> Kevin Foltinek <firstname.lastname@example.org> said: > >Associativity (and commutativity and distributivity) are needed and > >used much sooner than senior high. For example, distributivity is > >used extensively in the "long multiplication" algorithm. > > You don't need to bother with the property, which is merely a > formalization of the intuition. At this level, the intuition of > addition, through counting, is enough. Long division is a shorthand > for repeated subtraction, which, again, is intuitive.
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.
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.
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.
> The associativity you need to worry about operates at an intuitive > level, no need to bother about formalizing it.
Once again, you are ignoring the two points that much of intuition is actually learned and that intuition can be wrong.
> >There's a little trap there: "1+2+3" does not even make sense until > >you know that addition is associative. > > In intuitive terms, 1+2+3 makes all the sense in the world: count one, > than count two more, than count three more.
Why not count three, then two more, then one more? Why not two, then three, then one more? Why not (etc.)?
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.
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).
> The fact that later > on mathematics models addition as a function that takes two arguments > and barfs out a result is a consequence of the mathematical MODEL of > addition, not of the intuitive concept of counting or adding.
The intuitive concept of adding is *precisely* take two things and barf out a result (that happens to result from "joining" those two things).
> So, I come back to my point: before the restricted model is taught, we > must sediment 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.
> >The trouble with intuition is that it easily can be, and often is, > >wrong. Indeed, it is intuitive to many students that (1/2) + (2/3) = > >(1+2)/(2+3). > > It is not intuitive, and it's a consequence of the perversity of the > archaic notation we use to represent arithmetic.
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]).
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 point is, we use a CONVENTION that division has higher precedence > than addition.
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.
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 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).
> The student who understands the way the operations themselves work > will have little problem fathoming the ins and outs of notation. The > issue here isn't the understanding of how the operations work, but the > understanding of what the notation is trying to say. Two different > issues !
Yes, two different issues; no, the issue is not notation, as I said above.
> Arithmetic is intuitive. Arithmetic notation is contrived. There are > two problems here, one is to understand arithmetic, the other is to > grasp our notation of arithmetic. We mustn't confuse the two issues !
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".
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.
> I still remember my little first and second grade colleagues > counting in their fingers to figure out the result of an addition, > because that was intuitive enough - and my teachers would bother > them nonstop to get them to stop using their fingers instead of > leveraging on that intuitive impulse.
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.
> It is obvious, to me at least, that digital feel is a lot more exact > than analog feel. So, length is not a good thing to use, unit is. > Don't glue anything, just count - and here, cubes or rods are as good > as one's own fingers. The difference is, our fingers are at the tips > of our hands all the time.
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.
> And if nothing else, in this age of computers, digital intuition is a > whole lot more useful than its analog counterpart.
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.