Mathematics-IT is supported by a set of apps at App Store which can be found by searching for “newmath”.

The Swedish version is presented at Matematik-IT

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- Counting1: NewMath
- Calculus1: NewMath
- Mechanics1: NewMath
- Mechanics2: NewMath
- Biology1: New Math
- Running: NewMath

collecting some of the examples presented on Mathematics-IT. The review process takes 1-2 weeks.

This is the first step to a realisation of my master plan to present a version of Mathematical Simulation Technology for basic school in app form. Here are the icons of the first apps:

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The plan of the program takes the same form independent of level, from basic through high-school to university:

**Goal:** Interactive digital simulation of real and imagined worlds = computer games.

**Language:** Formal mathematics + programming.

**Tools:** Phantasy, brain, eyes, fingers och computer/iPad.

**Content:**

- Construction av natural numbers by iteration x = x+1 with start x=0.
- Computation with natural numbers: addition, subtraction, multiplication.
- 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. 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+adt (or dv=adt) 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 of elementary functions by construction. Generalisation to more complex f, 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.
- Constructi0n/programming av computer games building on 1-11.
- Calculus as solution of dx=f(x)dt.
- Linear algebra as generalisation av 10.

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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:

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 of time such that

- for with , (*)

where is a given function of and and a given initial value, by successive time stepping according to

- = with ,

with small. This is formally a finite time step version of or with vanishingly small time step , and .

If depends only on , the solution is the integral

- .

If and , the solution is .

More generally, with simple dependencies of and 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 and a multi-dimensional space coordinate and depending on partial derivatives of with respect to space coordinates, which is solved by time stepping after finite element discretization in space.

Constructive Calculus can thus be summarized as solved by time stepping . 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.

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BodyandSoul is now returning to Chalmers as its web version MST appearing as supplementary material in the mathematics program for mechanical and design engineering students.

MST thus welcomes Chalmers students with the hope of opening a window to the wonderful world of simulation on the web constructed by computational mathematics.

MST is being developed for the quickly expanding MOOC market which is bringing revolutionary changes to education, in particular to education in mathematics, science and engineering.

MST is banned at KTH which gives Chalmers students the opportunity to acquire specialist competence which is hard currency on the job market: What is asked for is unique knowledge and skill and not what everyone knows. The more MST is banned at KTH, the more the interest in MST at Chalmers may increase.

But MOOC and MST will eventually break the walls also at KTH, and so it is better for Chalmers students to now hurry up into the new brave world of mathematical simulation technology.

But the prospects for MST at Chalmers are after closer inspection, not so clear:

but the wheel turns and tomorrow is a new day with new students eager to learn the tricks of the trade.

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

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