We're looking at the second sample problem set solution. The answer to question one, this person gives is perfectly okay. Observes that you are, first of all, makes a note that this thing is false which is correct. And says you have to look at these values of m for that reason. Look at those values of m from that reason. And then says, if you calculate the values for all of those six possible pairs, you'll find that the answer is never 12 and this proves the result. So it's logically correct, 4. It's perfectly clear. There's a great opening. Conclusion is certainly stated. Reasons are given. Overall, 4 max. So this is a 24. Incident it is no need to actually carry out these calculations. For this audience or for any audience, for someone who is able to read a proof like this. You can assume that they're able to carry out simple whole number calculations like this. So there's no need to actually carry out those calculations. You can simply share it. So, no need to do the calculations, it's enough to state that the calculations can be done. Saying that you're doing the calculations is different from doing the calculations. If you hadn't said that, then you're leaving the reader in the lurch. But simply by saying that it follows by calculation, the reader can fill in the details. Perfectly okay. Well, let's look at number two now. This person thinks it's false. Well, something's gone wrong because this is actually true, okay. Let's see what's gone wrong. This is okay, that's a good representation of an arbitrary sequence of five consecutive integers. Then you add them all together, that's fine. You've got 5n + 1 + 2 + 3 + 4, that's 5, you got 5ns and you got a one and a two and a three and a four. So you got 5n but, aha, there's a problem. 4 + 3 + 2 + 1, it's actually 10. Dear. That is a shame. How do we deal with this? Because this is clearly an utterly trivial mistake. The person has simply made a little arithmetical mistake. So I'm not going to take too many marks off that, because clearly anyone who can produce an argument like this, which is fine. Is capable of adding four numbers together and getting the right answer. They just made a slip. And the experts make slips like that all the time. I make slips like that all the time. And the world doesn't come to an end. Of course, if you're an engineer building a bridge or something, then the consequences of a simple arithmetical slip can be quite severe. But at the moment we're looking at people's ability to do mathematical thinking. And I'm not going to take an awful lot out of that result because they've made a simple arithmetic slip. I'm going to give 2 for logical correctness, because it's actually not logically correct. Because adding those four together is part of the logical argument. So the argument is not correct. So I'm going to [INAUDIBLE] that down to 2. It is however clear. There's a nice, clean opening. Clear statement of a conclusion. Reasons are given, okay? Even talking about the Division Theorem. Overall though, I'm going to have to drop down to 2 for that, because it's wrong, the result is wrong, okay? And there's an arithmetical mistake. Yes, it's tiny little slip, simple arithmetical slip, we know this person could have gotten the right answer if they were fully awake at the time. But end of the day it's wrong, so this person only get's 20. I say only, 20, 24 is actually this kind of material is a good resource. So we're just making note of the fact that there is an arithmetical mistake. Otherwise, we'll congratulate the person on a nice elegant little proof. In question three, what does this person do? First of all, they don't say whether it's going to be true or false. And we can't immediately infer that without reading through the thing. I'm going to be a little bit hard on this one. So this person is claiming that it's true, okay? Well, they're not claiming it but they're assuming it's true. That was their conclusion, they simply didn't make a claim. I think on this occasion, given that you have to sort of read through to the end. To find out whether they think it's true or false, I think that's not a good start. Okay, let me see. This is good, this is really the key to this thing. It's the observation that a pair of successive integers multiplied together is always even, because one is even and the other one is odd. Around here is a typo, because that's meant to say +1. So there's some carelessness in writing this one down. But the idea about this is there's some carelessness. In fact, it's even worse because that's really not what was claimed. What was claimed is this n squared + n + 1 is odd. So what you would have to say is that this guy equals that guy plus one. So you might have to say hence, cross that out there, is odd. As claimed. Okay, now we've sorted this guy out, okay? This shows a particular proof. So there are some problems with this. The person sort of obviously made the key insight, knew what the answer was. So the issue with this person isn't that he or she isn't able to do the mathematics, they're not able to communicate it well. This is not a proof. We're looking for a proof and this isn't one for a whole variety of reasons. So what am I going to do in the, well, it's logically correct. I'm going to give 4 for that. I'm only going to give 3 for clarity, once we got into it, things were fairly clear. There were some typos and so forth, but I think it's sufficiently clear. Opening [LAUGH] well, in terms of the proof, there's a good opening. I'm going to give 4 for the opening. I'm going to solve separate sheet. Stating it's true is not really the opening, it's just sort of announcing the answer. And then for the opening saying that for any n this is true, that's fine. Stating the conclusion, I'm going to give 3 because he really didn't state this part of the conclusion, he left it in that other form. Giving reasons, that I think is fine, reasons are given. Overall, I'm going to go down to 2 on that one, I think. So I've got a total of 20. There's nothing really wrong. It's just niggling little bits. And in fact, if I was grading this one out of headache, I might have even knocked it down a little bit more. Because I think this is a lot of carelessness here. They know what they're doing in a sense but they just need to figure out how to learn, how to acquire the ability, you know what to do. They know what they're doing. What they need to do is learn how to express what they've done. In a way that makes it a good argument, it's a good proof. If I was grading an actual paper from a student, when that student had got marked into the 20's for the first three questions, I would not have expected to see an answer like this. And in fact this is a consequence of the fact that I've put these solution sheets together from the kind of answers I've seen over the years. And so this would have be produced by a student very different from the kinds of students who would have produced the answers to questions one, two, and three we just looked at, because this is just totally wrong. It's a sort of trying to prove some kind of a converse. The person is presumably trying to show that if you start with an integer, then you get an odd natural number. You start by doing this, and the they've said it's not true because of there are circumstances you get negative numbers, which are not natural numbers. So it's just a total failure to understand what's required here. And when you've got that degree of failure the grading sort of becomes easy in terms of giving a numerical grade. You put 0s all the way through. Now if you're faced with a physical student, and your job is to help that student do as best they can, which is what we do with our students on the campus, then you're going to have to spend some time with that student, because this is just totally confused. And it certainly seems to be based on misunderstanding the implication. You're supposed to start with any odd natural number and deduce that it's of one of these forms, not take an integer and look at the way these numbers follow from that integer. And by taking a negative integer get the result that these things are not natural numbers at all. So in a sense this is an easy one to grade numerically. It's just 0s across the board. In number five, what's going on here? There's a bold statement here n can be expressed in one of the forms, can it? Well, indeed it can. The reason we know that is, because there's a very important theorem, called the Division Theorem. And without stating that, we're just making a bold statement and leaving it up to the reader to figure out why, okay? So already we've got problems in this one. A proof is meant to be an explanation. It's meant to be a story, right? With a beginning, a middle, and an end. This doesn't have a beginning, and so we're just jumping straight in. So it's not a proof, it's not an argument to convince someone. It's a challenge to someone. It's a puzzle to someone to figure out what's going on, and that's not what proofs are about. Okay. We'll address that when I give the marks, but let's read on through. It's certainly the case that by the Division Theorem you do get that. The remainder is either 0, 1, or 2 when you got 3 as the divisor. In the first case, it's divisible by 3. The second case, n+2 is divisible by 3. And in the third case, n + 4 is divisible by 3, okay. So, logical correctness? Yeah, it's logically correct okay? Everything this person does is what's required in the terms of the logical structure. Clarity, I'm going to knock that down one in terms of clarity, because that makes it somewhat unclear. I'm going to take more account of that in a minute. And I don't want to sort of double penalize someone, and so this is an issue of reasons here and openings. Nevertheless, having that missing means it's unclear. It's also, there's a bit of unclarity, because there's no sort of conclusion at the end saying, you might need to say something like this. What can we say? This takes care of all three possibilities. Something like that, just to sort of say that there are three possibilities. And we've taken care of them. Namely, possibility one, possibility two, [INAUDIBLE]. That alone, I mean this is the end of the story if you like. And so we could say, and that proves the result. What could be 2 or 3 open about that. But I'm not too worried about that one, because if this was a long argument if there were ten cases, the reader could have forgotten where they were, but there are only three cases there and three lines. They're right in front of the reader. So the reader who's read through this, having had it established that there are three cases, the reader would instantly, I think, recognize that that's the end of the proof. So I'm not too worried about this one. But coupled with the fact that it didn't have an opening, we've now got a story. This has just got a middle, no beginning and no end. So I'm going to knock it down for clarity here. I'm also going to knock it down for opening, because it doesn't have a good opening. In fact, there really is no opening, this is just an in-your-face statement. Ditto stating the conclusion, it was left a little bit vague. Reasons, I'm going to put a 0 there. I mean, this is a huge reason. This is the key to the whole result. This is a consequence of the Division Theorem. So if you don't mention the Division Theorem, you haven't really said what's going on, you haven't said why this thing is true. So this is huge, there's no way I could give more than zero for that. Of course I couldn't give less than zero. So overall, I'm going to put a two here. And I don't think I'm double penalizing here, because there's no beginning and there's no end. And arguments are supposed to have beginnings, middles, and ends. At this level in mathematics when we're talking about proofs, it's not enough to be able to do the algebra. It's to be able to use the algebra In the course of giving an argument to establish a result. Okay, so I've got what, I've got what looks like a 15 I think. Right? Yes so this person lost quite a few marks, but I think in terms of proofs these are big, big errors. Yep, I think that's a fair mark, 15. Okay, number 6, proves that the only prime triple is 3, 5, 7. So, we want to take 3 successive integers and show that they're not all primes. Where n is bigger than 3. I show that 3 divides, this is a very good start, very good start, very clear. Look at three numbers, well except this person clearly meant to say, odd, because it's about odd numbers, and they've picked them two apart. So there's a little bit of carelessness here. Okay, well, I'm inclined to let that go. I mean, I think it's clear, given the way this starts, that that's a typo. If [INAUDIBLE], n = 3 + 1, 3q + 1 or 3q + 2 for some q. Well, again, the person maybe should've said integer q here for some integer q. That would've been nice, but it's really not required because we are doing number theory. And in number theory you can assume, unless otherwise stated, that everything is an integer, because that's what number theory is about. So, well, at least this kind of number theory, there's something else called analytic number theory, but that's something different. But in this kind of number theory, basic number theory, if it's not stated, you can assume something's an integer. If it's a positive integer, you need to say so, but integers are what it's about. So let's see, in the first case, n + 2 = 3 q + 1, and I think the person really wanted to put in here and missed it out, 3 times q + 1, okay, because they want to say that, good lord, there's another typo. This has to be a typo. The person surely meant to write n + 2 there, because they've shown that n + 2 equals 3 times something. And it indeed, they want to show that 3, the whole point is you want to show that one of these things is divisible by something like 3, and so is not a prime. And 3 is what's going to stop it being a prime, by the way. So this was wrong. I mean, given the kind of argument that's going on, I'm sure this is a typo, but this is getting to be a bit of a habit, okay. In the second case, n+4 equals, and again, I think we should have put in here 3(q+2), so, deary me, because then you've got n+4 as a multiple of 3. So that should now say n+4. Okay, this 3 must divide one of the three numbers, that's right. This person was absolutely doing the right thing. Which means they cannot all be prime, okay. So the logic is very good, but boy, this person is careless. So let's see how I'm going to grade it up. Logically correct, yes. Clarity, I'm going to put 4, because I think these were so transparently typos, so transparently typos, all of them. In fact, if I was grading a pile of papers, and I wasn't thinking about recording myself going through the grading, I would've just read through these and not even noticed these things, I would have probably read what I thought I was reading, because the structure is so good. So I think I'm only noticing these because I'm recording it for posterity. Okay. The opening is wonderful. Start with 3, they've absolutely stated the way they're going to prove it, okay? The conclusion is stated correctly. The only key reason to state, really, is the Division Theorem. And that's stated, so I'm going to give 4 for that. On the other hand, this thing is littered with typos. I'll overlook one typo, and maybe two, but there are typos everywhere here. Now, that just is way too careless. I'm going to go right down to zero here. You know, you can't get by if you're constantly making typos. Though, incidentally, you might have thought that I should have been a bit harsher here. The reason I was very tolerant of typos is that those of us who are doing mathematics, when we're proving things, we do make typos all the time. We miswrite, we mistype, and the reason is, we're concentrating on the hard stuff. We're concentrating on the logical flow. I mean, I do the same thing when I'm speaking. If you play some of these tapes over again, you'll realize that I often don't say quite what I'm writing, or I misspeak, and that's because I'm focusing on the depth of the things, I'm focusing on the concepts, not on what I'm saying. And after, all this is a course about mathematical thinking and communication. But in the case of communicating, if you wanted to see my attempts at writing things down correctly with all the typos eliminated, then you read my book. The book that goes with the course, there are maybe still one or two types, but I worked out and I took feedback from other people to eliminate the typos. So when we're producing papers and books, those of us in the business do take care to get rid of the typos. But when we're actually generating a proof, when we're thinking, when we're doing something, then typos crop up all the time. And yes, we'll deduct marks, because they do detract from being a proof. But in themselves, typos are secondary at this stage in the game. And so I just took the marks off right at the end. Okay, now that docks it down to 20. Still a remarkably good result, because this person clearly knew what to do, and it's actually a very good proof. It's a good proof that's spoiled because of total carelessness, resulting in a lot of typos. And so I don't think I could have given more than 0 there. It was just too many of them. Okay, let's take a look at number 7. So number 7, prove that for any natural number n this guy equals this guy. Obviously, I think it's fairly obvious that you were going to do this by induction. But in any case, this person is doing it by induction and begins very appropriately by stating that. Let's check that the proof is correct. For n = 1, the left-hand side reduces to just 2 to the 1, which is 2. The right-hand side is 2 to the 2 minus 2, which is this, which is also 2, so that's true. Assume it holds for n, then, well, now we've got a puzzle on our hands. How, I mean, where did this come from? What is the person doing? Where's the induction hypothesis being applied? Well, if we stare at it for a few minutes, we realize that the induction hypothesis is what lies behind what's going on here. Because what this person's doing is saying, this is the left-hand side of the identity we're trying to prove. And this is the right-hand side of the identity. So we're taking the original identity, which is that this equals this, and we're adding 2 to the n+1 to both sides. So we're taking an assumed equation and adding the same thing to both sides. And then we're doing a little bit of algebraic simplification, and we're going to come down to this, which is the identity for n+1. And the result follows by induction. Okay, so that's all fine, but we have to puzzle it out, because what this person should have said was something like, then adding 2 to the n+1 to both sides of the assumed identity. If we do that, then it's clear what's going on. We're taking the assumed identity, and were adding the same thing to both sides. We don't really need to explain this. It's fairly obvious that this is just algebra. You could say, by algebraic manipulation, by elementary algebra, say what you like, but this is the key thing, because this relates to the structure of the whole argument. And that's the identity at n+1, and the result follows by induction. Okay, so this is the problem, and I'm not going to be able to overlook this, because it was a big miss. Question is, what am I going to say in terms of the grade? Well, it's certainly logically correct. It wasn't clear, I had to work. I had to sort of figure it out, what was going on. So it's not a clear proof. It certainly had a good opening, prove it by induction. That was good. Okay. Conclusion was there. In fact, the inner conclusion, the conclusion of the induction step was there, and then the result follows by induction. That's wonderful, I really like that. This is where the person steps to the front of the stage and takes a bow, takes the applause of the audience. This is the end of the thing, it's the grand finale, I've gotta give a 4 for that, wonderful. Reasons, woo. Well, I mean, the big one was there, okay, the one that's stating it's by induction. But this one is really important, and I'm going to take it down to 2 for that. Yeah, I think 2 is about right. You may want to go a bit lower than that, you may say this is really significant, I'm inclined to think 2's okay. I've sort of taken some account of it here. But it is a major hole. And as a result, I'm only going to give 2, because this didn't read that smoothly as a result of that. So the total is 18. And I think that's a good mark in a sense. I mean, that's the mark it should be because this was just, it just left it too difficult. Okay, let's see if number 8 brings any more joy. [SOUND] This is interesting. This expression is one that you'll see professional mathematicians use, especially when they're sort of working among themselves. Most of us I think when we're teaching discounted material to an undergraduate audience we would avoid saying that. Because that could be interpreted by a beginner as meaning just pick one that you like, or pull one off the shelf or something. And that's not really what it means, unless you're saying pull it off the shelf randomly. It's much better to say let epsilon greater than 0 be given. In other words, you have no choice whatsoever in that epsilon. It's literally given to you arbitrarily. Now mathematicians use this. So when they do that, they're using elliptically to represent pick epsilon greater than 0 arbitrarily. They don't articulate the way they are arbitrarily. It's understood to mean pick it arbitrarily. But I'd say for a beginner, it could be confusing. And since proofs are meant to be devices that convey reasoning, convey the truth of something to a particular audience. What classifies as an acceptable proof in one group of people, doesn't necessarily classify as an acceptable proof in another. We're not talking here about rigged proofs in mathematical logic that can be read by computers or any of that. We're talking about one human talking to another. The focus of the course is both humans reasoning mathematics they're not just about mathematics, but about things in the world and communicating their reasons. That's what we're looking at. That's why the course is called mathematical thinking. It's not called mathematics or mathematical logic. And so we have to be always conscious of who it is we're trying to convince and to convey some meaning too. And this I think is not a good idea. And I'm going to deduct a little bit for that even though mathematicians wouldn't do that. Okay. Since that tends to limit this, there is an N switch that does, that's true. So this is actually fine and if this was a graduate course in real analysis, I would give full marks for this. Absolutely. Because I would know by that stage in their development, everyone in the class would know what that meant. And they would be able to read this. This is actually, I think, difficult for a beginner to read partly because we're using lots of words rather than symbols. This is one of those occasions where I think some absolute value symbols and some minus signs and some less than signs would make it clearer. To the outsider to mathematics, mathematical symbols seem to make things more obscure. But once you're in mathematics, I think we all found that the symbols are much clearer to read. And if you don't believe that you should get hold of a medieval manuscript in mathematics where they used almost no symbols and it was all written in words, and they're almost impossible to make sense of. In fact, when I do that, when I look at ancient manuscripts, and I do from time to time, I translate them into modern symbols in order to understand them. Okay, how am I going to assign points to this? Well, it's logically correct. Absolutely, it's logically correct, so four for that one. I'm going to say a three for clarity. A couple of reasons, one is that I don't think is clear. I think that just makes it obscure. The student might well think, well what if I just randomly picked a number for which the proof worked but it didn't always work? That would not be at all unjustifiable for a beginner to say that. So I think there's some level of unclarity here. And I also think this hence would need some kind of a explanation like, by definition of limits. Just a pointer to the student as to what's going on, hence. Well why hence? Well because it's by definition of limits. The stuff in the middle, if it was written out with absolute value symbols, minus signs, less than signs, everywhere, inequalities in a standard way, I think this would be relatively east to follow the logic for a class at this level of development, sort of undergraduate level of development. So I think the confusion, if there is any, is probably caused by the fact that it's written in words. So I think I'll just take the one down here. The opening I'm going to take one off there as well because although this is something that professionals do, they do it unknowingly and I think in this group with this audience, that actually is not a good opening. State the conclusion, I think I'll get four, I mean, I worry about that part. Look, I mean, the conclusion was absolutely stated. Now arguably, this is part of the conclusion. But I'm going to take care of that when I talk about not all the good reasons there. I can't take this down by too many marks because this is technically all correct. I'm just sort of nudging it because of audience design issues really. And likewise, I'm going to take that down to 3. I think it's just a little bit too brief, succinct, too crisp, in an advanced mathematical sense. It's a function well as a proof in the community we're now working. So all together, I've lost, which is 20. I've taken four marks off all together. So it's a good mark. And all of the deductions were based on the fact that this is not a good argument for the audience for which it's intended. And in some the sense, audience for which it's intended, other students in this class who will be grading each other on the final exam. So, one has to sort of provide a good example, a good illustration to a typical student of the class. Well, maybe not a typical student, a student who's still here at the end of the class. And as we all know, that's usually between 5 and 10% of the students who start the MOOC. So with that intended audience, I think I'm justified in taking four marks off. Okay. Well this is interesting. The answer is what shows on the question, but of course the question contains the definition of this thing. Okay, what does this person do? Well, they obviously begin by a typo there I guess, should be a comma. This person begins by sort of writing some explicit intervals. There's a unit interval, open unit of interval, then first half of that, then the first half of that then the first half, now some of the checking in a open interval between 0 and 1. Then they're taking just the 1 between the 1/2. Then they're going down to a quarter. Then they're going down, down, down. And they're going to squeeze in on the origin there, okay. And then they write down the general definition of the nth interval. That's good. [SOUND] Okay, right. Well, I did my PhD in set theory. I worked solidly in researching set theory for ten, 15 years or so. So when I read this, I have no trouble with this. I know exactly what's going on. This works as a brilliant proof for me. So if I'm the intended reader, yup, full max. But the intended reader for this proof, this particular one of this clause isn't me. It's other students in the course. And for that audience this is just not good. It's just too obscure. So I'm going to start taking some marks off for the person making it just too difficult to follow. I can't take marks off of logical correctness. Because it's absolutely logically correct. I saw that the instance I looked at it. I'm going to take a mark off for clarity because it's really not clear. It's obscure. Opening, in so far as there is an opening, that's fine. I mean, this person actually sort of even motivates it by looking at the sequence that we're taking the definition for. So, I think that's a good opening. What about the conclusion? Well, the conclusion was stated absolutely crisply. So, the conclusion was definitely stated. We have no doubt that this person claims at this point to have proved the result. Okay, what about reasons? Boy, I can only give 2 for that. As I say, if I'm the intended reader, this is all the reasons I need there. But for students in this class, I think we need a little bit more than that. We need some assistance as to why that tending to 0 implies that. Well, maybe I'm even being too generous with 2. But I'm inclined to sort of go with 2 here. And overall, I'm going to take it down by one. Because I just think, it's just too obscure for the intended audience. So I'm down to 20, still a good mark. But I think, just way, way too brief for this audience. As I say, for a graduate class in set theory, absolutely fine. For an undergraduate introductory course like this, where set theory is just an example at the end, not fine. Unfair on the reader, and we've taken account of that. Okay, and finally, we've got this one, very similar, a sort of variant of the previous one if you like. And in this case, the person has taken the closed interval from 0 to 1 over 2n. [SOUND] By the same argument as in question, that's fine. If you've got an argument that you've already used and you've established, you don't have to repeat the argument. You can simply refer back to it. It's a bit like calling a subroutine in a computer program, or pulling up a module or something like that. You're entitled to do that. That's a perfectly valid thing to do, providing the material is accessible and within the context of the same exam, that's fine. And then it follows that that equals that. Okay, so it's essentially the same argument as before. With a closed interval, then you get a slightly different outcome. So I'm going to approach this. I should mention one thing I've just noticed. Neither in the last case, nor in the present case, did the person prove that. Well, you could say they should have proved that, because you are supposed to show that it has a stated property. And arguably, that is part of the stated property. I mean, I think most of us probably read it as part of the set-up, but it really has to be verified. On the other hand, when you're looking at these kind of intervals, it's obvious that they're nested if you start drawing the intervals. So I'm not going to ding any marks for that. I'm not going to go back to the question 9 and take a mark off. I think given that there's a degree of complexity in the reasoning here, that just pales into insignificance. So in other circumstances, maybe we'd need to take more account of that. But for here, I think that we can just focus on the half of the thing. Since it's essentially the same argument as before, I'm going to approach the grading as before. It's logically correct, yes. I'll take one off for clarity, because it's not clear to the intended reader of this argument. There's a good opening. There's a good, clear statement of the conclusion, some missing reasons. And overall, we need to take a point off because it was just too slick, and show off, and brief, for the intended audience. So that takes me down to 20 in that one. So altogether on this paper I'll give, what did I give? Now, I give a total of 24 for the first one, 20 to the second, then another 20. Then there was that 0, that disastrous question, that disastrous fourth question. Then there was 15, then there was a 20. There was an 18, then there was a 20, and another 20, and another 20! There were 20s all the way through at the end, right. Okay, so I've got a total of 177 out of 240, which is 74%. So, although this wasn't the work of a single student, it was an aggregate. By and large, most of the questions were done pretty well. We've got lots of results in the 20s and the 18s. So, if this had been a single student, and it's possible that it was even a single student and there was an aberration. Who knows what will happen then. But that would have been a pretty good result. 74% for a beginner in this kind of material is really a very strong result.