Computer Algebra in Mathematics Education:
Curriculum information

System/jurisdiction

Qualification/study

CAS status

Use in assessment/examinations

Comments/links

Denmark

(Danish Ministry of Education: Undervisnings Ministeriet)

Baccalaureate

(Studentereksamen)

Matematik A

Matematik B

Assumed

One 1 hour technology free ‘pencil and paper’ examination; and one 3 hour (B-level) or 4 hour (A-level) technology active examination, with assumed student access to a CAS (calculator or software)

Students may use any CAS tool, hand-held device and/or software for the technology active examination.

http://www.eng.uvm.dk/

http://us.uvm.dk/Uddannelse/Gymnasiale%20uddannelser/Proever%20og%20eksamen/Centralt%20stillede%20skriftlige%20opgavesaet%20stx%20og%20hf.aspx

http://www.uvm.dk/Uddannelse/Gymnasiale%20uddannelser/Om%20gymnasiale%20uddannelser/Politiske%20oplaeg%20og%20aftaler/Gymnasiereformen/Reform%20paa%20engelsk.aspx?menuid=152510

France

(National Ministry of Education: ministère education nationale)

Baccalauréat Général
(séries Scientifique)

Permitted

One 4 hour examination, an approved graphics calculator or an approved CAS calculator permitted.

 http://eduscol.education.fr/D0056/accessujets.htm

Norway

(Ministry of Education and Research)

Certificate of Completion of Upper Secondary School Examination

Permitted

One 2 hour technology free ‘pencil and paper’ examination; and one 4 hour technology active examination, with assumed student access to suitable technology, including CAS.

Mathematics for the Natural Sciences, 2 programme subjects Mathematics R1(mainstream) and Mathematics R2 (advanced). R2 is complementary to R1.

http://www.udir.no/upload/larerplaner/Fastsatte_lareplaner_for_Kunnskapsloeftet/english/Natural_science_mathematics/Mathematics_for_the_natural_sciences.rtf

competency aims include:

• execute and analyse constructions defined by straight lines, triangles and circles in the plane, with and without the use of dynamic software

• draw graphs to functions with and without digital means, and interpret the basic characteristics of a function using the graph

• transform and simplify complex rational functions and other symbolic expressions with and without the use of digital aids

• find and analyse recursive and explicit formulae for numerical patterns with or without digital means, and implement and present simple proofs linked to these formulae

• sum finite series with or without digital means, derive and use the formulae to the sum of the first n members in arithmetic and geometric series, and use this to solve practical problems

• solve differential equations and draw vector diagrams and integral curves, and interpret them using digital tools

Sweden

(Skolverket – National Agency for Education)

Upper Secondary School Leaver Certificate

Permitted

National assessments (tests) which comprise a technology free part; and a technology active part. A test generally takes four hours, with a time of around one hour recommended for the technology free part.

There are no centrally set national examinations. Schools are required to offer national assessments (tests) to students. These are not compulsory, however the majority of students undertake them.

http://www.skolverket.se/sb/d/190

http://www3.skolverket.se/ki/eng/pgm_eng.pdf

http://www3.skolverket.se/ki03/front.aspx?sprak=EN&ar=0809&infotyp=8&skolform=21&id=MA&extraId=

http://www3.skolverket.se/ki03/front.aspx?sprak=EN&ar=0708&infotyp=4&skolform=21&id=M&extraId=

http://www8.umu.se/edmeas/np/information/np-tidigare-prov_eng.html (this link provides access to past national assessment tests).

Austria

(Federal Ministry for Education, Arts and Culture)

Maturazeugnis

Teachers can choose to preclude, permit or require student use of CAS as applicable.

Final Mathematics examinations are locally set by schools, include an external examiner.

Mathematics is compulsory for all students undertaking the Matura (Reifeprüfung). Related materials can be accessed from:

http://www.bmukk.gv.at/enfr/school/index.xml

http://www.acdca.ac.at/

http://www.acdca.ac.at/material/vortrag/boehm_came2007.pdf

Switzerland

(Swiss Conference of Cantonal Ministers of Education: EDK)

Matura (Maturité)

Teachers can choose to preclude, permit or require student use of CAS as applicable.

Final Mathematics examinations are locally set by schools, and are made available for public viewing.

In Switzerland, the main responsibility for education and culture lies with the cantons. They coordinate their work at the national level: http://www.edk.ch/dyn/11586.php; http://www.edk.ch/dyn/11553.php

Schools/teachers choose enabling technology for senior secondary mathematics curriculum and assessment, many (~ 60 – 70 %) choose to use CAS which may be calculator or software. Some teachers use a technology free (minor component) and technology active (major component) structure for examination assessment.

New Zealand

(New Zealand Qualifications Authority)

National Certificate of Educational Achievement (NCEA), standards-based

Assumed for external assessment of relevant standards

An approved CAS calculator allowed for Level 1: AS90799 and AS90800 standards (available from 2007); Level 2: AS90806, 90807, 90808 standards (available from 2008); and Level 3: AS90833, AS90834, AS90835 standards (available from 2009) ; Externally assessed in one 3-hour examination.

Levels 1, 2 and 3 typically correspond to standards (involving study of algebra, functions and calculus for those listed here) undertaken in the final three years of secondary schooling, respectively.

http://www.nzqa.govt.nz/ncea/acrp/secondary/4/47.html

CAS are allowed for all internally assessed standards. The externally assessed CAS allowed standards for Levels 1, 2 and 3 are mutually exclusive with parallel non-CAS (but graphics calculator allowed) standards. For Levels 2 and 3 there are technology free externally assessed standards AS90809 and AS90836 respectively. Standards are detailed individually, for example:

http://www.nzqa.govt.nz/nqfdocs/ncea-resource/achievements/2009/as90835.pdf

and can be accessed by searching the NZQA for achievement standards at the relevant level (1, 2 or 3).

Victoria, Australia

(Victorian Curriculum and Assessment Authority)

Victorian Certificate of Education (VCE): Mathematical Methods(CAS)

Assumed

One 1-hour technology free examination; and one 2 hour technology active examination, with assumed student access to an approved CAS (calculator or software).

 Mainstream function, algebra, calculus and probability course:

http://www.vcaa.vic.edu.au/vce/studies/mathematics/cas/casindex.html

http://www.vcaa.vic.edu.au/vce/studies/mathematics/cas/casexams.html

http://www.vcaa.vic.edu.au/vce/studies/mathematics/calculators.html

There is a parallel graphics calculator enabled Mathematical Methods study, which has its last year of implementation in 2009. The technology free examination is common to both studies.

From 2010, all students enrolling in this type of study will take Mathematical Methods (CAS).

Examinations accounts for 66% of the final student study score. School based-assessment in accordance with VCAA specifications, and statistically moderated with respect to the examination component accounts for the other 34% of the final student study score. This also includes technology free and technology active components of teacher devised tasks.

Victorian Certificate of Education (VCE): Specialist Mathematics

Permitted

One 1-hour technology free examination; and one 2 hour technology active examination, with assumed student access to an approved graphics calculator or an approved CAS (calculator or software).

 Advanced pure and applied mathematics calculus based course:

http://www.vcaa.vic.edu.au/vce/studies/mathematics/specialist/specialmathindex.html

http://www.vcaa.vic.edu.au/vce/studies/mathematics/specialist/exams.html

http://www.vcaa.vic.edu.au/vce/studies/mathematics/calculators.html

From 2010, as with Mathematical Methods (CAS) – which is an assumed pre- or co-requisite study – the technology examination will assume student access to an approved CAS (calculator or software).

Examinations accounts for 66% of the final student study score. School based-assessment in accordance with VCAA specifications, and statistically moderated with respect to the examination component accounts for the other 34% of the final student study score. This also includes technology free and technology active components of teacher devised tasks.

Victorian Certificate of Education (VCE): Further Mathematics

Permitted

Two 1½ hour technology active examinations with assumed student access to an approved graphics calculator or an approved CAS (calculator or software).

Non-calculus based course with a data analysis and discrete mathematics practical applications numerical and graphical emphasis:

http://www.vcaa.vic.edu.au/vce/studies/mathematics/further/furthermathindex.html

http://www.vcaa.vic.edu.au/vce/studies/mathematics/further/exams.html

http://www.vcaa.vic.edu.au/vce/studies/mathematics/calculators.html

Examinations accounts for 66% of the final student study score. School based-assessment in accordance with VCAA specifications, and statistically moderated with respect to the examination component accounts for the other 34% of the final student study score. This also includes technology free and technology active components of teacher devised tasks.

Western Australia

(Curriculum Council)

WA Certificate of Education (WACE): Mathematics (penultimate year units 2AMAT - 2DMAT; final year units 3AMAT - 3DMAT)

Assumed

One 50 minute technology free examination; and one 100 minute technology active examination, with assumed student access to an approved CAS calculator.

3AMAT - 3DMAT examination from 2010:

http://www.curriculum.wa.edu.au/internet/Senior_Secondary/Courses/Mathematics

WA Certificate of Education (WACE): Mathematics Specialist (units 3AMAS – 3DMAS)

Assumed

One 50 minute technology free examination; and one 100 minute technology active examination, with assumed student access to an approved CAS calculator.

Advanced pure and applied mathematics calculus based course:

3AMAS - 3DMAS examination from 2010:

http://www.curriculum.wa.edu.au/internet/Senior_Secondary/Courses/Mathematics_Specialist

Assumes Mathematics 3AMAT - 3DMAT as a pre- or co-requisite study.

Queensland, Australia

(Queensland Studies Authority)

Queensland Certificate of Education (QCE) Mathematics B (course covers the final two years of schooling)

Permitted

Assessment is school-based and externally moderated. Structure of exit assessment is prescribed, this includes demonstration of achievement with and without the assistance of enabling technology. Using technology is one of seven key competencies

Mainstream function, algebra, calculus and probability course:
http://www.qsa.qld.edu.au/downloads/learning/snr_maths_b_08_syll.pdf

A range of technological tools must be used in the learning and assessment experiences offered in this course. This ranges from pen and paper, measuring instruments and tables through to higher technologies such as computers and graphing calculators, including those that allow for algebraic manipulations. The minimum level of higher technology appropriate for the teaching of this course is a graphing calculator.

To meet the requirements of this syllabus schools should consider the use of … hand-held (calculator) technologies designed for mathematics teaching and learning, e.g. graphics calculators with and without algebraic manipulation or dynamic geometry facilities. Complete dependence on calculator and computer technologies at the expense of students demonstrating algebraic facility may not satisfy syllabus requirements for Knowledge and procedures.

Queensland Certificate of Education (QCE) Mathematics C

(course covers the final two years of schooling)

Permitted

Advanced pure and applied mathematics calculus based course:
http://www.qsa.qld.edu.au/downloads/learning/snr_maths_c_08_syll.pdf

Study of Mathematics B is a co-requisite and the same technology requirements apply to Mathematics C as for Mathematics B.

Tasmania, Australia

(Tasmanian Qualifications Authority)

Tasmanian Certificate of Education (TCE): Mathematical Methods

Permitted from 2010

One 3 hour examination.

Mainstream function, algebra, calculus and probability course:

http://www.tqa.tas.gov.au/4DCGI/_WWW_doc/008567/RND01/MTM315109.pdf

Students are assumed to have access to graphics calculators, including algebra-capable calculators, and become proficient in their use “Students studying MTM315109 Mathematics Methods  MUST, as a part of their course, develop skills in the use of CAS and other forms of computer based technologies as a part of their learning program.”

Programs of study derived from this course need to embrace the range of technological developments that have occurred in relation to mathematics teaching. Students should have access to graphics calculators, including algebra-capable calculators, and become proficient in their use. Graphics calculators can be used in all aspects of this course in the development of concepts and as a tool for solving problems.

Tasmanian Certificate of Education (TCE): Mathematics Specialised

One 3 hour examination.

Advanced pure and applied mathematics calculus based course:

http://www.tqa.tas.gov.au/4DCGI/_WWW_doc/008569/RND01/MTS315109.pdf

Students are assumed to have access to graphics calculators, including algebra-capable calculators, and become proficient in their use “Students studying MTS315109 Mathematics Specialised 3 MUST, as a part of their course, develop skills in the use of CAS and other forms of computer based technologies as a part of their learning program.”

Programs of study derived from this course need to embrace the range of technological developments that have occurred in relation to mathematics teaching. Students should have access to graphics calculators, including algebra-capable calculators, and become proficient in their use. Graphics calculators can be used in all aspects of this course in the development of concepts and as a tool for solving problems.

New South Wales, Australia

(Board of Studies)

Higher School Certificate (HSC): General Mathematics

HP40G can be used - equivalent in terms of enabling functionality for this examination to a graphics calculator

One 3 hour examination, students permitted to used Board approved calculators with bivariate statistics functionality.

Non-calculus based course with a data analysis and discrete mathematics practical applications numerical and graphical emphasis:

http://www.boardofstudies.nsw.edu.au/manuals/calculators_hsc_gen_maths.html

Graphics or CAS calculators are not approved for other HSC mathematics examinations, only scientific calculators with approved functionality:

http://www.boardofstudies.nsw.edu.au/manuals/calculators_hsc_features.html

The College Board, Advanced Placement Program, US

AP Calculus AB

AP Calculus BC

AP Statistics

Permitted

For each of the AP Calculus AB and AP Calculus BC courses, there is one 3 hour 15 minute examination in two sections each with two parts: in total 100 minutes technology free; and 95 minutes technology active, with assumed student access to an approved graphics calculator or an approved CAS calculator.

For AP Statistics there is one 3 hour examination with assumed student access to an approved calculator (graphics or CAS) with statistical capabilities.

While developed in the US, AP examinations are administered around the world.

http://apcentral.collegeboard.com

 http://www.collegeboard.com/student/testing/ap/calculus_ab/topic.html?calcab

http://www.collegeboard.com/student/testing/ap/calculus_bc/topic.html?calcbc

http://www.collegeboard.com/student/testing/ap/calculus_ab/exam.html?calcab

http://www.collegeboard.com/student/testing/ap/calculus_bc/exam.html?calcbc

http://www.collegeboard.com/student/testing/ap/calculus_ab/calc.html?calcab

http://www.collegeboard.com/student/testing/ap/statistics/topic.html?stats

http://www.collegeboard.com/student/testing/ap/statistics/exam.html?stats

http://www.collegeboard.com/student/testing/ap/statistics/calc.html?stats

Ontario, Canada

(Ministry of Education)

Mathematics Year 12

Advanced Functions (MHF4U)

Calculus and vectors (MCV4U)

One of several technologies to be used.

Assessment is school- based in accordance with Ministry requirements for assessment, grading and reporting

MHF4U and MCV4U are complementary university preparation studies, the latter requiring the former as a pre- or co-requisite.

http://www.edu.gov.on.ca/eng/curriculum/secondary/math1112currb.pdf

The overview statement on technology: applications such as databases, spreadsheets, dynamic geometry software, dynamic statistical software, graphing software, computer algebra systems (CAS), word-processing software, and presentation software can be used to support various methods of inquiry in mathematics. Technology also makes possible simulations of complex systems that can be useful for problem-solving purposes or when field studies on a particular topic are not feasible is supported by a comprehensive set of specific expectations and sample problems in the curriculum. These specify what is required without and with access to various technologies, including CAS, for example:

• determine, through investigation, and compare the properties of even and odd polynomial functions [e.g., symmetry about the y-axis or the origin; the power of each term; the number of x-intercepts; f(x) = f(– x) or f(– x) = – f (x)], and determine whether a given polynomial function is even, odd, or neither Sample problem: Investigate numerically, graphically, and algebraically, with and without technology, the conditions under which an even function has an even number of x-intercepts

• make connections, through investigation using technology (e.g., computer algebra systems), between the polynomial function f(x), the divisor
  x – a, the remainder from the division , and f(a) to verify the remainder theorem and the factor theorem
  Sample problem: Divide f(x) = x⁴ + 4x³ – x² – 16x – 14 by x – a for various integral values of a using a computer algebra system. Compare the remainder from each division with f(a)

• solve simple rational equations in one variable algebraically, and verify solutions using technology (e.g., using computer algebra systems to determine the roots; using graphing technology, to determine the x-intercepts of the graph of the corresponding rational function)

• determine, through investigation with technology (i.e., 3-D graphing software) and without technology, that the solution points (x, y, z) in three-space of a single linear equation in three variables form a plane and that the solution points (x, y, z) in three-space of a system of two linear equations in three variables form the line of intersection of two planes, if the planes are not coincident or parallel.