Mathematics has a bad reputation … … implies that teaching and learning should … Haapasalo...

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

Lenni Haapasalo / ACTM 20072

Lenni Haapasalo / Bratislava 20.04.20072

What can be learned from the history of mathematics?

apply

calculate construct

order

argue invent

playevaluate

proofs Heuristics(“Analysis”)

axiomatics

beliefs, values,

esthetics

algorithms architecture, geometry

gametheorygametheory, , stochasticsstochastics

modeling


The importance of procedural knowledge

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 syntaxof its representation (”touch of doing”)

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)


On the other hand, eduactional needs to emphasize

- seeing links between things

- to be able to make choices

- thinking abilitises


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.139-157.

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)

Is there a way to combine these opposite directions?

C generated by P

logical reason G or SA

Genetic principle


Quasisystematic model:Dynamic interaction and simultaneous activation

Example:(freely downloadable from

http://www.joensuu.fi/lenni/programs.html


Proportionality - Linear Dependence - Gradient of a Straight Line through Origin


Simultaneous activation


Novice learner (“Alien”)


Expert learner

http://www.joensuu.fi/mathematics/MathDistEdu/Animations2MentalModels/SavonlinnaLETTET2005/LeTTET_Figure4JSP.html

Utilizing SA method

with ClassPad

http://www.classpad.org

Main Application work areaCurrently 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: Geometry window:

Linear equation in x and y An infinite line

Equation of circle in x and y A circle

2-dimensional vector A point or vector

2 2 matrix A transformation

Equation y = f(x) A curven 2 matrix A polygon (each column represents a vertex)


Example 1: Gradient / Orientation

1. Alg: Write y= 0.Then copy it.

2. Geom: Paste, a horizontalline 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 lineappears.


Example 2. Conics

1. Draw a circle

2. Drag-and-drop it into albegraic window

3. Manipulate equation

4. Drag-and-drop it into geometric window


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!

SA method

to foster

mental links

students developed themselves

Construction of informelmathematics:

Versatility of newtechnologies

vs

minimalist Instruction

Systematic planning

vs

Minimalsit Instruction

Minimalist Instruction

Carroll, J. M. 1990. The Nurnberg Funnel: Designing Minimalist Instruction for Practical Computer Skill. Cambridge, Massachusetts: The MIT Press, 7–10.

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.”


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;


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 pre-specified, 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. 47-48) vs. Types of learning cannot be identified independent of the content and context of learning.


Methods (Syst vs Min) 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;

... 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.

What does modern technology imply for the assessment and

problem posing?

Mathematics has a bad reputation … … implies that teaching and learning should … Haapasalo...

Documents

Transcript of Mathematics has a bad reputation … … implies that teaching and learning should … Haapasalo...