Posted in Mathematics education

Theories and ideas behind the math lessons in this blog

I have put together in this post some of the ideas behind the kind of mathematics teaching I promote. As I stated in the subheadings of this blog, the articles and lessons I write here are not about making mathematics easy because it isn’t but about making mathematics makes sense because it does. Before reading any of the articles below, I suggest read the About page first and what I think mathematics is. I hope I make sense in the following articles. Click here for the list of math lessons.

To understand mathematics is to make connection

Mathematics is an art

Teaching through Problem Solving

Mathematical habits of mind

What is mathematical investigation?

Exercises, Problems, and Math Investigations

Why it is bad habit to introduce math concepts through their definitions

What is reasoning? How can we teach it?

What is the role of visualization in mathematics?

Making generalizations in mathematics

What is abstraction in mathematics?

I also have a new blog about research studies in mathematics teaching and learning.

Posted in Algebra, What is mathematics

Love and Tensor Algebra

Come,  let us hasten to a higher plane

Where dyads tread the fairy fields of Venn,

Their indices bedecked from one to n

Commingled in an endless Markov chain!

Come, every frustrum longs to be a cone

And every vector dreams of matrices.

Hark to the gentle gradient of the breeze:

It whispers of a more ergodic zone.

In Riemann, Hilbert or in Banach space

Let superscripts and subscripts go their ways.

Our asymptotes no longer out of phase,

We shall encounter, counting, face to face.

I’ll grant thee random access to my heart,

Thou’lt tell me all the constants of thy love;

And so we two shall all love’s lemmas prove,

And in our bound partition never part.

For what did Cauchy know, or Christoffel,

Or Fourier, or any Bools or Euler,

Wielding their compasses, their pens and rulers,

Of thy supernal sinusoidal spell?

Cancel me not – for what then shall remain?

Abscissas some mantissas, modules, modes,

A root or two, a torus and a node:

The inverse of my verse, a null domain.

Ellipse of bliss, converge, O lips divine!

the product o four scalars is defines!

Cyberiad draws nigh, and the skew mind

Cuts capers like a happy haversine.

I see the eigenvalue in thine eye,

I hear the tender tensor in thy sigh.

Bernoulli would have been content to die,

Had he but known such a2 cos 2 phi!

Love and Tensor Algebra is from the book The Cyberiadwritten by  Stanislaw Lem. Stanis?aw Lem was a Polish writer of science fiction, philosophy and satire. The Cyberiad is one of his best work.

The Cyberiad (Polish: Cyberiada) is a series of short stories. The Polish version was first published in 1965, with an English translation appearing in 1974. The main protagonists of the series are Trurl and Klapaucius, the “constructors”. The vast majority of characters are either robots, or intelligent machines. The stories focus on problems of the individual and society, as well as on the vain search for human happiness through technological means. The poem Love and Tensor Algebra found its way to this blog because of its mathematical flavor. And I love reciting it.