[ver] 4 [sty] _DEFAULT.STY [files] [charset] 82 ANSI (Windows, IBM CP 1252) [revisions] 0 [prn] Lexmark Optra R [port] LPT1: [lang] 1 [desc] 833940762 8 832987409 347 5 0 0 0 0 1 [fopts] 0 1 0 0 [lnopts] 2 Body Text 1 [docopts] 5 2 [GramStyle] Academic Writing [ParaNum] 1 [tag] Body Text 2 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 216 0 [spc] 33 288 1 0 72 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Text 0 0 [tag] Body Single 3 [fnt] Times New Roman 200 0 49152 [algn] 24 1 288 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Single 0 288 [tag] Bullet 1 5 [fnt] Times New Roman 240 0 49152 [algn] 1 1 288 288 288 [spc] 33 288 1 0 144 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*5> 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Bullet 1 0 0 [tag] Number List 6 [fnt] Times New Roman 200 0 49152 [algn] 17 1 0 0 0 [spc] 33 288 1 0 144 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*:>. 360 1 1 0 16 0 0 [nfmt] 272 1 2 . , $ Number List 0 0 [tag] Subhead 7 [fnt] News Gothic MT 240 0 16385 [algn] 1 1 0 0 0 [spc] 33 288 1 216 0 1 100 [brk] 4 [line] 8 0 1 0 1 2 1 20 10 1 [spec] 0 2 0 1 1 0 0 0 0 [nfmt] 272 1 2 . , $ Subhead 0 0 [tag] Header 9 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 10 0 1 0 1 1 4 10 80 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Header 0 0 [tag] Footer 11 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 9 0 1 0 1 4 1 80 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Footer 0 0 [tag] Footnote 12 [fnt] Times New Roman 200 0 49152 [algn] 1 1 0 0 0 [spc] 34 360 1 0 72 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*?> 0 1 1 256 0 0 0 [nfmt] 280 1 2 . , $ Footnote 0 0 [frm] 1 25166464 1440 1177 8080 6688 0 1 3 4 0 0 0 0 0 12779519 0 16777215 3 [frmname] Frame3 [frmlay] 6688 6640 1 0 0 1 1249 216 0 2 0 0 0 0 1 1440 7864 0 [txt] <:s> <:s> <:s> <:s> <+B><:f360,BArial,0,0,0><+"><+!>Quo Vadis <-">Algebra?<-!><:f> <+B><:s> <+B><:s> <+B><:s> <+B><:s> <+B><:s> <:s> <:f240,BArial,0,0,0>By Lin McMullin <:f><:f240,BArial,0,0,0>Burnt Hills - Ballston Lake High School<:f> > [frm] 1 8389184 1440 1206 8092 2372 0 1 3 4 0 0 0 0 0 12779519 0 8421504 8 [frmlay] 2372 6652 1 144 0 1 1278 144 0 2 0 12 97 32780 1 1584 7953 0 [txt] <+A><:s><:f360,BArial,255,255,255><+!>The Fifth Conference on <+A><+!><:f360,BArial,255,255,255>the Teaching of Mathematics<-!><:f> > [frm] 1 42009088 1440 3452 8064 5080 24 1 3 3 0 0 0 0 0 12779519 0 16777215 7 [frmlay] 5080 6624 1 0 0 1 3596 0 0 2 0 12 97 32780 1 1440 8063 0 [txt] <+@><:I0,0,0,0><:f280,BArial,>The TI-92 can make the analytic perspective<:f><:f280,BArial,> a full partner<:f><:f280,BArial,> with the numerical and graphical. <:f><:f280,BArial,>We have a machine that<:f><:f280,BArial,> not only can do symbolic manipulation, but can also teach it.<:f> > [frm] 1 704 8033 1163 10777 6141 0 1 3 4 0 0 0 0 0 12779519 0 16777215 6 [frmlay] 6141 2744 1 0 0 1 1235 0 0 2 0 12 97 32780 1 8033 10777 0 [txt] > [lay] Standard 513 [rght] 15840 12240 1 1440 1440 1 1440 1440 0 1 0 1 0 2 2 1440 5962 12 1 720 1 1440 1 2160 1 2880 1 3600 1 4320 1 5040 1 5760 1 6480 1 7200 1 7920 1 8640 6278 10800 0 [hrght] [lyfrm] 1 11200 0 0 12240 1440 0 1 3 1 0 0 0 0 0 0 0 0 1 [frmlay] 1440 12240 1 1440 72 1 792 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] @Header@<:s> @Header@<:s> @Header@ > [frght] [lyfrm] 1 13248 0 14400 12240 15840 0 1 3 1 0 0 0 0 0 0 0 0 2 [frmlay] 15840 12240 1 1440 792 1 14472 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] @Footer@ <:P12,2,> > [elay] [l1] 0 [pg] 5 8 88 42 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 27 246 45 0 0 0 0 65535 65535 Standard 65535 0 1 1 216 17 0 0 0 65534 65535 1 1 65535 0 0 0 0 0 38 473 41 0 0 0 0 65535 65535 Standard 65535 0 2 1 216 17 0 0 0 65534 65535 2 1 65535 0 0 0 0 0 49 268 46 0 2 0 0 65535 65535 Standard 65535 0 2 1 113 61 0 0 0 65534 65535 2 2 65535 0 0 0 0 0 70 0 43 1025 0 0 0 65535 65535 Standard 65535 0 2 1 113 61 0 0 0 65534 65535 2 2 65535 0 0 0 0 0 [edoc] <:s><:#1008,4522> <-!><+!><:f360,2Times New Roman,0,0,0>B<-!><:f>efore I venture to predict where algebra is going, a brief look backwards: <:s><:#1728,4522>I told my mother, who is 92, about the work I had been doing with the TI-92 and explained to her very briefly what it could do. She immediately asked me If it does everything for the kids how are they going to learn to do their math? <:#1728,4522>We have heard this question before, first when the original arithmetic calculators came out (the ones that could add, subtract, multiply and divide real fast and cost (only) $50). Wont the kids be at a great disadvantage if they cant do their arithm etic by hand? A lot of us had trouble answering that question or at least convincing ourselves that we believed our answer. When scientific calculators came on the market and we didnt have to teach computations with logarithms and interpolation in Trig tables an y more, we could point to all the other, better stuff we could teach instead. Then came graphing calculators: now we could go on the offensive and show our critics how the study of mathematics benefited from the ability to quickly produce many accur ate graphs and tables of values. Finally we could really look at the problems from multiple perspectives: numerically and graphically (and, oh yes, analytically). <:s><:#1440,4522>I think the analysis was a little neglected in the excitement generated by graphing calculators. Perhaps this was only fair since numbers and graphs had, of necessity, been neglected for so long. <:s><:#2304,4522>We are now at the threshold of the next generation of technology. I am talking especially of the TI-92, but also of the various symbol manipulation programs available for computers. This new generation of machines gives us and our students the ability to do the analytic part the manipulation of symbols quickly and accurately. <:#1152,4522>And we come full circle to the question: Is nothing sacred? Surely students <+">must<-"> be able to simplify, to factor and to solve equations on their own, mustnt they? <:#2592,4522>Im sorry, I dont know the answer to that. I do know that the learning and teaching of algebra will change again. It will change because of these machines. I think it is a change for the better. I would like to discuss, however briefly, the new direc tion in the teaching of mathematics that this machine presages. But first .... @Subhead@<:s><:#331,4522><:f280,,>What is Algebra?<:f> <:s><:#1584,4522><-!> <+!><:f360,2Times New Roman,0,0,0>M<-!><:f>ost people associate algebra with manipulating symbols. Here are some definitions of algebra from current, non-mathematical, sources: @Body Single@<:#2160,4522>Algebra branch of mathematics in which letters are used to represent basic arithmetic relations.... Classical algebra, which is concerned with solving equations, uses symbols instead of specific numbers and employs arithmetic operations to establish pr ocedures for manipulating symbols .... <[>I]n its most general form algebra may fairly be described as the language of mathematics. <[>8] @Body Single@<:s><:#240,4522> @Body Single@<:s><:#720,4522>Algebra <+"> <-">a branch of mathematics dealing with properties of numbers and quantities by means of letters and other general symbols. <[>3] @Body Single@<:s><:#240,4522> @Body Single@<:s><:#960,4522>A generalization of arithmetic in which symbols, usually letters of the alphabet, represent numbers or members of a specified set of numbers.... <[>1] @Body Single@<:s><:#240,4522> <:s>To most people Algebra is symbol manipulation. This is what the dictionaries and encyclopedia say, this is what they remember from school and college.<:F @Footnote@One of my calculus students, after giving a very nice answer based on the calculator graph (nicely sketched on the paper) added This also works when tested mathematically. She then went on and reworked the problem mathematically. > <:s><:#1440,4522>This being the case one modern journalist, writing for teens, quipped Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra. <[>4] <:s><:#1152,4522>With the introduction of graphing utilities the interest and emphasis shifted. A recent beginning algebra text states that its version of algebra @Body Single@<:#1680,4522>... emphasizes mathematical models and representations, variable and functions, symbolic reasoning rather than symbolic manipulation .... conceptual rather than procedural knowledge .... <[>S]tudents develop their understanding of the<[>se] central concept s and processes<[>:] @Body Single@<:s><:R1,12,1,720,1,1620,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#480,4522> Identifying variables and relationships among them @Body Single@<:s><:R1,12,1,720,1,1620,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#720,4522> Representing the relations among variable in numerical, graphical and symbolic forms @Body Single@<:s><:R1,12,1,720,1,1620,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#480,4522> Drawing inferences about modeled relations @Body Single@<:R1,12,1,720,1,1620,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#720,4522> Recognizing limitations in applications of mathematical models to real-life situations. <[>3,xii-xii] @Body Single@<:s><:R1,12,1,720,1,1620,1,2160,1,2880,1,3600,1,4320,1,5040,1,5760,1,6480,1,7200,1,7920,1,8640,><:#240,4522> <:s><:R><:#4608,4522>It is not that one cannot learn mathematics from a strictly analytic point of view, but rather that one can learn <+">better<-"> from multiple perspectives. For years only the symbolic approach was used. In a practical sense it was the only approach that <+">could<-"> be used. When graphing calculators became available (barely six years ago) there was a big shift in emphasis to studying problems numerically and graphically. This was both because it could not be done previously (at least not easily) and beca use the new calculators did not do the symbol manipulation. Graphing calculators allow us to study problems much more easily from a numerical and graphical approach. It was a practical way to go. <:s><:#3744,4522>The change in direction in mathematics instruction was welcome. Symbol manipulation is tough, detailed, complicated and allows no leeway. It required lots of drill. To teach and learn symbol manipulation required breaking the material into many, many small pieces. There were so many pieces that we needed years to teach it and students needed years to learn it. Because of that the big picture was lost; algebra became <+">only<-"> symbol manipulation in the minds and experience of most people. In real life there was no such thing as algebra. @Subhead@<:s><:#331,4522><:f280,,>Today<:f> <:#4896,4522><-!><+!><:f360,2Times New Roman,0,0,0> W<-!><:f>e can get tables of values and lots of very accurate graphs. Where does that leave the symbols? In the text cited above symbolic work (not really symbol manipulation) is presented in the final two chapte rs and there, only with technological approaches to develop a <+">symbol sense<-">, that is graphical, numerical and symbolic meaning for the algebraic symbols. <[>3,xii] In a report in <+">The Mathematics Teacher<-"> two of the authors of that text report that some teachers have followed <[>this program] with some <[>instruction in] traditional symbolic manipulation skills for six weeks to two months. <[>5,655] <:s><:#3168,4522>That is one possible approach. The high-school student of today, armed with a graphing calculator can learn, understand and appreciate the value of algebra by working and doing mathematics without the abstraction, sophistication and hard work required to ma nipulate symbols. You really do not need to know how a motor works before you drive a car. Those (few) who do go on to become mathematicians can learn the symbolic work later. @Subhead@<:s><:#288,4522>But.... <:s><:#3024,4522> <:f360,,><+!>I<-!><:f> am not sure it is necessary to segregate the symbol manipulation instruction from the rest of beginning (high school) algebra. The TI-92 and symbolic algebra computer programs can also be used to teach symbol use and symbol man ipulation <+">from the first days<-"> of algebra instruction. The challenge is to find ways to make and keep analysis an equal partner with the numbers and graphs. <:s><:#2016,4522>I think this can be done. I want to share with you a few ideas on how to do it. Here are just a few examples showing how the use of symbols can be handled in a beginning algebra curriculum using a TI-92. The appendix contains the four discovery exercises th at are discussed below. @Subhead@<:s><:#288,4522><:f240,,>Example 1: Adding the Quick Way<:f> <:s><:#4176,4522> <:f360,,><+!>O<-!><:f>ne possible approach is demonstrated by the first example Adding the Quick Way. (See below p. 6 - 8)<[>6] The students are asked to enter various simple expressions on the TI-92: in this case adding a string of identical variables. The TI-92 simplifies the expressions and the student is asked to figure out what the machine is doing and why. This is followed by work with adding and subtracting like terms, then the distributive property and finally writing sums as products (factoring). Students are asked to discover what the rules are and to express their unde rstanding in brief sentences. <:s><:#576,4522>A similar approach may be used with the rules of exponents. @Subhead@<:#288,4522>Example 2: Step-by-Step <:f360,,><+!>A<-!><:f>nother approach is illustrated by Step-by-Step. (See below p. 9 - 12) <[>6] This exploration leads students to solve equations on the TI-92 as they would on paper. After the equation is entered, the TI-92 allows the stud ents to subtract 3x from both sides, then add 15 to both sides, then finally divide both sides by 4. The student still needs to know what to do, but the TI-92 does it. The student tells the TI-92 what to do next. <:F @Footnote@ The first time I tried this approach I was teaching a ninth grade pre-algebra class. One of my students became quite incensed the he should have to think <+">for<-"> the TI-92! > W all know that most of the time spent solving equations is really spent being sure the students are doing the arithmetic correctly. With this approach, students can solve equations with decimals, fractions and radicals from the start. They must know the pr ocedure, and the procedure is, after all, what we are trying to teach them. The TI-92 is used to practice the procedure. @Subhead@<:s><:#576,4522>Example 3: Factoring to Solve Equations <:#3312,4522> <:f360,,><+!>T<-!><:f>he third example shows how a similar idea can be applied to equations that require factoring. (See below p. 13 - 16) <[>6] In the first exploration the students learn the relationship between the TI-92s <+">expand(<-"> and <+">factor(<-"><+!> <-!>operations <-"><-">(note question 12 that asks the student to explain the relationship in his or her own words). Then, in the next exploration, the use of the factored form is demonstrated by using it to find the solution of equations. <:#1440,4522>In both of the examples the TI-92s <+">solve(<-"> operation is purposely <+">not<-"> used. The purpose is not to solve the equations but to develop an understanding of what a solution really is, where it comes from, and what to do to get it. @Subhead@<:s><:#288,4522>Example 4: Limits <:s><:#4464,4522> <:f360,,><-!>L<:f>ater in the curriculum the TI-92 can be used in a similar way to learn about limits. The symbolic approach and the numerical and graphical approach enhance on another. This final example concerns limits and is intended for third year high school or pre-calculus students. (See below p. 17 - 22) <[>7] After explaining how to use the various forms of the <+">limit(<-"> operation on the TI-92, the student is asked to compare the <+">limit <-">of a function with its <+">value<-"> its <+">graph<-">. The student is asked not only to find the limit but to discuss (write) what the <+">limit<-"> tells about the <+">graph<-"> and what the <+">graph<-"> tells about the <+">limit<-">. @Subhead@<:s><:#288,4522>Ways to use a calculator. <:s><:#1296,4522> <:f360,,><+!>T<-!><:f>here are three basic ways the TI-92 or a symbolic processor may be used in an instructional setting: @Bullet 1@<:s><:#2016,4522>As a tool: Use the built in operations to get an answer add, find the sine, draw a graph or find a derivative. The student needs only relate the problem to the proper buttons. The calculator is simply a tool and its full potential in instruction and l earning is not realized. @Bullet 1@<:s><:#2592,4522>To explore a situation or problem the built in applications are used together to explore some idea or concept. Several applications are used together in a true multiple perspectives approach. The exploration is directed by the user. This can only be do ne after the student has some understanding of the various applications and their interrelationships. @Bullet 1@<:s><:#1728,4522>As a source of data about symbols: The TI-92 uses the symbols correctly and the student uses this information to learn about the use of symbols. This can be done from the first days of beginning algebra. <:s><:#2592,4522>At first blush the calculator-as-tool may be all people see. The latter two approaches are really where the TI-92 has its place in the teaching of mathematics. It is the two latter approaches that I have tried to develop in the two books from which the exa mples were taken: <+">Discovering Mathematics with a TI-92 Beginning Algebra<-"> and <+">Discovering Mathematics with a TI-92 Functions<-">. <:s><:#2304,4522>The student is asked to discover and explain the results on the screen with the hope he or she will develop and understanding of what is happening (as in Adding the Quick Way). The machine handles the details and gives the correct result. The student uses the results as data to develop an understanding of the concept. <:s><:#2592,4522>The Limit example illustrates how the symbol work and resulting symbol sense can and should be developed along with the numerical and graphical sense. The numerical, analytic and graphical viewpoints lead the student to a fuller understanding of the concept. Having correct data about symbols lets the student con centrate on the concepts behind the symbols. <:#3168,4522>The other two examples (Step-by-Step and the comparison of <+">expand( <-">and <+">factor(<-"> ) show the student how to do the problem, not necessarily by the quickest method, but in a way that will make the procedure and the ideas behind the problem clear. The student directs the work the TI-92. This means the st udent must know what to do even though the machine is doing the work. The user can concentrate on the mathematics and let the machine handle the symbol manipulation. @Subhead@<:s><:#288,4522>Summary NAG the Problem <:#3024,4522> <:f360,,><+!>W<-!><:f>e all know that mathematics is best learned when problems are<+!> NAG<-!>ed: that is when they are studied from the <+!>N<-!>umerical, <+!>A<-!>nalytic and <+!>G<-!>raphical perspectives. Graphing calculators are excellent tools for numerical and graphical study. Now we have available good, inexpensive machines and computer programs for the manipulation of symbols making the analytic viewpoint equal with the others. <:s><:#2304,4522>We considered here how to use them to help young people learn algebra. The TI-92 can be used to let students discover the simplest and most fundamental procedures of symbolic manipulation. Much higher level work can be done on the TI-92, but do not overlook the beginning work. The TI-92 can be used to teach and learn basic algebra also. <:s><:#288,4522> <:s><:#288,4522> <:s><:#288,4522> <:s><:#288,4522> @Footer@<:s><:#368,4522> @Subhead@<:s><:#288,4522>Sources @Subhead@<:s><:#288,4522> @Number List@<:s><:#960,4522><+">The American Heritage Dictionary of the English Language<-">, 1992 by Houghton Mifflin Company. All rights reserved. Microsoft Bookshelf 1987 - 1994 Microsoft Corp. @Number List@<:s><:#720,4522><+">The Concise Oxford Dictionary of Current English <-"> The Oxford University Press, 1982, Microsoft Bookshelf 1987 - 1994 Microsoft Corp. @Number List@<:#1440,4522>Fey, James T, Heid, M. Kathleen, <+">et al.,<-"> <+">Concepts in Algebra A Technological Approach<-">, 1995, Janson Publications, Inc. While I have not taught from this text or used the CIA program I do not mean to denigrate it. It is, in my opinion, an excellent approach. @Number List@<:#960,4522>Fran Lebowitz <+">Social Studies<-">, Tips for Teens (1981) quoted in<+"> The Columbia Dictionary of Quotations, <-">Microsoft Bookshelf 1987 - 1994 Microsoft Corp. @Number List@<:#720,4522>Heid, Kathleen M, and Zbiek, Rose Mary, A Technology-Intensive Approach to Algebra, <+">The Mathematics Teacher,<-"> November 1995. @Number List@<:s><:#720,4522>McMullin, Lin,<+"> <-"><+">Discovering Mathematics with a TI-92 Beginning Algebra<-">, D & S Marketing Systems, Inc. Brooklyn New York. @Number List@<:s><:#720,4522>McMullin, Lin,<+"> <-"><+">Discovering Mathematics with a TI-92 Functions<-">, D & S Marketing Systems, Inc. Brooklyn New York. @Number List@<:#240,4522>Microsoft Encarta 1993 Microsoft Corp.<-!> > [Embedded] 00024414