Saturday, July 05, 2008

What do Mathematics students do?

The following questions are taken from "Some myths and analyses on learning and teaching mathematics" by George Brown (Teaching methods and Educational Technology, Nottingham University). I have written the questions and then tried to give my answer to some in another colour. I welcome others to answer the questions too.

1) Do students learn mathematics during lectures?

We do learn mathematics from lectures, but only the "chapter zero" on some instances. Obviously we can't answer this question on a general level because it depends on the type of courses we take. For courses which are more about computation and about applying certain methods, we learn from lectures. (How much we learn is debatable, but most importantly we learn some things). For more conceptual courses, like algebra say, the lectures are there to put the stepping stones out. We get a general gist of what the course is about, but we do learn something. So if one listens then one can learn, however some students don't listen and don't learn in this way.

2) What proportion of the lecture class is devoted solely to copying notes from a blackboard or overhead projector?

Once again this varies depending on the course and lecturer. Some are able to balance the two nicely, which is a "Eureka" moment!

3) What are lectures in mathematics for?

Lectures are there to tell a story. The more fascinating the plot (determined by the author/lecturer) the more attentive we are. I can't think of a better way to describe lectures in mathematics. The lecturers are obviously the story tellers and I have met some fantastic ones during my lifetime! (hehe)

I did write a post about this and I will link it when I can find it. The stories have a plot

4) How do students work in problem solving classes?

Erm... I will take this to mean the example classes we can attend, which are classes of at least 30 people (on a good day), where you work through the problems and members of staff are around if you become stuck. Some students work in groups during this time whereas others work alone. Me... I don't tend to do any work during example classes, but instead I have a nice conversation with the lecturer present about a variety of things.

Well I take my problems to the example classes and try to get them solved, rather than working there. If I don't have any problems to take I can sometimes work on a problem I've partially attempted, and get on with it. However it is very rare for me to actually sit through the fifty minute slot and work through the question sheet. I can't really explain this, but sometimes (if I'm being brutally honest) I get annoyed when the people present loiter over my shoulder watching me attempt the question. That is my cue to strike a conversation you see. I understand where they are coming from, but if I'm stuck I will put my hand up and ask for assistance. I don't need to be asked if everything is OK every two seconds. (Once is enough for they might assume that my worried face requires assistance, but I have a permanently worried face! Also some people don't like putting their hands up so that gives them a chance to say "yes I'm not OK", but as I said, once or twice is OK).

You see what's the point in doing a maths problem without any struggle. I only like to ask for help as a last resort, but sometimes during example classes I find that they are opportunity for me to avoid that struggle.

Don't get me wrong, if I attempt the questions before hand and take my issues then they are great! Well that's what I found with Dr. Coleman's example classes for sequence and series and Real Analysis, because I always enjoyed them. (Ha, that was perhaps the only course that I actually worked hard in, which maybe explains why I always had problems!)

Anyway, this isn't a very good thing that I do because I don't benefit from the classes if I have fallen behind in the module. For them to be useful I need to be on top of my studies which is never the case.

5) Do students learn from each other in problem solving classes?

Possibly. I hate being told the answer and sometimes in a group, it is very easy for someone to tell you what to do. But what exactly does it mean by "learn"? "Learn" what? Working with others is a good way to check your answer and perhaps discover another method of solution. (Its best purpose is that someone can tell you that you've copied the question out incorrectly!)

6) What are problem solving classes for?

To do problems? (Oh and talk to the lecturers and "persuade" them to attend a Galois Group lecture!)

7) How do "able" and "less able" students tackle problem solving?

I would say that I'm a middle type of student, and that I can't answer this question! Fine I lied, but once again it depends on the problem we're working on. Sometimes I try to specialise or generalise the question to something I am familiar with to see if I can get elsewhere. I always try to start by writing down what we have and what we want. Then I write the "what we have" line out again and try to make deductions.

A picture ALWAYS helps me so I try to sketch things out. Another thing I sometimes do is to actually answer the question in a long winded and tedious way, just to make sure that my answer is right, and then write the short cut down.

If I see inequalities, I first have a cry to myself and then as I have learnt, I stick blimmin' values inside the daft things to make sure the signs are the right way round! Yes, I'm not the best of students but this semester it has all been my fault.

Oh yeah, and thanks to certain lecturers I try to ask questions or question every step I do. (If I can't answer a question I use one of 36 methods of proof to convince myself, but it doesn't seem to work on others...)

8) How can problem solving be taught?

Well I think reading books like Polya and the one by John Mason on my to read list, are quite beneficial. Although I sometimes did "specialise" or "generalise" I wasn't properly aware of that "technique" until I read about it in a book. I also think you can learn a lot from your tutors. My original PT and Dr. C have moulded me into the annoying git I am. Well you'll have to thank Dr. Coleman for the way I write basically every step out on most occasions, and that I try to explain next to everything. I still can't prove things, but it will dodgily come together I hope. (It's a shame that Dr. C and PS will no longer be teaching me. :( )

9) How do "able" students read mathematical texts?

NA on both parts! But perhaps I could say something on the second part. If I am stuck on a topic and my notes are hazy too, I look through more than one book from the library and try to make sense of the topic. It always has to be at least two books, and sometimes I am lucky to discover a really nice book this way. Sadly one thing I haven't done at university (which I used to do at school and college) is work through problems in the text books. That's the most important thing and I'm not doing it! There is no sense in reading the books if you can't answer the darn questions. And NO, I'm not going to say that hopefully next year it will be different.

10) How can one improve a student's reading skills in mathematics?

Erm design mini projects on small topics like the Euclidean Algorithm, where a student has to read about the subject and hand a written report in. A bit like our workshop module but for that we were already given knowledge of the topic and didn't have to do much reading ourselves. Our module was namely about solving problems in groups, but it would be more worthwhile if had been sent on missions to explore certain topics further. (I would have enjoyed that!) I guess I still could have read into the topic at hand, but the problems were demanding!

Alternatively, we have supervisions in our first year which only consist of eight students and a lecturer. If individual students were given a topic to read about and then later present to the class, it would be very beneficial. Once again, it could be something as trivial as numbers in different bases but students would have to read about this themselves. This would also give the students to present in front of an audience. Actually I like this idea and might run it past someone at University. I mean the "presentation" element in the workshop was sort of ridiculous but that I already posted about last year.

11) How can one improve the study and learning skills of students?

I need to think about this but I think I have sort of answered it in my answers above. Students need to be made aware of how they can help themselves to improve. Watching videos like the one on the Internet on by Jean-Pierre Serre, and reading other study guides can be useful. We are very very naive (with the two dots on top!) when we start university and some of us are seriously clueless about Mathematics. It is a steep gradient that we must climb and help would be useful.

Perhaps there could be an element of "oral exams" in each module which requires students to actually do some reading themselves. This assessment won't be based on answering questions but on how well students are able to communicate mathematical ideas that they have read about.

Our lecturers are gifted mathematically and it is obviously hard for some to understand what it's like for us dodos. (Or they understand but find it hard to empathise). However, the great story tellers are those who are able to communicate effectively with us and students should be given the opportunity to do this too. Some students understand things straight away but are unable to communicate their methods to others.

I don't really know what I'm getting at here but communicating maths is very helpful for students. I found that when I blogged about "proper maths" (once upon a time!) it actually was useful. Hmm, perhaps students should create blogs to help them?

Anyway I'm going to shut up now because I am being called, but there is a follow up to this post soon. (Oh - I forgot about the different colours!)


Beans said...

Rather than editing the post, if I have more to add I will just comment here.

A continuation on question 2:

2) What proportion of the lecture class is devoted solely to copying notes from a blackboard or overhead projector?

I think when you are copying with the lecturer, you are learning too (links to question one). Could you have a maths lecture where nothing was written (?) on the board?

Anonymous said...

I loveee Maths too...please give me your adress... this is my adress: *edited out* i would like chating with you:X.

Beans said...

Erm hi! You can always talk to me through this blog and you will also find my email address on my profile. Hope that helps.

(For those wondering, I didn't publish the comment as it was because it had an email address.)