Lucia Bostrom – Why Study Geometry?

Your paper is very well-written on the whole. I think the first paragraph could do a better job of introducing the relations between the different ways of thinking about the mathematics, though (and not just “lay them out” the way you do.) See the markings for a few corrections and suggestions. I think the conclusion is good and, of course, I agree with what you are saying.

Specific Comments

1. For the Greeks generally, and Proclus in particular, the “clearing the eye of the soul” referred to ideas pretty far outside of mathematics itself. The study of Euclidean geometry was often seen as the pathway to philosophy, so it was not just clearing the way for further “mathematical endeavors.”

2. The phrase “knowledge of universals” here seems also to hint at Plato’s concept of eternal, unchanging “ideas” or “forms” behind the objects and relationships in the physical world.

3. As you point out later in the paragraph, Pascal is not really agreeing with the idea that developing logical thinking (at least of the sort used in Euclidean geometry) is that valuable. So you should indicate somehow that Pascal’s ideas are going in a different direction here by saying “however” or something similar. This would be a clearer signal what the main point of this paragraph is going to be.

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 13

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 13 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

13 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 12

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 15

4. (10) Lacks incomplete sentences – 10 Totals: Content and Structure – 94; Style and English Usage – 97.

Catherine Caracciolo

Your essay is a good response to the basic question. I have a question about one thing you said in the first paragraph, though. How is doing mathematics “for its own sake” different from doing mathematics “as busywork.” One might think that those two reasons are almost the same thing, so it would be good to be clearer about the distinction.

1. The title was the Elements right from the beginning, so it sounds a bit strange to say that that is the way people refer to it today.

2. I would disagree here. I think Millay really does mean beauty in the esthetic sense. And it is “beauty bare” because it is beauty in its simplest, purest form. All externals and temporary features are “boiled away,” so to speak, leaving only the most basic ingredients of beauty, which are all the more dazzling for that. Also, be careful of the word “simplistic.” It has a pejorative connotation that I don’t think you mean here. You say something is “simplistic” if it is too simple and omits things that should be there because they are important.

3. Does “having the worst circumstances” mean “with the weakest possible assumptions (hypotheses)?” This was not clear, and the phrase “worst circumstances” has connotations that do not really fit with this discussion.

4. This is an excellent point!

5. It would probably be good to identify the source of this quotation.

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 12

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 14 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 9

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 12

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15 6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 11

Totals: Content and Structure – 96; Style and English Usage – 92 Sarah Cahn – Why Study Geometry?

This is a very good essay on the title question. The only serious criticism I have is that you have not really addressed any of the quotations in the topic description. I think that would have added some interesting dimensions to what you are saying here. For instance, on the basis of the Pascal quote, do think he would have agreed with the idea that the kind of thinking one does in geometry is really that useful in the law? (I think not, for the reason that he would argue it is too tied to clearly stated postulates, and does not take into account the complexities of events and situations in the real world.)

1. Were these casual comments? Or did you do some sort of more formal survey of opinions about geometry as a part of this class in high school? The way you describe this makes it sound like a more formal survey, and if so, it would have been good to mention the circumstances. If not, it would be good to “set up” these comments by saying that you had conversations at several points with classmates, and they shared their opinions with you in this way.

2. This is a good point, but I think it could even have gone closer to the beginning of the paragraph. I have the feeling that this particular rationale for studying geometry is very specifically Catholic and perhaps even specifically a part of the tradition of Jesuit education, so it would be good to place the whole discussion of this aspect of studying geometry in that context.

3. Say “legal profession” here – “Lawyer profession” is not correct English. Also, isn’t it the logical reasoning skills (and not the geometry itself) that is useful to lawyers?

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 8

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 8

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13 4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 10

6. (15) Uses correct punctuation – 10

7. (15) Is free of colloquialisms, cliches, and misused words – 12

Totals – Content and Structure: 91; Style and English Usage: 95.

Daniel Finn – Why Do We Study Geometry?

You are actually very right in saying that (another) reason geometry is so important is that geometric ideas are used all the way through calculus and other more advanced mathematics courses. This is not a reason for teaching geometry to all students, of course, since many students (at least traditionally in the U.S.) never get to those more advanced courses. But it is also a very important reason.

I think your first paragraph is a bit too rushed. It would be good to take some more time to lay out the reasons you will discuss in slightly more detail.

A minor technical point – ordinarily, one would not capitalize “geometry” unless it was the title of a specific course or a book.

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 15

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 13 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 8

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 14

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 14

6. (15) Uses correct punctuation – 14

7. (15) Is free of colloquialisms, cliches, and misused words 12

Totals – Content and Structure: 98; Style and English Usage: 92.

Muhammad Javaid – Why Study Geometry

You have a lot of good ideas and examples here. But the writing is a bit rough. See the comments and the corrections on the paper.

0. Usually, you don’t capitalize the word geometry unless it is the title of a course or a book.

1. Actually, I think the story is trying to make the point that knowledge is good in and of itself and that one should not need any prospect of reward to pursue it. One can disagree with this, of course, but the scornful way Euclid treats the student comes across pretty clearly in the “give the man three obols since he must profit from what he learns.”

2. These are all great examples of the ways the Greeks used geometry to develop knowledge about the physical world and new technologies. I think it is questionable that Euclid could have foreseen all of this. And the quote seems to indicate that he would not have agreed that this was the main reason to study geometry in any case.

3. I think I know what you mean here, but this is a somewhat convoluted sentence (and progress is not a transitive verb – you cannot progress something).

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 13

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

13 c. (10) Transitions between paragraphs are logical – 8

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 16

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 10

4. (10) Lacks incomplete sentences – 9

5. (15) Lacks run-on or overly complicated sentences – 13

6. (15) Uses correct punctuation – 13

7. (15) Is free of colloquialisms, cliches, and misused words – 11

Totals: Content and Structure – 94; Style and English Usage – 82 Kyle Kadlick – Why Study Geometry?

Your paper is well written for the most part and has some good ideas. However, I don’t think you have really appreciated the main point of the quote from Pascal (see comment 1 below). The first paragraph could also say a bit more about the reasons for Euclid’s point of view. I do not think the discussion of the quote from Stobaeus is completely correct either. It is not only a question of ignorance, but also a question of incorrect motivation. Finally I’m not sure about how useful SAT-type problems are for real-world problem solving. They seem pretty artificial to me!

1. You are correct that Pascal is referring to the Euclidean style of deductive mathematics with its apparatus of definitions and postulates. However, the main point that Pascal is trying to make is that this style of thinking can be quite limited if it is taken too far. To paraphrase what Pascal is getting at: This approach to problem-solving does not take into account that situations in the real world humans must deal with are rarely so clear-cut and orderly. So mathematicians who try to rely on this sort of thinking when it does not really apply are frequently drawn into incorrect conclusions. I have to say that from my experience, Pascal is perfectly correct here!

2. This is a good point. It is also one of the main reasons why the de-emphasis of geometry in high school mathematics is a real problem! However, the sentence at the start of the following paragraph does not really fit with what you are saying the rest of the way. For Euclid, the reason to study geometry was definitely not the practical applications. (I’m not sure if that is what you meant to say, but that’s what the sentence seems to be saying.)

3. Sorry, but I have no idea what the “rubber pencil trick” you are referring to is! Explain, please.

4. It might be somewhat more correct to say that Plato’s view was studying geometry can lead one to appreciate the existence of those eternal forms, and hence to the study of philosophy. (This is the element of the propaedeutic in the study of geometry – it is a preparation for the even higher study of philosophy.)

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 11

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 13 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

12 c. (10) Transitions between paragraphs are logical – 8

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15 Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 14

7. (15) Is free of colloquialisms, cliches, and misused words – 13

Totals: Content and Structure – 89; Style and English Usage – 97.

Matt Kasuba – It’s Hip To Be Square

Your essay does a good job of addressing the issues raised by the Stobaeus and Proclus quotations. But I don’t think that it is the “merit” of studying geometry that is really the issue. I would say it has more to do with the motivations or reasons for doing that. There are a few minor corrections of writing issues that I have marked on the paper itself – please take a look, in particular at the difference between “its” (possessive) and “it’s” (contraction for “it is”). I would advise you to avoid the contraction in formal writing like this. It is fine to use that if you are reporting everyday speech, but spell out “it is” otherwise.

1. Your examples of the “barriers caused by the senses” seem more like errors of logic in construction of mathematical proofs to me. When philosophers say this, they are usually thinking of much more concrete examples, such as taking a mirage for an actual object, or mistakenly interpreting the motion of the Sun across the sky as a result of the Sun revolving around a stationary Earth.

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 12

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 13 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 11

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 13

Totals: Content and Structure – 95; Style and English Usage – 94.

Astrid Ludwig – Geometry: Our World

You have certainly grasped most of the point of the story about Euclid and the student from Stobaeus. However, note that the person who asked the question “what shall I get from studying geometry?” was not a slave. He was presumably a reasonably well-to-do person in Alexandria who could afford to spend the time to attend Euclid’s lectures. There is a slave in the story, though. Namely, Euclid calls to his slave and says to give the student who asked the question the insultingly small amount of money. Note that there is an added level of scorn involved here, since it is not even Euclid himself who “forks over” the three obols. It is Euclid’s slave who is asked to do the demeaning money transfer(!)

1. Technical comment: When identifying quotations like this, even though you have only one item in your bibliography, it would still be standard practice to indicate the source of the quote by more than just the page number – for instance, say (Bloom, 206).

2. Careful, cells have much too much complicated interior structure to be represented by geometrical points. Also, the other examples of geometric shapes in nature are only approximately of the shapes you are describing. For example, a tree ring is never a perfect circle. (See the next comment too.)

3. This sounds more like “comprehensive” or “inclusive” than “bare” to me. I think it is closer to Millay’s intention to say that it is “beauty bare” because beauty in geometry is beauty in its simplest, purest form. All externals and temporary features are “boiled away,” so to speak, leaving only the most basic ingredients of beauty, which are all the more dazzling for that (the “nothing, intricately drawn nowhere ... ” also refers to the insubstantial but luminous geometrical relationships).

4. Careful here too – mathematical induction is a very different idea from the sort of inductive reasoning discussed in philosophy or psychology (reasoning from the evidence seen in examples to a general conclusion). From the point of view of those disciplines, in fact, mathematical induction is a purely deductive form of reasoning.

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 11

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 19

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 12

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 14

6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 12

Totals: Content and Structure – 95; Style and English Usage – 93.

John Macomber – Hidden Beauty

In your first paragraph, I am somewhat doubtful that Euclid is really a “rubric for solving real life problems.” Euclidean geometry is much too abstract for that to be possible. Including an example to illustrate Pascal’s point about the insufficiency of pure logic to solve some practical problems was a good idea. See point 2 below, though, for another angle on the specific example you used.

1. Picky comment – while modern students certainly might ask why they need to learn about degree measure of angles, recall that Euclid does not use that idea at all. Indeed, the only specific angles that he uses at all are right angles and straight angles.

2. I think it might be more accurate to say that President Bush and his advisers were arguing that we had to invade Iraq to stop Saddam Hussein from continuing to develop his weapons programs and to eliminate the repressive features of his regime. But of course they were also misinterpreting (or fabricating) a lot of the evidence for the existence of those weapons programs. So it is not clear that they were really doing the sort of reasoning Pascal was talking about. The thing that comes closest to that sort of thinking to me was Defence Secretary Donald Rumsfeld’s approach to the military issues – superior air power, “shock and awe,” and the rest were supposed to eliminate Saddam’s military and then the Iraqi people would welcome us with open arms. Needless to say, it didn’t quite work out that way.

3. I am not sure exactly what you mean by “dynamic” here. Are you thinking of statements that are more involved (or less “obvious”)?

4. Yes, Euclid might have said that you should have just been happy to learn Latin for its own sake, or for the intellectual challenge of learning a differently-structured language.

5. This sentence does not really make sense. You have the connection to the real world in both “places” here. I also suggested a rearrangement of this sentence because the way you have it, “The Elements” should be the possessive “The Elements’.” That would be correct, but awkward. (The same construction also occurs in the last paragraph.)

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 13

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 13 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

14 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15 Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 12

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 12

7. (15) Is free of colloquialisms, cliches, and misused words – 12

Totals: Content and Structure – 95; Style and English Usage – 91.

Andrew Marzo – Why Study Geometry

Your essay goes into a lot of very detailed discussion of some of the philosophical implications of studying geometry and also the way that study supposedly prepares one to think logically in other settings. The discussion of the role of logical thinking on the SAT and other standardized tests seemed to be a bit “off topic” to me, although I grant that there is a connection.

My main commen is that I don’t think you have really appreciated Pascal’s objection to these ideas, though. See comment 2 below for what I mean about this.

1. You are using “math” here in the restricted sense that often comes up in colloquial phrases such as “do the math.” This refers, of course, mostly to simple arithmetic. There is much more to mathematics as an intellectual discipline! So I do not really agree with what you are saying here.

2. You are correct that Pascal is referring to the Euclidean style of deductive mathematics with its apparatus of definitions and postulates. However, the main point that Pascal is trying to make is that this style of thinking can be quite limited if it is taken too far. To paraphrase what Pascal is getting at: This approach to problem-solving does not take into account that situations in the real world humans must deal with are rarely so clear-cut and orderly. So mathematicians and others who try to rely on this sort of “strictly Euclidean” thinking when it does not really apply are frequently drawn into incorrect conclusions. I have to say that from my experience, Pascal is perfectly correct here!

3. Really, there is no question of trying to prove them false. The postulates are to be taken as the starting point for the process of deducing new propositions.

4. This is the idea of the propaedeutic that I mentioned in the paper topic description. For Proclus and other Greek thinkers, the study of geometry can lead the student to the study of deeper issues in philosophy by introducing him or her to the idea of pure forms.

5. But see point 2 above. Isn’t this training ground really much different from the real world that the training is supposed to prepare students for?

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 12

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 13 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 8

4. (10) Is concise, clear, and not too repetitive – 10 5. (15) Has a clearly identifiable conclusion that draws together the ideas presented

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 12

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 12

7. (15) Is free of colloquialisms, cliches, and misused words – 11

Totals: Content and Structure – 93; Style and English Usage – 90 Xuan Ha Nghiem – Why Study Geometry?

Good essay on the “why study geometry question!” There are still a few places where minor rewordings would improve the readability. But I am impressed at how hard you worked to revise and polish this essay. Good work.

A Comment

1. I like this interpretation of what Proclus was getting at in the quotation I put in the paper assignment very much. It was really not just pure intellectual curiosity that led the Greeks to study this subject. There was another side to it as you say – the motivation was to study mathematics because it could provide insight about the natural world.

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 15

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 14

Totals: Content and Structure – 100; Style and English Usage – 97 Kaitlin O’Hagan – Why Study Euclidian Geometry?

Good essay on the way Euclid embodies the pure, logical form of beauty that Millay was getting at in her poem. The first paragraph really only addresses part of what you say later, though, so that could do a better job of setting up the rest of the essay. Your writing is generally very good. I have marked a few corrections and comments on the paper. (Very minor comment – I know the spelling “Euclidian” you use is correct too, but I think “Euclidean” is more common.)

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 15

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 13 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 15

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 19

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 12

Totals: Content and Structure – 98; Style and English Usage – 96.

Max Pettinelli – No Title

I like the fact that you were (I think) pretty honest about your opinions of mathematics here. Your writing is also generally good. See the paper for a few corrections and suggestions for smoothing things out.

1. I don’t quite understand what you meant here. Did you mean something like “process of argumentation?” I agree that there is an element of truth to this. But as a prospective lawyer you should be aware (and you will certainly learn) that there is much more to being a good lawyer than this. As either a prosecutor or a defence attorney, you also have to understand the psychology of jurors, and that is something that is often far from logical. (My brother is a lawyer who worked for years as a DA and he and I have had a lot of interesting conversations about this point.) Many prosecutors, in fact, do not want mathematicians on the juries they have to deal with because they don’t really “get” the idea of “beyond reasonable doubt.” This is, of course, a part of what the Pascal quotation is getting at. The strict Euclidean style of thinking can be quite limited if it is taken too far. To paraphrase what Pascal is getting at: This approach to problem-solving does not take into account that situations in the real world humans must deal with are rarely so clear-cut and orderly. So people who try to rely on this sort of thinking when it does not really apply are frequently drawn into incorrect conclusions. I have to say that from my experience, Pascal is perfectly correct here!

2. Actually, almost all engineers need to use calculus in one way or another.

3. See comment 1 above. I think there is an aspect of Pascal’s ideas that you are not appreciating completely.

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 13

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10 3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of

pronouns) – 12

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 10

6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 12

Totals: Content and Structure – 98; Style and English Usage – 94.

Tom Santa Maria – Why Study Geometry?

Very strong essay on the reasons for studying geometry. There are a few minor punctuation issues that I have marked on the paper, but overall your writing is also very good.

1. Don’t capitalize “honesty” here since it is not the start of a new sentence.

2. Yes, this is the key idea that Pascal is trying to express. The strict Euclidean style of thinking can be quite limited if it is taken too far. To paraphrase what Pascal is getting at: This approach to problem-solving does not take into account that situations in the real world humans must deal with are rarely so clear-cut and orderly. So people (especially mathematicians) who try to rely on this sort of thinking when it does not really apply are frequently drawn into incorrect conclusions. I have to say that from my experience, Pascal is perfectly correct here!

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 15

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 15

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 15

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 13

7. (15) Is free of colloquialisms, cliches, and misused words – 15

Totals: Content and Structure – 100; Style and English Usage – 98.

Kylee Sullivan – Studying Geometry: The Key to Human Survival

Your essay is very good. I’m not entirely sure about the title, though. I think this is overstating (and overdramatizing) things a bit. You are right that we need to understand the environment we live in in order to survive. However, it is not obvious that geometry is the best or most important way to derive that sort of knowledge.

1. You should be aware that the emphasis on philosophy here is probably not all that common. It is a reflection of the traditional structure of education in Jesuit schools. (Up until the late 1960’s, in fact, every student here had effectively to do a double major in a subject like the ones we have now, together with a second collection of required courses in philosophy and theology.)

2. These shapes appear only approximately in the physical world of course. And it is only the trunk of a tree that is approximately cylindrical. If you take the crown (leaves) and the roots into account, the shape is much more “interesting(!)”

3. I am not entirely sure why you said “abstract emotions” here. How did you mean the “abstract?” It would have been good to make the connection here with the quotation from Pascal because, of course, that is exactly what he is saying too. The strict Euclidean style of thinking can be quite limited if it is taken too far. To paraphrase what Pascal is getting at: This approach to problem-solving does not take into account that situations in the real world humans must deal with are rarely so clear-cut and orderly. So people who try to rely on this sort of thinking when it does not really apply are frequently drawn into incorrect conclusions. I have to say that from my experience, Pascal is perfectly correct here!

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 12

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10 3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of

pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 14

Totals: – Content and Structure – 97; Style and English Usage – 97.

Kevin Sutman – Why Study Geometry?

Your essay on the title question is very good. I like that you have drawn on your own experiences (struggles?) with mathematics for this. The writing is also generally very good, although there are a few places where I have marked some corrections and suggestions for rewordings. There is also a place toward the bottom of page 4 where I think that something got mistakenly erased or truncated.

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 15

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 10

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 13

4. (10) Lacks incomplete sentences – 10

5. (15) Lacks run-on or overly complicated sentences – 15

6. (15) Uses correct punctuation – 15

7. (15) Is free of colloquialisms, cliches, and misused words – 14

Totals: Content and Structure – 100; Style and English Usage – 97.

Peter Zona – The Importance of Geometry

1. This is somewhat repetitive since you have already repeated the quote. It is not necessary to go through the story again in this way.

2. This was certainly true in the past. But I think it is becoming less true today. Or, at least, it is becoming more common to mix geometric topics into the high school mathematics curriculum each year without devoting a whole year to geometry as was done in the past. This has the unfortunate effect that students who take this sort of “integrated mathematics curriculum” in high school are not introduced to the idea of proof in mathematics to the same extent as was true in the past.

3. Careful here – Eratosthenes’ calculation gave an estimate for the circumference of the Earth (essentially assuming that it had a spherical shape). This is not actually a proof that the Earth has that shape, though. I think you could also have said a bit more about the meaning of the Pascal quotation.

You are on the right track there, but there is another aspect of this. Pascal is also saying that the strict Euclidean style of thinking can be quite limited if it is taken too far. To paraphrase what Pascal is getting at: This approach to problem-solving does not take into account that situations in the real world humans must deal with are rarely so clear-cut and orderly because they involve things like feelings and attitudes that may not be logical. So people who try to rely on this sort of thinking when it does not really apply are frequently drawn into incorrect conclusions. I have to say that from my experience, Pascal is perfectly correct here!

Content and Structure – The paper:

1. (10) Has a clearly stated topic or question that it will address – 10

2. (10) Does address the stated topic or question – 10

3. (15) Presents a good understanding of the issues raised in the quotations – 13

4. Has a clearly articulated and logical structure:

a. (15) First paragraph lays out the topic or question – 15 b. (15) Later paragraphs have topic sentences and clearly identifiable main ideas –

15 c. (10) Transitions between paragraphs are logical – 10

4. (10) Is concise, clear, and not too repetitive – 8

5. (15) Has a clearly identifiable conclusion that draws together the ideas presented – 15

Style and English Usage – The paper:

1. (20) Is clear and direct in its use of language – 20

2. (10) Is free of misspellings – 10

3. (15) Is free of grammatical mistakes (e.g. subject/verb agreement, proper use of pronouns) – 14

4. (10) Lacks incomplete sentences – 10 Totals: Content and Structure – 96; Style and English Usage – 99.