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The case of technology in senior secondary mathematics: Curriculum and assessment congruence?

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Technology

• systematic application (logia) of an art (tekhne)

• art - a skill, especially a human skill in contrast to nature

• artefact - a product of human art and workmanship

• synergy of artefact and art

• knowledge, understanding and use of a particular artefact as a tool for a purpose

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Aspects of technology

• human beings are technological creatures, and technology plays a central role in how our species, societies and histories have evolved and developed, and continue to evolve and develop

• by and large technologies make our lives easier -they enable us to do certain things more readily and reliably with less effort

• when technologies are familiar to us they are almost invisible and their application is taken for granted

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Some aspects of technology (ctd)

• sometimes newer technologies replace older technologies, in other instances they subsume them or exist alongside them

• it is mainly when a new technology is developed, or a new application of an existing technology emerges, that we pause and reflect on the benefits and suitability of its application

• whenever mathematics moves beyond the private mental acts of an individual into the public domain, this process is invariably mediated by some kind of technology

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Some different computationaltechnologies from antiquity to 1942

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Some different computational technologies 1942 - ?

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Finger counting and calculation (dactylonomy)

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Counting board

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Abacus

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Leibniz mechanical – stepped reckoner (arithmetic)

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Slide rule - analog mechanical scientific calculator

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Mechanical calculator (Leibniz several centuries on)

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Electro-mechanical calculator (relays)

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Casio electronic (arithmetic)

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Electronic scientific calculators

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Graphics calculators

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First generation CAS calculators

• While arithmetic and scientific calculators only operate in the numeric mode, and graphics calculators operate in numeric and graphics modes, first generation CAS calculators operate across all three modes

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First generation CAS calculators

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Nested relationship between calculator functionality

CAS

Graphics

Scientific

Arithm etic

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2nd generation CAS calculators

• calculators, such as the Casio Classpad and TI-nspire (CAS), developed during the 2000’s

• they support multiple functionalities: spreadsheet, graphic, numeric, symbolic manipulation, statistical, dynamic geometry, text and e-presentation

• students can select and use a functionality in its own right, or in natural combination with other functionalities, as applicable to the task at hand

• Symbolic manipulation is only one of several functionalities

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Web enabled CAS

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Congruence

• as a metaphor

• alignment between curriculum, assessment and pedagogy

• beliefs, values and preferences (choices)

• complementary mental, by hand and technology enabled approaches to working mathematically

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Working and memory

• mental – short/long term brain cells

• by hand – written marks on paper

• digital technology – electro-magnetic

• different types of filing structures and systems biological, material and electronic

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Functionality and platform

• Functionality – numerical, graphical, statistical, geometrical, symbolical, textual

• Platform – hand-held, software, web based

• Platform ≠ functionality• Multi-modal input, keyboard and palette, stylus,

voice

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Assumed technology for Year 12 examinations in Victoria 1970 - 1997

• Pre-1978: four-figure logarithm tables and/or an approved slide rule

• 1978 – 1996: Scientific calculator

• 1997: Scientific calculator - graphics calculator permitted but not assumed

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Assumed technology for Year 12 examinations in Victoria 1998 - 2005

• 1998-9: graphics calculator for Mathematical Methods and Specialist Mathematics Examination 1 and 2. Scientific calculator with bi-variate statistical functionality or approved graphics calculator for Further Mathematics Examination 1 and 2

• 2000 – 2005: graphics calculator for Further Mathematics, Mathematical Methods and Specialist Mathematics Examination 1 and 2

• CAS* for Mathematical Methods CAS pilot study, 2002 – 2005 (both examinations)

* calculator or software

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Assumed technology for Year 12 examinations in Victoria 2006 - 2009

• graphics calculator or CAS* for Further Mathematics Examination 1 and Examination 2

• common technology free Examination 1 for Mathematical Methods/Mathematical Methods CAS

• graphics calculator for Mathematical Methods Examination 2

• CAS* for Mathematical Methods CAS Examination 2• technology free Examination 1 for Specialist

Mathematics• graphics calculator or CAS* for Specialist

Examination 2 (technology active but graphics calculator/CAS neutral)

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Assumed technology for Year 12 examinations in Victoria 2010 - 2013

• CAS* or graphics calculator for Further Mathematics Examination 1 and Examination 2

• Mathematical Methods (CAS) and Specialist Mathematics each have a technology free Examination 1

• CAS* for Mathematical Methods (CAS) and Specialist Mathematics Examination 2

* calculator or software

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Assumed technology for end of Year 12 examinations in Victoria 2014 - ?

• (Draft) Australian curriculum has four senior secondary mathematics studies: Essential mathematics (Course A); General mathematics (Course B); Mathematical methods (Course C) and Specialist mathematics (Course D), currently under consultation

• If things proceed well, 2014 could be the first year of implementation in Victoria

• Assessment remains the province of states and territory jurisdictions for the interim (first accreditation cycle)

Outcome 1

• On completion of each unit the student should be able to define and explain key concepts as specified in the content from the areas of study, and apply a range of related mathematical routines and procedures.

• To achieve this outcome the student will draw on knowledge and related skills outlined in all the areas of study.

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Outcome 2

• On completion of each unit the student should be able to apply mathematical processes in non-routine contexts, and analyse and discuss these applications of mathematics.

• To achieve this outcome the student will draw on knowledge and related skills outlined in one or more areas of study.

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Outcome 3

• On completion of each unit the student should be able to select and appropriately use a computer algebra system and other technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.

• To achieve this outcome the student will draw on knowledge and related skills outlined all in the areas of study.

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Australian curriculum

Shape of the Australian Curriculum: Mathematics (ACARA, May 2009):

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Australian curriculum (ctd)

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A great research opportunity!

• Eight states and territories implementing the same senior secondary curriculum

• Different assessment regimes

• Similarities and differences of pedagogical culture and philosophical beliefs between and within states and territories

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Potential benefits

A range of potential benefits for the use of CAS are typically

articulated by those who favour or are open to the potential for

its use, including, for example:

• the possibility for improved teaching of traditional mathematical topics;

• opportunities for new selection and organisation of mathematical topics;

• access to important mathematical ideas that have previously been too difficult to teach effectively;

• extending the range of examples that can be studied;

Potential benefits (2)

• emphasising the inter-relationships between different mathematical representations (the technology allows students to explore mathematics using different representations simultaneously);

• long and complex calculations can be carried out by the technology, enabling students to concentrate on conceptual aspects of mathematics;

• the technology provides immediate feedback so that students can independently monitor and verify their ideas;

• situations and problems can be modelled in more complex and realistic ways.

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ConcernsThese benefits can be contrasted with various concerns:

• the extent to which the use of CAS may reduce students knowledge and skills with important and valued conventional by hand or mental techniques;

• how students, including those who may be less mathematically inclined, will cope with a more conceptually demanding curriculum;

• a diminished role for teachers in terms of traditional (and valued) pedagogy; and whether appropriate cognisance has been given to the role of by hand approaches in the development of important mathematical concepts, skills and processes.

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A CAS active system (1)

The decision for a system to move towards a CAS active

curriculum, pedagogy and assessment therefore necessarily

takes place within a policy framework that has a positive

orientation with respect to the role of technology in society and

education and its potential benefits, as well as being informed

by various research and practical implementation data.

In short, there needs to be a clear articulation of the practical

and in principle basis on which such a move is made, and

monitoring of it’s effects during implementation.

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A CAS active system (2)

This monitoring looks closely at two related and critical

questions:

• have the mental and by hand capabilities of students learning with access to CAS been compromised compared with the standard cohort?

• have there been any discernable improvements in terms of mathematical understanding and/or problem solving capacity?

Essentially a robust on balance no answer to the first

question and yes answer to the second question is sought.