Welcome to the *iconic math* site. Please don’t be put off by the word “math”. The site is about how to ** change math** so that it is easy to understand and easy to use. For about a century, common mathematics has been dominated by a

*symbolic style*. Symbols do not look like what they mean. For example,

**. This site is**

*return common intuition and direct experience to the teaching and learning of mathematics**not*about brilliant mathematicians and the profound contribution of mathematics to intellectual thought. Certainly advanced math should be thoroughly supported in our culture. But the discipline of math is a very specialized field, not something that every school child should know. There are about three thousand professional mathematicians in the US and no more than two million scientists and economists, many of whom may use math extensively. This site is written from the perspective of the 99%, the rest of us. The goal of the site is simple: to show that math does not need to be disembodied for it to be useful or even for it to be rigorous. Iconic math is both anchored to physical experience and formal.

*Formal*means that it really is mathematical: structured, rigorous, built from axioms or first principles, and provable. The site shows that

*formal*is entirely consistent with humane and intuitive. But first we have to get rid of those meaningless symbols.

The * Image* is from Gregor Reisch’s university textbook,

*Margarita Philosophoca*, 1503. It shows the

**Allegory of Arithmetic**as a woman conducting an arithmetic contest between a happy-looking Boethius who is using the “modern” Hindu-Arabic numerals, and a sad-looking Pythagoras who is using the traditional counting beads laid out on a counting table similar to an abacus. Five hundred years ago, universities began to take math out of our hands and put it into our heads. That turned out to be a good idea until the advent of ubiquitous computing. Now, in the twenty-first century, it is time to take math out of our heads and put it into computers. That frees up our minds for

*thinking*rather than memorizing what math symbols mean and how to juggle them around to do arithmetic. We need to reconnect math with our bodies and with our common reality so that it will make sense. It’s time for Pythagorus to grin again.

## The Iconic Math Site

We encounter all the math content on this site before college. The site addresses simple arithmetic, simple algebra, and simple logical thinking. There’s an iconic calculator. There are some quite technical pages and many quite obvious pages. The site includes many examples of iconic math. Animations show the physical process of doing math, while stills show the steps. Every demonstration on the site (arithmetic, algebra, logic) obeys the iconic principles that return math to the world, to our bodies, and to reality. The entertainment value of the site comes from two sources. The site is *visual*, you can look at concepts and examples without necessarily reading any words. And the site provides many new and different ways to think about and to do simple math.

I’m preparing a book (*Humane Mathematics*) that will explore the organic principles of mathematics in more detail. For now, please enjoy!

## Iconic Mathematics

**2 + 3 = 5 **

**•• + ••• = ••••• ***computers* were people who computed with numbers as a job. There is no particular reason why we should learn to do math with paper-and-pencil, other than in 1950 paper-and-pencil were at the front of the technological revolution. (I still have a callous on my finger from being required in school in the mid-1950s to write with a *nib-pen*!) And everybody knows in this century there is no particular reason why we should know how to do long division, unless, of course, you are trapped in a car without electricity or batteries and suddenly need to compute the car’s miles-per-gallon.

Math should look like what it means. Doing math should show us how math works, as we do it. We *add* by putting things together; putting things together shows us how adding works.

**Using symbols makes common math too difficult**. Symbols must be memorized to be understood. The algorithms for addition, subtraction, multiplication and division take years of effort to learn. Symbolic thinking is unnatural. By design, symbolic math is independent of physical grounding and actual experience. Symbolic arithmetic provides no information about the processes of addition and multiplication. When disconnected concepts are combined with disconnected instruction, students end up hating symbolic math.

**Using icons makes math very easy.** *Iconic math* returns math to our bodies and to our common sense. It is immediate, visual, interactive, and grounded in physical experience. Icons can be directly recognized. Doing arithmetic with iconic forms is intuitively simple. Iconic arithmetic dynamically *shows us how arithmetic works*.

## New Ideas

Here are a few of the ideas that you will be able to find on this site.

• It’s time to stop teaching symbolic math and to stop teaching math classes (Math Education).

• The universal place-value notation for numbers is outdated and inefficient (Depth-value Notation).

• Numbers can be written in one, two and three dimensions (Varieties).

• Network numbers add by stacking side-by-side and multiply by stacking top-to-bottom (Network Numbers).

• The Iconic Calculator *shows* us how arithmetic works (Calculator).

• Arithmetic does not need zero (Container Algebra).

• One operation can take the place of subtraction, division, and roots (James Algebra).

• There is a *new* imaginary number, J, that is simpler than the imaginary number i (James Imaginary).

• Deduction is visceral (Logic).

• Logic does not need the concept FALSE (Boundary Logic).