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The Role of Technology in Teaching Mathematics

- Bernhard Kutzler
- ACDCA
- (Austrian Center for Didactics of Computer

Algebra) - RISC / University of Linz

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Technology

- Mathematics (intellectual) Moving (physical)

Technology

- Mathematics (intellectual) Moving (physical)
- walking

Technology

- Mathematics (intellectual) Moving (physical)
- mental calculation walking

Technology

- Mathematics (intellectual) Moving (physical)
- mental calculation walking
- cycling

Technology

- Mathematics (intellectual) Moving (physical)
- mental calculation walking
- paper pencil calculation cycling

Technology

- Mathematics (intellectual) Moving (physical)
- mental calculation walking
- paper pencil calculation cycling
- driving a car

Technology

- Mathematics (intellectual) Moving (physical)
- mental calculation walking
- paper pencil calculation cycling
- calc/comp calculation (automation) driving a

car

Technology

- Mathematics (intellectual) Moving (physical)
- mental calculation walking
- paper pencil calculation cycling
- calc/comp calculation (automation) driving a

car - using a wheel chair

Technology

- Mathematics (intellectual) Moving (physical)
- mental calculation walking
- paper pencil calculation cycling
- calc/comp calculation (automation) driving a

car - calc/comp calculation (compensation) using a

wheel chair

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Technology in Education

- (1) Tool for trivialization
- (2) Tool for experimentation
- (3) Tool for visualization
- (4) Tool for concentration

Trivialization

- New basic operations
- Scientific (numeric) calculator made 3.9876

5.1238 equal to 4 5 - Graphic calculator / graphing software

made Graph 2sin(x/2)cos(x) equal to

Graph x1 - CAS made diff ln(sin(cos(tan(?(x²-x1)))))

equal to diff x²

Trivialization

- Solve (more) complicated examples
- Solve (more) realistic examples
- Buchberger White-Box/Black-Box-Principle

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Experimentation

Experimentation

example

algorithm

Experimentation

observe

example

conjecture

algorithm

Experimentation

observe

example

conjecture

algorithm

theorem

prove

Experimentation

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

Experimentation

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

Experimentation

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

Buchbergers Creativity Spiral

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

Phase of Experimentation

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

Phase of Experimentation

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

Phase of Exactification

Phase of Experimentation

Phase of Application

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

Phase of Exactification

INDUCTION

Phase of Application

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

Phase of Exactification

INDUCTION

Phase of Application

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

DEDUCTION

INDUCTION

PRODUCTION

new example

apply

observe

example

conjecture

algorithm

new algorithm

theorem

prove

implement

DEDUCTION

Experimentation

- Learning by doing an observing (Goethe)
- Learning by trial and error
- Genetic learning

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Visualization

- Psychology of learning
- Reinforcement works best if it follows the

action immediately.

Visualization

- Learning through immediate feedback
- Demana/Waits Power of visualization

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Concentration

- 5x - 6 2x 15

Concentration

- 5x - 6 2x 15
- Choose equivalence transformation - 2x

Concentration

- 5x - 6 2x 15
- Choose equivalence transformation - 2x
- Simplify 3x - 6 15

Concentration

- 5x - 6 2x 15
- Choose equivalence transformation - 2x
- Simplify 3x - 6 15
- Choose equivalence transformation 6

Concentration

- 5x - 6 2x 15
- Choose equivalence transformation - 2x
- Simplify 3x - 6 15
- Choose equivalence transformation 6
- Simplify 3x 21

Concentration

- 5x - 6 2x 15
- Choose equivalence transformation - 2x
- Simplify 3x - 6 15
- Choose equivalence transformation 6
- Simplify 3x 21
- Most typical beginners mistake

Concentration

- 5x - 6 2x 15
- Choose equivalence transformation - 2x
- Simplify 3x - 6 15
- Choose equivalence transformation 6
- Simplify 3x 21
- Most typical beginners mistake
- I want to get rid of the 3 - 3

Concentration

- 5x - 6 2x 15
- Choose equivalence transformation - 2x
- Simplify 3x - 6 15
- Choose equivalence transformation 6
- Simplify 3x 21
- Most typical beginners mistake
- I want to get rid of the 3 - 3
- I got rid of the 3 x 18

Continuous change of levels

Concentration

Choose equivalence transformation

Simplify

Continuous change of levels

Concentration

Choose equivalence transformation

Simplify

Continuous change of levels

Concentration

Choose equivalence transformation

Simplify

Continuous change of levels

Concentration

Choose equivalence transformation

Simplify

When using algebraic calculators

Concentration

Choose equivalence transformations

Simplify

The house of mathematics"

The Scaffolding Method

arithmetics

The house of mathematics"

The Scaffolding Method

elementary algebra

arithmetics

The house of mathematics"

The Scaffolding Method

equations

elementary algebra

arithmetics

The house of mathematics"

The Scaffolding Method

.....

elementary trigonometry

equations

elementary algebra

arithmetics

How we teach learn today step 1

The Scaffolding Method

topic A

How we teach learn today step 1

step 2

The Scaffolding Method

topic B

topic A

topic A

How we teach learn today step 1

step 2

The Scaffolding Method

topic B

topic A

topic A

?

- As long as topic A is not perfected, step 2 does

not work

Building a house" with technology (cf. Solve

a linear equation)

The Scaffolding Method

The Scaffolding Method

topic A

Example topic A Simplify topic B Choose

equivalence transformation

The Scaffolding Method

topic B

topic A

topic A

Example topic A Simplify topic B Choose

equivalence transformation

The Scaffolding Method

topic B

topic B

topic A

topic A

topic A

Example topic A Simplify topic B Choose

equivalence transformation

The Scaffolding Method

- If it is not necessary to use CAS, it is

necessary not to use CAS. (H. Heugl, Austrian

DERIVE Project) - Scaffolding Method Pedagogically justified

sequence of using and not using technology for

trivialization, visualization, experimentation,

and concentration in the sense of an automation

or a compensation tool. (B. Kutzler)

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automation, compensation

trivialization, experimentation, visualization,

concentration

automation, compensation

scaffolding method

trivialization, experimentation, visualization,

concentration

automation, compensation

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Mathematics

- the principal language of science technology

(the book of nature is written in the language

of mathematics - G Galilei) as such it is the

key for understanding and shaping the world

around us ? problem solving training - the principal means of educating the human mind

(C F Gauss) as such it is the key for mental

wellness and health and a prerequisite for

developing problem solving skills ? mental

training

Bernhard Kutzler, Austria, 2001

Mathematics Mental Training Problem Solving

Training

- This model
- helps to develop appropriate curricula
- helps to plan the proper use of technology

Bernhard Kutzler, Austria, 2001

Mental Training

- (Physical) sports guarantees physical wellness

and health - Increase in automation is a threat for our mental

wellness and health - What must a person with a healthy mind be able to

do? - We needed the school subject sports
- Now we need mental training ( mental

sports) - as part of mathematics !

Bernhard Kutzler, Austria, 2001

Problem Solving Training

Model M

Real World

PM

P

translate

Bernhard Kutzler, Austria, 2001

Problem Solving Training

Model M

Real World

PM

P

translate

calculate

S M

Bernhard Kutzler, Austria, 2001

Problem Solving Training

Model M

Real World

PM

P

translate

calculate

S M

S

translate

Bernhard Kutzler, Austria, 2001

Problem Solving Training

- Problem solving in todays classrooms turns

into exercises for those manual skills which are

needed for calculations. - Most students are afraid of translations -

because nobody teaches them. - When using technology for calculations we have

(more) time for teaching translations.

Bernhard Kutzler, Austria, 2001

Teaching Goals

- Goal of mental training mental wellness/health

preparation for problem solving technology may

be used as a training tool (mathematical home

trainer) - Goal of problem solving training skill of using

given tools for solving given problems technology

should be used as much as possible. - Mental training I go jogging.
- Problem solving I need to be in London tomorrow

noon.

Bernhard Kutzler, Austria, 2001

We need BALANCE

Bernhard Kutzler, Austria, 2001

We need BALANCE

Problem Solving Training

Mental Training

Mathematics

Bernhard Kutzler, Austria, 2001

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Assessment

- For testing the achievements of the mental

training no tool is permitted, not even a simple

four function calculator. - For testing the achievements of the problem

solving training every tool is permitted, in

particular powerful computer tools. - Mental training compulsory exercise (in ice

skating) - Problem solving voluntary exercise / freestyle

Bernhard Kutzler, Austria, 2001

Assessment Two-Tier Exams

- For testing the achievements of the mental

training no tool is permitted, not even a simple

four function calculator. - For testing the achievements of the problem

solving training every tool is permitted, in

particular powerful computer tools. - Mental training compulsory exercise (in ice

skating) - Problem solving voluntary exercise / freestyle

Bernhard Kutzler, Austria, 2001

Concluding Remarks

Shakespear Nothing is either good or bad. Only

thinking makes it so.

Corollary Calculators and computers are neither

good nor bad teaching tools. Only using makes

them so.

If technology is used properly, it leads to ...

- ... more efficient teaching and learning
- ... more independent productive student activity
- ... more student creativity
- ... an increased importance of the teacher
- ------gt Teacher Training !!!

B.T. ( before technology)

Good Students

Weak Students

A.T. ( after technology)

Good Students

Weak Students

Should we use technology for teaching?

Should we use technology for teaching? Absolutely

yes, because we owe it to our students!

b.kutzler_at_aon.at http//b.kutzler.com contains pap

ers going with this presentation, a list of

relevant conferences, ...

Bernhard Kutzler, Austria, 1999