1. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 1 'Didactical principles of integrated learning mathematics with CAS' Peter van der Velden, M.Sc. Netherlands p.vd.velden@compaqnet.nl

2. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 2 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.

3. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 3 “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)]

4. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 4 Technology generates new didactical possibilities:

5. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 5 Technology generates new didactical possibilities: order of learning content can change {1}

6. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 6 Technology generates new didactical possibilities: order of learning content can change stud's urged to think about their actions [2] [3] [2] [3]

7. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 7 Technology generates new didactical possibilities: order of learning content can change studs urged to think about their actions the didactic approach can change [4] [4]

8. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 8 Technology generates new didactical possibilities: order of learning content can change studs urged to think about their actions the didactic approach can change [4] [4] studs stay focused on the essentials {5}

9. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 9 Technology generates new didactical possibilities: order of learning content can change studs urged to think about their actions the didactic approach can change [4] [4] studs stay focused on the essentials interesting didactic approach realizable [6] [6]

10. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 10 Technology generates new didactical possibilities: 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

11. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 11 Technology generates a lot of learning aims and activities but students can easily become confused…

12. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 12 Technology generates a lot of learning aims and activities Students can easily become confused: is it?: learning a new theory (supported by technology) or is it?: learning to use the technology tool

13. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 13 Technology generates a lot of learning aims and activities 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?)

14. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 14 Technology generates a lot of learning aims and activities 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

15. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 15 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)

16. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 16 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 {8}

17. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 17 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][10]

18. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 18 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}

19. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 19 2. "use to learn“ (learning math supported by technology) goals: understanding and / or practising new math. subjects

20. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 20 2. "use to learn" didactical constraints and needs: not interfere tool learning with math learning

21. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 21 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] [13] [14]

22. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 22 didactical constraints and needs: not interfere tool learning with math learning new subjects based on preknowledge; (avoid a “closed” black box) {13} [13] [14] [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"

23. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 23 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] [16] 2. "use to learn" (guided explorative learning)

24. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 24 3. "learn to apply" (learning applications with the aid of the tool) goals: Systematic problem solving

25. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 25 3. "learn to apply" goals: Systematic problem solving Dealing unexpected tool results {17}

26. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 26 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

27. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 27 3. Learning applications occurring faults of course analysis of the problem can be wrong faults while modeling the problem

28. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 28 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

29. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 29 3. Learning applications conquer these possible faults preventive: working secure & accurate

30. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 30 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

31. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 31 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

32. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 32 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}[18a] [18] [19] [20]

33. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 33 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] [22]

34. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 34 summary 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)

35. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 35 summary 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.

36. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 36 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)

37. 20-07-20067TH DERIVE & TI-CAS CONFERENCE 37 '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