“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.

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

“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.

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

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

After learning that the client was on a tight budget which (he hadn't known at first), Joe made a revised plan for the project.

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I'm going to start a brand new blog focusing on the Civil War (in addition to my blog about the Revolutionary War).

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Jane made so many good friends during her time at university (but more importantly, she also obtained her degree). She would go on to use her degree to get a banking job.

“Justice and Parents”

We tend to think of justice as a matter of strict equality.  For example if someone wants to buy an item, they are understandably expected to pay an amount that is roughly equal to its value.  Likewise, when a law declares that the penalty for speeding is $150, it is considered just that one who breaks this law pays the fine.  However, justice can also pertain to matters that are beyond mere equality.

An obvious example of this is the case of the relationship between children and their parents.  Unlike the cases discussed earlier, children will have had little opportunity to repay they’re parents for all that they have done for them.  Technically speaking, strict “equality” would require the child to give birth to the parents.  This is an absurd thing to suggest.  Similarly absurd is the suggestion that children should directly repay the rearing offered by parents.  Once again, strict equality cannot occur in the child-parent relationship.

However, there is a kind of justice between children and his or her parents.  For example, when a parent ages and is in need of assistance, if the adult child can provide assistance but refuses to do so.  This is a kind of injustice to the parents.  On a more positive note.  For example, an adult child might send his parents on a cruise in memory of their anniversary.  Such an action would be a kind of “justice” done to the parents—at least in that it recognizes the immense debt that is owed by the child to the parents in question.

In these (and many other cases), justice is less about equality than it is about paying what one can pay.  That is, in such cases, both justice or injustice is determined by worthiness that surpasses mere equality.  This kind of justice could be called justice above justice.  That is, it is a kind of justice that surpasses the “normal” justice of equality.  Such acts of justice ensure that society does not become reduced to the cold calculation of equality but, instead, has the warmth of true affection and loving gratitude.

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“What is Leisure?” 

It would likely surprise [1] 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. [2] 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, [3] it could be considered the classic Western position about the very meaning of life. [4] This, of course, lies in contrast to all other global positions, but that’s a topic for another day. 

[5] 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. [6] On the one hand, there are many activities that are for the sake of something else. [7] 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. [8] 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.” [9] 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. [10] Indeed, even the eating of dinner is for the sake of something else, namely attaining adequate nutrition. 

[11] In contrast to these examples, leisurely activities are those that are done for they’re own sake. [12] 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. [13] Some types of leisure are better than others and likely are more preferable.

Choose the answer that best corrects section [9].