Re: [MATHEDCC] RE: Curriculum Guide for H.S. Algebra 1
David E Miller (dmiller@IVY.TEC.IN.US)
Thu, 5 Mar 1998 12:45:10 +0500
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>Date: Tue, 03 Mar 1998 18:47:24 -0800
>From: The Old Pro <jgajniak@IX.NETCOM.COM>
>To: RWW Taylor <RWTNTS@RITVAX.ISC.RIT.EDU>
>Cc: math ed cc <mathedcc@archives.math.utk.edu>
>Subject: Re: [MATHEDCC] RE: Curriculum Guide for H.S. Algebra 1
>References: <01IU8B7MSHNAEE6XLO@ritvax.isc.rit.edu>
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>Reply-To: The Old Pro <jgajniak@IX.NETCOM.COM>
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>
>RWW Taylor wrote:
>>
>> dir *.wpc
>>
>> 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.
>
>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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