“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.

To what is the underlined "this" referring?

“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 sovling 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!

To what does the underlined section refer?

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

Delving into the topic of quantum physics with a great deal of enthusiasm, the research team went on to make a number of startling claims about the relationship between time and space.

“Bach and German Hymnody”

[31] The great German composer, Johann Sebastian Bach was a member of a prodigiously talented musical family. [32] A significant number of Johann Sebastians ancestors and descendants were musicians of various levels of talent. [33] Although he was known for music of a variety of forms, one of his most enduring legacies is the repertoire of chorale music by which he improved and solidified the world of German congregational singing.

During the period after the Protestant Reformation, a number of hymns were written for use in the common worship of the Christians of the time. [34] The structured form of these compositions were well suited for congregational singing although they would sound strange to our contemporary ears. [35] The harmonies and meters of these hymns are very close in character to the music with which we are familiar. They lack some of the standard structural elements that we take for granted in this form of organized [36] Western music, these small differences would stand out to our sensibilities.

It was the great glory of Johann Sebastian Bach to have harmonized a great number of these hymns, often penning multiple such harmonies. [37] One solitary single melody might be made by Bach into three, four, or even five different harmonic compositions. [38] This amazing feat of musical prowess is no small addition to the original behest of Lutheran hymns that Bach had inherited from his fellow religious brethren. [39] Many of these harmonies remain to this day as classic renditions of these songs. [40] They are sung not only in the German world but in Protestant and Catholic services. [41] Indeed they are even sung in wholly secular concerts thanks to their great beauty!

Interestingly enough, Bach was more recognized as a talented performer, rather than composer, of music in his own lifetime. While he was still composing, knowledge of Bach’s work was limited to music connoisseurs who happened to be physically near places [42] he lived and worked. It was not until the early 19^th century, when the first biography of Bach was published, that academic and popular interest [43] truly picked up steam. In the two centuries that followed, his works have continued to proliferate in both religious and purely musical contexts.

This great diffusion of one mans’ work is a testament to his prodigious talent. [44] It also stands as a testament to the fact that Bach’s work came at a pivotal time when the Protestant hymnody was crystallizing, as well as when Western harmonies were coming into a particular expression that is known as the Baroque. [45]

Choose the answer that best corrects section [38].