Implementing CAS into Implementing CAS into Teaching, Learning and Teaching, Learning and Assessment:Assessment:An Australian ExperienceAn Australian Experience

Peter FlynnPeter Flynn

The University of The University of MelbourneMelbourne

Melbourne, AustraliaMelbourne, Australiaflynnpj@unimelb.edu.auflynnpj@unimelb.edu.au

Computer Algebra Systems-Computer Algebra Systems-Curriculum, Assessment and Curriculum, Assessment and Teaching Project (CAS-CAT)Teaching Project (CAS-CAT)(2000-2002)(2000-2002)

• Explore the effect of CAS on Explore the effect of CAS on teaching/learning/assessmentteaching/learning/assessment

• Research partners including TIResearch partners including TI

• Presented at many teacher and Presented at many teacher and research conferences and research conferences and published many paperspublished many papers

• http://extranet.edfac.unimelb.edu.au/DSMhttp://extranet.edfac.unimelb.edu.au/DSME/CAS-CATE/CAS-CAT

Why CAS?Why CAS?

• Make students better users of Make students better users of mathematicsmathematics

• Closer link between ‘real’ and Closer link between ‘real’ and school mathematicsschool mathematics

• Achieve deeper learning by Achieve deeper learning by studentsstudents

• Promote a less procedural view of Promote a less procedural view of mathematicsmathematics

• Introduce new topics into the Introduce new topics into the curriculumcurriculum

Mathematics in VictoriaMathematics in Victoria

• Victorian Certificate Education (2 Victorian Certificate Education (2 years)years)

• State-wide examinationsState-wide examinations

• 3 mathematics subjects 3 mathematics subjects

• Mathematical Methods (MM)Mathematical Methods (MM)

• ‘‘Middle subject’ in terms of difficultyMiddle subject’ in terms of difficulty

• Coordinate Geometry, Functions, Coordinate Geometry, Functions, Calculus, Algebra and ProbabilityCalculus, Algebra and Probability

CAS-CAT Project 2000-CAS-CAT Project 2000-20022002

• Started with 3 schools (N=78)Started with 3 schools (N=78)• School A: TI-89School A: TI-89• School B: CASIO FX 2.0School B: CASIO FX 2.0• School C: HP 40GSchool C: HP 40G• 2 CAS-permitted examinations2 CAS-permitted examinations• 80% common questions between 80% common questions between

Mathematical Methods and Mathematical Methods and Mathematical Methods (CAS) Mathematical Methods (CAS)

2003-2005 Extended 2003-2005 Extended PilotPilot

• 2003: ~ 13 schools offering 2003: ~ 13 schools offering Mathematical Methods (CAS) Mathematical Methods (CAS)

• 2005: ~ 40 schools2005: ~ 40 schools

• 2003-: Various hand-held or 2003-: Various hand-held or computer-based CAS permittedcomputer-based CAS permitted

• 2006-: Calculator free and 2006-: Calculator free and calculator permitted examinationcalculator permitted examination

CAS-CAT Project FindingsCAS-CAT Project Findings

• Effect on TeachingEffect on Teaching

• Effect on Student learningEffect on Student learning

• Effect on Examination AssessmentEffect on Examination Assessment

Effect on TeachingEffect on Teaching

• Took longer than expected to learn CASTook longer than expected to learn CAS

• Used CAS as an ‘add-on’ initially but Used CAS as an ‘add-on’ initially but with experience and confidence, CAS with experience and confidence, CAS became a greater part of teachingbecame a greater part of teaching

• Provided more solution methods for Provided more solution methods for doing mathematicsdoing mathematics

• Used more for learning mathematicsUsed more for learning mathematics

• Promoted greater dialogue between Promoted greater dialogue between teacher and studentsteacher and students

• Create changes in teaching philosophyCreate changes in teaching philosophy

By-Head/By-Hand/By-CASBy-Head/By-Hand/By-CAS

• Balance is importantBalance is important

• One teacher:One teacher:– simple cases (eg derivative of sin(simple cases (eg derivative of sin(xx)) ))

completed by-head/by-handcompleted by-head/by-hand– recognise when CAS was requiredrecognise when CAS was required– decisions made on efficiency and decisions made on efficiency and

accuracyaccuracy

• Some students were worried that CAS Some students were worried that CAS would ‘steal’ their skillswould ‘steal’ their skills

• Some students became too CAS-Some students became too CAS-dependent dependent

Effects on Student LearningEffects on Student Learning

• Promoted greater discussion Promoted greater discussion between studentsbetween students

• Liked to use CAS in different Liked to use CAS in different ways (eg combining operations, ways (eg combining operations, switching representations)switching representations)

Promoting Functional Promoting Functional ThoughtThought

• Students became more Students became more comfortable working with comfortable working with functionsfunctions

Explosion of MethodsExplosion of Methods

• Increase in solution approachesIncrease in solution approaches

• Different compositions of Different compositions of methodsmethods

• Given Given ff((xx)=)=axax33-44-44xx22++bxbx-12 find -12 find aa and and bb if if ff(-1)=0 and (-1)=0 and ff(2)=0 (2)=0

Communicating Communicating MathematicsMathematicsLynda Ball (2003, 2004)Lynda Ball (2003, 2004)• CAS solutions generally shorter CAS solutions generally shorter

and contain more instructionsand contain more instructions

• CAS students tend to use more CAS students tend to use more function notationfunction notation

• Students had to find a and b given ( ) log ( )ef x a b x and worded information leading to the identification of (1, 0.5) and (1.5, 0.3)

Solve (1) 0.5f and (1.5) 0.3f for a and b.

Hence 1 1

and 32 5log2e

a b

Algebraic KnowledgeAlgebraic Knowledge

• Teachers maintained and valued Teachers maintained and valued basic by-hand algebra skillsbasic by-hand algebra skills

• Algebraic knowledge improved but Algebraic knowledge improved but not more than normalnot more than normal

• Algebraic knowledge required to Algebraic knowledge required to use CAS was underestimated use CAS was underestimated initiallyinitially

• Entering expressions correctly and Entering expressions correctly and recognising equivalent formsrecognising equivalent forms

• Given more attention by teachers.Given more attention by teachers.

You have learned about factorising quadratics and cubics.

What about quartics, polynomials of degree of 4?

We haven’t learned about that but they would have 4 factors.

Can you factorise ?

There will be 4 factors.

That’s what I expected. There are 4 factors.

Oh wow! How come I got that?

That’s what I expected, not my answer because with quadratics it’s two factors, with cubics it’s three, therefore with the pattern in quartics it’s four.

If you look here, Yes but not in the simplest form. Factorising simplifies.

Are both answers correct?

How can we tell if both are correct?

We could expand it or try to simplify it.

That makes more sense.Oh. This has an extra four.

We started with different expression.

Equivalence of FormEquivalence of Form

• CAS outputs was a critical issueCAS outputs was a critical issue

• Students need algebraic Students need algebraic knowledge of a different flavourknowledge of a different flavour

• Students with a more well-Students with a more well-developed by-hand/mental developed by-hand/mental algebra seemed to adapt to CAS algebra seemed to adapt to CAS more readilymore readily

AssessmentAssessmentCAS Features Influence CAS Features Influence DesignDesign• "I can do a 3 hour Mathematics "I can do a 3 hour Mathematics

Examination in 20 minutes with a Examination in 20 minutes with a CAS“ CAS“

• CAS can change what mathematical CAS can change what mathematical knowledge a question is testingknowledge a question is testing

Examination AssessmentExamination Assessment

• If CAS is permitted in examinations, If CAS is permitted in examinations, assessment needs to changeassessment needs to change

• Reluctance to construct questions Reluctance to construct questions that really require the symbolic that really require the symbolic power of CASpower of CAS

• Examiners tended to set questions Examiners tended to set questions that only require numerical/graphicalthat only require numerical/graphical

• A common response was to set A common response was to set questions with parametersquestions with parameters

Student Performance in Student Performance in CAS ExaminationsCAS Examinations

• Questions thought as 'trivialised' Questions thought as 'trivialised' may not be so. 69% of CAS may not be so. 69% of CAS students correctly identified the students correctly identified the linear factors oflinear factors of xx44++xx33-3-3xx22-3-3xx

Some General Some General ConclusionsConclusions• Increasing number of schoolsIncreasing number of schools• General positive attitude towards CASGeneral positive attitude towards CAS• Time required to learn effective CAS useTime required to learn effective CAS use• Students can be puzzled by CAS outputsStudents can be puzzled by CAS outputs• Students (and teachers) require good Students (and teachers) require good

algebraic knowledge to work with CAS algebraic knowledge to work with CAS effectivelyeffectively

• Assessment will take time to evolveAssessment will take time to evolve• CAS can cause major change to CAS can cause major change to

teaching methods and beliefsteaching methods and beliefs• Teachers must be well-supportedTeachers must be well-supported

Danke SchDanke Schöönn

The The University of Melbourne’s CAS-University of Melbourne’s CAS-CAT Research Project websiteCAT Research Project websitehttp://http://extranet.edfac.unimelb.edu.auextranet.edfac.unimelb.edu.au

/DSME/CAS-CAT/DSME/CAS-CAT