first central- and eastern european conference on computer algebra and dynamic geometry systems in...

First Central- and Eastern European Conference on Computer Algebra and Dynamic Geometry

Systems in Mathematics Education

Herschel’s heritage and today’s technology integration: a postulated parallel

Kenneth RuthvenUniversity of Cambridge



“… so very useful for a great variety of purp-oses that every person engaged in… comput-ations … or physico-mathematical inquiries of any description will [find it valuable to have] always at hand.”

Herschel (1833): On the investigation of the orbits of revol-ving double stars



It had also been taken up by an educational movement aiming to give pupils ‘practical’ experience and encourage their ‘self-activity’

Such experience was suited to ‘mathematical laboratories’ aimed at greater ‘correlation’ between mathematical and scientific study

By the end of the nineteenth century, the price of the technology had fallen by two orders of magnitude and its use spread widely



In the “laboratory method” teachers work with pupils individually or in small cooperative groups, “to develop on the part of every student the true spirit of research, and an apprec-iation, practical as well as theoretic, of the fundam-ental methods of science.”

Moore (1903): On the Foundations of Mathematics


Systems in Mathematics EducationMyers (1903): The Laboratory Method in the Secondary School• drawing instruments• cross-ruled paper• logarithmic slide

rules• tape measures• weighing scales• surveying equipment• barometers and

thermometers• pendulums and

gyroscopes• cords and pulleys• spherical blackboards



In 1912, ICMI held a study conference on ‘methods of intuition and experiment in mathematical teaching in secondary schools’



Board of Education (1912): Suggestions for the Teaching of Arithmetic“Squared paper is to be found in any well-equipped elementary school”

Godfrey (1912): Methods of intuition and experiment in secondary schools“The use of graphical methods in elementary algebra teachings is universal and entirely a 20th century development”

Smith (1913): Intuition and experiment in mathematical teaching in secondary schools“Of the value of squared millimetre paper there is no question anywhere”


Systems in Mathematics EducationBoard of Education (1909, 1914)

“Little danger to health is likely if no paper with rulings less than one-tenth of an inch apart is used.”

“The use of paper ruled in squares for the working of arithmetic examples has no real educational value”Clarified proliferating interpretations of ‘graphic(al) algebra’



“symptomatic of a much wider transformation of mathematical curricula in response to… demands … outside the academic mathematical community…the growth of new educational philosophies; the development of science teaching and the associated need for mathematics correlation; the growing demands of engin-eering and technical education.”

Brock & Price (1980): Squared Paper in the Nine-teenth Century: Instrument of Science and Engineering, and Symbol of Reform in Mathematical Education


Systems in Mathematics EducationKlein (1908, 1909): Elementary Mathematics from an Advanced Standpoint“the function concept at the very centre of instruction”

“introduced as early as possible with constant use of the graphical method”

“far-reaching fusion of arithmetic and geometry”



“Numerical evaluation of algebraic expressions, accurate construction of geometrical problems, plotting of curves, graphical solutions, use of logarithms in computation, in fact the bulk of the methods which have been adopted in the class teaching of Mathematics… these to me do not mean Practical Mathematics. Such operations can be conducted in a class-room without the use of further apparatus than a box of instruments, some squared paper, and a table of logarithms”.

Fawdry (1915): Laboratory Work in Connection with Mathematics



Disciplinary congruence with influential contemporary trend in mathematics

External currency across wider mathem-atical practice within and beyond school

Adoptive ease as regards incorporation in classroom practice and curricular activityPedagogical value in terms of range of beneficial uses outweighing antagonisms

Graph paper in early 20thC



Disciplinary congruence formerly seen in terms of place of ‘algorithmic thinking’

External currency widely acknowledged but not well understood in substance

Adoptive ease helped by trends towards personal access and classroom availabilityPedagogical value perceived in terms of facilitating routine, accentuating features, helping investigation, aiding consolidation

Computer tools in early 21stC



Disciplinary congruence with mathematics as computational and modelling discipline

External currency plausible but not salient

Adoptive ease problematic because adapted rather than designed for educational use, notably in respect of uncontrolled outputPedagogical value seen in terms of taking over routine, aiding experimentation, raising attention, providing multiple representations

Computer algebra systems


Systems in Mathematics EducationFey (2006): Review of The Didactical Challenge of Symbolic CalculatorsSome developments may have “focused too narrowly on the applications of CAS to traditional algebraic symbol manipulation problems and have looked too hard to find subtle problems that are not well handled by CAS functions”

“Learning how to use CAS functions to support applied problem solving is not as complicated or as fraught with the potential for mistakes as learning how to use the same tool for more general algebraic reasoning”



Disciplinary congruence with unrealised dynamic elaboration of classical geometry

External currency less plausible because primarily associated with educational use

Adoptive ease seen as problematic when conceived primarily as episodic visual aid rather than as pervasive disciplinary toolPedagogical value seen in terms of disciplined enquiry by main advocates, but of guided discovery by ordinary teachers

Dynamic geometry systems



Hölzl (2001): Using Dynamic Geometry Soft-ware to Add Contrast to Geometric Situations “In the literature… the reader is provided with numerous examples of how DGS can support the heuristic phase of problem solving… However, a closer look at various examples [questions whether] the software is used effectively to support a methodical and an active style of knowledge acquisition. Often…[the] DGS is used only in a verifying manner: that is, learners are just supposed to vary geometric configurations and confirm empirically more or less explicitly stated facts. ”



Adoptive ease reframed by naturalising tools and minimising their proliferation

Disciplinary congruence strengthened by developing functional, dynamic and algorithmic thinking as pervasive themes

Pedagogical value recognised and elabor-ated for spectrum of educational purposes

External currency better established

Towards stronger integration

first central- and eastern european conference on computer algebra and dynamic geometry systems in...

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