Prémio de Melhor Artigo de Jovem Investigador atribuído pela empresa Timberlake, apresentado na 1ª Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação -
CSEI2012, que decorreu no IST nos dias 2 e 3 de Abril.The main goal of this paper is to present some interactive tools, F-Tools, designed by us and implemented with the computer algebra system Mathematica, which we hope will improve the teaching-and-learning process by providing teachers and students alike with new ways to explore some of the main mathematical subjects, at the secondary and university levels, speci cally in the
areas of precalculus and di erential calculus. We believe that these new tools are an important contribution to Mathematical Education, providing new ways for teaching and learning. We intend to make available several F-Tools, such as F-Linear, F-Quadratic, F-Exponential, F-Logarithm, and F-Trigonometric, at the Wolfram Demonstrations Project site

Conceição, Ana C.

Pereira, José

Silva, Cátia

Simão, Cristina

Sapientia

MATHEMATICA IN THE CLASSROOM:NEW TOOLS FOR EXPLORINGPRECALCULUS AND DIFFERENTIALCALCULUSAna C. Conceic¸a˜oDepartamento de Matema´tica, Faculdade de Cieˆncias e TecnologiaUniversidade do Algarve – Campus de Gambelas8000-810 Faro, PortugalJose´ C. PereiraDepartamento de Engenharia Electro´nica e Informa´tica, Faculdade de Cieˆncias e TecnologiaUniversidade do Algarve – Campus de Gambelas8000-810 Faro, PortugalCristina R. Sima˜oDepartamento de Matema´tica, Faculdade de Cieˆncias e TecnologiaUniversidade do Algarve – Campus de Gambelas8000-810 Faro, PortugalCa´tia M. SilvaDepartamento de Matema´tica, Faculdade de Cieˆncias e TecnologiaUniversidade do Algarve – Campus de Gambelas8000-810 Faro, PortugalAbstractThe main goal of this paper is to present some interactive tools, F-Tools, designed by us andimplemented with the computer algebra system Mathematica, which we hope will improve theteaching-and-learning process by providing teachers and students alike with new ways to exploresome of the main mathematical subjects, at the secondary and university levels, specifically in theareas of precalculus and differential calculus. We believe that these new tools are an importantcontribution to Mathematical Education, providing new ways for teaching and learning. Weintend to make available several F-Tools, such as F-Linear, F-Quadratic, F-Exponential, F-Logarithm, and F-Trigonometric, at the Wolfram Demonstrations Project site.Preprint submitted to Elsevier 1 February 2012Key words: F-Tool, Precalculus, Differential Calculus, Teaching-and-learning process,Symbolic Computation, Wolfram Mathematica1. IntroductionIt is not possible to achieve the objectives and skills of a modern mathematics courseat the secondary school without resorting to graphic concepts. These concepts can bemore easily apprehended when the students work with a large number and variety ofgraphics, in an interactive way, with the support of the appropriate technology. Obviously,calculations with the support of technology are not a replacement for paper and pencilcalculations, and they should be properly combined with other methods of calculation,including mental calculation. Students should be prepared for an intelligent dialogue withthe tools they have available.The main goal of this paper is to present some interactive tools, designed by us with thecomputer algebra system Mathematica 1 , which will improve the teaching-and-learningprocess precisely by providing teachers and students alike with new ways to explore someof the main mathematical subjects, at the secondary and university levels, specifically inthe areas of precalculus and differential calculus.The use of the Mathematica software system in the classroom (in Precalculus and Cal-culus) and in research, by some of the authors of this paper (see (Conceic¸a˜o, Kravchenko,Pereira, 2010), (Conceic¸a˜o, Kravchenko, Pereira, 2011), and (Conceic¸a˜o, Kravchenko,Pereira, 2012)) motivated the construction of new tools for exploring the concept of realfunctions and their principal properties, such as the domain, range, existence of zeros,existence of asymptotes, existence of critical points, existence of maximums and mini-mums, existence of inflection points, invertibility, symmetries, and the first and secondderivatives.We created the concept of F-Tool, an interactive Mathematica notebook, specificallydesigned to explore the concept of real functions and their graphics, by analyzing theeffect caused by changing parameters present in the corresponding analytical expression.Going forward, we intend to make available online to the general public, via the WolframDemonstrations Project 2 several F-Tools, such as F-Linear, F-Quadratic, F-Exponential,F-Logarithm, and F-Trigonometric (with the functions sine, cosine, and tangent). Thecorresponding source codes will also be available at the Wolfram Demonstrations website.We believe that these new tools are an important contribution to Mathematics Edu-cation, providing new ways for teaching and learning Mathematics. They also open thepossibility for the development of other interactive tools, based on the symbolic compu-tation capabilities of modern computer algebra systems such as Mathematica, for use atthe elementary, middle, secondary, and university levels.? This research was partly supported by CEAF at Instituto Superior Te´cnico (Portugal), under theproject PEest-OE/MAT/UI4032/2011.Email addresses: aicdoisg@gmail.com (Ana C. Conceic¸a˜o), unidadeimaginaria@gmail.com (Jose´ C.Pereira), cristinasimao@sapo.pt (Cristina R. Sima˜o), catiamasilva@hotmail.com (Ca´tia M. Silva).1 Available at http://www.wolfram.com2 In this project (http://demonstrations.wolfram.com) the creators of Mathematica promote and di-vulge globally the innovations designed by its users.2The rest of this paper is organized as follows.In Section 2 we discuss briefly the modern concept of computer algebra system andexplain the reasons behind our choice to use Mathematica for the construction of ourtools.In Section 3 we present the concept of the F-Tool. We discuss all the functionalitiesavailable within each F-Tool, and how the dynamism and interactivity of these tools playa major role in the illustration of many precalculus and calculus concepts.In Section 4, we discuss the designing of the F-Tools and present the main charac-teristics specific to each of the F-Tools. We also explain some of the major symboliccomputation aspects of the code itself.In Section 5, we make some remarks on how we intend, going forward, to extend theconcept of F-Tool to other school levels and to other areas of Science. We also discusshow we plan to try and generalize the use of these tools in the classroom.2. The Computer Algebra System MathematicaIn recent years, several software applications with extensive capabilities of symbolicand numeric computation were made available to the general public. These applicationsreceive the generic designation of Computer Algebra Systems (CAS).Symbolic manipulations supported by CAS typically include, but are not restrictedto, the simplification of mathematical expressions in reference to a given canonical for-mat, the analytical resolution of several types of equations and inequalities, symbolicdifferentiation and integration, or operations with vectors and matrices.Also, in alignment with other major evolutions of Information Technology, many CASinclude additional functionalities such as the use of a numeric system with infinite pre-cision or the use of their own programming language to enhance the design of perfectlyintegrated applications.All these capabilities make CAS a tool with increasing usage in many areas of Sci-ence and Technology. Today there is a real and significant effort to try and delegate todigital computers many of the decision making processes that mathematicians and otherresearchers usually perform with the traditional tools pencil and paper.Wolfram’s Mathematica is a powerful CAS used in scientific, engineering, and math-ematical fields and in other areas of technical computing. In what concerns the workpresented in this paper, Mathematica graphics are completely integrated into its dynamicinteractive language. Any visualization can immediately be animated or made interactiveusing a single command and developed into sophisticated, dynamic visual applications.Creating interactive visual models with Mathematica allows students to explore hard-to-understand concepts, test theories, and quickly gain a deeper understanding of thematerials being taught firsthand. The students can explore changes to text, functions,formulas, matrices, graphics, tables, or even data.In our work, the concept of F-Tool arose naturally from the dynamic, interactive, andvisual capabilities of Mathematica. The F-Tool provides rigorous analytical informationon various classes of functions and concepts associated with their study, allowing simulta-neously students to visualize several features, change values and graphics, and see resultsin real time.As the Wolfram Education Group wittingly states: “Calculators are for calculating,Mathematica is for Calculus”.33. The F-Tool conceptThe F-Tool is a new concept that allows to explore in a dynamic, interactive, andvisual manner various classes of real functions and features associated with their study.In Figure 1 we can see a usage example of the F-Exponential, one of the alreadyexisting F-Tools, designed to explore a class of functions of the exponential type. Sinceall the F-Tools share a common design, we will use this example to present the mainfeatures of the F-Tool concept.Fig. 1. Usage example of the F-Exponential tool.A F-Tool is divided in three main panels:Left Panel : In this panel the user can vary the values of one or more parameters,choose which “transformations” of the main function are to be presented,and wether or not to show the tangent line in a chosen tangent point.In addition, the user has the option to see the results in the exact orapproximate forms.Middle Panel : In this panel all the functions are plotted, according to the options madeby the user in the left panel. For the main function it is also possible tosee, in an interactive way, the coordinates of the corresponding graphicalpoints (click-and-drag the ⊕ locator placed over the function line).Right Panel : In this panel it is presented all the analytical information concerning themain function and its “transformations”, again in accordance with theoptions chosen by the user in the left panel.4Precalculus and differential calculus, are concerned with the study of functions andtheir main properties. The functionalities of each F-Tool, along with the displayed in-formation, are all designed to conform with the way those subjects are presented in theclassroom, at the secondary and university levels. One of the main contributions for thisdesign option was the sharing of ideas with two of the co-authors of this paper, whoare currently master students in Mathematics Education and are actively enrolled inpreservice teaching.As mentioned in Section 2, all the graphical and analytical information presented inthe middle and right panels is displayed in real time, thus rendering the F-Tools aneminently dynamic tool. In particular, the Mathematica controls allow the automaticcontinuous change of the parameter values (the I button depicted, for instance, inthe tangent line option in Figure 1). When choosing this option, the user will then seethe corresponding graphics move continuously and the analytical information change inaccordance.We believe that this kind of dynamic interaction, by providing teachers and studentsalike with new ways to explore some of the main subjects in the areas of precalculusand differential calculus, renders the F-Tool concept an important contribution to Math-ematics Education, improving the teaching-and-learning process.4. Designing the F-ToolIn this section we discuss how the symbolic computation capabilities of Mathematica,and its own programming language, along with the pretty-print 3 functionality, allowedus to implement on a computer, and in a rather straightforward manner, all the ideasthat go into the F-Tool concept.Each and every F-Tool has a common general code structure, depicted in Figure 2,which is actually very simple.Fig. 2. General code structure of the F-Tool3 Pretty-print is a functionality that allows to write mathematical expressions on the computer usingthe traditional notation, as if on paper.5The code consists of some initial definitions that pertain to each particular class ofreal function, followed by a single command Manipulate. This command is responsiblefor creating the interactive object that contains the three panels described in Section 3.In particular, Manipulate generates all the functional controls, such as sliders for the pa-rameter values and checkboxes for the plot options. Also, integrating this command witha DynamicModule[· · ·] is what allow us to create, for instance, the locator functionality,available in the middle panel, by enabling the dynamic update of selected variables.In summary, each F-Tool is essentially created by a single Manipulate command,whose output is not just a static result but a running program that we can interact with.4.1. F-LinearThe class of linear functions is the simplest family of functions to become acquaintedwith. Accordingly, F-Linear is, code wise, the simplest of the existing F-tools.Linear functions model constant rates of change of a given quantity and can be writtenin the general form f(x) = ax + b, a, b ∈ R.This class of functions includes the constant functions, for when a = 0 (see Figure 3).In spite of their simplicity, constant functions have to be dealt separately because theyhave no inverse function, and their range and set of zeros are completely different fromother linear functions.Fig. 3. The case a = 0 for the F-Linear tool.We note that, constant functions are a common element to all the classes of functionsdiscussed in this paper. This means that, in all F-Tools, the constant case had to be codedseparately, in order to generate the correct analytical information for those functions.4.2. F-QuadraticQuadratic functions can be written in the standard form f(x) = ax2+bx+c, a, b, c ∈ R.However, when studying quadratic functions there are many advantages in using thederived vertex form f(x) = a(x− h)2 + k.F-Quadratic uses the latter form mainly because the parameters a, h, and k have amore direct relation with the graphical transformations that we intend to explore withthis tool. Nonetheless, because of its general importance the quadratic standard form is6always displayed in the right panel of F-Quadratic , as depicted in Figure 4. We notethat the vertex form does not include the class of polynomials of degree one. In fact, thiscase can be more properly explored with the F-Linear tool.Quadratic functions have a multivalent inverse function, which is not a topic discussedin most courses at the secondary and university levels. Therefore, the F-Quadratic tooldoes not make available the inverse function option.Fig. 4. Usage example of the F-Quadratic tool.In Figure 5 it is depicted the block of code that generates the analytical informationabout the vertex and extreme values which are characteristic of any quadratic function.Note the need to deal separately with the constant case (a = 0), as already mentionedin Subsection 4.1.Fig. 5. Code snippet of the F-Quadratic tool. This block of code generates the analytical infor-mation about the vertex and extreme values.74.3. F-ExponentialThe F-Exponential tool explores the class of functions of exponential type, that canbe written in the general form f(x) = a eb(x−c) + d, a, b, c, d ∈ R.One good usage example for this tool is when we choose to plot the inverse function,as depicted in Figure 6. In this case, when using exact arithmetic, the exact analyticalexpressions of the function and its inverse are presented. The user can then easily verify,for instance, the composition law f−1 ((f(x)) = f(f−1(x))= x. The dashed line seenon the plot of Figure 6 has the equation y = x and corresponds to the symmetry axis ofthe inverse transformation.Fig. 6. Usage example of the F-Exponential tool. Note the computation of the inverse functionin the exact form.In Figure 7 it is depicted the block of code that generates the analytical informationabout the zeros of the exponential type function. Note the several conditionals, necessarydue to all the degrees of freedom that the general expression possesses.Fig. 7. Code snippet of the F-Exponential tool. This block of code generates the analyticalinformation about the zeros of the exponential type function.4.4. F-LogarithmThe F-Logarithm tool explores the class of functions of logarithmic type, that can bewritten in the general form f(x) = a ln(b(x− c)) + d, a, b, c, d ∈ R8In Figure 8 it is depicted the block of code that generates the analytical informationabout the domain of the logarithmic type function. Contrary to the other classes offunctions already discussed, this logarithmic class does not possess the trivial domain R.Fig. 8. Code snippet of the F-Logarithm tool. This block of code generates the analytical infor-mation about the domain of the logarithmic type function.Note that the domain is only computed when b 6= 0. In fact, for b = 0 the correspondingfunction is not well defined and, in this case, the user will receive the error messagedepicted in Figure 9. Simultaneously the user will hear the sentence “Sorry. I cannot dothat!”, which is generated with a simple command Speak. As a consequence, all optionswill be unavailable until any other value is chosen for the parameter b.Fig. 9. Error message for the case b = 0 in the F-Logarithm.4.5. F-TrigonometricThe F-Trigonometric tool explores the class of functions of trigonometric type, thatincludes the functions Sine, Cosine, and Tangent. Because of the similarity between the9F-Tools that explore those three kinds of functions, we will only discuss the F-Sine toolin this section.The F-Sine tool explores the class of functions that can be written in the general formf(x) = a sin(b(x− c)) + d, a, b, c, d ∈ R.In Figure 10 it is depicted a usage example for the F-Sine tool where both the func-tion and its first derivative are plotted. Also worth noting is the analytic informationpertaining the inflection points, which is not relevant for any of the previously discussedF-Tools. For the same reasons considered in the F-Quadratic tool, the inverse functionoption is not present in F-Sine.Fig. 10. Usage example for the F-Sine tool.5. Final remarksWe believe that the F-Tool concept is an important contribution to MathematicsEducation, providing new ways for teaching and learning Mathematics. It also opensthe possibility for the development of other interactive tools, based on the symboliccomputation capabilities of modern computer algebra systems such as Mathematica, foruse at the elementary, middle, secondary, and university levels.Going forward, we would like to see the usage of the F-Tools in the classroom becomeincreasingly more common. One important step to achieve this goal would be the real-ization of a study in loco, statistically rigorous, to estimate the real pedagogical value ofthe F-Tools.10AcknowledgementsThe authors thank Ricardo Pereira (Agrupamento de Escolas Jose´ Belchior Viegas,Portugal) for his early contributions to the genesis of this project.ReferencesConceic¸a˜o, A. C., Kravchenko, V. G., and Pereira, J. C., 2010. Factorization Algorithmfor Some Special Non rational Matrix Functions. Oper. Theory Adv. Appl. 202, 87-109.Birkha¨user VerlagConceic¸a˜o, A. C., Kravchenko, V. G., and Pereira, J. C., 2011. Computing some classesof singular integrals with Mathematica software. Submitted in Adv. Comput. Math.Conceic¸a˜o, A. C., Kravchenko, V. G., and Pereira, J. C., 2012. Rational Functions Fac-torization Algorithm: a symbolic computation for the scalar and matricial cases. Sub-mitted in J. Symbolic Comput.11

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