Course: PreCalculus
School: UCIrvine, through Coursera (free).
Instructors: Dr. Sarah Eichhorn, Dr. Rachel Cohen LehmanQuote: This course is designed to prepare you for a collegelevel Calculus course. Through this course you will acquire a solid foundation in algebra and trigonometry. Emphasis is placed on understanding the properties of linear, polynomial, rational, radical, piecewise, exponential, logarithmic, and trigonometric functions. You will learn to work with various types of functions in symbolic, graphical, numerical and verbal form.
[Addendum: This course has been converted to the new Coursera platform; it’s divided into two parts. Content may have changed, and the experience is likely to be different]
That might’ve been the intent… but for me, the experience was a lot different. This course reminded me that I hate math. I forgot for a while there.
It sounds churlish to bitch about something offered for free, but considering the other math MOOCs I’ve taken over the past year and a half – courses I loved – this was a disappointment. It’s ironic that my disappointment is partly due to the fact that I’ve had the good fortune to encounter some spectacular math courses through MOOCs (most of which I’ve talked about at length), with instructors who put thought and effort into their planning and execution. Granted, I struggled through some of them, and sometimes felt a little beatup by the time they ended, but I always recognized their value. Like most silver linings, it comes with a cloud: ordinary math courses just don’t cut it any more.
This was a very ordinary math course. It isn’t what MOOCs can be. It isn’t even what math can be.
Things started out with great promise. Week 0 – a terrific idea – allowed everyone to review some concepts from prereq algebra, to get used to the (very clumsy, grr) system of answer entry (come on, Coursera, can’t you figure out how to integrate LaTex? And why so stingy with the Preview buttons?), and do the usual “Hi, I’m suchandsuch from Hereabouts and I’m 16/30/75 years old and I love/hate/never took math” forum posts. I noticed a number of acquaintances from other math MOOCs, all of them seemingly beyond the need for precalculus (most of us met in various calculus classes, in fact) but someone unfamiliar with my difficulties with algebra probably thought the same thing about me. I was looking forward to the course, and to finding out what I’d been missing all along, whatever it was that was keeping me from understanding algebra enough to recognize what I encountered in other settings. I was also looking forward to being able to help out others; I love answering questions, giving hints, working on explanations, and I figured I could do more of that than I typically can in Calculus.
But things went downhill fast.
Each lecture video, delivered by a disembodied voice, started out with, “Let’s look at… solving rational equations/evaluating logarithmic expressions/using half angle identities” and ended with “And this is how we… solve/evaluate/use.” In between, one or two problems was worked out step by step in great detail. That’s great – and in one case, I discovered why I have so much trouble solving inequalities – but nothing related to anything else; no particular reason for looking at rational equations, or logarithmic expressions, or half angle identities, was given. It was back to 10th grade, when I thought what mathematicians did all day was solve problems out of a book, never even thinking about who wrote the problems in the book or why they needed to be solved; it was what math was about: here’s a problem, find the answer, next.
A PDF textbook was included in the course materials, and it seems the idea was: if you want to know why synthetic division, or the quadratic equation, or the halfangle identity works, go look it up. Now, there’s a lot to be said for doing personal research, but if I could learn math from a textbook, I wouldn’t be taking MOOCs. It seems to me even a few videos explaining key concepts would’ve gone a long ways. And, for pete’s sake, the MathIsFun website was used as a major resource. It’s not that I have any problems with the website (except the name) – I love their “interactive unit circle” – but it shows an attitude of “Why teach? Just link. We have better things to do.” Maybe that’s the idea behind the course: it’s not about increasing understanding, it’s about listing resources, and after that, you’re on your own. Just like real life.
The instructors, who seemed active on the boards in Week 0, disappeared completely after that. Now, that happens in lots of MOOCs (though less so in math courses), but usually staff or CTAs (community teaching assistants, students who took the class before and did well, and showed some ability and interest in helping other students on the forums) are on hand to provide expertise. Not so here. At various points, even the Coursera technical staff seemed to abandon us, and issues of missing videos, outoforder videos, and inaccessible elements went unaddressed. Even the strangest element of the course – they announced the discussion forums would be “closed” once the exam was released – never happened; I’ve never heard of that being done before, it seems ridiculous to me, but to announce it and not do it just shows how unconnected the people running this are. It’s like they’ve converted this course to “remote control/selfpaced” while keeping the time limits. I don’t like the selfpaced approach (and, unfortunately, that’s where MOOCs are going), but even I recognize each approach has advantages and disadvantages; still, combining the worst of both makes no sense.
Some of us tried to expand beyond the “here’s a problem: do it” mentality. We had a rollicking discussion of positive and negative square roots, but I still could use some expert guidance on this; it seems sometimes the primary square root is always assumed, and sometimes it isn’t, and I don’t feel confident that I’ve nailed down the possibilities. I would’ve liked to have done a lot more work on logarithms and exponentials, one of the main reasons I took the class; I got more out of my random wanderings through AoPS and Khan than I did from this course, which covered howtodoit but not that elusive whyitworks. Trig identities was the biggest disappointment. Someone asked about the connection between the unit circle and the traditional Cartesian graph of trig values, and while I could point them to lots of interesting graphics, I realized I have no idea how to explain it. I should, at this point. I should be able to create those graphics (well, except for the programming part). So the takeaway is this: I took yet another trig class and all I got was a list of identities. I have that already. I wanted to understand them, how they fit together, why they work.
I also brought in a couple of goofy “how would you solve this” puzzles from other sources. In both cases, that led to wonderful explorations with one or two other students. Most of it didn’t have much to do with material in the course (though it was the first time I’ve ever been motivated to actually use logarithms by anything other than a math test), but it was enjoyable; maybe, for me, learning to “enjoy” math is the most important lesson. There wasn’t much interest in this, however (only one other person ever joined in), so I stopped doing it.
Another great experience was in helping another student, through email. He’d missed a week way back, and was struggling with a few questions. Going through his work and figuring out 1) what the answers should be, and exactly why, and 2) where he went wrong, felt like a very profitable use of what turned into a significant chunk of time. The old “you don’t understand something until you can explain it to someone else” is very true. The discussion forums provide some opportunity for this, but mostly people are looking for answers. Also, I’m so slow in coming up with explanations, I’m usually too late, and while I’m working out the details (and discovering what I don’t understand about the underlying principles), someone else has answered the question and everyone’s moved on. I’ve always been too slow for real time – even the real time of a message board. Discussions tended to deadend without any feedback from the original questioner. Math course message boards are usually terrific (I still refer to old Calculus and Mathematical Thinking posts occasionally) but not here; I’m not sure why. The students seemed younger; lots of high schoolers, maybe that had something to do with it.
Maybe I have unrealistic expectations. Maybe I’m missing something so obvious to everyone else, no one needs this stuff. Maybe I’m lazy and I should continue to research it myself (which hasn’t, to now, been a screaming success, and is laden with misconceptions that don’t always become evident until a unique set of circumstances exposes them – at which point I’m back to square one). Maybe this course was too easy for me – though that strikes me as a ridiculous notion. Maybe the course I want doesn’t exist, or I’m looking in the wrong place. But I expected a lot more in the way of understanding, and instead got a lot of “this is how we graph parabolas.”
I did well on the weekly quizzes, scorewise, which surprised me. Nearly every quiz, I was shocked when I scored 4/4, 5/5 on the first attempt. Here’s what still concerns me: if I don’t know whether or not I’ve got the right answer, does it count?
Then there was the final exam. The timed final exam.
I’ve always said I don’t care about grades, and to a large degree that’s true; at my age, I’m over grades. But with this course, it was a matter of pride; if I’m going to slam something, it doesn’t look good to flunk. Also, since I’ve been taking Calculus for a year and a half (and I’ll keep taking it for the next year and a half, until I feel like I understand it), I should be able to do precalc. So I felt some pressure to pass. But I don’t do math quickly, and the time limit worked out to about 4 minutes per question. That’s barely enough time for me to set things up so I’m ready to do the math. See, I work in multiple media: I’ve become quite adept at using the Word Equations function for algebraic calculations, which eliminates handwriting mistakes (but allows typos; nothing’s perfect). But sometimes I use a whiteboard, for drawing unit circles or graphs or just putzing around diagramming a number line or graph. I also have piles of paper, which is nice if I want to work standing at the window instead of sitting at my computer. Sometimes I start in one medium, then realize another is better suited. Sometimes I just use the completely wrong approach, and don’t realize it until halfway through; I have to start over. And sometimes (often), I make “boneheaded mistakes” – drop a minus sign, calculate 4*8=36, that sort of thing. Sometimes I have to stop and think about adding and subtracting negatives and positives. Sometimes I need a walk around the block, or a cup of coffee, or just a rest break. This all adds up to a lot more than 4 minutes per question.
Pressure!
Suddenly it became all about getting a grade – a pat on the head, approval, performing the tricks I’ve been trained to do – instead of about doing math. I realized: this is why I always hated math classes. And this is why I’ve loved the math MOOCs I’ve been taking, even when I didn’t do well on tests: I was still learning something, failing at something worth doing, something worth trying for again (which is why I take so many math classes more than once).
For the record: two attempts at the final were allowed. On the first, I only got to 24 of 35 questions, got 2 of them wrong, for an overall score of about 60. I went through every test question (some of the questions I’d skipped because they looked scary turned out to be quite simple, if I’d just taken the time to actually read them and think about them), checked a few procedures (I know how to find the inverse of a function, I just don’t always remember that I know), lined up my ducks in a row (do I have my whiteboard? Calculator? Coffee? Teddy bear? Halfangle identities cheat sheet? Because, no, I’m not going to memorize that). And I ended up with a perfect score. Yes, a few of the ones I’d already seen were repeated, but most were new.
But did I learn anything?
Well, of course I did – I got some muchneeded practice in trig identities, for instance. But mostly, I learned what I wanted to be doing instead.
I identified some concepts I want to understand better. The primary square root, for example, versus the square root function. Derivations of trig identities. One question from the final intrigued me: it turns out tan^2(x) – sin^2(x) = tan^2(x)sin^2(x). How can the difference of two functions, equal the product of those same two functions? I know the identities; I can solve the problem – I got the question right, so I know the procedure – but there’s a relationship there that I don’t grasp just from knowing tan = sin/cos… What is it? But it was the final, I had four minutes; so I saved it, and I looked at it more closely once the timer stopped ticking away – but I’m a mathematical idiot. I’ll look again.
During the introductions of Week 0, it seemed to me that the course was taken by more firsttime MOOCers than most. I felt like going around apologizing to them, telling them, “This isn’t what a MOOC can be.” But who am I to decide for someone else? Maybe it was exactly what they needed. It wasn’t what I was hoping for, but maybe I can spin a silk purse out of a sow’s ear anyway. Maybe that’s what I needed to learn.
= One question from the final intrigued me: it turns out tan^2(x) – sin^2(x) = tan^2(x)sin^2(x). How can the difference of two functions, equal the product of those same two functions?
– Hum… We left that unanswered 🙂
– Well, Karen, let’s see first, if this is a general property or not. Take a couple, please, of those f, g functions you like, and check if they satisfy this property: f – g = f * g. What is the simplest case? Probably, the one with two constant functions, both equal to zero for every x, that is f(x) = 0, g(x) = 0. Evaluate the left side: f(0) – g(0) = 0 – 0 = 0. Evaluate the right size: f(0) * g(0) = 0 * 0 = 0. Conclusion: There is at least a couple of functions, the functions f(x) = 0 and g(x) = 0, such that their difference equals to their product indeed. Fine. Now, what? A couple of functions which they do not satisfy this property. As previously, probably the next “immediately” less simple case than the simplest one is: f(x) = 1 and g(x) = 1, then f(x) – g(x) = 1 – 1 = 0 and f(x) * g(x) = 1*1 = 1, but 0 =/= 1. Conclusion: There is, indeed, at least a couple of functions, the functions f(x) = 1 and g(x) = 1, such that their difference does not equal to their product. Hence, the property f – g = f * g is not a general property of functions.
= Rule of the thumb?
– Yes! Try to start checking by using 0 and 1 🙂
– Now, it seems that there is no other ghost to discourage us from trying to prove the given identity in the special case of f(x) = tan^2(x) and g(x) = sin^2(x), wherever it is defined, of course.
= Why “wherever”?
– Because, tan(x) is not defined for all x; we have to exclude all that infinite number of values of x, which are odd integer multiplies of pi/2. Therefore, we have to check the validity of the given property “only” at the remaining, uncountably many, “permitted”, “defining” values of x.
= How?
– Well, after all that preparation of the frame of the discussion – “the context” 🙂 we said – it is enough to check the validity of the given property at some, random, permitted value x – here, we silently – that is tricky, as usual, we roil the water to look deep 🙂 – replace the selected name x of the independent variable with some value x. Thus:
tan^2(x) – sin^2(x) = { tan(x) definition } :
[ sin(x) / cos(x) ]^2 – sin^2(x) = { squaring the fraction terms } :
[ sin^2(x) / cos^2(x) ] – sin^2(x) = { common factor sin^2(x) } :
sin^2(x) * ( [ 1 / cos^2(x) ] – 1 ) = { adding fraction and 1 } :
sin^2(x) * ( [ 1 – cos^2(x) ] / cos^2(x) ) = { by identity sin^2(x) = 1 – cos^2(x) } :
sin^2(x) * [ sin^2(x) / cos^2 (x) ] = { fraction of squares is square of fraction } :
sin^2(x) * [ sin(x) / cos(x) ]^2 = { definition of tan(x) } :
sin^2(x) * [ tan^2 (x) ] = { drop the brackets } :
sin^2(x) * tan^2 (x) = { commutativity of the product } :
tan^2 (x) * sin^2(x) = { at last } 🙂
– Have fun!
Petros! What a lovely surprise! You are the second Courserian to follow me to my lair, I am always so honored that someone would go to such trouble… and I see you have deleted your posts (perhaps your account?) from the course, which is a shame; I got so much out of our discussion there. But things move on, I suppose.
Thank you for providing such a detailed process – I like the idea of checking “is this a general function” first, since that is just something I assumed. I had no problem doing the problem and getting the right answer, using exactly the same two identities you used. I got the question correct on the exam.
But… isn’t it a rather monumentous special case, when the product of two functions equals their difference (within restrictions, as you mentioned, which I’d also forgotten to consider… details matter, Karen!)? Doesn’t the specialness of this, have to mean something about triangles, or angles, or circles, or… something? We could just say the Fibonacci numbers are the result of a little game of adding numbers in a certain way, but it turns out they have greater significance – wouldn’t the unusual nature of this coincidence indicate a similar greater significance beyond finding the right answer?
I played with unit circles for a while, and for me what it came down to, was an insurmountable obstacle: what is tan? Physically, on the unit circle? Cos is the distance on the x axis (yes, I know, it is not that simple, but on the unit circle, that is how it is represented) and sin is the distance on the y axis, and tan is… the slope of the radius that forms the angle?
then I started wondering what does it mean, physically on the graph, to multiply tan^2 by sin^2… or to subtract sin^2 from tan^2…. is this where linear algebra comes in, or is that in a whole different ball park (I know next to nothing about linear algebra except that it involves multiplying vectors that look to me like segment, but I would assume the actual significance of a segment is far more complicated than assigning the value to sin to one and manipulating it)? Or am I on the wrong track entirely?
It’s a lovely puzzle…
Do you like geometry? I have a project going, Blogging Euclid, at another blog (this blog is primarily for what I’m reading, at least these days it is); I’ve just finished Book II today (I have a new set of MOOCs starting and will be proceeding very slowly for a while). Company is always welcome! Or, I’ve also done some documentation of a geometry game I am quite fond of at yet another blog. If you enjoy that sort of thing, you might like the game.
Are you taking any further Coursera courses? I’m about to take my second run at SV Calculus, which I failed last time (the only course I’ve actually failed, I just stopped moving forward at week 4 and went back to go over the material again, I hope to get to week 6 this time). And a bunch of other things. I can’t resist, I see courses and I sign up, then I end up frantic with too much work! But I am afraid all this will disappear in a puff of smoke – my computer will break, the internet will become too expensive, Coursera will start charging fees, I will lose my mind or get sick or become homeless – so I am trying to fit in everything while it is available to me!
Thank you so much for stopping by, you’ve made me very happy! You remind me of another friend I made this summer in another math course. You both make me think very hard, and I like that.
Dear Karen,
Although, I understand perfectly that your blog is definitely not a technical one, I can not leave this provocative subject unfinished…
Therefore, I am writing this “addendum” *before* I proceed with the complete reading of your reply above, because I am not satisfy with my “answer”, since that “answer” is not really faced your question – a question which in the meantime was “adopted” by me too 🙂 – that is:
“How can the difference of two functions, equal the product of those same two functions?”
Or, equivalently:
“For which ordered [1] couple of functions [2] (f,g) we have f – g = f * g ?”
where, for “the context” or “the frame” we have to discriminate:
[1]: “ordered”: first f, second g, due to the order of their appearance in the difference: f – g
[2]: “functions”: from any domain to any range, as long as, all the “needed” operations are defined, that is initially the difference “” and multiplication “*”, and then any other notion that “will appear in the following” – that is the addition “+”, which precedes the difference ““, as well as, the division “/”, the constant “0”, the constant “1”
Well, how stupid I am!
We had to “attack” our question *directly*! That is:
(a) f – g = f * g f = f * g + g f = g * ( f + 1 ) g * ( f + 1 ) = f
or
(b) f – g – f * g = 0 f – f * g – g = 0 f – f * g = g f * ( 1 – g ) = g
Now, to proceed further with the division, we have to take the usual precaution: to not divide by zero 0, that is:
(a’) If f + 1 =/= 0 f =/= 1, then we have from (a):
g = f / ( f + 1)
or
(b’) If 1 – g =/= 0 1 =/= g g =/= 1, then we have from (b):
f = g / ( 1 – g)
But, we also have to see what happens to “our condition”:
f – g = f * g
when f = 1 or g = 1.
Well, if f = 1 then our condition becomes:
1 – g = 1 * g 1 – g = g 1 = g + (g) 1 = 0
while, if g = 1 then our condition becomes:
f – 1 = f * 1 f – 1 – f = 0 f – f – 1 = 0 0 – 1 = 0 1 = 0
therefore, if f = 1 or g = 1 then our condition can not be true.
Now, we are ready to conclude with the answer to our question, that is:
For every
f =/= 1 and g =/= 1
we have
f – g = f * g
if and only if
g = f / ( f + 1) or f = g / ( 1 – g)
= And then what?
– Well, we are now in two 2 “advanced” positions 🙂
1. We can solve quizzes, like the given one, as soon as, we recognize that the appearing functional equation has the given form f – g = f * g – that is to do as you done first 🙂
2. We can construct quizzes, like the given one, and ask the people to solve them 🙂 – that is like as “they” done to us 🙂
= How?
– By taken as f almost * a n y * function, that is only watching to either not take that “excluded” 1 value: f =/= 1 or to explicitly exclude it from the domain of f, and then as g, the always defined then function: g := f / f + 1, that is without any need to check if g can take its excluded value 1, because this happens “automatically”, since if f =/= 1 and g := f / ( f + 1) and we suppose that it is possible to have g = 1, then g = f / ( f + 1) = 1 f + 1 = f 1 = 0 : impossible.
= Any construction example?
– Which is the simplest one? That one with f = g = 0 ! We have f = 0 =/= 1 and then g = f / ( f + 1 ) = 0 / ( 0 + 1) = 0 / 1 = 0 and f – g = 0 – 0 = 0 = 0 * 0 :
It works!
= These are tricky constants. Any other?
– That one you already observed: f := tan^2(x), wherever is defined, then since it is a square we always have f >= 0 that is f =/= 1. Then we define as g := f / ( f + 1 ) = tan^2(x) / ( tan^2(x) + 1), but then
g = { as we done with: tan(x) = sin(x) / cos(x) and sin^2(x) + cos^2(x) =1 } :
sin^2(x).
= But, It seems here, that g takes the “forbidden” value 1, e.g. when x = pi/2 or pi/2 or when, in general, x is an integer odd multiplier of pi/2.
– Yes, it is true, but it does not bother us because these values of x are excluded from the domain of tan(x).
= 🙂
– 🙂
= Any additional example beyond these we already seen?
– That rule of thumb wants the simplest to take first, therefore the identity function:
f(x) := x ; x =/= 1
g = x / ( x + 1) ; x =/= 1
We check that “our” property:
f – g = f * g
which, in the “frame”  “context”: x =/= 1, becomes:
{ the “deepest” ordered couple of parentheses: “(” , “)” has “application priority” over anything else, as well as: “/” over “” , by that “universal convention” } :
x – x / ( x + 1 ) = x * x / ( x + 1)
{ fraction addition and then equal denominators omission } :
x * ( x + 1) – x = x * x (x * x + x) – x = x * x + x – x = x *x
“oper edei deixai”:
http://en.wikipedia.org/wiki/Q.E.D.
to never forget our ancestors…
🙂
– Have fun!
Petros
P.S. Now, I am relieved from all that 🙂 and I can read your previous reply: more confessions to come 🙂
Ok, yes, wordpress (this free version, anyway) is not really conducive to math. Some of it is my fault: I incorporated a different font when I set this up, and I’ve put backgrounds; all of this has an impact on some subcoding level. But I like my curvy font. It never occurred to me in my wildest dreams, when I set up this blog, that I would someday want to post math. I finally figured out how to use LaTex on my Euclid blog, but I haven’t tried it here – I did notice that the superscript tap ^{ } doesn’t work, which is disappointing, but maybe it means I need to look around at other hosts and see who’s math friendly, should I decide to blog more math.
But going through your solution was very educational – I phrased it as a function problem, then ignored the function aspect, so thank you for addressing it as a function. I’ve written out my own wording (which in places is less precise than yours, since it’s designed for my level of understanding). The takehome is that any function that meets the conditions, would have this productequalsdifference property. tan and sin just happen to meet the conditions (I did wimp out at the end, and took W/A’s word for it, but fiddling with trig identities is something I can do anytime; the important thing is the process and the plan).
You are going to love the Mathematical thinking course if you’re still taking it. You will, of course, ruffle some feathers, but I suspect you will find many like minds. It almost makes me want to sign up for a third run, just to see what happens, but starting in 2 days, the next 14 weeks of my life are devoted to SV Calculus: The Second Match. I will emerge, bloody but unbowed.
My notes (I have never included images in a comment before, I hope this works):
Dear Karen,
I carefully read a great portion of that you have written, but not exhaustively since I stopped, when I realized that we already put a lot of information in various discussion frames, possibly all interconnected by intuition.
Therefore, in order to proceed having some perspectives of success in our common understanding, we have to strictly exercise our selfdiscipline, otherwise, we will be lost, as usual, and disappointed once again. Hence, I propose to continue by dealing with the part “1/3” of your notes, specifically, with the reasoning which drove your downstairs, that is to reveal what exactly justifies any particular descent pass, beginning with the very first one, that is, for example:
{
from the top stair: [ f – g = f * g ], to the next down one: [ f = ( f * g ) + g ], is a pass, which is logically permitted by first invoking the law of addition: “for every a, for every b, for every c, if a = b, then a + c = b + c”, and then by applying it to our special case of any function values:
a := [ f(x) – g(y) ], b := [ f(x) * g(y) ]
–
that is the results of the subtraction of g(y) from f(x), as well as, of the multiplication of f(x) by g(y) insides of an ordered fromlefttorightinthewesternworld coupled, of left and right parentheses
–
c = :g(y), that is:
“for every x, y”
–
Our attention, please! From now on, this it will be accepted by a common agreement as abbreviation
–
–
Again: Our attention, please ! This will not exclude any instance of y = x ! On the contrary! It includes all the cases x = y, and thus it can be invoked to step down once more, in a more particular, but still general enough case of y = x or indiscriminately x = y, because of this, commonly accepted by us, property of equality which is called: “symmetric”
–
“if x = y then, if f(x) – g(y) = etc
–
Once more: Our attention, please! A consecutive/cascade/second,/deeper “if” second f(x)
–
or simply, by another commonly accepted abbreviation agreement: with just one “for every”, instead of these two ones plus the “antecedent of the hypothetical proposition” of second if:
“for every x, if f(x) – g(x) =” etc.
or even more simple, by once more commonly accepted abbreviation agreement: with just f instead of f(x) and hence without the implied “for every x”, that is only with a “bare” formula:
“If f – g = f * g then ( f – g ) + g = ( f * g ) + g ”
or by using the “implication: if – then” sign “>” “dash” “greater than”, which I hope will not be filtered out by that free version of wordpress parser, as you said, as well as will be not formed badly in the font (otherwise: very nice) you selected, as you also said, the simplest symbolization, “without any word”, of the implication:
f – g = f * g > ( f – g ) + g = ( f * g ) + g
}
and so on, carefully keeping the symbolism consistent, e.g. [ f * g ] always, instead of mixing it with [ fg ],
= How long these hard, tedious, laborious, painful explanations have to be continued ?
– Well, since we declared that we want to be hunters of the deep, fundamental truth, until, we will be indeed sure that, we will be indeed in place to exactly understand and explicate any such pass, whenever it will be needed in the future. Until then. Of course, we always will be still free to desert at any moment, from the service of such Cause 🙂 It is up to us, as usual 🙂
Have fun!
Aha. I’m afraid you left me behind fairly quickly here. I fear you have underestimated my inability to understand 😉 and this is (albeit slightly) complicated by typography and language.
From the beginning:
***
{
from the top stair: [ f – g = f * g ], to the next down one: [ f = ( f * g ) + g ],
is a pass, which is logically permitted by first invoking the law of addition:
“for every a, for every b, for every c, if a = b, then a + c = b + c”,
and then by applying it to our special case of any function values:
a := [ f(x) – g(y) ], b := [ f(x) * g(y) ]
–
***
I follow this more or less (with one major problem I’ll address in a moment) but have a few detail questions:
First: What is this “law of addition”? Is it known by some other phrase? I google “law of addition” and I get “associative law of addition” and “commutative law of addition” but no “law of addition.”
I do not doubt the veracity, of course; in fact, the statement you attribute as the Law of Addition seems remarkably similar in spirit to Euclid’s Common Notion 2: “If equals are added to equals, then the wholes are equal.” I simply wonder if the nomenclature “Law of Addition” might be known to me under different phrasing….
Second: what I see rendered on my screen is: “a :=” or “aspacecolonequals” or, later, other placements of colon and equal sign. Is this how you intended it or is that the WordPress transliteration? in any case, I don’t know what the colon means, or why it is there, and since it appears later I feel like I had better take the opportunity to understand, rather than shrug it off as “one of those mathy things.” Or maybe I am just focusing on unimportant details which are easy to decode, rather than on what is significant and harder to comprehend.
And that would be the next step where I lose my mind:
***
that is the results of the subtraction of g(y) from f(x), as well as, of the multiplication of f(x) by g(y) insides of an ordered fromlefttorightinthewesternworld coupled, of left and right parentheses
–
c = :g(y), that is:
“for every x, y”
***
Wait! So we are adding g to both sides, and thus the c of the Law of Addition is g? But… where does the y come in? Because in the initial problem, and in my mind throughout all this, we were dealing with f(x) and g(x), not g(y), and suddenly (actually, in the last step, but I deferred the question until now) we’ve got g(y)… no no no!!
Now, I will admit that I’ve lost the thread of your argument from here forward – is the point to show that we start from f(x) and g(y) and then proceed to show that this is the case only if x = y, so we have f(x) and g(x)?
If I know where we are going (and if this is where we are going), I might find it easier to follow. At this point I’m so alarmed by the inclusion of “y” that I curl up in a ball and let the drowning begin (a frequent posture when I get “in over my head” in math, or any subject, really; like mathematical philosophy for instance).
Dear Karen,
We do not fear, when we are following our heart, irrelevantly of in the world anyone’s else opinion about our inabilities or of our definitely limited abilities. Fortunately enough, we are still free to freely think and to freely discuss our free products of our free thoughts over the free yet internet – etc etc all free 🙂
Now, could you remember, please, that I already promised you more confessions to come? I admit now that I deliberately tried to pull out some of them in my previous response 🙂 Well, let’s see what happen:
 First: What is this “law of addition”?
– Well, since in many cases “they” are calling “laws”, some fundamental [ do you see now? The Word is Appeared 🙂 ] properties of operations, like the addition one, expressed by formulas like the one we discuss:
a = b > a + c = b + c
which is well known as the (logical) “Additive Property of Equality”.
Thus, of course you are right about “my” “law of addition” !
I am terribly sorry for the inconvenience. Specifically, I had in my mind the very close related property to that one, which is commonly known as the (mathematical) “Cancellation law of addition”, that is the reverse (mathematical) implication of the above (logical) property:
a + c = b + c > a = b
Mea culpa (:
What do you think now about my overestimated “abilities”? 🙂
 Second: […] “a :=” […] what the colon means, or why it is there […]
 Or maybe I am just focusing on unimportant details which are easy
 to decode, rather than on what is significant and harder to comprehend.
– Well, we are indeed at “The Heart of the Matter” now, so, we do have to indiscriminately focus on all of the details themselves, and putting aside, at least for now, any such, moreorless invented, distinction between “important” and “unimportant” ones.
Anyway, in this particular case, we use a colon with an equal sign to stretch that our aim is to replace, in a given formula, *every* occurrence of the symbol which is in the same side of the equal sign as is the colon, with that symbol which stands alone on the other side of the equal sign, that is: “a := [ f(x) – g(y) ]” means that since the “colon” is on the left side of the “equal” we will replace every occurrence of “a” in the Additive Property of Equality, with [ f(x) – g(y) ], and so on for “b”, and “c”.
 Wait! So we are adding g to both sides, and thus the c […] is g?
– Yes, “but not exactly” 🙂 since “c” is the name/symbol of some “thing” (number) and g is the name/symbol of a function, that is of a special (functional) relation between “things” (numbers) of numbers), so we have to do the following:
First, we replace every occurrence of that “some” “c”, with some “thing” (number), which especially is the “thing” (number) that results by “applying” the function to some “thing” (number) y, that is: ” c := g(y) “. Next, by a series of generally agreed “conventions”, we finally arrive to “f” and “g” as stand alone names.
= Why, we don’t also set “x”, instead of “y”, inside the parentheses of “g”, from the very beginning?
– Well, given this occasion, we would like to emphasize that the replacement of the “a”, “b”, and “c” “things” (numbers) can be done by *independent*in*any*way* “things” (numbers), that is in this particular case, by “things” resulting from *independent*in*any*way* “things” “x” and “y”, including the case of their equality.
– More confessions to come 🙂 as long as, you are enduring them 🙂
– Hum… I observe that the margins, of the nested frames containing our consecutive replies, are of constant width, thus, I am wondering if, when, and who of us will have to sandwich text in a column of zero 0 characters wide… In addition, I also see that once I wrote a response, I can not edit it, so, there is no way, for example, to restore above the meaning of that silly “(numbers) of numbers)” to just “(numbers)”…
((
“– More confessions to come 🙂 as long as, you are enduring them :)”
))
I am not only enduring them, I look forward to them! This is like having my own private math/logic teacher, it’s a privilege – I only hope I can keep up, not prove so dense you get annoyed and go find more promising talent to work with.
By the way, as for the nested comments growing ever narrower, I noticed last night that the “walls” appeared to be closing in from the sides, mimicking large iron bars of a medieval prison… that is the wrong connotation! So I have decreased the setting for comment nesting from 8 to 3 (I don’t remember why I increased it so much in the first place; I’m guessing there’s an old post somewhere with multiple participants and comments, probably one of the Reality TV posts, since those were the only posts anyone ever paid attention to anyway…) in any event, this is better, things seem less claustrophobic now, yes?
It is as if you have thrown me a life preserver (Schwimmring?) and now I can keep my head above water (the first translation for “life preserver” I found was “Totschläger” but I’m glad I took the extra step to verify it by image search, because to use that word would have been catastrophic!) and swim a bit further…
I understand the Additive Property of Equality and the Cancellation Law of addition (new toys!) and I follow the subtitution of a “thing” for a name. I am still worried about x and y, but I will trust that this too will become clear, and if not, you will throw me another Schwimmring.
To verify the expansion of the replacement, I will rephrase as I understand it:
We have a := [ f(x) – g(y) ], b := [ f(x) * g(y) ] and c = :g(y)
so to work on the problem at hand, we do
a = b
a+c = b+c then we do the replacements and this comes to:
f(x)g(y)+g[y)=f(x)*g(y) + g(y) which simplifies to:
f(x) = f(x)*g(y)+g(y)
To verify my understanding of that process: This is what I would normally, when doing algebra, would call “move g(y) to the other side” by adding it to both sides of the equation; it’s nice to know the official principles for this, and to know the names of these principles. I’m sure I was told what they were back in 1968 or so, but I have been told so many things since then, some things just didn’t stick.
New material: Then we get to:
(((
“Again: Our attention, please ! This will not exclude any instance of y = x ! On the contrary! It includes all the cases x = y, and thus it can be invoked to step down once more, in a more particular, but still general enough case of y = x or indiscriminately x = y, because of this, commonly accepted by us, property of equality which is called: “symmetric””
“if x = y then, if f(x) – g(y) = etc
)))
Question: When you say “it can be invoked to step down once more” do you mean now we are stipulating that x=y, and then continuing on to work with the functions f(x) and g(y), thus we have in effect restricted those functions to using the same input? I hope so, because this is the step I was waiting for. I think we are on the same page. But maybe this is not what you meant, maybe you mean only that the instance of x=y is included, but so are instances of x not = y…?
To Verify:
(((
“A consecutive/cascade/second,/deeper “if” second f(x)”…
)))
I think this is what I would call a “nested if” (in a former life I was a computer programmer, though a mediocre one)… but I’m not sure if that term crosses languages, or if my understanding is correct. Oh, here it is – “verschachtelte WENNFunktion”…yes?
Next:
(((
“or simply, by another commonly accepted abbreviation agreement: with just one “for every”, instead of these two ones plus the “antecedent of the hypothetical proposition” of second if:
“for every x, if f(x) – g(x) =” etc.”
)))
YES! So because we have stipulated x=y, we now just make it f(x) and g(x). “antecedent of the hypothetical proposition” is a little hard for me to parse; “if x=y” is an antecedent, yes? but so is “if f(x)g(x) = etc”, yes? Or is that considered a consequent? Is it both a consequent (of the first antecedent) and an antecedent (of the second nested level)? I think I follow the general flow, I’m just not sure of the exact details, and I’d like to be.
Moving forward, the next section I believe I understand, if I have the above correct:
(((
“or even more simple, by once more commonly accepted abbreviation agreement….
and so on, carefully keeping the symbolism consistent, e.g. [ f * g ] always, instead of mixing it with [ fg ],”
)))
I believe I follow to the end – I am rather sloppy about notation, switching from f*g to fg as you have noted. Can I assume either is correct, that they are identical (as they are considered identical in common use), it’s just a matter of sticking to one or the other? I think I tended to keep it as “clean” as possible, without the * sign, unless I was nervous about clarity, resulting in the mishmash. It is good for me to be reminded that neatness counts. 😉 It was the x=y that turned everything else into a pile of confusion to me, and once I saw the Schwimmring, it was manageable again. I am so grateful to you!
Whew! Amazing to peel away level after level, and still find more levels beneath, an onion that just keeps peeling…
Comment: This parser, beyond that of Coursera, has problems too 🙂 It filters out, at least, all the equivalences from the previous comment, that is strings of consecutive characters: “” and displays just “white space”.
Comment 2: I see. Well, once again, this time in words: strings consisting of “lowest than”, “equal to”, “greater than”, as well as, isolated “semicolons”. It is a fiction literature parser; definitely 🙂
Hello, Petros – Well, you have given me quite a wonderfully busy morning, between deciphering what WordPress does to math (yes, I know – you don’t know what I go through to get things to look close to the way I’d like them to look) and deciphering the math itself – thank you so very much, you have no idea how much I appreciate such detailed explanations…
Before getting to the math, however, there is something else I need to tell you. I think I now understand why you remind me so much of my other math friend from this summer – you = him. I don’t know if you intended to reveal this or not – I uncovered several clues through my blog’s “stats” page, which tells me 1) what links visitors have clicked on, and 2) a general tally (not an identification) of what countries “hits” come from. I’ve noticed a spike from a certain European country in the past week, and yesterday I saw a “click” on a link I did not recognize, because I did not include it – turns out, you did, when you left a comment, and someone (you?) clicked on it. Even I could put 2+2 together with a certain conversation from Calculus, and realized, Aha! In a way, it’s too bad, I’d been thinking suddenly Coursera was replete with people willing to share knowledge via a unique verbal style. But I’m very happy to know I have not lost touch with someone who made my struggle with Calculus so much easier to bear, and someone who is so willing to share knowledge and experience.
Now to math… but that’s another comment.
“Karen Carlson says: September 8, 2014 at 7:54 am”
Dear Karen,
your starting question was about the possibility they had two real valued functions f and g to verify the relation f – g = f * g. We also considered that question as an opportunity to dive into the onion of “the fundamentals” and grope for some logical and mathematical of them. I think that it is now the time to conclude.
Well, although, you implied that in your question these two functions were functions of one and the same independent variable taking values from a subset of reals and we answered that specific question, I have to confess that we attempted in fact, either implicitly or explicitly, some restrictions and extensions of this question regarding the dependency of these two functions on the number and sameness of the independent variables “iv” , as follows (where, instead of unreproducible, by the free wordpress parser, equivalence symbol we use the abbreviation “iff” : ifandonlyif or equivalently):
1. Trivial:
Two functions without iv or constants or constant functions with any number of either the same or not the same ivs:
f() – g() = f() * g() iff f() =/= 1 and g() = f() / [ 1() + f() ]
or in either notation:
a – b = a * b iff a =/= 1 and b = a / ( 1 + a)
– Answered.
2. Usual:
Two functions of one and the same iv x:
f(x) – g(x) = f(x) * g(x) iff f(x) =/= 1 and g(x) = f(x) / [ 1 + f(x) ]
– Answered.
3. Uncommon:
Two functions of one but either the same or not the same iv:
f(x) – g(y) = f(x) * g(y) iff f(x) =/= 1 and g(y) = f(x) / [ 1 + f(x) ]
We discriminate two 2 cases:
3a. x = y:
– Answered.
3b. x =/= y:
We have a bunch on (new) subcases to examine regarding the sameness or not of the domains of these two functions in conjunction with their cardinality, that is the “number” of their elements: only one and the same element, only one but not the same element, or at least two elements.
Have fun!
–
Quite a conclusion you have brought to this adventure – I am so grateful to you for the time you have spent working through this with me! I would imagine the cases where x and y can be different would be limited. I still hope some day to look at this from a purely trigonometric point of view, perhaps – brainstorm – through Taylor series!!! – SV Calc has just restarted, and I need to focus some attention there (today is a “lost” day, for logistical reasons, and tonight will be a “lost” night due to an event at my local library if I can find the courage to venture out, hermit that I am) but I just realized, tan and sin can be expanded, and those expansions manipulated… I think I have made a connection!
Again, so many thanks! I hope we will continue to cross paths, perhaps find ourselves in another course at some point.
I hope for that, too – Internet pseudohermits, in fact, like all of us – Thank you, Karen.
* virtual hermits
😉
My hermitation far preceded the internet; in fact, the internet has made much more human contact available to me.
Dear Karen,
It seems that you are insisting to search for “The Fundamentals”, now as a fellow student in coursera’s mooc: “Introduction to Logic”, but this time, I can assure you for that, there is no more depth than that to dive in 🙂
Sincerely,
Petros Zimourtopoulos
P.S. As of your motivating question about: f – g = f.g , surprisingly enough, I met it in “An Introduction to Functional Analysis” – a coursera’s mooc too – as: p + p’ = p.p’, resulting from the form: 1/p + 1/p’ = 1 (in Hoelder’s Inequality), but I was fully prepared to handle it, since I was already fully trained on this subject from our discussions here 🙂
P.S.2 Finally, as you are the only one mooc Journalist – or more precisely: a Reporter perhaps, and/or even better, a Chronicler one – I know – a volunteer one, I suppose, and a rather successful one, in my humble opinion – I think that it must be of your interest the real environment consisting of our virtual classmate scholars in such massive oocs, as it is emerged after a little but appropriate “pushing” 🙂 This was repeatedly my thought, which was resulting from the revealing content of the following posts regarding their “social aspects” and far apart from their “technicalities”, of course, I collected to bring in your attention – although you will have to register of course, even temporarily, to “An Introduction to Functional Analysis” class, in order to read them:
Grading – expectations about precision of proof in this course:
https://class.coursera.org/functionalanalysis002/forum/thread?thread_id=195
Example of sigmaAlgebra:
https://class.coursera.org/functionalanalysis002/forum/thread?thread_id=239
Axiom of choice:
https://class.coursera.org/functionalanalysis002/forum/thread?thread_id=292
To whom it may concern:
https://class.coursera.org/functionalanalysis002/forum/thread?thread_id=268
Petros – lovely to see you again! I am terribly pressed for time at the moment (too many MOOCS!) but I am very eager to delve into all you have presented here in the next day or so… I’m so glad to hear you are in Logic, I have somewhat neglected that message board, but now I will keep my eyes peeled! (uh oh… will that American idiom survive the translation?)
Two people have made me happy today, and you are one of them!
My goodness, Petros, you have been busy stirring up hornet’s nests… I can’t really tell what is going on over there, except that things started out just fine, you seemed to find a kindred spirit in Whackamole, and then… the downvotes, on every post. And a cryptic remark about “before posting, see this thread”, the “to whom it may concern” thread, which seems to be… gone! I’ve never seen a thread totally disappear before – several instructors have indicated it’s impossible, even the title can’t be changed, but you seem to have created a thread that disappeared! The interesting question is: who disappeared it? That alone might go a long way towards answering the obvious question: why was it disappeared? Purgy, my friend, just how much mischief have you been making?!?!?
Are you familiar with Richard Bartle’s hypothesis (from the 90s) about different “player types” in multiuser computer games? I used to play such a game, and I found his insights highly accurate. I always classified myself as a social achiever, but once I’d taken a longer view of things, I very much enjoyed hanging out with the “killers” and very much wanted to be an “explorer” – but I lacked the ability. I see similarities. 😉
I love that people in that class are freaking out over peerassessment of proofs. Yep. Been there, done that, in Mathematical Thinking. I was lucky, though – I don’t know if the system was rigged to give people like me “easier” proofs to grade, but I got three spectacularly wellwritten assignments to grade. I just feel sorry for whoever had to slog through mine. People do take these grades so seriously! I take the courses very seriously, but not the grades.
I love that you’ve quoted Wittgenstein in your Functional Analysis comments! But of course you would! For the record, I love Wittgenstein. I don’t always understand what he’s saying, but he’s fun anyway. Great movie, too.
I was, alas, dismayed to see students advocating for charging fees for MOOCs to increase completion rates. Some day I willl do a whole rant on this. I took Dan Ariely’s “irrational behavior” course, I understand his reasoning, but I disagree STRONGLY that coercing payment increases student engagement, and even a small fee would eliminate a great many participants (including, sadly, me, so I do have bias in this matter). But the joke is: THE NUMBER OF STUDENTS COMPLETING THE COURSE WOULD NOT INCREASE! ONLY THE PERCENTAGE! The effect would be to reduce the population, and whom does it hurt if I did not finish a course because (as in SVCalc) I only could handle part of it? I STILL LEARNED SOMETHING! Keep those capitalist elitist hands off my lifeline!
Oh, dear, it seems I ranted… but in an undeveloped, unsupported fashion. I can do better. In the meantime, I am too busy taking classes I am FULLY engaged in, to write about how important those classes are and why they must remain accessible to everyone.
Like SV Calc, for instance… the second Quiz started yesterday, which is why I was feeling overwhelmed by time pressure. I just fininshed. Didn’t do well, though much tbetter than last time (3.5 improved to 7.5) – but it’s clear to me I have a much better (though imperfect) grasp of the concepts presented, than I did last summer. Having said I don’t take grades seriously, I do take this grade seriously, because it’s a measure of if I have improved my understanding, or if I am just wasting mmy time. I hope to get through Integration in this run, but I probably won’t “complete” the course.
And, I was right in my comments a few months ago – it is not as much fun without you. In fact, the message boards are very quiet! Then again, I have been quiet, focused – was I the only one making noise last summer, so that it just seemed busier?
Dear Karen,
Due to my browser’s cache “issues”, I didn’t realized that those virtual “bullies”, beyond they massively “redmarked” my posts, they had bypassed too the coursera’s security and deleted my posts, so thank you very much, indeed, my Friend, since you drove my attention to that fact.
Very Interesting Indeed !
😀
Well, in any case, I took my countermeasures after all that: I deleted all the rest of my “survived” posts
–
Do you remember and understand now why I deleted, all of my posts, in “PreCalculus” mooc? – Same reasons!
–
as well as, I uploaded those, I hardly worked for, in a safer place. Could you, please, be so kind to read there, that one titled:
Professors and Students – “teachers” and “scholars”:
because, I think this is the most appropriate end to put in the tale you begun 🙂
Sincerely yours,
Petros Zimourtopoulos
I am afraid I understand less than before!
Well, in the case you are still interested enough, you have a new chance now,
and perhaps the last one, because the whole related thread with title
“Legal Issues” [*] may be evaporated, as it happened before with my opening
post of this thread, which I have to repeat it below, in order to get some sense
the rest, currently existing there 15 = 16 – 1, posts, as well as, to eliminate
any possibility for further excuses:
 “Legal Issues”

= xxxx xxxxxx:

 “You are strongly encouraged to discuss the class material using the forum”

 Fri 12 Sep 2014 10:00 AM CEST
 https://class.coursera.org/functionalanalysis002

 Hello.

 I am terribly sorry for the inconvenience, but I just noticed that

 [ screen capture of a typical coursera email ]

 coursera puts its Copyright Stamp on all emails it sends. Therefore,
 I had to delete all of my Open Access contributions here.

 Sincerely,

 Petros Zimourtopoulos
Farewell, Karen.
* https://class.coursera.org/functionalanalysis002/forum/thread?thread_id=437
I still don’t really understand the problem (then again, I’m remarkably sanguine about the legal quagmire inherent in online life) but I wish you a splendid journey! I myself will be traveling to Hell with Dante and Virgil, as of tomorrow…
May we both joyously get where we are going.
Is that you driving up my stat count from a certain European country? If so, I’ve left a message in another post…
Pingback: Discovering A Lot Besides Precalc MOOC  A Just Recompense
Checking your blog if you’re into “Minds and Machines” on edX just now, I stumbled, not getting immediately aware about the time lapse, into this conversation with that Petros guy, who certainly isn’t me. I faintly recall some really quirky encounters with his posting on Coursera, and do not remember if we two collided in any way in the forum. BTW, WhackAMole is a nice guy in my perception.
I do not expect to be much on this “Minds and Machines”forum, it being some really philosopher’s forum with overwhelming lots of wise singletons, but no constructive discussion of a more concrete, sufficiently specified topic. If I find enough leisure time, and the forum develops to my taste, I also intend to “contribute” to “Philosphy and Critical Thinking”, which appeares to be somehow in your schedule.
As an other exchange, MOOCs made me sad lately: some have intolerable lecturers, some videos are really bad in didactical perspective, the plague of “certification” gets increasingly annoying, but the worst, imho, is that the numbers of students per MOOC seem to drop dramatically. Each forum is just stuffed with over and over the same really old guys and dolls, and no fresh meat anymore to meet. Gone are the glorious days of types like that maniac W. Lewin who was a real star at the blackboard (but sadly also in real life) and now is kicked off MIT.
I shiver a bit, how a perfectly literate person mistakes my crooked style of writing for Petros’. I must be really a sadddening distance from writing a solid English idiom ;(.
You can easily extract a “certain European country” from the top level domain of my mail address, it is from my home contry and no fake. 🙂
Feel free to use it! Purgy
Purgy! So glad to see you – oh, my, I have been thinking of you so much. I will tell you why in a minute, but first things first.
Yes, I am in the Minds & Machines course, it snuck up on me after I made my “list”. I haven’t posted anything except one shoutout to Damien from Paradox/Infinity. I do have a question to pose re the Searle article but I’m still working on phrasing. Sometimes properly phrasing a question provides the answer. 😉
I am not hopeful about Philosophy and Critical Thinking (I peeked into a different Phil course from that university, was not impressed), but I will give it a shot.
As for MOOCs and certification –I agree with you. In fact, may I quote your post in a couple of places? I will not use your name unless you explicitly give me permission. I am in a Google Group with a bunch of Rogue MOOCers, and we have been mourning the demise of Coursera as we know it, into another Udacity, aimed at teaching vocational skills. I am also “mentoring” a Coursera class (I like to help, and it involves more technical/administrative work than subject knowledge) and they keep rolling out these changes – requiring payment to take quizzes? Are you insane???? – and I keep screaming EDUCATION ≠ VOCATIONAL TRAINING! but no one listens to me, they are trying to find a profitable business model and apparently that means students who are willing to pay, so those of us who are merely trying to educate ourselves are out of luck. So the Powers that Be at Coursera view me (and my friends, I would imagine) as the Old Guard, who will fade away eventually, leaving only our smiles. (you have read Alice, yes?)
As I said above, I have been thinking of you a lot lately, because of a precalc I took (it is still running, I’m still covering the message boards). I wrote it up here; if you had been there, we would have had a lot of fun, but it turned into a dud because no one wanted to play! Then a student showed up just the other day, and we argued for a while about a question, it was fascinating, I finally saw his point and he saw mine, but it was a great conversation, worthy of those you and I have had. So I have just been sitting here wishing you had been part of that course.
Heads up: edX also has a course titled “Fun with Prime Numbers” coming up in January. It might be too simple for you, but take a look.
I have been noticing a “certain European country” in my stats recently, and had hopes it might be you. I am glad I was right. 😉
Hi,
according to your first things first, I follow the occurences above:
1. The M&M course grew quite absorbing to me, I did not expect to spend so much time in it. Nevertheless, I’m somewhat disappointed by the lecturer and his put forward content. As I wrote already in a comment, his style might remind of an auctioneer, and being more evil, perhaps of a used car seller. Imho, he is after his own presentation more than after clearly presenting cognitive content. Some costudents (not the best) complain in the forum openly about this lack of precision. My grades for lecturer and lecture content are: not satisfactory to me. The forum is quite lively, but I know the best part of the contributors already. You know, I estimate highly Whacky, so do I for marnane, Galaxian and a few others, there are some morons like slate, polofourme and some weirdos, like fdesilva. djr does a fabulos job, mostly avoiding clear positions (like the master, like the slave), but keeping discussions going, always friendly, everywhere.
2. Philo&Critthink is a surprise to me. Reading your judgement and seeing the first clips coincided in an OMG! Blockheadedly continuing made me really enjoy the lecture snippets meanwhile. They appeared sooo childish at first glance, now I believe to experience some decent humor. Him being perhaps better, but that’s maybe just because of her voice (sic!). I do like the written content! It’s not a fancy philoshow, but an admirably neutral presentation of classical philosophy, apt for a relative beginner like me. Personally, I can’t stand that modern, arrogant behaviour of the atheist clique, laughing about religions, dualism, transcendentalism, …. To me they can’t defend their religion no iota better than all the other believers can argue theirs. Listen, it’s me who’s talking! 😉
3. The big smile of the vanished cat enjoys being cited, even when not given as source, perhaps the source is still somewhere and someone else. As a child I only knew the W.Disney version, and did not like that, it was too fast for me and the Queen of Hearts with the Club Soldiers way too evil, upbringing by calm soul. In my youth I did not like to read that many unscientific rubbish, and later I got to know that Carrol was a mathie, not a literature guy, so why read something beyond his formulas.
4. Postponing calculus for being my main reason for writing.
5. Will see if I have a look at prime numbers. I’m not very engaged in number theory, and “fun” as showcased attribute of math topics make me rather shy away. I have experienced so many moments of heartfelt joy when grasping a new math concept, that I do not want to spoil this possibility by some ephemerical “party fun”.
Calculus finalis.
1. I apologize for having shot you down in the MVT thread. I did not intend so. Please, feel free to mail to me any question about this, and I will try to answer aside from the forum. Considering your constant effort in conquering this concept, I truly think you deserve guidance. Please, let me know if and how I can offer remedy for my misbehaviour.
2. This current course about integration is not a really good course, neither methodically, nor for its rigor, nor for its friendliness. It is a rather plug and chug as many problems as possible course. Constantly mentioning how difficult integration were, is no rewarding access to this fantastic concept, giving the basis for all measurement on earth. This difficulty, celebrated in the exercises, is in days of Matlab, or Maple, or other crutches a long surmounted one. I’m looking forward to what integration concepts they will be presenting beyond antiderivatives. This is not to say that integration does not have its quirky sides.
3. I hope I make no failure in pointing you to an online integrator ( http://integrals.wolfram.com/ ) leading you also further to general tasks (deriving, solving, finding roots, … by wolframalpha.com. It’s free to use, and alleviates you from the imho nowadays ridiculous task of finding cheap integration tricks. Certainly, the use of this tools deprives you of exercising hard labour, and you will miss fluency in timed closed book exams, but I am convinced that integrating by hand is totally old school. May it help you to suck maximal gain from this course. Wolfram remark: there is even a mouse over button (A) to get the result ready for copy and paste, you just need to insert the dreaded “*”s in the right place.
Nice snowing you have around here! Cheers, Purgy
Hiya –
1) I’m quite enjoying M&M. I’m delighted the primary documents are included (Searle’s article, Turing’s paper). They’re so often omitted in courses like this, I’m very happy to see them there. I do appreciate that they’re somewhat edited, since it’s easy to get lost in technical material like that. I haven’t been doing much in the forums just because I haven’t had time to spend on the course (time pressure!) but I do have something I want to bring up, probably later today, re digital vs analog. It’s the sort of class where showoffs will be at peak preeniness.
2) I had much the same reaction to the Critical Thinking course as you did – The Glossary Fairy? Really? Must you? This is what happens when “Education must always be entertaining and fun in the pop culture sense” is your primary value – I find going through ideas to be very entertaining and fun. I’ve done several logic courses at this point, so I’m feeling pretty confident, though I get lost in the endless parade of “isms”. I’m particularly enjoying that a friend of mine is taking the course – he’s thinking of continuing his realworld education in an online program and is taking a few MOOCs to see if he can manage it and stick with it. He’s a lot of fun, though we’ve only had one conversation (he isn’t a fan of forums).
3) I ran into Dodgson/Carroll’s work when playing with Euclid. Some day I’m going to seriously read Alice (I am, at heart, literaturebased). Right now, I’m readng Flatland – the 19th Century Edwin Abbott work about a twodimensional world, and the implications for us of a 4th dimension – in an annoted version (by Ian Stewart) which is just wonderful, again, the original is right there, but with explanations for those of us who aren’t quite able to handle the serious stuff without help. What’s most surprising is that the first half is a satire of Victorian England, particularly in regard to the role of women and their right to education.
Calculus finalis
1) You didn’t shoot me down. I appreciate your insistence on precision and accuracy in all things, and I find it beneficial in many cases, but right now i’m feeling enormous time pressure. I haven’t even finished the first section! I haven’t even reached integrals yet! So I’m going to need to make some compromises, and one of them is moving on before completing that exercise. I do get the general concept of the “cone” containing the function, which serves as bounds of a sort, and I hope I can return to it when I’m not feeling so harried. By the way, I preferred your comments to Karene’s. I fear I’m not on her wavelength. I miss Stephen (I think that was his name) from the first course. And I’m glad Jen is still around. Did you know she emailed me to be a CTA? I declined, of course; I can’t even imagine why she would ask (she asked another friend of mine, neither of us had made more that 2 posts outside of “hello” posts). But I did get a kick out of everyone giving complicated advice to the guy who was using hours instead of minutes!
2) I’m quite fond of these MIT courses. They approach things from angles I haven’t seen before, which is something I need, otherwise I just rely on “oh, I know that” and don’t really learn anything.
3) I typically have at least one Wolfram window open at any time I’m doing math. 😉
The “snow” always surprises me, every year. It’s a WordPress feature: they turn it on right after Thanksgiving, and turn it off around January 6. As a user, I can only turn it on or off. I like it, so I keep it on. It varies in speed, density, and direction (towards mouse pointer, away from mouse pointer) over time. It’s fun to play with. I have simple tastes.
Oh, wow, did you see this:
Bite sized chunks. I might be able to do that. Unfortunately, it’s on the “new” platform – oh well, everything’s going to be on the “new” platform pretty soon.
Hi,
I just came here to tell you about this – evidently my mail alert did not tell me about your message. Where do you have this detailed motivations from? I only got a Coursera mail about it, I immediately enrolled, misunderstood the term “preview”, looked at the video, did the tests and the homework, or whatever, only to get afterwards the real meaning of “preview”, which I took for prerelease. Oh, it was really good to be reminded of his graphic style and his distinctive voice. Nobody else does it so ladidah. 🙂
I did not like the content of the preview very much, I have no grasp on why he uses this admittedly! circular definition of derivative, especially, since I explicitely dislike the bigO and its siblings. I’m strongly looking forward to the announced differential maps.
I really do not understand, why webdesign in general has to go all the way down to rockbottom dysfunctionality! If this is really going to be the “new” Coursera looks, they have degraded dramatically their ambiente. As is typical for me, I looked for the forum, but I was not a bit seduced to open even one thread – it was too hard on my eyes to single out one title. Ah, yes, and the pressure to “verified”: stongly increased! You hardly can get to the exercises before subscribing to put down your money. Arrrgh!
Meanwhile the META101course has come to a disappointing end, the last week being the weakest. I even visited the fairy in my dispair: must we, really?! But I’m still doubting in ranking it against M&M. Honestly, I start to increasingly dislike the lecturer for his pronounced secondhandcarseller behaviour, he is strongly after convincing his audience of one aspect, only to introduce triumphantly the contrary in the next minute, as latest, breaking, not news, but truth. You know my aversion against sidestepping precision, he seems to me a grandmaster in this art to me. He is a gifted rhetorician in my estimation, but people using this talent to move me in their direction annoy me.
The forum grew boring, imho, lately, the offending atheist moron slate aggressively driving his four horsemen across the land, you having no time, WhackAMolE being only reducedly active, people under heavy suspect of being teachers, pointing to Youtube videos with pittoresque longdivision, … And worst of all, but understandably, djr is busy running around, and cannot devote his time to me, only. 😉
Meanwhile, time is also ripe to wish jolly days to the holidays, so all the seasonal best and greetings!
Purgy
“Detailed motivations”? You mean prof/g’s tweet? I know a lot of people think Twitter is evil, but it’s quite a marvelous way to peek over the shoulders of academics. I’ve discovered some very helpful resources that way. And of course, there are the cute cat pictures and the political outrage. And keeping in touch, unintrusively, with friends. But at this point, I have more MOOC professors than anything else in my feed, and many of them are heavyduty mathematicians and scientists. Makes me feel smart. 😉
I’m definitely going to take the SV calc series again, but I want to finish the MIT 1B on integration (which I love, very well explained, lots of staff support though the input interface is seriously verrückt) first. I can only do one heavyduty math class at a time.
Neither philosophy course really makes me happy. I loved the Logic courses from The University of Melbourne, but those were mostly symbolic logic rather than an examination of philosophical positions such as dualism or functionalism. I’m still not sure if M&M/MIT is trying to teach the basics of logic via AI, or if it’s trying to teach the basics of AI using logic. It seems to me it’s not doing either particularly well, though I suppose if I were the type of student who would qualify for admission to MIT, I might feel differently. Critical Thinking would be pretty good if it weren’t for the fairies, grotesques, and referees. I dearly love whimsy and goofiness, but the same shtick over and over just falls flat after a while. If it were an 8 week course I would’ve dropped, but after 3 weeks, I can do one more. I can’t imagine what a nightmare it’s like for someone who’s native language is other than English, however – it’s a very “talky” class, with lots of idioms and inferences, some of which I don’t get (what the hell is a bush turkey? red cordial?).
Frohe Weihnachten und ein gutes neues Jahr zu Ihnen auch!
“PROSIT” for the New Year for you.
Reason for me privatly contacting is your comment on liking those FTC fairies. It totally took me by surprise, considering your fully understandable reservation against the “grotesques”, the fairy, and my personal horror the fallacy referee. In my “perfect” sensitivity I did not want to mention my position to this skit, but just to reprimand speculations about pregnancy, I experience as unallowed intimacy.
Evidently, judgements on educational content, intended to be “funny”, strongly depend on prerequisites, me being not so abhorred by the grotesques and you liking the FTC fairies, or are those whitches? And do you belong to one kin? ;p
I hope you’re doing well, not only in the Calculus, and please, don’t be negatively touched by me being “continuously” the “certified trouble maker” and “constantly” cheeky.
Best wishes, yours Purgy
It’s funny, I almost mentioned the Glossary Fairy in that post on the FTC Fairies (they were personifications of the Differentiation Operator, and the Integration Operator, as indicated by the little hats). And I’ll admit: I have no idea what the difference is, making one “meh” and the other adorable. It’s very possible it just boils down to connection: I feel very connected to Jen (this is the second course I’ve taken with her) and Karene, I so appreciate their efforts (I really have noticed how on top of the boards they are, on weekend evenings, on holidays, in this “deadzone” between Christmas and New Years, they and Phaine have done a superb job of staff support, beyond the student support that you and others have been providing (sometimes staff is necessary, to check grader errors, etc). And I’ll admit, Jen’s “endorsed” two of my posts as answers, which also makes me more likely to adore her. And they were genuinely adorable and happy, whereas the Glossary Fairy looked like she’d had root canal surgery that afternoon and was reciting poetry she’d memorized for her 4th grade teacher.
So I was surprised at my reaction, too. And yes, I’d say it had less to do with the quality of the schtick, than with my predisposition towards the players.
I was a lot less impressed with the “speeding ticket” at the beginning of the course, though that was for a more concrete reason: all I could think of was, if the professor/driver had been black, he would’ve been shot. And when the dashcam video was played, half of America would’ve declared he deserved it for mouthing off to a cop. Also, while I’m sure the prof is a nice guy and an excellent classroom teacher, I have no connection to him at all, and I find he frequently assumes facts not in evidence – that is, that I know what he’s talking about when he skips 13 steps. So he gets less slack with me on his shtick. Just like the cops give less slack to someone with dark skin, except my not being particularly fond of a calc teacher is not fatal.
The above has been yet another bitter rampage. Forgive me. You may not have any idea what I’m talking about, but a grand jury just decided a cop did nothing wrong when he shot and killed a 12 year old after <2 seconds of deliberation, and I'm sick of this.
As far as the Calc 1B is going: I'm probably not going to pass (I find the B questions unfathomable, and the way the grading is set up, it seems impossible to pass without them) but I consider it an excellent course and I'm very happy i'm taking it; it really makes me work! I also have noticed, for the first time, my background work to build up what i call the "fundamentals" – algebra, trig – are paying off. I spend a lot less time notrecognizing things like trig identities and factoring tricks, and that allows me to focus my energy on the calculus. Turns out, there is a reason for prerequisites! Who knew?
By the way, I think I've noticed a small change in you as well. While you have retained your "troublemaker" status and have no patience for namedroppers, you also seem to be enjoying taking a genuinely helpful an supportive role with those who are struggling, offering not only gentle and welltargeted hints, but inquiring as to their effectiveness. And as you've seen, it's been greatly appreciated. I've come across several of your exchanges after the fact and have found them quite helpful. It's been quite nice to see!
Hi,
some themes accumulated that hint me to write to you. First of all, since I estimate your interest in general MOOC matters very high, I had to recognize, that edX is stopping to award *any certificates* at all. There are now offers out, which award *verified certificates* only – of course with some fee. Luckily, as you will have noticed, I do not care at all about certificates, but I skimmed a 5part AMOcourse (Atomic and Optical Physics), running since Nov.2015, where 3 parts were finished a while ago, and where they *NOW* took down the links to the already earned honor certificates. Imagine, how some students complain about their vanished download links! With Coursera I have now two courses with no link to some “Statement of Acomplishment”, only giving some percentage. Taking this together with the empty fora at Ghrist’s, the generally seemingly declining numbers of course attendants, is this the dusk/fall of MOOCs?
And I did it again … made trouble with that youngster P. Haine. True heroes wouldn’t have done that. Perhaps, hopefully, we managed to reach calm waters again. You are perfectly right, I lack patience, and so it slipped out of my keyboard that someone should not *propagate rubbish*, I took the just reprimanding, edited and apologized, but reading my apology being called *rude*, saw me calling for the ice bucket, to suppress my burning hot desire for revenge. And all of this, because him becoming insanely jealous, with me saving some virgins/princesses from the dragon, when he couldn’t see the experienced real danger and stumbled across his own eagerness. Honestly, he was as ridiculous as I am. (But I was right!) 🙂 I try to take his LaTeX offer as a calumet.
Perhaps you should not put so much hope in me being genuinely helpful. Possibly its just my meager assortment of flowery closing phrases. 😉
I hope you are doing well with the math. Imho, the third part does not offer very much math, just rolled out concepts of “integration yields areas/volumes”. This is the heart of many exam problems, the joy of every teacher, and something to feel some afiliation with reality, but nothing really mathy, like the problem with ln(1/x). I seriously hope you got a grasp on the latter, it is much more important than this integration of the difference of two functions.
I don’t want to take any fun you might have out of the other chapters, but their appearance is owed more to the “common core” but to true, intrinsic math content. So just do what you like, you do not miss much math in case you do not fully engage. Just to spread some fragrance of reason: numerical math, done at this level, is a task for computers, they are better at it, and otherwise, numerical math is tedious, and mostly applied in specific topics; the propability, as tought in these lectures, is really best forgot, it’s similarly flawed as all the physic’s references in the lecture (see my “Food for Thought”), and the averages seem to be his hobby horse; perhaps google “higher moments” if you find some interest in them.
May I tell you about my compassion with every victim of the apparently trigger happy American police. Grown up in European culture with almost no weapons in private hands, I do not dare to critisize your rights to carry a gun, and I am also far from any strict believe that giving way to Obama’s wishes would reduce the number of gun rampages. I’m simply clueless, so I prefer not to discuss this, and feel lucky not to be in charge, but certainly wish for a salvaging(?) development.
Returning to math, I won’t miss to assure you my preparedness to help you in passing this course, just ask, and please, in case you feel confused by my answers, let me know of this circumstance, too, since I strictly want to stop immediately and at best avoid this at all cost. I do not know if I’ll go over the problem sets, but I have seen no really new problems in them. Doing them will award you with another step of getting payed off for having acquired routine and fluency.
I somewhat lauded in the forum your idea of having ungraded problems to discuss, but then I recalled the SV calculus, where two guys fought for getting as first the list of complete solutions to the public, and that I harvested a lot of downvotes just for slightly questioning this manner.
Talking about Ghrist, I admire your spirit to cling to this. May I humbly recommend to combine the lot of more simple material of the Calculus B course and the definitely more substantiated view of SV calculus. I’d never explicitely state this in the forum, but I feel obliged to tell only the truth to you: the “Pure Math” level of this course is “very poor”. It is dominated by the Eds, and them being the low end of the professional hierarchy. I know the FLCT, and I truly like it from the esthetical point of view, but I cannot judge, if it is apt to learn calculus from it. As always with Ghrist, I can’t get rid of the feeling that the inital step he requires to take, is a damned steep one for beginners. This also applies to the Calculcus Blue. He excellently points to the important material, but maybe one must search for additional exercises to get a firm hold on the subjects. I think that this Calculus B course does a really reasonable good job on providing exercises, and the “Pure Flaws” are of minor interest to the vast majority of attendants.
My interest in the M&M course is a big rollercoaster. I totally let go to even look at the exam, because I’m sick of “best guesses” based on finding no definitions for notions, which satisfy my requirements of strictness, but the forum grew to be the best I’ve ever seen in a philosophy MOOC. The new kids on the block Fabien and *saph* contributed some more solid ground for me to stand on while listening to the “grand story” told by A. Byrne, but I still do not want to buy a car from him. Damien is really a great Elisa, supplying that many students with “Is that so?”, “How does this sound?”, only scarcely with a “No”, and additionally with really an admirable amount of substance in his replies, too. Congrats to you for having had a better second half in the midterm. As said, I do not even know, where you’re better in. 🙂
All the best, Purgy
This will be a long response, so I’ll break it into numbered sections.
1) Would you be willing to take this to email? If I’m going to start slandering people/organizations, I should probably do it in a less accessible place. You can email me at sloopie72 (it’s a long story – based on an American song from the 60s, “Hang on Sloopy”) at gmail, if you wish.
2) I am heartbroken at the direction moocs are taking. Coursera in particular has done everything it can to destroy the magic they had three years ago: the new platform is terrible, the focus is clearly on business and computer science, the “on demand” idea is disastrous for any kind of interaction, and the paywall around exams is just mean. I served as a Mentor in a philosophy course last Fall, and so got to hear some “behind the scenes” rationale – none of which made me feel any better. In fact, I don’t think they have a clue what they’re doing, but they’re doing it at a reckless pace nonetheless. I mourn for KD’s Maththink, and for ModPo (a poetry course), which simply cannot exist in ondemand; I don’t know what will happen to those. I grieve that outstanding creative people like prof/g, dr.noor, Jim Fowler, and many others, who have put care and effort into their courses, have given way to “McMoocs” by committee. As for edX, it’s just a matter of time, I suppose. Right now I’m very much enjoying their offerings, though the quality is uneven. I guess the betatesting period of moocs is over, and it’s time to monetize, which means people like me, who provided the betatesting, will be thrown aside for alpha users who can fork over money for careeroriented certificates.
3) I missed your exchange with phaine – I will look for it. I take it you went beyond constructive criticism? 😉
4) I remember the people posting long, detailed solutions to ungraded questions in SV. At the time, I thought they were helpful. I have since revised my opinion, to think it is so much better to let someone ask – for many reasons, not the least of which is, in order to ask a question, you must have some formulation of the problem, and sometimes just doing that helps. Also, I’ve become a big fan of “speaking math” – learning how to communicate clearly and precisely (an average level of precision, not yours!). Knowing how to ask requires understanding. Getting “hints” rather than “here are the steps” is so much more conducive to learning.
5) I have a different perspective from you on all the “math ed” topics. Some mathy people inspire me to press onward. prof/g and KD fall into that category, as do many others. I can’t really learn math from prof/g, he’s above my level, but I so adore that he put Chaucer and Milton and art and humor into his works, I want to understand them. The understanding will have to come from a level I can follow, but that makes his course the “prize” I have my eyes on. In other words, my yardstick for measuring someone’s pedagogical value may have less to do with the mathematical rigor you look for, and more to do with whatever just keeps me going.
6) I’m quite disappointed in my peformance in the M&M course (disappointed in myself, that is, not in the course). I have no excuse; I just can’t keep it all straight. I have the basics of dualism and functionalism and this and that, but the devil is indeed in the details, and a simple detail can take hours to unravel. I will make my usual post about the course, and will wonder if the format did the material justice – if lectures are really the best way to go for this, if so much material should be crammed into a single mooc, if there’s some other way to handle these “thoughtintensive” courses. I do love all the slightly annotated source reading that’s included; so many courses skip over that. The subject is indeed fascinating, but the lectures are sometimes a buzzkill. I haven’t been dealing with the forums much, simply because I have no time – between Calculus and the Neuroscience course I’m taking, I’m overloaded.
7) A new course is starting tomorrow – “Fun with Prime Numbers” on edX. I can’t figure out if it’s a goofy thing, or serious – the preliminary quiz (the “am I in the right place” quiz) seemed much more substantive than the intro video, but they claim to require only high school algebra. Just mentioning it, if it’s of any interest to you as a recreational endeavor.