Michele's Made-up Math Grammar
Not terribly long ago, someone asked "Why is it that when you
divide by a fraction, you have to turn it upside down and multiply?"
I had a lot of trouble trying to figure out how to explain that and a
good explanation unraveled from the dialog, both on the list and what
she and I e-mailed each other privately. At one point she exclaimed
"So, you mean it really is The Same Thing and not some kind of
'trick'?!" And I said "Yes, it really is the same thing." It never
occurred to me that anyone would view a shortcut as a 'trick'. I
ultimately showed that you CAN directly divide by a fraction, it is
just very convoluted and difficult to keep it all straight in your
head, even for me, and that is why this is something where they teach
you the Rule in place of teaching you the long process. Usually,
schools teach the long process first and only let you use a shorter
version if you have thoroughly mastered the long method first.
One example of my own unique thought process is that when you turn the
fraction upside down and also change the division sign to a
multiplication sign, it is kind of like 'magnetic poles': if you
reverse it twice, it is the same thing. Using the reciprocal of the
number and also using the reciprocal function (that is probably not an
accurate term -- but I mean multiplication and division are 'opposite'
in a similar fashion as reciprocals are) is just like reversing the
poles on two magnets and thereby having them still do the same thing
they were doing -- either attracting or repelling. If you only
reverse one magnet, you change what happens. If you reverse both, you
get the same result even though it all may be 'turned on its head'.
Another way to think of this would be that if you turn yourself around
180 degrees and face in the opposite direction, and then do it a
second time, you will find yourself facing back in the same direction
you started with. Using the opposite of the fraction (the reciprocal)
and the opposite of the function (multiplication instead of division)
gets you turned around twice, so you find yourself still going in the
right direction. That is extremely Obvious to me, which is why I had
trouble even explaining Why this technique Really is The Same Thing as
simply dividing by the original fraction.
I use the concept of 'reciprocals' for a lot of things in math where I
have sort of defined my own 'reciprocals' and the 'rules' for how they
interact. For example, since we have a base 10 number system, if you
think of numbers 1 through 9 as 'reciprocal sets', it makes addition
and subtraction much faster. The 'reciprocal' in this case is the
number you get when you subtract it from 10. Thus, 1 and 9 are
'reciprocal', 2 and 8, 3 and 7, 4 and 6, and 5 and 5. This helps me
enormously because I never really have to add and subtract some
things, I just have to know the 'reciprocal'. I do not have to
subtract 60 from 100, I can instantly 'see' that the 'reciprocal' of
60 is 40. It is obvious and I don't have to do any calculating. I
cannot, at the moment, think of a more complex example but I can tell
you that such ideas make me much faster than average at doing math in
my head and also mean that I seem to have 'memorized' large amounts of
data when I may not really have it memorized, it just takes me a very
short time to 'get the answer' because of some Rule I have made up.
I wind up mentally lumping together various 'patterns' that I 'see'
that we never got taught were related. All of the rules for how
numbers work wind up having some similar rules and some opposite rules
and they appear to me to follow a pattern. When you add two even
numbers, you get an even number. When you add two odd numbers you get
an even number. When you add an even and odd, you get an odd. So, I
then tend to think of negative numbers as 'odd' numbers and positive
numbers as 'even' numbers and then their rules for multiplying follow
the same kind of pattern: 2 negatives gives you a positive product, 2
positives gives you a positive product, and one of each gives you a
negative product. So, if you have 2 of the same, you get either even
or positive and two different, you get 'odd' numbers (and negative
numbers strike me as rather odd creatures indeed, so I feel I have
covered that with the word 'odd' without explicitly saying 'negative
numbers included'). The word 'Odd' often is used to mean 'different'
in our culture, so when two things are different in math, the result
is usually 'odd' and when two things are similar the result is the
'norm' (even or positive -- the 'not odd' things in my mind).
Similarly, I group addition and multiplication together as being the
same thing and following basically the same rules and I group
subtraction and division together and recognize that they largely
follow the same kinds of rules as well. For example, both addition and
multiplication have commutative properties (it does not matter what
order you add or multiply 2 numbers in) and subtraction and division
do not (it matters very much which number you are subtracting or
dividing by). The fact that I make so many correlations and see a
kind of 'grammar' going on here means that I have to learn a lot fewer
rules. I learn one rule, I remember what other kinds of things work
the same way, and then I only really need to be very clear about the
'exceptions' -- like the ways in which subtraction is different from
division or the ways in which negative numbers are not at all
'similar' to odd numbers in how they 'behave' themselves.
Actually, that is a terrible example of an Exception for my way of
thinking. In fact, my Default Setting is to group all things that are
similar. So, I would be inclined to group traits of negative numbers
that are NOT similar to odd numbers with something completely
different rather than continuing to compare them to odd numbers and
have to list differences. Listing the differences would have me
remembering a bunch more Rules, which would defeat the purpose of
thinking about it My Way. So, really, My Way is more like building a
web: I knot together the things that are similar and this one
'thread' will be similar to several others and will get 'tied' to each
of those other things in the places where they are Close. Exceptions
would really be where there is a Gap in my web or maybe a big gnarled
knot that keeps Tripping Me Up and I cannot seem to Get Around it
without remembering the actual Rule that my math teacher taught me.
Since I do lump together addition and multiplication as being the
exact same thing and think of multiplication as 'the shortcut', when I
think that even and odd numbers add in the same 'pattern' that
negative and positive numbers multiply, I don't even clearly
differentiate that these patterns are for different functions.
Multiplying IS a form of adding so I don't really care about the
difference. It is close enough for my logic and for giving me an
overview of how all of it hangs together. When I really need to be
specific, I can dig around in my brain and come up with the exact
rule. The rest of the time, the general principle lets me get by
without having a huge catalog of rules uppermost in my mind at all
times.
I think that was why Algebra came easily to me and Geometry did not:
Short cuts get you in trouble in geometry. They want you to really
remember every little rule and write down every little step and I
cannot shorten it. Shortening it is actually against the rules. The
fact that I mentally make those connections and my mind feels no need
whatsoever to remember all the steps -- certain things are 'obvious'
to me and I can't imagine why on earth I would have to 'prove' it --
the manner in which I tend to truncate mathematical processes makes it
very hard for me to dredge up all the stuff I have 'skipped' and
remember every last step so I can put it into a geometric proof. Why
on earth would I want to do that?
I often got low marks in geometry when I would shorten a proof from 25
steps to 8 or 10 because the other steps seemed extraneous to me. I
guess it would sort of be like reciting the alphabet as "a, b, c, d,
l, m, n, o, p, x, y and z" I often just could not see why my teacher
insisted that all those other things were necessary. I got to the
end, didn't I? And I did it all in the right order, didn't I? And "A
comes at the beginning, Z comes at the end and there is some stuff in
between. So why are you being so hard on me?" (giggle) Or so I felt
at the time.
I also routinely reduce stuff at the beginning rather than the end
when working with fractions. This is not the way it is supposed to be
done but it is actually much more efficient and I make a lot fewer
mistakes, in part because it keeps all the numbers as small as
possible. If I have to multiply 1/7 and 7/8, I take the 7 out of both
numbers and skip to the end: the answer is obviously 1/8. Why on
earth would I want to torture myself by saying it is 7/56 and then
trying to find the simplest form? Multiplying it first and THEN
reducing just unnecessarily complicates the whole thing and has you
doing 'extra' steps because first you multiply two numbers by 7 and
then you divide two numbers by 7. Why bother? Just pull the 7 out of
there at the very start and you are much less likely to make a dumb
mistake. I mean, I am not going to think it is 1/7 if I reduce first,
a mistake I could make if I do it the 'long' way and get confused or
distracted for some reason.
I often make dumb mistakes when I make the problem all convoluted by
doing it the long way they teach in school. So I simplify first, then
calculate. It removes many calculations that are simply 'redundant'
in my mind and thus removes additional chances for making a mistake.
Often, the shortest route is the one by which you are least likely to
get lost, or at least that is how it works for me. I guess it is sort
of like that game where one person whispers something in someone's ear
and they 'pass it down' and by the time it goes through 20 people it
isn't at all what the first person actually said. If I put in all
those 'extra' steps, I find that it is much harder to get to the end
without making a simple mistake in division, adding or subtracting or
some other very basic function.
When doing algebra and above, the vast majority of mistakes are that
type of 'dumb' mistake: where you know any kid in elementary school
could have performed that particular piece of the problem (multiplied
by 7 and then divided by 7, for example) since it is basic arithmetic.
But it has you pulling your hair out, trying to figure out where you
went wrong, only to notice you forgot to carry the negative sign or
you subtracted wrong in the second line of this page-long maze of
numbers. To me, the Real Reason those mistakes happen is because
these algebra problems take forever. Shortening them gives you better
odds of getting through it all with a few hairs still remaining on
your head. (This saves on the cost of wigs.)
Anyway, that is a few examples of how I shorten stuff and find
'grammar' and 'syntax' in the language of math. And that is why I
sometimes have trouble trying to explain all the steps: I didn't use
them to solve the problem. I took a different mental path, and that
mental path would, in and of itself, be very hard for me to articulate
for a math teacher, if only because it isn't any kind of mathematical
rule that my math teacher would admit exists anywhere on planet earth.
He certainly never taught me any such thing and what kind of nonsense
is this girl spouting? It sounds like fairy tales, not math! It is
actually kind of hard for me to really remember the names of the
rules, like commutative property, or the difference between natural
numbers and whole numbers, etc. But I can work the problems and
I actually was an excellent tutor when I was 16 and I have been a
godsend for my oldest son, who just never, ever could wrap his brain
around math the way they taught it in school but could kind of
understand some of mom's explanations.
On his own, he also concluded that reducing fractions at the start is
simply more efficient. And I totally agreed, much to his astonishment
since he was trying to Justify doing it that way and got no flack from
me but had expected to get told he couldn't do that.
If you are interested, the book, "Math Magic" strikes me as the Rules
that some
Math Geek made up for himself and then made a career out of teaching
to others because they just work better for a lot of people than what they got
taught in school.
To Top
|
|
Free Curriculum on the Web
The Math Files
|
