DigiMat from School to Pro

DigiMat  is new reformed mathematics education for the digital world (see Scope and Method) consisting of:

DigiMat is a unified program with varying depth and scope over all levels with computation as leading principle, where all mathematical objects are constructed by computation according to computer programs as mathematics expressed  in symbolic form (see The World as +1).

DigiMat gives new answers to the key questions of what?, why? and how? of school mathematics, with the traditional answers no longer convincing in the digital society.

DigiMat combines formal mathematics with programming into a synthesis, which in different forms can serve many different students.

DigiMat meets new school curricula asking programming to be part of mathematics education.

DigiMat is understandable because a principle of construction can be understood, and useful by turning principle into output by powerful computing.

DigiMat is learning-by-constructing. DigMat offers a rich program unified over levels to a wide variety of students (and teachers) with different interests and capacities.

DigiMat opens to understanding of basics of the digital society and gives tools to meet possibilities and challenges. Read For Whom? and Comparison with Music!

DigiMat is an expansion of the BodyandSoul program.

DigiMat Pro is launched as MOOC on edX as High Performance Finite Element Modeling Part 1 and 2.

From School to University

Mathematics is unique as science in that the most basic directly connects to the most advanced. In DigiMat this comes to expression by the fact (which will become apparent as you proceed) that essentially the same things are done on all levels just with different depths and complexity, just like in cooking and music, from school to university.

DigiMat Pro at top offers a rope securing meaningful possible climb for anybody to any level.

The fact that in mathematics top connects to base is illustrated in a proof of the statement 2 = 1 + 1 filling two full dense pages in the monumental three-volume treatise Principia Mathematica by Russell-Whitehead, something which is clear to any child. So DigiMat starts by constructing the natural numbers as 1, 2=1+1, 3=2+1, 4=3+1,…,simply by repetition of the basic operation of +1.

Get Started

The menu items IntroductionBasicsModel Workshop  and Game Workshops show a path from the most simple over less simple to more complex.  If you follow the path, you will with satisfaction discover that you have learned to master powerful mathematics, which in traditional teaching is viewed far too advanced to serve as school mathematics. It is the computational-constructive aspects of DigiMat which allows you to take this leap even if you are not a math freak. Be confident that the path is open to you!

Books gives the foundation of the path and lead into a wider world as computational mathematics.

Start by clicking around in Model Workshop and Game Workshop (also in the menu) to get an idea of what it is all about.

Continue to explore DigiMat School by watching Introduction and then proceed to Basics, which will take you step by step from the simple to more advanced following a basic principle to construct-simulate-understand-control the world by computation (combine with Perspective on Math Education):

  1. First the natural numbers are constructed by repetition of the basic operation of +1 according to a basic prototype of all computer programs of DigiMat of the form n = n + 1, which starting with n = 1 generates 2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1, and so on. Similarly the negative integers are constructed with repetition of the operation -1.
  2. Once the integers have been constructed, representation in binary or decimal form is specified and the basic algorithms of addition, subtraction, multiplication and division are defined and then expressed as computer programs taking the form of a pocket calculator programmed by the student, which in a next step is extended to rational numbers in binary or decimal form.
  3. With rational numbers at hand Newton/Leibniz’ world of Calculus and Linear Algebra, can be constructed as spatial and temporal variation expressed by numbers. The prototype program then takes the form x = x + v*dt and v = v+a*dt, where x is spatial position, v is velocity and a is acceleration, which can depend on x and v, and dt is a time step. This is the time stepping version of Newton’s Laws dx =v*dt, dv=a*dt and a=f/m.
  4. The next step is to extend to cover all aspects of the world as solid/fluid/quantum mechanics, thermodynamics, electromagnetics, biology, geoscience, economy, and more, all in constructive form based on computer programs of the prototype form.

To Browse

Here you find more material in the form of earlier versions of DigiMat:

connecting to

Constructive vs Symbolic Calculus

The basic mathematics courses at university level consists of Calculus and Linear Algebra, which are closely connected and can be thought of as one subject with the objective of mathematical modeling of real and imagined worlds, which consists of (i) formulating and (ii) solving mathematical equations.

A Standard Calculus text book, like Calculus: A Complete Course by Adams and Essex, is filled with symbolic formulas covering more than 1000 small print pages and is difficult for the student to digest and heavy to carry along. A typical page may look like this:

adams1

The objective of a Standard Calculus text book is to convince the student of the usefulness of Calculus through mass demonstration by presenting so many specific problems, which can be solved with pen and paper using Symbolic Calculus, that there can be only a few left which cannot be solved this way. In short, the objective is to show that Symbolic Calculus works by presenting very many specific examples. But the massiveness is misleading since in fact very few problems can be solved symbolically with pen and paper.

The essence of the BodyandSoul approach as Constructive Calculus presented here as the essence of Mathematical Simulation Technology, is the opposite: Instead of many specific problems solved by symbolic mathematics with pen and paper, one general problem containing all the specific problems of Standard Calculus and many more, is considered. The essence of the theory is then to show how and why any given instance of the general problem can be solved by the computer, as expressed in a Fundamental Theorem of Calculus.

The one general problem of Constructive Calculus, in one variable to start with, is the Initial Value Problem (IVP): Construct a function u(t) of time t such that

  • \frac{du}{dt} = f for t > 0 with u(0)=u_0,         (*)

where f=f(u,t) is a given function of u and t and u_0 a given initial value, by successive time stepping according to

  • u(t+\Delta t) = u(t) + f(u(t),t)\Delta t with u(0)=u_0,

with \Delta t > 0 small. This is formally a finite time step version of du = fdt or \frac{du}{dt}=f with vanishingly small time step dt, and du \approx u(t+\Delta t) - u(t) =f\Delta t\approx fdt.

If the function f(t) depends only on time t, the solution u(t) is the integral

  • \int_0^t f(s)\, ds + x_0.

If f=u and u_0=1, the solution is u(t)=\exp(t).

More generally, with simple dependencies of u and t all elementary functions (exponential, trigonometric, Bessel, …) are constructed this way and their properties follow from the specifics of the IPV they solve.Calculus in one variable can thus be reduced to a study of the IVP (*).

Similarly, Calculus in several variables can essentially be reduced to an IVP of a generalization of (*) with u=u(t,x) and x a multi-dimensional space coordinate and f depending on partial derivatives of u with respect to space coordinates, which is solved by time stepping after finite element discretization in space.

Constructive Calculus can thus be summarized as \frac{du}{dt} = f solved by time stepping du=fdt. Constructive Calculus combines simplicity with generality, which is a prime goal of (computer) science and mathematics, to be compared with the difficulty of all the specific cases of Symbolic Calculus.

We may summarize as follows:

  • Constructive Caculus is simple and general.
  • Symbolic Calculus is difficult and special.

Inspiration: Jumbojet

A milestone of mathematical fluid mechanics was reached in 2013 as Direct Numerical Simulation (DNS) of the turbulent flow  of air around a  jumbo jet in landing configuration described by the (incompressible) Navier-Stokes equations, which gives a quality assessment of Mathematical Simulation Technology as an efficient combination of mathematics and physics e.g. opening to the development of new realistic flight simulators and games:

The student can use DNS as a signum of the present educational program enhancing inspiration and motivation.

Inspiration: D’Alembert’s Paradox

  1. Watch the Resolution of D’Alembert’s Paradox formulated in 1752 which was accomplished in 2008, after 256 years of fruitless struggle, by solving the Euler equations on a computer.
  2. Listen to d’Alembert formulating his paradox in 1752.
  3. Listen to what Nobel Laureate Sir Cyril Hinshelwood says about theoretical (mathematical) fluid mechanics vs practical  (experimental) fluid mechanics.
  4. Listen to what Thomas Kuhn says about Scientific Paradigms.
  5. Listen to what Garret Birkhoff says about modern fluid mechanics.

Video Introduction 2

Here is the first in a series of videos introducing basic concepts, methods, tools and visions of Mathematical Simulation Technology in a short presentation of constructive calculus:

Summary: Constructive calculus is classical calculus restricted by the limits of finite precision computation and augmented by the new capacities of automated computation by the computer. In constructive calculus, the elementary functions are computed solving differential equations by time stepping, which decodes the reality behind symbols like sin(1) and exp(2) and shows how these symbols are to be used. In constructive calculus the student actively constructs models of the world from scratch in the form of computer programs (like a composer writes a partiture), which is given life by a computer running the program (like an orchestra performing the composed music). Constructive calculus is today booming with the computer, while classical calculus was perfected in the 19th century.