```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 mathstudents, I have run across some ideas lately related to my work.1. Clyde Greeno's WorkClyde Greeno on the Mathedcc discussion list has been working on adevelopment 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 toarithmetic before. My book has been developing an algebraic view of arithmetic by focusing on various algebraic properties ofthe different systems of numbers in arithmetic. There is stilla conceptual view of arithmetic, of course, but there is alsoa focus on structure as well. I suppose I'm not surprised thatI started taking this view (and I still might) since one of mymathematical specialities is commutative algebra. Other posts of his are scattered in several of the recent discussionthreads on Mathedcc. The archives for Mathedcc can be found athttp://mathforum.org/kb/forum.jspa?forumID=184.One wonderful idea that Clyde Greeno has mentioned lately is thatmathematical language as commonly used in K-12 and in remedialand elementary college mathematics is that the mathematical languageis a mess! One of the biggest messes with arithmetic language isthe confusion between the words "fraction," "ratio," and "proportion."Ratios in recent years have been equated with fractions--that is, asquotients of two numbers--and proportions as equations expressingequality of two quotients. He believes this problem of confusingratios with fractions causes much confusion among students andteachers when it comes to learning and teaching fractions andproportional reasoning. I believe the MIRA issue contributes tothis mess quite a bit as well, but working with messy languagedoes not exactly help students or teachers either. One post of his on Mathedcc where he mentions some of theseproblems with arithmetic language can be found athttp://mathforum.org/kb/message.jspa?messageID=7210776&tstart=0.As for algebraic language, Clyde has mentioned that standardcurricular language makes the terms "algebraic expression" and "equation" confusing for most students because these terms areoften used without specifying what they are expressing orwhat things are being equated. He also mentions that schoolalgebraic language does not distinguish the terms "variable" and"parameter." That is, according to most algebra books, allletters are variables (even if some algebra books point out that this is not the case, they are still not clear on what determineswhen 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 notvariables in this equation. A post of his mentioning some ofthe mess with algebraic language can be found athttp://mathforum.org/kb/message.jspa?messageID=7218288&tstart=0. Those who may interested in serving as advisors for the AMPS (AdultMathematical Preparation System) may contact Clyde Greeno atgreeno[at]malei.org.Though the focus is on remedial college mathematics, I can see thatmany of the flaws Clyde has identified are flaws with K-12 matheducation, one of these flaws being that mathematical languageis a mess. Alain Schremmer on this same Mathedcc list has identified someproblems as well, one of those problems of not working withnumber phrases--that is, being sloppy with the distinctionbetween 16 and 16 apples, for example. He believes we start to losestudents when we become sloppy in this way, and I have no doubtsabout that. For instance, the distinction between ratio and fractionis lost when we do not distinguish between abstract numbers such as16 (that do not represent any real-world measurement or count) andconcrete numbers such as 16 apples (which represents a count). In the fraction A/B, A and B are abstract numbers whereas in theratio A to B, the quantities A and B can be concrete or abstractnumbers.Another problem he has identified, though not mentioned lately,is the problem with context-free language. He uses context-freelanguage and notation in his books and teaching until studentsbecome fluent enough with the language and concepts to use theusual notation and language of mathematics. Some of these recent discussions on Mathedcc accidentally worked theirway into Math-Teach when some of my own posts on Mathedcc somehow gotaccidentally copied over to Math-Teach.2. Paul A. Tanner III's WorkPaul A. Tanner III is working on writing a teacher development ofarithmetic using an algebraic approach as well but instead developingthe properties of arithmetic on cancellative groupoids (all we assumeis that the set is closed under a single binary operation and thatthe cancellation property holds: ab = ac implies b=c) and otheralgebraic structures that possess the bare minimum of algebraicproperties to develop the properties of numbers we see in arithmetic.This book of his is not meant to be a way to teach arithmetic tostudents 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 similarwork into a single unified document. That is, it appears that allthe work so far on this approach to developing the properties ofnumbers we see in arithmetic are scattered over many papers invarious journals. Paul's original post on Math-Teach can be found athttp://mathforum.org/kb/thread.jspa?threadID=2143964&tstart=0. Clyde Greeno believes Paul is onto something here. His reply toPaul can be found athttp://mathforum.org/kb/message.jspa?messageID=7212278&tstart=0.I know this is a lot to look over and digest. I'm still lookingit all over and trying to digest it myself. One of Dom's postson Mathedcc sometime earlier this month had encouraged muchrecent discussion lately on developing completely newapproaches to developing arithmetic and algebra for remedialmath students. Mentioning my own book has also encouragedmuch recent discussion lately, and I believe we are onto somegood ideas here. And no doubt that some of these ideas canbe incorporated into or modified appropriately for K-12 math education as well. Jonathan Groves
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