Harking back to my posts of August 11 and 15, which were about mathematics, I want to express some thoughts about math and me, and the teaching of math in general.
Pardon me for saying so, but I was an exceptionally intelligent child, and yet almost from the beginning I became apprehensive when numbers were mentioned in a classroom. If the chalk squeaked on the blackboard and words appeared, I perked up, but if the chalk formed numbers, I cringed. I might be able to explain that better if I could remember the very first experiences, but I can’t.
Try as my arithmetic teachers might to make numbers interesting, I was a lost cause. One problem was that I’ve always needed to understand the foundation, the most basic principles, of anything I set out to learn. While a teacher was setting us to memorize multiplication tables, I was still asking questions like, “What is a number?” I didn’t really know how to formulate the questions, but I felt that I was being thrown into the middle of a chaotic mystery rather than starting to build from the electrons, protons, atoms, and molecules.
Whether not the need to begin with basic principles is a defect in my thinking I don’t know, but it’s there, and very insistent. . . and in my first twelve years of school, very inconvenient. I suspect that it was related to my dislike of hypothetical practical applications.
One way the schoolbooks of the 1940’s and 50’s tried to interest students in math was to pose “problems” in everyday practical terms. “If a farmer has twelve bushels of wheat. . .” “If train A leaves station A at 11:12 a.m. and travels at 60 miles an hour, and if train B. . .” “You have $12.10 in your pocket, and you . . .” Even at this moment I can feel my stomach shrink as I write those words. Why is that?
Oddly enough, I begged to have number manipulation taught to me in abstract terms, for the numbers’ own sake so to speak, and for the interest that resulted in wondering what would happen if this or that mathematical operation were applied. I literally pleaded for abstraction, but my feeling was that the teacher didn’t comprehend what I was talking about and didn’t know how to answer me. The result was that I took almost no mathematics courses.
Leaving my attitude aside, it makes sense that most children will be more interested in something they can use than in something they’re just required to memorize and parrot back, but practical applications arising from a farmer’s truckload of produce or a train schedule were not things we needed to know then. What I have learned in suffering from my ignorance of math in “Second Life” (“SL”) today is that the motivation to learn about numbers (if an exciting abstract approach isn’t taken) comes from a present desire to do something which one really wants to do out of personal enthusiasm, and cannot do without knowledge of mathematics.
To take a simple example which reveals my abysmal ignorance: The basic size of a parcel of land in SL is 512 square meters. If the parcel of land is square, how long will the boundary lines be in meters? I have no idea, but I need to know, now, because I’m looking into buying a piece of land today. How do I even estimate the boundary lines of a rectangular piece of land of 3320 sq. meters? It’s not an activity that sounds fun, but it is necessary in light of my enthusiasm.
And if I want to put a roof on an SL house whose frontal width is 9.3 meters, using two pieces of roof which will rest on the existing house sides and meet in the center, how long and wide should those roof sections be? Even I soon recognized that it depends in part on the angle of the roof, which gave rise to many interesting abstract questions in addition to the practical ones.
See what I mean? As far as practical applications are concerned, a present need to know, based on personal enthusiasm for accomplishing something now, seems to me the key to motivation in learning math.
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The last time my math lethargy was dispelled was to gain advantage in an on-line game. Hilarious.
ReplyDeleteBasic principles. Only recently did it occur to me how the symbology for numbers probably evolved.
In the neolithic as now, the ancients must have commonly used fingers for counting. But as transactional complexity grew, they needed a better means of keeping sums straight while they argued over value. So they carried small sticks or bones of equal length. Probably 12 of them. As substitutes for fingers, they arranged these sticks into rows, racks, and stacks.
Eventually they standardized stick arrangement, allowing them to save time by simply drawing the scratches onto a surface. Then for yet more speed, they stylized the 2, the 3, the 4, naturally rounding off the gaps in the scratches for speed. The 5 is a four-rack with a stick across the top. The 6 is a five-stack with one stick poking off from it. The 7 is a five-stack with two sticks hanging off it (and a cross-bar missing). The 8 is a double 4-rack. The 9 is a five-stack and a tail enclosing the abbreviated presence of a 4-rack below it. A ten is one double-five stack. A twenty is two such stacks, and so on.
So our numerical system isn't really 10-based, it's 5-based, then doubled. Why carry twelve sticks? There was an older system based on double-sixes, the enduring popularity of which can be inferred from the sheep's knee knuckles which have been found in many stone age archaelogical sites. The knuckles are well-worn, with a mark upon each side. The cave people played games of chance with dice, and thought in terms of dozens. Eggs are still traded by that measure.
Till you wrote this, I had absolutely no interest in these computer simulation games. But what you said about maths struck a very positive chord in me, for I had almost exactly the same reactions. It seems to me that there is a vast untapped response to numbers and relations between abstract things, which is trampled upon by ordinary teaching, which tends to use the stick before the carrot has ever been tried.
ReplyDeleteI clearly remembered my first encounter with algebra, when we were introduced to x and told it was not a letter but a number, a number whose value we did not know but had to discover. Actually it was quite fascinating but I could hardly listen to the teacher because so many unanswered questions had to sit and wait their turn, in this case for many weeks: such as "Why?" "Whoever first bothered with this? Of all the silly questions which could be raised, of all the interesting childish games we could play, like each child dropping a stick off one side of the bridge and then running to the other side to see which stick was going to win the race, why invent a game of "x" where we had to find out what "x" is? A bit like "I spy with my little eye, something beginning with p." I don't know if that game is played in America.
Or why is it "math" in America and "maths" in England?
Yet numbers were interesting, if only they were not presented as a kind of punishment.
Marc, this is brilliant. You should publish a paper.
ReplyDeleteVincent, you can't imagine how fascinating, and moving, it is to me to learn that you had the same reactions to maths teaching that I did. (Why "maths" and "math"? I have no idea, but "maths" is more rational, isn't it?)
ReplyDeleteYour description of your reaction to algebra describes my own reaction. We were presented with "basics" which were not really basic principles, weren't we? I had no idea in the world why algebra existed, what it did that couldn't be done with numbers before it existed, and who cared anyway. . . all those kinds of things I needed to know in order to get a grasp of the subject.
I must say that I am finding math much more meaningful and interesting when it is associated with geometric shapes and their interrelationships than I did when it was associated with Jane's pennies or the amount of fuel in a tank.
Thanks!
Link, thanks for the comments on mathS and lithping. I'm glad to hear that I’m not the only person who needs “to understand the operation of something in its entirety before I can start to comprehend it in its parts.” You expressed that so much better than I did. Judging from the people who have commented on this post, my learning defect is a badge of brilliance!
ReplyDeleteI will be seeing you in Second Life soon because it costs nothing, so how can you resist? For the expense of zero (I did learn some math) you can do everything that can be done in SL except own land. You can learn to create objects and build in public areas called “sandboxes”. (All materials used to create objects in SL are free.) The only catch is that if you want to own land, you must become a “Premium” member for about US$10.00 per month.
I swore I would never spend a penny on imaginary land, but, to answer your question, I am now forking out $10 a month for the Premium membership. You can buy 512 sq. meters of land (on the open market, around $20-22 real money) without incurring monthly “maintenance fees” which the SL ownership imposes depending on the amount of land one owns. If you buy a total of 1024m of land, add $5 per month to the $10. That is plenty of land for building your dream house. . . as might be 512m.
Information: http://secondlife.com/whatis/
Download: http://secondlife.com/community/downloads.php
Link, if you are going to give SL a spin, which I hope you will because I’d like to share ideas with you, I can save you some wasted time at the beginning by giving you a few tips and information sources.
As a math teacher, I've seen many types of learning styles. My personal preference are the students who enjoy the abstract. That is usually an indication of high math ability which should be nurtured and developed rather than suppressed and silenced as was the case for you, Fleming.
ReplyDeleteElementary mathematics has become a farce over the past 15 years. Students are often marked wrong if they indeed have the correct number answer but are unable to explain, with words and illustration, how they arrived at that answer. This is true for the simplest of problems, ie. 16 + 25 or 3 x 4.
This approach to math guarantees students with the greatest math ability will run far away from mathematics in the future. I think that is what happened to you, Fleming. You did not process numbers and equations the way the other students did. Your needs were different and were, therefore, ignored. Had you been able to learn math on your terms, your relationship with numbers would be very different today.
As far as the roof problem goes, that is a classic trigonometry problem. However, you need to declare one of two things: the height of your roof or its angle.
The width of the building is 9.3 m. If you want a symmetrical building, you'll want the peak of the roof to be at 4.65 m. If you imagine a line drawn from the top of the highest point of the roof to its base, you will form two right triangles, each with a base of 4.65m. I'll simplify the problem by making the assumption that the height of the roof is the same as the base. To find the length of the longest side of the triangle (the hypotenuse), just use the Pythagorean theorem:
the sum of the squares of the legs of a right triangle must equal the square of the hypotenuse.
4.65^2 + 4.65^2 = hyp^2
The longest side should be 6.576 meters.
The way that SL has sparked your interest in math makes a wonderful argument for the use of virtual worlds in education. I am taking notes on you, Fleming!
Thank you, teacher, I actually understand! Colleen, it was actually fun to follow your explanation about the roof -- although I needed smelling salts when I first read "the sums of the squares". . . not having remembered what the square of a number is. Can you believe that?
ReplyDeleteYour comments about learning are very valuable to me and I'm sure others. So I wasn't just a defective piano with a dumb note for math(s); the potential was there. That's encouraging, although it's sad to think how much I missed.
For awhile I was having doubts about FLIGHTS OF PEGASUS, but when we include all the comments submitted for this post, I feel that something really worthwhile has been accomplished here.
Thank you, Colleen.