“I
Just Don't Get Algebra”
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The Big
Picture
Many children (and adults) fail to “get
algebra”. It is commonly heard “I did well at arithmetic, but just couldn’t
make sense of algebra”. Yet algebra is nothing more than the distilled essence
of the laws of ordinary arithmetic. The truth is that before failing algebra
they failed to understand arithmetic. Unfortunately in the scramble to meet
curriculum driven requirements for the year teachers are often forced to
substitute rote memory and formulas for understanding. We get “result
arithmetic”, arithmetic which gets results but without the understanding. This
is good enough for bookkeeping (as it should be), but fails to measure up when
it comes to using arithmetic as a handrail for understanding the nature of
reality, how the universe and the mind work. And often it is not even enough to
understand simple algebra!
We’ll demonstrate an example wherein
readers may note their own particular training may well have been deficient. If
you have covered all the points you are one of the lucky few. Clarifying the
issues will definitely involve going slower: the quick formula approach most of
us have been brought up on is definitely faster and more suited to the
curriculum. The fully clarified version while slower will make more sense if
you work with it (don't just read it, do some examples yourself).
The clarification process immediately makes
it clear that there are major difficulties associated with using the straight
counting line as one’s only basis for explanation. The difficulty is in fact,
insurmountable. We will briefly
indicate why at the end.
The Details
Here's a simple
algebra problem:
3x
= 6
đ
3x/3 = 6/3
đ
x =
2 (answer)
Simple stuff. But some children just don't get it. Let’s look at the underlying
principles. We’ll look at division and
how it is commonly taught using pie arithmetic:
6/2 = 3 6/3 = 2
Figure 1
We can see six divided into 2 parts is 3
and 6 divided into 3 parts is 2. But a lot is missed out here. For example, here is a pie with 323 parts.
Can the student divide it into 19 equal parts?
Figure 2
They can if they use a calculator. But we
are talking about children learning division for the first time here. Can they
accomplish the task without resorting to division algorithms, which come later
in the educative process? Can they
accomplish it from first principles? [NB: calculators use division algorithms
internally].
To make it clear: if the task were to
determine 19 x 17 then children could use the calculator, or an advanced
algorithm, but they also have the option of simply applying the definition of
multiplication and, using ruler and pen, measuring out 17 lots of 19 from zero
and then counting and confirming the result came to 323.
This example illustrates the point that the
above “pie” approach to division is insufficient. It doesn’t allow the child to
use first principles, and it doesn’t link division back to multiplication. It’s
a partial teaching job and it gives a partial understanding. Lets do the job properly now.
Division From First Principles
First comes counting, counting up and then
counting down.
Thereafter everything based on counting
inherits this dual nature.
Addition follows and its other side is
subtraction.
We add up and we subtract down.
Finally we reach multiplication. With it
comes division.
We multiply up and divide down. Division
should be shown to be the logical reverse of multiplication exactly as
subtraction is the logical reverse of addition.
To multiply is to take by addition a given number, a given number of times.
Demonstrating: 2 x 3 = 6 means (starting from zero!) that two is added to itself
three times to reach six.
Figure
3
Division is the logical opposite of
that. We start not at zero, but at the
result of the multiplication. We start our division at 6. Then we subtract, not add, three lots of two
before we arrive at zero where our division stops. The answer is the number of
steps we took (or parts we broke the 6 into). The answer is 3.
Figure
4
Given a line 323 units long a child can
actually take a 19-unit ruler and count backwards along the line (or anticlockwise
around a circle) until they reach zero. It will take them 17 steps. It is a big
task, but one that can be carried out using only the ability to count and
subtract. No fancy algorithm needed. No resort to calculator required. It can be
done from first principles.
Clarification
This doesn’t mean that the simple “divide a
pie” method is wrong. It is correct. It was just missing a step. The above is
the missing link. One method involves counting a length backwards, the other
consists of dividing a length into equal parts. When beginning, we should
discern for the child when we are using which. To do this we introduce a
(temporary) “home-made” device. We will
use the superscripts L and P:
L = Length
P = Parts
This way, if we begin at 6 and (adhering to first principles) subtract three lots of two to reach zero, breaking the 6 into 3 equal parts, we would write the equation like so: (shown both along a line AND around a pie):
6 = 3P
2L
Figure 5
If, on the other hand, we immediately
trisect the 6 into 3 equal parts perhaps by simply cutting it up from line of
sight as we might do for a pie, we write it like so (again diagrams show both
line and pie versions):
6 = 2L
3 P
Figure 6
Our task then is to determine the length of
each part – which we could count or work out from our prior knowledge that
2x3=6. [The arrows show our answer,
however either portion could have been counted, all things being equal.]
Commutativity of Division
When we examine the above two diagrams we can
see they are essentially the same. The difference is merely a matter of
direction of flow (i.e. of timing): in each case we have broken the line (or
circle) into 3 equal parts.
In the first case we began with a known length and determined a number of
parts. In the second case we began with a known number of parts and determined
the length. Its very important to slow these processes down to examine them in
detail. Here’s the detail:
In the first case we didn't know when we
began that there would be 3 parts. What we knew was that we would be using a
measure of 2 units and thereafter just kept blindly subtracting 2 at a time.
Starting from 0 (or “6”) we went back to 4 and made a mark. We went back 2 more
to reach “2” and made a mark. We went back 2 more and reached zero, the end.
Finally we had to count the number of parts (remember a larger division might
have many more parts). To do so we draw the lines which show the slices of the
pie, and number each segment: 1, 2 and 3. The answer is 3, our new knowledge.
When complete we have a trisected circle sitting on the table.
Lets assume we walk out of the room and a
new person enters. They know nothing of what we have done. They see a 6 circle that has been neatly
trisected. They need to know what measure we were using (the “2”). They could call on their existing knowledge
that 2x3=6 or count a segment, thereby determining the length of the original
measure to be 2 units. Their new knowledge is that which we began with as given
in the first case above. They have reversed our procedure, producing a cycle of
knowledge.
In both cases we considered a total of 6,
which was broken into 3 equal parts of length two.
The equations, stripped of the L and P indicators, are commutative inverses:
6/2 = 3 and 6/3 = 2
The equations are different, however they
both describe 6 cut into 3 equal parts of length 2. What the equations describe
differ only in the direction of knowledge flow (a matter of timing).
There are actually FOUR such linked
equations, which are necessary to make up the full picture. Each of those two equations can be written
two ways, and we have seen but one way.
Here are all four:
|
6/2=3 |
6/3=2 |
|
6 = 3 P 2 L |
6 = 2 L 3 P |
|
6 = 3 L 2 P |
6 = 2 P
3 L |
Figure 7
I'm just going to diagram that using circles (“pies”) but you can add the line version if you wish:
A B
D C
Figure 8
Notice the contradictions here. Going down the columns the equations (if we
strip away the L and P superscripts) are the same. Yet their meanings are
commutative opposites. Going across the
rows we have commutative equations (bottom row for example shows 6/2=3 and
6/3=2) yet essentially their meanings (stripping away from direction) are
identical. This breaking into opposites
is a natural phenomenon. It also creates a cycle.
I’ve labelled the quadrants ABCD. A is related to B as an opposite flow, as
already described. A begins with a known length and leads to a trisection
result which is B. B then reverses this process and leads back to the known
length. B relates to C because, if we
strip away the like L’s and P’s, we find they both have the same equation: 6/3 = 2.
C then leads to D in the same way that A lead to B. Finally D and A have the same equation when
stripped of L’s and P’s (6/2=3) and this brings us full circle. The cycle is a
dynamic movement of reversals going ABCD ABCD ABCD…
I would be quick to point out to a child
that we view the world in both time and space. The observer, as memory, is
primarily time oriented. The outer world as “objects outside objects” is
primarily spatially oriented. But we can conceive of space in our mind, and we
have time in the world (although we only know it exists if we use memory of some
form: for example a clock would be useless if we forgot from moment to moment
what the time was). I might also mention that all time
measurement is based on circularity and that the Earth, Sun, moon stars and
planets are spherical in space.
The divisions where we count backwards (A
and C) are in time primarily. They are a temporal activity. The divisions with
the lines through them dividing them into parts (B and D) are the same
divisions but completed and set out spatially, geometrically. They exist in
space primarily. (For those into Hegel, you may note this models the observer
as process, and the observed as the same process but completed. The object then
immediately turns back into the observer via the reverse process.)
The equations and the diagrams show a label
and the thing. Or the label and its inner meaning in this case. Notice each
algebraic label (without the Ls and Ps) depicts two opposed meanings (6/2=3 doubles as 6 counted backwards two at
a time taking 3 steps, and also that 6 cut into 2 parts results in each part
being 3 long). The truth is only to be found in the whole (taking all four at
once as a cycle).
We have shown that commutativity of
equations arises from following a temporal process of division and then
comparing it to its end result. Commutativity could obviously be explored
further. We haven’t considered negative numbers, nor have we looked at inexact
divisions. But this is a start. Now we will re-address our original
algebraic problem.
Back to the Algebra
3x
= 6
đ
3x/3 = 6/3
The statement 3x/3 can best be appreciated
by using the spatial model. Three groups (each x long) are divided into 3
portions. The result is x:
3x
= x L
3 P
Figure 9
Dividing into parts it is clear that each
part is x long. The division is easy.
Now we consider the right hand side. It is
“6/3” which we say is 2. But it could
have been 6729833869669 / 9944955 and we would need to reach for our
calculators. That is, we would need to work out our unknown by some temporal process.
Effectively the right hand side of the equation should be determined by the
process of counting backwards (the algorithm inside the calculator does just
that in effect). Thus it is appropriate
to use the following form:
6
= 2 P
3 L
Figure 10
Now you can see the difficulty a child
might have. They have to reconcile 3x/3 = 6/3:
=
Figure 11
Looked at that way the equation is not so
simple! The child has to understand
division and grasp it in all its nuances before they can properly appreciate
the algebra. To really grasp it entails an understanding of the commutativity
of division together with the nature of division as a process in time and as a
result in space. However, a good grasp of these fundamentals then leads to a
superlative understanding of algebra, the self-movement of maths and
furthermore it relates maths to both world and mind. [You may also note that
children have now been mentally prepared for the concepts of radicals in
chemistry and the transitory lifetimes of particles found in quantum physics.
Not that you would cover them here, you would keep the arithmetic and
explanation simple, but in fact it has exposed them to these concepts.]
Compare this to the result-based curriculum
paced teaching method with its hand-waving two liner:
3 x 2 = 6, hence 6/2 = 3 and 6/3 = 2
(because multiplication is commutative so
is division).
A much faster result, but most of the
understanding has been lost in transit!
The
Teacher’s Nightmare
Normally if we ask what is 6/-2 the answer
is -3. We are free to choose the (-2)
as being a length. And we can equally choose to interpret the (-3) result as a
length.
But if we pin this problem down using our notation, we notice that there is no instance when a division has the form:
x = (-z) L
(-y)
L
Rather we always divide by a length to
produce a number of parts or vice versa.
Lets now pin down a specific equation for demonstration:
6 = (-3) L
(-2)
P
The task is, to demonstrate in clear
numerical terms suitable to a child’s understanding, how dividing 6 into (-2)
parts gives an answer which is (-3) in length.
The (-3) in length might seem comprehensible, but what sensible
interpretation can be given to dividing 6 into (-2) parts?
It doesn’t help to switch the sum around into:
6 = (-3) P
(-2)
L
as in this case we are met with the impossibility of making sense of our answer, the (‑3) “parts” which we must break the 6 into. In fact this demonstration cannot be done if we rely only on the straight counting line to infinity as our foundation for number. A circular approach to number is necessary. The good news is that given the circular approach, the negative number nightmare no longer exists. It just goes away. We’ll show that in the next article.
***
This
article was written by R.M. Taylor and is based on the CircleMath concepts
taught to him by SW Taylor.
Robert
Taylor (MSc Hons)
16-10-2003
© Copyright RM & SW Taylor
for
Octanary Publications 2003
All
Rights Reserved
You can contact me by:
email:
robeetay
@ slingshot.co.nz
[get rid of the spaces around the @ sign first! We do
this to prevent search engines sending our email address to mailing lists]
Auckland
New Zealand