> Kevin Foltinek <firstname.lastname@example.org> said: > > >One of the classic incorrect intuitions deals with a finite sequence > >of coin flips. > > But probability and intuition part ways very early. Probability is not > about intuition; probability is a measure of things past.
Probability is not a measure of things past, it is a measure of things future (or not currently known).
In any case, the reason I mentioned this was because you claimed that "Unless we're dealing with infinite objects, intuition is seldom wrong". If you now claim that "probability and intuition part ways very early", then either intuition is often wrong, or we need mathematics, rather than intuition, to understand probabilistic situations (including every time we don't have perfect information).
> Furthermore, the likelihood that HHHHTTTT and HTTHTHHT occur with the > same frequency can only be really verified with a very high number of > tosses, that is, the measure isn't really exact until we tend to > infinity.
That's the nice thing about probability - it lets us talk about the probability that things will occur, without having to perform very large numbers of experiments. In this particular case, we don't even need an experiment, we just need a convincing argument that subsequent tosses of even a biased coin are independent.
> >If you want to add something, whether "intuitively" or formally, you > >need some things to add. > > How about fingers ? They're always with us.
This is not relevant to the context (which you conveniently deleted) in which I made this statement, that adding is a function of two (or more) things to be added.
> Our innate intuitions are not nearly as negotiable as literature > suggests
Have you read the literature?
> - if the literature was right, the talent factor would not exist.
There are plenty of explanations of "the talent factor" other than born intuition. For example, it could be related to an early ability to learn the (relevant) intuition (or even to early environmental conditions that create this learning ability).
> >Developing pattern recognition skills, engaging in discussion about > >evidence and conclusions, and the side effect of developing social > >interaction skills (such as listening to others and thinking about > >what they say), are not wasted bandwidth. > > Pattern matching cannot occur without a wealth of patterns being > available on instant recall. Discussions about evidence may be good in > law school,
Once again you deleted the context - the "evidence" and "conclusions" are things like "hey, 2+5=7 and 3+4=7 and 4+3=7 and 2+2+3=7... maybe there's something going on here". Your definition of "evidence" must be almost as narrow as your definition of "mathematics".
> but in learning mathematics, specially of the applied > sort, what's important is the ability to use it,
One cannot use a subject that one does not know.
> >Without the skill of abstracting into a general rule, you are not > >building understanding, you are building regurgitation. Figuring out > >general rules should be something done quite often in K-12 (with > >appropriate guidance, of course, so that the correct rules are > >discovered). > > Whoa, abstraction is not about rules. Abstraction is about dropping > out unnecessary factors from our model.
You cannot drop things from a model until you have a model, and since a model is (by definition) not the same thing as the reality, a model is a way of relating something that is understood to something else. The way of relating, and the understanding itself, can be and often are rules.
> understanding without the ability to use is by and large irrelevant to > most of us who do applied and practical stuff,
The ability to use without understanding what you are using is worse, because you cannot know if you are using properly.
> >Mathematics is the study of the (logical) consequences of systems of > >rules (axioms). > > Systems of rules are nothing but models.
Systems of rules can be anything you want them to be. They don't have to model anything.
> And if you take the tack that > mathematics is merely the study of such models, then I'm going to be > forced to say that we need to teach our students a whole lot more than > just mathematics as you have defined it.
I don't disagree that we need to teach students things other than mathematics, but I think we should be teaching them *some* mathematics (which often we are not). I have already stated my reasons.
> Because we need way more than just study the consequences of systems > of axioms. We need to develop an associated set of computational and > algorithmic skills,
If you look closely at your computations and algorithms you will see that the rules or axioms are used extensively. The algorithms are consequences of the axioms.
> and we also need to develop in our students the > capability of generating THEIR OWN models
So why did you object to the "can you find a general rule" question about discovering commutativity-type properties?
> - in case you miss it, it's called "computer programming" by some of > us.
Most call it "modelling" and recognize that it is quite distinct from "computer programming".
> A fraction is just a shorthand notation for a division not yet made: a > promise of a future division. The sequence 1 2 /, when operated upon, > either gives you 0.5 or 0, depending on whether we're dividing in the > realm of reals or in the realm of integers.
The sequence 1 2 / is not defined in the integers. Your computer implementation of ((int)1)/((int)2) is not an implementation of the integers.
The fraction 1/2 is something called a "rational number" - I know you've heard of them because you have mentioned them. The rational numbers form a field; algebraists call it the fraction field formed from the integers. (Fraction fields can be formed from many other objects.) The fraction 1/2 is not a promise of a future division; it is an object all on its own.
The rationals are a subfield of the reals.
> Here, again, I see you > confusing concept with notation - fractions are notational > conveniences, nothing more.
You would do well to educate yourself. I'd suggest learning some algebra.
> People in computer science have an > expression for this kind of thing: "syntatic sugar". Tastes sweet, but > it's unnecessary at best, and it may be hamful to your health !
Probably the standard example of "syntactic sugar" is a[i] used in place of *(a+i). The analog of this is the different notations that we have been discussing: 2/3, 2 3 /, [2;3], etc. The fundamental object, analogous to *(a+i), is [(2,3)], the equivalence class of the pair (2,3).
This equivalence class is far from syntactic sugar; on the contrary, the notion of a fraction is analogous to a class in object-oriented programming.
> >You still don't get it - "divide 1 by 2" requires the notion of a > >fraction; the result is this object that I have called [1;2]. The > >problem that people have is in adding the two objects [1;2] and [2;3]. > > No, Kevin, "divide 1 by 2" does not require the notion of a fraction. > It may require the notion of a real number,
Very few people understand what a real number actually is, and it is extremely difficult if not impossible to talk about real numbers without first talking about rationals (fractions).
> and that only if we're > dividing in the set of reals. I could easily state that 1 divided by 2 > is zero, if I'm operating on the set of integers.
I could just as easily state that the sky is green and the sun rises in the west.
> >Very many more people have problems with (1/2)+(2/3) than with > >(6/2)+(8/4) (rewritten with whatever notation you want). > > Again, that's a problem of notation being taken for concept. It should > be painfully obvious that (1/2)+(2/3), in the realm of reals, is the > same as 0.5 + 0.6, or 0.50 + 0.66, or 0.500...+0.666..., to whatever > many digits of precision you may want
There is no excuse for this level of ignorance. Equality is not the same as approximation.
(1/2)+(2/3) in the realm of reals, is exactly the same(*) as (1/2)+(2/3) in the realm of the rationals, neither of which is the same as 0.5+0.6. You can approximate it in both the reals and the rationals by 0.5+0.6, or 0.50+0.66, or 0.50+0.67, or (etc.). (*) By "exactly the same", I mean that I am identifying the usual subset of the reals with the rationals.
> AND THAT BECAUSE OUR NOTATION RULES SAY, DIVIDE FIRST, ADD LATER.
No, the parentheses make it clear which to do first, you don't have to use this rule. In any case, the only thing the notation does is tell you that we need to add two fractions.
> When we tackle the other > expression, it's intuitively simpler because both 6/2 and 8/4 reduce > to integers, hence, it boils down to 3+2.
It's not just intuitively simpler, it *is* simpler, because 6/2 and 8/4 are both defined in the integers.
> But the fact still is, if we > emphasize concept and not notation, they will learn that fractions are > merely a notational convenience, that 1/2 and 2/3 stand for rational > numbers,
See, you know that rational numbers exist, but for some reason you don't recognize that a fraction is a rational number.
The notation does not matter - the concept of fractions or rationals is what matters.
> >The former requires adding the fractions [1;2] and [2;3], while the > >latter is just 3+2. > > The former requires divide first, hence, add 0.5 to 0.666.
When you divide first, you get the rational numbers [(1,2)] and [(2,3)] (usually denoted 1/2 and 2/3, respectively).
By the way - what does the notation "0.5" mean?
> The issue > of adding 1/2 + 2/3 and getting 7/6 is one of manipulating notation to > get more notation: a side trip away from arithmetic concept. And here, > again, 1/2+2/3 ain't the same as 3/5 BECAUSE IT DOESN'T WORK - Just do > the operations with rationals and that's easily seen.
You seem to be completely oblivious to what I have been saying for some time now - "do the operations with rationals" is where the students have problems.
> Kevin, a fraction is a mere notation convenience, there's no concept > involved.
A fraction (with integer numerator and denominator) is a rational number. A rational number is a fraction of integers. It's a very big concept. The set of rational numbers *is* the fraction field over the integers.
> The real number obtaining by doing that division is the concept.
Nowhere do we need the notion of a real number to compute (1/2)+(2/3). We do need the concept of fraction or rational number.
> A fraction is merely the deferment of a division.
Fraction = Rational.
> The addition of fraction is not defined by [a;b]+[c;d]=[ad+bc;bd],
On the contrary, that is *exactly* how it is defined.
> that's a notation consequence of the fact that [a;b] is a rational number, so > is [c;d],
And we add rational numbers how?
> and we can INFER by arithmetic common sense that if we add > those two rational numbers that are the result of doing the two > divisions a/b and c/d, we get the same value as we would get by > dividing ad+bc by bd.
Again, how do we add the result of doing those two divisions, and how do we "do" those two divisions? (Make sure you've told me what "0.5" means.)
> But notice that if we compute a/b+c/d directly > we need three operations,
Directly presumably meaning by "doing those divisions"?
> >That is the only difference. By definition (of the notation), > >(a+b)(c+d) means to compute a+b, compute c+d, and then multiply them. > >Whether you remember what a+b was, or you write it down, or you put it > >in register "p", makes no difference. > > You have to put it somewhere. If you don't put it in an intermediate > variable, you have to mentally push it into your stack, or jot it down > in a piece of paper. The use of the "register", as you put it, merely > brings out to light what you and I are really doing when we compute > the expression.
Often one writes it down "in place" - that's what's involved in "showing your work". For example, (2+3)*(4+5) = (5)*(9) = 45 . The (5) and the (9) are "in place".
The use of "registers" or pieces of paper or whatever is a complication.
The same applies to postfix stack operations: 2 3 + 4 5 + * = (2 3 +) (4 5 +) * = (5) (9) * = 45 (I introduced the parentheses only for easier direct comparison with the infix notation).
> But while a simple case like (a/b)*(c/d) is easy to > handle, other expressions may not be nearly that simple. Hence I > strongly believe that splitting long expressions into steps is a good > way to go.
Sure, and often we do that. Big ugly expression; isolate parts of it, say "let A=(...) and B=..., then expression=A*B" and proceed to calculate A and B.
> For example, it's one thing to teach them the quadratic formula, even > if we back it up with proof - and proof here is achieved by algebraic > manipulation whose meaning will not sink in the minds of most > students. It's a totally separate thing to tell them to do it in > steps: > 1. compute d = b^2-4*a*c > 2. if d is negative, there's no real solution > 3. if d is zero, the one solution is s = -b/(2*a) > 4. if d is positive, > 4a. compute r = sqrt(d) > 4b. one solution is (r-b)/(2*a) > 4c. the other is (-r-b)/(2*a) > Now, first, the steps are clearly delineated, and they're easy to > memorize
I would strongly disagree that these steps are easier to memorize than the traditional formula.
You'll also run into problems when you start talking about complex numbers - your algorithm rolls over and dies at step 2, whereas the traditional formula is (with a only small amount of interpretation about what "sqrt" means) perfectly applicable.
> and even easier to throw into a computer program. The issue > of the sign of b^2-4*a*c is clearly handled, and the reasons for one > or two equations are obvious and they're part of the mainstream of the > process.
The *reasons* for the one or two equations are not at all obvious. They're just sitting there in the mysterious algorithm (which is just as mysterious as the traditional formula, until you've seen a derivation of it).
> But the other way, which is, derive the formula then analyze it for > factfinding, I find it confusing and hard to teach.
Perhaps because of a lack of familiarity with rules and their consequences, or perhaps because the "factfinding" is better done in the derivation itself.
> >Unfortunately 0.6666 (or any finite string of 6's) is not the same as > >2/3. (And if you change base so that you can represent 2/3 as a > >finite string 0.****, I can find a new fraction which cannot be > >finitely represented.) > > Here, again, we part ways with the mathematicians. Whatever object you > represent by 2/3, it doesn't exist in the real world.
Neither does the object I represent by -1.
> What does exist > is 0.6, 0.66, 0.666, 0.6666,
What does "0.6" mean, and why does it exist but not 2/3?
> Moreover, 2/3 is only such a number if we're in > the realm of reals, if we're doing integer division 2/3 is zero, > period.
Again, 2/3 is rational, and division of 2 by 3 in the integers is not defined.
> In applied reality, 2/3 is just a shorthand for a value that > we may not be able to represent unless by approximation.
2/3 is a perfectly acceptable representation.
> So, here the issue of mathematics being a model for number sense > springs out again. The fact that 2/3 can be a shorthand for many > different values isn't easily handled by the mathematical model,
Probably because it's not a fact, it's a lie.
> >There is more to a fraction than an operation. The operation is not > >the problem, nor is the notation; as I said above, the students have > >no trouble recognizing that the notation is representing an object > >called a "fraction", they have trouble understanding and/or applying > >the rules that apply to such objects. I don't think it is > >unreasonable that this problem is partly solved when the students have > >already become familiar with the application of rules to abstracted > >objects with which they are already familiar. > > We gain nothing by treating a fraction as an object.
We gain the rational numbers, from which we can build the real numbers, the complex numbers, the quaternions... Did you know that the quaternions can be used to model rotation in space?
> It adds nothing to the students' concept of number,
Nothing except for a whole realm of abstract stuff beyond finger-counting.
> nor does it add much to their capacity of using the stuff in other > disciplines such as physics
Physics would be nothing without calculus, which (by definition) requires the real numbers - which as I've said, requires the rationals.
> or computer science.
You do know that the IEEE floating point standard uses a subset of the rational numbers extensively, don't you?
> Now, if we teach them that 1/2 + 2/5 is the same as > 0.5 + 0.4, or that 1/2 + 2/3 is the same as 0.5... + 0.6..., that's > useful.
What is 0.4? What is 0.6....?
> [snip] > any musician, will intuitively know how to skip > count in twos, threes, fours, sixes and eights, the more educated will > skip count in fives and sevens too, and they'll intuitively know that > counting in fives is the same thing as alternative counting in > two-then-three or in three-then-two.
But in music, two-then-three and three-then-two are not really the same thing (5/4 time may have emphasis on the 1 and 3 or the 1 and 4).
> [snip] > >Everyone who wants rigorous analysis instead of fuzzy intuition cares. > > In other words: math majors.
Lawyers, accountants, political and economic analysts, bankers, engineers, managers, all directly use rigorous analysis, and I'm sure there are many more.
> This has been my point all around. I'm > not saying the stuff is irrelevant, I'm saying however that there's > other things that have higher priority in the teaching of non-math > majors.
One of my points is that mathematics is far broader than you think it is, and often people do mathematics without realizing that they are. Another of my points is that learning this mathematics helps in areas other than mathematics. I am not suggesting that all other subject be abandoned; I am suggesting that math class should be about broader mathematics, not about number crunching.
> > > But the intuition we need is digital. > > > >Talk to somebody who does visualization work on the computer, and see > >how false this is. > > I happen to do a fair amount of it, and yes, it's digital.
This explains why so much of visualization involves plotting, the use of colours and shapes, animation, and various other approximations of analog things.
> Again, in > the end, there's no essential difference between 3.14159 and 314159, > it's all a question of scale.
If there's no essential difference between those two, then there is no essential difference between 1 and 2, between 10 and 20; really there are only three things, negative, zero, and positive.
> > > Computers are digital, so is everything that happens inside them. > > > >Talk to user interface specialists and see the difficulties this > >raises. > > And let me put it this way: math here helps zilch.
Arithmetic probably doesn't help, but more general mathematics might (probabilistic models, for example). In any case, the point was to dispute your claim that our intuition is digital.
> >Lack of precision does not imply digital. > > Lack of precision implies discrete transitions, therefore digital.
Lack of precision does not imply discrete transitions. Lack of precision might mean that a value is 5.3 plus or minus 0.5; the plus or minus should often be interpreted as standard deviations of a Gaussian density.
> [snip - 3.141592 is essentially the same as 3141592] > If you are as old as I > am, and if you were as I was, grown up with a slide rule, you would > readily figure out what I'm trying to convey. The only thing that > decimal point gives us is a scale factor, which can be, and is, easily > abstracted when we need to.
I guessed that's what you were talking about. I understood the scaling given by the decimal point before I learned to use a slide rule; it's something I learned when I learned about "scientific notation" and computing with it by hand. Of course, one need merely change base to see how stupid it is to say that 3.14 is essentially the same as 314: 3.14=3.16 would be essentially the same as 316=380 (that's why I said above that all positive numbers would be essentially the same).
This decimal point business is extremely dependent on the base we use, and has nothing to do with either the numbers themselves or any reality we might be modelling with them.
> >If you mean that they're the same modulo multiplication by a power of > >10, the only thing this is a cornerstone of is details of some > >computational algorithms. > > Which is by far what most of us non-math majors need. And then, > 3.14159 kilograms is the same as 3141.59 grams, so, the difference > between 3.14159 and 3141.59 is one of which unit we're using.
But when we write 3.14159 we're not using units. When we write 3.14159 kg, we are using units.
> I'm sorry, Kevin, I'm an applied guy. One kilogram is 1000 grams. One > meter is 100 centimeters.
But one kilogram is not 1000 kilograms.
One cup is eight ounces. So 1 and 8 are essentially the same. As are 1 and 440 (furlongs to cubits) and 1 and 2/3 (feet to cubits). (Hey, look, 2/3 does exist in the real world!)