1. the White Box/Black Box principle
Learning Algebra
by using
the White Box/Black Box principle
Helmut Heugl

2. Worries about the use of technology
The reality?
Our hope
Sustainability
Quelle: Bärbel Barzel
My answer:
A strategy against „Black-Box-Teaching and Learning“

3. The White Box/ Principle 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
Phase 1: The White Box Phase -
phase of recognizing exploring
and consolidating
Reasonable selecting algorithms and concepts developed in further white boxes
Calculating by using technology as a black box
Testing and interpreting
Phase 2: The Black Box Phase -
phase of phase of knowledgeable
application

4. The White Box/ Principle Black Box
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
The White Box/ Principle
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
Phase 1: The White Box Phase -
phase of recognizing exploring
and consolidating
Reasonable selecting algorithms and concepts developed in further white boxes
Calculating by using technology as a black box
Testing and interpreting
Phase 2: The Black Box Phase -
phase of phase of knowledgeable
application

5. Termbox white Learning Algebra using the White Box/Black Box Principle
Using CAS
for testing and experimenting
Investigating the equivalence of terms
Generating a formula
Calculating with terms
Interpreting terms
Competence
of structure recognition
Ex 1
Ex 2
Ex 3
Ex 4

6. Ex 1: Structure recognition
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

7. Equationsbox White Termbox black
Without CAS
Equationsbox
White
Termbox
black
With CAS
Investigating several sorts of solutions; testing the correctnes of solutions; acquiring tool competences
For necessary term opartions using CAS
as a black box
Developing strategies for solving equations e.g. equivalence transformations
Ex 5
Ex 6
Ex 7

8. 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
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.

9. Box of systems of euqations –
white
Ex 8
Equationsbox black
Ex 9
For solving singular equations using CAS as a black box by using the „solve“ commans
Termbox
black
Ex 10
For necessary term operations using CAS as a black box
Investigating several sorts of solutions; testing the correctnes of solutions.
Developing strategies for solving systems of equations e.g.
the substitution method
the equalization method
the addition method

10. White Box Systems of Equations
Changes of cognition caused by technology
Working with the names of the equations
With technology
Technology
changes cognition
With technology
Working with the equations
Without technology
Working into the equations

11. Ex 13: Equivalent transformations of inequalities
Inequality Box
White
Termbox
black
For necessary term opartions using CAS
as a black box
Investigating several sorts of solutions; testing the correctnes of solutions.
Developing strategies for solving inequalities e.g. equivalence transformations
Ex 13: Equivalent transformations of inequalities

12. Applicationsbox – white Box of systems of equations black
Analysisbox black
Box of systems of equations black
….box
Equationsbox black
Termbox black

13. 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

14. 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

15. A personal advertising
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, 1991
A personal advertising

17. Working with the equations
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
Technology allows a direct translation of the verbal formulated activities into the language of mathematics

18. Working with the names of the equations
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

19. Mathematics Education with Technology
Online Materials:
TI-Nspire-Files of the book you can find at the web site:
Mathematics Education with Technology
a didactical handbook
for teachers