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'Didactical principles of integrated learning mathematics with CAS'PowerPoint Presentation

'Didactical principles of integrated learning mathematics with CAS'

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### 'Didactical principles of integrated learning mathematics with CAS'

### 'Didactical principles of integrated learning mathematics with CAS‘by Peter van der Velden, M.Sc.

Peter van der Velden, M.Sc.

Netherlands

p.vd.velden@compaqnet.nl

7TH DERIVE & TI-CAS CONFERENCE

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.

7TH DERIVE & TI-CAS CONFERENCE

“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)]

7TH DERIVE & TI-CAS CONFERENCE

Technology generates new didactical possibilities: 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.”

7TH DERIVE & TI-CAS CONFERENCE

Technology generates new didactical possibilities: 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.”

- order of learning content can change {1}

7TH DERIVE & TI-CAS CONFERENCE

Technology generates new didactical possibilities: 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.”

- order of learning content can change
- stud's urged to think about their actions[2] [3]

7TH DERIVE & TI-CAS CONFERENCE

Technology generates new didactical possibilities: 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.”

- order of learning content can change
- studs urged to think about their actions
- the didactic approach can change[4]

7TH DERIVE & TI-CAS CONFERENCE

Technology generates new didactical possibilities: 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.”

- 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}

7TH DERIVE & TI-CAS CONFERENCE

Technology generates new didactical possibilities: 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.”

- 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
- interesting didactic approach realizable[6]

7TH DERIVE & TI-CAS CONFERENCE

Technology generates new didactical possibilities: 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.”

- order of learning content can change
- 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

7TH DERIVE & TI-CAS CONFERENCE

Technology generates a lot of learning aims and activities 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.”

but students can easily become confused…

7TH DERIVE & TI-CAS CONFERENCE

Technology generates a lot of learning aims and activities 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.”

Students can easily become confused:

- is it?: learning a new theory (supported by technology)
- or is it?: learning to use the technology tool

7TH DERIVE & TI-CAS CONFERENCE

Technology generates a lot of learning aims and activities 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.”

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?)

7TH DERIVE & TI-CAS CONFERENCE

Technology generates a lot of learning aims and activities 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.”

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

7TH DERIVE & TI-CAS CONFERENCE

Three prominent roles of technology 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.” :

- "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)

7TH DERIVE & TI-CAS CONFERENCE

"learn to use“ 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.” (learning the tool)

goals:

- skilled in using the tool
- confidence but also awareness of limitations
- skilled in reading and interpreting results

{8}

7TH DERIVE & TI-CAS CONFERENCE

1. "learn to use" 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.”

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]

7TH DERIVE & TI-CAS CONFERENCE

1. "learn to use" 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.”

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}

7TH DERIVE & TI-CAS CONFERENCE

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

goals:

- understanding and / or practising new math. subjects

7TH DERIVE & TI-CAS CONFERENCE

2. "use to learn" 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.”

didactical constraints and needs:

- not interfere tool learning with math learning

7TH DERIVE & TI-CAS CONFERENCE

2. "use to learn" 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.”

didactical constraints and needs:

- not interfere tool learning with math learning
- new subjects based on preknowledge; (avoid a “closed” black box) {13}[13] [14]

7TH DERIVE & TI-CAS CONFERENCE

2. "use to learn" 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.”

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

7TH DERIVE & TI-CAS CONFERENCE

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

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]

7TH DERIVE & TI-CAS CONFERENCE

3. "learn to apply" 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.” (learning applications with the aid of the tool)

goals:

- Systematic problem solving

7TH DERIVE & TI-CAS CONFERENCE

3. "learn to apply" 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.”

goals:

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

7TH DERIVE & TI-CAS CONFERENCE

3. "learn to apply" 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.”

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

7TH DERIVE & TI-CAS CONFERENCE

3. Learning applications 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.” • occurring faults

of course

- analysis of the problem can be wrong
- faults while modeling the problem

7TH DERIVE & TI-CAS CONFERENCE

3. Learning applications 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.” • 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

7TH DERIVE & TI-CAS CONFERENCE

3. Learning applications 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.” • conquer these possible faults

- preventive: working secure & accurate

7TH DERIVE & TI-CAS CONFERENCE

3. Learning applications 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.” • conquer these possible faults

- preventive: working secure & accurate
- by reflection on (final and sub) results:
- first a rough but critical judgmentthen - if necessary -
- checking steps of solving process
- checking details precisely
- final check

7TH DERIVE & TI-CAS CONFERENCE

3. Learning applications 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.” • 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

7TH DERIVE & TI-CAS CONFERENCE

3. Learning applications 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.” • 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}

7TH DERIVE & TI-CAS CONFERENCE

An advice about 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.” 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]

7TH DERIVE & TI-CAS CONFERENCE

summary 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.”

learning 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)

7TH DERIVE & TI-CAS CONFERENCE

summary 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.”

learning 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.

7TH DERIVE & TI-CAS CONFERENCE

conclusions 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.”

- 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)

7TH DERIVE & TI-CAS CONFERENCE

If you have any question about didactics or about my textbooks, please contact me:

p.vd.velden@compaqnet.nl

7TH DERIVE & TI-CAS CONFERENCE

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