Maths Frameworking 3rd Edition Answers Homework Book 3

[Phoebe Sibilio]

Asmuch as I dislike to search for answers on the internet, I am often forced to by time constraints if I even expect to complete the homework in time for submission. (I am taking 2 other modules and writing an undergraduate thesis too).

It destroys our ability to calibrate the course difficulty. Twenty hours of homework a week is very high for a math course; higher than I would expect from any course that was not promoted as a "boot camp" style course. Either you are falling behind the rest of the class, or other people are turning in much scantier work than you are, or everyone is googling the problems. The first two situations are obvious, and your professor should be adjusting to it. The last situation is invisible. We had an analysis course at MI last year pedagogically ruined because everyone kept solving the homework problems, so the professor kept increasing his pace, until an in class test revealed that no one was actually doing the homework themselves.

It forces us to use more obscure, and often not as good, problems. There are some fields where there are computations every student should do -- and, as a result, they are written up in books and online sources everywhere. It hurts my ability to design good problem sets if I can't put this fundamental problems on the problem set. Even in fields where there are not such key problems, there are often only so many ways to set up an example so that it is doable in a reasonable amount of time. If I can't use the examples which are already online, then I need to pick larger and stranger values for my parameters, which makes the problem set harder.

I do not believe that students will learn as much from reading a solution as finding it themselves; this is probably uncontroversial. Moreover, I think that hearing a solution from a classmate with whom you have been discussing the problem together is better than hearing it from a classmate who solved it separately; hearing it from a classmate is better than hearing it from a faculty member; and hearing it from a faculty member is better than reading it in a textbook or here on math.SE. I think that the more interactive and the less polished the presentation, the more you have to engage your own understanding to process and take in the answer. This is why I almost never leave full answers to questions that look like homework here; I think it is harmful.

Homework Policy: You are welcome to consult each other provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. If you seek help from other mathematicians/math students, you should be seeking general advice, not specific solutions, and must disclose this help. I am, of course, glad to provide help!

I don't intend for you to need to consult books and papers outside your notes. If you do consult such, you should be looking for better/other understanding of the definitions and concepts, not solutions to the problems.

You MAY NOT post homework problems to internet fora seeking solutions. Although I know of cases where such fora are valuable, and I participate in some, I feel that they have a major tendency to be too explicit in their help. You may post questions asking for clarification and alternate perspectives on concepts and results we have covered.

Personal anecdote. In the late 1970's i was taking topology from Munkres, Topology: a first course. The professor was Joel Spencer, a wonderful teacher, who is up for an AMS Trustee position, see the current Notices. In particular, he made up his own assignments that might not be questions in the book, which takes extra care and work. We had gone through compactness and the more intuitive sequential compactness and limit point compactness. We did most of the proof in class, that the product of just two compact spaces was also compact. the homework was to complete the proof for compactness, and throw in proofs that the product of two sequentially compact spaces was also sequentially compact, and the product of two limit point compact spaces was also limit point compact. Two of them were easy enough, but i struggled with the limit point one for at least a couple of days. Eventually I handed in a paper saying just that "I couldn't do this one." It came back from the grader with "Excellent" written on top, because the supposed fact is false. I was mystified, I asked Prof. Spencer what was so great about it. It took years for me to understand that not being able to prove something false was exactly right.

Secondly, I think this process can actually be helpful to your learning, provided you spend a reasonable amount of time thinking about the problem first, as you are likely to collaterally learn other things while looking for the information you want. I would also recommend talking to other people on your course (and/or the professor) about the problem before you search the net.

Thirdly, if you don't understand what you read online, then don't hand it in as a solution. It's usually better to give whoever is reading your homework assignments an accurate idea of what you do and don't understand.

As an aside, there are a number of classical theorems proved by mathematicians like Gauss that are not unreasonable to set as homework exercises. You will likely have been presented with a completely different theoretical framework to the one that existed historically, which can make these results much easier to prove than they would have been at the time.

My opinion is that there is nothing wrong at all with posting homework questions here, particularly interesting ones, and I find much of the negative reaction to homework-question posters to be somewhat strange, alien to my way of learning mathematics in a give-and-take exchange of mathematical ideas. Surely posting questions here and studying the answers is not much different than studying hard in the library, talking mathematics with one's colleagues at math tea or talking to one's professor, which are all excellent ways to learn mathematics. In particular, I expect that students who post questions here might learn just as much if not more from the resulting answers as from their professors---we have a number of talented mathematicians, who are very good at explaining things---and that math.SE provides a valuable service to students having unapproachable professors, having professors who do not explain well, or who have few colleagues able to help them. Furthermore, the math.SE community strongly benefits from the questions and the insightful answers that might be posted.

(...)In particular, I hereby give all of my own students complete permission to post any and all their homework problems here, and indeed I encourage them to post their questions here and to study the answers well and thereby to learn some mathematics. I will be testing them on their understanding at the exam.

I would also encourage all mathematics professors to adopt a policy of encouraging collaboration on homework among their students, as talking about mathematics with one's colleagues is assuredly one of the best ways to learn mathematics. Indeed, I recommend that all professors should actively encourage their students to form study groups in order to work on their homework problems together. Learning as a group, they will go very far.

In my opinion, restricting study materials is counterproductive (particularly if no computer-searchable version of the course textbooks exist.) I realize that blindly copying answers is bad, but cheating on coursework has always been a problem and it is an issue that is independent of the Internet.

One common complaint is that students will learn less by Googling than they will by reading the textbook. This may or may not be true, but being able to search gives the learner access to much more targeted information. The difference between needing to skim through fifty pages you already understand in hopes of finding a paragraph you didn't, and being able to immediately enrich yourself on the topic desired, is phenomenal.

The thing is, the anti-Google teachers are right about one thing - you aren't going to remember how to use it practically if you don't actually use it. One answer here said that the degree of the problem became apparent when an in-class test revealed that the students, who up until then had been passing relentlessly difficult questions with ease, knew next to nothing.

This is actually a really useful thing to know, because armed with that knowledge the real problem becomes apparent - the students aren't using their research, which is why it isn't 'sticking.' A great option would be to hold a brief, three- or five-question test before each class - placing numerous, smaller checkpoints along the way will teach the students how to learn the material and retain it for use far better than either cramming or Googling together a paper.

I'm going to go one step further, though, and say that this also illustrates a deeper need for education to evolve. We don't live in the dark anymore - we live in an age of effulgence, where learning of any sort is a phrase away. To educate successfully, it will become necessary to embrace this by teaching more applied mathematics and asking more questions. To wit, if the course itself demands knowledge, the students will learn.

First of all, be relax and take things easier. If some problems are hard and you cannot solve them then I see no problem to ask for help as long as you want to understand and learn the tackling ways of the problems, not only to hand in a solution. The real fact is that some teachers do really poor at their classes and expect a lot from their students. I don't know how things in your

case are, but if you like mathematics you may learn a lot on your own. Moreover, if you study the problems and the solutions posted on this site you'll learn a lot!

The aim of doing homework is practicing what you theoretically learned at school. When you do your homework, you gain experience in the subject. If you are having difficulties on doing your homework, that is the sign of your lack of comprehension in the subject.

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