A worthy observation about " ..."expressions with variables" as objects in their own right ...." It truly puts a finger on a major cause for the American disaster in mathematics education. Much curricular talk about "algebraic expressions" ... and almost no talk about what they actually *express*!
All too often, students have no idea of what such "expressions" are talking about. So they are forced to view such formalisms as meaningless text with they are expected to perform data-processing tasks as prescribed by authors and teachers.
In the 1960's, the SMSG Committee on Algebra (Swain, Dolciani, et al) came up with the phrase "algebraic expression" because they thought that "formulas" should be *equations.* Apparently, none had ever worked in manufacturing industries ... or in a kitchen ... or looked the word up in a dictionary.
>>>>>>From dictionary.com >>>>>> for·mu·la ? ?[fawr-myuh-luh] noun, plural for·mu·las, for·mu·lae ?[-lee] Show IPA. 1.a set form of words, as for stating or declaring something definitely or authoritatively, for indicating procedure to be followed, or for prescribed use on some ceremonial occasion. 2. any fixed or conventional method for doing something: His mystery stories were written according to a popular formula. 3. Mathematics . a. a rule or principle, frequently expressed in algebraic symbols. b. such a symbolic expression. 4. Chemistry . an expression of the constituents of a compound by symbols and figures. Compare empirical formula, molecular formula, structural formula. 5. a recipe or prescription <<<<<<<<<<<<<<<
Introductory algebra is all about formulas for performing calculations. Like any kitchen recipe a mathematical formula consists of "ingredients" and instructions on how to combine those ingredients. Use such and such ... in this and that way ... to *produce* so and so. For the perimeters of rectangles, one formula is "2L+2W." It tells how to combine L's and W's to get perimeters. In more technical vernacular, formulas are combinations of functions ... which act on their entries.
The "Algebra Committee" would argue the "formula" is the "equation", 2L+2W=P ... where "2L+2W" is one "expression" and "P" is another "expression." Part of the ignor-ance was ignoring the "gives" use of "="... which now is how the [=] key is used on all calculators. Instead, the Committee regarded "=" as always meaning that the "expressions" on the two sides are synonyms.
Another part of the ignor-ance was that, as students, the Committee members had not been taught about functions until calculus. So they ignored the concept of functions, believing that it was too "advanced" for introductory algebra. [Meanwhile, another committee was teaching "function machines" to elementary school children.] The eventual outcome is that we now have an "algebra" curriculum that is all about manipulating text expressions which have very little meaning.
True, the "2+3=5" equation can be taken to mean that the "2+3" phrase means 5. But the *reason* that "2+3" expresses 5 is that "2+3" is a *formula* ... and if you execute that formula ... by [=] ... you will get to 5. As combinations of functions, formulas have definitive meanings. If "algebraic expressions" are to make any mathematical sense at all, they must be revealed as expressing formulas.
Even very young students learn to regard formulas as objects ... and later to manipulate those ... far more easily than doing so with "expressions" that are seen only as text.
- -------------------------------------------------- From: "Joe Niederberger" <email@example.com> Sent: Friday, November 09, 2012 1:26 PM To: <firstname.lastname@example.org> Subject: Re: How teaching factors rather than multiplicand & multiplier confuses kids!
> Robert Hansen says: >>One last thing Joe. If you agree with Clyde then you must understand >>Clyde, correct? > > Please - in this thread I meant that one specific observation. I often > find points of agreement with him though, in general, but other times he > loses me as well with his full-blown theories and terminology. > > I will note that he does know the difference between "vector space" as > commonly used in abstract algebra and his particular appropriation of that > word. I don't have the URL handy, but he says so right in the first > paragraph or so. > > Robert Hansen says: >>The counting/adding/apple phase ends pretty quickly in the second grade. I >>shed a tear every time I remember my son adding on his fingers. They grow >>up so fast. > > Absolutely true for counting and adding. However, when you first start to > learn about "functions" perhaps you have a very concrete idea of functions > of a real variable. To an advanced mathematician these days, that's like > counting on your fingers - their are so many more uses for mappings in > general. > > That's why I like "meta" better than "symbolic". There are meta levels > upon meta levels, not just two levels. The distinctions are always > relevant no matter how far one gets. "Meta" also captures better the idea > of studying "expressions with variables" as objects in their own right, > rather than simply viewing them as ways of expressing things *about* > integers or rationals or what-have-you. I agree with you fourth graders > are usually not at that level. Nor, sadly, are a lot of 10th garders. > > Joe N