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The Role of Technology in Teaching Mathematics

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mental calculation walking. Technology. Mathematics (intellectual) Moving (physical) ... exercises for those manual skills which are needed for calculations. ... – PowerPoint PPT presentation

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Title: The Role of Technology in Teaching Mathematics


1
The Role of Technology in Teaching Mathematics
  • Bernhard Kutzler
  • ACDCA
  • (Austrian Center for Didactics of Computer
    Algebra)
  • RISC / University of Linz

2
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3
Technology
  • Mathematics (intellectual) Moving (physical)

4
Technology
  • Mathematics (intellectual) Moving (physical)
  • walking

5
Technology
  • Mathematics (intellectual) Moving (physical)
  • mental calculation walking

6
Technology
  • Mathematics (intellectual) Moving (physical)
  • mental calculation walking
  • cycling

7
Technology
  • Mathematics (intellectual) Moving (physical)
  • mental calculation walking
  • paper pencil calculation cycling

8
Technology
  • Mathematics (intellectual) Moving (physical)
  • mental calculation walking
  • paper pencil calculation cycling
  • driving a car

9
Technology
  • Mathematics (intellectual) Moving (physical)
  • mental calculation walking
  • paper pencil calculation cycling
  • calc/comp calculation (automation) driving a
    car

10
Technology
  • Mathematics (intellectual) Moving (physical)
  • mental calculation walking
  • paper pencil calculation cycling
  • calc/comp calculation (automation) driving a
    car
  • using a wheel chair

11
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

12
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13
Technology in Education
  • (1) Tool for trivialization
  • (2) Tool for experimentation
  • (3) Tool for visualization
  • (4) Tool for concentration

14
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

15
Trivialization
  • Solve (more) complicated examples
  • Solve (more) realistic examples
  • Buchberger White-Box/Black-Box-Principle

16
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17
Experimentation
18
Experimentation
example
algorithm
19
Experimentation
observe
example
conjecture
algorithm
20
Experimentation
observe
example
conjecture
algorithm
theorem
prove
21
Experimentation
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
22
Experimentation
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
23
Experimentation
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
Buchbergers Creativity Spiral
24
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
25
Phase of Experimentation
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
26
Phase of Experimentation
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
Phase of Exactification
27
Phase of Experimentation
Phase of Application
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
Phase of Exactification
28
INDUCTION
Phase of Application
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
Phase of Exactification
29
INDUCTION
Phase of Application
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
DEDUCTION
30
INDUCTION
PRODUCTION
new example
apply
observe
example
conjecture
algorithm
new algorithm
theorem
prove
implement
DEDUCTION
31
Experimentation
  • Learning by doing an observing (Goethe)
  • Learning by trial and error
  • Genetic learning

32
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33
Visualization
  • Psychology of learning
  • Reinforcement works best if it follows the
    action immediately.

34
Visualization
  • Learning through immediate feedback
  • Demana/Waits Power of visualization

35
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36
Concentration
  • 5x - 6 2x 15

37
Concentration
  • 5x - 6 2x 15
  • Choose equivalence transformation - 2x

38
Concentration
  • 5x - 6 2x 15
  • Choose equivalence transformation - 2x
  • Simplify 3x - 6 15

39
Concentration
  • 5x - 6 2x 15
  • Choose equivalence transformation - 2x
  • Simplify 3x - 6 15
  • Choose equivalence transformation 6

40
Concentration
  • 5x - 6 2x 15
  • Choose equivalence transformation - 2x
  • Simplify 3x - 6 15
  • Choose equivalence transformation 6
  • Simplify 3x 21

41
Concentration
  • 5x - 6 2x 15
  • Choose equivalence transformation - 2x
  • Simplify 3x - 6 15
  • Choose equivalence transformation 6
  • Simplify 3x 21
  • Most typical beginners mistake

42
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

43
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

44
Continuous change of levels
Concentration
Choose equivalence transformation
Simplify
45
Continuous change of levels
Concentration
Choose equivalence transformation
Simplify
46
Continuous change of levels
Concentration
Choose equivalence transformation
Simplify
47
Continuous change of levels
Concentration
Choose equivalence transformation
Simplify
48
When using algebraic calculators
Concentration
Choose equivalence transformations
Simplify
49
The house of mathematics"
The Scaffolding Method
arithmetics
50
The house of mathematics"
The Scaffolding Method
elementary algebra
arithmetics
51
The house of mathematics"
The Scaffolding Method
equations
elementary algebra
arithmetics
52
The house of mathematics"
The Scaffolding Method
.....
elementary trigonometry
equations
elementary algebra
arithmetics
53
How we teach learn today step 1
The Scaffolding Method
topic A
54
How we teach learn today step 1
step 2
The Scaffolding Method
topic B
topic A
topic A
55
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

56
Building a house" with technology (cf. Solve
a linear equation)
The Scaffolding Method
57
The Scaffolding Method
topic A
Example topic A Simplify topic B Choose
equivalence transformation
58
The Scaffolding Method
topic B
topic A
topic A
Example topic A Simplify topic B Choose
equivalence transformation
59
The Scaffolding Method
topic B
topic B
topic A
topic A
topic A
Example topic A Simplify topic B Choose
equivalence transformation
60
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)

61
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62
automation, compensation
63
trivialization, experimentation, visualization,
concentration
automation, compensation
64
scaffolding method
trivialization, experimentation, visualization,
concentration
automation, compensation
65
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66
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
67
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
68
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
69
Problem Solving Training
Model M
Real World
PM
P
translate
Bernhard Kutzler, Austria, 2001
70
Problem Solving Training
Model M
Real World
PM
P
translate
calculate
S M
Bernhard Kutzler, Austria, 2001
71
Problem Solving Training
Model M
Real World
PM
P
translate
calculate
S M
S
translate
Bernhard Kutzler, Austria, 2001
72
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
73
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
74
We need BALANCE
Bernhard Kutzler, Austria, 2001
75
We need BALANCE
Problem Solving Training
Mental Training
Mathematics
Bernhard Kutzler, Austria, 2001
76
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77
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
78
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
79
Concluding Remarks
80
Shakespear Nothing is either good or bad. Only
thinking makes it so.
81
Corollary Calculators and computers are neither
good nor bad teaching tools. Only using makes
them so.
82
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 !!!

83
B.T. ( before technology)
Good Students
Weak Students
84
A.T. ( after technology)
Good Students
Weak Students
85
Should we use technology for teaching?
86
Should we use technology for teaching? Absolutely
yes, because we owe it to our students!
87
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
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