I'm afraid even the students we already have enrolled don't really know what mathematics is!
This is a first effort to put this discipline, I enjoy so much, in some kind of wrapping for my students.
Arithmetic is part of it, but mathematics doesn't end there. I just went to check the Mathematics Genealogy Project, looking for Morris Klein, and I found out that, Courant was one of his ancestors. There was an influx of European mathematicans to the United States that is rather recent. It is not surprising that the US doesn't have many historic names in mathematics yet. Mexican mathematicians are even a smaller bunch. Obviously both our countries are new to this discipline. This makes it harder to invite high school students. They don't even know the answer to: What is Mathematics?
Simply Mathematics encompasses, arithmetic, geometry, algebra, and calculus. Why those? Well those subjects are taught in Mexican professional schools.
I can write a few more characters on this subject: Numbers, shapes, variables, and change. I want to add another name that students do not put together with these, but it belongs there very much. Computer Programs.
Wolfram|Alpha produces this:
Mathematics
do basic arithmetic
125 + 375
do exact arithmetic with fractions
1/4 * (4 - 1/2)
compute a decimal approximation of a specified number of digits
pi to 1000 digits
convert a decimal number to another base
219 to binary
plot a function
plot x^3 - 6x^2 + 4x + 12
plot a region satisfying multiple inequalities
plot x^2+y^2<1 and y>x
solve an equation
x^3 - 4x^2 + 6x - 24 = 0
factor a polynomial
factor 2x^5 - 19x^4 + 58x^3 - 67x^2 + 56x - 48
compute eigenvalues and eigenvectors of a matrix
eigenvalues {{4,1},{2,-1}}
calculate a derivative
derivative of x^4 sin x
compute an integral
integrate sin x dx from x=0 to pi
solve an ordinary differential equation
y'' + y = 0
compute properties of a geometric figure
annulus, inner radius=2, outer radius=5
plot a conic section and identify its type
2x^2 - 3xy + 4y^2 + 6x - 3y - 4 = 0
compute properties of a polyhedron
dodecahedron
compute a prime factorization
factor 70560
solve a Diophantine equation
solve 3x+4y=5 over the integers
compute a possible formula and continuation for a sequence
5, 14, 23, 32, 41, ...
analyze a graph specified by adjacency rules
1->2, 2->3, 3->1, 3->4, 4->1
solve a recurrence
f(n)=f(n-1)+f(n-2), f(1)=1, f(2)=2
minimize or maximize a function
maximize x(1-x)e^x
numerically integrate functions that cannot be integrated symbolically
integrate sin(cos x) from x=0 to 1
compute a truth table
P && (Q || R)
generate a Venn diagram
(complement S) intersect (A union B)
do computations with special functions
d/dx Si(x)^2
do computations with number-theoretic functions
phi(110)
find representations for a function
gamma(x) integral representation
compute properties of a function of a complex variable (use the variable z)
e^z
do vector analysis computations
div (x^3 y, y^3 z, z^3 x)
compute properties of a knot
8_1 knot
I leave it there for today.
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