DigiMat fills new school curricula for mathematics+programming with concrete meaning and content, which can be seen as a realisation of the classical Quadrivium now in the age of the computer, along the following principles:
- Mathematics is constructed by the student in short computer codes with rich output.
- Properties of mathematical objects follow by construction and as such can be understood and used by the student.
- Text programming starts early along with learning to read and write.
- The material is unified over all levels differing only in scope and depth.
Here is a School Syllabus which with increasing depth and width has the same formulation for basic school and high school into University Syllabus, thus expressing unity over levels. Detailed Syllabuses follow below. Each Syllabus is filled with concrete meaning by its Course Material, which mainly takes the form of self-explicatory computer codes supported by menu-listed Pages with full account under Books.
In mathematics, as a deductive science built on axioms of simple form, the most basic directly connects to the most advanced, which is an essential feature of DigiMat. As an example, the three-line code of Cosmic Interaction can be written and understood by a 10-year old, while it captures the essence of the Millennium Simulation as frontier research. It illustrates the short code – rich output principle of DigiMat as digital mathematics.
On the other end of the spectrum representing long proof – short output, you find Fermat’s Last Theorem as an example of formal/symbolic mathematics, the statement of which again can be understood by a 10 year old, but which requires a proof of 500 pages which is understood by very few. (No three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2).
Goal: Interactive digital simulation of real and imagined worlds = computer games.
Language: Formal mathematics + programming.
Tools: Fantasy, brain, eyes, fingers and computer/table/mobile.
- Construction av natural numbers by iteration x = x+1 with start x=0. Properties of natural numbers by construction.
- Computation with natural numbers: addition, subtraction, multiplication, division.
- Programming av operations according to 2. Digital representation in different bases.
- Digital construction av rational numbers as solutions x to the equation px=q with p and q natural numbers. Properties by construction.
- Programming av operations for rational numbers: addition, subtraction, multiplication och division.
- Digital representation of spatial position x: coordinate system, computer screen/pixels.
- Motion: change of position: x=x+vdt (or dx=vdt) with v velocity and dt time step.
- Change of velocity: v = v+a*dt (or dv=a*dt) with a acceleration.
- Newton’s 2nd law: a = f/m where f is force and m mass.
- Digital construction av elementary functions (polynomials, sin, cos, exp, log, et cet) by programming of time stepping of a=f/m, v=v+adt, x=x+vdt with simple f. Properties by construction.
- Generalisation to more complex forces, e. g. Newton’s law of gravitation.
- Digital representation of and operation on geometrical objects i 2d och 3d.
- Digital representation of and operation on image and sound, or other data.
- Calculus as solution of dx=f(x)dt.
- Linear algebra as generalisation av 13.
- Construction/programming av computer games building on 1-16.
- National curriculum in England: computing programmes of study
- Programmeringen kommer 2016 (i Finland) – är skolorna redo?
- Survey of schools: ICT in Education
Goal: Digital simulation and control of multi-physics systems (the World) (fluid-structure-gravitation-electromagnetics-biology-chemistry-quantum).
Method: Symbolic + Digital Calculus:
- Mathematical modeling of multi-physics as systems of PDE (geometry, calculus in 3d)
- Discretisation of systems of PDE (finite elements)
- Digital solution of discretised systems of PDEs (time-stepping, linear algebra)
- Applications to simulation of given systems, parameter identification in systems, control of systems, robotics, automated modeling of systems, machine learning, AI.
Here follows more detailed specifications for different levels. The Course Material defining the syllabus takes the form of self-explicatory computer programs from very simple to more complex building on a few basic principles of counting and time-stepping in a user-friendly p5.js web browser setting.
The unification over levels is expressed e g by short codes for Poisson’s equation in simple geometry (5-point scheme), which can be mastered by students from grade 4. This is followed by using FEniCS for complex geometry, then not as black box but with understanding and experience from simple geometry. This brings top level into school to give inspiration, motivation and meaning to learning.