The following points are taken in order from Vic's `answers' to my questions (which I have copied for everyone below):
1. You are a hypocrite. With one hand you say that you are not going to infringe on ZimÂs beliefs about something that is verifiably false, yet with the other you try to infringe as much as possible on JackÂs ideas which are based on reason. Its okay for Zim to have opinions but its not okay that Jack has opinions too.
2. I am in no fear of your accusations. I would merely find it very humorous if you had. By the way, mere quantity has absolutely nothing to do with quality. Zim could probably fill pages of unintelligible chatter (Zim: go back to school Â you will thank me later for the suggestion). Zim may have the minor, but I would hire Jack.
3. Let me assure you that those were not professional mathematicians (where they math ed?). A function is a mapping from a set A to a set B with the property that each element in A is mapped onto exactly one element in B. Now to debate whether sets A or B are numerical is simply silly and ignorant. By the very definition A and B can be any two sets. For example, A could be U.S. senators and B could be U.S. states. No numerals there. It is so basic that every high school algebra teacher knows this. Why would `professional mathematiciansÂ debate a definition which is already general enough to handle what they were debating about?
4. Based on (3) the discussion is not possible. Mathematics is a logical system. Once inside the system, everything in mathematics has a well-defined meaning derived from set theory. Therefore, all definitions are written `in stoneÂ inside that system. The reason why we choose to use this system (and not another logical system) is because it is the most useful of all logical systems. It is so useful, for instance, that we are within a 100 years of understanding the basic set of equations (Grand Unified Theory) which describe the universe. The body of mathematics is one of the crowning achievements of mankind and should be passed down through the generations much as you describe.
5. EuclidÂs books `The ElementsÂ were in fact the primary textbooks for elementary students for some 2000 years. They truly are masterpieces. The reason why we stopped using them is not the reason you describe above (MATHEMATICS IS DONE THAT WAY). It was because the great mathematician David Hilbert reorganized the textbooks (around 1899) to bring them up to modern standards. We know more than Euclid did and professor Hilbert used that knowledge to upgrade EuclidÂs text - Professor Hilbert basically wrote the second edition.
6. The problem with constructivism is that it has many definitions (remember me talking about ill-defined concepts?). So if you disagree with my use of the word constructivism, you can always choose one of the other definitions and then proceed to call me ignorant. Let me clarify for you my use of the word `constructivism.Â Obviously I wasnÂt using the definitions such as trivial constructivism, or radical constructivism, or social constructivism (oopsÂ .maybe old Newty isnÂt so ignorant about this stuff). If you want a critique of constructivism at that level, I highly suggest you read the paper by the Nobel piece price winner I mentioned in a previous post. I was referring to `Applied ConstructivismÂ or how constructivism is applied in the classroom (and yes, it is often referred to in the math ed. literature as a way of teaching). This constructivism is what Jack was a victim of and what you tried to distance yourself from by saying his teacher was misguided. Here is the part I always love to rub in constructivismÂs face: Since constructivism (philosophically) says that students must construct their own knowledge it implies immediately that no one can instruct a teacher how to use constructivism in the classroom (if you did, you would be contradicting the theory). They must be allowed to construct their own knowledge just as their students must. For instance, JackÂs teacher constructed his own knowledge of constructivism and therefore should be looked upon as a valid form of applied constructivism (since it was constructed) whether you think it is misguided or not. If this upsets you (and I am sure it does), then you are starting to feel the rage that most mathematicians get when they hear stories of children who are allowed to construct inherently wrong mathematics in classrooms run using constructivism. What is so humorous is that the constructivist philosophers are not even immune to their own theory. But since it is their theory, they shouldnÂt wine when it is applied correctly to produce principles which they detest - unless, of course, the theorist are willing to admit that their theory is wrong.
Now that you know the definition of constructivism I was using it is clear what I meant when I said `one way.Â Applied constructivism is a way of teaching which includes several teaching methods just like Direct Instruction is a way of teaching which includes several teaching methods. You were associating `teaching methodsÂ with `wayÂ to get that there was one teaching method called constructivism. I hope this clears up that misunderstanding.
7. Many cloudy and illogical things can be said in a scant six lines (yours was an example). The amount of lines does not tell the reader if the body of the text is logical or not, much like the volume of mathematics does not imply anything about the quality of the mathematics (see #2 above). To paraphrase a character in a recent movie about conflict in space, "knowing how to write does not imply the writer is intelligent." While brevity is a virtue in writing, sometimes it takes time to fully develop an idea (like #6 above).
8. I will stay anonymous for my own reasons. For all you know I could be a senator, doctor, lawyer, mathematician, teacher, parent, or like Jack, just a concerned citizen. I am not interested in polite private responses and prefer to keep the dialogue as open and logical as possible. If that means attacking the credibility of a philosophy by critically analyzing the writings of its proponents then so be it. But lately this seems to be as easy as shooting fish in a barrel. Therefore I have no more questions for you Vic. Give me some time and in a couple of weeks I will write up some really good questions I have about the NCTM standards and why it is tied to ways of teaching (like constructivism) instead of being tied to the content students should learn.
On 17 Aug 99, Victor Steinbok wrote re. The GRNAD FINALE: A Jack in every stable:
> I guess, there is no time like the present and I must respond to the > latest round of lunatic rants before this forum gets shut down by > divine interference (I wish!). Zim's nonsense notwithstanding, I > believe anyone who got a CS degree in 1979 (math minor or not) got a > better math education than quite a few people with math degrees since > then. I cannot judge the actual quality of Zim's mathematical > knowledge, but for me it suffices that the mere quantity is > considerably greater than Jack's, and Jack should have the curtesy of > butting out from discussions where he does not belong--not as a > teacher, not as a math practicioner of any sort and not as a parent. > If he wants to take issue with the course he apparently detested, this > is not a proper forum for that. > > Zim's "math" is not dangerous. It is not anything I would consider > teachable, nor is it "art" as Zim seems to insist. I am not about to > infringe on his beliefs--he can hold whatever etherial opinions he > wishes. There is little danger they will pursuade any students to drop > whatever they doing and join the Creative Math cult. And as long as > Zims of this Cult do not ask for support money to develop teaching > materials, I am not worried. > > However, the discussion that followed the original post was not any > better. I've known a few Euclids and Newtons, but none of them were > also named Leibniz (some of their pets were, but that's another > story). So I am working on the assumption that Newton Leibniz, or > simply Newty, wants his name hidden for whatever purposes. In this > case either his pseudonim has gone to his head or he expects us all to > be telepathic. However, his fear of me accusing him of little > knowledge were unfounded--my jab indeed was to Jack alone (perhaps Zim > deserves it too, but Jack's ignorance is considerably more obvious). > > On 15 Aug 99, Newton Leibniz wrote re. Huh? Whatchu talking about > Willis?: > > > Vic, > > > > In the past three days I have seen postings which do not make any > > sense: Zim's mathematics and now yours. Please answer these > questions > > as clearly and to the point as possible: > > > > 1. Which person are you referring to that doesn't know any math: Zim > > or Jack? While Zim has an undergraduate minor in mathematics, I > would > > say that Jack has a better understanding of the mathematics he > knows. > > Or are you referring to me? That would be down right hilarious. > > I believe I've answered this question above. > > > 2. What do you mean by "doing what math is all about"? Do you even > > know what you just said? I am really interested in your answer. > > I know exactly what I said, and I did not imply that Zim in any way is > doing math. Just last week several professional mathematicians argued > about whether a "function" can have non-numeric values (all seem to > have accepted the possibility of non-numeric arguments). The > invocation of Bourbaki made some cheer and some shudder. The issue was > settled simply by sayng that the Bourbaki required definition does not > limit the scope of values, but, for practical purposes, most of the > time when we refer to functions, we actually mean numerically-valued > functions. So, for example, an X-->R2 function can always be > interpreted as a pair of X-->R1 functions and satisfy both > definitions. Now, people like Jack and the clique down in San Diego > would have us believe that such a discussion is not POSSIBLE in > mathematics because the Divine Immutable Mathematics has been handed > down to Moses to Plato to Newton and Leibniz (who had a debate that > would rival any current scientific disagreement) to Newton Leibniz, > Wayne Bishop (or what have you) and it is to be transmitted unchanged > to Jack and his ilk. All mathematical definitions are written in > stone. In fact, it was a great loss when we stopped teaching Euclidean > Geometry in schools and simply started calling it Geometry. > > Take a look at the old Cambridge schoolarship examinations, say from > mid-1800. You will find that half the questions would look like "State > Proposition X" or "Prove Theorems V and IX". When I first saw this, I > was puzzled, but it soon dawned on me that the references were to the > Elements. With time, however, these types of questions have > disappeared from the examinations because that is NOT HOW MATHEMATICS > WAS DONE and perceived. > > > 3. People who think math can be taught and learned in only one way > are > > probably dangerous. However, I have never seen Jack post any such > > statement. What I have seen Jack post is many many statements about > > the dangers of teaching mathematics in ONE particular way -- > > constructivism. > > This statement to me is a display of ignorance--constructivism is NOT > a way of teaching mathematics. At best, it is a learning theory or a > cognitive philosophy. Jack has been ranting and raving about his > "constructivist" class for nearly a year now. I can only assume that > both he and his instructor are deeply misguided. There is indeed a > great danger posed by people who profess to criticize something they > know nothing about. The really scary part is that a large number of > teachers interpret "constructivism" to be what people like you tell > them it is, but instead of agreeing with you, they proceed to teach > following exactly the principles you detest. > > It is particularly ignorant to talk of ONE WAY OF TEACHING under > constructivism, since ANY flavour of constructivist philosophy will > tell you that it is not possible. Instead of reading the fluffy > textbooks that are churned out by the likes of Addison Wesley, you > should brush up on the basics and go back to the research sources > rather than to their less-than-stellar textbook incarnations. > > > 4. What do you mean by finding allies with the Kansas state board of > > education? The state board removed the teachings of evolution from > > the curriculum so that teachers could construct creationism if they > so > > chose. I do not think he would find any allies, only enemies since > > Jack is proposing that we teach science that is based on the > > observable physical world. Under that premise, one must conclude > that > > evolution is the best working model we have and any other working > > model is vastly inferior. > > While Jack's self-professed materialism would greatly offend their > sensibilities, Jack and the KBE share an intolerance for things they > do not understand (or, more accurately, refuse to understand). > > > Thank you Vic for providing another prime example of cloudy, > > confusing, non logical writing. Once again I leave you with > questions > > to ponder, but this time they are directed at Vic and I expect > > answers. > > You want cloudy? Gee... My original post took up scant six lines. > Since then Jack has provided voluminous irrelevant drivel in multiple > copies with Newty chiming in his 2c-worth. For what it's worth, if > Newty expects a polite private response to his accusations, he should > provide his electronic coordinates and a real name. > > VS-) > >