Mathematics has a bad reputation. implies that. teaching and learning should be somehow related with the psychological situation of the learner

Transcription

1 Mathematics has a bad reputation implies that teaching and learning should be somehow related with the psychological situation of the learner Lenni Haapasalo, University of Joensuu

2 What can be learned from the history of mathematics? proofs axiomatics order Heuristics ( Analysis ) beliefs, values, esthetics evaluate argue invent play gametheory, stochastics algorithms calculate construct apply modeling Lenni Lenni Haapasalo Haapasalo / Bratislava / ACTM architecture, geometry 2

3 The importance of procedural knowledge Lenni Haapasalo / ACTM

4 Procedurale knowledge (P ) means knowledge about dynamic and successful application of specific rules, algorithms or procedures by using one or more different representations Therefore it is (normally) not only necessary to have specific knowledge about the corresponding objects, but also about the syntax of its representation ( touch of doing )

5 Hippocampus "If you can't do it with the hippocampus you can't do it with anything (Theodore Berger of the University of Southern California in Los Angeles)

6 On the other hand, eduactional needs to emphasize - seeing links between things - to be able to make choices - thinking abilitises Lenni Haapasalo / ACTM

7 Conceptual knowledge (C ) means understanding of elements, of their connections as semantic network, and knowledge about dynamic changes between different representations. These elements of the network can be e. g. concepts, rules (algorithms, procedures etc.), even problems (a solved problem can generate a new concept or a new rule). Haapasalo, L. & Kadijevich, Dj. (2000). Two Types of Mathematical Knowledge and Their Relation. Journal für Mathematik-Didaktik 21 (2), pp Lenni Haapasalo / ACTM

8 Educational principle Invest for C and develop ability to reflect (metacognition). Logical background DI or SA. Anderson (1983) Carpenter (1986) Byrnes & Wasik (1991) Hiebert & Carpenter (1992) Haapasalo (1993)

9 Is there a way to combine these opposite directions?

10 Genetic principle C generated by P logical reason G or SA

11 Quasisystematic model: Dynamic interaction and simultaneous activation Lenni Haapasalo / ACTM

12 Example: (freely downloadable from

13 Proportionality - Linear Dependence - Gradient of a Straight Line through Origin Lenni Haapasalo / ACTM

14 Simultaneous activation Lenni Haapasalo / ACTM

15 Lenni Haapasalo / ACTM

16 Novice learner ( Alien ) Lenni Haapasalo / ACTM

17 Expert learner Lenni Haapasalo / ACTM

18 SavonlinnaLETTET2005/LeTTET_Figure4JSP.html

19 Utilizing SA method with ClassPad

20 Main Application work area Currently displayed screen (Graph Editor, Graph, Conic Editor, Table, Sequence Editor, Geometry, 3D Graph Editor 3D Graph, Statistics, List Editor, and Numeric Solver). Main Application window: Linear equation in x and y Equation of circle in x and y 2-dimensional vector Geometry window: An infinite line A circle A point or vector 2 2 matrix A transformation Equation y = f(x) n 2 matrix A curve A polygon (each column represents a vertex)

21 Example 1: Gradient / Orientation 1. Alg: Write y= 0. Then copy it. 2. Geom: Paste, a horizontal line appears. 3. Geom: Rotate 45 degrees. The matching line appears. 4. Geom: Copy it. 5. Alg: Paste, equation y=x appears. 6. Alg: Change the equation to y=2x. Then copy this equation. 7. Geom: Paste it, the matching line appears. Lenni Haapasalo / ACTM

22 Example 2. Conics 1. Draw a circle 2. Drag-anddrop it into albegraic window 3. Manipulate equation 4. Drag-anddrop it into geometric window Lenni Haapasalo / ACTM

23 Example 3: Transformations 1. Construct a segment CJ. 2. Drag-and-drop it to algebraic window. 3. Construct a line perpendicular to CJ. 4. Drag-and-drop it to algebraic window. 5. Make hypotheses concerning the gradients. 6. Make any general transformation. 7. Two matrices appear in algebraic window, 8. Fill them and look what happens! Lenni Haapasalo / ACTM

24 Construction of informel mathematics: SA method to foster mental links students developed themselves

25 Versatility of new technologies vs minimalist Instruction

26 Systematic planning vs Minimalsit Instruction

27 Minimalist Instruction Carroll, J. M The Nurnberg Funnel: Designing Minimalist Instruction for Practical Computer Skill. Cambridge, Massachusetts: The MIT Press, Carroll noticed that learners often tend to jump the gun avoid careful planning resist detailed systems of instructional steps be subject to learning interference from similar tasks have difficulty recognizing, diagnosing, and recovering from their errors.

28 Characteristics (Syst vs Min) Pre-established goals vs. Goals determined from authentic tasks; Identified prerequisites vs. On-going assessment of learner needs; Step by step sequenced instruction vs. Processes of learning modelled and coached for students with unscripted teacher responses; Elimination of error vs. Use errors for instruction Comprehensive coverage vs. Learners construct multiple perspectives or solutions; Emphasis on tutorial pacing vs. Emphasis on learning by doing and exploring; Feedback for correct responses vs. Criterion for success is transfer of learning; Lenni Haapasalo / ACTM

29 Presuppositions (Syst vs Min) Learning causes an observable change in the learner vs. Learning causes a change in perception and action potential; Learning outcomes can be pre-specified vs. Specific content and outcomes cannot be prespecified, although a core knowledge domain may be specified; Skills should be learned one at a time vs. Skills are learned within social contexts; Learned skills should build on previously acquired skills vs. Learning focuses on the process of knowledge construction and development of reflexive awareness of that process; Learning and knowledge are hierarchical in nature vs. Constructing knowledge through discussion and collaboration; There are five types of learning (verbal information, intellectual skills, cognitive strategies, attitudes, and motor skills; see Gagne, 1985, pp ) vs. Types of learning cannot be identified independent of the content and context of learning. Lenni Haapasalo / ACTM

30 Not (according to Gagne 1985) (1) Gaining attention (2) Informing learners of objective (3) Stimulating recall of prior learning (4) Presenting the content (5) Providing learning guidance (6) Eliciting performance (7) Providing feedback (8) Assessing performance (9) Enhancing retention and transfer; Methods (Syst vs Min)... but (according to van der Meij & Carroll (1998, p. 2) (1) Choose an action oriented approach, (2) Anchor the tool in the task domain, (3) Support error recognition and recovery, and (4) Support reading to do, study, and locate. Lenni Haapasalo / ACTM

31 Lenni Haapasalo / ACTM

32 What does modern technology imply for the assessment and problem posing?