

From arithmetic to algebra - tokenadult
http://math.berkeley.edu/~wu/C57Eugene_3.pdf

======
btilly
The mathematical points are correct. But miss the key educational fact that
the ability to learn abstraction seems to be tied to puberty in most children.
Therefore drawing a "gentler slope" of slowly increasing abstraction pre-
algebra will cause problems for many children due to developmental
limitations.

By all means lay as correct a foundation as feasible. But be aware of the
limits that children have, and do not set them up for failure by ignoring
those limits. (This was one of the failings of the "new math" movement.)

BTW none of this is to say that I'm remotely happy about the disaster that is
math education. I just don't think that this is the right solution to it.

~~~
tokenadult
_key educational fact that the ability to learn abstraction seems to be tied
to puberty in most children_

What is the evidence for this, in view of the fact that children in several
other countries around the world learn the same things the author suggests
that American children learn at much earlier ages?

[http://www.aft.org/pubs-
reports/american_educator/fall99/ame...](http://www.aft.org/pubs-
reports/american_educator/fall99/amed1.pdf)

[http://www.cbmsweb.org/NationalSummit/Plenary_Speakers/ma.ht...](http://www.cbmsweb.org/NationalSummit/Plenary_Speakers/ma.htm)

<http://nces.ed.gov/timss/>

~~~
btilly
For evidence of the age at which children improve on abstraction, look at
Piaget's work. The onset of puberty is when children usually move from
Piaget's concrete phase to formal operations. Before that age you need to keep
things concrete.

The links you provide don't contradict this in any way, shape or form. My
complaint about the proposed curriculum change is that it attempts to teach
arithmetic proceeding from definition to result. That goes over kids heads.
Children learn better by repeatedly doing concrete operations that exercise
the underlying concepts. The examples of Chinese teachers demonstrate good
ways of doing that.

In short I'm complaining about pedagogy, not content.

Re-read
[http://www.cbmsweb.org/NationalSummit/Plenary_Speakers/ma.ht...](http://www.cbmsweb.org/NationalSummit/Plenary_Speakers/ma.htm)
with this in mind. I don't see children being challenged with abstraction.
Instead I see concrete steps that are ignored in the USA being broken out and
done repeatedly. Children may not be able to describe composing a mathematical
expression in abstract terms, but they can learn to do it. And having that
skill makes word problems a lot easier.

And, of course, the concrete skills and connected concepts give them a better
foundation to build abstractions on when they are old enough to do that.

A random incidental note. Your links lead me to
<http://nces.ed.gov/timss/results07_math07.asp> which suggests that the USA is
not as bad in international comparisons as is often claimed. That said, the
USA clearly could improve, and Taipei would seem to be a particularly good
country to learn from.

------
spectre
Abstract thinking is sadly lacking from our school curriculums.

I've seen students even at university level who are bewildered by equations
like w^2-2w+4=0 or d/dm(m^3-m^2+m) just because the they haven't been taught
that the variable x is just an abstraction.

------
RiderOfGiraffes
And more from the same author ...

<http://math.berkeley.edu/~wu/EMI2a.pdf>

