“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."

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

She moved the jar of pickles from their place on the condiment table.

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Each of the performers has an idea for their performance.

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A teacher who fails to grade essays and exams on time can expect to field many complaints from their students at the end of each grading period.

“On the Nature of Belief”

Belief and faith often are critiqued in a scientific culture.  It can seem that mere belief is a replacement for science made available to soothe the ignorant masses.  There is some truth to such accusations, and many people do use belief as a screen to cover their own ignorance about the truths of reality.  Everyone should be aware, however, that almost every single human being have these kinds of “blind spots.”  We all live with many things that we merely believe, all of which are so central to ones world view.

Even if we set aside all such types of beliefs, there still remains a broad terrain of human life in which faith and belief remain—even if we ignore all religious matters whatsoever.  Imagine the scientist who’s work on brain neurons depends upon many discoveries made by many other people.  Yes, if it were possible, it would be better for such a person to know all of the details that they accept merely at the word of other scientists.  In all cases, seeing directly is more fulfilling than merely hearing about something.

However, is the scientist better off when he or she knows only what they have experienced directly.  Although it is preferable that he or she knows such facts. However, it is impossible to investigate everything.  Sometimes, one must extend one’s own vision with the vision of someone else.  In a way, the person who thus “takes it on faith” gains a further vision.

Such faith always relies upon the credibility of the person who shares the experience, of course.  For one person to believe on faith what another person says, it is presupposed that the other person is not a liar and actually could have experienced the matter in question.  This means that even the “scientific believer” must take the risk of placing credibility in someone who has witnessed things that he or she has not seen.  While this does not vindicate every kind of faith that people have had, it does provide a telling sign that faith, as such, is not always the refuge of the ignorant.  Indeed, it is an important part of all of our lives, even the lives of scientists, who’s daring and investigative work is rarely criticized as being a refuge for ignorance!

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

“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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After the unbelievable fame of J.K. Rowling the author of the Harry Potter series hundreds of idealistic authors struggled to achieve similar success. Because of Rowling's legendary accomplishments, we have invited the author, herself to the book signing so that she can share her experience to anyone that wants to listen. Whether you're an amateur writer, expert author, or simply an eager fan, all kinds of people can benefit from her insight. After all, if a person wants to be successful, you have to be willing to listen to the advice of others. If anyone are interested in attending, please contact the office before the end of the month.

Choose the answer that best corrects the bolded portion of this passage. If the bolded portion is correct as written, please select NO CHANGE. 

Over winter break, our AP Spanish class took a trip to South America to practice the language and learn more about different Spanish-speaking cultures.  Before we left, every student had to get their vaccinations for yellow fever and typhoid. Our first stop was Bolivia. When we landed at El Alto airport the highest in the world, we could see the city of La Paz sprawling up the hillsides in the shadow of the Andes.   The first day in La Paz, we went to the Witches’s Market, run by local yatiri. The yatiri are medical practitioners who perform healing rituals in they’re communities. At the market, we saw dried frogs, llama fetuses, herbs, and seeds, used in various rituals. After we visited La Paz, we traveled to Cochabamba.  Cochabamba is Bolivia’s culinary capital, with delicious specialties such as salteñas and rellenos. In Cochabamba, we also climbed 1400 steps to see the Cristo de La Concordia statue, the tallest of it’s kind in South America. Next, we visited the salt flats of Uyuni.  Several lagoons dotted the high desert landscape, some of who were full of flamingos!

On our way out of Bolivia, we stopped at Lake Titicaca on the Peruvian border. Because the lake’s elevation is over 12,000 feet, some of my classmates experienced a bit of altitude sickness.  Despite the altitude sickness, everyone said they loved seeing the beautiful scenery and visiting the mysterious “floating islands” made of reeds.   When we crossed the border into Peru, one of my classmates thought she had lost her passport. I would of had a panic attack! She ended up finding it buried in a pocket of her backpack and we were on our way. 

My favorite part of Peru was our trek to the ancient Incan ruins of Machu Picchu. We spent four days hiking in the Andes on a trail to the ruins. Our guides, whom spoke Spanish, English, and Quechua, were very knowledgeable about the history, flora, and fauna in the region.  Upon waking up the next morning, a beautiful sunrise greeted us and cast the mountains in its soft pink glow. After we explored the ruins, we took a train ride back to Cuzco.  The next day, we flew to Lima and prepared to return home. I was sad to leave, but I know I would be back some day!

Select the answer that best corrects the bold underlined portion of the passage. If the sentence is correct as written, select NO CHANGE.

Over winter break, our AP Spanish class took a trip to South America to practice the language and learn more about different Spanish-speaking cultures.  Before we left, every student had to get their vaccinations for yellow fever and typhoid. Our first stop was Bolivia. When we landed at El Alto airport the highest in the world, we could see the city of La Paz sprawling up the hillsides in the shadow of the Andes.   The first day in La Paz, we went to the Witches’s Market, run by local yatiri. The yatiri are medical practitioners who perform healing rituals in they’re communities. At the market, we saw dried frogs, llama fetuses, herbs, and seeds, used in various rituals. After we visited La Paz, we traveled to Cochabamba.  Cochabamba is Bolivia’s culinary capital, with delicious specialties such as salteñas and rellenos. In Cochabamba, we also climbed 1400 steps to see the Cristo de La Concordia statue, the tallest of it’s kind in South America. Next, we visited the salt flats of Uyuni.  Several lagoons dotted the high desert landscape, some of who were full of flamingos!

On our way out of Bolivia, we stopped at Lake Titicaca on the Peruvian border. Because the lake’s elevation is over 12,000 feet, some of my classmates experienced a bit of altitude sickness.  Despite the altitude sickness, everyone said they loved seeing the beautiful scenery and visiting the mysterious “floating islands” made of reeds.   When we crossed the border into Peru, one of my classmates thought she had lost her passport. I would of had a panic attack! She ended up finding it buried in a pocket of her backpack and we were on our way. 

My favorite part of Peru was our trek to the ancient Incan ruins of Machu Picchu. We spent four days hiking in the Andes on a trail to the ruins. Our guides, whom spoke Spanish, English, and Quechua, were very knowledgeable about the history, flora, and fauna in the region.  Upon waking up the next morning, a beautiful sunrise greeted us and cast the mountains in its soft pink glow. After we explored the ruins, we took a train ride back to Cuzco.  The next day, we flew to Lima and prepared to return home. I was sad to leave, but I know I would be back some day!

Select the answer choice that best corrects the bolded, underlined portion of the passage. If the sentence is correct as written, select NO CHANGE.