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Let's say you're a university lecturer who regularly teaches remedial and/or college algebra courses. A standard textbook for such a course usually starts out with a series of facts about real number properties, but doesn't distinguish between facts that are definitions, versus facts that can be proven from more basic assumptions. A small number of examples:

  • What is the definition of subtraction and division in use?
  • Is the fact that $(-1)x = -x$ a definition, or a theorem?
  • Is the fact that $a^0 = 1$ a definition or a theorem?
  • What about the fact that $a/b = c/d \Leftrightarrow ad = bc$?
  • Is the fact that we can add a value to both sides of an equation and maintain equivalence a definition, or provable from simpler assumptions?

Over the years I've spent a good amount of time disentangling these issues, but I'm wondering if there is an existing book that I could recommend to others. Say: We start by taking the real numbers as a roughly undefined term, and define the equality relation clearly. Explicate the definitions of the various operations in use, and the order of the operations. Establish the properties of commutativity, association, and distribution as axioms. From that point forward, clarify what facts are definitions, versus what are theorems, and present explicit proofs for the latter. Even if we don't present the information to our students exactly this way, I think it would be a relief for a mathematically-trained instructor to know what really must come first in the logical sequence, and what can really be proven to an inquisitive student, versus the few things that are effectively "just because".

Is there such a resource that formally develops topics from the start of a basic algebra course?

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    $\begingroup$ Have you looked through HH Wu's texts? I mentioned both the Pre Algebra and Algebra texts back in MESE 1857... $\endgroup$ – Benjamin Dickman Aug 30 '17 at 4:35
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    $\begingroup$ Any textbook on abstract algebra develops algebra formally. "Linear algebra and geometry"/"Algèbre linéaire et géométrie élémentaire" (1969) by Jean Dieudonné is an interesting case. It is intended for teachers (see "Introduction"). As you wish, it deals with the field axioms in Chapter 1. AFAIK, it is an outcome of a failed attempt of Bourbaki to do in education what you want to do. $\endgroup$ – beroal Aug 30 '17 at 6:07
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    $\begingroup$ A point you may want to consider is that field is a complex structure, a conglomerate. It consists of additive and multiplicative parts. Each of these parts may be taught in progression: associative operation (semigroup) → monoid → group. $\endgroup$ – beroal Aug 30 '17 at 6:12
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    $\begingroup$ My opinion: all of these things are way beyond a "remedial or college algebra" course. You want them to know and use $(-1)x = -x$. But telling them "it is a theorem" or "it is a definition" will not help. These topics may be interesting for eager, well-motivated math students. But those students have already been removed from the "remedial or college algebra" class. $\endgroup$ – Gerald Edgar Aug 30 '17 at 12:47
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    $\begingroup$ @BenjaminDickman: Thanks for those links. Wu's philosophy in his General Introduction (to Prealgebra) exactly matches mine here. In fact, his example of the definition of equivalent fractions is something I was particularly rattled to realize just this year, which partly motivated this question. $\endgroup$ – Daniel R. Collins Aug 30 '17 at 17:55
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Short answer: (migrated from comments by request)

Hung-Hsi Wu (Berkeley) has a home page full of links to writings of this ilk.

Two specific examples that may fit the bill are ones I mentioned back in MESE 1857 (April 2014); in particular, links to a text on Pre-Algebra (pdf) and an Intro to School Algebra (pdf).


Further comments:

Some of H.H. Wu's materials are intended more for pre-service/in-service teachers to bolster their mathematical content knowledge. The question of how to enact these ideas in one's classroom practice is a tough one. As an example, I have seen (simple) fractions presented as pairs of numbers with a horizontal line (vinculum aka fraction bar) between them, for which the top number is an integer and the bottom number is a non-zero integer. One can then define two fractions $\frac{a}{b}$ and $\frac{c}{d}$ to be equivalent under (some condition) and then define the rational numbers as being the fractions modulo this equivalence relation (whatever that means). Of course, the notation using a horizontal line is merely a convention; one can also think of rational numbers as being ordered pairs in $\mathbb{Z}\times\mathbb{Z}_{\neq 0}$ modulo the appropriate equivalence relation.

I mention this example since the post above contains:

Say: We start by taking the real numbers as a roughly undefined term, and define the equality relation clearly.

Incidentally, I recently encountered a secondary school mathematics teacher who defined two fractions $\frac{a}{b}$ and $\frac{c}{d}$ to be equivalent if there exists an $r \in \mathbb{R}$ for which $ar = c$ and $br = d$. This definition is plausible (in some sense) if, as in the quotation pull above, one begins by allowing for something called the "real numbers." Formally, though, it can turn out to be problematic: Usually the development of $\mathbb{R}$ proceeds in some way from the development of $\mathbb{Q}$, so defining the latter by using an element of the former becomes circular.

The standard condition given for fraction equivalence is, indeed, one that is also contained in this post: equality under cross multiplication, i.e., $a/b = c/d$ if and only if $ad = bc$. Another approach is to talk about "simplifying fractions," and then one can say that two fractions are equivalent if, when fully simplified, each yields the same ordered pair of integers.

Nevertheless, motivating this level of formality is an expected challenge irrespective of the population with whom one works. I do not know of a mathemagical way to excite everyone around distinguishing between theorems and definitions but, as remarked in the comments above (in response to another comment), I think we should avoid concluding that an enrollee in a remedial course on college algebra will find these ideas un-interesting, or that such courses are devoid of "eager, well-motivated math students." It is my sense that students can end up in these courses because, all too frequently, they have experienced mathematics in earlier years as a nebulous collection of assertions that lack the cohesion/interconnectedness that may be more apparent when the formal definitions are clearly articulated. And the corresponding lack of sense-making in previous math courses can, indeed, take its toll on one's eagerness/motivation to pursue mathematical understanding. Under such conditions, I sincerely doubt that continuing to teach by fiat is the right response.

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    $\begingroup$ +1 Thanks for writing that up. In particular, I fully agree with the last paragraph. Even if remedial college classes are not taught fully axiomatically, I would think it the height of tragedy for college students to never hear the words that mathematics involves definitions, theorems, and proofs. In fact, this discussion bolstered my belief to add another introductory slide on that topic for my remedial course this fall. $\endgroup$ – Daniel R. Collins Sep 2 '17 at 6:49
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The best careful textbook you can find on elementary algebra that I'm aware of is the classic by Gelfand. It is beautifully written and completely careful-it's the book you wish you'd had in high school or grade school. I think you'll find it VERY helpful for both yourself and your students.

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This is a possibly arch answer but I think it warrants being documented here. The formal development of the topics of elementary algebra is embedded in, and historically the motivation for, any text in abstract algebra. And perhaps also the principal reason why high-school math educators should take such a course. The first few sections of a typical elementary algebra text are effectively presenting the field axioms for the real numbers, and then proceeding to topics that can be proven from those.

Thanks to comments by beroal and DRF for making me think about that.

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