0:05

Well, we're looking at the first sample problem set solution.

And in this one, the student's answer is wrong because the statement is false.

The problem is that this answer clearly assumes that zero is a natural number.

And by definition the natural numbers are one, two, three on,

that's historically correct as well so this is actually a false assumption.

So I'm not going to give full max logical correctness.

On the other hand, you will find some mathematicians who do include zero in

the natural numbers so I'm not going to be too strong in the penalization here.

0:43

But we're a community and

mathematics takes place in the community when we're writing proofs.

We do so, using the norms and practices of that community.

We design our proofs to match the audience that they're intended for.

So, writing a proof has to take account of which community you're in and

who you're writing for.

When I write for other mathematicians I do them very differently from the kind of

proofs I write for students.

When I write proofs for graduate students, I do it different.

We have to adjust the kind of arguments we're giving

depending on the community we're in and the people we're writing for.

So this is a community aspect to write its true, it's not a matter of right and

wrong always it's a matter of which one we communicate to.

So what i'm going to do is I'm going to put two for logical correctness.

I want to give some credit for

saying well if you do assume 0's a natural number then this is the solution.

But in fact we're in a community and we brought some accepted practices now.

We've established a convention that 0 is not an natural number and

in fact that is the standard convention within mathematics.

So I think I've been pretty generous here in giving two, but I think that's fair.

2:12

Overall, I think I would have to give a zero.

Because this is, from the point of view of what we're doing in this class, and

the audience we're trying to communicate to when we write proofs,

this is just not good.

It really doesn't answer the question within the framework that we're working.

Okay so that means I'm going to give a total of 18,

I think that's a pretty good result.

In fact I think that's generous given the students made

a sort of fundamental mistake within the context of this course.

2:47

For number two, the student gives the right answer, but

the proof is completely wrong, there is a fundamental misunderstanding here.

The student has assumed that this is an existential quantifier,

that there are five consecutive integers whose sum is divisible by 5.

In fact this is a universal quantifier.

3:11

What it means is that take any five consecutive integers then their sum is

divisible by 5.

So this is just not good, student doesn't really understand universal

quantification, so we're going to have to give zero for everything here.

This is just a basic misunderstanding,

there's nothing we can give any credit for, we have to give a zero.

When you really misunderstand something fundamental like what it

means to have a universal quantifier as opposed to an existential quantifier,

3:57

Okay, number three now.

Well first of all, the student doesn't actually say whether it's true or

false, not strictly speaking.

On the other hand, the student does start out and say I'm going to prove it, so

it's obvious that this person believes it's true, which it is.

So I'm not going to deduct any marks for not saying whether it's true or false,

that's just a slip.

It's got nothing to do with mathematical thinking I would say,

4:19

they just missed out saying it.

So there's something missing here but we're not going to worry about it,

going to prove it by induction.

Okay let's see, For n=1, true okay,

suppose this is the induction hypothesis

following this through this is all okay.

Beginning to look a bit suspicious this one because this argument doesn't

actually use the induction hypothesis.

Which means it's not an induction, it's not an induction for everyone.

4:57

Let's see, but one of n + 1, this is good,

this is good, we'll come back to talk about that in a minute.

Now that's good, so that's even.

Well, certainly this is to prove the results, but

it's not by induction because there was no use of the induction, that was never used.

And you know there's another problem here, this actually says for

any integer that includes the negative integers,

induction only works with the positive integers.

You can actually extend induction to work with the positive and negative integers,

we haven't done it yet in this course but you can do that.

5:38

But if you do that, then you're going to have to sort of take account for the fact

that it's a little bit different for the positive and the negative integers.

So as we've said, things are induction, and this is really standard.

Induction is usually taken to deduce to mean the positive integers and

the natural numbers.

But the fact that you can use it for

positive and negative is an extensions of it that's not what we're talking about.

So there's a problem here, the induction as we've got it standard induction,

standard mathematical induction doesn't prove things for all integers.

6:31

Logical correctness, I'm going to give one because there is a logical flow to this,

it's not an induction proof except in some strictly formal vacuous sense but

I'll give one.

Okay, clarity, I'll give it two, in and of itself,

it's clear but as a demonstration of of the truth of n, it's not clear.

7:04

Opening, again I'm going to give two because it sort

of begins fine it's just that it's not relevant.

It's not totally irrelevant, this is not completely off the wall but

it's not really addressing the issue, so I'm going to give two for that one.

State of conclusion, that was well done, I'll be 4 on that one,

the conclusion was stated.

7:31

Reasons, boy this is not an easy one.

I'm going to give two for that, sort of, I'm on the fence here,

reasons were given but they just weren't always the right reasons.

On the other hand, this guy really,

this is remarkable because if you up to here,

the real key to proving this is to take that guy.

And factor it as n times (n + 1) + 1.

And then observe that one of these is even, and the other one is odd.

Hence their product is even.

So when you add 1, it's odd.

So the proof of the result really just amounts to factoring this as n times (n+1)

+1 and observing that that product is even, and then when you add 1, it's odd.

That is the proof.

Now the person has spotted that here.

8:51

But everything else was just off the wall, almost off the wall, okay.

Okay, overall,

I'm going to have to give 0 because this simply doesn't get at the issue.

The student has sort of seen it, but wasn't aware that he or she had seen it.

So I can't give more than that.

Okay, so I've got a total of what?

4, 6, all right, let's say, 11.

9:13

Okay, just somewhat less than half marks.

Okay, I'd alter it, but I think that's about right.

I mean, I couldn't even get 50% for this one.

I think that's as good as I could give, because I'm really just giving credit for

this one particular observation.

9:47

But if this had been an actual student paper, if these questions were all

answered by the same student, then at this point I'd actually get rather suspicious

because questions one and two were answered very badly, I mean really badly.

And then suddenly a student comes up with a key idea.

At this point I'd begin to suspect that the student had actually copied something

from another student, either copied it wrongly or

copied something that been done done wrongly by the other student.

Because making this observation just seems so

strange given what we've seen in questions one and two.

And as I said, this is not an exam solution from a single student.

It's a compilation to illustrate points of grading and evaluation,

but in reality if I was grading a paper, I'd be suspicious.

I would still give the grades the same way.

You grade what's in front of you, but your suspicions are often aroused.

And sometimes of course with a real course where there's actual credit being given

for a certification and for a degree, say college credit,

then you often end up calling the student into the office and going through it and

double checking if the student really knows what they're doing.

Okay, we've given 11 for that,

I think the student shouldn't have any grounds to complain.

They've actually got as good a grade as they're likely to get for this one.

11:05

Okay number 4, we're proving by

induction, what's going on?

It looks as though the student is trying to prove some sort of a converse,

namely that every number of one of these forms is odd,

namely that 4n + 1 is odd, and 4n + 3 is odd.

Well that's trivial.

4n is even, so you add 1, it's odd.

You add 3 it's odd.

And yet that's what the student seems to be proving,

12:15

So the student is not even answering the question.

They're answering the complete converse of the question which is, in fact, trivial.

So, no way I can give any credit for that one, okay.

Let's just check and see if this is a valid argument.

If it was induction, let's just see.

That's okay if it's true for n.

12:44

Well, okay, but since it's got nothing to do with what's been asked for,

I'm not going to even look for giving partial credit for that.

In a different context I may well have looked for partial credit

if this had something to do with what was being asked for, but it's not.

This is just completely irrelevant to what's been asked for.

And as I've indicated when we set up the rubric,

you can give partial credit so long as it's relevant to the question.

13:24

Where number 5, I have no idea what the student's trying to do here.

It looks as though he or she has used that technique you often developed at high

school of sort of trying to match to a template, looked at the question, looked

through his or her notes to see something where this kind of thing is dealt with,

and then throwing down expressions that seemed relevant at that time.

In other words, using the template method,

trying to match the question to a template, and then apply the template,

which is as I've said many times in this course in the lectures already,

templates tend not to work for this kind of material.

You've got to think of the actual question.

So forget trying to match a template and apply a method that you've seen and

that you've got in your notes.

Think about the actual question, and

this has nothing to do with the actual question here.

These might seem like relevant statements, but they're not.

This simply doesn't really address it at all.

There's nothing I can give here except 0s throughout, again,

as in the previous question.

You can't look for

partial credit when what's written bears no real relationship to what's required.

14:38

Clutching at straws, hoping that by scattering some sentences down that were

copied from notes or from somebody else's notes you're going to get some grade, no,

isn't going to work.

Well, this is very much like the last one.

It's a clutching at straws argument, putting things down that look as

though they might be relevant on the hope that they'll get some partial credit.

But it really just doesn't address the, wait a minute.

There's something I kind of like about this.

Suppose p, q is a, Let's look at this first bit.

This is the opening part.

There's a period missing.

I'm not going to deduct any marks for punctuation, but let's put a period in.

This actually shows some mathematical sophistication.

There's something clever going on here.

This makes sense as an attempt to start to prove something.

Take a pair of twin primes and then show, with p greater than 5, and

show that you can't extend them.

So in terms of establishing a method whereby you're going to try and

prove something, this actually is kind of impressive.

Surprisingly so I've given the The other answers we've seen from this students.

And remember this is actually a compilation, it's not a single paper.

But if it was from a single student's paper then this would make me,

I'd want to talk to the student because a student who starts off

by saying this actually indicates some mathematical ability.

So I'd want to determine if it was that student's own work, because if it was

I would work with him or her in order to try and improve their overall performance.

Because this actually shows some ingenuity.

In fact, what I'm going to do is I'm going to give marks.

I'm going to give full marks for the opening because this is impressive.

I will give credit for a very good opening as a way of starting.

I approve.

Now, it turns out that the student's done nothing with this.

It makes no sense.

And so it's not logically correct.

It's not clear at all what's going on.

I have no idea what's going on here.

Okay. I mean, I can't give credit for

stating the conclusion because it's not addressing the question.

I can't give credit for that either.

And I'm certain not going to give credit for that.

I'm just going to acknowledge the fact that this indicates some

good mathematical sense, some ingenuity, and I want to give credit for that.

17:14

Okay so, you know, it sounds as though I'm contradicting what I;ve said earlier.

I'm actually not.

I'm trying to make a fine distinction

because this is not totally irrelevant to proving this kind of thing.

It's the kind of thing you might want to do.

It makes to start that but then it's not delivered,

it doesn't convert into anything sensible.

So it's not an irrelevant thing to say.

18:04

That's okay.

Assume it hold for n.

Adding, now this is nice, explaining what's going on.

So I'm certainly going to be giving partial credit for this one.

[INAUDIBLE] 2n+1 brings them together, simplifies that way.

Seventh.

Okay, so there's some really nice stuff here.

On the other hand there's some glaring holes in what's supposed to be a proof.

18:52

So the opening simply isn't there.

That's why part of the opening and last part of the opening is just a.

We're going to prove this by mathematical induction and that's completely missing.

Okay, this assumption here is part of the actual mechanics of the induction, okay.

Stating conclusion,

I'm going to go for 0 there and maybe I'd been a little bit harsh,

20:15

Reasons.

I'm simply going to give two max, because locally reasons were given.

But the big thing that was missing was induction,

and that's a big reason that's involved in the argument.

But within here, reasons were nicely given, so I'm going to give two for that.

And that means overall I'm going to give two.

20:45

The trouble is though the things that are missing are big things and so

you're going to lose a lot of marks without induction proofs,

draw upon a very important principle.

And you really have to state that you're using induction.

And you have to mention when you're using it.

Okay. Incidentally there was no mention

of where the induction proof used the induction hypothesis.

22:03

This is good.

This is good.

We can find an n so it's at that, by the assumption, by the given assumption.

Then whenever that scales, that all follows.

That works fine, which shows that, I'd be inclined to here want the student to put

something like, by the definition of limits.

23:10

Okay, a professional mathematician can look at this and

fill in the missing blanks and know what's going on.

But my guess is, is most of you, the students in this class,

would have to struggle to follow the student at this stage and

to see what they're doing, especially given the fact that there's nothing

by way of an opener to the thing and there's no reasons given.

23:43

I'll worry about that when I talk about giving marks for reasons I think.

But absolutely the conclusion is stated.

Arguably this is part of the conclusion.

And so when I look at reasons That I'm just going to give 2,

because there's no reasons given anywhere in here, and

in particular, it's not explained why it shows it.

Which shows that?

Well, at this level, a student is reasonably entitled to say,

how come it shows up, why does it show up, what's going on?

There's something missing.

So some reasonable beginning.

But at the very least, you need to alert people that this is simply the definition

of a limit that's been applied here.

So overall, [SOUND] I'm going to go with three.

I could've gone two, but I'm going three, because this is actually correct.

This is very slick, it's written very briefly and simply.

There's explanations missing that could help a beginner, but

this is actually pretty slick.

25:24

Well, the first thing I want to do here is get a picture of the student's answer.

So let's see, A1 is going to be 1 over 2 to 1.

A2 is going to be a third, to a half.

A3 is going to be a quarter to a third, etc.

So let me get a picture here.

And let me draw the unit interval from 0 to 1.

A1 is a half to 1, so

this guy here is A1, okay.

A2 is a third to a half, so let's put a third in.

And there's A2.

A3 is a quarter to a third, so let's put a quarter in.

Well, the scale is off here, but nevermind.

We're just going conceptually.

A3!

Already now, we see that the student's example doesn't work,

because it doesn't satisfy the condition that An+1 is a subset of An.

In fact, these are completely disjoined.

So, the example is plain wrong.

Dear, what a shame.

Okay, so it's going to have to be 0 here.

The example is wrong.

It doesn't satisfy the requirements.

26:44

Well, remember, we are trying to be helpful to a student.

The point of evaluation is to provide feedback, to give credit for

relevant work that is appropriate.

So, having seen that there's some stuff here that looks good,

I'll see if I can give credit.

Because what's going on here is the student's carrying out

the kind of argument that would be required to prove this result.

27:12

Now, admittedly, they have made a mistake here, and we've dinged them for

that quite significantly, they've got a 0 for that, the example's incorrect.

But the rest of this is a kind of argument that you would need to give, or

you could give, if the example was correct.

If the students had given intervals that satisfy this requirement,

then this is the kind of argument that you would need to do.

So if these An's satisfied that requirement,

if they'd been different definitions, this is the kind of argument you'd want to do..

So this is not irrelevant work here, there's a fundamental mistake,

29:40

But the argument itself is absolutely logical.

It's rock solid.

So, given the quality of the argument,

I certainly wouldn't penalize for doing that.

You may differ on that one, but as is pretty clear by now I'm sure,

grading, even when you've got a rubric, grading is pretty subjective.

Now it is the case that professionals, by their subjective grading, actually,

achieve an objective result, because professionals end up remarkably similar in

the grades they give, not exactly, but they're pretty close.

So, there is this, it's a mixture of art and science going on here.

But this is absolutely correct.

I'm going to give, I'll put the marks down, first of all.

I'm going to give 4 for everything.

30:23

Again, if this was a single student's submission, I would be very suspicious and

want to talk to that student.

Because to have gone from some really bizarrely wrong answers,

that suggests the student's got no understanding,

to produce something of this level of sophistication,

defies credibility as being a legitimate piece of work by one student.

30:56

Also, observe that anything other than 0 is eventually going to get missed out.

And so, anything other than 0 won't be in the intersection, so

the intersection consists purely of 0.

This is slick, it's slick and it's as brief and succinct but

correct as it is possible to be.

So grading what's in front of me, for me, it's straight 4s all the way through.

Because given the slick nature of this proof,

I would be forgiving for having missed that.

Yeah, that should be in.

I mean, you should make that statement,

especially given the audience that this is intended for,

which is an introductory class, so you should really observe things like that.

31:43

Okay, well, that's the first sample exam solution done.

What have we given for this one?

Let's see, we gave the totals, we've given for this one, for

the whole sheet we've done, we gave 18, we gave a 0.

Looking back on my list now, we've got an 11,

we gave a 0, we gave a 0, we gave a 4,

a 12, a 15, and 8, and a 24.

That's what we gave, so altogether,

we've given 92 out of a possible 240.

So overall, for this paper, we have given 38%.

Slightly more than one-third, okay?

32:45

Remember that the grades are meant to indicate how close you

are to being able to produce at the level of a professional.

And no one expects someone in an introductory course

to produce the kind of results that a professional does.

So, for this kind of course, or at this kind of level,

anything over the sort of 30, 35% mark is actually acceptable.

And it would be unusual for

a student to score sort of 70 or 80% if they were really genuinely a beginner.

You do find students like that.

You sometimes find students who score 90 and 100% at the beginning.

They tend to end up becoming professors of mathematics.

Okay, that's the first problem set solution done with.