learning algebra by using the white box/black box principle helmut heugl
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Learning Algebra
by using
the White Box/Black Box principle
Helmut Heugl
Sustainability
Our hope
Quelle: Bärbel Barzel
My answer:A strategy against „Black-Box-Teaching and Learning“
The reality?Worries about the use of technology
The White Box/ Principle
Phase 1: The White Box Phase -
phase of recognizing exploring
and consolidating
Phase 2: The Black Box Phase -
phase of phase of knowledgeable
application
Black Box
The learning process proceeds in two phases when teaching mathematics according to this recursive model:
Developing of concepts and algorithms.
Experimenting and proving. Calculating without the use of
technology. CAS supported usage of Black
Boxes which were explored in earlier White Boxes
Reasonable selecting algorithms and concepts developed in further white boxes
Calculating by using technology as a black box
Testing and interpreting
The White Box/ Principle
Phase 1: The White Box Phase -
phase of recognizing exploring
and consolidating
Phase 2: The Black Box Phase -
phase of phase of knowledgeable
application
Black Box
The learning process proceeds in two phases when teaching mathematics according to this recursive model:
Developing of concepts and algorithms.
Experimenting and proving. Calculating without the use of
technology. CAS supported usage of Black
Boxes which were explored in earlier White Boxes
Reasonable selecting algorithms and concepts developed in further white boxes
Calculating by using technology as a black box
Testing and interpreting
Content of the black box is only the execution of the operations and not the understanding of the mathematical concepts and strategies that is to say the mathematical thinking technology
Learning Algebra using the White Box/Black Box Principle
Termboxwhite
• Generating a formula• Calculating with terms• Interpreting terms
Competenceof structure recognition
Investigating the equivalence of terms
Using CASfor testing and experimenting
Ex 1 Ex 2 Ex 3 Ex 4
2 x y 2 x 2 y
2 x y 2 x y x 2 y
??????
Ex 1: Structure recognition
2
2
7 21 3 304
6 5 252
.( )
.
xx
x xx
Enter the following expression
by using the math templates of TI Nspire
by using brackets
Ex 2: Investigating the equivalence of terms
Three strategies:
Entering the expressions: Simplifying by CAS allows some decisions
Using algebra commands like „factor“ or „expand“
Calculating the difference or the quotient of terms
Ex 4: Structure recognition when calculating in Analysisi
The results which CAS tools offer sometimes differ from the expected structure of solutions => students need structure recognition competence
EquationsboxWhite
Termboxblack
Developing strategies for solving equations e.g. equivalence transformations
For necessary term opartions using CASas a black box
Investigating several sorts of solutions; testing the correctnes of solutions; acquiring tool competences
Ex 5 Ex 6 Ex 7
3 x 12 3
x 9
3 x 12 3
3 x 3 9
Without CAS
With CAS
Ex 5: Equivalence transformations for solving equations
By experimentimg with equivalent transformations students try to develop their own strategies for solving euations
Ex 6: Solving equations with higher degrees
Step 1: Solving by factorizing
Step 2: Using the „solve“ command
Ex 7: Visualizing of equivalence transformations
1x 1 x 5
2 a) Solve the equation by equivalence transformations
You can interpret the left and right expressions of the equations as functions le and ri. Draw the graphs of the functions le and ri after every equivalence transformation and describe the result.
b) Multiply both sides of the equation by x. Is it an equivalence transformation? Draw the graphs of the generated functions le and ri.
Box of systems of euqations –white
Equationsbox black
Termboxblack
Developing strategies for solving systems of equations e.g.- the substitution method- the equalization method- the addition method
For necessary term operations using CAS as a black box
For solving singular equations using CAS as a black box by using the „solve“ commans
Investigating several sorts of solutions; testing the correctnes of solutions.
Ex 8
Ex 9
Ex 10
White Box Systems of Equations
Without technology Working into the equations
With technology Working with the equations
Working with the names of the equations
With technology
Technology
changes
cognition
Changes of cognition caused by technology
Inequality BoxWhite
Termboxblack
Developing strategies for solving inequalities e.g. equivalence transformations
For necessary term opartions using CASas a black box
Investigating several sorts of solutions; testing the correctnes of solutions.
Ex 13: Equivalent transformations of inequalities
Applicationsbox – white
Box of systems of equations black
Equationsbox black
Termbox black
Analysisbox black
….box
The /White Box
Principle
Black Box
The learning process proceeds in two phases when teaching mathematics according to this recursive model:
Phase 1: The Black Box Phase -
phase of experimental and active learning to come to
suppositions by using technology as a black box
Phase 2: The White Box Phase -
phase of justifying and proving, of developing
algorithms and defining new concepts
The Black Box/White Box Principle in differential calculus
The central thinking technology of calculus is the idea of limits. Students must experience, calculate themselves and interpret the derivative as the limit of the quotient of differences.
Examples for experimenting by using technology as a black box:
Investigating the derivative of power functions
Finding conjectures about “continuity” and “differentiability”
Ex 11
Ex 12
Why learning Algebra with CAS
Expert
the teacher
Expert
the cognitive system student&tool
Change of the role
If we understand cognition as a functional system which encompasses man and tools and the further material and social context, then new tools can change cognition qualitively and generate new competences. Learning is then not simply the development of existing competences but rather a systematical construction of functional cognitive systems
The computer and computer software must therefore be seen as an expansion and a strengthening of cognition.
W. Dörfler, 1991A personal advertising
Ich solve the 1st equation with respect to y
I substitute y in the second
equation
and solve this equation with
respect to x
Then I use this result in the
first equation and solve with
respect to y
Working with the equations
Technology allows a direct translation of the verbal formulated activities into the
language of mathematics
I store the two equations with the names gl1 and gl2 and operate with the names of the
equations
Working with the names of the equations
Technology supports the development of new mathematical language lements
Online Materials:
TI-Nspire-Files of the book
you can find at the web site:
http://mathe-mit-technologie.
veritas.at
Mathematics Education with Technology
- a didactical handbookfor teachers