Category: MS221: Exploring mathematics


I had a really great study timetable worked out for June and July, which included a nice weekend off (for a little holiday in the Peak District), followed by a week of gentle catching-up to get back on schedule. Unfortunately for me, I came back from my holiday with a nasty cold, and ended up missing an extra week of studying. So for the last three days I’ve been trying desperately to get back up to speed with MS221, which has basically meant studying from breakfast til bedtime, with short breaks for meals. If anyone is wondering whether it’s possible to do a unit of MS221 in 24 hours, I can confirm that it is! Whether it’s possible to do it well is another matter entirely…

Anyway, despite having to rush through it, I’ve enjoyed Block C of MS221 for the most part. I was a bit bored by Taylor polynomials at first, but now that I’ve finished unit C3, I do find them quite satisfying to work with – like the presenter in the associated Algebra Workout program said, it’s a bit like “splitting the atoms” of functions, and getting to look at the building blocks that they’re composed of (or at least, that a good approximation might be composed of).

I’ve just about finished TMA03 now, so this afternoon I’ll be getting back to Block 4 of M366, and then making a start on TMA03 for that course – I’ve heard that it’s a very tough assignment, so I’m really quite worried that I won’t get it finished in time. Still, if the last few days are anything to go by, fear of failure is an excellent motivator when I’m pushed for time!

One thing I really like about Block C of MS221 is that it reiterates the basics of calculus, which were originally covered in MST121, before going on to the more complicated stuff. Normally I’d be annoyed by the repetition of a topic across two courses, but with calculus I’m actually really glad the course team decided to do it this way. For some reason, the MS221 chapters on differentiation and integration seem way more accessible and understandable to me – I remember having a bit of trouble with Block C of MST121, but so far I haven’t had any similar problems with the MS221 version. Perhaps the material is presented or explained in a different way (I should really dig my MST121 books out and compare the relevant chapters, but they’re all the way up in the spare room and I’m too lazy to get them!), or maybe the preceding two blocks of MS221 make for a better preparation for calculus? I’d be interested to hear from other students who have done both MST121 and MS221, to see whether anyone else has found calculus easier or more interesting in the latter.

In any case, it’s really gratifying to actually enjoy calculus this time around – it’s such an important part of mathematics, I’d feel like I was really missing out on something if I still couldn’t find anything beautiful or fascinating in it. So far, the section of Chapter C1 about graph sketching has been the most satisfying bit to me – the fact that we can figure out the shape of a graph, along with its long-term behaviour, just by studying the equation of the graph is amazing! Ooh, and the Newton-Raphson method, that was pretty awesome too – the speed at which you can get a good approximation of the solution to a particular equation, and the ease with which you could implement the approach on a computer, is brilliant! I can honestly say that the Newton-Raphson bit of the Mathcad work for Chapter C1 was the most fun I’ve had with Mathcad in my OU career. I really do wish calculus had been this enjoyable in MST121!

You’ll notice there isn’t the characteristic exclamation mark on the end of this post’s title – I’m a bit embarrassed of these two assignments, since both got grades quite a bit lower than the previous ones. For M366 I got 84% (just 1% short of a distinction, dammit!), and for MS221 it was 89%. I’m actually more ashamed of the MS221 score, because almost all of the lost marks were due to stupid errors, which I absolutely should have picked up at the proofreading stage. My tutor’s comments were very nice, but I think he must be getting a bit frustrated with the trivial errors I keep making. Basically, his general feedback was:

  • Factorise expressions where possible
  • Check your work (Mathcad is useful for this)
  • Read the question carefully!

All very good advice, which I have been striving to apply today while working on TMA03. I’ve been checking everything, every little calculation, either in Mathcad or Wolfram Alpha – so far, I’ve caught some really stupid slip-ups, so hopefully I’ll be able to manage more than 89% for this assignment!

The M366 assignment got a lower mark, but I’m happier with that one because the lost marks were due to a genuine lack of understanding about various little details of the subject matter, rather than plain old carelessness. Actually, I get the impression that most of my lost marks were caused by me not fully understanding the questions themselves; some of them wanted slightly different or fuller answers than I expected. For example, in the question about the NetLogo rabbits simulation, I lost three out of six marks because I didn’t mention the details of the code behind the features I was talking about – I described the simulation at too high or abstract a level, without backing up what I was saying with a description of the “nuts & bolts” side of things. Which is an odd mistake for a programming student to make! Still, hopefully by the time the exam comes around I’ll be much better at interpreting the questions correctly. And at least there’s no NetLogo coding to mess about with in the exam!

I know I’m late to the party with this, but Wolfram Alpha really is a brilliantly useful tool for checking your maths work. Today I’ve been doing the exercises in Section 2 of the unit on differentiation, and I’ve already made good use of the site to investigate some problems that I just couldn’t grasp. In particular, I found Exercise 2.1(i) to be almost impenetrable – my solution for it was very different to the solution in the back of the book, and I just couldn’t see where I’d gone wrong.

This is where Wolfram Alpha has been coming in handy. The brilliant thing about it is that as well as calculating the value of a mathematical expression, it also displays the steps you could take to work it out yourself. This is brilliant when the solutions in the course text tend to be a bit terse! For example, the solution for the exercise mentioned above is this:

Not terribly helpful, if you’re as stuck as I was. But check out the Derivative section of the Wolfram Alpha page for the expression, and click on the “Show Steps” link – it gives much more detail about the solution, and after reading through that explanation I could see exactly where I’d gone wrong with my original answer. I suppose there’s a danger with sites like these that people might just enter TMA questions into it and copy the answers, but then again the OU apparently uses anti-plagiarism software that would probably flag up TMA solutions that were word-for-word copies of existing internet resources.

Anyway, plagiarism aside, I think Wolfram Alpha is going to be a very useful tool for me (especially if it means I can avoid phoning my tutor when I get stuck!).

I’m kind of taking it slowly this week, in preparation for the M257 revision marathon that I’m planning on starting on Friday, but I’ve been dipping my toes back into calculus a bit by making a start on Chapter C1 of MS221, which is about differentiation. Now, calculus is something I had trouble with in MST121, so I’ve not really been looking forward to Block C; on the other hand, I feel like I’ve got unfinished business with calculus since I didn’t get a good grasp of it the first time I encountered it, so despite my uneasiness about the topic, I’m determined to do a good job of it this time!
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I wasn’t expecting to get this assignment back for at least a week, since the cut-off date was only three days ago – my tutor must be very efficient indeed! I almost couldn’t open the envelope, though; I’m enjoying MS221 a lot, and I’m hoping to do a lot more maths in the future, so a low score on this first TMA would have left me feeling really crushed. But thankfully it was a nice 98%, hooray!

My tutor gave me some good, detailed feedback which was along the lines of:

  1. Don’t include so much unnecessary detail.
  2. Check results carefully to avoid silly slips.
  3. Draw diagrams, as they can help with your understanding of the problem.

Point (2) is a very familiar bit of advice, as I seem to make ridiculous slips in most of my courses, but point (1) really surprised me – I’ve always worried about not including enough detail. But it seems that there’s no need to be quite so longwinded about showing my working for MS221, which will certainly save me a lot of time in the other TMAs, and especially in the exam!

Point (3) is advice that I often give to Alex about his courses, but I have trouble actually following the advice myself. I don’t know whether it’s just that I find thinking visually/geometrically difficult, but I’m generally a bit reluctant to draw diagrams, even if it would help clarify the problem. I suppose it will get easier with practice, though.

And speaking of diagrams, I was very pleased to see that I got full marks for my hand-drawn ellipse diagrams – so their wobbly, lemony appearance didn’t really matter after all!

I think I might have been a bit too harsh on M366 in my post a couple of weeks ago; I’ve been working through TMA02 this week, and I particularly enjoyed writing about subsumption architecture in the question about McSCOR, the “M366 course Subsumption COntrolled Robot”. I’ve become quite interested in the vertical decomposition way of doing AI systems, but having to write about the drawbacks of the SENSE-ACT cycle in Question 4 certainly stopped me from getting overly smitten with the approach.
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Chapter B1 of MS221 is a bit of an odd one. Most of the chapter is about iterating functions, fixed-points, etc, but the final section is about binomial expressions, combinations and permutations – much more interesting! Not that there’s anything particularly dull about fixed-points, but puzzles like “How many six-letter permutations can be formed from the letters A, B, C, D, E, F, G, H, I, J?” grab my attention a lot more.

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I got my Exam Timetable mailing for the June/July exam period yesterday, and have been freaking out a little bit this morning about the prospect of fitting M257 revision in alongside M366 and MS221. The main thing that was worrying me was the fact that the TMA02 cut-off dates for both M366 and MS221 fall in the couple of weeks before the M257 exam (which is on the 17th of June). Last year during the summer exam period, I put MT262 on hold for about a month while I revised for M255 and M263, which seemed to work fairly well; but this year, I’ve got two courses to put on hold while I revise for a third, and that feels a bit trickier to negotiate.

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I spent a bit of time on Saturday proof-reading TMA02 for M257, but apart from that I’ve been mainly working on Chapter A3 of MS221, which has introduced me to the fascinating world of transformations and isometries. I found the accompanying “Visualising isometries” program on the course DVD a little bit overcooked, but on the whole I’m really quite enjoying this chapter. To be honest, I expected A3 to be quite tough for me, since I’ve always had a bit of trouble with trigonometry; I used to find it hard to visualise triangles and particularly the unit circle, so I ended up just learning a lot of things by rote rather than actually understanding them.

I’m not sure whether the MS221 course texts are better than the MST121 books, or whether I’m just better at this kind of maths these days, but one way or the other I found myself actually enjoying the trig section of A3! Having a printout of the Trig Cheat Sheet from Paul Dawkins’ very useful Online Maths Notes next to me is definitely a big help, too – I’ll probably end up writing an even further reduced version of it onto some blank space in my MS221 Handbook, along with some bits of the Algebra Cheat Sheet, ready for the exam.

It’s weird, really – so far, the MS221 activities and assignment don’t seem to be designed to get us to rote-memorise the material, they seem to be more about understanding how to apply various techniques and how to choose the right approach for a particular problem. But since I’ve heard that the exam is quite a challenge to finish in the 3 hours allotted, I wonder whether it might be worth putting some revision time aside to memorise various formulas, to save time, rather than looking things up in the Handbook? I suppose doing a few practice papers will give me the answer to that – I could well end up spending September memorising stuff from flashcards…

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