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

http://mathforum.org/kb/message.jspa?messageID=7186350&tstart=0.

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

http://mathforum.org/kb/forum.jspa?forumID=184.

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

http://mathforum.org/kb/message.jspa?messageID=7210776&tstart=0.

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

http://mathforum.org/kb/message.jspa?messageID=7218288&tstart=0.

Those who may interested in serving as advisors for the AMPS (Adult

Mathematical Preparation System) may contact Clyde Greeno at

greeno[at]malei.org.

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

numbers.

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

arithmetic.

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

http://mathforum.org/kb/thread.jspa?threadID=2143964&tstart=0.

Clyde Greeno believes Paul is onto something here. His reply to

Paul can be found at

http://mathforum.org/kb/message.jspa?messageID=7212278&tstart=0.

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