teaching algebra to elementary school students

http://www.washingtonpost.com/wp-dyn...501423_pf.html
Elementary Math Grows Exponentially Tougher
Students, Teachers Tackle Algebra
By Maria Glod
Washington Post Staff Writer
Wednesday, December 26, 2007; A01

This article suggests that many kids are capable of learning algebra
at earlier ages than it has traditionally been taught, and that poor
math preparation of elementary school teachers hampers instruction.

At the Gifted Exchange, Laura Vanderkam says

http://giftedexchange.blogspot.com/
"The simple economic solution is to pay math teachers (or any other
teachers with high-demand credentials) more. In reality, this has been
beastly hard to do. The culture of teaching -- to say nothing of union
contracts -- often undermine this. But I recently came across a
program in NYC and a few other cities called Math for America that
pays promising math teachers (defined pretty much as math majors who
did not major in education; the program pays for a master's degree in
teaching) an additional almost $20,000 per year above the salaries
they would earn as regular teachers. The America Competes Act has a
provision modeled on this program that will establish National Science
Foundation fellows around the country and boost their pay."

Beliavsky wrote:
http://www.washingtonpost.com/wp-dyn...501423_pf.html
Elementary Math Grows Exponentially Tougher
Students, Teachers Tackle Algebra
By Maria Glod
Washington Post Staff Writer
Wednesday, December 26, 2007; A01

This article suggests that many kids are capable of learning algebra
at earlier ages than it has traditionally been taught, and that poor
math preparation of elementary school teachers hampers instruction.

I hardly think this is esoteric. Seems like nearly every
math text my kids have had introduces elementary algebraic concepts
starting in 1st or 2nd grade--even the bad math texts (and for the
most part they have used fairly common texts). That's not
to say it's always taught well on the ground, or that it's implemented
well in the texts, but it seems to be in there. And I really don't
consider that "learning algebra." I suppose it is in a way, but it's
quite limited. It just puts some basic concepts in place to build on
later. It's not like they're solving simultaneous equations. I think
it's probably helpful in some ways, but I wouldn't make more out of it
than it is.

Best wishes,
Ericka

On Dec 27, 8:23 am, Ericka Kammerer wrote:
Beliavsky wrote:
http://www.washingtonpost.com/wp-dyn...07/12/25/AR200...
Elementary Math Grows Exponentially Tougher
Students, Teachers Tackle Algebra
By Maria Glod
Washington Post Staff Writer
Wednesday, December 26, 2007; A01

This article suggests that many kids are capable of learning algebra
at earlier ages than it has traditionally been taught, and that poor
math preparation of elementary school teachers hampers instruction.

I hardly think this is esoteric. Seems like nearly every
math text my kids have had introduces elementary algebraic concepts
starting in 1st or 2nd grade--even the bad math texts (and for the
most part they have used fairly common texts). That's not
to say it's always taught well on the ground, or that it's implemented
well in the texts, but it seems to be in there. And I really don't
consider that "learning algebra." I suppose it is in a way, but it's
quite limited. It just puts some basic concepts in place to build on
later. It's not like they're solving simultaneous equations. I think
it's probably helpful in some ways, but I wouldn't make more out of it
than it is.

Best wishes,
Ericka

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.'

I'm not clear where the additional $20k/teacher would come from, and
how 'math' teachers would be identified at an elementary level -- our
model has all teachers in k-2 teaching all subjects, then some slight
differentiation from 3-5 when kids start having lab-based courses. At
the HS level, it seems like splitting hairs to say the math teachers
should be more highly compensated than the physics/English/AP whatever
teachers, and likely not to really do much except strain an already
stressed budget.

Caledonia

In article
,
Caledonia 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? The kids learn to structure
algebraic problems by doing this, and that you solve these problems by
manipulation. The problems might be linear and the manipulation purely
arithmetic, but it's a long way from those "boring worksheets" the teacher
originally used to present arithmetic, and teaches much more sophisticated
skills than chanting "two twos are four" etc.

I had a look at the Algebra articles in Wikipedia and we certainly learnt all
the pre-algebra by the end of primary school (before age 12), and the use of
simple formulas (area and perimeter of regular shapes, for example). I think
we started quadratic equations and polynomials in Year 8 (age 13, our second
year of high school). Functions came a bit later.

The man going to St Ives has nothing to do with algebra that I can see; it's a
riddle.

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

http://chookiesbackyard.blogspot.com/

In article ,
Beliavsky wrote:
http://www.washingtonpost.com/wp-dyn...501423_pf.html
Elementary Math Grows Exponentially Tougher
Students, Teachers Tackle Algebra
By Maria Glod
Washington Post Staff Writer
Wednesday, December 26, 2007; A01

This article suggests that many kids are capable of learning algebra
at earlier ages than it has traditionally been taught, and that poor
math preparation of elementary school teachers hampers instruction.

The most important part of algebra is the general use of
variables; they are not merely for numbers, but for
ANYTHING. It is their lingusistic use which is most important.

With variables, complicated expressions become simple
statements or groups of statements. These are for setting
up problems, which is the most important part, and which is
not stressed. For SOLVING problems, the rule of equality,
which is that the same operation done on equal entities
gets equal results, is the main tool. With these simple
facts, very much is easy.

Mathematics is much easier with this basis; one can
concentrate on concepts, rather than memorizing facts and
rules which are given by fiat.

Teachers need to know the concepts, and be able themselves
to PROVE their statements. The concepts and understanding
of them, not just stating things as true for any other
reason, is the whole idea. Understanding the integers does
not require computational skills; they interfere.

My late wife was sickened by the attitude of prospective
teachers. My one experience with prospective high school
teachers of mathematics, taking probability, was that only
5 of the 21 I had could formulate problems involving
calculus similar to what had been done in class.

Learning how to solve problems is NOT learning mathematics.


At the Gifted Exchange, Laura Vanderkam says

http://giftedexchange.blogspot.com/
"The simple economic solution is to pay math teachers (or any other
teachers with high-demand credentials) more.

The problem is that the standards need to be set by
subject matter scholars, not subject to the bounds set
by the schools of education, whose members do not
understand their subjects, and only CLAIM to know how
to teach. They do not.

In reality, this has been
beastly hard to do. The culture of teaching -- to say nothing of union
contracts -- often undermine this. But I recently came across a
program in NYC and a few other cities called Math for America that
pays promising math teachers (defined pretty much as math majors who
did not major in education; the program pays for a master's degree in
teaching) an additional almost $20,000 per year above the salaries
they would earn as regular teachers. The America Competes Act has a
provision modeled on this program that will establish National Science
Foundation fellows around the country and boost their pay."


--
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 ,
(Herman Rubin) wrote:

This article suggests that many kids are capable of learning algebra
at earlier ages than it has traditionally been taught, and that poor
math preparation of elementary school teachers hampers instruction.

The most important part of algebra is the general use of
variables; they are not merely for numbers, but for
ANYTHING. It is their lingusistic use which is most important.

With variables, complicated expressions become simple
statements or groups of statements. These are for setting
up problems, which is the most important part, and which is
not stressed. For SOLVING problems, the rule of equality,
which is that the same operation done on equal entities
gets equal results, is the main tool. With these simple
facts, very much is easy.

Mathematics is much easier with this basis; one can
concentrate on concepts, rather than memorizing facts and
rules which are given by fiat.

I'd agree with this; I still struggle with arithmetic (which was delivered by
fiat, I suspect) but have not trouble with other branches of maths that I have
tried. I never understood what lay behind "carrying" until DH and I were
discussing Tom Lehrer's song "New Math" -- and I was over 30 then!

Teachers need to know the concepts, and be able themselves
to PROVE their statements. The concepts and understanding
of them, not just stating things as true for any other
reason, is the whole idea. Understanding the integers does
not require computational skills; they interfere. snip

Learning how to solve problems is NOT learning mathematics.

True; but well-composed problems are a major way that students learn to apply
concepts and logic -- and they are skills that require practice. Real-life
problems have their place, but can often require computations or techniques
that are out of the child's league. They are often worth exploring
nonetheless, but composing problems that are unambiguous and whose results are
easily obtainable using simple methods is much harder than it looks.

snip

The problem is that the standards need to be set by
subject matter scholars, not subject to the bounds set
by the schools of education, whose members do not
understand their subjects, and only CLAIM to know how
to teach. They do not.

I'm not convinced that subject matter scholars *alone* are the best people to
determine either curriculum content or teaching methods for younger children.
Certainly the trajectory of the curriculum needs to lead to university
entrants having a good grasp of the basics of the discipline, but that does
not necessarily imply that particular concepts need to be introduced in a set
order or taught in one particular way. For the first few years at school,
many children think "concretely". Introducing symbolic notation (for example)
will be completely unproductive until their brains have developed further --
and how and when this happens is a matter for a developmental specialist, not
a subject specialist.

Going by DS1's teacher's approach, the thinking styles among 5- and 6-yos are
sufficiently different that a range of teaching methods must be used to ensure
that all children understand even addition.

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

http://chookiesbackyard.blogspot.com/

In article ehrebeniuk-3B511E.23342628122007@news,
Chookie wrote:
In article ,
(Herman Rubin) wrote:

This article suggests that many kids are capable of learning algebra
at earlier ages than it has traditionally been taught, and that poor
math preparation of elementary school teachers hampers instruction.

The most important part of algebra is the general use of
variables; they are not merely for numbers, but for
ANYTHING. It is their lingusistic use which is most important.

With variables, complicated expressions become simple
statements or groups of statements. These are for setting
up problems, which is the most important part, and which is
not stressed. For SOLVING problems, the rule of equality,
which is that the same operation done on equal entities
gets equal results, is the main tool. With these simple
facts, very much is easy.

Mathematics is much easier with this basis; one can
concentrate on concepts, rather than memorizing facts and
rules which are given by fiat.

I'd agree with this; I still struggle with arithmetic (which was delivered by
fiat, I suspect) but have not trouble with other branches of maths that I have
tried. I never understood what lay behind "carrying" until DH and I were
discussing Tom Lehrer's song "New Math" -- and I was over 30 then!

Teachers need to know the concepts, and be able themselves
to PROVE their statements. The concepts and understanding
of them, not just stating things as true for any other
reason, is the whole idea. Understanding the integers does
not require computational skills; they interfere. snip

Learning how to solve problems is NOT learning mathematics.

True; but well-composed problems are a major way that students learn to apply
concepts and logic -- and they are skills that require practice. Real-life
problems have their place, but can often require computations or techniques
that are out of the child's league. They are often worth exploring
nonetheless, but composing problems that are unambiguous and whose results are
easily obtainable using simple methods is much harder than it looks.



The problem is that the standards need to be set by
subject matter scholars, not subject to the bounds set
by the schools of education, whose members do not
understand their subjects, and only CLAIM to know how
to teach. They do not.

I'm not convinced that subject matter scholars *alone* are the best people to
determine either curriculum content or teaching methods for younger children.
Certainly the trajectory of the curriculum needs to lead to university
entrants having a good grasp of the basics of the discipline, but that does
not necessarily imply that particular concepts need to be introduced in a set
order or taught in one particular way. For the first few years at school,
many children think "concretely". Introducing symbolic notation (for example)
will be completely unproductive until their brains have developed further --
and how and when this happens is a matter for a developmental specialist, not
a subject specialist.

The educationists do not understand concepts, and how they
are learned. It is unclear what educational development
is, but it is clear that it is not restricted to the modes
used in the schools, nor that those are even a good way.
Among others, they assume that concepts must be done by
examples first; this does not work with me, and I am
supposed to be quite good at concepts. I look at
understanding general concepts by examples are research.

This does not mean that examples should not be given, but
AFTER the concepts has been expressed, and they should be
sufficiently varied that the student can separate the
examples form the special cases. A former student told me
that the biggest problem that he had with general topology
was that he had metric space topology first, the usual
situation. It is the general concept which is often
easiest to understand, as with the use of variables.

Going by DS1's teacher's approach, the thinking styles among 5- and 6-yos are
sufficiently different that a range of teaching methods must be used to ensure
that all children understand even addition.

Quite a few older children and teachers do not understand
addition, except as an algorithmic process. There is now
clear evidence that not only very young children, but apes
(Asian "monkeys" were used) are able to understand that the
sum of two small previously displyed numbers of objects is
different from a displayed number of objects.

Learning the addition tables does not in any way teach the
concept of addition, nor does learning the multiplication
tables teach the concept of multiplication. The concepts
can be learned without these; learning the concepts first
helps. Machines can do the manipulations, but not use the
concepts, except as programmed, and this is not that much.

Nor are the educationists at all familiar with anything
other than observations or manipulatory processes in the
science, history, mathematics, or statistics. The use
statistical terms, and highly inappropriate statistical
procedures, such as converting to a normal distribution.
and often base their "results" on such.

Elementary school children can learn mathematical concepts
which seem to be unable for teachers to understand. The
same seems to be true in science, history, and language.


--
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

Concepts of algebra can be taught as early as 1st grade, using color
coded sticks of integer lengths. In one of John Holt's books, either
How Children Fail or How Children Learn, there is a brilliant essay
about this.

In high school I suffered for months in a geometry class taught by a
woman who did not grasp the geometry concepts herself. She knew that
about herself, and told us many times that she could teach us only the
mechanics because the concepts were beyond her. It was agony just
watching her try (and fail!) to draw a cube on the blackboard.

Pologirl

http://en.wikipedia.org/wiki/John_Caldwell_Holt

In article ,
(Herman Rubin) wrote:

I'm not convinced that subject matter scholars *alone* are the best people
to
determine either curriculum content or teaching methods for younger
children.
Certainly the trajectory of the curriculum needs to lead to university
entrants having a good grasp of the basics of the discipline, but that does
not necessarily imply that particular concepts need to be introduced in a
set
order or taught in one particular way. For the first few years at school,
many children think "concretely". Introducing symbolic notation (for
example)
will be completely unproductive until their brains have developed further --
and how and when this happens is a matter for a developmental specialist,
not a subject specialist.

The educationists do not understand concepts, and how they
are learned. It is unclear what educational development
is, but it is clear that it is not restricted to the modes
used in the schools, nor that those are even a good way.

Dear me, how much contact have you had with educationists to know all this --
and especially at primary school level?

Among others, they assume that concepts must be done by
examples first; this does not work with me, and I am
supposed to be quite good at concepts. I look at
understanding general concepts by examples are research.

Thought I'd discussed this with you previously, but it's one of the
difficulties of gifted education in the general classroom. Average students
learn better when examples are presented before the general concept, but for
gifted students it is the other way around. There are ways around it, of
course. One can express the concept in a single sentence before beginning
examples, then recapitulate. One can get students to work out the concept
inductively or deductively. One can teach the students in sub-groups. It's
only a problem for the unimaginative.

This does not mean that examples should not be given, but
AFTER the concepts has been expressed, and they should be
sufficiently varied that the student can separate the
examples form the special cases. A former student told me
that the biggest problem that he had with general topology
was that he had metric space topology first, the usual
situation. It is the general concept which is often
easiest to understand, as with the use of variables.

Unless this problem is very common, I'd imagine that this student's
difficulties arose from absent-mindedness. From what I can make out from the
Wikipedia article on metric space topology, it's a simplified form of topology
where you can make assumptions about certain properties. That probably does
make it a good starting point for beginners. Other lecturers in topology
should have made clear that MST *was* a specialised form and that those
assumptions could no longer be made -- probably by demonstration.

Certainly, special cases should always be discussed well after the general
rules are established, and their nature thoroughly explained. I'm having
trouble thinking of special cases in primary maths, though.

Going by DS1's teacher's approach, the thinking styles among 5- and 6-yos
are
sufficiently different that a range of teaching methods must be used to
ensure
that all children understand even addition.

Learning the addition tables does not in any way teach the
concept of addition, nor does learning the multiplication
tables teach the concept of multiplication. The concepts
can be learned without these; learning the concepts first
helps.

Exactly. My son began addition with concrete materials. They then moved to,
um, pictorial materials? The "John has two apples" stage, with pictures, then
to the purely symbolic, then to activities that encouraged memorisation. I am
oversimplifying, of course, because there were children at all stages of this
process and an immense variety of materials and activities. In fact, I'd
expect that there was more going on than this, but as I'm not a primary
teacher I simply didn't identify it.

Nor are the educationists at all familiar with anything
other than observations or manipulatory processes in the
science, history, mathematics, or statistics. The use
statistical terms, and highly inappropriate statistical
procedures, such as converting to a normal distribution.
and often base their "results" on such.

Well, lying with statistics is hardly uncommon, alas. OTOH I haven't noticed
conversion to normal distributions occurring in DS1's classroom. They have
done a few bar charts (eye and hair colour; transport to school), and had to
answer very simple questions -- most and least common categories. Inferential
reasoning is a bit uncertain at this age level, and that's the sort of thing
that a specialist in child development should be able to tell you before you
embark on a frustrating attempt to teach a child something they are not yet
capable of understanding.

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

http://chookiesbackyard.blogspot.com/

In article
,
Pologirl wrote:

Concepts of algebra can be taught as early as 1st grade, using color
coded sticks of integer lengths.

Is that what you call Cuisenaire rods when you can't use the trade name? :-)

Gee, they were fun. DS1's teacher doesn't use them; I wonder does his school
have a dusty pile of Cuisenaire boxes in a corner somewhere?

In one of John Holt's books, either
How Children Fail or How Children Learn, there is a brilliant essay
about this.

I'm always a trifle suspicious of books with titles like these. It sounds too
much like the educational equivalent of Growing Kids God's Way.

In high school I suffered for months in a geometry class taught by a
woman who did not grasp the geometry concepts herself. She knew that
about herself, and told us many times that she could teach us only the
mechanics because the concepts were beyond her. It was agony just
watching her try (and fail!) to draw a cube on the blackboard.

The real question is why she was permitted to teach you! I just can't imagine
that in a high school, where the teachers are specialists. OTOH I can't
imagine why she was drawing cubes on the blackboard; our geometry was largely
applied logic of the "Prove these triangles are congruent" variety.

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

http://chookiesbackyard.blogspot.com/