Why Is the Sequence Sequenced?
In seeming contradiction of Pasteur, John Malone published in 2002 his It Doesn’t Take a Rocket Scientist: Great Amateurs of Science, a series of wonderful stories of scientific discoveries made by interested amateurs.
Michael Faraday, for example, was the son of an impoverished blacksmith and received very little formal education. Nonetheless, he discovered electromagnetic induction (that a changing magnetic field produces an electric field) and so became the father of the technological age. Albert Einstein and Jack Horner almost failed elementary school, but Einstein revolutionized fundamental physics, and Horner redefined our understanding of the appearance, social lives, and methods of locomotion of dinosaurs. David Levy wasn’t a professional astronomer, but he was the co-discoverer of Shoemaker-Levy 9, the comet that plowed into Jupiter in July of 1994, creating the largest explosion our solar system is likely to see (we hope) in our lifetimes.
Inspiring stories, all. However, the more one learns of these stories, the more one realizes that the leading characters were all amateurs in the root sense of the word (Latin, amo, amat), they lacked professional credentials and training, but they made up for their deficiencies by spending years in pursuit of the objects of their affections. To a person, these great scientific amateurs were tenacious individuals, driven to read and learn and think about their subjects almost to the exclusion of everything else. The point is that stories of scientific discoveries by amateurs, wonderful as they are, rarely provide counterexamples to Pasteur’s general rule: Chance favors the prepared mind.
E. D. Hirsch, Jr., reminds us time and time again that “Knowledge builds on knowledge.” Newton famously remarked in a letter to fellow scientist Robert Hooke that he [Newton] was able to see far because he stood on the shoulders of giants. Those are different ways of saying the same thing that Pasteur said. In effect, every school child, at every point in that child’s school career, stands on the shoulders of his or her former self.
One cannot do differential equations without having learned one’s algebra. One can’t learn algebra without having learned arithmetic. The application of Newton’s and Pasteur’s idea is clear enough in mathematics, but it can be applied with equal, if less obvious, appropriateness in other curricular areas. One cannot understand what a Eb+7 chord is without knowing, first, what an Eb major chord is, and one can’t understand what a major chord is without understanding that the chord is built on a major scale and is built up from intervals (3 and 5) of that scale. If you want to teach students what abstract painting is all about, it’s a good idea to start by teaching them the basics elements of which compositions are made: hue, saturation, value, shape, and line. Then introduce them to representational painting. Next show them how representations can be progressively stylized, becoming less and less pictorial and more and more abstract.
Then you can talk about the Russian artist Wassily Kandinsky, the founder of abstract expressionism, and how he thought that a pure element could “represent” an emotion without being a picture of anything at all: Blue is sad. Sharp corners are angry. (See Kandinsky's delightful little book “Concerning the Spiritual in Art.”) By building knowledge upon knowledge, you build genuine understanding.
The key to a great lesson and to a great curriculum is cumulative sequencing. When teachers design a lesson, they look at what students are doing on a given day and ask, “Does the student have to know today, in order to do this, what he or she learned yesterday?”
Here’s why: Knowledge builds on knowledge. There are two ways in which new learning depends upon prior knowledge, and they are worth distinguishing:
First, studies in cognitive psychology and in child language acquisition have demonstrated conclusively that we are much more likely to retain new learning (to transfer it from short-term to long-term memory) if we have a preexisting memory framework to which the new learning can be attached.
So, if you already know something about baseball, and you encounter theterm infield fly rule, the new term connects to existing neural machinery that encodes your memories of what an infield is and whata fly ball is. But if baseball isn’t hockey to you, the new memory will find no place, in your brain, to call home. One might teach astudent to memorize, by rote, the definition of infield fly rule, but he or she isn’t likely to transfer that learning to long-term memoryunless his or her brain already contains a semantic framework for baseball. In such a case, the prior knowledge is prerequisite to retentionof the new knowledge. It’s for this reason that people tend to learn new words in batches. You take a painting class at a localarts center, and in a few days’ time, without being aware of it, you have learned not only what the instructor taught, specifically,about graffito and pointillism, but also a great deal that you picked up incidentally , the meanings, for example, of terms like sablebrush and gesso and blocking in. Furthermore, you retain this incidental learning not because you’ve worked at doing so but becausethe learning has taken place in the context of other learning about the same subject, and quite without your being aware of it,you’ve rehearsed the learning again and again.
When you learn the new word, it is connected to a whole network of memory relatedto being in that art class and doing that painting and interacting with the other art students and thinking about art in general. This isthe reason why kids learn new words at the astonishing rate of about ten per day up through early adulthood.
Second, in other cases, the prior knowledge is prerequisite to understanding, at all, what is being communicated in the learning situation.
If you know absolutely nothing about baseball, not even that it is a game, and you read that “The shortstop caught theinfield fly,” then that sentence might as well be written in Coptic. What educators realize is the extent to whichany act of comprehension is dependent upon hundreds, even thousands, of bits of prior knowledge. Definitions on theterms shortstop and infield fly will be insufficient. Of course, the child who has tossed a whiffle ball back and forth with Dad or Momin the backyard needs no such instruction. He or she has had the prerequisite course.
Cumulative or incremental sequencing ensures that the student will not miss learned information that is essential to later learning. It also ensures that new material will be learned in the way that the brain is set up to learn, by adding to existing memory. For the existing memories, the prior knowledge, are like flypaper. They make the new learning stick.
Mr. D. Beharry