In a recent posting we read:
> Date: Tue, 03 Mar 1998 05:48:12 -0800
> From: "The Old Pro (Jim C. Gajniak)" <jgajniak@IX.NETCOM.COM>
> Subject: [MATHEDCC] Curriculum Guide for H.S. Algebra 1
>
> To the learned educators on this list:
>
> I've been asked to list the main topics(sort of a Curriculum Guide) that
> should be taught in High School Algebra 1.(sort of a Curriculum Guide)
>
> I'm aware that most teachers on this list are teaching
> on the College Level; from reading so many reasoned conversations, whatever
> input you could provide would be appreciated and included in this Guide.
>
> I'm sure there are more, but so far I've come up with:
>
> 1. Solving Equations in one variable
>
> 2. factoring polynomials
>
> 3. solving simultaneous equations
>
> 4. graphing linear and quadratic equations
>
> 5. Basic Definitions of: "term", "coefficient", etc.
>
> 6. graphing "equalities" and "inequalities"
>
> 7. working with fractional exponents.
>
> 8. translating English sentences into algebraic expressions
>
>
> I've been requested to come up with about a two to three
> page document.
>
> I'm particularly interested in those algebra topics where you see "more
> than a few" students coming into college poorly prepared. That is those
> areas requiring remediation.
>
> Thanks,
>
> The Old Pro
I guess I'm with Ed Laughbaum in not seeing the logical order in the posted
list. Certainly there are more fundamental understandings, not so easily
described by traditional topic areas, that are what need attending to in
order to prepare students for higher-level (college) mathematics, and certainly
the _context_ for teaching has changed. Grasp of the idea of a function (in a
real sense, not just playing with functional notation and talking about
domain and range in toy examples) is by all means an understanding that
should start to be developed as early as possible.
Relating algebraic calculations to numeric calculations is also important.
The first thrust of mathematical education is learning to see the order
and structure in calculation with _numbers_. Attainment of perspective in
this respect is important, since medium-sized calculations are no longer
appropriately performed by digital algorithms on paper. Being able to sort
calculations into those which are appropriately performable "by inspection"
(in steps, say), those which should be passed along for naive electronic
calculation (for reliability and efficiency reasons), and those that are
large enough to require special treatment (worrying about round-off error,
squeezing out extra precision, etc.) is really a major part of the
mathematical maturity relevant to technical employment.
This concern has it's symbolic echo. Learning algebra is really learning
to see the order and structure in calculation with _symbols representing
numbers_ (it is only later, in "abstract algebra", that the symbols are
freed from any necessary numeric interpretation). The very first topic
addressed in any algebra course has to be the concept of a _variable_.
This is by no means a natural step, and the puzzlement and frustration of
students who are expected to see this as "no big thing" is often evident.
A big aid in swallowing this concept has been the introduction of
calculators with named memories. Using calculations with X on a graphing
calculator to support and justify calculations on paper with x is a very
powerful approach! Of course, nowadays it is also possible to perform
calculations with x on a _symbolic_ calculator such as the TI-92, but it
will be a while before any significant portion of the educational community
is ready to consider making use of this capability in conjunction with
the teaching of elementary algebra.
Manipulating simple algebraic expressions, getting a _feel_ for how they
work (yes, order of operation and use of parentheses), evaluating
expressions for given values of the variable, are all bread-and-butter
topics that have to be developed early on. There's no reason you can't
introduce functions right here, though. Again the graphing calculator
is a big help (misnomer -- no graphs yet, but plenty of home-screen work,
and probably tables). Not too early to start working with _lists_ as a
data structure.
Here is also where translating English into algebraic expressions comes in
also. The "quaintness" often perceived in traditional mathematical language
goes back to the days when all mathematics was done rhetorically. We're
mostly past this now, but there are lots of hidden language-related issues
involved in developing symbolic representations for verbally-expressed
ideas, even using everyday language. Many functional relationships have
common translations, which need to be recognized by students. Developing a
ready command for this sort of thing, and _maintaining_ this command over
future topics, is important in establishing the relevance of the work in
student eyes.
Only then is it time to take up conditional statements -- equations and
inequalities. The fact that an equation may be able to be _solved_ by
formal methods is also a major discovery, due adequate respect and
consideration. The fact that English _sentences_ also may correspond to
equations is also a key discovery, not an incidental adjunct ("word
problems"). This is the real pay-off, _why_ one is learning algebra in
the first place -- to develop a cleaner way of expressing relationships.
A way that lends itself to formal manipulation and the arrival at
conclusions that would have been difficult to arrive at otherwise. Not to
be overlooked at this point is the translation of the result _back_ into
English -- students seem to get left at the x=.. stage, without ever
getting to the "payoff". Developing a level of comfort with these verbal
skills is also a good part of the mathematical maturity relevant to
technical employment.
Of course I've slighted the _mechanics_ of solution -- the skill of solving
a linear equation, for example. That's a perspective issue again. I'd
say that _formulating_ the equations, _characterizing_ them as to basic
structure and predicting the nature of the solution set, and being able to
_check_ the correctness or acceptability of a proposed solution, are all
more important considerations than being able to jump in and mechanically
tug at this or that term in order to wrestle the equation into solved
form. Of course you've _got_ to do this for a collection of simple
equations in order to develop the basic principles involved. But
traditionally text books would go on from there by making the coefficients
tougher, adding nests of parentheses, etc. to the limit of student patience.
That's just soaking up the available enthusiasm pointlessly! Real examples
don't involve such pointless formalism, but also don't involve the "baby"
numbers that are useful instructionally, and don't succumb to the "solution
by inspection" techniques that are successful in these artificially simple
cases. It would be better to teach real methods of solving real equations,
with all of the messy complexities.
This also goes for the topic of _inequalities_. At this point it would be
possible to address inequalities, but again the _context_ should be
situational. A great many common every-day statements boil down to
inequalities, and some manipulation of the underlying inequality (by simple
algebraic processes) is often involved in reaching a conclusion. If the
appropriate mechanism for mapping between English and algebra has been
adequately developed, this is another big payoff available at this point.
Real methods include the use of graphing, particularly the use of the
powerful electronic graphing utilities available nowadays. Having worked
with linear equations in one variable, it's time to move up to equations
in _two_ variables, and the idea of ordered pairs, and their representation
in the coordinate plane. The general idea of a _relation_, and the idea
that a function can be expressed as a relation (and that it has a graph),
come in. The fact that an equation expressing a relation can (sometimes)
represent a function can be picked up on. General work with linear
equations in two variables, including solution of simultaneous equations,
fits here. And of course the use of multiple variables to represent
multiple unknowns in verbal situations.
Now we can move faster, as we are getting closer to the traditional order
of topics. It is now possible to consider successive classes of algebraic
functions, starting with quadratics, and their special properties.
Powers, roots and exponents (including negative and fractional exponents)
have to be addressed. There would seem to be little doubt that calculator
use here is relevant and helpful in establishing basic understandings.
Somewhere in here it is time to take a more general look at polynomials
(expressions built with addition and multiplication). A certain amount
of facility with manipulation of polynomial expressions by inspection
(including factoring) needs to be developed, but the perspective that a much
greater capability is generally needed, beyond the naive scratching out of
results on paper, has to be maintained.
It would be wrong (IMO) to not include solution of general (too tough to
solve by simple algebra) equations by electronic means (graphing, use of
solvers) even in a first-year algebra course. Understanding of the fact
that equations may or may not have solutions, and some _control_ of this
fact (including knowing what to do with a solution once you have obtained
and verified it), is perhaps the single most valuable understanding I would
hope students to develop in their technical mathematics courses.
That word "technical" -- I am of course speaking from my own educational
corner, which involves teaching students mathematics that supports and
enhances understandings they gain in pursuing career studies in technical
fields. Unlike educators in more traditional settings, where it is
_of course_ understood that students should be learning algebra, and
trigonometry, and calculus, etc. my colleagues and myself are faced with
a real and immediate challenge -- why should students waste any time
studying mathematical techniques when you can do all that nowadays by
pushing a few keys on a $200 device? Of course I have an answer (I think
it is implicit in what I said above), but these same questions are going
to be coming home to roost soon in a strong challenge to the traditional
high-school algebra curriculum. I think I have indicated why I believe
that many of the assumptions on which the content and sequencing of the
traditional curriculum are based are no longer valid. There are other
pespectives that are needed to complement the ones I expressed,
particularly as I haven't addressed any connections with geometry or
preparation for advanced _formal_ mathematical computations, but surely
there are many hands at work on this, and in a decade or two (or three?)
the standard HS mathematics curriculum will hardly be recognizable to
those brought up in this century. I sure hope I am around to see this.
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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