Didactical principles of integrated learning mathematics with CAS

20-07-2006 7TH DERIVE & TI-CAS CONFERENCE

'Didactical principles of integrated learning

mathematics with CAS'

Peter van der Velden, M.Sc.

Netherlands

p.vd.velden@compaqnet.nl

You can find explanations in the note section below the slide.

To get most profit of this presentation you need Derive (DfW5 or higher).

Most of the examples are linked in the slide and are marked with [ ] to a doc-file, a DfW-file or to a jpg-file.

You will find examples marked with { } in the note section.

“Only few statements and principles have to be acquired by the learner and the teacher from the CAS and then they can visualize, make animations, modify quickly the program data, perform symbolic and numeric calculations step by step and in the whole, and verify deductions on their own.”

[Mihály Klincsik, 2003 (ZDM)]

Technology generates new didactical possibilities:

• order of learning content can change {1}

• order of learning content can change• stud's urged to think about their actions

[2] [3]

• order of learning content can change

• studs urged to think about their actions• the didactic approach can change [4]

• studs stay focused on the essentials {5}

• studs stay focused on the essentials• interesting didactic approach realizable

• studs urged to think about their actions

• the didactic approach can change

• studs stay focused on the essentials

• interesting didactic approach realizable

• studs get experiment-/test-opportunities

Technology generates a lot of learning aims and activities

but students can easily become confused…

Students can easily become confused:• is it?: learning a new theory (supported by

technology)• or is it?: learning to use the technology tool

Students can easily become confused:• is it?: learning a new theory (supported by

technology?)• or is it?: learning to use the technology• is it?: learning or practising new manual skills• or is it?: practising manual skills and may I use

a standard calculator • or is it?: practicing applications

(and may I use the technology, or not?)

Because students have to be aware of

what is meant to be learned,

it is essential all those learning activities are separated, recognizable and supported by learning aims

Three prominent roles of technology:

• "learn to use" (learning the tool)

• "use to learn" (learning math supported by technology)

• "learn to apply" (learning applications with the aid of the tool)

1. "learn to use“(learning the tool)

goals:• skilled in using the tool• confidence but also awareness of limitations• skilled in reading and interpreting results

1. "learn to use"

goals:• skilled in using the tool• confidence but also awareness of limitations• skilled in reading and interpreting results

didactical constraint:• Only math. activities which are directly

connected with their knowledge {9} [10]

1. "learn to use"

didactical needs:• start with what math will be done with the tool • encourage security, accuracy and control {11}

• give enough exercises• show limitations if there are any• show cases the tool is not appropriate (if

there are any) {12}

2. "use to learn“(learning math supported by

technology)

goals:

• understanding and / or practising new math. subjects

2. "use to learn"

didactical constraints and needs:• not interfere tool learning with math learning

2. "use to learn"

didactical constraints and needs:• not interfere tool learning with math learning • new subjects based on preknowledge;

(avoid a “closed” black box) {13} [13] [14]

didactical constraints and needs:• not interfere tool learning with math learning

• new subjects based on preknowledge; (avoid a “closed” black box) {13} [13] [14]

• guided explorative learning implies:• learning aims indicated: explicit & in advance• only successful with guiding questions {15}

• ask explicit answering questions (reflection)• ask conclusions & offer possibility to verify

2. "use to learn"

didactical constraints and needs:• not interfere tool learning with math learning • new subjects based on preknowledge;

(avoid a “closed” black box)• learning aims indicated: explicit & in advance• only successful with guiding questions• ask explicit answering questions (reflection)• ask conclusions & offer possibility to verify • challenge to experiment (trials) [16]

2. "use to learn"(guided explorative learning)

3. "learn to apply" (learning applications with the aid

of the tool)goals:

• Systematic problem solving

3. "learn to apply"

goals:

• Systematic problem solving• Dealing unexpected tool results {17}

3. "learn to apply"

goals:

• Systematic problem solving

• Dealing unexpected tool results

• Effective use of the tool: – when for what / in which cases– be secure and accurate– do checks and reflections

3. Learning applications• occurring faults

of course

• analysis of the problem can be wrong

• faults while modeling the problem

3. Learning applications• occurring faults

of course

• analysis of the problem can be wrong

• faults while modeling the problem

extra faults using technology

• using the technology incorrect

• misinterpretations of the solutions

3. Learning applications• conquer these possible faults

• preventive: working secure & accurate

3. Learning applications• conquer these possible faults

• preventive: working secure & accurate

• by reflection on (final and sub) results: – first a rough but critical judgment

then - if necessary - – checking steps of solving process– checking details precisely – final check

3. Learning applications• learning reflection

Especially when working with technology tools it is really important that students learn how to reflect on and to have control on their activities and to get a critical attitude on their own work

3. Learning applications• learning reflection

Especially when working with technology tools it is really important that students learn how to reflect on and to have control on their activities and to get a critical attitude on their own work

An example how a student can work {18} [18a] [18] [19] {20} [20] {21}

An advice about collaborative learning

collaborative learning can be more efficient and effective

with such complicated activities (i.e. a continual alternation of thinking, doing and reflecting)

which are involved with integrated learning [22]

summarylearning activities in math courses and textbooks• existing and lasting

– learning a new theory – learning and practising manual skills

(with and without a calculator)– practising applications

(with and without a calculator)

summarylearning activities in math courses and textbooks• existing and lasting

– learning a new theory – learning and practising manual skills

(with and without a calculator)– practising applications

(with and without a calculator)

• in courses and textbooks which integrate CAS– learning a new theory supported by CAS – learning and practising the CAS– practising applications with CAS– applications become more sophisticated– opportunity: experimenting on students own level – opportunity: testing own work or the work of others.

conclusions• While students are doing varying activities,

they can easily become confused and wander, so they loose the purpose of their activity. That is why

• All those different activities must be separated and recognizable and supported by explicit intentions so that the student is aware of what of what is meant to be learned.

• And with every example, problem or exercise he or she must know which tool is meant or not (i.e. a calculator or a CAS or any tool or no tool at all)

'Didactical principles of integrated learning mathematics with CAS‘

by Peter van der Velden, M.Sc.

If you have any question about didactics or about my textbooks,

please contact me:

p.vd.velden@compaqnet.nl

Didactical principles of integrated learning mathematics with CAS

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