>There's also the problem that students have with expanding >(a+b)*(c+d). Many students have memorized the pneumonic "FOIL" for >"First, Outer, Inner, Last", meaning to group things as (a,c), (a,d), >(b,c), (b,d), and then multiply and add.
Again, this is a notation issue, and the need for the mnemonic illustrates how irrational the notation can be. The difference between x = (a+b)*(c+d) and a sequence such as
p = a+b; q = c+d; x = p*q;
is that the second shows the desired sequence of operations, the first doesn't. The problem, as I see it, is that we confuse the semantics of the notation with the semantics of the operations, and teach our students one for the other.
>For these students, this is the kind of "magic" to which I was >referring, because they do not see any reason why one would do this; >it seems arbitrary. They see neither that this is simply three >applications of distribution of multiplication over addition, >(a+b)*(c+d) = a*(c+d)+b*(c+d) = a*c+a*d+b*c+b*d, nor that they are >doing (nearly) the same thing as they did when they were multiplying >multi-digit numbers.
This is because people approach the problem from effect to cause. The issue is this, we want to express the fact that the result we need is obtained by
" add a to b, save it; add c to d; then multiply the result by that value you saved."
I hope you notice that this has nothing to do with the semantics of the operations themselves, it's about sequencing operations, and then, it's a question of precedence: do I add before I multiply, or do I multiply before I add ? I could just as wee have said,
"take a, take the product of b and c, take d, and add them"
which leads you to a+bc+d. The problem is, students confuse the notation and many will believe that when you write (a+b)(c+d) what you're asking for is a+bc+d. The issue here is not the application of the distributive property, the issue here is one of notation.
The point about knowing that (100 + 4)(10+6) can be computed by the application of the distributive property is well downstream from the semantics of multiplication. It has to do with the obvious intuition that (100+4) anythings is the same as 104 anythings; it has also to do with the fact that 104 (apples+oranges) is obviously 104 apples plus 104 oranges. You can call it "the distributive property", but that IMO doesn't help teaching elementary mathematics; intuition here can be used much more effectively.
More, even if you go the distributive property way, it's way easier to teach them how to compute (a+b)(c+d) if you limit a and c to be powers of 10 in the beginning. Because, here again, you are anchoring your teaching on intuition, and most everybody has that basic intuition.
>Memorizing the "FOIL" pneumonic fails to teach these students that the >basic properties have consequences. Obviously this fails to prepare >them for more advanced mathematics, but it also obviously fails to >prepare them for a whole lot of other things.
Memorizing is a need that arises from the teacher failing to anchor the issue on intuition. One can either memorize or rationalize it through using the formalism embedded in the property, yet that doesn't solve the fundamental issue, which is, how to give our students an INTUITION of multiplication. Because, in the end, what we really want is that they can compute without thinking, and inference is a bad substitute for automatism when all we want is to be able to use something en-passant.
>Once again, you are ignoring the two points that much of intuition is >actually learned and that intuition can be wrong.
Unless we're dealing with infinite objects, intuition is seldom wrong. When it is, it'll be a consequence of mismanaged teaching. More, lack of intuition on anything learned - not just math - is a serious handicap in life. Fact is, our inference engines are way too slow for us to rely on them in our daily lives.
>Why not count three, then two more, then one more? Why not two, then >three, then one more? Why not (etc.)?
Indeed, why not - that's where intuition comes. If I have six birds flying in the sky in front of me, it doesn't matter how I count them, as long as I get to six.
>The "intuitive sense" that you use here is an application of what you >have learned: first, that we work from left to right, and second, that >it doesn't matter if we work from right to left, and therefore the >above is well-defined.
It's the other way around, the intuition is upstream from all that. And then, we figure out ways to work that strenghten the intuition and allow us to effectively compute.
>For students who have not seen many expressions of the form a+b+c, the >above two facts are quite likely not at all intuitive (although the >first one might appear more natural, because that is the order in >which we read text).
Again, you throw in notation where it doesn't belong. The concept of "expression" is a notation concept, and notation is, in the end, up for grabs. And the sequence "take a, then add b, then add c", is only intuitive in that order to western people anyway. But I can take something like a+b+c+d and parse it as
x = b+d; y = a+c; r = x+y;
and I'm not doing it either left to right nor right to left. It's back to the old intuition, if I have four fruits and I want to eat them all, it doesn't matter which sequence I do it, the end result will be that there's no fruit left.
>The intuitive concept of adding is *precisely* take two things and >barf out a result (that happens to result from "joining" those two >things).
The intuitive concept of adding is to count then count again. So, if you count twice you add two numbers; if you count n times you add n numbers. The "take two things and barf out a result" is already a formalization of a subset of the intuition.
>We must teach the intuition. Whether we do this by explicit use of >words like "commutative" or by questions like "What do you notice >about 1+2 and 2+1, and about 5+8 and 8+5? Can you extend this to a >general rule of some sort?" is a matter of teaching methodology.
Intuition cannot be taught. It is inborn within ourselves. The fact that 2+1 is the same as 1+2 is so intuitive that the problem here is to take the question seriously: why waste precious time on such obvialities ? What is it that the teacher really wants ? And when you pose such an open question as "what can you observe", you will not get two students talking to you about the same point; it is, as I see it, a gross waste of bandwidth. Nowhere in K12 we are interested in generalizing intuition into figuring out some "general rule", we're interested in developing skills that can shore up future understanding. Those skills must be solidly anchored on intuition, or they're not going to come out the way we need it. For example, I'll probably be way more interested in having the students know that 5+8 is more than 1+2 without them having to compute it, and I'll probably be more interested that they see a+b as adding two numbers - and from there all the intuitive properties of arithmetic can be applied to algebra.
Further, we may not be interested in teaching the methodology, it's possibly only of interest to math majors. Coming back to your (a+b)(c+d) example, the reason why "add a to b, add c to d, add the two results together" isn't the same as a+bc+d is painfully obvious: it ain't so because it don't work, try it if you doubt it. Mathematics isn't a collection of universal truths, mathematics is just a set of models that work ! If it doesn't work, we throw it away. So, why is it that (a+b)(c+d) is the same as ac+bc+ad+bd ? Do it with a couple of examples, replacing a, b, c, and d, with real values, and the intuitive reasons clearly spring, automatically. Specially if we make a and c to be powers of 10, because then we can easily anchor it on the students' innate number sense.
>Nonsense. If we were to write it in prefix or functional notation, >+(/(1,2),/(2,3)), we would still require the notion of a fraction, >which we could represent by, say, [1;2] rather than 1/2 (to avoid >abuse of the "/" notation), and then we would have +([1;2],[2;3]).
That's why I said postfix, it throws away those confusing parentheses. One issue that we computer people bump into, over and again, is that not that many people are good at handling recursion. Both infix and prefix notations demand people to have recursion skills, and that's not a very good thing. The way to represent it should rather be
"divide 1 by 2" "divide 2 by 3" "now add"
and you can invent your own notation for that, it makes little if any difference. This comes back to what I said before, that looking at things with a computer eye sort of changes the way we see the world.
>In any case, if we were to completely change notation (say, to the >above), and teach nothing but "intuition", then the student would >still be faced with the same problem, but perhaps of a different form; >the student might be able to correctly compute +([1;2],[2;3]), but >would struggle with /(+(1,2),+(2,3)), because their "intuition" is >telling them that "things work nicely" (where "nicely" means "the way >that things I already understand work").
The issue is not whether we should change notation. The issue is whether it's a good idea to spend so much energy on notation. What precisely are we gaining by teaching our students to compute (a+b)(c+d), as opposed to telling them to compute p=a+b;q=c+d;r=pq ?
>The problem with (1/2)+(2/3) [you'll note that I explicitly included >parentheses, so that there would be NO DOUBT about order of >operations] is not the order of operations rules, it's about an >understanding of fractions. At some point, you will need to know how >to compute (1/2)+(2/3), or +([1;2],[2;3]), or however you choose to >represent it.
The problem, again, is that (1/2)+(2/3) is not an obvious notation, and requires one level of recursion, with the consequent intellectual effort of saving one intermediate result. To clearly see this, reach for your calculator and compute it: you punch 1, punch /, punch 2, you get 0.5. Now what ? You have to erase that result from your display in order to punch 2, punch / again, punch 3, to get 0.6666. Now you have to punch +, then you must go fetch that 0.5 from your memory, punch it in, to get the result. Or you can use memory store and memory retrieve, but the issue is the same: THE SEQUENCE OF OPERATIONS WE DO TO COMPUTE THE RESULT IS NOT THE SEQUENCE OF THE OPERATIONS AS EXPRESSED BY PARSING THE EXPRESSION LEFT TO RIGHT, and THAT is what confuses most students, because it's anti-intuitive.
>Indeed, the problem is not with order of operations; the incorrect >intuition is that there is a property (perhaps axiomatic) of division >and addition, this property being that (a/b)+(c/d)=(a+c)/(b+d). >Whatever notation you use to write it, this is the false property that >is believed. If the problem were with order of operations, then the >incorrect intuition would be something like (a/b)+(c/d)=a/(b+c)/d.
The problem is with what the notation expresses as far as a sequence of operations goes. To see that, remove the parenthesis: you get a/b+c/d. If Addition has higher precedence than multiplication - that is, if it must be computed before - the expression is the same as a/(b+c)/d. If division has higher precedence, the result is (a/b)+(c/d). The problem is, STUDENTS NOT FAMILIAR WITH MATHEMATICAL NOTATION PARSE THE EXPRESSION LEFT TO RIGHT, and that causes them to get it wrong. The problem here is not the understanding of what the operations' semantics is, but what the notation is supposed to convey.
And then we have the issue of associativity: is this operation left or right associative ? For example, 5-3-1, is the result 1 or 3 ? If we consider subtraction left associative, we get the traditional result, which is 1. If however we treat subtraction as right associative, the result is 5-(3-1), which is 3. Here, again, the issue is notation and not arithmetical concept. The same thing happens with division, what's the meaning of the expression 8/4/2 ? These things are notational issues, not arithmetic issues.
>(The right hand side of the above equation, by the way, illustrates >the problem with an intuitive expectation of associativity: a/(b/c) is >not the same as (a/b)/c).
This is not about an intuitive expectation of associativity, it's about what the notation is meant to convey. Which division you do first ?
>It is rather difficult to even do arithmetic without some notation. >Indeed, if you say "count to 3, then count 2 more", you are using the >entire "notation" of the English language, in addition to the >numerical notations "2" and "3".
Maybe - but language is taught well before arithmetic.
>The only way to do this sort of thing without some notation is to >revert to finger-counting, and the whole point of arithmetic is to >have something more useful/powerful than that.
But that more powerful machinery can leverage on finger counting.
>Again, the whole point is to get beyond finger-counting; in this case, >this means to perform addition in a more efficient manner than >counting.
The efficient way to perform addition is to use a computer. But does that help learning math ? No. One could easily make the point that now that we have computers we don't need to bother with arithmetic. Why is it that reality tells us otherwise ?
>Unfortunately, you can't pick your fingers up and rearrange them like >you can with manipulatives such as the Cuisenaire rods. The ability >to do this allows you to literally see that addition is commutative >and associative. Moreover, it is extremely difficult to use your >fingers to visualize multiplication by anything larger than 2 (unless >you are doing a group project), which you can do with (enough) >Cuisenaire rods.
But you don't need to rearrange anything when you use fingers. And again, who cares to "see" that addition is commutative, in fact, who cares for that concept to be thrown in until wel after ?
>Most interaction with computers (other than using them as word >processors) involves visualization and representation or modelling of >analog things. We evolved in an analog world, and almost certainly >our most fundamental intuitions are analog.
But the intuition we need is digital. Computers are digital, so is everything that happens inside them. Our intuitions are digital, because they can't go beyond a certain level of precision. The understanding that, essentially, 3.141592 is the same as 3141592, is one that's hard to come by, yet it's a needed cornerstone of applied number sense: numbers, like musical notes, repeat in a cyclical pattern.