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Suddenly the dog ran into the cafe to steal a piece of bread.

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My daughter was born on Monday, May 13, 1989.

“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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We went for a walk on the beach, the sunset was really beautiful.

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I love my dog, I fed her a treat.

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Marian never could have admitted it to herself, she really admired her rival's skill in the competition.

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. 

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

You are taking too long, and there's no way we will ever finish the project before the due date.

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