>One of the classic incorrect intuitions deals with a finite sequence >of coin flips. Most people feel that the sequence "HHHHTTTT" is less >likely to occur than the sequence "HTTHTHHT" (Kahneman and Tversky, >1972) and there is nothing infinite about this. Most of the time we >need to make decisions based on incomplete information, and where >possible the best way to do this is using probabilistic techniques. >Failing that, we rely on intuition - which is demonstrably wrong.
But probability and intuition part ways very early. Probability is not about intuition; probability is a measure of things past. The point is that "likely" to occur is not a sound way to look at it: either the phrase means that the two sequences DID occur with equal frequency, or it is making an ad-hoc prediction that has nothing to do with intuition. Our intuition that something has a higher "chance" of happening in the very near future is not at all addressed by statistics.
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.
>If you want to add something, whether "intuitively" or formally, you >need some things to add.
How about fingers ? They're always with us.
>Lots of literature suggests otherwise. Also, I will remind you of my >previous mention of the difficulties that many people had a few >hundred years ago accepting the number zero, a number which is very >intuitive to most people today. There is almost certainly no >biological basis for this change, so this intuition is not something >with which we are born.
Our innate intuitions are not nearly as negotiable as literature suggests - if the literature was right, the talent factor would not exist. Whether or not there is biological basis for this change is a thing for the future to tell, today we just don't know. What we DO know is that different people have different levels of intuition to different things, and in many cases that intuition shows up very early in the individual's life. Many of us who can do some math learned it jolly early in life.
>The time is not wasted. There are several reasons to teach this fact. >Perhaps the two that are most obvious are to teach kids about the >concept of rules (i.e., axioms, properties, laws of nature, etc.) and >to teach kids about abstraction (and modelling).
There's time for that, later on, after the intuition is well sedimented. Abstraction is taught, in part, by not using props and forcing students to use their minds instead of their senses. In early stages, OBSERVATION is the key, not rules: let's look at it, let's do it, let's live it, let's let it permeate into our intellects, and see where it leads us. The converse, which is, to establish a track inside our minds and force everything from the outside into that track, can be pretty limiting.
>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, but in learning mathematics, specially of the applied sort, what's important is the ability to use it, and individual learning is fundamentally important. There is a level of interiorizing that is well superior to just being able to discuss, reason and analyze, a level where we rather FEEL the issue and its environments - and that feel can strongly condition us out of lines of inference that are unproductive. This feel is develop out of grooming one's intuition with a thorough knowledge of the intellectual landscape, and that knowledge is achieved not by inference, but by absorption and experience.
>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. Furthermore, understanding is only the first and rather small step that starts a whole chain of events that lead to mastery and automatization. Furthermore, understanding without the ability to use is by and large irrelevant to most of us who do applied and practical stuff, so, it is of paramount importance that the understanding gets carried out well beyond armchair contemplation and into what you call "regurgitation" : the ability to recall things automatically and without the need to engage our thinker at the inferential level.
Figuring rules is irrelevant at elementary levels, and rules come out as consequences of things more fundamental: rules are models, but here we're not just worrying about the model, we're worrying more about the stuff we're modeling. Here we have yet another difference between the mathematician and the rest of us: to you guys those definitions and rules just are, to us, they're just models of something that none of your definitions and rules can really trap. Even in this superficial conversation I already detected one or two cases where your definition of something, or your rule for something, falls well short of what our observation of the real world needs to cover, and that's why there's so many fields in real life that aren't that amenable to the use of mathematics.
In the case of operations, the only rule is counting. Addition is counting, multiplication is repeated addition and hence counting, subtraction is adding the complement, division is repeated subtraction: there is only ONE arithmetic operation, the others reduce to it. If you doubt it, try implementing a long division on a Turing Machine !
>Yet this (or similar) errors are made very often.
And that is so because many of us teachers fail to make things obvious: again, no rule and no rationalization or inference replaces experimentation, just DO IT, the result comes out wrong, hence, it doesn't work. If it doesn't work, drop it, and let's go find something that does.
But to get to say it doesn't work, we have to have a standard to compare against. That standard is intuition and counting.
>Mathematics is the study of the (logical) consequences of systems of >rules (axioms).
Systems of rules are nothing but models. 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.
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, and we also need to develop in our students the capability of generating THEIR OWN models - in case you miss it, it's called "computer programming" by some of us. So, I may not care to know that a+b*c+d means "multiply b by c then add a then add d", I may want to build a model where b*c may mean "find the first occurrence of string c in string b+c and drop the rest", understanding that b+c means append string b to string c, with the proviso that, yes, + has higher precedence than *.
Furthermore, even if we are within elementary arithmetic, being able to tell how much a+bc+d happens to be may be very important - calculators and computers notwithstanding. If I'm teaching physics, for example, the last thing I want is students with zero number sense who have to abstract things into mathematical rules everytime they face a computation, and who don't know the difference between a 10 degree and a 190 degree angle on the fly.
>I explicitly said that we still require the notion of a fraction. >Let's do it in postfix: > 1 2 / 2 3 / + > = [1;2] [2;3] + . > >The "+" operator operates on the two objects [1;2] and [2;3].
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. Here, again, I see you confusing concept with notation - fractions are notational conveniences, nothing more. 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 !
>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, 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.
>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 - AND THAT BECAUSE OUR NOTATION RULES SAY, DIVIDE FIRST, ADD LATER. 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. 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, and that we divide before we add because the notation tells us to.
>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. 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.
>Exactly, because the problem is with the abstract notion of a >fraction, and the rules that *define* them. Though it is very >debatable whether we want to teach young children the following level >of abstraction, a fraction is an equivalence class of a pair of >integers, the second of which is required to be non-zero, and the >addition of fractions is defined by [a;b]+[c;d]=[ad+bc;bd]. No amount >of notation will change this definition or the troubles that students >have with it.
Kevin, a fraction is a mere notation convenience, there's no concept involved. The real number obtaining by doing that division is the concept. A fraction is merely the deferment of a division. The addition of fraction is not defined by [a;b]+[c;d]=[ad+bc;bd], that's a notation consequence of the fact that [a;b] is a rational number, so is [c;d], 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. But notice that if we compute a/b+c/d directly we need three operations, while if we use your rule we need five operations; where's the economy ? Why bother ?
>What we are gaining is three more letters that students won't forever >remember as "registers" (just as x is the independent variable, and >(a,b,c) are the variables in the quadratic equation).
What you call "variables" are merely symbols that stand for values. The fact is that the computation of a/b+c/d requires that the individual or the machine doing the computation is capable of storing at least one intermediate value plus the current partial result of the computation - and there's nothing in nature that will alleviate this issue. The problem here is, do we keep on trusting our kids' memories, or do we give them some real way of handling it ? The difference between saying "a b / c d / +" and saying "a b / @p c d / @q p q +" is that the storing of intermediate values is made clear. I see little point in combining multiple computation steps into one simple expression, again, why bother ? It makes things harder to understand and easier for errors to creep up.
>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. 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.
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 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.
But the other way, which is, derive the formula then analyze it for factfinding, I find it confusing and hard to teach.
>As I said, the problem is with an understanding of fractions. That >you discuss reaching for a calculator to divide 1 by 2 suggests that >you don't understand fractions any better than the students. Witness:
The problem is with understanding the notation, because fractions are just notation. A fraction is nothing but the promise of a division, a notational convenience.
>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. What does exist is 0.6, 0.66, 0.666, 0.6666, and so on, depending on to what precision we're computing it. 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. In applied reality, 2/3 is just a shorthand for a value that we may not be able to represent unless by approximation. So, we use our generalization engine to pretend that something like 0.666... does actually exist, with the implicit assumption that, given that there ain't no such thing as infinity in real life, we'll have to stop writing those sixes at some point in time, and that point will depend on our precision. So, to complete, 1/2 ain't really 0.5: it can be 0.5, but it can also be 0.50, 0.500, 0.5000, 0.50000, again, as many zeros as we need to fill up our precision. Because there's a difference in real life between 0.5 and 0.50: when I say 0.5, what I'm really saying is that what I have is 0.5x, and I don't know how much x is; but when I say 0.50, I'm saying that what I have is 0.50x, where I do not know how much x is, but I definitely know that the digit that follows 5 is 0.
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, which demands the creation of a hypothetical rational number with an infinite number of digits. That's ok, mathematically, but not ok in real life !
>Once again, the thing that confuses students is the rule for adding >fractions. The fact that most students see "1/2" and shudder while >uttering the word "fraction" shows that they are very comfortable with >the (1/2) and the (2/3) as being the objects that they have to add.
Again, the issue is that fractions are taught as real life objects, when they're merely notational convenience. The real result of 1/2 + 2/3 is obtained by dividing 1 by 2, jotting down the intermediate value, dividing 2 by 3, and adding that jotted down intermediate value in to get the result. THAT is arithmetic - the obtaining of 7/6 by applying the formula is a consequence, and true enough, you can model fractions and division by defining fractions as you did, but that's a departure from what I'd call real world number sense: it's nice to know, but relatively lower priority.
>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. It adds nothing to the students' concept of number, nor does it add much to their capacity of using the stuff in other disciplines such as physics or computer science. 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.
>That's my point - relying on the intuitive expectation of >associativity quite possibly leads the student to expect subtraction >or division to be associative. (After all, everything else behaved in >this nice intuitive way, nobody made a big deal of it, it's nothing >special, it's quite ordinary.)
Again you confuse concept with notation. The intuitive expectation of associativity can be boggled down by an aggressively unnatural notation. The operations DO behave in intuitive ways, the notation doesn't.
>The leverage is partly through motivation (this thing we do with our >fingers, it's called "counting", etc.) and partly through early >learning of results (memorization of addition and multiplication >tables, though as I've said, it's quite difficult to multiply, for >example, 7 and 9 using your fingers). Beyond that, the use of finger >counting is a hindrance to the leverage of the machinery.
Multiplying is the same as skip counting. To multiply seven by nine, I count skip-seven nine times:
all while counting nine times with my fingers. Is it necessary to memorize the sequence ? It sure helps. But then, we can do that with rhythm, or music: 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.
>One reason is because non-computer arithmetic skill is often useful: >sometimes a computer or calculator is not immediately at hand; >sometimes a quick mental check gives a (rather weak) verification that >the computer was used correctly and gave the correct answer (more >precisely, it provides evidence that the answer was not obviously >wrong and that no huge errors in data entry were made).
Ah, but this is why we need to teach them intuitively, so that they can use their native baggage to do this kind of thing. I couldn't agree with you more that every student needs to have a fair amount of number sense, and that includes being able to compute things on the fly.
>Another reason is because the teaching of arithmetic teaches us many >other things: the ability to abstract or model things, the ability and >discipline to follow standard procedures, the understanding that these >procedures have consequences (including the consequence that there can >be more than one way to arrive at the correct answer) that can have a >very rich structure.
Now, here I'm not too sure I agree. Abstraction can be learned in many other ways, and the reason we learn arithmetic is not to learn how to abstract, but to learn what to do after we abstract. Following procedures is sort of irrelevant: it should be about how we can set up our own procedures, and then teach machines how to follow them. And there's no such thing as a "correct" answer in practice, it all depends on your precision.
>Everyone who wants rigorous analysis instead of fuzzy intuition cares.
In other words: math majors. 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.
>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. Again, in the end, there's no essential difference between 3.14159 and 314159, it's all a question of scale.
>Talk to user interface specialists and see the difficulties this >raises.
And let me put it this way: math here helps zilch.
>Lack of precision does not imply digital.
Lack of precision implies discrete transitions, therefore digital. If I have six digits of precision, 3.14159 transitions to 3.14160 - it's the same transition as if I was going from 314159 to 314160.
>That's good, because it's very far from true.
But it isn't. It's a question of scale factors. 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.
>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. Which brings me back to that length of cubes thing you brought in.
>Based on your "same as" stuff above, I think you're talking about >multiplying numbers by 10 and getting the same sequence of digits. >Too bad this has nothing to do with numbers themselves, and everything >to do with their representation.
I'm sorry, Kevin, I'm an applied guy. One kilogram is 1000 grams. One meter is 100 centimeters. Numbers DO repeat in a pattern. Otherwise modulo arithmetic wouldn't work.