0:20

And says you have to look at these values of m for that reason.

Â Look at those values of m from that reason.

Â And then says, if you calculate the values for all of those six possible pairs,

Â you'll find that the answer is never 12 and this proves the result.

Â So it's logically correct, 4.

Â It's perfectly clear.

Â There's a great opening.

Â 0:42

Conclusion is certainly stated.

Â Reasons are given.

Â Overall, 4 max.

Â So this is a 24.

Â Incident it is no need to actually carry out these calculations.

Â For this audience or for any audience, for

Â someone who is able to read a proof like this.

Â You can assume that they're able to carry out simple whole number calculations

Â like this.

Â So there's no need to actually carry out those calculations.

Â You can simply share it.

Â So, no need to do the calculations,

Â it's enough to state that the calculations can be done.

Â Saying that you're doing the calculations is different from doing the calculations.

Â 1:38

Well, let's look at number two now.

Â This person thinks it's false.

Â Well, something's gone wrong because this is actually true, okay.

Â 1:58

Then you add them all together, that's fine.

Â You've got 5n + 1 + 2 + 3 + 4, that's 5, you got 5ns and

Â you got a one and a two and a three and a four.

Â So you got 5n but, aha, there's a problem.

Â 4 + 3 + 2 + 1, it's actually 10.

Â Dear.

Â That is a shame.

Â 2:54

And the world doesn't come to an end.

Â Of course, if you're an engineer building a bridge or something,

Â then the consequences of a simple arithmetical slip can be quite severe.

Â But at the moment we're looking at

Â people's ability to do mathematical thinking.

Â And I'm not going to take an awful lot out

Â 3:12

of that result because they've made a simple arithmetic slip.

Â I'm going to give 2 for

Â logical correctness, because it's actually not logically correct.

Â Because adding those four together is part of the logical argument.

Â So the argument is not correct.

Â So I'm going to [INAUDIBLE] that down to 2.

Â It is however clear.

Â There's a nice, clean opening.

Â 3:35

Reasons are given, okay?

Â Even talking about the Division Theorem.

Â Overall though, I'm going to have to drop down to 2 for that, because it's wrong,

Â the result is wrong, okay?

Â And there's an arithmetical mistake.

Â 4:43

I'm going to be a little bit hard on this one.

Â So this person is claiming that it's true, okay?

Â Well, they're not claiming it but they're assuming it's true.

Â That was their conclusion, they simply didn't make a claim.

Â I think on this occasion, given that you have to sort of read through to the end.

Â To find out whether they think it's true or false, I think that's not a good start.

Â 5:32

In fact, it's even worse because that's really not what was claimed.

Â What was claimed is this n squared + n + 1 is odd.

Â So what you would have to say is that this guy equals that guy plus one.

Â So you might have to say hence,

Â cross that out there, is odd.

Â As claimed.

Â Okay, now we've sorted this guy out, okay?

Â This shows a particular proof.

Â So there are some problems with this.

Â The person sort of obviously made the key insight, knew what the answer was.

Â So the issue with this person isn't that he or

Â she isn't able to do the mathematics, they're not able to communicate it well.

Â This is not a proof.

Â We're looking for a proof and this isn't one for a whole variety of reasons.

Â So what am I going to do in the, well, it's logically correct.

Â I'm going to give 4 for that.

Â 6:41

Opening [LAUGH] well, in terms of the proof, there's a good opening.

Â I'm going to give 4 for the opening.

Â I'm going to solve separate sheet.

Â Stating it's true is not really the opening,

Â it's just sort of announcing the answer.

Â And then for the opening saying that for any n this is true, that's fine.

Â 7:23

So I've got a total of 20.

Â There's nothing really wrong.

Â It's just niggling little bits.

Â And in fact, if I was grading this one out of headache,

Â I might have even knocked it down a little bit more.

Â Because I think this is a lot of carelessness here.

Â They know what they're doing in a sense but they just need to figure out how to

Â learn, how to acquire the ability, you know what to do.

Â They know what they're doing.

Â What they need to do is learn how to express what they've done.

Â In a way that makes it a good argument, it's a good proof.

Â 8:03

If I was grading an actual paper from a student,

Â when that student had got marked into the 20's for the first three questions,

Â I would not have expected to see an answer like this.

Â And in fact this is a consequence of the fact that I've put these solution sheets

Â together from the kind of answers I've seen over the years.

Â And so this would have be produced by a student very different from the kinds of

Â students who would have produced the answers to questions one, two, and

Â three we just looked at, because this is just totally wrong.

Â It's a sort of trying to prove some kind of a converse.

Â The person is presumably trying to show that if you start with an integer,

Â then you get an odd natural number.

Â 9:08

Now if you're faced with a physical student, and

Â your job is to help that student do as best they can,

Â which is what we do with our students on the campus, then you're going to have to

Â spend some time with that student, because this is just totally confused.

Â And it certainly seems to be based on misunderstanding the implication.

Â You're supposed to start with any odd natural number and

Â deduce that it's of one of these forms,

Â not take an integer and look at the way these numbers follow from that integer.

Â And by taking a negative integer get

Â 9:59

In number five, what's going on here?

Â There's a bold statement here n can be expressed in one of the forms, can it?

Â Well, indeed it can.

Â The reason we know that is, because there's a very important theorem,

Â called the Division Theorem.

Â 10:30

So already we've got problems in this one.

Â A proof is meant to be an explanation.

Â It's meant to be a story, right?

Â With a beginning, a middle, and an end.

Â This doesn't have a beginning, and so we're just jumping straight in.

Â So it's not a proof, it's not an argument to convince someone.

Â It's a challenge to someone.

Â It's a puzzle to someone to figure out what's going on, and

Â that's not what proofs are about.

Â 10:56

Okay. We'll address that when I give the marks,

Â but let's read on through.

Â It's certainly the case that by the Division Theorem you do get that.

Â The remainder is either 0, 1, or 2 when you got 3 as the divisor.

Â In the first case, it's divisible by 3.

Â The second case, n+2 is divisible by 3.

Â And in the third case, n + 4 is divisible by 3, okay.

Â So, logical correctness?

Â Yeah, it's logically correct okay?

Â Everything this person does is what's required

Â in the terms of the logical structure.

Â 11:42

And I don't want to sort of double penalize someone, and so

Â this is an issue of reasons here and openings.

Â Nevertheless, having that missing means it's unclear.

Â It's also, there's a bit of unclarity, because there's no sort of

Â conclusion at the end saying, you might need to say something like this.

Â 12:16

And we've taken care of them.

Â Namely, possibility one, possibility two, [INAUDIBLE].

Â That alone, I mean this is the end of the story if you like.

Â 12:42

if this was a long argument if there were ten cases, the reader could have forgotten

Â where they were, but there are only three cases there and three lines.

Â They're right in front of the reader.

Â So the reader who's read through this,

Â 12:54

having had it established that there are three cases, the reader would instantly,

Â I think, recognize that that's the end of the proof.

Â So I'm not too worried about this one.

Â But coupled with the fact that it didn't have an opening, we've now got a story.

Â This has just got a middle, no beginning and no end.

Â So I'm going to knock it down for clarity here.

Â I'm also going to knock it down for

Â opening, because it doesn't have a good opening.

Â In fact, there really is no opening, this is just an in-your-face statement.

Â 13:20

Ditto stating the conclusion, it was left a little bit vague.

Â Reasons, I'm going to put a 0 there.

Â I mean, this is a huge reason.

Â This is the key to the whole result.

Â This is a consequence of the Division Theorem.

Â So if you don't mention the Division Theorem,

Â you haven't really said what's going on, you haven't said why this thing is true.

Â So this is huge, there's no way I could give more than zero for that.

Â Of course I couldn't give less than zero.

Â 13:48

And I don't think I'm double penalizing here,

Â because there's no beginning and there's no end.

Â And arguments are supposed to have beginnings, middles, and ends.

Â 14:06

In the course of giving an argument to establish a result.

Â Okay, so I've got what, I've got what looks like a 15 I think.

Â Right?

Â Yes so this person lost quite a few marks, but

Â I think in terms of proofs these are big, big errors.

Â 14:26

Yep, I think that's a fair mark, 15.

Â Okay, number 6, proves

Â that the only prime triple is 3, 5, 7.

Â So, we want to take 3 successive integers and show that they're not all primes.

Â 14:46

Where n is bigger than 3.

Â I show that 3 divides, this is a very good start, very good start, very clear.

Â Look at three numbers, well except this person clearly meant to say,

Â odd, because it's about odd numbers, and they've picked them two apart.

Â So there's a little bit of carelessness here.

Â Okay, well, I'm inclined to let that go.

Â I mean, I think it's clear, given the way this starts, that that's a typo.

Â If [INAUDIBLE], n = 3 + 1, 3q + 1 or 3q + 2 for some q.

Â Well, again, the person maybe should've said integer q here for some integer q.

Â 15:26

That would've been nice, but

Â it's really not required because we are doing number theory.

Â And in number theory you can assume, unless otherwise stated,

Â that everything is an integer, because that's what number theory is about.

Â 15:37

So, well, at least this kind of number theory, there's something else called

Â analytic number theory, but that's something different.

Â But in this kind of number theory, basic number theory, if it's not stated,

Â you can assume something's an integer.

Â If it's a positive integer, you need to say so, but integers are what it's about.

Â So let's see, in the first case, n + 2 = 3 q + 1, and

Â I think the person really wanted to put in here and missed it out,

Â 3 times q + 1, okay, because they want to say that, good lord, there's another typo.

Â This has to be a typo.

Â The person surely meant to write n + 2 there,

Â because they've shown that n + 2 equals 3 times something.

Â 16:20

And it indeed, they want to show that 3, the whole point is you want to show that

Â one of these things is divisible by something like 3, and so is not a prime.

Â And 3 is what's going to stop it being a prime, by the way.

Â So this was wrong.

Â I mean, given the kind of argument that's going on, I'm sure this is a typo,

Â but this is getting to be a bit of a habit, okay.

Â 16:40

In the second case, n+4 equals, and again, I think we should have put in here 3(q+2),

Â so, deary me, because then you've got n+4 as a multiple of 3.

Â So that should now say n+4.

Â Okay, this 3 must divide one of the three numbers, that's right.

Â This person was absolutely doing the right thing.

Â Which means they cannot all be prime, okay.

Â So the logic is very good, but boy, this person is careless.

Â So let's see how I'm going to grade it up.

Â Logically correct, yes.

Â Clarity, I'm going to put 4, because I think these were so

Â transparently typos, so transparently typos, all of them.

Â In fact, if I was grading a pile of papers, and

Â I wasn't thinking about recording myself going through the grading,

Â I would've just read through these and not even noticed these things, I would have

Â probably read what I thought I was reading, because the structure is so good.

Â So I think I'm only noticing these because I'm recording it for posterity.

Â Okay.

Â 17:52

The opening is wonderful.

Â Start with 3, they've absolutely stated the way they're going to prove it, okay?

Â The conclusion is stated correctly.

Â The only key reason to state, really, is the Division Theorem.

Â And that's stated, so I'm going to give 4 for that.

Â 18:12

On the other hand, this thing is littered with typos.

Â I'll overlook one typo, and maybe two, but there are typos everywhere here.

Â Now, that just is way too careless.

Â I'm going to go right down to zero here.

Â 18:27

You know, you can't get by if you're constantly making typos.

Â Though, incidentally,

Â you might have thought that I should have been a bit harsher here.

Â The reason I was very tolerant of typos is that those of us who are doing

Â mathematics, when we're proving things, we do make typos all the time.

Â We miswrite, we mistype, and the reason is, we're concentrating on the hard stuff.

Â We're concentrating on the logical flow.

Â I mean, I do the same thing when I'm speaking.

Â If you play some of these tapes over again, you'll realize that I often

Â don't say quite what I'm writing, or I misspeak, and that's because I'm focusing

Â on the depth of the things, I'm focusing on the concepts, not on what I'm saying.

Â And after, all this is a course about mathematical thinking and communication.

Â But in the case of communicating, if you wanted to see my attempts at

Â writing things down correctly with all the typos eliminated, then you read my book.

Â The book that goes with the course, there are maybe still one or two types, but

Â I worked out and I took feedback from other people to eliminate the typos.

Â So when we're producing papers and

Â books, those of us in the business do take care to get rid of the typos.

Â But when we're actually generating a proof, when we're thinking,

Â when we're doing something, then typos crop up all the time.

Â And yes, we'll deduct marks, because they do detract from being a proof.

Â But in themselves, typos are secondary at this stage in the game.

Â 19:56

Okay, now that docks it down to 20.

Â Still a remarkably good result, because this person clearly knew what to do,

Â and it's actually a very good proof.

Â It's a good proof that's spoiled because of total carelessness,

Â resulting in a lot of typos.

Â 20:14

And so I don't think I could have given more than 0 there.

Â It was just too many of them.

Â Okay, let's take a look at number 7.

Â So number 7, prove that for any natural number n this guy equals this guy.

Â Obviously, I think it's fairly obvious that you were going to do this

Â by induction.

Â But in any case, this person is doing it by induction and

Â begins very appropriately by stating that.

Â Let's check that the proof is correct.

Â For n = 1, the left-hand side reduces to just 2 to the 1, which is 2.

Â The right-hand side is 2 to the 2 minus 2, which is this, which is also 2,

Â so that's true.

Â Assume it holds for n, then, well, now we've got a puzzle on our hands.

Â How, I mean, where did this come from?

Â What is the person doing?

Â Where's the induction hypothesis being applied?

Â Well, if we stare at it for a few minutes,

Â we realize that the induction hypothesis is what lies behind what's going on here.

Â Because what this person's doing is saying,

Â this is the left-hand side of the identity we're trying to prove.

Â 21:18

And this is the right-hand side of the identity.

Â So we're taking the original identity, which is that this equals this, and

Â we're adding 2 to the n+1 to both sides.

Â So we're taking an assumed equation and adding the same thing to both sides.

Â 21:34

And then we're doing a little bit of algebraic simplification,

Â and we're going to come down to this, which is the identity for n+1.

Â And the result follows by induction.

Â Okay, so that's all fine, but we have to puzzle it out,

Â because what this person should have said was something like,

Â then adding 2 to the n+1 to both sides of the assumed identity.

Â 22:08

If we do that, then it's clear what's going on.

Â We're taking the assumed identity, and were adding the same thing to both sides.

Â We don't really need to explain this.

Â It's fairly obvious that this is just algebra.

Â You could say, by algebraic manipulation, by elementary algebra, say what you like,

Â but this is the key thing,

Â because this relates to the structure of the whole argument.

Â And that's the identity at n+1, and the result follows by induction.

Â Okay, so this is the problem, and

Â I'm not going to be able to overlook this, because it was a big miss.

Â Question is, what am I going to say in terms of the grade?

Â Well, it's certainly logically correct.

Â It wasn't clear, I had to work.

Â I had to sort of figure it out, what was going on.

Â So it's not a clear proof.

Â It certainly had a good opening, prove it by induction.

Â That was good.

Â Okay.

Â Conclusion was there.

Â In fact, the inner conclusion, the conclusion of the induction step was

Â there, and then the result follows by induction.

Â That's wonderful, I really like that.

Â This is where the person steps to the front of the stage and

Â takes a bow, takes the applause of the audience.

Â This is the end of the thing, it's the grand finale, I've gotta give a 4 for

Â that, wonderful.

Â Reasons, woo.

Â 23:21

Well, I mean, the big one was there, okay, the one that's stating it's by induction.

Â But this one is really important, and I'm going to take it down to 2 for that.

Â Yeah, I think 2 is about right.

Â You may want to go a bit lower than that,

Â you may say this is really significant, I'm inclined to think 2's okay.

Â I've sort of taken some account of it here.

Â But it is a major hole.

Â And as a result, I'm only going to give 2,

Â because this didn't read that smoothly as a result of that.

Â So the total is 18.

Â And I think that's a good mark in a sense.

Â I mean, that's the mark it should be because this was just,

Â it just left it too difficult.

Â 24:24

This expression is one that you'll see professional mathematicians use,

Â especially when they're sort of working among themselves.

Â Most of us I think when we're teaching discounted material to

Â an undergraduate audience we would avoid saying that.

Â Because that could be interpreted by a beginner as meaning just

Â 24:43

pick one that you like, or pull one off the shelf or something.

Â And that's not really what it means,

Â unless you're saying pull it off the shelf randomly.

Â It's much better to say let epsilon greater than 0 be given.

Â In other words, you have no choice whatsoever in that epsilon.

Â It's literally given to you arbitrarily.

Â 25:03

Now mathematicians use this.

Â So when they do that,

Â they're using elliptically to represent pick epsilon greater than 0 arbitrarily.

Â They don't articulate the way they are arbitrarily.

Â It's understood to mean pick it arbitrarily.

Â 25:19

But I'd say for a beginner, it could be confusing.

Â And since proofs are meant to be devices that convey reasoning,

Â convey the truth of something to a particular audience.

Â What classifies as an acceptable proof in one group of people,

Â doesn't necessarily classify as an acceptable proof in another.

Â We're not talking here about rigged

Â proofs in mathematical logic that can be read by computers or any of that.

Â We're talking about one human talking to another.

Â The focus of the course is both humans reasoning mathematics they're not just

Â about mathematics, but about things in the world and communicating their reasons.

Â That's what we're looking at.

Â That's why the course is called mathematical thinking.

Â It's not called mathematics or mathematical logic.

Â And so we have to be always conscious of who it is we're trying to convince and

Â to convey some meaning too.

Â And this I think is not a good idea.

Â And I'm going to deduct a little bit for

Â that even though mathematicians wouldn't do that.

Â Okay.

Â 26:22

So this is actually fine and if this was a graduate course in real analysis,

Â I would give full marks for this.

Â Absolutely.

Â Because I would know by that stage in their development,

Â everyone in the class would know what that meant.

Â And they would be able to read this.

Â This is actually, I think, difficult for

Â a beginner to read partly because we're using lots of words rather than symbols.

Â This is one of those occasions where I think some absolute value symbols and

Â some minus signs and some less than signs would make it clearer.

Â To the outsider to mathematics,

Â mathematical symbols seem to make things more obscure.

Â But once you're in mathematics,

Â I think we all found that the symbols are much clearer to read.

Â And if you don't believe that you should get hold of a medieval manuscript in

Â mathematics where they used almost no symbols and

Â it was all written in words, and they're almost impossible to make sense of.

Â In fact, when I do that, when I look at ancient manuscripts, and I do from time

Â to time, I translate them into modern symbols in order to understand them.

Â Okay, how am I going to assign points to this?

Â Well, it's logically correct.

Â Absolutely, it's logically correct, so four for that one.

Â 27:32

I'm going to say a three for clarity.

Â A couple of reasons, one is that I don't think is clear.

Â I think that just makes it obscure.

Â The student might well think, well what if I just randomly picked a number for

Â which the proof worked but it didn't always work?

Â 27:48

That would not be at all unjustifiable for a beginner to say that.

Â So I think there's some level of unclarity here.

Â And I also think this hence would need some kind of

Â a explanation like, by definition of limits.

Â 28:07

Just a pointer to the student as to what's going on, hence.

Â Well why hence?

Â Well because it's by definition of limits.

Â The stuff in the middle, if it was written out with absolute value symbols,

Â minus signs, less than signs, everywhere, inequalities in a standard way,

Â I think this would be relatively east to follow the logic for

Â a class at this level of development, sort of undergraduate level of development.

Â 28:30

So I think the confusion, if there is any,

Â is probably caused by the fact that it's written in words.

Â So I think I'll just take the one down here.

Â The opening I'm going to take one off there as well because although this is

Â something that professionals do, they do it unknowingly and

Â I think in this group with this audience, that actually is not a good opening.

Â 28:52

State the conclusion, I think I'll get four, I mean, I worry about that part.

Â Look, I mean, the conclusion was absolutely stated.

Â 29:02

Now arguably, this is part of the conclusion.

Â But I'm going to take care of that when I talk about not all the good reasons there.

Â I can't take this down by too many marks because this is technically all correct.

Â I'm just sort of nudging it because of audience design issues really.

Â 29:19

And likewise, I'm going to take that down to 3.

Â I think it's just a little bit too brief,

Â succinct, too crisp, in an advanced mathematical sense.

Â It's a function well as a proof in the community we're now working.

Â 29:34

So all together, I've lost, which is 20.

Â I've taken four marks off all together.

Â So it's a good mark.

Â And all of the deductions were based on the fact that this is not a good

Â argument for the audience for which it's intended.

Â And in some the sense, audience for which it's intended,

Â other students in this class who will be grading each other on the final exam.

Â So, one has to sort of provide a good example,

Â a good illustration to a typical student of the class.

Â Well, maybe not a typical student,

Â a student who's still here at the end of the class.

Â And as we all know, that's usually between 5 and

Â 10% of the students who start the MOOC.

Â So with that intended audience, I think I'm justified in taking four marks off.

Â 30:26

Okay, what does this person do?

Â Well, they obviously begin by a typo there I guess, should be a comma.

Â This person begins by sort of writing some explicit intervals.

Â There's a unit interval, open unit of interval, then first half of that,

Â then the first half of that then the first half,

Â now some of the checking in a open interval between 0 and 1.

Â Then they're taking just the 1 between the 1/2.

Â Then they're going down to a quarter.

Â Then they're going down, down, down.

Â And they're going to squeeze in on the origin there, okay.

Â And then they write down the general definition of the nth interval.

Â That's good.

Â [SOUND] Okay, right.

Â Well, I did my PhD in set theory.

Â I worked solidly in researching set theory for ten, 15 years or so.

Â So when I read this, I have no trouble with this.

Â I know exactly what's going on.

Â This works as a brilliant proof for me.

Â So if I'm the intended reader, yup, full max.

Â 31:22

But the intended reader for this proof,

Â this particular one of this clause isn't me.

Â It's other students in the course.

Â And for that audience this is just not good.

Â It's just too obscure.

Â So I'm going to start taking some marks off for

Â the person making it just too difficult to follow.

Â 31:43

I can't take marks off of logical correctness.

Â Because it's absolutely logically correct.

Â I saw that the instance I looked at it.

Â I'm going to take a mark off for clarity because it's really not clear.

Â It's obscure.

Â Opening, in so far as there is an opening, that's fine.

Â I mean, this person actually sort of even motivates it by looking at the sequence

Â that we're taking the definition for.

Â So, I think that's a good opening.

Â 32:08

What about the conclusion?

Â Well, the conclusion was stated absolutely crisply.

Â So, the conclusion was definitely stated.

Â We have no doubt that this person claims at this point to have proved the result.

Â Okay, what about reasons?

Â 32:24

Boy, I can only give 2 for that.

Â As I say, if I'm the intended reader, this is all the reasons I need there.

Â But for students in this class, I think we need a little bit more than that.

Â We need some assistance as to why that tending to 0 implies that.

Â 32:41

Well, maybe I'm even being too generous with 2.

Â But I'm inclined to sort of go with 2 here.

Â And overall, I'm going to take it down by one.

Â Because I just think, it's just too obscure for the intended audience.

Â 32:56

So I'm down to 20, still a good mark.

Â But I think, just way, way too brief for this audience.

Â As I say, for a graduate class in set theory, absolutely fine.

Â For an undergraduate introductory course like this,

Â where set theory is just an example at the end, not fine.

Â 33:32

And in this case, the person has taken the closed interval from 0 to 1 over 2n.

Â [SOUND] By the same argument as in question, that's fine.

Â If you've got an argument that you've already used and you've established,

Â you don't have to repeat the argument.

Â You can simply refer back to it.

Â It's a bit like calling a subroutine in a computer program, or

Â pulling up a module or something like that.

Â You're entitled to do that.

Â That's a perfectly valid thing to do, providing the material is accessible and

Â within the context of the same exam, that's fine.

Â And then it follows that that equals that.

Â 34:04

Okay, so it's essentially the same argument as before.

Â With a closed interval, then you get a slightly different outcome.

Â So I'm going to approach this.

Â I should mention one thing I've just noticed.

Â Neither in the last case, nor in the present case, did the person prove that.

Â Well, you could say they should have proved that,

Â because you are supposed to show that it has a stated property.

Â And arguably, that is part of the stated property.

Â I mean, I think most of us probably read it as part of the set-up, but

Â it really has to be verified.

Â On the other hand, when you're looking at these kind of intervals,

Â it's obvious that they're nested if you start drawing the intervals.

Â So I'm not going to ding any marks for that.

Â I'm not going to go back to the question 9 and take a mark off.

Â 34:54

that just pales into insignificance.

Â So in other circumstances, maybe we'd need to take more account of that.

Â But for here, I think that we can just focus on the half of the thing.

Â Since it's essentially the same argument as before,

Â I'm going to approach the grading as before.

Â It's logically correct, yes.

Â I'll take one off for clarity,

Â because it's not clear to the intended reader of this argument.

Â There's a good opening.

Â There's a good, clear statement of the conclusion, some missing reasons.

Â And overall, we need to take a point off because it was just too slick, and

Â show off, and brief, for the intended audience.

Â So that takes me down to 20 in that one.

Â 35:38

So altogether on this paper I'll give, what did I give?

Â Now, I give a total of 24 for the first one, 20 to the second, then another 20.

Â Then there was that 0, that disastrous question, that disastrous fourth question.

Â Then there was 15, then there was a 20.

Â There was an 18, then there was a 20,

Â and another 20, and another 20!

Â There were 20s all the way through at the end, right.

Â Okay, so I've got a total of 177

Â out of 240, which is 74%.

Â So, although this wasn't the work of a single student, it was an aggregate.

Â By and large, most of the questions were done pretty well.

Â We've got lots of results in the 20s and the 18s.

Â So, if this had been a single student, and

Â it's possible that it was even a single student and there was an aberration.

Â