(1) Problem Identification:

What serious problem or challenge with broad significance does your use of technology address? Explain your context and the existing conditions that you are trying to improve or rectify.

There are two serious problems that my web site addresses, and I'll discuss them both briefly here. The first involves displaying math properly on the web, and the second involves math literacy.

Displaying mathematics properly on the web is hard; trying to work with it dynamically (as in randomly-generated exercises) is even harder. HTML alone just can't ‘do’ math. Without the ability to easily create and post math that looks the way it's supposed to look, there can be no serious advancement of math instruction on the web. A solution does exist—MathML (Math Markup Language)—but its use in the math educational community is virtually nonexistent, largely due to lack of exposure. If you're not aware of a better way to do something, then it won't get done the better way. Furthermore, change is difficult, and educators are busy. If someone is going to take the time to try something new, they'd better have convincing evidence of its worth. MathML is worth it—definitely—and one purpose of my site is to spread the word.

So, let's talk a bit about dealing with mathematics on the web. Take fractions, for instance.

• A ‘diagonal fraction’ like   1/3  (using the forward slash)—HTML can handle that. No problem.
• Now, put a sum in the numerator: say we want  1+2  divided by 3 .
Still not too bad:   just say (1+2)/3 , putting parentheses around the  1+2 .
(With the ‘horizontal’ fraction $\,\frac{1+2}{3}\,$ those parens aren't needed, but it isn't a big deal here.)
• Continuing to add to the complexity, take a look at the first step of a typical problem in my multiplying and dividing fractions web exercise:

[(x^2-9)/(5x^2 + 20x +15)][(x+1)/(x+4)] = [(x-3)(x+3)]/[5(x^2+4x+3)][(x+1)/(x+4)]

It's not so easy to make sense of that without proper mathematical display, and this is just beginning algebra.
By the way, here it is with proper mathematical display: $$\frac{x^2-9}{5x^2 + 20x + 15}\cdot\frac{x+1}{x+4} = \frac{(x-3)(x+3)}{5(x^2+4x+3)}\cdot\frac{x+1}{x+4}$$

How do math educational web sites overcome this problem? Most of them post the math as images. They use other software to create the math, take a picture of it, and then embed that picture in their HTML. Serious problems emerge with this approach. One hundred “pieces” of mathematics mean one hundred images. Pages are slow to load. File directories are bloated. More importantly, images don't contain searchable information. Trying to find a web site that talks about the famous equation $\,\text{e}^{i\pi} + 1 = 0\,$? Good luck finding it, if that site has posted it as a picture.

That's not, however, the worst problem with posting math as images. To be good at math, students need to practice, practice, practice—and you can't realistically create randomly-generated practice with images. Let's try it, by (theoretically) constructing a tiny online exercise—a litle bit of “dynamic math”. Suppose we want students to practice an addition of fractions problem: $\,\frac ab + \frac cd\,$. We want those to be horizontal fractions, so we'll use pictures. Let's be content with just a smidgeon of randomness—say, numerators randomly chosen from $\,1\,$ to $\,5\,$, and denominators from $\,6\,$ to $\,10\,$—so that a typical problem might be $\,\frac17 + \frac38\,$.

How many pictures would that require? Answer: $\,5^4 = 625\,$, if every sum is an image. Ah, but we can be clever: just create $\,5^2 = 25\,$ horizontal fraction pictures: $\frac16, \frac26, \frac36, \frac46, \frac56, \frac 17, \ldots\,$ and then (try to) vertically align them properly between the ‘$\,+\,$’ sign. Or, some web authors use a table with three rows for each and every fraction—the numerator in the top row, the horizontal fraction bar in the second row, and the denominator in the third row. These are just some of the “somersaults” that web designers do to try to get mathematical notation that, in the end, still doesn't end up looking quite right.

Let's keep going. Pictures don't re-size when you change your text size (MathML does). Pictures can't “read themselves” to you (MathML can). If a web exercise was created with pictures, then teachers can't easily take it and customize it—they'd need the entire set of image files. With MathML, however, a teacher with just a bit of HTML-savvy can grab the source code, adjust, and serve to students.

Students need offline practice, too. How about a worksheet of addition-of-fraction problems, so that work can be done away from the computer? Many sites offer *a* worksheet (i.e., precisely ONE), created, of course, with math software that can do the job. Need more practice? Sorry, that's it. A teacher wanting to give a quiz probably won't want to use exactly the same worksheet that's posted online. Dynamic worksheet creation—a different worksheet every time you click a button—just can't be done with pictures. It CAN easily be done with MathML; I do it for each and every one of my almost 250 web exercises.

Please don't take my word for it. Go to (say) Google, and type in “addition of fractions”. Of the top ten sites that came up under Google when I did this, $\,5\,$ achieved their horizontal fractions with the “table” hack I talked about above, $\,3\,$ used images, and $\,2\,$ used Java applications. (To test if math is just a picture, right-click on it and see if it says “open image...”) Repeat this exercise with a variety of math phrases: arithmetic with radicals, function notation, whatever. If you're lucky, you'll come up with at least one site in the mix that uses MathML$\ldots\,$ and, chances are, it will be one of my pages. To fully appreciate MathML, you need to see it in action. At the end of this introduction, I offer an efficient route to get the broadest view of what my site is all about.

Now, onto the second serious problem that my site tackles—math literacy. Go back in time, and imagine yourself sitting in a classroom (it doesn't have to be math). The teacher passes you a piece of paper containing several paragraphs that you're supposed to read. Upon glancing at the paper, however, you see that it is written in a language that you don't understand. Is the teacher being fair? Of course not. Indeed, the instructor is probably trying to make a point. Although the ideas in the paragraph may be simple, there is no access to the ideas without knowledge of the language in which the ideas are expressed. This situation has a very strong analogy in mathematics. People frequently have trouble understanding mathematical ideas, and often it's not because they're particularly difficult, but instead because they're being presented in a foreign language—the language of math. Cast yourself into a situation where you don't understand the language, and things can get pretty stressful. That's what happens to lots of people with math.

Authors realized, a long time ago, that students couldn't read their math books if they were written the “math” way, so they started taking out the “cryptic” math and replacing it with English. A problem emerged: the language of mathematics is designed to say the kinds of things that mathematicians need to say—English isn't. English “translations” had to make sacrifices in precision (the ability to make fine distinctions), conciseness (the ability to say things briefly), and power (the ability to express complex thoughts with relative ease). Books got bigger and fatter and heavier. (Kids started developing back problems carrying all those books around.) With the “English” approach, people do not learn to read, write, or speak mathematics. The “English” approach fosters dependence, not independence—people always need a math translator whenever any “real” math comes along.

There are many features of the math language that can be presented early on, and then reinforced throughout K-12 education. Simple things—like distinguishing between the “nouns” of math (expressions) and the “sentences” of math (things like equations and inequalities). Things like understanding that numbers have lots of different names, and the name you use depends on what you're doing with the number. Think of $\,5\,$ pieces of candy: $\,3 + 2\,$ (three for now, two for later); $\,2.5 + 2.5\,$ (break one piece in half, share equally with a friend); $\,6 - 1\,$ (I actually started with six, but one got dropped and stepped on). I wrote a little book, called One Mathematical Cat, Please! that introduces lots of language ideas, for middle-school up. Brooks-Cole Publishers loved it, and said that if you can find the right place to put it, this book will do a great service to the mathematical educational world. However, it was just a supplement, and supplements don't sell. So, instead, I put it on the web, filled it out to a complete Algebra I course, and gave it to the world$\,\ldots\,$ and that was just the beginning.

I'll talk more about the details of my work in the other essays, but I think everything will be best appreciated if you've seen a bit of my site. Here's one possible scenic route:

1. Go to my homepage (or, from anywhere, just Google “math cat fisher”).
2. Check out my MathJax-updated files to see dynamic math-on-the-web at its best.
(Lots more on MathJax in the upcoming essays.)
3. Watch Video #1: Welcome (8.5 minutes).
4. Watch Video #2: Power Users to see dynamic MathML in action and learn about Algebra Pinball.
5. Want to learn about the history of math-on-the-web and the development of my site in particular?
Watch the interactive slide show from my homepage.

So$\,\ldots\,$ play around a bit, if you have the time, and the remaining essays should be more meaningful.