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Tools and Notation: A Symbiotic Relationship for the Future. J. Dale Burnett University of Lethbridge
NCTM Canadian Regional Conference Edmonton, Alberta November 20 - 22, 2003 |
A Feynman Quote
Let me begin with two short stories of my life as a mathematics student. The first story occurred during my first year of undergraduate studies in a course called Finite Mathematics. One of the lectures was about place value notation. This was the first time that I was aware of the underlying meaning of an expression like 326. I distinctly remember both the surprise and the excitement. Surprise because it had never occurred to me, and excitement because it was such a neat idea. Today this idea is discussed well before a student comes to university, and I think this suggests both that the curriculum has changed in the last 40 years and that the change has been for the better. My second story occurred seven years later when I was just beginning graduate studies. A new computer language has just been introduced at the University of Alberta called APL. There was no documentation at the time and we just played with it to see what happened. My first few minutes went something like this: 2 + 3 the result was typed back 5 2 x 3 the result was 6 2 - (-1) the result was 3 -2 - 1 the result was -1 but the small horizontal line was near the top of the number. That caught my attention. The clue was in the horizontal line. It was located higher than the one I had typed. There were two different types of horizontal line: one in the middle of the space and one at the top of the space. They represented two different ideas. The small horizontal line in the middle of the space represented the operation of subtraction. The small horizontal line at the top of the space represented the minus sign and was "part of the number". This was the first time I had seen this distinction. Both of these examples are about notation. I have seen some books that use this latter distinction, but it is rare. Why? I think a major reason is technology. It was technology that made revolutionary systems possible in the first place. At that time almost all computing was done using punched cards. The program was then read into the computer and a new code was generated by a program called a compiler. This new code was then executed and the results printed out on a "high-speed" printer. APL was one of the first languages that bypassed the compiler step, printing out the result on a typewriter almost as soon as the finger had pressed the Return key. Today we take this instantaneous feedback for granted. But some progress is very slow. Take the keyboard. It is still QWERTY, even though it is recognized that there are more efficient key placements. And it is the keyboard that caused the difficulties for APL. APL used a wide variety of new symbols, many of them Greek letters, as well as two different horizontal lines, one representing subtraction and the other representing the minus symbol. Looking at my keyboard today, there is a key with two horizontal lines, but while one is the familiar subtraction/minus symbol, the other is a longer line used for underlining. Thus the desire for a character that permits us to change the appearance of the type is preferred to one that makes a useful mathematical distinction. One point for the wordsmiths. The technology story is not over. I am having difficulty writing this paper since I do not have a way of creating most familiar mathematical expressions. I am using some of the most sophisticated software available today for creating web pages, but I am not able to do even simple mathematics with this system. I am not even able to replicate the simple APL expressions that I was using 30 years ago. Another point for the wordsmiths. Let's have a quick look at the idea of positional notation in our present culture before continuing with the primary thread of the symbiotic relation of notation and tools. Consider the expressions 12 and 21. Now consider the expressions at and ta. With the letters, an "a" is always an "a", regardless of where it appears in a word. But this is not so with numbers. In the case of 12, a "2" is "two"; but in the case of 21, a "2" is "twenty". For young children, and even for many adults, this appears to be inconsistent. And if one has even a subconscious suspicion that mathematics is inconsistent, then the edict that one should memorize the "facts" takes on an aura of reasonableness, since meaning is not part of the equation. Our emphasis on memorizing basic number facts is inadvertently reinforcing the arbitrariness of the topic. Once the individual forms the idea the mathematics is just a set of arbitrary rules, the game is over. We are in the midst of a (very) rapidly evolving technology, yet our ability to integrate this technology with our conventional notational standards is lagging, and even our ability to establish new notational standards is painfully slow. This has always been so. Let's have a quick look at mathematical history. It is often surprising to people to realize that the way we write numbers is fairly recent, and that a universal global standard is more recent still. The earliest artifacts indicate a form of tally system of marks. The ancient Sumerians had a system of indentations on clay tablets. The Romans had a system of letters for representing numbers, which one can still see on some clocks. The Chinese and Japanese had a different system, that is occasionally still in use in nonmathematical contexts. The Mayan civilization of Central America had a beautiful set of symbols, pretty to look at but almost undecipherable. Even the present system of Arabic numerals has had a slow and controversial past. Incorporating a symbol for the number zero took centuries. The Greeks were the first to realize that not all numbers could be expressed as a ratio of integers. This caused great consternation as it had strong spiritual connotations. Today the conflicts that arise when discussing notation are usually about whether we can express it with a keyboard. However any discussion of mathematical notation should recognize that there is much more to the topic than simply how we write our numbers. Euclid (325 - 265 BC) is famous for bringing together much of what was known about the relationships between lines and curves into a field called geometry. Nicole d'Oresme, around the middle of the fourteenth century, is credited with inventing the first graph. Descartes (1596 - 1650), only a few centuries ago, showed a way to integrate two separate branches of mathematics. geometry and algebra, bridging two notational approaches with a field that is now called coordinate geometry. We now move easily between the visual graphical notation and the symbols of algrebra. It is the rosetta stone of mathematics. Today, computer graphics and dynamic displays are at the cornerstone of much mathematical thought. From a cognitive perspective, the strategy of "draw a picture" is often an important early step in understanding a situation. As biological organisms with eyes, we have evolved with a strong visual sense, and it makes sense to use this in developing an understanding of anything that has a spatial dimension to it. While geometric ideas have been around for at least two thousand years, and likely much longer, the idea of a graph to represent data is more recent. Whether it is a graph of an equation such as y = 2x + 3 or a histogram of the heights of a class of grade 5 students, the meaning of the image is not to be taken for granted. For many, even today, the first example is just a picture with three intersecting lines, and they may think of it as a poorly drawn triangle. In the latter it may be preceived as a collection of rectangles and that is all. Perhaps the most important notational convention in mathematics is the one that is so pervasive as to be invisible: natural language. This is the primary means for translating the images and symbol systems from meaningless marks to meaningful ideas. Try conveying the idea that a three sided figure is called a triangle without using language. There have been a number of well intentioned efforts to stress the importance of precision in the use of language when describing a new idea in mathematics, particularly at the elementary and middle school level. Sometimes this emphasis has been reduced to another form of rote activity. "You must say it this way .... " without an adequate explanation of why. Thus mathematics again is viewed as an arbitrary subject where the rules have been largely established by others. Another notation that is rarely invoked today is sound. Pythagorus is credited with first recognizing the relationship between various harmonic sounds and the length of a string (mathematics). The Greek idea of the music of the spheres was integral to their ideas of cosmology and the universe. The relationship of music to technology has seen the emergence of a new field of synthetic music, which once again, has shown the tight relationship between notation and technology, as new ways of representing music are being continually tried to better harness the capabilities that the new eletronic devices permit. There is the possibility that this may feed back into new forms of mathematics as well as new forms of music. That brings us back to the present. It is 2003 and we live in a world which includes the telephone ( and the newest incarnation, the cell phone), television, the Internet and its special new component the Web (less than a decade old!), powerful desktop computers, laptops and pda's. The near future suggests that all of these will soon be integrated into one (small) device. These developments are incredibly impressive. But they have a bias, perhaps more than one. The one I would like to point out is a bias toward communciation. But the communication is natural language and visual acuity. This makes sense. It is where the demand lies, and it is the demand that drives the development. However these devices have a paradoxical weakness. The devices are mathematically illiterate. One of the frustrations in preparing this paper is the difficulty of expressing a mathematical notation. But notation with technology is analogous to notation with people. It can be rote or it can be meaningful. Rote notation with technology is simply a typeface without the power. Thus I may be able to type 2 + 3 , but the technology may interpret this as only a pixel pattern. I seem to be in a situation that is the worst of both worlds. I can not, with the software I have at my disposal, either display mathematical expressions of anything but the simplest sort, nor can I manipulate symbols in any manner whatsoever. At the same time I am able to obtain sophisticated data analyses including elaborate data displays. But I cannot factor a simple algebraic expression. Andy Clark makes a persuasive case in his new book, Natural-Born Cyborgs, that we are tool users and tool creators, that our natural tendency is a symbiotic one of both influences. As he points out, Tools-R-Us. Yet these tools are not yet capable of elementary mathematics. Why? Is it, as many mathematicians have said, because mathematics is about ideas, and it cannot be reduced to some form of binary representation. Or is it, as I suspect, because we have yet to formulate a notation that allows us to create and manipulate symbols. Notation, as a tool, allows us to have ideas that we cannot have otherwise, and once we have those ideas, we may be able to modify our notations so that still other ideas become posssible. One of the difficulties at present is that it is not clear which strategy is likely to be the most successful. One extreme is a strategy that begins with our current paper-and pencil notational conventions and tries to find a way for the technology to interpret and utilize the expression. This possibility is showing more promise as we begin to create devices that recognise handwriting. The other extreme is to invent a totally new notation that is easy to implement on technology and that requires that we learn this new system. The best exemplars of this are the two software packages Maple and Mathematica. Neither package is seen in the k-12 system of education, and they are rarely encountered at the university level. There is no groundswell to have everyone learn these notations, in spite of the obvious power they lend to the user. The author of Mathematica, Stephen Wolfram, has just published a controversial book called "A New Kind of Science" that illustrates some of the new ideas that are possible when one becomes fluent in the notation. Wolfram's approach is different in nature from both our mathematical and scientific activities of the last two millenium. The emphasis is on simulation and modeling rather than analysis. Simulations are not new. Role-playing has been well recognized as an important strategy when thinking about interpersonal relationship for some time. Games such as chess and Go are recognized as abstractions of warfare, although in both cases they have become independent domains of strategy and thinking. Computer based simulations are now common in the sciences, although most models are presently based on an analytic conception of the problem. With Wolfram, the Mathematica notation provided the environment that led him to develop simulations. With others the technology has provided the environment for creating the simulations, leading to a desire to develop more powerful notations and simulation languages. Fascinating. A third possibility is a very slow, relatively speaking, evolution of natural notation as the culture gradually accepts the value of small incremental improvements. For example, we may adopt a couple of the ideas from APL such as having a different symbol for subtraction and for the minus sign. APL also does not permit implicit operations such as 2(2 + 3). This would be written as 2x(2+3), where the multiplication symbol is explicit, and clearly visible (and is not identical with the letter "x"). APL also adopts a different rule for the order of operations, one that is initially strange ( evaluate all expressions from right to left) but which offers incredible advantages when one becomes familiar with it. None of these conventions require more technology than a paper and pencil. I would like to expand, if only for a moment, with the idea that all operations should be explicit. I recall tutoring a student a few years ago who was having difficulty with solving two equations in two unknowns. Once it was clear to me that it was not clear to him what was happening, I began to work backwards to find a solid footing from which to venture. We were soon working with one equation and one unknown and the water was still deep. Moving ever closer to shore, we tried the expression 2x. This was still problematic. A few more questions and it became apparent that this was viewed as analogous to taking the word "at" and placing the letter "c" in front of it to get "cat". Toox was a new word, but not one in his vocabulary. If we had been writing this as 2 x x, or even 2*x where the middle symbol was a multiplication symbol, we would have been home free. He was quickly back to the road to understanding and to competence. Notation was the problem, not algebra or problem solving. History would suggest that this third possibility of a gradual evolution of notation (ignoring the specific example of APL) as the most likely. However we now have command of new mathematical disciplines such as seen with fractals (imagery), chaos theory (popular press) and nonlinear dynamics. This has given rise to the idea of a tipping point (Gladwell, 2000). Perhaps we are near a tipping point with respect to the relationship between mathematical notation and technology, and that the future will be much more exciting and chaotic. And schools will play an important part in this unfolding drama, since it is the young minds of youth that will be the players on the stage, and the writers of the script. But we may not recognize the language. References: I have just finished reading "Consciousness and the Novel" by David Lodge (2002) and am in the middle of "Nothing Remains the Same" by Wendy Lesser (2002). Both books consist of reviews and commentaries of other books. As a result I have a list of books to buy, some which I have purchased in the last couple of weeks, and some which I hope to buy in the near future, or find under the Christmas tree. In the same spirit as Lodge and Lesser I would like to offer a few other titles that I hope others may find equally enjoyable. They are grouped in a manner consistent with the tetrahedron model for Mathematics Education. Enjoy!
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E-mail: dale.burnett@uleth.ca |