New video series: Visual Algebra

Algebraic notation is broken.

By that I mean that it is a major hurdle for many who struggle with mathematics, a major source of typical mistakes, and still way less consistent than we teachers would care to admit. It has features such as:

  • invisible brackets: $x+y\cdot z$, $\frac{a+b}a$
  • ambiguous use of brackets: $a(b+c)$ vs. $f(x+1)$
  • function notation without brackets: $\sin\, x$
  • triple meaning of the equal sign (equation, identity, definition)
  • right-to-left function notation: $g\circ f(x)$

These examples may seem trivial to anyone already fluent in algebra, but they can be real stumbling blocks for novices. Worse yet, math tends to become more and more symbolic as school progresses, with drawings serving only as informal illustrations of the "actual" math. We are drowning students in notation. Huge swaths of exercises consist of purely symbolic calculations, such as simplifying fractional expressions, completely devoid of any geometric meaning.

(This follows the historical path, but something got lost on the way, as a price to be paid for the power of algebra. Symbolic notation was introduced to help grapple with geometric figures – the "actual" math back in the day –, but later it took center stage while relegating figures to the status of mere illustrations. What got lost – intuition – was scoffed at for being error-prone and misleading. This of course culminated in the formalist school of mathematics, where even proofs were considered a form of computation.)

But we are visual creatures. Even mathematicians find their ideas most often by doodling, before casting their thoughts into formulas. Geometric figures are intuitive, accessible, self-explanatory and friendly. Why have we cast aside the power of drawings, deemed "informal", in favor of dense, one-dimensional strings of (sometimes arcane) symbols? A good notation should make computations obvious, and mistakes almost impossible to make. In other words, it is intuitive. Any good programmer can tell you that. We need a better notation for algebra. Better than the historically grown amalgation that people mistake for the essence of algebra now. Something that brings intuition back to algebra.

Let's reclaim the power of visualization over computation.

I have therefore started a new video series called "Visual Algebra" ("Doodle Algebra" to its friends). It is a novel approach to school algebra using a more intuitive, graphical notation that can supplement the standard notation not only for deeper understanding, but also for more thoughtful manipulation.

Visual Algebra is not merely a collection of illustrations. It can be used to perform entire calculations completely within the visual representation! Not only that, but you can even mix and match it with the standard notation to selectively visualize only parts of an expression. Even fractional expressions find their most natural geometric meaning, and this in turn informs their simplification.

The full playlist in embedded below. I will add videos in the coming weeks, covering these topics:

  • addition and subtraction
  • multiplication and division
  • fractions and fractional expressions
  • powers and roots
  • logarithms