How To Write An Essay

Part 8 - Examples of Good and Bad Writing

Learning to write often works best by example. The following are excerpts from nine first-year student essays. Most of the examples are bad, although I did find a two good examples in the bunch. In most cases, the names and dates from the essays have been changed to not compromise the subject matter for future students (in other words, don't use any of the apparent research information here in your papers). I have tried to categorize the errors as best as I could. Errors or bad portions are usually bolded to help you identify them.

• Good Examples
• Bad Examples
  • Bad Grammar
  • Awkward Wording
  • Hyperbole
  • Mismatched Words, Phrases, and Pronouns
  • Misused Words and Phrases
  • General Sloppiness
  • Several Errors Combined

Good Examples

Smith was a religious, Christian man. His notion of monads included contextual references to God. He believed that God controls the harmony of life through these monads.

The essay then goes on to discuss these monads in a Christian context. Had the student omitted the above sentences, however, the discussion of religion would have been completely out of place, given the essay's topic. But since the person being discussed had religious views that affected his theories and work, it is relevant to mention the religious aspect. Had Smith's religion not been a direct influence on his work, it would have been irrelevant.

Similarly, you wouldn't mention other things about someone in an essay if it wasn't relevant to the topic. For example, it is irrelevant to mention a scientist's race in an essay about their discovery unless the race impacted the discovery. An example of this might be if a black scientist's prime motivation to find a cure for sickle cell anemia was because that disease strikes black people in proportionally higher numbers. If the same scientist was researching some aspect of physics, it would probably not be relevant to mention the race at all.

An introductory paragraph:

On March 4, 1849, John Smith was born to Anna Bradcock Smith and James Smith. Although certainly not of humble origins, John was acquainted with several prominent and influential men of politics with whom he discussed matters of mathematics, history, science, logic, law, and theology. Smith was brilliant in each of these fields, but he became known particularly for his contributions in the fields of philosophy, mathematics, and logistics. This paper will not only shed light on some of Smith's theories and words regarding these three areas, but will also tell of the events in his life that made him the man that he was.

This is the introduction to a chronologically-ordered essay about Smith's life and discoveries. As such, the choice to begin with his date of birth is a good one. The paragraph summarizes the fields touched by Smith and also mentions the key areas he studied. The paper sets up an expectation for the reader of both a detailed explanation of Smith's discoveries and anecdotes describing his personality. The sentence structure is grammatically sound and flows well.

Bad Examples

Bad Grammar

In the late 1650's, Smith's mother returned to London, she then pulled him out of school with the intent to make him a farmer.

1. Apostrophes indicate possessiveness or contractions, not plurality. The decade is the 1650s.
2. The sentence is a run-on. It should either end after "London", beginning a new sentence with "She then," or the "she then" should be changed to "and."
3. To make someone a farmer is to create a farmer for them. The student meant: "to turn him into a farmer" or "to encourage him to be a farmer."

Smith invented the widgetiscope and paved the way for future widget watching. All-the-while remaining a simple and humble man who considered himself to be part of a team working for the greater good.

1. The bolded part is not a complete sentence. The entire thing should be one sentence.
2. "All-the-while" does not require hyphenation.

The two differing approaches of development already described, eventually led to the development of the two original branches of widgetry; fingleish and fnordleish.

This sentence is mispunctuated. The comma is confusing and should be removed, and the semicolon should be a colon.

Awkward Wording

Another of Smith's ideas was the method of differentiation. The university re-opened after the plague in 1667. Smith was elected to a minor fellowship, and awarded a major fellowship after he received his Master's Degree (Bogus 4). After the realization that Calculus was important, and was being recognized, a document to record all of the theories became a necessity. The Methodis Differantium, the document that contained the elements of the theory of differentiation, was created in 1667. Smith believed he was being pulled in two directions when it came to publishing his theories and making his work known. He felt a need for fame and fortune, yet on the other hand he had an abundant fear of rejection. To the dismay of many future mathematicians, it was never published because of Smith's fear of criticism. Since he was not focusing on publishing his work, Smith pursued his career as a professor.

This so-called paragraph is an utter mess. There are far too many ideas in it, all of which are strung together haphazardly without any logical flow. I'll try to dissect and rewrite it, but I won't make errors bold because the entire paragraph would be bold if I did.

First, let's pick out the different topics being addressed:

1. the method of differentiation
2. the university re-opening after the plague
3. Smith's ascension through the university ranks
4. the need of a document detailing differentiation, which was eventually created
5. Smith's mental state, desires and fears

Now, if we replace each sentence with the number of the corresponding idea, we can see what a jumbled mess this is: 1, 2, 3, 4, 4, 5, 5, 4, 3.

Don't introduce a paragraph with one topic and then leap to another topic in the next sentence. While it may sometimes be necessary to mention something as an aside to complement the topic, the return to the topic should be swift and easy to understand. Don't bounce around within the paragraph as this student has done.

Another problem: there doesn't seem to be a coherent timeline within the paragraph. Did the university re-open in 1667, or was the plague in 1667? Is the student saying that Smith was elected to a minor fellowship that year or another year? Similarly, when did the major fellowship and Master's Degree come in? It's unlikely to have all happened in one year, though it is possible. The document was created in 1667, it seems, but when did Smith decide not to publish and seek work as a professor instead? Also 1667? It sounds like that was a very busy year for poor Smith!

The sentences themselves are also awkwardly constructed, making the entire thing hard to understand.

I'll make some assumptions regarding the confusing date information. Here is how this information should have been presented:

Smith's ideas on the method of differentiation were gaining recognition in the mathematical community, which made it necessary for him to produce a document detailing all of his theories on the subject. Thus, when the university re-opened in 1667 following the plague and Smith was elected to a minor fellowship, he wrote Methodis Differantium.

Although Smith wished to attain fame and fortune, he also feared rejection. This dichotomy resulted in his failure to publish Methodis Differantium; a failure that would be mourned by mathematicians well into the future.

Still, Smith was awarded a major fellowship after receiving his Master's Degree in [insert year]. Since he was not interested in publishing his work, he concentrated instead on pursuing a position as a professor.

Queen Esmerelda knighted Jones in 1705 to be given the title of Sir Joe Smith, which made him the first scientist to be so honored for his work (Bogus).

1. The phrase "to be given" is awkward here. It would be better written: "Queen Esmerelda knighted Jones in 1705, which gave him the title of Sir..."
2. Who else could be honoured for Smith's work other than Smith? It should say: "...which made him the first man to be honored for scientific work."
3. There probably should be a page number listed in the citation.

Jones had a main idea of analytic geometry.

What does this mean? Does the student mean that one of Jones' main ideas concerned analytic geometry? Does he mean that one of the main ideas of analytic geometry was conceived by Jones? Or does he mean something else entirely? This makes little sense and is very awkward.

Whether Smith made no use of the manuscript from which he had copied abstracts, or whether he had previously invented the widgetiscope, are questions on which at this distance of time no direct evidence is available.

1. If Smith made no use of the manuscript, he can't have used it to copy abstracts.
2. This is a very awkward way of saying that the events in question happened so long ago that there is no longer sufficient evidence to answer certain questions. It would be better written:

Questions as to whether Smith made further use of the manuscript from which he copied abstracts or whether he had previously invented the widgetiscope are rooted so far in the past that it is impossible to gather sufficient direct evidence to provide answers.

This is still a bit awkward. It's best when broken up into smaller sentences:

There are still questions as to whether Smith made further use of the manuscript from which he copied abstracts or whether he had previously invented the widgetiscope. Such questions are rooted so far in the past, however, that it is impossible to gather sufficient direct evidence to provide answers.

Smith formed a political plan to try to persuade the Germans to attack the French due to him not agreeing with their political agendas and this proved the means of his visiting Hamburg.

1. "Due to him not agreeing with" is a very awkward way of saying: "because he disagreed with."
2. The second bolded part should be a separate sentence.
3. "Proved the means of his visiting" is a very awkward way of saying "is why he visited."

Hyperbole

Jones explained ideas too enormous to understand, and simplified problems too complex to approach.

Not only is this hyperbole, it's also logically impossible. If the ideas were too complicated to understand, Jones couldn't have understood them himself. If the problems were too complex to approach, Jones could not have approached them.

More samples of hyperbole can be found in the collection of items with several errors.

Mismatched Words, Phrases, and Pronouns

After marrying Elizabeth, Smith's father fell ill for several months. After no sign of recovery, a lawyer was summoned to the manor. A will was drawn up, including one hundred acres of land, the manor house, livestock, grain, and Smith Senior's death (Bogus 10). His mother gave birth to Smith three months after Smith senior died. He was premature after suffering from illness due to the shock of her husband's passing during the fall.

1. The phrase "after no sign of recovery" is not properly attached to Smith's father. Instead, it is saying that the lawyer did not recover from something.
2. A will does not include land, a house, etc. It states to whom such things are bequeathed. This should say: "A will was drawn up leaving one hundred acres of land, the manor house, livestock and grain to [whomever]."
3. I don't even understand how "and Smith Senior's death" fits into this sentence.
4. "His" in the sentence "His mother gave birth..." refers to the antecedent "Smith Senior." Thus, Smith Senior's mother gave birth to Smith Senior's son. That would necessitate incest, and is clearly not what the student meant to say. They should have simply said "Elizabeth gave birth..."
5. Who else but someone's mother gives birth to them anyway?
6. Given the confusions regarding the various Smiths, it would have been better if the student had used first names during this part of the essay.
7. There is inconsistency in capitalization. It is Smith Senior once, and Smith senior another time.
8. The "he" in "he was premature" again refers to the wrong antecedent. Smith Senior was not premature.
9. Smith did not suffer illness due to the shock of Smith Senior's passing. Elizabeth did. This sentence says that Smith suffered the illness.
10. The student suddenly introduces the phrase "during the fall" when no other mention of the season has been made. This could be confused with Smith Senior dying from a fall.

Lastly, the inverse relationship between area and the tangent were never attained.

"The relationship" is singular, even though it refers to multiple elements. Thus, the verb "were" should be singular as well, and changed to "was."

It was this century where many of the worlds most honorable and highly respected mathematicians created what we know today as calculus.

1. A century is not a place, it is a section of time. Say it is a "place where..." or a "time when..." In this case, "It was this century when..."
2. Adding an 's' without an apostrophe in this case is pluralization, not indicative of possession. The student means "world's."

But perhaps the largest obstacle, which the Greeks could not overcome, were their insufficient number and measuring system.

"Were" is plural, but "obstacle" and "system" are singular. It should be "was."

Tragically at the age of six, Smith's father died.

This says that Smith's father died at the age of six. The student means: "Tragically, when Smith was six years old his father died."

Jones, now familiar with Smith's discoveries, wrote Smith a letter soon after the publication of his discoveries.

After the publication of whose discoveries: Jones' or Smith's?

Misused Words and Phrases

Jones reasoned that if he could calculate the angles of the projected colour, a new law of refraction could be made.

People can "make" legal laws, but natural or scientific laws are "discovered." To "make" a new law of refraction, Jones would have to alter physics.

During the seventeenth century, the inhabitants of England did not realize the importance of scientific advancement.

1. "Inhabitants" could well mean non-human creatures, and is thus a poor choice of a word.
2. Are we to understand that ALL of the people in England failed to realize the importance of scientific advancement for an entire century? It would have been better if the student had said "most people in England."

At the current time, the dominant belief was that light traveled in wave.

1. The current time is the moment the reader is reading the sentence. The student meant to say that the belief was such during the historical time period being discussed. "Current" should be omitted.
2. The phrase "in wave" has an error. It should either be "in waves" or "in a wave." Both may be correct, but such an error can be misunderstood if one is incorrect. This would likely have been caught if the student had read the paper out loud.

Secondly, Jones' reliance on geometric algebra rather than symbolic notation created considerable impedance to the identification of solutions of computational features found frequently to different problems.

Here is an example of a student not knowing the proper meaning of a word. Impedance means opposition to the flow of electric current. It does not mean the same as to impede, which is to be an obstacle. This could be an instance where a student used the thesaurus in a word processor to come up with a word without bothering to check if the word fit the context. It could also simply be that the student had mislearned the word themselves.

Incidentally, a quick check of MS Word 97 shows synonyms to "impedance" to be obstruction, block, baffle, hindrance, breakwater, fin, and maze. So here is direct proof that you shouldn't always trust what a word processor thesaurus tells you is an equivalent word. Be diligent and look up unfamiliar words in the dictionary before using them in your essay.

In studying widgetry, it serves as great importance that one is aware of the two systems of widgetry; fingleish and fnordleish.

Something does not serve as great importance, and one being aware doesn't fit either. This is a student trying to sound fancy but instead making no sense. The sentence should read:

In studying widgetry, one should be aware of the two systems of widgetry; fingleish and fnordleish.

General Sloppiness

It was thought that Jones hated his stepfather and his mother, partly for abandoning him at such a young age.

1. Who thought so?
2. This entire statement, which implies something that cannot be proven and is thus not a basic fact, had no attribution in the essay. Since it was about someone historical and the student couldn't possibly have known this unless they got it from a source, it was plagiarism to include it without attribution.

Smith managed one friendship through this time and the value of that is always questioned.

1. Who is questioning the value? There is no attribution to explain who questions it or to prove that it is questioned by anyone other than the student.
2. What precisely is being questioned? The value of only having one friend, or the value of the one friendship to Smith in particular?

...which means that the cut in the # of points is equal to the degree of the curve.

Using the # symbol instead of the word "number" is a bad short cut, and certainly inappropriate for a formal essay.

Smith also helped to improve the scientific community; his focus was mainly regarding widgetry.

How does a focus on a subject help to improve a community? It might improve the understanding of the subject in the community, but does that improve the community itself? This is a badly worded assertion. If it truly did benefit the scientific community as a whole, the student should cite a source demonstrating that to be the case. No attribution was present.

In one day, John's attitude towards school changed for the better. A boy ranked just above him kicked him in the stomach. At the end of the day John challenged the boy to a fight. Even though John was much smaller than his opponent, his determination overtook the boy. Winning the fight was still not enough. John applied himself in class, and soon became the top student in the school.

1. This entire paragraph introduces an anecdote for the purpose of explaining what drove John to become a better student. Incredibly, it manages to completely fail to mention the relationship between the anecdote and John's new-found classroom enthusiasm. The relationship is implied and the reader can guess that John wished to beat the boy in more than just a physical fight, and thus worked hard to outrank the boy in the classroom, but that is not stated.
2. The paragraph is very choppy and the sentences do not flow well. Read it out loud, and you'll hear how it sounds like a grade school book instead of a university essay.

During this time, Smith constructed a water clock. He constructed the clock out of an old box.

This is choppy. It could be easily combined into one sentence.

Jones became began to study motion.

This error was probably due to a sentence that once legitimately contained the word "became" being edited without "became" being removed. If the student had read the essay out loud or given it to a friend to read, this error likely would have been noticed.

Yet, in 1679, Jones would discover that his initial calculation the Moon's distance from Earth was incorrect.

Here is another example of a simple error of omission that could have been caught if the student had read the essay aloud or given it to a friend to read. The word "of" should be between "calculation" and "the." That one small error makes the entire sentence awkward and confusing. If the instructor has to reread the sentence to try to understand its meaning, the flow of the essay is interrupted. If this happens often enough in the essay, it gives an overall bad impression on what otherwise might be a very good paper in terms of research.

More examples of errors that could have been caught if the students had bothered to read their essay:

According to hi diary...

One of Smith's main contribution was his use of...

Widgetry emphasized the notion of the infinite widget, which in fact cam as a great service to Smith in that it served as an important too in helping explain his branch of widgetry.

Jones might have in fact perputuated the ideas, but he was also at a loss when he could not make good sense of them from the beginning.

Admiration for Smith grew in the filed of widgetry.

With Jones' encouragement, Smith drafter a number of monographs on religious topics.

Smith considers out universe to be a gravitational system...

On August 10, 1777, Jones was ent a letter from...

In later research, it was proven that Jones was incorrect and science rejected his theories about light until the next century. Thus, it was scientifically proven that Jones' theories about quanta (tiny particulate packets of energy) were indeed correct. The wave formulation was also correct.

1. When was this "later research?" Who performed the research? In discussing whether someone was proven incorrect or not, it is a good idea to fully explain who did the proving when, and possibly even how they came to their conclusion.
2. These sentences contradict each other. Was Jones proven incorrect or correct? Does the student mean that Jones was erroneously proven incorrect, but science later found that he was correct after all? Or was Jones correct about some things and not others?
3. The use of "Thus" implies causality. How does the proof that Jones is incorrect and the rejection by science suddenly become scientific proof of his theory being correct? Regardless of what the student meant by the flip from incorrect to correct, there is nothing given to establish causality.

Regardles of whether...

It's disappointing to see such sloppiness as this in an essay. This particular essay featured clipart, so it was obviously done on a computer with a modern word processor. It clearly wasn't spell-checked. Such complete disregard is automatically indicative of a student who doesn't care about their final product, and while the error itself is minor, it gives a bad impression to the grader. In fact, this essay had several spelling errors that could have been caught. That's inexcusable at the university level.

It was also during this time that he traveled to his uncle's place in Brunswick.

"Place" is colloquial. Use "home," "apartment," "residence" or other such appropriate word instead.

Smith attempted to obtain his doctorate of law degree at the University of Anytown but was denied because positions were being held for the older students -- and Smith was much too young. Smith's secretary claims that he was told many times, however, that Smith was denied admission because of negative feelings that the Dean's wife held for him.

1. Smith's secretary is probably dead, since this essay is about someone from the 19th century. Therefore, they no longer claim anything. It should be past tense.
2. Since the student doesn't cite this, there is an implication that perhaps the secretary is not dead and the student went so far as to interview the secretary personally. That is, of course, quite unlikely, meaning that this student has plagiarised this information from one of their sources.

The following are a few concepts that form the basis of Leibnizian calculus: [followed by three bulleted paragraphs comprised mostly of direct quotation]

Using bullets in a formal essay is rarely appropriate. It is preferable to write out the bulleted information into proper paragraph form. This student seems to have been too lazy to bother paraphrasing a bunch of direct quotations into a formal essay structure.

Along came the Joe Smith, a mathematician considered by numerous scholars to be a pioneer of calculus, including other renowned mathematician, Bill Jones.

1. "The" Joe Smith? There has only been one?
2. The student means "another," not "other." Sloppy.

The first page of the essay starts with:

have been developed (5).

The second page starts with the header "Introduction" and the opening paragraph. Clearly, the student stapled the pages out of order. What a sloppy mistake! Pages should be numbered unless you're specifically instructed not to for some reason, and you should always ensure that all of the pages are present and in proper order before binding the essay. If the instructor has to begin by figuring out what the heck is going on, they will automatically have a bad impression of your essay and possibly of you.

Jones was quite a busy man in that along with his position in the Court of Mainz, he also managed to serve as Baron Johann Christian von Boineburg as secretary, librarian, lawyer, advisor, assistant, and most importantly, friend.

1. "Quite a busy man" is a bit colloquial. "A busy man" would do.
2. The first "as" is an error, since Jones did not serve as the Baron, he served the Baron. This may have been caught if the student had read their essay out loud.

Several Errors Combined

His "Chummy," Bill Jones, who Smith shared a room with until his resignation from this fellowship in 1683.

1. "Chummy" should only be included if it was Smith's actual word for Jones. If this is the case, it is a quotation from a source and should be cited. If not, it is colloquial and should just say "His friend Bill Jones..."
2. "Who" should be "whom" in this case. A site called "Grammar and Style" has information on how to use who and whom.
3. This isn't even a complete sentence.

Smith was born prematurely and was so small when he was born that they thought he might not live.

1. Repeating that he was born is redundant.
2. Who does "they" refer to? Doctors? Parents? Relatives? Townsfolk? It is a pronoun without an antecedent.

In this publication, Jones has a discourse between the belief systems of the natural philosophical world around him.

1. "Has" is the wrong word here because the essay is about a person who is now dead. Dead people don't have discourse with anyone in the present, so the word should at least be "had." But even "had" is awkward, and a better word would be "wrote."
2. "Discourse" means to converse, especially orally. One does not speak orally in a publication. It is written. This word should be omitted.
3. "Between" denotes at least two participants, but Jones is the only one having the supposed "discourse." This too should be omitted.
4. "Natural philosophical world" is confusing. Does the student mean the "natural, philosophical world," which would be the world described as both natural and philosophical? Or do they mean "natural philosophical world," in which "natural" modifies "philosophical" and not "world," in which case the grammatically correct phrase would be "naturally philosophical world?"

This would be better written as:

In this publication, Jones wrote of the belief systems of the natural, philosophical world around him.

or, depending on the answer to the fourth point:

In this publication, Jones wrote of the belief systems of the naturally philosophical world around him.

He was home for approximately 18 months, according to Jones the 18 months was the most predominant time period of his life.

1. This is a run-on sentence. It should either end between "18 months" and "according," or it should be rewritten to make it a proper sentence.
2. "18 months" is repeated for no reason.
3. "18 months" is plural, so it should be "18 months were" not "18 months was."
4. "Predominant" means superior especially in power or numbers. Something cannot be "most superior." "Most" should be omitted.
5. "Predominant" is not the best word in this case anyway. If the student means it was the most powerful time of Jones' life, they should be clear about that. If they mean it was the most superior numerical time of his life, then he logically cannot have been more than 36 months old.

Simpson was content after his ability to reproduce Smith's experiment. Jones was not that easy, the two men fought constantly.

1. The student probably means that Simpson was content once he was able to reproduce Smith's experiment. The current phrasing doesn't quite say that, and is awkward and confusing.
2. Jones was "not that easy" to what? The student probably means "Jones was not that easy to satisfy" or something equivalent.
3. This is a run-on sentence. It should end after "easy," or be rewritten to be grammatically correct.
4. Which two men? Simpson and Jones or Smith and Jones?

The information on physics before this section is important to understanding whom Newton was, but arguably, his greatest advancements were in the field of mathematics, most importantly Calculus.

1. Incorrect use of "whom." Should be "who." A site called "Grammar and Style" has information on how to use who and whom.
2. There should not be a comma between "arguably" and "his."
3. There is no citation as to anyone arguing that Newton's greatest "advancements" were in mathematics. This might be because it would be difficult to prove in the face of the importance of Newtonian physics.
4. "Advancements" is probably the wrong word. "Achievements" or "discoveries" would be better. Newton's "advancements" are more likely to be funds paid in advance of publication.
5. The addition of "most importantly" is awkward. "Particularly" would have been a better word.
6. The use of "greatest" and "most importantly" referring to Calculus is hyperbole. Given that this essay was for a Calculus class, it sounds like a kiss-up. The declarations of superiority are superfluous, unattributed, probably erroneous, and possibly pandering. It's all very ugly.

A concluding sentence:

Smith's great work, theories, and studies will continue to live on forever in the ever-changing world of science and mathematics.

1. How can the student know that Smith's work will "live on" forever? That's an impossible assertion to make.
2. Work, theories and studies don't "live." They exist, but they are not organic creatures.
3. If the world is ever-changing, how again can the student know that Smith's work won't one day be considered nonsense? Or lost entirely?
4. "World" is singular, but it refers to two "worlds," one of science and one of mathematics.
5. This conclusion reeks of hyperbole. (So does the phrase "reeks of hyperbole," but this is not a formal essay.)

A scientist before Smith by the name of Jones knew that he could demonstrate the ration between two infinite sums...

1. The phrasing here is a bit awkward. It would be better phrased: "Jones, a predecessor of Smith, knew that..."
2. "Ration" is the wrong word. The student meant "ratio." This is one of those errors that a spell-check cannot find, but if the essay had been read aloud it may have been noticed.

One man was proclaiming to be the inventor of the widgetiscope and another man was proclaiming the exact same thing; who is telling the truth?

1. The main problem here is the change in tense. You can't go from "was" to "is" if the subject remains fixed in time. Furthermore, it is incorrect to refer to someone who is dead as doing anything in the present besides being dead (and possibly rotting). A dead person is not telling anything right now, but they were in the past.
2. Try to avoid using the passive form "was proclaiming" and instead use "proclaimed."
3. This particular statement is also bad because of the subject matter. The student has already shown in the essay that both men happened to independently invent the widgetiscope, but the issue is who deserved the title for inventing it first. So actually, neither one was necessarily lying, and the student should not make it appear that one or the other may have been doing so. You must be careful not to libel people.
4. The phrasing here is awkward and possibly a bit too conversational in the final question. A better way of writing this would be:

   Two men proclaimed to be the inventor of calculus, but only one could be given the credit.

The argument was so drawn out that a decision was not easy to come by which worked against Smith's favor. Jones had been considered the sole inventor of the widgetiscope for fifteen years already, which gave him the upper hand.

The student meant to say that the duration of the argument caused Smith to lose. But because the student failed to put the necessary comma between the bolded words, this sentence actually says, by means of a complicated string of multiple negatives, that it was not easy to come to a decision against Smith, meaning he won. This sentence would be better worded this way:

Because the argument took so long, Smith lost.

But then, at the beginning of the next paragraph, the student writes:

The argument took years to unravel and never really came to a definitive decision.

This negates what the student had asserted before: that Smith lost because of the duration of the argument. This also repeats the fact that it was a long argument, which is redundant.

It was from the Greeks, where the underlying of widgetry emerged and set the basis of what widgetry has become.

1. The Greeks are a people, not a place, so things come from "whom," not "where."
2. The comma in this sentence should not be there. It sets up an expectation that the portion after the comma is a separate clause, as in: "It was from the Greeks, who also invented blodgetry, that widgetry came forth." Note that because the "who" is in the separate clause, it should not be "whom."
3. The underlying what? You can't just say the underlying of widgetry. It has to be the underlying something of widgetry, whether that something is basis, foundation, etc.

Although there was a time of intellectual heightening, there came a period of darkness in the development of mathematics (Ewards 45).

1. "Intellectual heightening" is an icky, awkward phrase. "Intellectual development" would have been much better.
2. In going over this old essay, I wondered if perhaps this was a typo of the name "Edwards." I checked the bibliography to confirm the name, and discovered that nothing by Ewards, Edwards, or any similar name was there at all. Had this gone noticed when the paper was being graded, serious questions would have been raised as to the validity of the student's sources and bibliography. Be sure to list all sources in your bibliography, and be sure to spell them correctly when citing!

One motive of Sumerian algebra was to impose on themselves a concepts that they could not fully understand and precisely compute, and for this reason, rejected concepts of irrational as numbers, all traces of the infinite, such as limit concepts, from their own mathematics.

1. "Motive" applies to "Sumerian algebra," not "Sumerians." Therefore, that motive cannot be imposed on "themselves." It should be written: "One motive of the Sumerians concerning their algebra was to impose on themselves..." although that is still an awkward phrase.
2. "Concepts" should not be plural. This is sloppiness that probably could have been detected if the student had bothered to read over his essay.
3. The sentence should end after "compute." A new sentence should begin, "For this reason..."
4. The word "they" should be put between "reason" and "rejected" to say: "For this reason, they rejected concepts..."
5. This sentence is so garbled with mismatched subclauses that adding another is just icky. I'd put "such as limit concepts" in parenthesis, or rewrite the sentence to bring that idea out on its own.

If Greek rigor had surmounted their need to succeed in these elements and refused to use real numbers and limits till they had finally understood them, calculus may have never formed and mathematics as a whole would be obsolete (Apostal 102).

1. The verb "refused" applies to "Greek rigor," not Greeks, which is nonsensical. Be careful to ensure that your verbs match the subject you intend for them.
2. Don't use "till" when you mean "until." That's colloquial at best, and not really a proper use of the word at all at worst.
3. The proper phrase is "have never been formed." To say something never formed begs the question: What didn't it form?
4. Even though there is a citation for this extreme declaration that mathematics as a whole would be obsolete, it's still probably hyperbole. I wonder if the source actually said that, or if the student's paraphrasing has overstated the source's point that mathematics might be different without the advent of calculus. Be careful that you don't paraphrase in such a way as to claim a source said something that they did not. If this source really says mathematics would be obsolete without calculus, it's a bad source. Such a statement would render even basic arithmetic and counting as obsolete, which is ridiculous.

Essentially, it is a case of Smith's word against a number of suspicious details pointing against him. He acknowledged possession of a copy of part of one of Jones' manuscripts, on more than one occasion he deliberately altered or added to important documents before publishing them, and a material date I none of his manuscripts had been falsified (1675 had been changed to 1673)(Bogus, 78)

1. "Essentially" isn't technically incorrect here, but students do have a tendency to use words like "essentially" and "basically" too often. It's somewhat conversational, and possibly colloquial. Try to avoid it unless something is truly essential.
2. "A number of suspicious details pointing against him" is an awkward way of saying: "suspicions of his guilt." But what the student means is not suspicions, but points of evidence.
3. When you list several examples of something you've indicated, the way to punctuate it is as follows (note the placement of the colon and subsequent semicolons):

   [Point being made]:[proof 1];[proof 2];[proof 3]; and[proof 4].

   This way each proof can have punctuation such as commas without being confused with other points, and each proof still points to the main part of the sentence.

4. The "a material date I none of" doesn't seem to make sense at all. I think the whole thing is there in error, but for all I know the student was trying to say something different. I can't believe the student read this over and found it comprehensible.
5. The parenthetical comment is important enough to be in the sentence properly. The student likely put the information in parentheses because the sentence was too awkwardly full of commas and clauses already. Had the student properly punctuated the list of evidence, they would have been able to put this date information in as part of a proof segment.
6. The sentence has no period, which is sloppy.

This entire thing should be rewritten to say:

It is a case of Smith's word against the evidence of his guilt: he acknowledged possession of a copy of Jones' manuscripts; on more than one occasion he deliberately altered or added to important documents before publishing them; and his manuscripts had been falsified by changing 1675 to 1673 (Bogus, 78).

After quoting a dictionary definition:

The editors of the famous dictionary are probably unaware of the fact that they have just committed a cardinal sin in the mathematical world, in that they only described fingleish widgetry, and failed to include an explanation of fnordleish widgetry.

1. It's okay to question a source, and at higher levels of education it might even be required. But if you're going to do it, be careful to do it well and with evidence. This just sounds presumptuous. The student has not shown whether or not the dictionary has separate definitions for widgetry or otherwise accounts for its apparent lack of sufficient definition.
2. Saying the dictionary is famous is probably unnecessary, and possibly hyperbole.
3. A "cardinal sin" is a sin of fundamental importance. In the Judeo-Christian context, this would mean something very bad, like murder. Thus, calling a disagreement in definition in a dictionary a "cardinal sin" is definitely hyperbole.
4. Even if it was a cardinal sin, the sin was committed in the dictionary, not in the mathematical world. The student meant "against the mathematical world."

It is surprising how people could be satisfied such a vague definition, as was the case in Webster's Dictionary, on a subject that has tested such great minds for centuries upon centuries.

1. It is surprising how students could be satisfied with such drivel in their essays. That sounds nasty, doesn't it? That's because it is. Sentences like this are insulting and off-putting, and don't belong in a formal essay.
2. "Such great minds" requires an example. The word "such" should be omitted.
3. "Centuries upon centuries" is redundant. Just say centuries and leave it at that.

Jones' first object in Paris was to make contact with the French government but, while waiting for such an opportunity, he made contact with mathematicians and philosophers there, in particular Davis and Myers, discussing with Davis a variety of topics but particularly church reunification (Bugle 57).

1. An "object" is a thing. The student means "Jones' first objective..."
2. This is a bad run-on. It should be broken up like this:

Jones' first objective in Paris was to make contact with the French government, but while waiting for an opportunity to do so, he made contact with mathematicians and philosophers such as Davis and Myers. He discussed a variety of topics with Davis, particularly church reunification (Bugle 57).

Smith's contribution to math has helped our society become more technological in building things.

1. In this particular case, Smith made many contributions, not just one.
2. "Math" is the colloquial version of "mathematics."
3. Did Smith's contributions only help "our society?" What about other societies?
4. "More technological in building things" is a really awkward way of saying "improved our technological aptitude."

Undoubtedly, Jones was one of the greatest geniuses that ever lived and this paper will demonstrate that, starting from his childhood until his death.

1. Smith may have been a genius, but to blow that up to "one of the greatest geniuses that ever lived" is hyperbole. Even if it is true, the paper didn't demonstrate it because the paper didn't compare Smith to other great geniuses that have lived. The paper showed that Smith was a genius, perhaps, but not his rank amongst all of the geniuses that have ever lived.
2. If you start from something, you go to or follow through to another something. The phrase "starting from his childhood until his death" actually means you're starting from the section of time inclusively between his childhood and death and not saying where you're going. Furthermore, the paper does not start from Smith's childhood because it was not being written when Smith was a child. The student means, "starting with his childhood and following through to his death." That is still awkward, and the sentence would be best written:

Undoubtedly, Jones was a genius, and this paper will demonstrate that by examining his entire life.

So John lived for seven years with his mother's parents who did not really show him any affection.

1. "So" in this context is colloquial and should be omitted.
2. This really should be cited. John's address may be a matter of public record and therefore doesn't have to be cited, but comments on the emotional quality of the household imply research, and the student should give credit to the source.
3. "Really" is colloquial, and should be omitted.

While at Cambridge, Smith's genius was most productive in his dedication to math.

1. Who is Smith's genius? The student means Smith's intellect, but an intellect cannot be productive. It facilitates productivity, but it is not productive itself. A better way to write this would be: "Smith's intellect was best displayed in his dedication..."
2. "Math" is colloquial. It should be "mathematics."

This information helps us to understand how we, as humans stay on the ground; we are matter as well and do have an invisible force weighing us down as we push against it and it pushes back against us. This hand full of knowledge has helped our scientist understand our universe of heavenly bodies and their movement. It has also allowed scientist to delve further in exploring our galaxy.

1. Does gravity only affect humans? Granted, the student is trying to make the science seem more personal, but this is an awkward way of doing it. It is also something that seems to indicate an essay geared to children. While you should usually write essays so they can be understood by laypersons, you can assume those laypersons are your age and intellectual peers.
2. The description of the invisible force is very awkward. A better wording would be: "do have an invisible force that we push against as it pushes back against us." Gravity does not, in fact, weigh people down. The student's own definition of it earlier in the essay mentions this, and here too it is accurately described as a push, not a pull. To add in the bit about it weighing us down is contradictory.
3. The student means "handful." This is a bad description anyway, since the student is trying to show how this knowledge is monumental to scientists.
4. Both instances of "scientist" should be pluralized.
5. One delves further into something, not in it.

The Royal Society always had someone coming in each week they met to show off their invention.

1. "Always had someone coming in" is colloquial and awkward. It should say: "The Royal Society hosted a guest each week..."
2. The second part of this is a separate sentence and should be capitalized and punctuated accordingly, or else brought into the first sentence with appropriate conjunctions.
3. "Show off" is colloquial. "Demonstrate" would be better.
4. Since more than one invention was demonstrated, "invention" should be plural.

A concluding paragraph:

Jones was a great man who made an impact in all of our lives. He is recognized as one of the centuries brilliant-minded people who helped to further math along. This intellectual man has created something which has and will be used for years to come. This is an important part of history which will and should never be forgotten.

1. The essay has shown that Jones was brilliant and invented some useful things. It has not, however, demonstrated that he was a "great man." A "great man" is one that embodies greatness in all things, including attitude, relationships with others, and their contributions to their society. Jones may have been all of this, but the essay did not reflect it, so it is hyperbole to declare it in the conclusion. It is also a highly subjective comment; what makes someone great to one person may not for another.
2. "Centuries" is the plural of "century," not the possessive. The student means "century's." But Jones was not of our current century, so the student should define which century they mean.
3. Impacts are made on, not in.
4. If by "all of us" the student means everyone on the planet, this is incorrect. Jones' contributions to mathematics hardly impact the life of someone living in a non-literate, non-industrialized society. Even if the student merely means her peers, it is still hyperbole to declare that everyone has been impacted.
5. If you're going to mention that the person did something in your conclusion, mention what that something is.
6. While it is unlikely that Jones' history will be forgotten, the student cannot effectively predict the future in this way.

Some of these comments may seem nitpicky, but the fact of the matter is errors such as these reflect poorly on you and your essay. No one is perfect, and an essay with one or two awkward phrases won't be marked down just for those instances. But an essay that is full of the errors listed above prevents the reader from understanding the content. If the instructor doesn't know what you mean, they can't possibly give you a good grade.