Probably these kids are smarter than I am in a sense, with their agile young brains and good memories, but I admit that now in my old age, I sometimes forget the algorithm for subtracting and borrowing, and it helps me to rethink that when I borrow one from the tens column, that is really a...
Another place I encountered a difference of approach was in multiplication. My class had only been told that multiplication is repeated addition, so they had trouble grasping how to multiply numbers that are not integers. I.e. is sqrt(2)x pi equal to sqrt(2) added to itself pi times? I myself...
I looked at the article and admit I find it somewhat depressing that parents are struggling with understanding that a number like 43 means we have 43 separate items that are grouped into 4 sets with 10 things in each set plus 3 remaining single times. They are just asking us to realize that the...
I agree with you 100% about the constructive proofs being more persuasive. It seems there are 2 ways of doing calculus rigorously, either assuming the reals are represented by infinite decimals, or just assuming they form an ordered field with the least upper bound property. This second...
yes, pardon me, the proof i gave yields half of that result almost immediately, with my c equal to your B. Replacing f by -f gives the existence of a minimum, say d, for f. Then the image interval is at least contained in [d,c].
The theorem they stated is actually the composition of two basic...
I think you chose very well. That book was not as well known as many others, such as Thomas, but was excellent. As parent stories go, my mom wanted me to practice the piano and be the next Liberace. Instead I was reading Theory of Sets, at the nearby university library, by Erich Kamke...
Thanks, yes that was considered a good book in my day (by me). My guess is your self taught course excels what many high school courses offer. I myself was not offered calculus in high school, (and did not study it myself), indeed I did not even learn trig. Upon entering college I was...
as i observed above however, it does make you too expensive for some universities to be willing to allow it. so i don't find it odd. it speaks to the commitment of your university to teaching. the size of the classes is also relevant; e.g. at the university I attended as an undergraduate...
It depends on the funding available to the department and the College. In my university in Georgia, the math department was treated as a cash cow by the administration, in that it was expected to earn double what it consumed in wages. This meant many lower level course, but not all, were...
I always wonder what it means to "do calculus", in say the 9th grade. The hard concept is mainly the definition of "completeness" the real numbers, and without such a definition one cannot prove the existence of the things calculus is designed to find, such as maxima and minima, and areas of...
Well, in his article he said this is an interesting subject that has been much neglected, and he thinks it deserves more investigation. Thus I would think you are exactly the kind of person he wrote the article for. So I would suggest asking him exactly for the information that you asked us for.
My google research shows some 3 eras of hand computation of pi. First, Archimedes' method approximating area of a circle by that of a many sided polygon, eventually yielding a few dozen digits. Second, Taylor series for arctan augmented by addition formulas, yielding over 100 digits. (Euler...
Sorry, I must have an account so that screen did not come up for me.
Briefly he says almost no one has thought about the foundations for some 50 years or more, perhaps ever since Bohr "bested" Einstein in the public relations arena debating in the 1930's, arguing that we don't need to...
One of those occasions occurred today in this thread, namely @BvU, in post #2, I think it's 358 not 359. I.e. I can remember roughly 3.14159 26535 89793 23846, but my granddaughter knows about 3 times as many decimal places.
Based on this article in the NYT, if I were in your situation, I would ask Sean Carroll of CalTech for advice:
https://www.nytimes.com/2019/09/07/opinion/sunday/quantum-physics.html
I predict this thread will easily surpass 100 posts. At least in the past, these discussions of pedagogy seem to live on forever. I taught for a long time in a college math dept and listened to many of these arguments. I noticed that no one ever agreed about how teaching should be done...
In both cases you hve a set with one element, hence there are two subsets, the subset containing that element and the one not containing it. It does not matter what the element is.
I am not sure of the answer to your question, but I have some experience. In my own math career I have found that basic topology is probably the most useful and fundamental language and knowledge there is, absolutely valuable to almost everyone, certainly to me. The introduction to John...
I am not sure I understand you. Are you trying to help the OP, or just argue that I myself have nothing to offer him and should keep my mouth shut? In the latter case, you would not be the first.
I always heard J. Robert Oppenheimer took 6 or more courses at a time at Harvard and aced them all, graduating phi beta kappa in 3 years. Since I found it challenging to get a C there in any one course, taking only the usual 4, I think this is impressive, but apparently possible.
I also think Bourbaki gets a bad rap that to me at least is not borne out by reading Bourbaki books. In the ones I have been consulting, not only is the theory explained well, but the problems seem excellent and there is even a very rare, perhaps unique, discussion of the history of the subject...
Greg's comment about understanding how a child's mind develops reminded me of reading works by Piaget where he pointed out that his experiments showed that at a certain age children do not realize that the amount of milk does not change when you pour it from one glass to another different shaped...
I am not a physicist and do not know all those topics, but I am a mathematician, and have taught manifolds, covering spaces, and discrete groups for years. In my opinion one reason you are having a hard time is the ridiculously large amount of material you are trying to cram in. So you should...
Thank you for the feedback. Since you appreciate Dieudonne', I recommend also a much more traditional book as a complement to it, namely vol. 2 (also vol.1), of Courant's Differential and Integral Calculus. When Dieudonne' says in his preface "It is clear students should have a good working...
The calculus of logic did exist a long time ago, (in fact I think it used to be taught in schools as I once had an old textbook of logic), but mathematicians then and now do not actually prove things by checking truth tables. Rather they internalize these rules and write the proofs in English...
Technically I agree, but it is hard to imagine someone who needs to use that method to multiply 10 by 10 (i.e. one digit at a time). I.e. I would have said up to 9x9, but I couldn't imagine someone not knowing what it means to move a decimal point.
By the way as I read your post I could not...
@Mark44: yes when our son came home he had this guilty look, like he had a license to steal, since there were no challenging expectations on him at that school where he got to essentially play all day long.
When I had to teach the grouping stuff, I tried to make up real world examples, like using old fashioned English money: i.e. given 5,000 pence say, how many pounds, shillings, and pence would that make, if you changed it all into the largest denominators possible? Or given 650 empty bottles...