didactical principles of integrated learning mathematics with cas

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'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). - PowerPoint PPT Presentation

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  • 'Didacticalprinciples of integrated learning mathematics with CAS'Peter van der Velden, M.Sc.Netherlandsp.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. [Mihly Klincsik, 2003 (ZDM)]

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates new didactical possibilities:

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates new didactical possibilities:order of learning content can change {1}

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates new didactical possibilities:order of learning content can changestud's urged to think about their actions [2] [3]

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates new didactical possibilities:order of learning content can changestuds urged to think about their actionsthe didactic approach can change [4]

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates new didactical possibilities:order of learning content can changestuds urged to think about their actionsthe didactic approach can change [4]studs stay focused on the essentials {5}

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates new didactical possibilities:order of learning content can changestuds urged to think about their actionsthe didactic approach can change [4]studs stay focused on the essentialsinteresting didactic approach realizable [6]

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates new didactical possibilities:order of learning content can changestuds urged to think about their actionsthe didactic approach can changestuds stay focused on the essentialsinteresting didactic approach realizablestuds get experiment-/test-opportunities

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates a lot of learning aims and activitiesbut students can easily become confused

    7TH DERIVE & TI-CAS CONFERENCE

  • Technology generates a lot of learning aims and activitiesStudents 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 activitiesStudents can easily become confused:is it?: learning a new theory (supported by technology?)or is it?: learning to use the technologyis it?: learning or practising new manual skillsor 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 activitiesBecause 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:

    "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(learning the tool)

    goals:skilled in using the toolconfidence but also awareness of limitationsskilled in reading and interpreting results{8}

    7TH DERIVE & TI-CAS CONFERENCE

  • 1. "learn to use"goals:skilled in using the toolconfidence but also awareness of limitationsskilled in reading and interpreting resultsdidactical constraint:Only math. activities which are directly connected with their knowledge {9} [10]

    7TH DERIVE & TI-CAS CONFERENCE

  • 1. "learn to use"didactical needs:start with what math will be done with the tool encourage security, accuracy and control {11}give enough exercisesshow limitations if there are anyshow cases the tool is not appropriate (if there are any) {12}

    7TH DERIVE & TI-CAS CONFERENCE

  • 2. "use to learn(learning math supported by technology) goals:understanding and / or practising new math. subjects

    7TH DERIVE & TI-CAS CONFERENCE

  • 2. "use to learn"

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

    7TH DERIVE & TI-CAS CONFERENCE

  • 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]

    7TH DERIVE & TI-CAS CONFERENCE

  • 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] guided explorative learning implies:learning aims indicated: explicit & in advanceonly 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"(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 advanceonly successful with guiding questionsask explicit answering questions (reflection)ask conclusions & offer possibility to verify challenge to experiment (trials) [16]

    7TH DERIVE & TI-CAS CONFERENCE

  • 3. "learn to apply" (learning applications with the aid of the tool)goals:Systematic problem solving

    7TH DERIVE & TI-CAS CONFERENCE

  • 3. "learn to apply"goals:Systematic problem solvingDealing unexpected tool results {17}

    7TH DERIVE & TI-CAS CONFERENCE

  • 3. "learn to apply"goals:Systematic problem solvingDealing unexpected tool resultsEffective use of the tool: when for what / in which casesbe secure and accuratedo checks and reflections

    7TH DERIVE & TI-CAS CONFERENCE

  • 3. Learning applications occurring faultsof course analysis of the problem can be wrong faults while modeling the problem

    7TH DERIVE & TI-CAS CONFERENCE

  • 3. Learning applications occurring faultsof course analysis of the problem can be wrong faults while modeling the problem extra faults using technology using the technology incorrectmisinterpretations of the solutions

    7TH DERIVE & TI-CAS CONFERENCE

  • 3. Learning applications conquer these possible faultspreventive: working secure & accurate

    7TH DERIVE & TI-CAS CONFERENCE

  • 3. Learning applications conquer these possible faultspreventive: working secure & accurate by reflection on (final and sub) results: first a rough but critical judgment then - if necessary - checking steps of solving processchecking details precisely final check

    7TH DERIVE & TI-CAS CONFERENCE

  • 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

    7TH DERIVE & TI-CAS CONFERENCE

  • 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}

    7TH DERIVE & TI-CAS CONFERENCE

  • An advice about collaborative learningcollaborative 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

  • summarylearning activities in math courses and textbooksexisting 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

  • summarylearning activities in math courses and textbooksexisting 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 CASlearning a new theory supported by CAS learning and practising the CASpractising applications with CASapplications become more sophisticatedopportunity: experimenting on students own level opportunity: testing own work or the work of others.

    7TH DERIVE & TI-CAS CONFERENCE

  • conclusionsWhile 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

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