Thursday, October 28, 2010

Spelling / Grammar - worth the trouble?

If you want to make a difference in the world, you have to understand the issues. That needs language - unambiguous, CORRECT language.

If you start accepting non-standard uses of language, if sloppy grammar and spelling goes unchallenged, the consequences are dire.

Labor contracts cannot be binding if the wording is ambiguous or wrong.
Worse, if you lack a proper understanding of grammar and accept incorrect substitutes such as "relative" for "relevant," "irregardless" for regardless, your thinking is handicapped and you will find it harder to succeed in this world, or even to survive.

Read more:

Tuesday, October 26, 2010

Gleepix: Storm in a Teacup

Glee is not aimed at tweens as the CBC Radio guest, Prof. Sullivan, said.
It is a nostalgic look at our experiences in our senior year at high school.
Tweens are not buying the products advertised on Glee or in GQ for that matter.

The prudery expressed by Prof. Sullivan when she remarked on Lea Michele sucking on a lollipop shows that she doesn't get the joke. Lea Michele is the antithesis of Rachel Berry. She (Lea) is enormously attractive and would be hugely intimidating to any high school boy but all actors in the show pretend that she is some dowdily dressed, ugly Betty - to borrow a metaphor from another show that casts a beautiful actress as a frump. And yes, it is a legitimate source of humour as it reveals how little we understood ourselves in our teenage years.

Glee holds up a mirror for us, as we try to decide whom we best can identify with from among the stereotypes the cast represents. Reductio ad absurdam.

GQ is not a public magazine as Prof. Sullivan stated. It is a privately-owned business just like the Alberta Report. If you don't like it, don't buy it. But even the Alberta Report can still teach us something. GQ sells to its customers what they want to buy (as does AR) and they are all adults, notwithstanding the mother who claimed last week on CBC news that she was afraid lest GQ end up in the hands of her 8 year old son. It would be funny if it weren't so pathetic. Real high school girls wear less on the beach. Better not let your kid go to the beach, mom.

Glee presents for our consumption, tongue-in-cheek references to truths from our youth that we can all relate to, such as when Puck tells us frankly, what he really wants from his life, unedited by taste or inhibition, when Britney & Santana misbehave like comic book caricatures of teenagers, or when Rachel speaks her mind, and reveals the crass insecurities we all experienced. The upside is we adults can discuss the matters raised because Glee broke the ice,

Thanks to you Glee, GQ and to Michele, Aragon & Monteith and to CBC Radio for giving us food for thought.

Monday, October 25, 2010

"Because I can" doesn't cut it.

Ultimately, I hope we participate in sport for the joy of it. Bruce Kidd, Canadian Olympian said, "Sport is a pleasure of the flesh." It feels good to push our bodies, and competition enables us to push harder than we could on our own.

One of the values of sport is that it can teach us something about ourselves.

Thus, if we find that our competitiveness causes pain to others, it is my hope that we would desist.

By extension if we realized that what we were doing was unfair, unseemly, without grace, that we would stop doing it.

Take my Nina Kraft example, if I am beating Natascha and I know I am doing it because I cheated, I hope that I would feel bad enough about it to stop doing it. Natascha was given the title after Nina was disqualified but Natascha never got to enjoy the win by receiving her acclaim on Alii Drive. Nina knew she was dirty.
Similarly, Oscar Periero never got the acknowledgement on the podium on the Champs Elysees at the finish - Floyd Landis stole that from him, and Floyd knew he was dirty.

The win was meaningless for Nina and Floyd because they were disqualified, all they did was steal the rightful glory from the other kid. I hope for better from our sports heroes.

If you are struggling with this issue, and you should, good. We all should. Fairness is a fundamental concern in sport. Something I emphasized to the rugby and basketball teams I taught as a schoolteacher.

"Just because we can" is a phrase we hear all too often when clear thinking escapes us. Think, "why do you do this?" "Because I can." is a pathetic response. It reflects a lack of understanding about ourselves, and is especially common from pro athletes. Hardly surprising, few of them earned their graduations from high school or college honestly, like the rest of us had to.

"Because I can" has become a de facto excuse for getting away with stuff even though the person concerned knows it's unfair. So my response to your question above is that if "outside help" is understood as being unfair in a triathlon, then an honorable person knows whether "teaming up" is fair or not without being told. Even though it may be impossible to enforce.

Friday, October 22, 2010

A Philosophy of Teaching Mathematics


Where does the idea of a philosophy of mathematics come from?
Some of the first questions we ask are: What is there? What can I Know? and What ought I to do? Often philosophy is seen not as a discipline that gives answers, but one that merely hopes to ask better questions. I have a goal in mind, and that is, to do away with an establishment that seeks to bury math in symbolism, literally full of Greek symbols, such as pi and delta, unfamiliar to almost everybody, to demystify the activities that hide behind unexplained processes, such as calculus and algebra, even long division, which have plagued students for more than a century, since the inception of public schools. What I mean to do is to decode this mathematical language into a spatial reality.

After observing the world and the people and things in it we then want to talk about it. For this we employ language, and the first task is to name things: mother, father, pig, tree, antelope, and so on. Then we might want to say something about the scope of the world - how big things are, how far away things are, how many things there are. We employ metaphor: "The buffalo are as many as there are fish in the sea," or, as John Cabot told his King of the fish on the Grand Banks: "They are so plentiful, a man could walk on their backs from Greenland to Nova Scotia." Primitive man had only a few numbers: one, two, three and many. This sufficed for a primitive life. It had survival value up to a point. As long as there were enough buffalo to eat who cares how many there actually are? Need dictates.

When the metaphor breaks down we must resort to counting.

To describe the world, to say what there is, and how many there are, a few wavy hand gestures and holding up some fingers, works up to a point.
"Many buffalo, two days ride, that-a-way!" just about does it, until the population outnumbers the resources, then we need the accountants.

To master bigger numbers than we have fingers for, we employ symbols and right there we lose almost everybody. It is a common complaint to say, "I can't even balance my cheque book!"

What happened at that point was that, by employing symbols, mathematics was removed from the world of the concrete and into the abstract. This is not a necessary transformation. It has, however, been used to separate understanding from practice for most people and resulted in frustration, a sense of powerlessness and exploitation. How many big money earners have been defrauded by managers who knew more about manipulating numbers than their employers? If the math was concrete and available to everyone this wouldn't happen, at least, not so often. It may be that a fool and his money are still parted - eventually.

What I said I wanted to do is to decode the mathematical language into a concrete reality. You see - a key phrase - what has happened is that symbols are introduced into the learning process before the pre-pubertal mind can comprehend them, and therefore, rote and process learning have been substituted for understanding. Instead of understanding numbers, we merely train the processes of manipulating symbols according to rules and formula which must merely be remembered in lieu of understanding. Recitation has taken the place of comprehension.

This program has gone unchallenged for decades, for so long that most people think that there is no alternative, that children must be forced into the memorization of facts, rules, formulae and process, that that is all there is to math. Yet the alternative has been in front of us all along. To what am I referring? Why the very books we use to teach the subject. Open the schoolbooks. The storybooks, the atlases, the geography books about foreign lands and the people who live there, the history books about colonial North America, cook books for HomeEc, shop manuals for woodwork and car mechanics, science books about heat and light and sound. What do you see Pictures. You see a picture and a story below it, saying something about the picture. A picture is worth a thousand words. In the first storybooks we use to teach children there is only a picture of a thing, a boy, a dog, a ball, and the name printed underneath. this is the pattern. Open the math text, what do you see?

Where are the pictures?

What has happened as I said above is that we jumped straight to symbols, 6, +, -, x, etc. leaping from a vision of the world to abstract notation, bypassing the pictures altogether. This has made understanding impossible for almost all children. Piaget explained that the human goes through stages of development from sensori-motor to concrete operational to abstract which takes 10-14 years or more and no amount of coercion can change that. So in order to save children from counting on their fingers for ten years until their cognitive; development renders them capable of grasping abstract concepts, to give them something to do from grade one through grade ten we have resorted to drill and recitation, endless practice of meaningless skills so that when their intellect is ready for algebra, so the thinking goes, they can understand (suddenly) what we have been going on about for the first decade and a half of their lives. The cost in terms of their patience and cooperation has been enormous. Many children resent the boredom of mathematics the seemingly senseless waste of their time doing endless reams of sums for ten years that by the time they get to Grade 10, where there is often an option to drop math or at least to take some practical course like shop math or bookkeeping that may lead to employment is possible.

That in a nutshell is the problem To put it in the form of a question, what is the point of training efficiency and competence in arithmetic for the first ten years of a child's life if the cost is that it puts them off ever wanting to do math again? If they can be persuaded to study, to memorize the facts, rules, formulae and process necessary to achieve competence in math for ten or twelve years of schooling, but at the end they just hope they can remember it long enough to pass the exam and hope they never have to do it again, what really was the point of doing math in the first place?

The solution?

Change the method of presenting and doing math from grade one or two say, up to grade 8 or 9 or 10. Grandma and the pre-school teachers and the grade one teachers are doing just fine. All a child needs to start with they are already getting: the ability to count to nine and to build a rectangle. Although we have something to say about the counting.

If little Johnny can say, "one, two, three, four, five, six, seven, eight, nine, ten." Can he count? No. Well, maybe. You see all he has demonstrated is that he has memorized the names of he numbers, as he has memorized his A,B,Cs. But there is more to counting than knowing the names of the symbols 1,2,3,4,5,6,7,8,9, and 10.

You have to know that 4 is larger than 3. If we assumed that the child would know that 4 was bigger than 3 because it came later in the sequence wouldn't he think that Z was 26 times bigger than A? All we know from the recitation is that he knows the names of the number symbols, just as he knows the names of the letter symbols.

We need to teach to concept of magnitude and nothing is simpler than to substitute a manipulative, a plastic block, say, for the symbols, initially.
Let us use a small cube, a little green block, attractive, non-threatening. Let us hold it up as the child holds his and say, "This is one."
Then let us hold up a small orange block - merely so that it is easily distinguishable from the one by looking, and say, "This is two." The two-block will be the same size as two green "ones" placed together.
Then let us hold up a pink block and say, "This is three." The pink block will be the same size as three ones placed together.
And so on.

We will lay them out on the table and say these are the numbers, one, two, three, etc. and we may write down the symbols below the blocks and say, "These are the names of the numbers."

The point here is to teach counting in concrete terms, rather than with symbols. Piaget has shown that it is pointless to employ abstract teaching concepts before the mind is ready but we can still teach the skills of mathematics. we can teach "kindergarten calculus."

All we do in math is count.

Friday, October 15, 2010

The Acquisition of Meaning

How does a thing acquire meaning?

How does “y = Mx+C” acquire meaning?

How does “the square on the hypotenuse equals the sum of the squares on the other two sides” acquire meaning?

How does “6” become meaningful?

One of my lifelong philosophical investigations has been the acquisition of meaning. There is a TV show that ends its credits with a boy pointing at a tree on a hillside and asking, ”What does that mean?” Is that a sensible question? Consider if the tree has meaning, if the tree on a hillside has a meaning, if a tree on a hillside against a cloudless sky has meaning. I think what this line of questioning reveals is that we look for patterns of relationships to elicit meaning. Humans are essentially pattern recognizers. I wrote an undergraduate thesis entitled, Intelligence as the ability to recognize, extend and create patterns.” In it I wrote that a pattern is a relationship between at least two things. For there to be a pattern there must be at least two things. In the case of a dot on an infinite field there is no pattern. But a dot on a field with finite boundaries must have a relation to at least the boundary, or in the more pedestrian case, the dot has a relationship with the edge of the paper. By extrapolation, an isolated event, a “random” event is not a pattern. For an event to have a pattern there must at least exist a context for the event. If there are two such events, then of course, we have a pattern.

One way to investigate the acquisition of meaning is to consider learning a foreign language as an adult. I have taught ESL. One typically begins with some vocabulary, some examples of familiar objects: dog, cat, man, car. This is just re-naming. The meaning for the learner is still the original word/object relationship they discovered growing up. My understanding of cat is not enhanced by learning that, in French, it is called “chat.”

In some languages I am told, there are concepts that do not exist in other languages. If a student should acquire the new concept in the new (for her) language as an adult, then she would be able to tell us something about the acquisition of meaning. Without her we have to introspect a bit more. If I learn a new word in a foreign language, say, in a vocabulary list, such as perro, broma, barato, etc. I may memorize the list and pair them with the equivalent words in my first language but it is a chore and I need many reviews, even then I have to translate the word to English to assign meaning to the object to which the word refers. I think what is missing to give the new Spanish word meaning for me is having the relevant experiences with the object that a native speaker might have when acquiring the object in the quotidian way. This is to say that there are schemas (pre-concepts) and actual physical experiences (sensory experiences) that someone would have in association with learning a word. Take “perro” for example. When a Spanish child first meets a perro his parents might use the word as it licks his face and that experience is what the child recalls when he hears the word subsequently. Also, when he experiences the face-licking again he might utter the word “perro.” Importantly there will be feelings associated inextricably with the use of the word, and vice versa – the word can elicit the feelings.

This is an important feature of meaning acquisition. To put it bluntly, for there to be meaning, there must be feeling. Could there be a rote recall of an equivalent meaning as in vocabulary lists? Yes, of course, but for there to be a visceral understanding of a word or concept, there must be a feeling associated with it.

Therefore when teaching, we must seek to evoke feelings in association with a concept in order to ensure that students acquire meaning viscerally.

A corollary of this is, that if there are no feelings associated with an experience then it is meaningless. An example of this may be when a student witnesses an experiment in science, chemistry, say, perhaps litmus paper turning colour when in contact with an acid or akali, and having no accompanying feeling, she simply shrugs and says, “so what?” In this case I am prepared to admit that it is an appropriate response and we as educators ought to respond positively to it by changing our approach, rather than, say, reproaching the student for having a bad attitude.

Again, to acquire meaning requires that there be feelings present. What kind of feelings? There is an amazing range of possible feelings that could be associated with an experience. Consider dissecting a frog. You can easily imagine the range of feelings this could engender, ickiness, amazement, wonder, fear, disgust, and so on. No doubt all of us remember that experiment vividly. For this query, it is not the one, intended, scientific feeling that I am concerned with. I wish only to observe that with strong feelings comes strong learning. Focusing the learning experience is another project. Thus, in the case of learning math, where meaning is often lacking and the learning is sometimes considered tedious, it is obvious that we must strive to present the opportunity for the students to experience feelings when learning math. One feeling we all would like them to experience is the joy that comes with success.

How shall this be done?

One way is to employ the "explore and discover" principle. In practice this looks a lot like play, but it is directed play.

Exploring and discovering are activities naturally satisfying curiosity. And like attaining understanding, directed play, or exploring and discovering, produces good feelings.

Understanding a concept is usually accompanied by the production of dopamine and endorphins in the brain, in short, pleasure. When that little light goes on, and you finally "get it," it feels good!

Jean Piaget and Maria Montessori have shown that understanding of concepts occurs naturally in children if the appropriate sensori-motor experiences can be had. That means the necessary opportunity for exploring and discovering concepts must be provided for the children to learn naturally; in other words, play.

But not just any play, if you want children to learn language you must provide a language-rich environment. If you want children to learn math concepts you must provide a math-rich environment. More on that later, let's get this straight: sensori-motor experiences of the right kind are necessary for the acquisition of schemas - groups of experiences, that can be assimilated and synthesized into concepts or accommodated by the child's mind. This sense of understanding is enjoyable and is all that is needed for successful learning.

It is our mission as teachers and parents to provide the appropriate materials and situations for this to happen, and sometimes just to get out of the way.

While that made me happy, by 1998 the demand on my schedule had become so heavy I knew had to find another way to reach more people. My solution was to record my workshop on video. I willingly sent the tapes to interested parties I couldn't teach in person. The problem with that scheduling solution was no one could ask questions. Extra explanations were missing because I wasn't there in person at the white board. That led me to the development of a handbook.

I'll tell you more about that in a minute. You probably want to know more about the Mortensen Math system first.

MORTENSON MORE THAN MATH employs manipulatives to enhance the child's ability to visualize math concepts, to decode the mathematical language into spatial reality.

The best way I know to explain the Mortensen Math system is to talk about memory first. How good is your short-term memory? More importantly, how good is your short-term memory with numbers? Suppose I gave you 12 numbers, each of them seven digits long. Do you think you could remember them for an hour? Five minutes? Do you think you could remember them long enough to write them down, even right after I told you?

Not likely. That's because you've been taught like everyone else to memorize the hard way. The hard way is how most students are taught math as well.

The truth is the entire math curriculum used in traditional teaching situations, employing textbooks, relies on memorizing nothing but FACTS, RULES, FORMULAE AND PROCESS!

Our job as educators is to decode this mathematical language of symbols into a concrete reality. This is what the method does.

Learn more at my webpages: Geoff White - Teaching Math with Manipulatives using the Mortensen Method