2 Actualities (for colaboration)

Let us improve the following arguments, please write me your better arguments, criteria, ... :

Computer Technology in Mathematics Teaching: Dangers and Limitations

The following attempt to improve awareness in the use of computers in (mathematics) teaching has been motivated by a very lively discussion at the Fifth International Conference on Technology in Mathematics Teaching held in Klagenfurt in August 2001. The arguments presented here were discussed in part by my working group on Dangers and Limitations at this conference.

This interest at once dispelled the discouragement that I felt after my examination, in two articles in the MNU journal in 1985 at the very outset of the development of computer programs for teaching, of the essential problems and questions in this field had produced only a meagre response. [Computer als Herausforderung - zur Sklavenarbeit? Fragen zur Computerwelt und möglichen Reaktionen durch allgemeinbildende Schulen. MNU, 38. Jg. Heft 1, Januar 1985 / Heft 2, März 1985]

"What may be crucial is whether society develops a self-awareness that in its ordinary mathematics usage it is arranging itself in certain ways, and hence is doing something to itself." [Philip Davis/ Reuben Hersh: Descartes' Dream. San Diego/Boston/New York (Harcourt Brace Jovanovich) 1986, p.304]

"The computer could make mathematics instruction, at least in sections, unnecessary, if we do not see instruction in a wide context and under a substantial requirement for education." [Bourguinon in: B. Engquist u.a. (Ed.): Mathematics Unlimited - 2001 and Beyond, (Springer) 2001]

Applying the words of Davis and Hersh to computer usage in mathematics teaching requires us first of all to be conscious of the problems we must face. Practice, however, is usually dominated by taking any opportunity without facing the problems that may perhaps arise. The new possibilities, in the end, may cause at best not necessarily a weaker situation in mathematics teaching but also not a better one. In considering the possible advantages of computer use in teaching it is essential to add the old question of Hans Freudenthal of what we may lose by our change of teaching. We may not necessarily be able to avoid losing something, and perhaps we can accept this, here and there, but we should at least do so consciously and should think about necessary compensatory efforts. We will tackle these points offering some examples of problems that have to be resolved under the headings of Pedagogical Problems, Dangers and Limitations and Criteria.

Pedagogical Problems (P) / Dangers and Limitations (D) / Criteria (C)

P: The formation of concepts: We know very little about how students build up new concepts in relation to their activities or to the framing and the tools used in the activities.
D: To lose possibilities of concept formation that had previously been in practice even if they were not known explicitly. / New software may lead students to new misconceptions.

P: Understanding actions as opposed to only executing actions.
D: Producing results without thinking, without having understood any of the details involved, is much easier using computers. What happens if actions are performed without understanding the underlying concepts and processes, as, for example, in statistics?
C: It therefore becomes even more urgent to develop criteria for understanding. In what way does a new software enhance understanding? What are signs of understanding? How may they be tested by exams? The development of deep understanding also means involving students emotionally in their work.

P: The acquisition of world is mediated by presentations, that is, by the presence of adequate imagination and the construction of appropriate new mental images.
D: To offer images before pupils have first learnt to construct these images mentally.
C: To permit the use of "finished products" instead of doing things "by hand" we need to know much more about the right situation in the psychological development of the pupil.

P: Dynamics on the screen = dynamics in the mind? Example: Promoting functional thinking is an old educational principle (Friedrich Herbarth, Felix Klein) that is basically undisputed but is hardly successful in practice. Many have the expectation that mobile pictures would bring about a break-through in comprehension. But we do not have any indications that "the functional relations of mobile geometrical figures become more effectively transformed into thinking habits, when visualised on the computer display" than with earlier methods. [Katja Krüger: Erziehung zum funktionalen Denken. Thesis, Frankfurt/Main 1999, S.288]
D: Without practical experience, the kinaesthetic sense of movement is not conceptually incorporated. "The pupil's experience of moving the linkages of a mechanical model is wholly different from that of the use of a (computer) program." [Gerhard Hofsäß]
C: What basic experience and comprehension is necessary to be able to use a program meaningfully? The emphasis of our attention must move from the satisfactory control of a didactical model by the pupil to asking what concepts the pupils have thereby formed. But take care not to introduce inappropriate bias: If, for instance, one introduces the integral as a calculation of lower and upper totals, the calculations are often so laborious that the pupil may meanwhile forget the original problem. Thus a CAS can be suitable for the first formation of a concept.

P: The most important thing is to ask questions. For self directed activity it is therefore necessary to provide meaning directed structures in advance. Thus also with other teaching aids.
D: DGS: Danger of mere actionism when drawing [Peter Bender]. CAS: Danger of meaningless production of charts: "The possibility of producing graphs by simply pushing a button provokes the continual production of new graphs. Some groups of pupils produced over 50 graphs in half an hour on the display. Reading and interpreting ... is of course then no longer possible." [Hans George Weigand: Eine explorative Studie zum computerunterstützten Arbeiten mit Funktionen. In: JMD, Jg. 20 (1999) number 1, S.47] Example: We usually do not achieve that the pupils experience algebra as help. (How simple problems turn up, if we can write an equation!) Probably, because they could not build up sufficient experience by practical working. (Ringkowski: all those who solved the physics problems mentally solved them correctly. Those who did it by transposing formulas, made mistakes.) Will CAS increase this alienation? (Pupils believe that they must use the aid and do not dare without; but they cannot succeed with it, because they are not really mastering the tool. It masters them.) / The satisfaction of functioning on both sides: pupils do not inquire any further, whilst the teachers overlook gaps of understanding. / More consciousness is required of the pupils. The application of each computer tool really requires thinking to and fro between usage and background theory. That this is often uncomfortable is not a reason to avoid it.
C: Does a tool enlarge mental activity or will it limit to a narrow path? / In what way does a new software invite interaction? / Most usefully one should start with a problem, and then use the appropriate software for that context.

P: The classroom should stimulate expanding activities, rather than contracting ones. / The existence of the computer only devalues the swotting and the computerizable. Do weaker pupils therefore need more "pre computer experience"?
Example 1: Specifically, how can we sufficiently train the necessary spatial perception and imagination for visualisations of all kinds before the pupil deals with the respective programs?
D: To try to do also this training with the computer. / Too early too abstract, and too little "educational activity" [Hans Jürgen Elschenbroich].
Example 2: "Successmaker", a swotting program for weak pupils: If such a program is used inappropriately, couldn't pupil's be enabled by it to proceed with apparent success but without adequate concepts, thus establishing in themselves not more but less mathematics? Can we get by, for example, without the concept of 2/3 of 60, which cannot be generated by such a program?
When different independent teachers inevitably make didactical errors, the consequence of these eventually even out. Isn't it a very dangerous bias if very few such programs are used by all pupils in the same way?
D: If we do not offer resistance here, it is possible that we will create new obstacles in spite of computer's assistance.

P: Achieving the intended correspondence between the intentions of the teacher (the syllabus) and the understanding of the students is always difficult. New tools often redefine a problem or cause new problems. DGS may lead pupils automatically to different questions.
D: Teachers see through traditionally tinted glasses. Since they get involved in the new possibilities, pupils perceive another world. "A further category of changes through the mediation of the tool deals with the new behaviour of the objects due to the mediation. A very good example is given by the behaviour of points in a dynamic geometry environment. In a static environment, there is no reason to question the behaviour of objects when some basic elements are moved because there is no such possibility. But now it is natural to ask what could be or should be the trajectory of a point on a segment AB when one of its endpoints is dragged? There is no answer in Euclidean classical geometry because the question is meaningless. Thus the mediation of this geometry in a dynamic geometry environment actually creates objects of a new kind. The "danger" for the user is that the new nature of the object may be not visible. They may be transparent for the users who may believe that the objects are identical to those with which they are familiar." [Inge Schwank (Ed): Europ. Res. in Math. Ed. I. Osnabrück (Forschungsinst. f. Mathematikdidaktik) 1999, Group "Tools and Technologies" S.185] The tool may thus allow a different knowledge to develop than the teacher expected: New gaps are possible between pupils' conceptions and teacher's conceptions and thus new communication difficulties or actual communication break down becomes a possibility.

P: Complete view, long-term consequences: "Unthinking visualisation causes imagination to be diminished. ... the circumstance is widely overlooked that we require just this imagination, to ... understand simulations as models rather than latitudes. Handling computer simulations, in particular visualisations, therefore requires a higher quality of imagination to avoid the identification with the actor within the simulation that one can shift by mouse clicks through data space." [Gernot Böhme: Bildung als Widerstand. In: DIE ZEIT 38/1999]
Correct media education could mean at the very least that pupils design simulations themselves and thus experiences the selective decisions that underly them. In mathematics that would mean assembling building blocks from basic components of programs (self developed tools in DGS etc.). / Very often simple novelty effects are touted as unique consequences of certain methods.

P: Enable students to act competently in respect of the realities of life.
D: Confusing model and reality. For instance, because of the normative power of daily computer usage, of the accommodating "exactness" and "correctness". Example: "the result looks accurate, therefore the underlying assumption must be correct", or "it looks correct, therefore the model must be correct." [F. Förster u.a., in: W. Herget u.a. (Hg): Standardthemen ... Proceedings, Hildesheim (Franzbecker) 2000, S.155f] This is no small problem. We have also seriously underestimated the socialization effect of television. More and more its pernicious influence becomes obvious: Televi-sion distorts reality. [viz. M. Myrtek: Fernsehen, Schule und Verhalten - Untersuchung zur emotionalen Beanspruchung von Schülern. (Hans Huber) 2000.] The degradation of emotional reactions found by the authors could also become a problem for instruction. Particularly in the constructivist view, learning presupposes the engagement of personal mental activity on an emotional level.
C: The central question is: what is carried over from school to later life? / Should we only prepare students for some professional use of electronic boxes? Whilst they don't always have to know how such a black box works in detail, they should have a good idea of what it can do and can't do. / A warning example is given by the accident at Chernobyl, where the scientists' understanding of exponential growth turned out not to be deep enough for safety.

P: Key role of language: "The computer can promote communication." No, that is generally due to the method of working. / When students write a diary they have to express their thoughts actively, whilst the teacher may derive some insight into the students' understanding.

P: We need to develop a curriculum based on solutions to these problem

2 Actualities (for colaboration)

 

Let us improve the following arguments, please write me your better arguments, criteria, ... :

 

Computer Technology in Mathematics Teaching: Dangers and Limitations

 

The following attempt to improve awareness in the use of computers in (mathematics) teaching has been motivated by a very lively discussion at the Fifth International Conference on Technology in Mathematics Teaching held in Klagenfurt in August 2001. The arguments presented here were discussed in part by my working group on Dangers and Limitations at this conference.

 

This interest at once dispelled the discouragement that I felt after my examination, in two articles in the MNU journal in 1985 at the very outset of the development of computer programs for teaching, of the essential problems and questions in this field had produced only a meagre response. [Computer als Herausforderung - zur Sklavenarbeit? Fragen zur Computerwelt und möglichen Reaktionen durch allgemeinbildende Schulen. MNU, 38. Jg. Heft 1, Januar 1985 / Heft 2, März 1985]

 

"What may be crucial is whether society develops a self-awareness that in its ordinary mathematics usage it is arranging itself in certain ways, and hence is doing something to itself." [Philip Davis/ Reuben Hersh: Descartes' Dream. San Diego/Boston/New York (Harcourt Brace Jovanovich) 1986, p.304]

 

"The computer could make mathematics instruction, at least in sections, unnecessary, if we do not see instruction in a wide context and under a substantial requirement for education." [Bourguinon in: B. Engquist u.a. (Ed.): Mathematics Unlimited - 2001 and Beyond, (Springer) 2001]

 

Applying the words of Davis and Hersh to computer usage in mathematics teaching requires us first of all to be conscious of the problems we must face. Practice, however, is usually dominated by taking any opportunity without facing the problems that may perhaps arise. The new possibilities, in the end, may cause at best not necessarily a weaker situation in mathematics teaching but also not a better one. In considering the possible advantages of computer use in teaching it is essential to add the old question of Hans Freudenthal of what we may lose by our change of teaching. We may not necessarily be able to avoid losing something, and perhaps we can accept this, here and there, but we should at least do so consciously and should think about necessary compensatory efforts. We will tackle these points offering some examples of problems that have to be resolved under the headings of Pedagogical Problems, Dangers and Limitations and Criteria.

 

 

Pedagogical Problems (P) / Dangers and Limitations (D) / Criteria (C)

 

P: The formation of concepts: We know very little about how students build up new concepts in relation to their activities or to the framing and the tools used in the activities.

D: To lose possibilities of concept formation that had previously been in practice even if they were not known explicitly. / New software may lead students to new misconceptions.

 

P: Understanding actions as opposed to only executing actions.

D: Producing results without thinking, without having understood any of the details involved, is much easier using computers. What happens if actions are performed without understanding the underlying concepts and processes, as, for example, in statistics?

C: It therefore becomes even more urgent to develop criteria for understanding. In what way does a new software enhance understanding? What are signs of understanding? How may they be tested by exams? The development of deep understanding also means involving students emotionally in their work.

 

P: The acquisition of world is mediated by presentations, that is, by the presence of adequate imagination and the construction of appropriate new mental images.

D: To offer images before pupils have first learnt to construct these images mentally.

C: To permit the use of "finished products" instead of doing things "by hand" we need to know much more about the right situation in the psychological development of the pupil.

 

P: Dynamics on the screen = dynamics in the mind? Example: Promoting functional thinking is an old educational principle (Friedrich Herbarth, Felix Klein) that is basically undisputed but is hardly successful in practice. Many have the expectation that mobile pictures would bring about a break-through in comprehension. But we do not have any indications that "the functional relations of mobile geometrical figures become more effectively transformed into thinking habits, when visualised on the computer display" than with earlier methods. [Katja Krüger: Erziehung zum funktionalen Denken. Thesis, Frankfurt/Main 1999, S.288]

D: Without practical experience, the kinaesthetic sense of movement is not conceptually incorporated. "The pupil's experience of moving the linkages of a mechanical model is wholly different from that of the use of a (computer) program." [Gerhard Hofsäß]

C: What basic experience and comprehension is necessary to be able to use a program meaningfully? The emphasis of our attention must move from the satisfactory control of a didactical model by the pupil to asking what concepts the pupils have thereby formed. But take care not to introduce inappropriate bias: If, for instance, one introduces the integral as a calculation of lower and upper totals, the calculations are often so laborious that the pupil may meanwhile forget the original problem. Thus a CAS can be suitable for the first formation of a concept.

 

P: The most important thing is to ask questions. For self directed activity it is therefore necessary to provide meaning directed structures in advance. Thus also with other teaching aids.

D: DGS: Danger of mere actionism when drawing [Peter Bender]. CAS: Danger of meaningless production of charts: "The possibility of producing graphs by simply pushing a button provokes the continual production of new graphs. Some groups of pupils produced over 50 graphs in half an hour on the display. Reading and interpreting ... is of course then no longer possible." [Hans George Weigand: Eine explorative Studie zum computerunterstützten Arbeiten mit Funktionen. In: JMD, Jg. 20 (1999) number 1, S.47] Example: We usually do not achieve that the pupils experience algebra as help. (How simple problems turn up, if we can write an equation!) Probably, because they could not build up sufficient experience by practical working. (Ringkowski: all those who solved the physics problems mentally solved them correctly. Those who did it by transposing formulas, made mistakes.) Will CAS increase this alienation? (Pupils believe that they must use the aid and do not dare without; but they cannot succeed with it, because they are not really mastering the tool. It masters them.) / The satisfaction of functioning on both sides: pupils do not inquire any further, whilst the teachers overlook gaps of understanding. / More consciousness is required of the pupils. The application of each computer tool really requires thinking to and fro between usage and background theory. That this is often uncomfortable is not a reason to avoid it.

C: Does a tool enlarge mental activity or will it limit to a narrow path? / In what way does a new software invite interaction? / Most usefully one should start with a problem, and then use the appropriate software for that context.

 

P: The classroom should stimulate expanding activities, rather than contracting ones. / The existence of the computer only devalues the swotting and the computerizable. Do weaker pupils therefore need more "pre computer experience"?

Example 1: Specifically, how can we sufficiently train the necessary spatial perception and imagination for visualisations of all kinds before the pupil deals with the respective programs?

D: To try to do also this training with the computer. / Too early too abstract, and too little "educational activity" [Hans Jürgen Elschenbroich].

Example 2: "Successmaker", a swotting program for weak pupils: If such a program is used inappropriately, couldn't pupil's be enabled by it to proceed with apparent success but without adequate concepts, thus establishing in themselves not more but less mathematics? Can we get by, for example, without the concept of 2/3 of 60, which cannot be generated by such a program?

When different independent teachers inevitably make didactical errors, the consequence of these eventually even out. Isn't it a very dangerous bias if very few such programs are used by all pupils in the same way?

D: If we do not offer resistance here, it is possible that we will create new obstacles in spite of computer's assistance.

 

P: Achieving the intended correspondence between the intentions of the teacher (the syllabus) and the understanding of the students is always difficult. New tools often redefine a problem or cause new problems. DGS may lead pupils automatically to different questions.

D: Teachers see through traditionally tinted glasses. Since they get involved in the new possibilities, pupils perceive another world. "A further category of changes through the mediation of the tool deals with the new behaviour of the objects due to the mediation. A very good example is given by the behaviour of points in a dynamic geometry environment. In a static environment, there is no reason to question the behaviour of objects when some basic elements are moved because there is no such possibility. But now it is natural to ask what could be or should be the trajectory of a point on a segment AB when one of its endpoints is dragged? There is no answer in Euclidean classical geometry because the question is meaningless. Thus the mediation of this geometry in a dynamic geometry environment actually creates objects of a new kind. The "danger" for the user is that the new nature of the object may be not visible. They may be transparent for the users who may believe that the objects are identical to those with which they are familiar." [Inge Schwank (Ed): Europ. Res. in Math. Ed. I. Osnabrück (Forschungsinst. f. Mathematikdidaktik) 1999, Group "Tools and Technologies" S.185] The tool may thus allow a different knowledge to develop than the teacher expected: New gaps are possible between pupils' conceptions and teacher's conceptions and thus new communication difficulties or actual communication break down becomes a possibility.

 

P: Complete view, long-term consequences: "Unthinking visualisation causes imagination to be diminished. ... the circumstance is widely overlooked that we require just this imagination, to ... understand simulations as models rather than latitudes. Handling computer simulations, in particular visualisations, therefore requires a higher quality of imagination to avoid the identification with the actor within the simulation that one can shift by mouse clicks through data space." [Gernot Böhme: Bildung als Widerstand. In: DIE ZEIT 38/1999]

Correct media education could mean at the very least that pupils design simulations themselves and thus experiences the selective decisions that underly them. In mathematics that would mean assembling building blocks from basic components of programs (self developed tools in DGS etc.). / Very often simple novelty effects are touted as unique consequences of certain methods.

 

P: Enable students to act competently in respect of the realities of life.

D: Confusing model and reality. For instance, because of the normative power of daily computer usage, of the accommodating "exactness" and "correctness". Example: "the result looks accurate, therefore the underlying assumption must be correct", or "it looks correct, therefore the model must be correct." [F. Förster u.a., in: W. Herget u.a. (Hg): Standardthemen ... Proceedings, Hildesheim (Franzbecker) 2000, S.155f] This is no small problem. We have also seriously underestimated the socialization effect of television. More and more its pernicious influence becomes obvious: Televi-sion distorts reality. [viz. M. Myrtek: Fernsehen, Schule und Verhalten - Untersuchung zur emotionalen Beanspruchung von Schülern. (Hans Huber) 2000.] The degradation of emotional reactions found by the authors could also become a problem for instruction. Particularly in the constructivist view, learning presupposes the engagement of personal mental activity on an emotional level.

C: The central question is: what is carried over from school to later life? / Should we only prepare students for some professional use of electronic boxes? Whilst they don't always have to know how such a black box works in detail, they should have a good idea of what it can do and can't do. / A warning example is given by the accident at Chernobyl, where the scientists' understanding of exponential growth turned out not to be deep enough for safety.

 

P: Key role of language: "The computer can promote communication." No, that is generally due to the method of working. / When students write a diary they have to express their thoughts actively, whilst the teacher may derive some insight into the students' understanding.

 

P: We need to develop a curriculum based on solutions to these problems.

s.