Indeed. Very often you are actually asked to teach a class, either a real class, or a mock class where other faculty and student volunteers pretend to be students, ask you questions, etc. That is in addition to a research talk.
There is already a number of better analogies posted as responses to this. Let me add another one.
Mr. Smith runs a hotel. One day, Mr. Jonson checks in and obtains a room key. After spending some time in town, he returns to the hotel and opens what he thinks is his room, only to find that he made a mistake and entered the room across the hall from his. Luckily, the guest in the room did not wake up, so Mr. Johnson quietly leaves and closes the door behind him. He wonders how is it possible that his key easily opened another room. A quick check reveals that he can use his key to open every single room on the floor. Further investigation shows that he can open every door in the entire hotel. He can even enter Mr. Smith's office where he finds a list of all addresses and credit card numbers of all guests, sitting on top of Mr. Smith's desk. During the investigation he comes to a conclusion that it is not just an accident, but a result of a systematic neglect, that his key was able to open every single door in the building. He then decides to inform all other guests about the problems he discovered. He also leaves a copy of the information at the front desk. Mr. Smith then has Mr. Johnson arrested for braking and entering. Mr. Johnson's defense is grounded in the fact that his privacy and safety, as well as privacy and safety of other customers was being recklessly put in danger by Mr. Smith's negligence.
I don't know. The first one seems pretty immature, but if he is mature enough to keep his immaturity out of the workplace, he may be good to have around. Having wild ideas is not always bad. The second one seems like pretty orderly family guy. Sounds good, but he may be boring like a brick. You may not want that on your team. The last one seems pretty capable, but you wouldn't want him to spend time moding game consoles when he should be doing his work.
I would check their references, see who they are and what they have to say. I would either ignore the pictures, or use them as a guide what to look for during the interview.
Are you talking about pstricks? That's pretty much the only important (commonly used) package that depends on postscript specials and that does not have an equivalent which works with pdftex. You can also produce pstricks illustrations as standalone postscript figures, convert them to pdf with pstopdf, and include them in your file. Many things that can be done with pstricks can also be done with pgf/tikz. And there is also something called pdftricks, although I have never tried it.
I am probably feeding a troll here, but I don't understand how this could have gotten "3, insightful". There are lot of things that can be said against LaTeX, and there are lot of reasons plain TeX can be preferable to LaTeX, but the parent post contains none of them.
First, LaTeX designers did not rewrite TeX. LaTeX is a macro package written on top of TeX, every time you use LaTeX, you in fact use TeX. It may not be the best designed package, and your claim that its creators failed to understand TeX does have some merit (the way they tried to hide some "complexities" of TeX and failed), at least you are not the only person claiming that, although others actually have produced some supporting arguments for that claim.
Second, could you perhaps elaborate on the source of your information that Knuth did METAFONT before TeX? It seems to be commonly accepted that he started working on METAFONT when he was already working on TeX, and realized that he will need a way to create fonts, as none of the existing computer fonts were suitable for TeX.
Most importantly, LaTeX generally does not require multiple passes to process a document. Again, LaTeX is just a macro package build on top of TeX, so if TeX can process a document in one pass, so can LaTeX. LaTeX will need multiple passes if your document include any floating material (figures, tables, table of contents, page headers, index etc). The reason is that some of this material must appear on paper before it appears in the source code, and there is simply no way to do that without multiple passes. Back when I was writing my master thesis, I did not know about LaTeX (it was very new then), and I used plain TeX. I wrote a package that produced things like table of contents, list of figures etc. It turned out I did it pretty much exactly the same way designers of LaTeX did (theirs is much more complicated and handles many things mine couldn't), as it is pretty much the only way to achieve this result in TeX: during the first pass, write all this "floating" information into a file, then include the file at a proper place during the second pass. Including this extra stuff changes pagination, which means that a third pass is required to fix that. Sometimes, if you are very unlucky, the change in page numbering or figure placement will cause another change in pagination, so you may need a fourth pass. If you are really unlucky, that is still not enough. Theoretically you can get into a situation where you will never have a document in which all the page numbers and references are correct. It has never happened to me, and I use LaTeX a lot. As far as I can tell, there is no way to avoid multiple passes for purposes of floating material.
Anyway, number of command line utilities and LaTeX editors can be used to completely hide this from you (unless you run in the very unlikely situation described above where an infinite number of passes is needed). They run LaTeX, detect if it needs to be ran again, and run it again until all is correct.
As far as writing core mods for TeX, number of people are working on that. There are PdfTeX, XeTeX, Aleph, LuaTeX at least. Lot of good work going into these projects, look them up. There are also other macro packages besides LaTeX: CONTeXT, eplain, lollipop,... But none of them will be able to "keep with the one pass elegance" if you want to have an automatically generated table of contents at the beginning of the document.
Going from tex to a camera-ready pdf is fairly nasty,...
This makes me curious, I had never actually had to produce a camera ready pdf myself, I just submit the tex fime, but is there anything wrong with pdf produced by pdflatex?
Graphics support is a pain...
Pdflatex can include pdf files and several bitmap formats. Is there any graphics format that cannot be easily converted to either png (for bitmaps) or pdf (for vector graphics)? Are there any drawbacks to such conversion?...editors may be more accustomed to submissions from Word users...
Luckily, not in my field. Editors in my field are definitely more accustomed to LaTeX users, to the point where most journals actually require submission in LaTeX....like over-eager hyphenation...
I never had trouble with that, but if you want to limit the number of hyphenated words in an easy way, try pdflatex with the microtype package. It tries to solve problems with microtypographic extensions before hyphenating, and I have seen pages and pages of beautiful output without a single hyphenated word when using it.
If the answer is "yes" to any of these questions, then you *should* know the pain of installing new latex libraries.
You mean like unzip or untar the package to/usr/local/share/texmf and run texhash? Yeah, that's really painful. In the worst case, you have to run tex to the.ins file, make a directory under/usr/local/share/texmf/tex/latex, copy your files there and run texhash.
Thesis and dissertation formats are sometimes difficult. Part of the reason is that some of them were created at the time when people were typing their thesis on a typewriter, or even handwriting them, and the requirements were instituted to ensure good readability. TeX will produce good looking readable document for you, but it will be a huge pain to make it follow some random word-per-line requirements.
The rest of the stuff is not that bad. Ratios, indentations, spacing, required empty pages etc can be handled quite easily, and the best thing is, once somebody does it once, if they do it well enough, everybody can simply use it. I wrote one of the several Ohio State dissertation document classes that are still floating around, it was my first document class, and it was definitely not hard. I know number of people who used the class to write their own thesis or dissertation, and all were telling me how easy it was to use.
I did not know that at the same time some guy from EE department also wrote a document class for dissertation, his was much better, with more features like watermarks and other options, I think most people these days use his, but the point is, it was not very hard, and once done, it was completely reusable. Luckily, OSU did not have a word-per-line requirement.
For me the major incompatibility is the equation editor. Most of the time I produce all my documents and presentations using TeX, but every once a while I need to work on a project where the documents need to be in MS Word format (that's usually when I do a job for the education department). That's a bit of problem for me, as I run Linux on both my home and office computer. Openoffice is fine for opening an occasional email that somebody clueless sent in a Word format although it really contains nothing but text, but when the document must have bunch of formulas and equations, Openoffice will not do it. I can create equations in Openoffice, and it will even open a Word document with equation objects in it and display them just fine, but an equation object I create in Openoffice will not display correctly in Word. I don't blame Openoffice for this, the problem is that Microsoft and DesignScience created a format that is even more cryptic and less portable than.doc, but the problem is there, and it makes it impossible for me to use Openoffice for these documents.
Do any of the web based wordprocessors have a decent equation editor?
Simple. 5 + 5 = 10, I know that, I have two hands, 5 fingers on each, that gives me 10 fingers. If you don't know that you have a serious problem. 6+6 means you have add two more fingers.
He'd have to at least memorize by ROTE learning (there's that dreaded word again) that 6 x 10 is 60.
6x10 is 60 by a simple definition of the number 60. That's what 60 is. What in the heck is there to memorize?
Ok, maybe not memorize since you can simply add a 0 to 6. But how does he figure out what 60 - 6 - 6 - 6 is without memorizing some subtraction tables?
That's addition up to 10. 10 - 6 = 4, I can do that on my fingers.
Oh I see, we do need some level of ROTE learning, but we must not be permitted to memorize a simple multiplication table so that we can be forced to go through the exercise of new math even if it requires a calculator.
1) I have never said that there should be no ROTE learning at all. I am just claiming that most of such memorizing as it is done in our schools is pointless, at least for some students.
2) I never said that people should not be permitted to memorize their multiplication tables. Some people don't find memorizing things hard. In my third grade class, I believe number of people simply memorized the tables and were done with it, but those of us who had problems memorizing things did not have to do that, since we were taught how to find other ways. And we probably learned more actual math while doing that. I am complaining about a typical American fourth grade classroom, (and I have seen a number of them) where the kids spend more than half a year drilling times tables, without ever realizing that there is more to multiplication that bunch of pointless facts. Than, two years later they are given a calculator and told that they can happily forget all that stuff they have learned before.
3) I don't know where you got the idea I am talking about any type of "new math". The way of learning multiplication I am talking about is over 100 years old. In fact, couple years after I finished the elementary school, it was declared too old, and was replaced by some sort of new method, where students spent first several years of school learning about something called "sets", and spent tons of times memorizing stupid terminology like "empty set" and so on. In first grade. Never mind that what they called "sets" had actually nothing to do with any notion of set in any possible set theory in existence.
4) Exactly how does anything I described here require a calculator? I was under the impression that what I said was that you don't actually need to even memorize the tables in order to be able to do multiplication reasonably fast. It would seem to me that a logical conclusion would be that one does not need a calculator even if one has trouble memorizing things, like myself. I don't let my students use calculators until they get to advanced calculus or diff eq. By that time most of them actually don't need it.
I'm sure this MIGHT be useful for some of the smarter types out there who are in love with numbers, but it scares the living daylights out of normal people and they end up going through life not knowing basic multiplication and hating/fearing math.
I don't have any statistics on this, but it does not appear to me that in the countries that use (or used in the past) the "method" (if you want to call it that, I personally consider the term "teaching method" idiotic) I described had significantly larger percentage of people hating or fearing math than the US.
Its true, that, when learning long division, all you do is "memorize the steps".
In fact, if you learn long division properly, there is very little memorization required. The algorithm follows very simple logic: You start with a large chunk of the dividend, ignoring all the digits at lower position to make it simpler, and see how many times can you fit the divisor in there. That will give you a partial quotient. Then you figure out how much of the dividend is covered by this partial quotient. You subtract that, as it is already covered, and repeat the process with the remainder. Nothing mysterious there.
As I have already explained in a reply to another post, I was not talking about memory capacity here.
You think memorizing multiplication tables somehow prevents someone from eventually becoming a good mathematician?
It will obviously not prevent anybody from becoming a mathematician, however, it will not help either. In fact, it may give you a slight handicap, as it will give you a completely wrong idea about what math is.
You're effectively advocating the use of higher-level algebra to derive a simple 8 x 7 problem. This might have positive effects for some elite math types though memorizing more tables is more harmful, but what is undeniable is that the methods you advocate is disastrous for the masses who end up hating/fearing math.
Come on, do you really believe this is higher-level algebra? Sure, you can formulate it in terms of group and field axioms etc, but in fact all you need to use is commutative, associative and distributive properties of real numbers. I don't know if you ever had a chance to teach introductory algebra, but pretty much the only problem with teaching these "laws of real numbers", how the textbooks like to call them, is to explain the students why they should even bother with such an obvious stuff, and why should we give these simple and obvious properties such fancy names. As a matter of fact, I have yet to see an introductory algebra textbook that manages to explain this in a good way.
Do you really think that 3rd graders will have trouble realizing that if you organize bunch or rocks in 7 rows with 8 rocks in each, you get the same number of rocks as if you do a 5 by 8 rectangle and another 2 by 8 rectangle?
As for hating or fearing math, I know a number of people who hate or fear math exactly because it evokes in their mind an image of endless memorization of useless facts. The funny thing is that stuff like that has actually nothing to do with math. If I was required to memorize my multiplication facts in third grade, I would probably ended up hating math as well.
Except that no one learns like that. Learning is not "just like" throwing objects into a box. Study after study has shown that humans learn by taking facts and drawing connections between those facts and other facts that they already know. In this sense, the more facts you have, the more connections you're able to make.
There are also number of studies that seem to suggest that people learn better by discovering things rather than being told facts to memorize. Also, there are more useful interesting facts and less useful and interesting facts. As far as multiplication goes, the facts that 3x5=15 is pretty useless by itself. The fact that you can arrange 15 pieces in a rectangle with 3 rows, and if you take 1 row away you will have 10 pieces left, is in my experience and opinion much more interesting and useful (and I don't mean useful for "practical" applications, I mean useful for drawing connections).
And you miss all of the connections that could be created by having a proper library of facts to draw from.
This view of teaching (simple facts + basic principles) is very attractive as a theoretical model, but there's very little data to support any view of that being the way people actually learn.
Having a larger library of facts is not necessarily helpful. I highly doubt that memorizing all known zeros of the zeta function will take you any closer to proving the Riemann hypothesis. I may be wrong, and you may discover a weird pattern that everybody missed so far, but I believe that general consensus is that to learn about the Zeta function, you need to learn much smaller number of much simpler, but more essential facts.
But "simple" multiplication is a decently complex algebra problem. And when I was in 3rd grade, what I described is how we have learned multiplication. We simply studied numbers and organized things in rectangular patterns, we discovered and subsequently memorized some rules, and eventually learned how to multiply. Some of us, those that were better at memorizing rather than derivation, memorized most of the facts, others learned ways similar to the way I described. Later, in 6th and 7th grade, we just formalized all the intuitive ideas about properties of numbers and operations we actually discovered and learned in the second and third grade.
But my point is, as far as I remember, I did not spend any significant time drilling multiplication facts at school. There was some small amount of drilling, but we spent far more time actually exploring numbers and discovering ways how to avoid drilling. Which is, at least at that level, what math is all about.
I definitely do not suggest that everybody learns my exact method of multiplication. I believe that the best way to learn multiplication is, with some guidance and help, figuring out your own way how to do it.
On the other hand, I agree that learning to memorize is a useful tool, which I have never really master. I am notoriously bad at remembering names, phone numbers, addresses etc. That's one thing I have never learned. Not that I was not really required to. In many subjects at school we were supposed to memorize whole bunch of stuff. Most of those subjects I barely passed. Fortunately, math was not one of them.
I know that half of 7 is 3 and a half, that's easy, because half of 6 is three. Multiplying by 10 is easy, very little memorization required there. That will give me 35 + 7, and I do have sort of memorized that 5 + 7 is 12, I am pretty sure about it because 5+5 is 10, and then there is 2 more.
I will probably remember 6x7=42 for several days now, but I guarantee you that after a short while I will forget it and have to derive it again.
I am not talking about space in terms of memory. I meant space in perhaps more abstract form. Perhaps better way to put is this: After memorizing the tables, you have this pile of pretty much useless facts, sort of jumbled together, and you have to go back and sort through them to discover any connection. If you instead memorize only some basic facts and several basic rules, you will have something that I would liken to a clean table with just few objects on it, with plenty of space to draw connections.
8 times 7 is a particularly tricky one. I would either do 70 - 14 (the 10 tables and 2 tables I have mostly memorized), or 40 + 16 (the 5 tables are rather easy, too, you first divide by 2 and then multiply by 10). So it seems that 8 times 7 is 56, but no, I am not able to recall that without doing one of the above (or some similar such) calculations.
Indeed it is... IF you've got the multiplication tables memorized...
I have a PhD in math, and I still don't have the multiplication tables memorized. I can multiply without problems, because these things are very easy to figure out. In fact, I thing that should my school require me to memorize the tables, I probably would not choose to study math. And if I did, I would probably be worse at it.
Learning is about making connections. Memorizing is about having the bits in place to connect. Education requires both.
True. However, after memorizing "the tables", how much space is there to make connections? There are number of fascinating connections related to multiplication that can be discovered after memorizing just a few simple rules. And after kids spend several months memorizing and drilling multiplication tables, how much time and how much desire is there to make connections?
Slashdot Mirror
Indeed. Very often you are actually asked to teach a class, either a real class, or a mock class where other faculty and student volunteers pretend to be students, ask you questions, etc. That is in addition to a research talk.
There is already a number of better analogies posted as responses to this. Let me add another one.
Mr. Smith runs a hotel. One day, Mr. Jonson checks in and obtains a room key. After spending some time in town, he returns to the hotel and opens what he thinks is his room, only to find that he made a mistake and entered the room across the hall from his. Luckily, the guest in the room did not wake up, so Mr. Johnson quietly leaves and closes the door behind him. He wonders how is it possible that his key easily opened another room. A quick check reveals that he can use his key to open every single room on the floor. Further investigation shows that he can open every door in the entire hotel. He can even enter Mr. Smith's office where he finds a list of all addresses and credit card numbers of all guests, sitting on top of Mr. Smith's desk. During the investigation he comes to a conclusion that it is not just an accident, but a result of a systematic neglect, that his key was able to open every single door in the building. He then decides to inform all other guests about the problems he discovered. He also leaves a copy of the information at the front desk. Mr. Smith then has Mr. Johnson arrested for braking and entering. Mr. Johnson's defense is grounded in the fact that his privacy and safety, as well as privacy and safety of other customers was being recklessly put in danger by Mr. Smith's negligence.
I don't know. The first one seems pretty immature, but if he is mature enough to keep his immaturity out of the workplace, he may be good to have around. Having wild ideas is not always bad. The second one seems like pretty orderly family guy. Sounds good, but he may be boring like a brick. You may not want that on your team. The last one seems pretty capable, but you wouldn't want him to spend time moding game consoles when he should be doing his work.
I would check their references, see who they are and what they have to say. I would either ignore the pictures, or use them as a guide what to look for during the interview.
I have once walled in a server in order to hide it. Unfortunately, I have accidentally sealed in a huge half blind black cat with it.
Go ahead, but it will help you about as much as putting a rouge on a pig.
I didn't mean entomology, I meant etymology...
Don't worry!
Languages change over time, word shift meaning.
In a few years, who know what will "entomology" mean.
...believe in fiscal responsibility, reduced government, etc.
So you are saying he is going to keep the current staff?
Are you talking about pstricks? That's pretty much the only important (commonly used) package that depends on postscript specials and that does not have an equivalent which works with pdftex. You can also produce pstricks illustrations as standalone postscript figures, convert them to pdf with pstopdf, and include them in your file. Many things that can be done with pstricks can also be done with pgf/tikz. And there is also something called pdftricks, although I have never tried it.
I am probably feeding a troll here, but I don't understand how this could have gotten "3, insightful". There are lot of things that can be said against LaTeX, and there are lot of reasons plain TeX can be preferable to LaTeX, but the parent post contains none of them.
First, LaTeX designers did not rewrite TeX. LaTeX is a macro package written on top of TeX, every time you use LaTeX, you in fact use TeX. It may not be the best designed package, and your claim that its creators failed to understand TeX does have some merit (the way they tried to hide some "complexities" of TeX and failed), at least you are not the only person claiming that, although others actually have produced some supporting arguments for that claim.
Second, could you perhaps elaborate on the source of your information that Knuth did METAFONT before TeX? It seems to be commonly accepted that he started working on METAFONT when he was already working on TeX, and realized that he will need a way to create fonts, as none of the existing computer fonts were suitable for TeX.
Most importantly, LaTeX generally does not require multiple passes to process a document. Again, LaTeX is just a macro package build on top of TeX, so if TeX can process a document in one pass, so can LaTeX. LaTeX will need multiple passes if your document include any floating material (figures, tables, table of contents, page headers, index etc). The reason is that some of this material must appear on paper before it appears in the source code, and there is simply no way to do that without multiple passes. Back when I was writing my master thesis, I did not know about LaTeX (it was very new then), and I used plain TeX. I wrote a package that produced things like table of contents, list of figures etc. It turned out I did it pretty much exactly the same way designers of LaTeX did (theirs is much more complicated and handles many things mine couldn't), as it is pretty much the only way to achieve this result in TeX: during the first pass, write all this "floating" information into a file, then include the file at a proper place during the second pass. Including this extra stuff changes pagination, which means that a third pass is required to fix that. Sometimes, if you are very unlucky, the change in page numbering or figure placement will cause another change in pagination, so you may need a fourth pass. If you are really unlucky, that is still not enough. Theoretically you can get into a situation where you will never have a document in which all the page numbers and references are correct. It has never happened to me, and I use LaTeX a lot. As far as I can tell, there is no way to avoid multiple passes for purposes of floating material.
Anyway, number of command line utilities and LaTeX editors can be used to completely hide this from you (unless you run in the very unlikely situation described above where an infinite number of passes is needed). They run LaTeX, detect if it needs to be ran again, and run it again until all is correct.
As far as writing core mods for TeX, number of people are working on that. There are PdfTeX, XeTeX, Aleph, LuaTeX at least. Lot of good work going into these projects, look them up. There are also other macro packages besides LaTeX: CONTeXT, eplain, lollipop, ... But none of them will be able to "keep with the one pass elegance" if you want to have an automatically generated table of contents at the beginning of the document.
Going from tex to a camera-ready pdf is fairly nasty, ...
This makes me curious, I had never actually had to produce a camera ready pdf myself, I just submit the tex fime, but is there anything wrong with pdf produced by pdflatex?
Graphics support is a pain...
Pdflatex can include pdf files and several bitmap formats. Is there any graphics format that cannot be easily converted to either png (for bitmaps) or pdf (for vector graphics)? Are there any drawbacks to such conversion? ...editors may be more accustomed to submissions from Word users...
Luckily, not in my field. Editors in my field are definitely more accustomed to LaTeX users, to the point where most journals actually require submission in LaTeX. ...like over-eager hyphenation...
I never had trouble with that, but if you want to limit the number of hyphenated words in an easy way, try pdflatex with the microtype package. It tries to solve problems with microtypographic extensions before hyphenating, and I have seen pages and pages of beautiful output without a single hyphenated word when using it.
If the answer is "yes" to any of these questions, then you *should* know the pain of installing new latex libraries.
You mean like unzip or untar the package to /usr/local/share/texmf and run texhash? Yeah, that's really painful. In the worst case, you have to run tex to the .ins file, make a directory under /usr/local/share/texmf/tex/latex, copy your files there and run texhash.
Thesis and dissertation formats are sometimes difficult. Part of the reason is that some of them were created at the time when people were typing their thesis on a typewriter, or even handwriting them, and the requirements were instituted to ensure good readability. TeX will produce good looking readable document for you, but it will be a huge pain to make it follow some random word-per-line requirements.
The rest of the stuff is not that bad. Ratios, indentations, spacing, required empty pages etc can be handled quite easily, and the best thing is, once somebody does it once, if they do it well enough, everybody can simply use it. I wrote one of the several Ohio State dissertation document classes that are still floating around, it was my first document class, and it was definitely not hard. I know number of people who used the class to write their own thesis or dissertation, and all were telling me how easy it was to use.
I did not know that at the same time some guy from EE department also wrote a document class for dissertation, his was much better, with more features like watermarks and other options, I think most people these days use his, but the point is, it was not very hard, and once done, it was completely reusable. Luckily, OSU did not have a word-per-line requirement.
all alike.
Actually, that is a pretty good description of slashdot.
Just turn off the light and wait.
For me the major incompatibility is the equation editor. Most of the time I produce all my documents and presentations using TeX, but every once a while I need to work on a project where the documents need to be in MS Word format (that's usually when I do a job for the education department). That's a bit of problem for me, as I run Linux on both my home and office computer. Openoffice is fine for opening an occasional email that somebody clueless sent in a Word format although it really contains nothing but text, but when the document must have bunch of formulas and equations, Openoffice will not do it. I can create equations in Openoffice, and it will even open a Word document with equation objects in it and display them just fine, but an equation object I create in Openoffice will not display correctly in Word. I don't blame Openoffice for this, the problem is that Microsoft and DesignScience created a format that is even more cryptic and less portable than .doc, but the problem is there, and it makes it impossible for me to use Openoffice for these documents.
Do any of the web based wordprocessors have a decent equation editor?
How does he know 6 + 6 = 12 without a calculator?
Simple. 5 + 5 = 10, I know that, I have two hands, 5 fingers on each, that gives me 10 fingers. If you don't know that you have a serious problem. 6+6 means you have add two more fingers.
He'd have to at least memorize by ROTE learning (there's that dreaded word again) that 6 x 10 is 60.
6x10 is 60 by a simple definition of the number 60. That's what 60 is. What in the heck is there to memorize?
Ok, maybe not memorize since you can simply add a 0 to 6. But how does he figure out what 60 - 6 - 6 - 6 is without memorizing some subtraction tables?
That's addition up to 10. 10 - 6 = 4, I can do that on my fingers.
Oh I see, we do need some level of ROTE learning, but we must not be permitted to memorize a simple multiplication table so that we can be forced to go through the exercise of new math even if it requires a calculator.
1) I have never said that there should be no ROTE learning at all. I am just claiming that most of such memorizing as it is done in our schools is pointless, at least for some students.
2) I never said that people should not be permitted to memorize their multiplication tables. Some people don't find memorizing things hard. In my third grade class, I believe number of people simply memorized the tables and were done with it, but those of us who had problems memorizing things did not have to do that, since we were taught how to find other ways. And we probably learned more actual math while doing that. I am complaining about a typical American fourth grade classroom, (and I have seen a number of them) where the kids spend more than half a year drilling times tables, without ever realizing that there is more to multiplication that bunch of pointless facts. Than, two years later they are given a calculator and told that they can happily forget all that stuff they have learned before.
3) I don't know where you got the idea I am talking about any type of "new math". The way of learning multiplication I am talking about is over 100 years old. In fact, couple years after I finished the elementary school, it was declared too old, and was replaced by some sort of new method, where students spent first several years of school learning about something called "sets", and spent tons of times memorizing stupid terminology like "empty set" and so on. In first grade. Never mind that what they called "sets" had actually nothing to do with any notion of set in any possible set theory in existence.
4) Exactly how does anything I described here require a calculator? I was under the impression that what I said was that you don't actually need to even memorize the tables in order to be able to do multiplication reasonably fast. It would seem to me that a logical conclusion would be that one does not need a calculator even if one has trouble memorizing things, like myself. I don't let my students use calculators until they get to advanced calculus or diff eq. By that time most of them actually don't need it.
I'm sure this MIGHT be useful for some of the smarter types out there who are in love with numbers, but it scares the living daylights out of normal people and they end up going through life not knowing basic multiplication and hating/fearing math.
I don't have any statistics on this, but it does not appear to me that in the countries that use (or used in the past) the "method" (if you want to call it that, I personally consider the term "teaching method" idiotic) I described had significantly larger percentage of people hating or fearing math than the US.
Its true, that, when learning long division, all you do is "memorize the steps".
In fact, if you learn long division properly, there is very little memorization required. The algorithm follows very simple logic: You start with a large chunk of the dividend, ignoring all the digits at lower position to make it simpler, and see how many times can you fit the divisor in there. That will give you a partial quotient. Then you figure out how much of the dividend is covered by this partial quotient. You subtract that, as it is already covered, and repeat the process with the remainder. Nothing mysterious there.
Exactly how big you think a human mind is?
As I have already explained in a reply to another post, I was not talking about memory capacity here.
You think memorizing multiplication tables somehow prevents someone from eventually becoming a good mathematician?
It will obviously not prevent anybody from becoming a mathematician, however, it will not help either. In fact, it may give you a slight handicap, as it will give you a completely wrong idea about what math is.
You're effectively advocating the use of higher-level algebra to derive a simple 8 x 7 problem. This might have positive effects for some elite math types though memorizing more tables is more harmful, but what is undeniable is that the methods you advocate is disastrous for the masses who end up hating/fearing math.
Come on, do you really believe this is higher-level algebra? Sure, you can formulate it in terms of group and field axioms etc, but in fact all you need to use is commutative, associative and distributive properties of real numbers. I don't know if you ever had a chance to teach introductory algebra, but pretty much the only problem with teaching these "laws of real numbers", how the textbooks like to call them, is to explain the students why they should even bother with such an obvious stuff, and why should we give these simple and obvious properties such fancy names. As a matter of fact, I have yet to see an introductory algebra textbook that manages to explain this in a good way.
Do you really think that 3rd graders will have trouble realizing that if you organize bunch or rocks in 7 rows with 8 rocks in each, you get the same number of rocks as if you do a 5 by 8 rectangle and another 2 by 8 rectangle?
As for hating or fearing math, I know a number of people who hate or fear math exactly because it evokes in their mind an image of endless memorization of useless facts. The funny thing is that stuff like that has actually nothing to do with math. If I was required to memorize my multiplication facts in third grade, I would probably ended up hating math as well.
Except that no one learns like that. Learning is not "just like" throwing objects into a box. Study after study has shown that humans learn by taking facts and drawing connections between those facts and other facts that they already know. In this sense, the more facts you have, the more connections you're able to make.
There are also number of studies that seem to suggest that people learn better by discovering things rather than being told facts to memorize. Also, there are more useful interesting facts and less useful and interesting facts. As far as multiplication goes, the facts that 3x5=15 is pretty useless by itself. The fact that you can arrange 15 pieces in a rectangle with 3 rows, and if you take 1 row away you will have 10 pieces left, is in my experience and opinion much more interesting and useful (and I don't mean useful for "practical" applications, I mean useful for drawing connections).
And you miss all of the connections that could be created by having a proper library of facts to draw from.
This view of teaching (simple facts + basic principles) is very attractive as a theoretical model, but there's very little data to support any view of that being the way people actually learn.
Having a larger library of facts is not necessarily helpful. I highly doubt that memorizing all known zeros of the zeta function will take you any closer to proving the Riemann hypothesis. I may be wrong, and you may discover a weird pattern that everybody missed so far, but I believe that general consensus is that to learn about the Zeta function, you need to learn much smaller number of much simpler, but more essential facts.
But "simple" multiplication is a decently complex algebra problem. And when I was in 3rd grade, what I described is how we have learned multiplication. We simply studied numbers and organized things in rectangular patterns, we discovered and subsequently memorized some rules, and eventually learned how to multiply. Some of us, those that were better at memorizing rather than derivation, memorized most of the facts, others learned ways similar to the way I described. Later, in 6th and 7th grade, we just formalized all the intuitive ideas about properties of numbers and operations we actually discovered and learned in the second and third grade.
But my point is, as far as I remember, I did not spend any significant time drilling multiplication facts at school. There was some small amount of drilling, but we spent far more time actually exploring numbers and discovering ways how to avoid drilling. Which is, at least at that level, what math is all about.
I definitely do not suggest that everybody learns my exact method of multiplication. I believe that the best way to learn multiplication is, with some guidance and help, figuring out your own way how to do it.
On the other hand, I agree that learning to memorize is a useful tool, which I have never really master. I am notoriously bad at remembering names, phone numbers, addresses etc. That's one thing I have never learned. Not that I was not really required to. In many subjects at school we were supposed to memorize whole bunch of stuff. Most of those subjects I barely passed. Fortunately, math was not one of them.
I know that half of 7 is 3 and a half, that's easy, because half of 6 is three. Multiplying by 10 is easy, very little memorization required there. That will give me 35 + 7, and I do have sort of memorized that 5 + 7 is 12, I am pretty sure about it because 5+5 is 10, and then there is 2 more.
I will probably remember 6x7=42 for several days now, but I guarantee you that after a short while I will forget it and have to derive it again.
I am not talking about space in terms of memory. I meant space in perhaps more abstract form. Perhaps better way to put is this: After memorizing the tables, you have this pile of pretty much useless facts, sort of jumbled together, and you have to go back and sort through them to discover any connection. If you instead memorize only some basic facts and several basic rules, you will have something that I would liken to a clean table with just few objects on it, with plenty of space to draw connections.
8 times 7 is a particularly tricky one. I would either do 70 - 14 (the 10 tables and 2 tables I have mostly memorized), or 40 + 16 (the 5 tables are rather easy, too, you first divide by 2 and then multiply by 10). So it seems that 8 times 7 is 56, but no, I am not able to recall that without doing one of the above (or some similar such) calculations.
Indeed it is... IF you've got the multiplication tables memorized...
I have a PhD in math, and I still don't have the multiplication tables memorized. I can multiply without problems, because these things are very easy to figure out. In fact, I thing that should my school require me to memorize the tables, I probably would not choose to study math. And if I did, I would probably be worse at it.
Learning is about making connections. Memorizing is about having the bits in place to connect. Education requires both.
True. However, after memorizing "the tables", how much space is there to make connections? There are number of fascinating connections related to multiplication that can be discovered after memorizing just a few simple rules. And after kids spend several months memorizing and drilling multiplication tables, how much time and how much desire is there to make connections?