Origin and purpose of this book

«Music is arithmetical work of the mind,
to which it remains concealed that it thinks in numbers».
Gottfried Wilhelm von Leibniz (17th century).

Since the classical antiquity music and mathematics have been described as a wonderful pair. Yet until now the lucky relation between the two disciplines has been associated neither with primary school mathematics nor with the topics of primary school musical instruction. Topics of an interdisciplinary discussion have rather been questions of harmonics, acoustics, and - for some years - the digitalization of sound or the discussion about the pitch of instruments and the mathematical conceivability of composition and interpretation.

The domains of physics, stochastics, combinatorics and informatics dealing with these topics do not have much in common with the curriculum of primary school. Therefore, it is not surprising that the kick-off to «Maths by Music», a project about educational material, did not develop out of this ancient relationship of the two disciplines but out of the observation of learning children and of successful instruction.

These observations have been supported and promoted by the concept of multiple intelligence formulated in 1983 by Howard Gardner and the sign-system oriented comprehension of musical instruction that has been based on it.

Proceeding from the experiences with the project "increased musical instruction" the work with the participating classes soon showed that a modified time-table also entailed quite some changes apart from musical instruction.

The intense work with music soon urged out of its topical borders and the musical perspective offered interesting practices and tasks in the instruction of languages, mathematics and general studies - or formulated the other way round - we realized that many topics and methods in school are basically full of music.

Regarding the concept of transdisciplinary teaching, it becomes apparent that - apart from the interdisciplinary subjects, which so to say lie "between" subject-matters - activities are important, which illuminate and explore the disciplinary work from the point of view of other subjects (transdisciplinary thinking and acting).

For the realization of this discovery we have created the term of "Music as a teaching principle". This led to the availability of lessons with much music for all interested teachers and independently of time-tables.

Practical work with many classes has shown since that the elementary connections between instruction in mathematics and music enrich experience-oriented, exploratory and pleasure-centered learning and teaching in many ways.

On the other hand, didactics of mathematics have since the beginning of the nineties developed in direction of John Deweys’ (1859-1952) concept of exploratory learning. Together with this new orientation, the point of view and way of thinking of children are more and more taken into consideration and approaches are postulated, which are linked to the context of teaching subjects. This change also implies "concentration on fundamental ideas of arithmetics and geometry" and "turning away from instruction in tiny steps in favor of a conceptual entireness of the learning situation". Based on this concept, the means and ways of visualization are basically discussed as well. The way from perception to mental conception is comprehended as idiosyncratic constructional process (which also depends on the person), which consciously puts up with individual interpretation and differential paths of solution. As a means of illustration, we have introduced some charts and special materials that must comply with a number of criteria. e.g.:

  • adequate representation of the structure of the mathematical fact
  • manifold possibilities of usage
  • possibility of continuation
  • simple handling and simple (easily surveiable) structure
  • simple possibility of transfer to graphical representations
  • easy practicability of mental operations
  • possibility of discovering individual and differential strategies of solution and of social interchange on it
  • continuous availability for all students and demo version for the class
  • low price
  • stability and environmentally harmless material

Sound and motion highly comply with these conditions. Due to the missing tradition of a corresponding approach, the two have until now hardly been brought into consideration as working or illustrative material - as tools for teaching and learning. If they appear in educational material, corresponding patterns of action are applied quite accidentally now and then and mostly as a means of decoration. This deficit is often compensated by music-minded teachers and their imaginative teaching and methodical skills. But this is no justification for the limitation of educational material to verbal, visual, haptic and mathematically-abstract approaches; acoustical (sound), kinesthetical (motion) and tactile (touch) impulses are of high value, especially with respect to acting and experiencing of children of primary school age. The integration of this type of experiences in mathematical instruction can make accessible important active learning paths of children in classroom; simultaneously, the insight of teachers in unexpected 'thinking paths' of their pupils is promoted.

The present website "Music by Maths" (based on the volume «Mathe macht Musik») wants to help to cultivate approaches to mathematics via sound and motion and to enrich musical instruction.

Training ideas and impulses for a "musical" realisation of actual subjects of the mathematical curriculum as "patterns and classifications", "numerical series and the concept of number", "duration and length", "addition and substraction", "throwing dice", "decimal system", "numerical series", "calculating with money", "forms and shapes in the world around us", "time" and "mathematical rows" are shown. Games of perception, instructions to motion, songs, rhythmical games, concentration and observation tasks and exercises for creativity are used to transdisciplinarily exploit the musical potential and to create a positive atmosphere for learning. This type of instruction also supports the relaxed access of children to music and transdisciplinary acting and thinking.

Many of the proposed musical impulses for mathematical topics and many of the musical exercises and applications of mathematical questions are by themselves not spectacular and they do not imply an extraordinary musical talent of the teacher. Yet the impact of their consistent application on individual children and on classes as a whole, especially on motivation and the learning climate, is remarkable. Due to their inborn musical potential, impulses are always fun to play with. The mathematical and musical processes of exercise and automation stimulated by them often can be combined with the tasks of daily mathematical training ("Blitzrechen").