Enjoying Mathematics

In GWS #77 we asked to hear from people who enjoyed math and did it in ways other than by using textbooks and workbooks. Our mailbox has been full of these letters ever since – see GWS #78 for the first set of responses.

Discovered Pattern

Cerelle Simmons (TX) wrote in the September 1989 issue on a local newsletter she used to publish:

One day the kids and I were fooling around with pennies when we stumbled across the pattern behind square numbers. Because I’m not much of a mathematician, it was new to me, too. The episode went like this. We had four pennies in a square. [Cerelle shows a picture of a 2 x 2 square.] To make a bigger square, we needed five more pennies [picture of a 3 x 3 square]. For the next square, we needed seven more pennies [picture of a 4 x 4 square]. At about that point, we realized that the first square number is 1, and the next square number (our original 4) is 3 more than that. Hmm… 3, and then the next one is 5 more, and then 7 more… how about that! Sure enough, 9 more pennies would give us the next square number, 25! From there, we could predict all the rest.

We were as pleased with ourselves as we could be, which is the whole point of using manipulatives. Memorizing math rules is no fun at all, but discovering that there really is an order amid seeing chaos is the stuff of enlightenment.

Building Lego Clock

Marian Bever (IN) writes:

After you printed my letter about Mark’s interest in bridge (GWS #74) and how we had started the hunt for a mathematical way to find out about probabilities, we received a very nice letter from a man in Canada. He filled us in on how to figure out the probabilities for the hand I had described in the letter. It was really exciting to get a letter in response to my letter in GWS.

Mark is still interested in bridge, though we have not found any young people who are also interested in it. I have a feeling that most people think bridge is too hard for children to learn, but I think in depends on their level of interest. In all fairness I must say that we have not searched very hard for other young bridge players, but I am not sure where to begin with this. He still reads the bridge column in the newspaper regularly and he received a book of bridge problems for Christmas. He has worked through chapter one of that so far. I suspect that for now, he has worked on the probability angle of bridge hands as much as he wants to and is satisfied at this point. Still, he knows that these techniques are available to him should he want to pursue it further.

His most recent adventures in math have come with the Lego bricks. He built an electric clock out of them! I asked him to explain to me the process he went through to get this accomplished, because I was completely in the dark about this project until I saw the completed clock running and merrily keeping time. I have no idea how he happened to think of trying to build a clock, and he doesn’t really know either.

The first stage was that he removed the motor from an old tape recorder. He said that he clipped the motor to a transformer with alligator clips. This reduced the voltage from regular household current to about six volts DC. This particular motor was not designed to run on household current. He tells me that he worked at figuring out the speed of the motor by rigging up some gears to reduce the speed by a ratio that he knew. He attached a Lego brick as a kind of arm and counted how many times it went around in a minute. Then he must have multiplied by the numbers represented by the gear ratios and came up with about 2,88O revolutions per minute for the tape recorder motor.

Some time after this he ordered two packages of the Lego gear wheel assortment. There were about twenty-one gears in each package, and various gear ratios were represented. He used the belt from the tape recorder to attach the motor to the Lego pulley and axle. From the motor to the pulley was approximately a 4.4 gear reduction. After he divided 2,88O by 4.4, he had about 654 revolutions per minute which he needed to reduce to one revolution for the second hand of a clock.

Then he asked his dad for some advice. His dad suggested that he find the prime factors of 654. These turned out to be 2, 3, and 109. I asked Mark how he arrived at this and he said he divided by 2, then by 3. This left him with 109. He worked with this for a while to see if it was divisible by anything and decided it must be prime. In the Lego gear wheel assortment, there are gears for making a 2:1 and 3:1 reduction. There remained the 109:1 to figure out. He found that the closest he could come to 109:1 with the gears he had was 112.5:1. He did this by arranging several gear wheels in a series. He used the 1.5:1, 3:1, and two 5:1 gears. He later streamlined this by using one 3:1 gear instead of the 1.5:1 and the 2:1. To reduce the speed for the minute hand, he found the prime factors of 60, which are 2, 2, 3, and 5. He used the Lego gears with those ratios. Then to reduce the speed further for the hour hand, he found the prime factors of 12 (2, 2, and 3) and used the gears with those ratios. With the help of his dad, he figured out that the clock had an error of about 3.1 percent. This was because of having to estimate the exact speed of the motor and because of not being able to reduce the speed by the ratio of 109:1 exactly.

When he was describing the Process to me, Mark several times used the expression, “I played around with the gears,” which reminded me of some things John Holt wrote about. He wrote that children need time to play with equipment, get familiar with it, before they are ready to build or do experiments with it. Perhaps something like this was going on with Mark. It is hard to quantify thinking time. As I talked with him, it seemed evident that he had done a lot of thinking about this before he actually began building the clock.

So much of my own math training was abstract that I was glad to see an actual reason for needling to know the prime factors of numbers. After this project, I think Mark will have a good understanding of prime numbers and their possible uses as well as an understanding of ratios.

We bought some math workbooks this year thinking that Mark might want to dig into algebra. He does not seem to like doing workbooks. He worked at them for a while, but they gradually began gathering dust. When he does something creative like this clock, I begin to ask myself, does he really need me to teach him math?

Loving Numbers

Wilt Clarke (MA) writes:

I’m writing in response to your request to hear from people who enjoy math. Our daughter Chelsea’s interest in, and ability with, numbers has always seemed to be slightly ahead of her progress with other skills. At 6 she reads numbers up to 100. With single-digit numbers she can add, subtract, multiply, and divide quickly, in her head, but she is only beginning to understand what we mean by those labels. She has a fair understanding of fractions, especially quarters and halves. I don’t know that her abilities are at all remarkable except that she has picked this up on her own. We have tried to answer her questions and have provided materials, like dot-to-dot puzzles, that she requests. She seizes opportunities to make calculations, such as dividing treats evenly or deciding how many outfits to take on a trip. In contrast, although she loves books and knows the sounds of the letters, she has no interest in sounding out words and when we have prodded her to do so she resists.

Measuring and estimating distances are also great fun for Chelsea. She is much more likely to ask for a piece of string five inches long than to indicate the length with her hands. She often mixes inches and centimeters (“four inches and one centimeter”), I suppose because so many rulers show both. She has not shown much interest in Cuisenaire rods, except as blocks, but she did enjoy playing with an abacus at a friend’s house recently. Her favorite “math toy” has been one of those little plastic money clickers for adding up pennies, dimes, and dollars.

The kinds of math she is exposed to are very ordinary: dividing recipes, balancing checkbooks, and so on. I use a fair amount of arithmetic in my business so she sees me using a calculator often (I find mental arithmetic tedious and, even though I didn’t grow up with them, have embraced calculators wholeheartedly). We occasionally try and milk the educational opportunities of activities like cooking and building, but we don’t play math games or anything like that. We do find ourselves talking her through calculations rather than simply answering the question, partly because it’s fun to see what she can do and partly because we imagine that it helps her learn. Mostly she doesn’t seem to mind this, but I have mixed feelings. I wouldn’t like to be treated like that, but it does seem that children recognize learning basic skills as part of their work and don’t necessarily resent adult help. The hard part is seeing whether she wants to learn more about numbers or just wants to know how many forks to get out.

I was fortunate enough to miss learning algebra in school, by changing schools, and had to learn it later in order to understand high school chemistry. I learned it by studying the chemistry text and with some help from the chemistry teacher, and because I had an immediate application for it, I enjoyed doing so. It seemed like this great trick, a wonderful tool for answering seemingly impossible questions. That’s what I like about math – what it can do – but Chelsea may love numbers just for themselves.

Whole Family is Interested

Ann Bingham of New Jersey writes:

We are a homeschooling family who has a very strong interest in mathematics. My husband and I met in graduate school, and we both have PhDs in math. We are homeschooling four children, ages 3 to 11.

We believe in unschooling, and do not use a formal curriculum in any subject. Of course, that means that in math we don’t use textbooks per se. I use them as a reference to see what incremental steps to take to present a new concept, and occasionally my eldest will sit down and do just the word problems. But usually the books have pages and pages of computation, which is just too boring to bear. It can take away much of the joy of the subject to have to do so much repetitive work, when we all know that calculators will do it faster. I must admit that my children do not know their “math facts” as well as some school children, but I take solace in the fact that even the National Council of the Teachers of Mathematics ranks computation seventh in a list of important math skills. The first six are problem solving, estimation, alertness to reasonable answers, applying mathematics to everyday situations, and communication of mathematical ideas.

Clearly all of these are easily included at home. We do all the usual hands-on math – counting the days ‘til birthdays, using money, telling time, and cooking with the younger children, balancing the checkbook with my older daughter and having her help at booksales and things like that.

A deck of cards provides any number of math games, as does a pair of dice. Our library has books on math games for children, most of which are inexpensive to make. We also have Cuisenaire rods and Unifix cubes, both of which are fun to play with on many different levels. Two math games we like are Dec-a-ten, a great game for practicing multiplication and addition (Ideal School Supply) and Fraction Games (Lauri).

The magazine Provoking Thoughts (4O4 NW 1st St, PO Box 1004, Mower County, Austin MN 55912) contains logic problems, arithmetic puzzles, music problems, palindromes, and more, and has all levels of difficulties. It is a great little periodical!

We also make use of the computer. The educational games we have for the computer have not been well-received by the children, but the 11 year old and the 7 year old both enjoy LOGO, a computer language that starts out as just graphics but can also be a true computer language. We do like Speak ’N Math and Computron for playing; in is mainly flashcard work with beeps, but those beeps seem to make it fun.

We don’t do “book math” every day. In fact weeks go by when we don’t sit at a desk to do math (but we do math in real life every day, of course), but we go through phases of being very interested in a particular subject, say long division or prime numbers. Then we may make posters (a favorite activity at our house) and try to incorporate that subject into more of our other activities. There are wonderful books at the library on all sorts of math concepts. The ones by Anno come to mind immediately.

Solving Puzzles

Barbara Needham of California writes:

I am an adult who enjoys math. I have two sisters who also enjoy it and three daughters who in varying degrees do not like it. At present I am not doing much with it, except trying to teach algebra to a ninth grader. I enjoy working the problems in the text (Saxon Algebra I) more than she does, even though she can do them.

I have worked as an accounting assistant and balance our checkbook and do our taxes. But the most enjoyable math I do right now is when Dell Puzzle Magazines issues a new copy of Math Puzzles and Logic Problems.

I graduated from college as a math major, with special interest in abstract algebra. The puzzle aspect of math has always interested me more than other parts – trying to solve an intricate or difficult problem. Once in college we were given an extra credit proof to solve. I was the only one who got it on my own, but I never did quite get the topic being taught in the class at that time because that extra credit problem was occupying my thoughts so completely.

An actuary who worked at my father’s office once told me he liked math only if it was practical and he could see what it was used for. I was – and am – the opposite. I don’t really care if in is useful or not, I just like things like number patterns, algebraic systems, etc. I enjoy stories of mathematicians engaged in “pure” research who only later learned that there was application in nature for what they had learned.

Other things I have enjoyed using math for are machine language programming on a Sinclair computer and figuring out knitting patterns.

Why I like math I don’t know. It wasn’t anything special to me in grade school and I had as hard a time with multiplications tables as anyone – my mother finally made me learn them. In seventh grade I started to like it but can’t remember anything specific that teacher did. In high school I took all the math offered (no calculus in high school then) and worked all the problems even if they weren’t yet assigned. The first two years of college were not too great, then after a break of five years I got all A’s in math.

Let me add by way of encouragement that I was never that great at ordinary computation – adding, subtracting, multiplying, and dividing (actually, dividing was OK because it was more of a process) – until about ten years ago when I worked on my mental arithmetic, and five years ago when I took a 10-key calculation course (self-taught, self-paced). And I am 49 years old.

Maybe She Missed Something

More from Lesley Westrum (OH):

Typical of me, I skipped over pages 19-22 of GWS #78 – I always skip the parts about math because I hate math. However, I jumped right in to the letter from Marion Cohen, because in was titled “Love at First Sight”, and I’m very much interested in love!

Now, for the first time in thirty-four years of intense math-avoidance, I begin to feel as if I may have just possibly missed something. If falling in love is like solving a polynomial equation, then I can see I need to find out what a polynomial equation is!

I think my hating math must have started in school, where the first thing I learned about the subject was that it was (or seemed) completely unpredictable. You never knew if your answers were right until the teacher graded the paper, by which time you couldn’t recall how you got the answer. Worse yet, sometimes you’d get the right answer but get it in the wrong way, so the answer would still be considered wrong.

Obviously, as a more or less functioning adult, wife, mother of four homeschooled kids, I do run into numbers rather often in my daily life. That isn’t a problem for me, because I long ago (two decades ago) taught myself a sort of mind game to deal with my “mathematical inability”. It’s a simple matter of self-delusion, really. When I’m cooking, or designing and drafting a pattern, or doing needlework or planning a trip with roadmap in hand, or planning an order from the food co-op, I simply pretend to myself that what I’m doing doesn’t “really” have anything to do with math.

At first glance you’d think a mother who hates math wouldn’t be able to teach in to her children. Since I’ve been reading as long as I can remember and love to read and write you’d expect me to raise lopsided children then, who are reading fiends and math-avoiders. Yes?

No! Madeline (10) was reading quite well at age 3. She now reads constantly, and says she hates math. Fine. James (9) is just now beginning to really read, but for a couple of years he’s been doing math in his head as fast as I can, and often faster. I don’t mean 2 + 2, either. As an example, one night his dad and I were sitting in bed with a road atlas, planning a trip to Grandma’s house. The kids were asleep, the house silent except for the scratch of Mike’s pen on the paper. He mumbled, occasionally under his breath, and finally concluded, “‘Well, I’d say about 360 miles”.

I swear not four seconds elapsed before a sleepy voice came wafting in from James’s side of the hall, saying, “Dad, at 60 miles an hour it’ll take six hours, right?”

Until I met this son of mine I never realized a person could actually LOVE numbers. Amazing!