Maple Computer Algebra System

Πρόλογος
Στόχος αυτού του μικρού φυλλαδίου  είναι να δώσει μια συμπαγή αλλά ουσιαστική εισαγωγή στις αναδρομές, με γέφυρες από τα σχολικά μαθηματικά προς βαθύτερες ιδέες. Περιλαμβάνονται πλήρεις αποδείξεις και λυμένες ασκήσεις, ώστε να μπορεί να χρησιμοποιηθεί σε μάθημα ή ως δωρεάν ηλεκτρονική διάθεση.

Οι συναρτησιακές εξισώσεις αποτελούν μία από τις πιο ενδιαφέρουσες περιοχές των μαθηματικών, καθώς συνδέουν άμεσα την αλγεβρική σκέψη με την αναλυτική μέθοδο. Στο παρόν σύγγραμμα συγκεντρώνονται βασικά θεωρήματα, χαρακτηριστικές κλασικές εξισώσεις (Cauchy, Jensen, d’Alembert), καθώς και μία πλούσια συλλογή λυμένων ασκήσεων σε κλιμακούμενη δυσκολία – από τις απλές έως τις «ολυμπιακού» επιπέδου. Στόχος είναι να παρουσιαστεί όχι μόνο η τεχνική επίλυσης, αλλά και η μεθοδολογία: χρήση ειδικών τιμών, επαγωγή, συμμετρίες (άρτια–περιττά), υπόθεση συνέχειας ή μονοτονίας, και αντιμετώπιση παθολογικών λύσεων. Το βιβλίο μπορεί να λειτουργήσει ως βοήθημα σε μαθητές, φοιτητές, αλλά και σε διδάσκοντες που αναζητούν παραδείγματα για την τάξη ή για μαθηματικούς διαγωνισμούς.

Η παρουσίαση είναι αυτοτελής: κάθε κεφάλαιο ξεκινά με τις κλασικές μορφές και καταλήγει σε πιο σύνθετα προβλήματα, ενώ στο τέλος παρατίθεται γλωσσάριο ορολογίας για διευκόλυνση.
ΓΙΑΝΝΗΣ ΠΛΑΤΑΡΟΣ

This paper describes the design and implementation of an algorithmic differentiation framework in the Axiom computer algebra system. Our implementation works by transformations on Spad programs at the level of the typed abstract syntax tree -Spad is the language for extending Axiom with libraries. The framework illustrates an algebraic theory of algorithmic differentiation, here only for Spad programs, but we suggest that the theory is general. In particular, if it is possible to define a compositional semantics for programs, we define the exact requirements for when a program can be algorithmically differentiated. This leads to a general algorithmic differentiation system, and is not confined to functions which compute with basic data types, such as floating point numbers.

We present an algorithm which, given a deformation of a reduced plane curve singularity, computes equations for the equisingularity stratum (that is, the µ-constant stratum in characteristic 0) in the parameter space of the deformation. The algorithm works for any, not necessarily reduced, parameter space and for algebroid curve singularities C defined over an algebraically closed field of characteristic 0 (or of characteristic p > ord(C)). It provides at the same time an algorithm for computing the equisingularity ideal of J. Wahl. The algorithms have been implemented in the computer algebra system Singular. We show them at work by considering two non-trivial examples. As the article is also meant for non-specialists in singularity theory, we include a short survey on new methods and results about equisingularity in characteristic 0.

Finding the symmetries of the nonlinear fractional di erential equations plays an important role in study of fractional di erential equations. In this manuscript, rstly, we are interested in nding the Lie point symmetries of the time-fractional Kaup-Kupershmidt equation. Afterwards, by using the in nitesimal generators, we determine their corresponding invariant solutions.

A polytope is called regular-faced if every one of its facets is a regular polytope. The 4-dimensional regular-faced polytopes were determined by G. Blind and R. Blind [2,. The last class of such polytopes is the one which consists of polytopes obtained by removing a set of non-adjacent vertices (an independent set) of the 600-cell. These independent sets are enumerated up to isomorphism and it is determined that the number of polytopes in this last class is 314, 248, 344.

Arithmetic systems such as those based on IEEE standards currently make no attempt to track the propagation of errors. A formal error analysis, however, can be complicated and is often confined to the realm of experts in numerical analysis. In recent years, there has been a resurgence of interest in automated methods for accurately monitoring the error propagation. In this article, a floatingpoint system based on significance arithmetic will be described. Details of the implementation in Mathematica will be given along with examples that illustrate the design goals and differences over conventional fixed-precision floating-point systems.

This work focuses on the derivation of composition methods for the numerical integration of ordinary differential equations, which give rise to very challenging optimization problems. Composition is a useful technique for constructing high order approximations, while conserving certain geometric properties. We survey existing composition methods and describe results of an intensive numerical search for new methods. Details of the search procedure are given along with numerical examples, which indicate that the new methods perform better than previously known methods. Some insight into the location of global minima for these problems is obtained as a result.

Runge-Kutta schemes are the methods of choice for solving nonstiff systems of ordinary differential equations at low to medium tolerances. The construction of optimal formulae has been the subject of much research. In this article, it will be shown how to construct some low order formula pairs using tools from computer algebra. Our focus will be on methods that are equipped with local error detection (for adaptivity in the step size) and with the ability to detect stiffness. It will be demonstrated how criteria governing 'optimal' tuning of free parameters and matching of the embedded method can be accomplished by forming a constrained optimization problem. In contrast to standard numerical optimization processes our approach finds an exact (infinite precision) global minimum. Quantitative measures will be given comparing our new methods with some established formula pairs. (~) 2005 Elsevier Ltd. All rights reserved.

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformation...

A systematic method to derive the nonlocal symmetries for partial differential and differentialdifference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2 × 2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.

Motivated by a constructive realization of dihedral groups of prime degree as Galois group over the field of rational numbers, we give an explicit construction of the Hilbert class fields of some imaginary quadratic fields with class numbers 7 and 11. This was done by explicitly evaluating the elliptic modular j -invariant at each representative of the ideal class of an imaginary quadratic field, and then forming the class equation. In an appendix we determine the explicit form of the modular equation of order 5 and 7. All computations were carried out on the computer algebra system MACSYMA. We also point out what computational difficulties were encountered and how we resolved them. In this paper we summarize the progress made so far on using the Computer Algebra System MACSYMA [8] to explicitly calculate the defining equations of the Hilbert class fields of imaginary quadratic fields with prime class number. Our motivation for undertaking this investigation is to construct rational polynomials with a given finite Galois group. The groups we try to realize here are the dihedral groups D p for primes p . These groups are non-abelian groups of order 2p and are generated by two elements σ = (1 2 3 . . . p ) and τ = (1)(2 p )(3 p -1) . . . ( 2 p +1 _ ____ 2 p +3 _ ____ ) _ ______________

BRUEN, JENSEN. AND YUI with certain Frobenius groups as Galois groups. 111.1. Preliminary results. 111.2. Realization of Frobenius groups of prime degree as Galois groups (general existence theorem). 111.3. Construction of polynomials with Galois group F,,, , bl over 0. 111.4. Construction of generic family of polynomials with Galois group F,c p I uz (p = 3 (mod 4)). 111.5. Special examples. 111.6. Remarks and problems. IV. Frobenius fields. IV.1. Frobenius fields over 0. IV.2. Frobenius fields over function fields.

We present an algorithm that generates automatically (algebraic) invariant properties of a loop with conditionals. In the proposed algorithm program analysis is performed in order to transform the code into a form for which algebraic and combinatorial techniques can be applied to obtain invariant properties. These invariants are then used for verifying partial correctness of imperative programs in the Theorema system (www.theorema.org). The application of the method is demonstrated in few examples.

We present a verification environment for imperative programs (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and specifications of auxiliary tail recursive functions. These methods use techniques from (polynomial) algebra and combinatorics, namely Groebner bases, variable elimination and symbolic summation (the Gosper algorithm, the technique of generating functions). The methods are demonstrated on several examples which have been treated automatically by our implementation.

Most of the properties established during program verification are either invariants or depend crucially on invariants. The effectiveness of automated verification of (imperative) programs is therefore sensitive to the ease with which invariants, even trivial ones, can be automatically deduced. We present a method for invariant generation that relies on combinatorial techniques, namely on recurrence solving and variable elimination. The effectiveness of the method is demonstrated on examples.

An approach utilizing combinatorics, algebraic methods and logic is presented for generating polynomial loop invariants for a family of imperative programs operating on numbers. The approach has been implemented in the Theorema system, which seems ideal for such an integration given that it is built on top of the computer algebra system Mathematica, has a theorem prover for first-order logic as well as for mechanizing induction. These invariant assertions are then used for generating the necessary verification conditions as first-order logical formulae, based on Hoare logic and the weakest precondition strategy. The approach has been successfully tried on many programs implementing interesting number theoretic algorithms. It is also shown that for a subfamily of loops, called P-solvable loops, the approach is complete in generating polynomial equations as invariants.

Rewriting induction ) is an automated proof method for inductive theorems of term rewriting systems. Reasoning by the rewriting induction is based on the noetherian induction on some reduction order and the original rewriting induction is not capable of proving theorems which are not orientable by that reduction order. To deal with such theorems, Bouhoula (1995) as well as Dershowitz & Reddy (1993) used the ordered rewriting. However, even using ordered rewriting, the weak capability of non-orientable theorems is considered one of the weakness of rewriting induction approach compared to other automated methods for proving inductive theorems. We present a refined system of rewriting induction with an increased capability of non-orientable theorems and a capability of disproving incorrect conjectures. Soundness for proving/disproving are shown and effectiveness of our system is demonstrated through some examples.

We present an algorithm that generates automatically (algebraic) invariant properties of a loop with conditionals. In the proposed algorithm program analysis is performed in order to transform the code into a form for which algebraic and combinatorial techniques can be applied to obtain invariant properties. These invariants are then used for verifying partial correctness of imperative programs in the Theorema system (www.theorema.org). The application of the method is demonstrated in few examples.

We present a verification environment for imperative pro- grams (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and of specifications of auxiliary tail recursive functions. These meth- ods use algorithms from (polynomial) algebra and combinatorics, namely Groebner bases,

When generating verification conditions for a program, one is faced with one major task, namely with the situation when some additional assertions are needed (e.g. loop invariants). These assertions have the property that either they are invariant during execution of the program, or they depend on some other invariant properties. Therefore, automated formal verification is sensitive to the automated generation of invariants. In this paper, we present an alternative to expecting programmers to fully annotate code with invariants, namely a method for automatic generation of invariants from the program itself, using combinatorial algorithms and equational elimination, as well as an ongoing research work with some results based on applications of polynomial ideal theory. The implementation and the verification process is done in a prototype verification condition generator for imperative programs, which is part of the Theorema system, a computer aided mathematical assistant that offers automated reasoning and computer algebra facilities for the working mathematician. The verification conditions for programs containing loops are generated fully automatically, in a form which can be immediately used by the automatic provers of Theorema in order to check whether they hold. We illustrate the effectiveness of our method by few examples.

We present a method that generates automatically algebraic invariant properties of a loop. The implementation and verification process is done in a prototype verification condition generator for imperative programs. This verification tool is integrated into the overall framework of the Theorema system, which is based on a version of higher order predicate logic and includes verification procedures for functional and rewrite algorithms but also for procedural programs. The main contribution of this paper is the algorithm that generates invariants for loops with conditionals. In the proposed algorithm program analysis is performed in order to transform the code into a form for which algebraic and combinatorial techniques (symbolic summation, variable elimination, polynomial algebra) can be applied to obtain an invariant property. The application of the method is demonstrated in few examples.

An approach utilizing combinatorics, algebraic methods and logic is presented for generating polynomial loop invariants for a family of imperative programs operating on numbers. The approach has been implemented in the Theorema system, which seems ideal for such an integration given that it is built on top of the computer algebra system Mathematica, has a theorem prover for first-order logic as well as for mechanizing induction. These invariant assertions are then used for generating the necessary verification conditions as first-order logical formulae, based on Hoare logic and the weakest precondition strategy. The approach has been successfully tried on many programs implementing interesting number theoretic algorithms. It is also shown that for a subfamily of loops, called P-solvable loops, the approach is complete in generating polynomial equations as invariants.

We present a verification environment for imperative programs (using Hoare logic) and for functional programs (using fixpoint theory) in the frame of the Theorema system (www.theorema.org). In particular, we discuss some methods for finding the invariants of loops and specifications of auxiliary tail recursive functions. These methods use techniques from (polynomial) algebra and combinatorics, namely Groebner bases, variable elimination and symbolic summation (the Gosper algorithm, the technique of generating functions). The methods are demonstrated on several examples which have been treated automatically by our implementation.

We present the design and the implementation of a prototype verification condition generator for imperative programs. The generator is part of the Theorema system, a computer aided mathematical assistant which offers automated reasoning and computer algebra facilities. We use Hoare Logic and the weakest precondition strategy, but in addition we propose a novel method for analyzing loop constructs by aid of algebraic computations: combinatorial summation and equational elimination. The verification conditions and the termination term for programs containing loops and procedure calls are generated fully automatically, in a form which can be immediately used by the automatic provers of Theorema in order to check whether they hold.

We present an algorithm for finding valid polynomial relations (i.e. invariants) among program variables for imperative loops. The algorithm is implemented in the verification environment for imperative programs (using Hoare logic) in the frame of the Theorema system (www.theorema.org). The algorithm uses techniques from (polynomial) algebra and combinatorics, namely Gröbner Bases, variable elimination, algebraic dependencies and symbolic summation (the Gosper algorithm, handling geometric series, C-finite solving). These methods are demonstrated on several examples which have been treated automatically by our implementation.

Most of the properties established during program verification are either invariants or depend crucially on invariants. The effectiveness of automated verification of (imperative) programs is therefore sensitive to the ease with which invariants, even trivial ones, can be automatically deduced. We present a method for invariant generation that relies on combinatorial techniques, namely on recurrence solving and variable elimination. The effectiveness of the method is demonstrated on examples.

Explicitly stated program invariants can help programmers by identifying program properties that must be preserved when modifying code. In practice, in most of the cases, however, these invariants are usually implicit. In this paper we present an alternative to expecting programmers to fully annotate code with invariants, namely a method for automatically generation of invariants from the program itself, using an implementation of a prototype verification condition generator for imperative programs. The generator is part of the Theorema system, a computer aided mathematical assistant which offers automated reasoning and computer algebra facilities. We use Hoare Logic and the weakest precondition strategy, and we propose a novel method for analyzing loop constructs by aid of algebraic computations: combinatorial summation and equational elimination. The verification conditions for programs containing loops are generated fully automatically, in a form which can be immediately used by the automatic provers of Theorema in order to check whether they hold.

The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynam...

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformation...

We consider several identities involving the multiple harmonic series v^ 1 which converge when the exponents /, are at least 1 and i\ > 1. There is a simple relation of these series with products of Riemann zeta functions (the case k = 1) when all the i } exceed 1. There are also two plausible identities concerning these series for integer exponents, which we call the sum and duality conjectures. Both generalize identities first proved by Euler. We give a partial proof of the duality conjecture, which coincides with the sum conjecture in one family of cases. We also prove all cases of the sum and duality conjectures when the sum of the exponents is at most 6. and l'l,Ϊ2,. ••,/*)= (so (1) is S(α, ft)). With this notation, S(i) = Λ(ι) = £(/). The relation between the S 's and A's should be clear: for example, Note that (2) implies ,4(2, 1) = C(3). It is immediate from the definitions that 5(11, h)+s(i 2 , ii) = c(ίoc(/2) + c(/i + ω and A(h , I 2 ) +^(/2, I'l) = C(/l)ί(l2) -C(/l + 12) whenever i\, i^> 1. More generally, if ίi, iι, ... , i^ > 1 the sums Σ ^(ίσίl) J » ίσ(ik)) and

Many applications of Mathematical Physics and Engineering are connected with the Laplacian, however, the most part of BVP relevant to the Laplacian are solved in explicit form only for domains with a very special shape, namely intervals, cylinders or domains with special (circular or spherical) symmetries. In recent articles [1]-[2]-[3]-[4]-[5], a solution of some canonical problems of Mathematical Physics was achieved by using the classical Fourier method. In this lecture a survey of our main results is given, including the Dirichlet problem for the Laplace equation in starlike domains of the space R 2 and R 3 , and the heat equation for a starlike plate.

The interior and exterior Robin problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so-called “superformula” introduced by J. Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained. The computed results are found to be in good agreement with the theoretical findings on Fourier series expansion presented by L. Carleson.

The paper deals with problems arising in the application of the computer algebra systems for the symbolic-numeric stability analysis of difference schemes and schemes of the finite-volume method approximating the two-dimensional Euler equations for compressible fluid flows on curvilinear spatial grids. We carry out a detailed comparison of the REDUCE 3.6 and Mathematica (Versions 2.2 and 3.0) from the point of view of their applicability to the solution of the above problems. We draw a conclusion that a preference should be given for Mathematica from the viewpoint of the execution of symbolic-numeric computations. We also describe in detail our new symbolic-numeric algorithm for stability investigation, which was implemented with the aid of Mathematica. The proposed method enables us to reduce the needed computer storage at the symbolic stages by a factor of about 20 as compared with the previous algorithms. A feature of the numerical stages is the use of the arithmetic of rational numbers, which enables us to avoid the accumulation of the roundoff errors. We present the examples of the application of the proposed symbolic-numeric method for stability analysis of very complex schemes of the finite-volume method on curvilinear grids, which are widely used in computational fluid dynamics.

Υλικά: Για την υλοποίηση του μαθήματος καλό θα ήταν να είμαστε στην αίθουσα με τους υπολογιστές. Αν αυτό δεν είναι δυνατό θα έπρεπε να έχουμε αίθουσα με διαδραστικό πίνακα.
Στη συγκεκριμένη διδακτική πρόταση ενδεχομένως να προκύψουν καθυστερήσεις στην υλοποίησή του καθώς οι προτεινόμενες δραστηριότητες και απαιτούν συντονισμό και προσπάθεια από όλη την τάξη. Επιπλέον οι δραστηριότητες με το geogebra όπως και η χρήση του phet ενδεχομένως να απαιτήσουν επιπλέον χρόνο καθώς οι μαθητές δεν αναμένεται να γνωρίζουν τη χρήση του και ίσως απαιτηθεί να δοθούν επιπλέον εξηγήσεις και παραδείγματα. Αφήνεται ένα περιθώριο μιας ώρας αν χρειαστεί ώστε να ενταχθούν όλες οι δραστηριότητες ή/και να προστεθούν άλλες ώστε η διδακτική πρόταση να ανταποκρίνεται και στις ανάγκες άλλων τάξεων όπως A΄ Λυκείου, αλλά και σε παρακάτω ενότητες. Με τη συγκεκριμένη πρόταση γίνεται προσπάθεια ένταξης ΤΠΕ σε ενεργητική διδασκαλία και προσπάθεια εκπαίδευσης – ενημέρωσης των μαθητών σε λογισμικά και προγράμματα που μπορούν να χρησιμοποιηθούν στα Μαθηματικά αλλά και διαθεματικά. Επιπλέον θα μπορούσαμε να εντάξουμε και δραστηριότητες γραφικών παραστάσεων με χρήση exel ή και άλλων λογισμικών.

This paper focus on a new blended root-finding algorithm to solve the given transcendental/nonlinear equations. This algorithm is based on the classical methods, namely Newton-Raphson method and trisection method. One of the advantages of the proposed algorithm is not only fast converging but also guarantee the root. The proposed algorithm is easy to understand and one can implement this algorithm in any mathematical software tools. Several numerical examples are presented to illustrate the algorithm and comparisons are made to show the efficiency of the proposed algorithm.

Approaching the problem of imperative program verification from a practical point of view has certain implications concerning: the style of specifications, the programming language which is used, the help provided to the user for finding appropriate loop invariants, the theoretical frame used for formal verification, the language used for expressing generated verification theorems as well as the database of necessary mathematical knowledge, and finally the proving power, style and language. The Theorema system (www. theorema.org) has certain capabilities which make it appropriate for such a practical approach. Our approach to imperative program verification in Theorema is based on Hoare-Logic and the Weakest Precondition Strategy. In this paper we present some practical aspect of our work, as well as a novel method for verification of programs that contain loops, namely a method based on recurrence equation solvers for generating the necessary loop invariants and termination terms automatically. We have tested our system with numerous examples, some of these example are presented at the end of the paper.

Modul ini membahas konsep dasar Sistem Persamaan Linear (SPL) beserta metode penyelesaiannya, baik secara manual maupun menggunakan perangkat lunak Maple 2021. Pembahasan mencakup berbagai teknik penyelesaian SPL, seperti metode substitusi, eliminasi, serta metode berbasis matriks seperti Eliminasi Gauss, Gauss-Jordan, dan Aturan Cramer. Selain itu, modul ini memperkenalkan metode numerik seperti Jacobi dan Gauss-Seidel untuk penyelesaian sistem dengan banyak variabel. Dengan menggunakan Maple 2021, modul ini memberikan panduan dalam menyusun, menyelesaikan, dan memvisualisasikan sistem persamaan linear secara komputasional. Penyajian dilakukan melalui contoh soal yang disertai langkah penyelesaian manual serta implementasi dalam Maple, termasuk visualisasi grafik dalam bentuk 2D dan 3D. Melalui modul ini, diharapkan pembaca dapat memahami konsep SPL, menguasai teknik penyelesaiannya, serta mampu membandingkan hasil perhitungan manual dengan solusi yang diperoleh menggunakan perangkat lunak.

From 2011 – 2013 the VCAA conducted a trial aligning the use of computers in curriculum, pedagogy and assessment culminating in a group of 62 volunteer students sitting their end of Year 12 technology-active Mathematical Methods (CAS) Examination 2 as a computer-based examination. This paper reports on statistical modelling undertaken to compare the distribution of results for this group with the standard cohort, and any differences in student response between the two groups at the item level.

Analyses and commentary for 2002-2005 Mathematical Methods (CAS) pilot examinations in Victoria, on student performance with respect to common items with the standard course have been reported at previous MERGA conferences. In 2006, both Mathematical ...

2009 was the final year of parallel implementation for Mathematical Methods Units 3 and 4 and Mathematical Methods (CAS) Units 3 and 4. From 2006-2009 there was a common technology-free short answer examination that covered the same function, algebra, calculus and probability content for both studies with corresponding expectations for key knowledge and key skills. There was also separate technology-assumed examinations comprising common and different multiple choice and extended response questions. In 2009 the two cohorts were of similar size, comprising 7000-8000 students each. This paper analyses student performance for both cohorts with respect to these common items.

Students undertaking mathematics courses in their final year of secondary education in Victoria are assessed using a combination of school based coursework assessment and examinations. Over the last decade, students have used technology such as graphics calculators, statistical software, spreadsheets, and computer algebra systems (CAS) extensively in tackling extended investigative, problem solving, and modeling tasks as part of their school based assessment. The use of graphics calculators has been permitted in calculator neutral mathematics examinations since 1997, while from 2000, these examinations will be set with assumed student access to an approved graphics, calculator and will also include some calculator active questions. This development has been supported by the implementation in 2000 of a revised senior mathematics study that includes a technology active learning outcome for all mathematics courses. The University of Melbourne, in partnership with Board of Studies, Victoria, and three calculator companies, has received a major Commonwealth government 3-year research grant 2000 -2 to undertake a pilot study () for a CAS active mathematics course and corresponding CAS active examinations, and to plan for its possible accreditation within a revised mathematics study. Congruence between* curriculum, pedagogy, assessment, and values is a key issue for the related discourse. Some important connections between these elements and associated issues for systems and policymakers are discussed. (Contains 18 references.) (Author/ASK) Reproductions supplied by EDRS are the best that can be made from the original document.

Analysis and commentary on the 2002 and 2003 Mathematical Methods (CAS) pilot examinations in Victoria, in particular with respect to common items with the standard course, were reported at the 2002 and 2003 MERGA conferences. This paper extends analysis to student performance on aspects of the 2004 examinations, with consideration of some emerging trends over the three years. Several possible areas for a broader research agenda with respect to implementation of a CAS enabled senior mathematics curriculum are proposed.