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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

Mathematics Education with Technology 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 handbook for teachers hheugl@aon.at