The personal technology of the graphics calculator is presently the only one likely to be available widely enough to influence curriculum design and implementation on a large scale. The algebra curriculum of the past is overburdened with symbolic manipulation at the expense of understanding for most students. But algebra is much more than just symbolic manipulation. Connections between some aspects of algebra: expressing generality, functions, equations and advanced algebra and some graphics calculator capabilities are briefly described. It is suggested that these kinds of connections need to be taken into account in developing the algebra curriculum as well as in classroom teaching

Kissane, B.

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  MURDOCH RESEARCH REPOSITORY  http://researchrepository.murdoch.edu.au/6333/            Kissane, B. (1999) Graphics calculators and algebra. In: 8th South East Asian Conference on Mathematics Education, 30 May - 4 June 1999, Ateneo de Manila University, Quezon City, Philippines.      It is posted here for your personal use. No further distribution is permitted.    Graphics calculators and algebraBarry KissaneThe Australian Institute of EducationMurdoch UniversityMurdoch WA 6150kissane@murdoch.edu.auAbstract: The personal technology of the graphics calculator is presentlythe only one likely to be available widely enough to influence curriculumdesign and implementation on a large scale. The algebra curriculum of thepast is overburdened with symbolic manipulation at the expense ofunderstanding for most students. But algebra is much more than justsymbolic manipulation. Connections between some aspects of algebra:expressing generality, functions, equations and advanced algebra and somegraphics calculator capabilities are briefly described. It is suggested thatthese kinds of connections need to be taken into account in developing thealgebra curriculum as well as in classroom teaching.The main purpose of this paper is to highlight some of the important connections between algebrain the secondary school and currently available graphics calculators. The focus is on theintroductory study of algebra, usually the province of the lower secondary school years in Australia,formally starting in either Year 7 or Year 8 (dependent on the state concerned). In fact, the study ofalgebra starts much earlier, in the primary school, with a focus on important mathematical ideasassociated with patterns and regularities.The most obvious difference between a graphics calculator and a scientific calculator is the smallgraphics screen on the former. One of the several uses of the graphics display screen is to drawgraphs of functions, so graphics calculator are sometimes called 'graphing' calculators, although thisdescription is too restrictive in outlook. Technology of this kind has been around now since the mid1980's. Although they are not yet in widespread use in South East Asian countries, graphicscalculators are now widely used in parts of Australia, North America and Europe. Two distinctivedifferences between graphics calculators and other technologies for school mathematics, such ascomputers, is that the devices were produced mainly for educational use and they are much moreportable. Indeed, it has even been suggested that graphics calculators are the first examples of agenuinely personal technology for school mathematics.Personal technologyAs the phrase suggests, 'personal technology' refers to the technology available to individual pupilson a personal and unrestricted basis. In most parts of South East Asia at present, so-called 'personal'computers are not examples of personal technology in schools, despite the use of the term 'personal'to describe them. Their availability in schools is generally at the collective level, such as in acomputer laboratory, rather than the individual level. In such situations, their use is controlled bythe teacher, rather than by the pupil. They are not permitted for use in examinations, particularlyimportant examinations external to schools. Although some pupils have individual access to apersonal computer at home, many others do not. They are still too expensive for any curriculumauthority to produce curricula based on the assumption of individual ownership. The situation issimilar in Australia, where the mathematics curriculum and common teaching practices of schoolshave been only slightly affected, if at all, by the increased availability of personal computers inschools.In contrast, there have been significant changes in school mathematics in a fairly short period oftime as a consequence of the availability of personal technology. In some Australian states, studentsare permitted – in fact, expected – to use a graphics calculator in important external examinations.A consequence of this is that the classroom experience is affected, with students needing to learnhow and when and why to use a graphics calculator to help them to think about or to domathematics.At present, and for the last twelve years or so, the most mathematically powerful examples ofpersonal technology are graphics calculators. These come in various forms, but at least three aredistinguishable when thinking about algebra. The most powerful are those containing ComputerAlgebra Systems (CAS), such as Texas Instruments' TI-89 and TI-92, Hewlett Packard's HP-48 andCasio's cfx-9970. Given their capabilities, including extensive symbolic manipulation in algebraand calculus, CAS calculators raise a number of significant issues for algebra teaching and learning,some of which are discussed in Kissane, Bradley & Kemp (1997). There is not space in this paperto deal with these issues, although it ought be noted that these sorts of calculators can readilyperform all of the symbolic manipulation expected of secondary school mathematics students. Thisobservation alone suggests that such technologies are worth a closer look.The next set, which I call 'high-end' graphics calculators, are designed to accommodate the needs ofstudents in the later years of schooling and the early undergraduate years. All four manufacturers,Casio, Hewlett Packard, Sharp and Texas Instruments make good examples of these, which aredeservedly becoming quite popular in many senior secondary schools in Australia. Algebraicallyspeaking, they are distinguished by having various automated equation solving capabilities and awider range of function representations (rectangular, parametric, polar and recursive) than the thirdset of graphics calculators, described below. They are probably the most popular graphicscalculators, since they span a wide range of uses over the spectrum of secondary and lowerundergraduate education. Students who acquire a graphics calculator of this kind in the secondaryschool will still find it of use some years later in the early undergraduate years.The third set might be described as 'low-end' graphics calculators. They appear to have beenproduced mainly with middle school pupils in mind and are manufactured by Casio, Sharp andTexas Instruments. Importantly for the notion of personal technology, they are very much lessexpensive than typical high-end calculators, usually somewhere between half and a third of theprice. Indeed, the price of this sort of technology is now comparable with that of scientificcalculators of the kind that have been routinely purchased by many, if not most, Australiansecondary school students for almost two decades. While still not cheap, low-end graphicscalculators are comparable in price with other adolescent purchases in affluent countries, such as apair of shoes or two or three modern CD's, and thus are already affordable to the great majority ofAustralian families. Many schools have found that the most affordable way of introducing graphicscalculators into their curriculum is through the purchase of a class set, so that each student (or pairof students) in a class has regular access to a calculator. In fact, the majority of Australian studentsdo not require a much more powerful calculator than a low-end graphics calculator to meet theirmathematical needs.There is an urgent need to reconsider the secondary school algebra curriculum in the light of whattechnology is potentially available, either through ownership or long-term personal loan, to everysingle pupil. For at least the next few years, it seems likely that only graphics calculators will fitthis description. Curriculum development in schools and school systems throughout South EastAsia ought to be informed about the relationships between this kind of technology and themathematics curriculum.AlgebraMost secondary school students have encountered some algebra in school. Evidence from manysources, over many years, from the anecdotal to the more carefully researched, suggests that themeeting has often been characterised by limited success and even dread (on the part of pupils andteachers alike). The algebra offered by schools, until very recently, appears to most students to havebeen preoccupied with routines for symbolic manipulation, of dubious utility and devoid of muchmeaning beyond the confines of the mathematics classroom. These routines have included the'collection of like terms’, 'expanding', 'simplifying', factorising expressions and solving (aremarkably small repertoire of) equations. Even in 1999, I have no doubt that many studentsinterpret algebra in such a procedural way. Although we have managed to produce a small subset ofpupils with technical competence at such manipulations, very few of these have gained muchinsight into what algebra is (and isn't), what it is for or why it is important. For most students, muchof the time, algebra mainly comprises a collection of symbolic manipulation procedures, rather thanalso including a richly intertwined collection of concepts and strategies. The noted mathematicseducator, Robert Davis, was less than complimentary about such an emphasis:At one extreme, we have the most familiar type of course, where the student is asked to master rituals formanipulating symbols written on paper. The topics in such a course have names like “removing parentheses,”“changing signs,” “collecting like terms,” “simplifying,” and so on. It should be immediately clear that a courseof this type, focussing mainly on meaningless notation, would be entirely inappropriate for elementary schoolchildren; many of us would argue that this type of course, although exceedingly common, is in fact inappropriatefor all students. (1989, p 268)A recent attempt to try to inject meaning into the algebra curriculum and to focus more carefully onthe important ideas of algebra was provided by Lowe et al (1993-4). In part, this work wasinformed by the seminal work in both A National Statement on Mathematics for Australian Schoolsand the National Mathematics Profile, which identified three broad dimensions of algebra andindicated how these might develop over the early years of the algebra curriculum. The three'substrands' of the 'Algebra' strand were labelled 'Expressing generality', 'Functions' and 'Equations'.Later refinements of these documents built upon the same structure, identifying for example that thestudy of functions involves both relationships and graphs, and that inequalities and equations oughtto be considered together.ConnectionsThe essential connection between personal technologies such as graphics calculators and thealgebra curriculum is that the technology provides fresh opportunities for pupils to learn aboutalgebra. The key to these is the capacity of the calculator to enable exploration of key concepts –related to the metaphor of the calculator as a laboratory (Kissane, 1995). Space precludes anexhaustive listing of the kinds of explorations made possible, many of which are contained inpublications such as Kissane (1997). However, a few of the connections are described brieflybelow.Expressing generalityGraphics calculators use the standard conventions of algebraic representation; for example, 2AB2calculator with alphabetic memories storing numbers seems likely to help students come to termswith the notion of variables as place holders. In the same way, 2(A + 1) and 2A + 2 will give thesame numerical value on a calculator, regardless of the value of A, while 2(A + 1) and 2A + 1 will(usually) give different values.Some of these characteristics are shown in the screens below, inwhich A has been given the value 5 and B the value 7. (These and other screens in this paper wereproduced on a Casio cfx-9850G calculator, a good example of a 'high-end' graphics calculator.) Equivalence transformations  such as factorising, collecting like terms and expanding can berepresented either numerically and graphically to enhance meaning. For example, graphs of y = x2 –1 and of y = (x + 1)(x – 1) are identical, as are their associated numerical tables of values. Thecritical concept of algebraic identity is representable both numerically and graphically; in contrast,student work with identities has often been restricted to the symbolic in the past. The followingscreens show some of these kinds of connections.FunctionsAspects of the study of functions are also positively affected by the capabilities of graphicscalculators to represent relationships symbolically, numerically and graphically – the so-called 'ruleof three'. Explorations of functions represented graphically or numerically (in tables) are easilyundertaken by pupils on graphics calculators with minimal prior experience. While movementsamong representations were available before personal technologies like this were invented, theywere frequently hindered by the time and error-prone complexity of producing them (by handgraphing or numerical substitution, or both). Thus, students can now readily produce the family ofgraphs shown below and focus on how and why the graphs differ.As well as movement among representations, personal technology permits pupils to readily explorefamilies of functions and thus the crucial notion of transformation. The screens below show anexample of a graphics calculator used as a function transformer (Kissane, 1997), to permit studentsto see the effects of transforming, in this case by the subtraction of a constant.A graphics calculator allows the algebra curriculum to focus on classes of relationships of obviousimportance that were traditionally neglected until later algebra study (such as exponentialfunctions). Although exponential functions are very important because of their usefulness to modelgrowth situations, they are usually not dealt with in introductory algebra courses, because studentsfind it difficult to deal with them. However, using a graphics calculator, exponential functions areno more difficult to graph or to tabulate than are quadratic functions.Automatic graphical exploration capabilities of calculators allow students to deal numerically withquestions which were previously not accessible until the calculus had been studied. For example,the screens below show how a calculator can locate a relative minimum point of a functiongraphically.With a graphics calculator available, students might be expected to learn to use graphs of functionsrather than merely to draw them; we might even expect that students will decide for themselveswhen and why a graph would be appropriate, rather than relying on teachers and textbooks to tellthem. The difference is of considerable practical importance to the algebra curriculum.Equations and inequalitiesElementary equations and inequalities can be explored profitably by pupils making use of calculatorcapabilities for graphing and numerical tabulation. Unlike the conventional equation-solvingalgorithms using symbolic manipulation, such explorations are not restricted to the linear and thequadratic. Armed with graphics calculators, students might be expected to explore in new waysrelationships between functions, equations and graphs and to develop a repertoire of ways ofdealing with equations and inequalities, rather than the 'one best way' characteristic of the past. (SeeKissane (1995) for an extended example of this.) It would be reasonable to expect that pupils willbe able to solve a particular equation in several ways, and will develop the acumen to choose themost appropriate method for a particular circumstance. To give an elementary example, the screensbelow show two ways in which (approximate) solutions of the cubic equation x3 – 3x = 1 can beobtained on a Casio cfx-9850G.It is clear that the significance of factorising quadratic expressions and of the quadratic formula isaltered by the availability of technology of these kinds.Advanced algebraSome aspects of the study of elementary algebra have traditionally not appeared until relatively latein students' education. However, access to technology may make them accessible much earlier. Anexample concerns dealing with sequences and series. Even the least sophisticated graphicscalculators allow for recursively defined sequences to be generated easily. The calculator screenbelow shows successive terms of an arithmetic progression with common difference 3. Each pressof the "Execute" key has the effect of generating the next term. This process makes it relativelyeasy for students to find directly a certain term or to find which term has a certain value.More sophisticated calculators such as the Casio cfx-9850G allow for series to be numericallyevaluated and even for Fibonacci series to be studied, as shown in the next pair of screens.To give another example, complex numbers are dealt with routinely by the Casio cfx-9850G, asshown below.ConclusionConnections of the kinds illustrated above between elementary algebra and graphics calculatorsdeserve attention in secondary school curriculum development, even in countries in which adequateaccess to technology is not yet available. Although computers are much more powerful forms oftechnology, graphics calculators offer powerful new ways of dealing with the problems traditionallyaddressed by secondary school algebra. It seems important to re-evaluate the significance ofdifferent aspects of school algebra, particularly the focus on symbolic manipulation. It might beargued, for example, that we concentrate more attention than in the past on expressing relationshipsalgebraically rather than on manipulating the expressions themselves; similarly, we may focus moreattention on formulating equations and interpreting solutions, rather than on the algebraicmanipulations required to solve equations. The development of the graphics calculator demandsthat we take a fresh look at the algebra curriculum, how it is taught and how it is learned, under anassumption of continuing and self-directed personal access to technology. Both Lowe et al (1993-4)and Kissane (1997) offer many specific examples of this kind of thinking.ReferencesDavis, Robert 1989. ‘Recent studies in how humans think about algebra’, in Sigrid Wagner &Carolyn Kieran (Eds) Research issues in the learning and teaching of algebra (Volume 4),Reston, VA: National Council of Teachers of Mathematics, pp 266-274.Kissane, B. 1995. Technology in secondary school mathematics – the graphics calculator aspersonal mathematical assistant. In Hunting, R., Fitzsimmons, G., Clarkson, P. & Bishop, A.International Commission on Mathematics Instruction Conference on Regional Collaboration(pp 383-392). Melbourne: Monash University.[Available at  http://wwwstaff.murdoch.edu.au/~kissane/epublications]Kissane, B. 1995. How to solve an equation. Australian Mathematics Teacher 51 (3): 38-42.Kissane, B. 1997. Mathematics with a Graphics Calculator. Casio fx-7400G. Perth:  MathematicalAssociation of Western Australia.Kissane, B. 1997. More Mathematics with a Graphics Calculator. Casiocfx-9850G. Perth:Mathematical Association of Western Australia.Kissane, B., Bradley, J. & Kemp, M. 1997, Symbolic manipulation on a TI-92: New threat orhidden treasures. In N. Scott & H. Hollingsworth (Eds), Mathematics: Creating the future,Melbourne, Australian Association of Mathematics Teachers, 388-396.[Available at  http://wwwstaff.murdoch.edu.au/~kissane/epublications]Lowe, I., Grace, N., Johnston, J., Kissane, B. & Willis, S. 1993-94, Access to Algebra, Melbourne:Curriculum Corporation.Original SourceThis paper is reproduced with permission from:Kissane, B. 1999. 'Graphics calculators and algebra'. In Ester B. Ogena & Evangeline F. Golla(eds),  8th South East Asian Conference on Mathematics Education, Technical papers:mathematics for the 21st century, Ateneo de Manila University (pp 211-219).

How to solve an equation.

Mathematics with a Graphics Calculator. Casio fx-7400G. Perth: Mathematical Association of

More Mathematics with a Graphics Calculator.

Original Source This paper is reproduced with permission from:

Recent studies in how humans think about algebra’, in Sigrid Wagner & Carolyn Kieran (Eds) Research issues in the learning and teaching of algebra (Volume 4),

Symbolic manipulation on a TI-92: New threat or hidden treasures. In

Technology in secondary school mathematics – the graphics calculator as personal mathematical assistant. In

Graphics calculators and algebra

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