0:05

We're turning to the third sample problem set solution, let me tell you up front

Â that all 10 answers are going to end up getting 24 marks.

Â This is, if you like, a model solution sheet.

Â I mean, I don't give model solutions because you can't grade this kind of

Â mathematics by comparing someone's work with a model solution.

Â You have to look at the work they've done and evaluate it on its own terms.

Â 0:28

Just as you can't do this kind of work by looking for

Â templates, you can't grade it by templates either.

Â You have to evaluate it in its own terms.

Â So I don't give out model solution sheets.

Â On the other hand, this particular video that you're watching will show you

Â a good solution for each one.

Â And in most cases, actually in all cases, a very slick solution.

Â So before we begin, we know that we're going to be able to congratulate

Â whoever it is that did this solution.

Â Now wait a minute, that was me.

Â I did the solution and I can't really congratulate me,

Â because I'm the one putting the course together.

Â Okay, let's just make some,

Â what I'll do is I'll go through and I'll just make some comments.

Â So the grading is going to be fairly straightforward.

Â It's going to be 4, 4, 4, 4, 4, 4, 24 and it's going to look the same throughout.

Â 1:44

So I can just go straight in and

Â put the numbers 4 in everywhere and a total of 24 for this one.

Â And this is very much like the solution we saw in sample exam solution two,

Â except in that case, the answer had an elementary mathematical mistake,

Â here an arithmetical mistake where the student put 8 rather than 10.

Â But other than that, this is essentially the same argument.

Â It's written extremely slickly and

Â everything's there that you need to be there.

Â I wouldn't normally expect a typical student in the class to produce something

Â quite as short and succinct as this, but it's absolutely a perfect proof.

Â All of the reasoning is there, everything you need there, so very nice little proof.

Â Looking at number 3 now, I'm not going to put in all of the 4s anymore,

Â I'll just jump straight in and put the 24, a beautiful proof.

Â Consider the two cases even and odd separately, very elegant.

Â If it's even, then that guy works out as being odd and

Â I've actually stated that it's odd.

Â I admit that I was the one that wrote this out.

Â If n is odd, same thing, you follow it through and you show that this guy is odd.

Â And then you wrap this thing up by concluding that in both cases n squared

Â + n + 1 is odd.

Â And the danger with this particular video is that it's going to go by too

Â quickly for you to sort of take it in.

Â So what I suggest you do is as we get to each one you freeze the video and

Â you just take a look at it and you see why it works.

Â Incidentally, some of the early research that was done at Stanford onto students

Â doing MOOCs showed that there was a very strong correlation between the number of

Â times the students used the pause and rewind button and

Â the success they had in the course in the long run.

Â And that's for a whole variety of reasons.

Â One is, of course, it shows that the student's engaged,

Â if you have to keep stopping and rewinding.

Â It shows you're engaged, it shows you're reflecting on the work,

Â it shows you understand when you don't understand and what's going on.

Â You know when you don't understand.

Â And you're taking control of your own learning.

Â So the correlation between using the pause and rewind button and

Â success in the course is actually quite high.

Â And indeed one of the advantages that MOOCs have over classroom lectures

Â is that you can do that.

Â It's very hard when I'm lecturing in the class to freeze me and rewind me.

Â In fact, I wouldn't recommend you try to do that, okay?

Â I certainly hope you wouldn't try anyway.

Â Okay, let's go on to number 4.

Â 4:30

Two of them are even, so that if the number m is odd,

Â the only possibilities are the other two.

Â And that's the proof, very slick argument.

Â Again, you might want to freeze the video at this point and

Â look at this, but absolutely perfect little proof.

Â Everything's there that you need to be there.

Â Number 5, again put down 24.

Â I think the only comment to make here is the fact that this solution doesn't

Â state explicitly that q is an integer as I've mentioned before in the grading.

Â When you're doing number theory you can assume that any variable that's not

Â explained or defined denotes an integer.

Â Integers are the default case.

Â So simply by looking at the context you can get by without saying it's an integer.

Â It's not wrong to say it's an integer,

Â you might want to do that just to play extra safe.

Â But in this context, that's going to be an integer.

Â 5:21

If we were trying to make a claim where we were saying something about there had

Â been a positive integer, then we'd need to say it.

Â But the default case when you're doing number theory is that

Â anything that's not specified can be taken to be an integer.

Â So there's no need to say that it's an integer, this is fine.

Â Okay, 24, we know that.

Â I think the only remark I want to make here, and I've made this remark before,

Â it's perfectly okay to cite previously established results,

Â just as it's okay to cite standard results.

Â So it's okay in the context of putting together a piece of

Â work to say by the answer to the previous question.

Â It's like using a level when you're proving a theorem.

Â So perfectly good, very short, very slick, definitely worth full marks.

Â 6:07

Well, the way most people prove this is by induction.

Â This person's done it a different way and this is very nice.

Â Look what they do. They say let S be the sum we're

Â interested in.

Â Then double S, and you multiply it through by 2, so you get a 2 squared there and

Â that last one becomes a 2n+1.

Â Then you subtract the first one from the second one, and everything drops out,

Â except you've got the 2n+1 at the beginning from the end.

Â And you've got the -2 from there.

Â So you've got 2n+1- 2, but 2S- S is just S.

Â So you've got S equals that guy.

Â S, this is S, remember, = 2n + 1- 2.

Â Isn't that nice?

Â That is beautiful.

Â 7:01

It's pleasing to see things like that.

Â Okay, so let me put the 24 down, and let's see how it works.

Â So epsilon greater than 0 be given, very nice, starting as one should,

Â by the assumption we can find an N such that this.

Â Why are we picking an N with this?

Â Well, because the whole thing is about multiplying by M, and so

Â in order to make it work out at the end, we're going to start by dividing by M.

Â Because we're looking ahead, we're going to say we're going to end up

Â with an M times something, and we want to end up with an epsilon.

Â Well, how can you take M times something and end up with an episilon?

Â Well, there's something you multiply it by, it's going to be epsilon over M.

Â So that where that came from, we get this by looking ahead to this last step.

Â 7:43

When you first meet this kind of structure,

Â these kind of proofs are hard to do.

Â But when you've seen a few of them, you get pretty good at looking ahead and

Â saying, yeah, I'm going to have to divide by M or I'm going to have to take

Â epsilon by two or whatever it is, and that gives you the limit.

Â So you almost certainly are having difficulty with this at this stage in

Â the game, but if you spent a few more weeks looking at this kind of argument,

Â you'd get very good at looking ahead.

Â It's not guess work, it's literally looking ahead to see what you want to do.

Â 8:24

This is about as slick as you can get.

Â Incidentally in the last sample exam solution,

Â this one was written out in words.

Â I think you'll agree with me,

Â [LAUGH] given that you're still in this course, that it's much clearer

Â to write it like this in symbols than to write it in terms of words, with saying

Â things like an is a distance less than epsilon over M of L, and so forth.

Â To the modern reader, symbols are much cleaner and

Â easier to understand than a lot of words.

Â 8:54

This is a particularly nice proof.

Â Admittedly, the person doesn't prove this part,

Â we've talked about that with the previous sample exam solution.

Â I think it's okay not to do that, it would sort of been nice to have it there,

Â but it's okay, given that this is so good.

Â And let's just, because it is obvious, I mean, these intervene,

Â arguably you should state that.

Â Okay, but we're pushing the limit of where that's really necessary now when we get to

Â this kind of sophistication.

Â And this is sophisticated, it's very slick.

Â You let those with the intervals, you observe that this is sort of almost,

Â it's getting very close to saying that, but it's a little bit beyond that.

Â You're saying it's a set, everything's within 0 to 1.

Â So everything in the intersection is going to be in there, and

Â then this is very nice.

Â If x is anything in there,

Â we can find a number such that it's going to be excluded from that interval.

Â 10:12

So this is almost exactly the same as the previous example except

Â that this is now left closed.

Â So 0 is an element of all of these things observed here, but now it's

Â okay to refer to the previous argument, the one for question 9, because that shows

Â that no element x strictly between 0 and 1 can be in the intersection.

Â So the intersection consists only of the point 0.

Â It really is the same argument as before at that point.

Â We've just included the initial point in there.

Â So, again, very elegant, very slick.

Â And that really completes the three sample solutions.

Â The first one was really a bit of a low end one.

Â There were all sorts of things wrong with that.

Â You often get papers like that.

Â In fact, you may get a fellow student's paper like that.

Â It's the kind of thing that you often get when students are really beginning

Â and struggling.

Â The second sample solution, that was the kind of thing that you would tend to get

Â from pretty good students in a class where we've got sort of 74, 75%.

Â That's probably as good as you'd get.

Â This solution set, the third one where everything was perfect,

Â now that's something that only a professional is likely to produce.

Â I mean, it really was a pro's job and

Â I would be very surprised to see that coming from a student.

Â It would be most unusual.

Â 11:32

So having done the three sample solutions,

Â your next task will be to grade three papers from your fellow students or

Â at least three papers from your fellow students.

Â Chances are, they're not going to look as slick and as elegant as this.

Â They're probably going to be a bit longer.

Â These are all compressed to fit on the slide, carefully chosen and

Â edited to make it look neat.

Â The real world is never as neat as that.

Â So you're probably going to find it challenging in different ways, but

Â I hope having gone through these three sample solutions

Â you've got some degree of confidence as to the kind of thing to look for.

Â It's clear that this is not an exact science.

Â By having a rubric, we can get some level of precision.

Â As I said, the professionals tend to agree by and large, but

Â you're going to get variations of one or two points absolutely, and

Â sometimes even more than that.

Â This is qualitative stuff, it's not quantitative, and assigning numbers

Â to something that's essentially based on judgments is not an easy thing to do.

Â Okay, well, good luck on grading your fellow students' work.

Â