Algebra & computer algebra systems

CAS working group report

12th ICMI Study: The future of the teaching and learning of

algebra

Outline of the presentation

1. On the learning of algebra using CAS

2. On the curriculum

Part 1

On the learning of algebra using CAS

Question 1

• Question 1a: How does CAS use influence student conceptualization?

• Question 1b: What is the nature of the students’ development and ability to work with CAS as an instrument?

• Question 1c: How does the way students work on tasks with paper and pencil inform their work in a CAS and vice versa?

How does CAS use influence students’ mathematical

conceptualization?

• Four CAS capacities: – Multiple linked representations that include

symbolics– Vary symbolic constraints and parameters so

that students can confidently generate and examine many examples

– Operate on variables and functions as “objects” and in a generic way

– Process symbolic language and perform symbolic manipulation

Multiple linked representations that include symbolics

• H0: Students will link representations

• H0: Visual confirmation leads to greater confidence

• H0: Coherence among results contributes to conceptual development

• Example: rotating and solving

Vary symbolic constraints and parameters so that students can

confidently generate and examine many examples

• H0: Experimentation, generalization and pattern recognition will be stimulated

• H0: Literal expressions afford ready access to generality

• H0: If students view results as reliable, they feel confident in reasoning from them

• H0: Immediate feedback may give students more confident in their reasoning

Operate on variables and functions as “objects” and in a generic way

• H0: Students will develop deep understanding of variable, function, expression and function families

• H0:Students will develop a better understanding of the different roles of literal symbols (e.g. variable - parameter distinction)

Process symbolic language and perform symbolic manipulation

• H0: When CAS is the primary generator of symbolic routines, solid conceptual understanding can develop and provide a basis for the learning of by-hand symbolic manipulation

• H0: Students will develop a deeper insight into the structure of formulas and a better symbol sense and symbolic reasoning

• CAS can allow more time to focus on concepts

Process symbolic language and perform symbolic manipulation

(cont)

• H0: CAS production of surprising results may enhance student ability to identify equivalent forms

• H0: CAS production of surprising results may afford students with an opportunity to develop symbolic reasoning

Question 2

What is the nature of the students' development and ability to work with

CAS as an instrument?

CAS capability: CAS can 'scaffold' students’ activity

• Hypothesis

– CAS compensates for weakness in previous learning (e.g. for algebra when beginning calculus)

– Offers relevant representations for thinking, that a student can internalize

CAS capability: CAS provides algebraic phenomena

• CAS can puzzle students and even teachers (unusual notations, different internal algebra..)

• CAS differs from paper and pencil, for which a student generally cannot be surprised by unexpected results

• Hypothesis – Creating a cognitive conflict is an opportunity for

interesting classroom discussion– It challenges the teacher ‘s mathematical

knowledge– It can be a source of difficulty in his (her)

classroom management

• Question– How do students learn to deal with these

phenomena?

CAS capability: CAS as a personal instrument

• A ‘partner’ in the students' everyday mathematical work– Individual interaction with CAS– Role of CAS in social interaction

Individual interaction with CAS

• Ability to decide on what CAS is useful for and what will be better done by hand

Individual interaction with CAS

• Control of the machine– Being aware of possibilities and constraints– Being aware of possible differences between

mathematical and CAS functioning : symbolic notations, internal algorithms…

Individual interaction with CAS

• Monitoring the operation (syntax and semantics of the input/ output, algebraic expectation..)

• Navigating between screens, menu operations

Role of CAS in social interaction

• Role in peer interaction, as a “third party”…

• Role in classroom discussion (reference to the CAS as a 'partner'…)

• Influence of the teacher (privileging of CAS…)

Question 3

How does the way students work on tasks with paper and pencil inform their work in a

CAS environment and vice versa?

What is the interplay between each of these and students' mathematical thinking and

understanding?

• This is a relatively new consideration which hasn't really been focussed on in the past.

• There has been little integration of paper and pencil and CAS approaches.

An integrated approach

• CAS• Paper and pencil (P&P)• Student mathematical thinking

understanding

Considerations

• Relating CAS methods to P&P methods• They may differ in:

– syntax – variable use– equivalence of functions– meaning of operations

• Choosing an approach drawing on the strengths of each

Construction of an overall task strategy combining CAS and P&P

• CAS requires a precise strategy• P&P may cause loss of sight of overall

strategy• CAS can provide opportunities to reflect on

algebra which P&P cannot, such as a system of equations

Input and output

• Requires mathematical understanding to formulate, check and interpret

• Added reliability may increase confidence and progress

• Mental formulation of expected output

Modelling

• CAS may facilitate ‘realistic’ modelling by helping students to:– focus and reflect on model formulation– interpret solutions– validate solutions

Teachers

• How gain:– confidence and ability to reconstruct teaching

using CAS and P&P?– ability to develop deep and rich CAS-active

tasks?

• Positive experiences with CAS will help

Part 2:On the curriculum issues

• ‘Curriculum’ has been interpreted broadly to include:– what is taught– what teachers do– what students do– the teaching and learning resources used– assessment structures

What algebra is taught?

• More exposure to more algebra– CAS will provide stimulus for and access to more

sophisticated algebra– eg parameters arise naturally at an earlier stage– eg, less constraints on the types of functions

and combinations of functions that can be used in real life modelling situations

– eg, new programming possibilities, especially at the tertiary level

Changes of emphases

• Increased discussion about the educational value of routine symbolic manipulation

• Should some symbolic manipulation be assigned to CAS?– Empirical evidence is still needed

• Applications of algebra and modelling rather than routines may become more prominent

Curriculum balance

• Access to CAS is unlikely to free up time– Interpreting CAS outputs and becoming CAS

competent takes time – students may explore algebra more deeply

• Addition of extra algebra topics may be problematic

Curriculum resources

• Curriculum documents & official guidelines– realistic expectations of students and teachers

are needed

• Algebra textbooks– CAS is not just an ‘extra’ or ‘decoration’– fundamental rethinking is needed– should reflect good CAS use for learning– new tasks, including rich tasks, are needed– platform neutrality is desirable, although

difficult; the best tasks are platform neutral

• Assessment materials & guidelines

Implications for students

• More algebra choices available– more decisions needed (eg, substitution)– more approaches to problems are supported

• Written communication– writing for a selected audience: self, peers,

teacher– both familiar and unfamiliar with CAS– teachers of other subjects, external audiences

such as examinations

New expectations of students

• Development of technical expertise– This takes some time– But it is essential for independent CAS use

• Using CAS in other subjects– Science subjects– Economics– Elsewhere at school and beyond

Teachers and CAS

• How do teachers who have themselves learned algebra without CAS make decisions about what students should do with CAS?

Towards the future

NOW … mostly enthusiasts

FUTURE … all algebra teachers

Teacher practice• Little is known of teachers’ views of what

algebra is but they value ‘techniques’ and textbooks. More research is needed.

• Changing teachers’ practice and beliefs is notoriously difficult and often doesn’t result in significant change of practice

• We don’t have ‘solutions’ but note:– PD must involve teachers doing a great deal of

algebra with CAS – Teachers should have positive experiences in their

own learning of algebra with CAS – It is important to allow teachers to create

problems and solutions.

A CAS-algebraic future?

• It requires a tradition for something to become a natural classroom tool

• This takes time, training and continuing dialogue with teachers, schools, governments, students, parents and even ourselves

Teacher knowledge

• Awaiting Teachers’ knowledge for teaching algebra report. Until then ...

• Although we expect a positive correlation between teachers’ subject knowledge of algebra and ‘good’ teaching with CAS this should not be assumed a priori

• Without warning, CAS can confront teachers’ deep understanding of algebra– Especially when teachers are not confident with

algebra– Teachers become learners of algebra with CAS

Knowing and teaching

• Knowledge of mathematics/algebra/CAS is one thing; using this knowledge in teaching is another– e.g. knowing how to factorise, solve and expand

with CAS does not necessarily lead to knowledge of how to provide students with learning activities that provide students with algebraic insight.

External influences

Government and school policies, attitudes and beliefs of parents, students, teachers and researchers impinge on the ways in which teachers actually use CAS for algebra

The influence of exams

• In most, but not all, countries, ‘high stakes’ exams have a profound influence on what is taught and the tools that may be used in teaching

• If changes in algebra are not reflected in exams it is unlikely that there will be significant change in teachers’ practice

• Teachers need familiarity with – algebraic forms and problems – use of CAS and the conventions the help students get

correct answers – past examination papers

A final unresolved issue

Is there a school level at which CAS should be first introduced to students?

CAS Working Group

• Lynda Ball• Roger Brown• Ivan Cnop• Paul Drijvers• David Driver• Peter Flynn• Kathleen Heid• Margaret Kendall• Ivy Kidron• Barry Kissane

• Jean-baptiste Lagrange• David Leigh-Lancaster• Giora Mann• John Monaghan• Robyn Pierce• Michael Thomas

• with advice from

• Michele Artigue• Kaye Stacey