be surprised if it happens pretty soon.