teaching algebra to elementary school students

Herman Rubin wrote:
In article , Penny Gaines wrote:
Herman Rubin wrote:

[snip]

If anything, you are wrong here. The problem with having
examples presented first is that the special properties of
the examples have to be unlearned. Unlearning is VERY
difficult. My encounters with "typical" students in
service courses is that their knowledge of algebra is only
how to solve well-formulated problems, and the same is true
for calculus.
[snip]

It is worth noting though, that Newton developed the concepts of
calculus originally by doing many examples, and then seeing that there
were common links.

This is definitely the case in most research. But should
it be the case in teaching?

I think when one is being taught advanced mathematics it is
worth knowing that the ideas did not appear fully formed to the
people who first developed the concept.

[snip]
I am unconvinced that Newton developed the concepts of
calculus by doing many examples; possibly a few. The
ideas were already there, and I believe the Greeks
would have had it if they had the general algebraic
ideas developed by Viete, leading to the representation
of functions by graphs by Descartes. Notation IS very
important, and the teaching of "mathematical notation",
not fully developed until the 20th century, should come
with beginning reading. THIS is teaching algebra;
variables are not limited to numbers, but can stand
for anything, including verbs, phrases, etc.

Newton spent about two years doing different tangent and area
problems before realising that they were inverse processes.

Viete and Descartes developed ideas that were important, and given
that both Liebniz and Newton arrived at calculus (although not quite
the modern understanding) at a similar time, it could be said it was
an idea whose time had come.

I don't think that the Greeks thought algebraically at all: they
thought in geometric terms.

--
Penny Gaines
UK mum to three

Barb Knox wrote:
In article , Penny Gaines
wrote:

Herman Rubin wrote:

[snip]

If anything, you are wrong here. The problem with having
examples presented first is that the special properties of
the examples have to be unlearned. Unlearning is VERY
difficult. My encounters with "typical" students in
service courses is that their knowledge of algebra is only
how to solve well-formulated problems, and the same is true
for calculus.
[snip]

It is worth noting though, that Newton developed the concepts of
calculus originally by doing many examples, and then seeing that there
were common links.

Even if so, it is very much easier to learn something that is already
well understood by one's teachers, than to invent something for the
first time. (This, BTW, is why educational "discovery methods" are so
inefficient; which is not to say that they are not valuable in certain
special circumstances.)

I quite agree: new methods in mathematics tend to be discovered because
that particular area is of interest to mathematicians.

--
Penny Gaines
UK mum to three

Caledonia wrote:
[snip]
There is no mathematical topic called "pre-algebra". That is a
US educational classification, but it isn't a mathematical
classification.
[snip]
Thank you! I was confounded regarding how to explain that calling this
algebra was a dilution of the term -- sort of like saying that DD2 is
studying phytomorphology in preschool when they're labeling parts of
plants and leaves. I'm content saying 'they're looking at plants and
leaves' and 'thinking about numbers.'

Caledonia

Is it the idea that the first grade worksheets are "algebra" that
bothers you and Ericka, or is that the children are told that they are
algebra?

--
Penny Gaines
UK mum to three

Penny Gaines wrote:
Caledonia wrote:
[snip]
There is no mathematical topic called "pre-algebra". That is a
US educational classification, but it isn't a mathematical
classification.
[snip]
Thank you! I was confounded regarding how to explain that calling this
algebra was a dilution of the term -- sort of like saying that DD2 is
studying phytomorphology in preschool when they're labeling parts of
plants and leaves. I'm content saying 'they're looking at plants and
leaves' and 'thinking about numbers.'

Is it the idea that the first grade worksheets are "algebra" that
bothers you and Ericka, or is that the children are told that they are
algebra?

I don't think the kids are usually told that they're
doing algebra (though sometimes the teachers mention that what
they're doing is forming the foundation for algebra). At least
that's been the case with my kids. I just think that it's a
misleading use of the term to say that 1st graders are "doing
algebra" in this case. Might be a regional language difference,
but at least in my experience, saying someone is "doing algebra"
means something rather different from what these kids are doing
in early elementary. My older son is "doing algebra" in the way
that phrase is commonly understood in the US. My younger son is
not, though he is developing concepts he will put to good use
when he does. I think it's making a mountain out of a molehill.

Best wishes,
Ericka

Penny Gaines wrote in
:

Newton spent about two years doing different tangent and
area problems before realising that they were inverse
processes.

Viete and Descartes developed ideas that were important,
and given that both Liebniz and Newton arrived at calculus
(although not quite the modern understanding) at a similar
time, it could be said it was an idea whose time had come.

are there any good books on the *history* of mathematics? i'm
suddenly intrigued...

lee who had all the really horrific math teachers. the
algebra teacher who insisted girls can't learn math, so i had
to be cheating... etc

On Thu, 3 Jan 2008 02:17:10 +0000 (UTC), enigma
wrote:

Penny Gaines wrote in
:

Newton spent about two years doing different tangent and
area problems before realising that they were inverse
processes.

Viete and Descartes developed ideas that were important,
and given that both Liebniz and Newton arrived at calculus
(although not quite the modern understanding) at a similar
time, it could be said it was an idea whose time had come.

are there any good books on the *history* of mathematics? i'm
suddenly intrigued...

lee who had all the really horrific math teachers. the
algebra teacher who insisted girls can't learn math, so i had
to be cheating... etc

You might like this website:

http://library.thinkquest.org/22584/

toto wrote in
:

On Thu, 3 Jan 2008 02:17:10 +0000 (UTC), enigma
wrote:
are there any good books on the *history* of mathematics?
i'm suddenly intrigued...

You might like this website:

http://library.thinkquest.org/22584/

oh! looks like a good start at least. thanks!
lee

In article ,
enigma wrote:
Penny Gaines wrote in
:

Newton spent about two years doing different tangent and
area problems before realising that they were inverse
processes.

I am still of the opinion that it was that hard.
This type of problem goes back to the Greek geometers,
who definitely had the idea of the Riemann integral.
That they could not compute many is because of notational
problems.

Viete and Descartes developed ideas that were important,
and given that both Liebniz and Newton arrived at calculus
(although not quite the modern understanding) at a similar
time, it could be said it was an idea whose time had come.

No, it was not an idea whose "time had come'. It was
something apparently first realized at the time.

Euclid used variables for points, lines, and planes,
but until Diophantus, nobody had used variables for
numbers. It was Viete who expounded on the use of
many variables, and it was this which was important
in the development of mathematics. Newton's calculus
notation was worse than that of Leibniz. Also, it
was Euler, in the next century, who introduced variables
for functions, but the general use of variables for
anything did not come until the 20-th century.

Try making a formal definition of limit and the Riemann
integral, both of which ideas the Greeks understood. It
is hard to proceed without reasonable notation; only
computers can be expected to process very long expressions,
and that only because they have a type of functional
notation themselves, namely, addresses.

If Archimedes had the use of variables, I believe he
would have developed calculus; he had what it took,
otherwise. But the earliest use of variables for
numbers was 500 years past his time.

are there any good books on the *history* of mathematics? i'm
suddenly intrigued...

lee who had all the really horrific math teachers. the
algebra teacher who insisted girls can't learn math, so i had
to be cheating... etc


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558

In article , Penny Gaines
wrote:

We've had the same experience -- I realized that some of the 1st and
2nd grade worksheets of word problems really were getting kids used to
solving for x, where x was expressed as a proportion of other
variables. Kind of like the question about the man going to St. Ives.
I cringe at the idea that teachers are billing this as 'algebra.'

But it *is* -- what else would you call it?

Pre-algebra, or some such similar thing.

From a mathematical POV it is part of algebra: it is very basic
algebra, admittedly, but it is still part of algebra.

There is no mathematical topic called "pre-algebra". That is a
US educational classification, but it isn't a mathematical
classification.

Indeed. I had no idea of pre-algebra until it was mentioned here.

(FWIW, I didn't do a topic labelled "algebra" until I got to
University: over here, most school courses are generally called
just 'maths', and the topics within it might be called pure maths,
probablity (or statistics) and applied maths (or theoretical
mechanics.)

Our school courses are also called Maths but within them we studied
sub-branches like Algebra, Trigonometry, Geometry etc (with the names). I
think the great advantage of Maths as I learned it was how it all came
together at the higher levels. For example, we might have learned how to plot
points on a Cartesian plane in Yr 7, then lines (y=2x). Later (Year 9, say?)
we would have encountered the notation f(x)=2x, plotted that on the graph,
then in our last two years of high school, that fed straight into calculus,
and working out areas/volumes under the curve. Similarly, the initial
understanding of trigonometry moves on to sine functions etc and again into
uses in calculus. (Older members may remember my contention that calculus is
a specialised type of algebra!)

Sorry if this sounds a bit vague, but it's 20 years since high school -- it's
only in retrospect that one realises that these convergences were deliberate.
The apparently rigid boundaries of the US subjects don't seem like a good idea
to me at all, especially if people are led to believe that the manipulation of
variables is not algebra!

--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

http://chookiesbackyard.blogspot.com/

In article ,
Ericka Kammerer wrote:

Is it the idea that the first grade worksheets are "algebra" that
bothers you and Ericka, or is that the children are told that they are
algebra?

I don't think the kids are usually told that they're
doing algebra (though sometimes the teachers mention that what
they're doing is forming the foundation for algebra). At least
that's been the case with my kids. I just think that it's a
misleading use of the term to say that 1st graders are "doing
algebra" in this case. Might be a regional language difference,
but at least in my experience, saying someone is "doing algebra"
means something rather different from what these kids are doing
in early elementary.

I'd describe it as "using variables" myself, but as we've said, that is
actually part of algebra. I imagine that when people are "doing algebra" in
Erickaville, they are understood to be "doing Algebra", ie, a course by that
name?

--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

http://chookiesbackyard.blogspot.com/