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Still looking for replies! Any opinions?

I am leaning towards taking courses during the coming school year, my senior year, as follows (with other courses from my second major): Fall: graduate real analysis i, modern algebra Spring: graduate real analysis ii, applied linear algebra, upper level theory of statistics course But instead of the modern algebra and applied linear algebra in the fall and spring, respectively, I could take honors abstract algebra i and ii as long as I substitute the graduate real analysis i and ii with any two of the following (two-course sequences count as two courses) graduate probability theory (i and ii) mathematical modeling complex analysis honors introduction to topology introduction to ordinary differential equations (I took a combined differential equations/linear algebra course) partial differential equations (i and ii) I am unable to take the honors abstract algebra sequence concurrently with the graduate real analysis sequence due to a schedule conflict. The modern algebra course is primarily an introduction to groups, rings, and fields. The honors abstract algebra course would cover much of Artin, 2nd edition. My question is "which of the courses would you rather have taken," not "which do you think I should take," so that I can avoid answers that imply strong dependence on my future plans. I am unsure of whether I will be going into academia or industry; it is true that I am considering going into engineering research and development or into a masters or PhD program in math. I would love to someday have a very highly developed knowledge of probability and statistics (to the graduate level). My second major is in engineering. I would also be interested in hearing responses to the sub question: do you think the honors abstract algebra two-course sequence would be noticeably superior than the course in modern algebra with regard to prep for the math GRE or for graduate school in math? I could follow up the modern algebra course with the one on complex analysis, thus killing two birds because they are both covered in the GRE. Thoughts appreciated.

Don't worry, I'm very pleased with your response and this expansion upon it.

Thank you for all of your feedback. I would reply to more of your questions/statements, but many seem to be rhetorical. Can you please give an example of how I could have tied the example to the thesis statement skillfully? I tried to do that; for example, in the second paragraph, I wrote "Therefore, the kid in this situation would not learn explicitly the moral rightness of his action." I agree that I could/should have justified that assertion about praising positive actions. Can you explain what you mean when you say that " 'utilize training' isn't idiomatic ..." (including the whole line, especially when you say "you are at a level of diction that is at odds with other expressions you have used") About the jarring mix of colloquialism and formality: I understand. My justification at the time was to emphasize that that reasoning was from the mind of a child. Less questions, more declarative statements. Got it. The bit about my hearing of a native village in Africa wasn't made up. I think the purpose of mentioning that was to increase the plausibility of the possibility that the children were lying in the interviews. I can see what you mean when you mention my use of "therefore," etc. , especially in the first essay. In particular, I ended the third paragraph by saying that "therefore ignoring the negative action is not the best approach," when it would have been better to say "therefore, ignoring negative actions can lead to unfortunate results," or something of that sort (I didn't show that ignoring the negative action wasn't the best approach but only that ignoring the negative action could have poor results). I can see how I should have more explicitly connected the paragraphs to the thesis statements.

True. They are GRE essays. I also want to say that I did indeed notice that I slightly misinterpreted the scenario for the second essay before I wrote this post. I read that the children said to the researchers that they talked with their parents more frequently than the other adult villagers, but the statement was that the children talked about their parents more in the interview.

The best way to teach is to praise positive actions and ignore negative ones. Write a response in which you discuss the extent to which you agree or disagree with the recommendation and explain your reasoning for the position you take. In developing and supporting your position, describe specific circumstances in which adopting the recommendation would or would not be advantageous and explain how these examples shape your position. I disagree that the best way to teach is to praise positive actions and ignore negative actions. This approach seems to be a misguided attempt to utilize Pavlovian-type conditional training in order to teach. The better approach is to praise positive actions and to point out better alternatives to negative ones. To praise positive actions on a consistent basis is to encourage the “students” to associate positive feelings with the actions that evoke the positive response. To refrain from praising such actions is, in many situations, to create a situation in which students are unsure of whether the things they did were praiseworthy or not. For example, a kid may find 100 dollars in his grandpa’s bedroom, and be proud of himself for finding such a treasure and proceed to return it to his grandpa; a parent who, being told of this discovery and generous return, does not praise the kid risks causing the kid to doubt whether his was the best course of action; consider that he may be wondering whether it wouldn’t have been wiser to keep the prize for himself so he can buy something nice for his daddy! Therefore, the kid in this situation would not learn explicitly the moral rightness of his action. Hence, it is wise to praise positive actions so that there is a concreteness to the goodness of an action. To ignore negative actions on a consistent basis is to encourage the “students” to associate whatever feeling they have when performing the action (or the feeling felt soon after) with the action itself. Suppose that a toddler is annoyed of his father’s talking on the phone. A proactive young girl as she his, she winds up and punches her father square in the tender parts. She even feels somewhat proud that his father immediately becomes silent, and walks away from her with strange expression on his face; she learned to cause something beneficial to happen without any help! She no longer must listen to his annoying grown-up talk. Hence, she associates the positive feelings with the action, and she becomes more likely to do such a thing in the future. Clearly, such an unintentional encouragement of such behavior is not a good thing. Therefore, ignoring the negative action is not the best approach. Neither is harsh criticism an effective approach, either. Consider a kid who, finding the 100 dollars as in the above scenario, decides that the best course of action to take is to keep the money for himself in order to do good to his daddy (buy something for him at the store). Upon hearing sharp, generalized criticism (for example, in the form of insults and spankings), the kid may feel that such good intentions are not appreciated, and future such intentions and actions associated with them may be discouraged; he may feel that to act on good intentions is not a good idea because of the risks they bring. As a result, he may grow up to be unnecessarily unconfident or become less altruistic. To praise the good intentions upon hearing them, and to point out better courses of action – for example, to return the money to its rightful owner – is a better approach than generalized criticism. In general, it seems to be a wise course of action to praise positive actions. On the other hand, it is not wise to ignore negative actions for the reasons pointed above. To point out alternative, superior courses of action is a better way to teach than to ignore or harshly criticize negative actions. The following appeared in an article written by Dr. Karp, an anthropologist. “Twenty years ago, Dr. Field, a noted anthropologist, visited the island of Tertia and concluded from his observations that children in Tertia were reared by an entire village rather than by their own biological parents. However, my recent interviews with children living in the group of islands that includes Tertia show that these children spend much more time talking about their biological parents than about other adults in the village. This research of mine proves that Dr. Field's conclusion about Tertian village culture is invalid and thus that the observation-centered approach to studying cultures is invalid as well. The interview-centered method that my team of graduate students is currently using in Tertia will establish a much more accurate understanding of child-rearing traditions there and in other island cultures.” Write a response in which you discuss what specific evidence is needed to evaluate the argument and explain how the evidence would weaken or strengthen the argument. First of all, Dr. Field’s observational research was conducted twenty years ago. Your research was conducted recently. If you have reasonable evidence to believe that the culture in the group did not change over the past twenty years, then any differences you found in the society compared to Dr. Field may indeed be due to superiority in one or the other research. However, the statements that you made do not, alone, give satisfactory proof that the Dr. Field’s conclusion is invalid or that the interview-centered method that your team is using will establish a more accurate understanding of the traditions followed in Tertia. I recently heard of a native village in Africa whose members actually lie to the researchers in order to prevent the researchers from learning their secrets, including knowledge of their language. Is it possible that the kids were lying for one reason or another? Is it not also possible that the quality of the interviews was lacking, for example, that the kids did not clearly understand those things that the interviewers were saying or did not know how to express themselves? If it is not possible that there was deception or misunderstanding in the interviews, is it possible that Tertia is an exception of your finding that children spend much more time talking with their biological parents – after all, your research says that the interviews were of children from the villages that only include Tertia; not all of the children were from Tertia, so the responses may give evidence unrepresentative of society in that specific village and so it may be true (even given your research) that the children do not speak with their biological parents often. Suppose evidence were presented that clearly indicate that it is true that the children do indeed talk with their biological parents more than the other adults; talking does not necessarily constitute “rearing;” therefore it is possible that by “rearing” you have something different in mind compared to Dr. Field. Is it possible that the other villagers, in aggregate, performed more obviously beneficial tasks for the kids than the parents? For example, did they teach the kids, offer them food and shelter? He may have witnessed the increased amount of talking between the children and their biological parents, but did not deem to constitute that as “rearing” them. Hence, you did not show that Dr. Field’s research was invalidated from these statements alone. You have not demonstrated that your research is “much more accurate” than Dr. Fields in that you did not show that the information acquired in the interviews was of high quality or (in the case that it was of high quality) that your research is in contradiction of Dr. Field’s.