No attempt was made to list the topics in any logical order.
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.
Yes, I'm tempted to invest about $400 in MultiMedia Math CDRom which is
supposed
to discuss the concept of function in great detail. Is this the kind of
concept
that you are thinking about?
>
> 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_.
Thank you. This is what should be item #1. 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!
I'm not sure I follow you here. Could you elaborate?
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.
I'm not sure that the TI-92 should be available in high school Algebra.
>
> Manipulating simple algebraic expressions, getting a _feel_ for how they
> work (yes, order of operation and use of parentheses),
I forgot to include the Order of operations and grouping symbols, but
they
will be added.
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.
By lists, I assume you mean a data table of x,y values, right??
>
> 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.
Sorry, but in my high school days, I was able to understand powers,
roots
and exponents and how to manipulate them; It just took me a little
longer
that's all. I don't think for a minute that a calculator would have,
in any way, been able to help me in understanding these ideas.
Just as a trig tables doesn't help one to understand the tangent ratio
any
better, I don't think a calculator can help me to understand Powers any
better.
The slide rule that I used in the fifties helped me calculate
quantities,
but it didn't help me to understand the process any better than if I had
to do the calculations by hand. I was always able to "hand calculate"
most
expressions and equations. It was fun to use the slide rule, but as to
aiding
in my understanding, I don't think so.
Am I in the minority here?
>
> 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.
Very well stated!
>
> 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?
By pushing "just a few keys" on the TI-92, do you honestly believe
that students are really learning and truly appreciating the power and
the
elegance of mathematics??
I don't think anyone could answer the above question in the
affirmative.
>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.
>
>
Mr. Taylor,
Very well summarized; I truly appreciate the time you took to write your
response.
It's worth "saving" and rereading.
Are you planning any monographs in the future; if yes, add my name to
your
potential list of subscribers.
Thanks again,
The Old Pro
Jim C. Gajniak
P.S. By the way, what is the plural of "mongoose"?
> 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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