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 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
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.