Date: Oct 4, 2010 1:11 PM
Author: Jonathan Groves
Subject: Ideas of Clyde Greeno, Alain Schremmer, Paul Tanner III on Arithmetic

Dear All,

As I continue to work on my arithmetic book for remedial college math
students, I have run across some ideas lately related to my work.

1. Clyde Greeno's Work

Clyde Greeno on the Mathedcc discussion list has been working on a
development of arithmetic using a vector-based approach.
One of his many posts can be found at

This is a new idea to me since I haven't seen this approach to
arithmetic before. My book has been developing an algebraic
view of arithmetic by focusing on various algebraic properties of
the different systems of numbers in arithmetic. There is still
a conceptual view of arithmetic, of course, but there is also
a focus on structure as well. I suppose I'm not surprised that
I started taking this view (and I still might) since one of my
mathematical specialities is commutative algebra.

Other posts of his are scattered in several of the recent discussion
threads on Mathedcc. The archives for Mathedcc can be found at

One wonderful idea that Clyde Greeno has mentioned lately is that
mathematical language as commonly used in K-12 and in remedial
and elementary college mathematics is that the mathematical language
is a mess! One of the biggest messes with arithmetic language is
the confusion between the words "fraction," "ratio," and "proportion."
Ratios in recent years have been equated with fractions--that is, as
quotients of two numbers--and proportions as equations expressing
equality of two quotients. He believes this problem of confusing
ratios with fractions causes much confusion among students and
teachers when it comes to learning and teaching fractions and
proportional reasoning. I believe the MIRA issue contributes to
this mess quite a bit as well, but working with messy language
does not exactly help students or teachers either.
One post of his on Mathedcc where he mentions some of these
problems with arithmetic language can be found at

As for algebraic language, Clyde has mentioned that standard
curricular language makes the terms "algebraic expression" and
"equation" confusing for most students because these terms are
often used without specifying what they are expressing or
what things are being equated. He also mentions that school
algebraic language does not distinguish the terms "variable" and
"parameter." That is, according to most algebra books, all
letters are variables (even if some algebra books point out that
this is not the case, they are still not clear on what determines
when a letter is a variable and when it is not). Yet, in the
slope-intercept equation for a line in the plane y = mx+b, m and b
are not variables yet m and b are still letters rather than specific
numbers. Most algebra books are not clear on why m and b are not
variables in this equation. A post of his mentioning some of
the mess with algebraic language can be found at

Those who may interested in serving as advisors for the AMPS (Adult
Mathematical Preparation System) may contact Clyde Greeno at

Though the focus is on remedial college mathematics, I can see that
many of the flaws Clyde has identified are flaws with K-12 math
education, one of these flaws being that mathematical language
is a mess.

Alain Schremmer on this same Mathedcc list has identified some
problems as well, one of those problems of not working with
number phrases--that is, being sloppy with the distinction
between 16 and 16 apples, for example. He believes we start to lose
students when we become sloppy in this way, and I have no doubts
about that. For instance, the distinction between ratio and fraction
is lost when we do not distinguish between abstract numbers such as
16 (that do not represent any real-world measurement or count) and
concrete numbers such as 16 apples (which represents a count).
In the fraction A/B, A and B are abstract numbers whereas in the
ratio A to B, the quantities A and B can be concrete or abstract

Another problem he has identified, though not mentioned lately,
is the problem with context-free language. He uses context-free
language and notation in his books and teaching until students
become fluent enough with the language and concepts to use the
usual notation and language of mathematics.

Some of these recent discussions on Mathedcc accidentally worked their
way into Math-Teach when some of my own posts on Mathedcc somehow got
accidentally copied over to Math-Teach.

2. Paul A. Tanner III's Work

Paul A. Tanner III is working on writing a teacher development of
arithmetic using an algebraic approach as well but instead developing
the properties of arithmetic on cancellative groupoids (all we assume
is that the set is closed under a single binary operation and that
the cancellation property holds: ab = ac implies b=c) and other
algebraic structures that possess the bare minimum of algebraic
properties to develop the properties of numbers we see in arithmetic.
This book of his is not meant to be a way to teach arithmetic to
students but to help teachers deepen their understanding of

Dave L. Renfro has mentioned that similar work has been done before,
but his search suggests that no one has yet compiled all such similar
work into a single unified document. That is, it appears that all
the work so far on this approach to developing the properties of
numbers we see in arithmetic are scattered over many papers in
various journals.

Paul's original post on Math-Teach can be found at

Clyde Greeno believes Paul is onto something here. His reply to
Paul can be found at

I know this is a lot to look over and digest. I'm still looking
it all over and trying to digest it myself. One of Dom's posts
on Mathedcc sometime earlier this month had encouraged much
recent discussion lately on developing completely new
approaches to developing arithmetic and algebra for remedial
math students. Mentioning my own book has also encouraged
much recent discussion lately, and I believe we are onto some
good ideas here. And no doubt that some of these ideas can
be incorporated into or modified appropriately for K-12 math
education as well.

Jonathan Groves