mtbb34@ms.uky.edu (Becky McEllistrem) (02/22/88)
In article <4497@garfield.UUCP> martyn@garfield.UUCP (Martyn Quigley) writes: >In article <660@cresswell.quintus.UUCP> ok@quintus.UUCP (Richard A. O'Keefe) >writes: > >>operator is. Talk about *boring*. Talk about *remote* from the interests >>of the children. GGGGGrrrrrrrrrr... you're hitting a sore spot with me in this discussion so sorry if it becomes a flame, it certainly wasn't meant to be... Boring is 1) in the eye of the beholder (most ed majors find computers boring too) and 2) is dependent on how well the teacher presents the subject.... if you were bored, then he/she presented it wrongly. >old "boring" stuff. > > >One reason so many people are weak in mathematics is that as it was taught >to them, it was NOT related to their prior experiences. Hands up everyone I'm not sure I understand this statement... Math word problems were constantly related to concrete experi There was nothing involving games or different drill "boards" or concrete ways of INTRODUCING the subject. Merely: this is how it's done, because I SAID so and WHY was not something a student was to worry about. My experience >of teaching mathematics majors is that none of them taught in this way can tell >me where to find a group in "the real world", what a group IS, or what it is >used for. Ask the nearest mathematics student what an eigenvector has to do >with his/her/its reflection in a mirror. And what do you do to help these students... I hope the classes at your math education department aren't pure lectures, because that's WHY they don't have an UNDERSTANDING of these subjects > >To paraphrase Piaget, teaching a child something it has not already met in the >course of its spontaneous development is a waste of time. You cannot teach a >concept from a definition. I agree with this. Any classes I've taught so far (and I'm not certified yet, but working on it) have gone much smoother, if I use a process-oriented curriculum rather than a lecture-teacher-oriented curriculum. The new math would have gone, I think if the students understood WHAT they were doing... and that's easier to bring across in a process oriented curriculum... > >>I think we need three things in elementary arithmetic teaching: >> principles: "This is WHY the addition algorithm works." And do you lecture that why or show them that why? Do you cut a square to prove two triangles make a square or do you give them some big formula that they won't remember? > >So why does it? My experience (as a teacher trainer) is that relatively few >elementary teachers know how it works. As for multiplication or division... Tell me about it... > >Drills are appropriate if what you want is high performance in a specific task >such as in piano playing, figure skating or mental arithmetic. However if >what you want is the ability to solve problems, then what you give is a lot of >problems, which, by definition, are not amenable to solution by routine >methods. Sigh, this is just what I'm working against in response to this article... What's the use of the drills if they don't understand what's happening BEHIND the process?!!! Becky P.S. Sorry again if this irritated someone, it wasn't meant to be a personal flame... -- -- "I ALways push the doors marked pull!"- (I don't know who said that.) -- Becky McEllistrem (Tadger) -- mtbb34@ms.uky.edu, mtbb34@ukma.bitnet, {rutgers,uunet,cbosgd}!ukma!mtbb34 -- University of Kentucky in Lexington Kentucky, USA
ok@quintus.UUCP (Richard A. O'Keefe) (02/22/88)
In article <8421@g.ms.uky.edu>, mtbb34@ms.uky.edu (Becky McEllistrem) writes: > >In article <660@cresswell.quintus.UUCP> ok@quintus.UUCP (Richard A. O'Keefe) > >writes: > >>I think we need three things in elementary arithmetic teaching: > >> principles: "This is WHY the addition algorithm works." > > And do you lecture that why or show them that why? Do you cut a square > to prove two triangles make a square or do you give them some big formula > that they won't remember? If you cut a square in front of the class, it's STILL a lecture. What I had in mind was something like this: a very important thing to get across is that addition is not something we made up, but something we discovered, that whether two bricks plus two bricks gives us four bricks or ten is not something we have the power to change, that whole numbers are defined by counting, that we can give whatever pictures we like to numbers and whatever names we like, but that doesn't change change the reality. Addition is based on counting. I think it's important to get it across that someone who adds 3 to 5 by putting down three fingers and then putting down five and counting the result is going to get the right answer, because that's an instance of what addition MEANS. So now we have a well defined operation, but getting answers is incredibly tedious. Finally getting to the point: you explain WHY the (usual, base-10) addition algorithm works, by showing how it is connected to counting. (What we'd be doing is giving an inductive proof that the diagram encode (NxN)---------> (pairs of base-10 numerals) | add by | usual base-10 | counting | addition algorithm V encode V N ----------> (base-10 numeral) commutes, but we don't have to use the language of category theory to do it!) What this should get across, even if the explanation is a bit muddled, is that all this carrying stuff is not an arbitrary game that somebody made up, but that addition means something however we do it, and that this particular method does that. There are several steps in this which can be done by the pupils. The other mathematical operations can be explained the same way. The really fundamentally important thing in this is really social: it's the lesson that things like addition *can* be explained, and that the teacher is telling you something that makes sense. It doesn't matter much if the pupils don't remember the explanation of why the addition algorithm works: the main thing is that they remember that there IS an explanation and the teacher is willing to show them. In my original message, it was the "New Math" with terms like "commutative" which are hard for a child to *say*, let alone relate to its experience, that I claimed was boring and remote. In my own schooling, which predated the adoption of "New Math" by 1 year in NZ, we started with coloured rods, worked up to play "shops" and recipes, and by the time I left high-school I had a solid grounding in calculus and elementary statistics. Now I have a 15-year-old cousin in Australia who was only taught how to draw a straight-line graph last year, and finds word problems a black mystery. She's not thick, either. Someone who *wants* to play the bagpipes will practice for hours, driving the neighbours mad with endless repetitions of simple changes. Someone who doesn't want to will find even picking the instrument up boring and meaningless. I don't believe that there's any problem with drills as such; the problem is pupils who have never really seen/been shown the *point*.
hes@ecsvax.UUCP (Henry Schaffer) (02/22/88)
and drills? Seriously - does anyone expect education to be completely free of boring times? Drill in arithmetic (to the point that whatever desired performance level is achieved and retained) is going to be boring, at least some of the time. There are valid differences of opinion on what arithmetic capability is appropriate (I, myself, tend to value good calculation/estimation skills.) but people who know better seem to have fallen into the boring == bad trap. *I* think that they really mean that (only boring) == bad but that sure is another story. Also, note that "boring" is not a constant - it may depend on what else is going on outdoors! (The same arithmetic problem or drill sheet may be much more boring on a warm day when one's companions are outside playing, than it is on a rainy day.) (This is not meant to disparage any of the very good previous comments on how to illustrate arithmetic and to motivate achievement in this area - I think that any inquiring teacher will have learned from the discussion.) Now let me make some comments which may be even more controversial. It is good when the students learn the principles behind arithmetic - but I will claim that it is even more important that they learn the arithmetic itself. This distinction is important when one is discussing students in percentiles 60 or so and lower. I think that this is one of the seldom mentioned reasons why the New Math failed. Students who did have the capability to learn arithmetic (well enough to make change, etc.) but who didn't have the capability to do both that *and* learn place notation/base seven/binary arithmetic - were spending all of their time trying to master wierdo bases - and therefore coming out with *nothing* of value. (I certainly believe that students who can master both should master both - but here we are discussing students who do not have the capabilities to master both, and what they should do.) --henry schaffer n c state univ
tlh@cs.purdue.EDU (Thomas L. Hausmann) (02/23/88)
> In my original message, it was the "New Math" with terms like > "commutative" which are hard for a child to *say*, let alone > relate to its experience, that I claimed was boring and remote. For me, that is what made math fun in elementary school -- was more than the usual arithmetic "stuff" -Tom
elg@killer.UUCP (Eric Green) (02/25/88)
in article <4643@ecsvax.UUCP>, hes@ecsvax.UUCP (Henry Schaffer) says: > Seriously - does anyone expect education to be completely free of boring > times? Drill in arithmetic (to the point that whatever desired performance > level is achieved and retained) is going to be boring, at least some of the > time. The question is whether some of the time devoted to rote memorization can be better used for other skills. Certainly there will be times that students feel bored (like, all the time in May :-). boring == bad only when boring == they're doing something that they know well enough that they could be moving on to new (more interesting?) topics. Personally, I have never learned anything by pure rote. There just isn't enough mental "hooks" to retain it for long. I learned multiplication tables by using multiplication in problems. I learned Spanish vocabulary by reading (and working) the exercises until I could translate the sentences "in my head". Reminds me of an article I read in, I think, _Phi Delta Kappan_, entitled "Dewey was right!". That is, we learn most when we are actually doing something. Or, from the Piagetian perspective, when we have a "cognitive framework" in which to place a fact... fancyspeak for "when we know what we're doing, and why we're doing it". > It is good when the students learn the principles behind arithmetic - but > I will claim that it is even more important that they learn the arithmetic > itself. This distinction is important when one is discussing students in > percentiles 60 or so and lower. In my experience with low-achievers, their biggest problem is that they really don't believe they can do it. That is, somewhere in their past, something important passed right over their head, and their teacher didn't notice it. So now, the subject seems mysterious, cryptic, and totally beyond their feeble abilities, because they've tried and tried and they just don't "get it". So they give up. At this point, even filling in the blank spot that gave them the problem won't help, because they truly believe that they are incapable of doing it -- and it's a self-fulfilling prophecy. The only way to break this vicious cycle is to show them that the subject isn't mysterious and cryptic, isn't difficult, and is really very simple... and that even they can do it. For example, giving an intuitive description of the counting process, and showing them how that relates to addition and subtraction, and so forth and so on until they have the self-confidence to sally forth & conquer it. One of the shameful things is that a low achiever usually won't ever get such help. Inevitably, it is the least-knowlegable teachers that get assigned to "low achiever" classes. Teachers who barely know the subject themselves aren't going to be much help in showing students how simple the subject is. > I think that this is one of the seldom > mentioned reasons why the New Math failed. Students who did have the > capability to learn arithmetic (well enough to make change, etc.) but who > didn't have the capability to do both that *and* learn place notation/base > seven/binary arithmetic - were spending all of their time trying to master > wierdo bases - and therefore coming out with *nothing* of value. I was one of the experimental dummies used in the "New Math". I remember 3rd grade, "clock arithmetic". "Duh, what's this?" But Mrs. Pear, the teacher, didn't know, either, alas. It took me 10 years to learn that the denotation of a number and the number itself were two different things, and that was only because I started programming 6502's in assembly language. All I ended up with out of that experience was the impression that math was something really weird and complex. Because the book was written in terms that a math major could understand -- not for a 3rd grader. Or even a 3rd grade teacher. I agree that place notation/modula 7 arithmetic/etc. probably aren't worthwhile topics to expect someone to learn in a vacuum. However, a brief explanation of the principles involved, and an actual APPLICATION (like the 6502 assembly language programming that I did 10 years later), can be quite useful when, later on, the student moves up to more complex subjects (I found my experience with computers to be invaluable when I took discrete math in college... it's easy to understand x V y, when you have experienced OR gates in real hardware). -- Eric Lee Green elg@usl.CSNET Asimov Cocktail,n., A verbal bomb {cbosgd,ihnp4}!killer!elg detonated by the mention of any Snail Mail P.O. Box 92191 subject, resulting in an explosion Lafayette, LA 70509 of at least 5,000 words
tlh@cs.purdue.EDU (Thomas L. Hausmann) (02/25/88)
In article <3482@killer.UUCP>, elg@killer.UUCP (Eric Green) writes: > One of the shameful things is that a low achiever usually won't ever get such > help. Inevitably, it is the least-knowlegable teachers that get assigned to > "low achiever" classes. Teachers who barely know the subject themselves aren't > going to be much help in showing students how simple the subject is. > > > I think that this is one of the seldom > > mentioned reasons why the New Math failed. Students who did have the > > capability to learn arithmetic (well enough to make change, etc.) but who > > didn't have the capability to do both that *and* learn place notation/base > > seven/binary arithmetic - were spending all of their time trying to master > > wierdo bases - and therefore coming out with *nothing* of value. > > I was one of the experimental dummies used in the "New Math". I remember 3rd > grade, "clock arithmetic". "Duh, what's this?" But Mrs. Pear, the teacher, > didn't know, either, alas. It took me 10 years to learn that the denotation of > a number and the number itself were two different things, and that was only > because I started programming 6502's in assembly language. What I found dissatisfying about "clock arithmetic" is that is was called "clock arithmetic". Why use modulo notation (4 congruent to 1 mod 3) and call it modulo arithmetic. I kept expecting hands to appear and tell me what time it was. As I said earlier and Eric pointed out. It is often a problem of the teacher not being comfortable with the material or they (the teachers) have spent to much time trying to come up with a cute example to keep the kids attention instead of telling the students EXACTLY what is happening. > Eric Lee Green elg@usl.CSNET > {cbosgd,ihnp4}!killer!elg .^.^. Tom Hausmann . O O . tlh@mordred.cs.purdue.edu ( ARPA ) . v . ...!purdue!tlh ( UUCP ) / | | \ ./ \. "Whooo do ya think you're foolin' " ______mm.mm_____ \_/
hes@ecsvax.UUCP (Henry Schaffer) (02/26/88)
In article <3482@killer.UUCP>, elg@killer.UUCP (Eric Green) writes: ... [lots of good discussion omitted - to get to the point I want to discuss] > In my experience with low-achievers, their biggest problem is that they really ^^^^^^^^^^^^^(1) ^^^^^^^^^^^^^^^^^^^^^(2) > don't believe they can do it. That is, somewhere in their past, something > important passed right over their head, and their teacher didn't notice it. So > now, the subject seems mysterious, cryptic, and totally beyond their feeble > abilities, because they've tried and tried and they just don't "get it". So > they give up. At this point, even filling in the blank spot that gave them the > problem won't help, because they truly believe that they are incapable of > doing it -- and it's a self-fulfilling prophecy. The only way to break this > vicious cycle is to show them that the subject isn't mysterious and cryptic, > isn't difficult, and is really very simple... and that even they can do it. ^^^^^^^^^^^^^^^^^^^^^(3) > For example, giving an intuitive description of the counting process, and > showing them how that relates to addition and subtraction, and so forth and so > on until they have the self-confidence to sally forth & conquer it. While elg is making very good points, with which I generally agree, there still may be something with which I can't agree. (2) and (3) together imply that difficult subjects aren't really difficult, and that the probem is really in the presentation. Of course this is in the context of (1) and arithmetic - but let me generalize. There seems to be the widespread idea that anybody can learn anything - and if they don't it is because it hasn't been presented properly. While this might be less wrong than the older idea that lack of learning is the fault of the studentI suggest that there are subjects which are difficult and which are generally beyond the grasp of a significant portion of the population. This goes against the egalitarian spirit, or at least seems to, and may even seem to be wrong when observing the overall college educated crowd that many people tend mix with. > ... > Eric Lee Green elg@usl.CSNET Asimov Cocktail,n., A verbal bomb Perhaps the real question in my mind - is how does one distinguish between the two types of situations? (inadequacy of the teacher vs. incapacity of the learner) --henry schaffer n c state univ