Representations, Conventions, Algorithmic Manipulations, and Logic
The written numbering system we use is merely conventional and totally arbitrary and, though it is in a sense logically structured, it could be very different and still be logically structured. Although it is useful to many people for representing numbers and calculating with numbers, it is necessary for neither. We could represent numbers differently and do calculations quite differently. For, although the relationships between quantities is "fixed" or "determined" by logic, and although the way we manipulate various designations in order to calculate quickly and accurately is determined by logic, the way we designate those quantities in the first place is not "fixed" by logic or by reasoning alone, but is merely a matter of invented symbolism, designed in a way to be as useful as possible. There are algorithms for multiplying and dividing on an abacus, and you can develop an algorithm for multiplying and dividing Roman numerals. But following algorithms is neither understanding the principles the algorithms are based on, nor is it a sign of understanding what one is doing mathematically. Developing algorithms requires understanding; using them does not.
But what is somewhat useful once you learn it, is not necessarily easy to learn. It is not easy for an adult to learn a new language, though most children learn their first language fairly well by a very tender age and can fairly easily use it as adults. The use of columnar representation for groups (i.e., "place" value designations) is not an easy concept for children to understand though it is easy for children to learn to read and to write numbers properly, and though it is fairly easy for children to learn color representations of groups, with practice.
And further, it is not easy to learn to manipulate written numbers in multi-step ways because often the manipulations or algorithms we are taught, though they have a complex or "deep" logical rationale, have no readily apparent basis, and it is more difficult to remember unrelated sequences the longer they are. Most adults who can multiply using paper and pencil have no clue why you do it the way you do or why it works.(21) And that includes most elementary school arithmetic teachers.
Now arithmetic teachers (and parents) tend to confuse the teaching (and learning) of logical, conventional or representational, and algorithmic manipulative computational aspects of math. And sometimes they neglect to teach one aspect because they think they have taught it when they teach other aspects. That is not necessarily true. The "new math" instruction, in those cases where it failed, was an attempt to teach math logically (in many cases by people who did not understand its logic) while not teaching and giving sufficient practice in, many of the representational or algorithmic computational aspects of math. The traditional approach tends to neglect logic or to assume that teaching algorithmic computations is teaching the logic of math. There are some new methods out that use certain kinds of manipulatives(22) to teach groupings, but those manipulatives aren't usually (merely) representational. Instead they simply present groups of, say 10's, by proportionally longer segments than things that present one's or five's; or like rolls of pennies, they actually hold 100 things (or ten things or two things, or whatever).
Students need to learn three different aspects of math; and what effectively teaches one aspect may not teach the other aspects. The three aspects are (1) mathematical conventions, (2) the logic(s) of mathematical ideas, and (3) mathematical (algorithmic) manipulations for calculating. There is no a priori order to teaching these different aspects; whatever order is most effective with a given student or group of students is the best order. Students need to be taught the "normal", everyday conventional representations of arithmetic, and they need to be taught how to manipulate and calculate with written numbers by a variety of different means -- by calculators, by computer, by abacus, and by the society's "normal" algorithmic manipulations(23), which in western countries are the methods of "regrouping" in addition and subtraction, multiplying multi-digit numbers in precise steps, and doing long division, etc. Learning to use these things takes lots of repetition and practice, using games or whatever to make it as interesting as possible. But these things are generally matters of simply drill or practice on the part of children. But students should not be forced to try to make sense of these things by teachers who think that these things are matters of obvious or simple logic. These are not matters of obvious or simple logic, as I have tried to demonstrate in this paper. Children will be swimming upstream if they are looking for logic when they are merely learning conventions or learning algorithms (whose logic is far more complicated than being able to remember the steps of the algorithms, which itself is difficult enough for the children). And any teacher who makes it look to children like conventions and algorithmic manipulations are matters of logic they need to understand, is doing them a severe disservice.
On the other hand, children do need to work on the logical aspects of mathematics, some of which follow from given conventions or representations and some of which have nothing to do with any particular conventions but have to do merely with the way quantities relate to each other. But developing children's mathematical insight and intuition requires something other than repetition, drill, or practice.
Many of these things can be done simultaneously though they may not be in any way related to each other. Students can be helped to get logical insights that will stand them in good stead when they eventually get to algebra and calculus(24), even though at a different time of the day or week they are only learning how to "borrow" and "carry" (currently called "regrouping") two-column numbers. They can learn geometrical insights in various ways, in some cases through playing miniature golf on all kinds of strange surfaces, through origami, through making periscopes or kaleidoscopes, through doing some surveying, through studying the buoyancy of different shaped objects, or however. Or they can be taught different things that might be related to each other, as the poker chip colors and the column representations of groups. What is important is that teachers can understand which elements are conventional or conventionally representational, which elements are logical, and which elements are (complexly) algorithmic so that they teach these different kinds of elements, each in its own appropriate way, giving practice in those things which benefit from practice, and guiding understanding in those things which require understanding. And teachers need to understand which elements of mathematics are conventional or conventionally representational, which elements are logical, and which elements are (complexly) algorithmic so that they can teach those distinctions themselves when students are ready to be able to understand and assimilate them.
Footnote 1. Mere repetition about conceptual matters can work in cases where intervening experiences or information have taken a student to a new level of awareness so that what is repeated to him will have "new meaning" or relevance to him that it did not before. Repetition about conceptual points without new levels of awareness will generally not be helpful. And mere repetition concerning non-conceptual matters may be helpful, as in interminably reminding a young baseball player to keep his swing level, a young boxer to keep his guard up and his feet moving, or a child learning to ride a bicycle to "keep peddling; keep peddling; PEDDLE!" (Return to text.)
Footnote 2. If you think you understand place value, then answer why columns have the names they do. That is, why is the tens column the tens column or the hundreds column the hundreds column? And, could there have been some method other than columns that would have done the same things columns do, as effectively? If so, what, how, and why? If not, why not? In other words, why do we write numbers using columns, and why the particular columns that we use? In informal questioning, I have not met any primary grade teachers who can answer these questions or who have ever even thought about them before. (Return to text.)
Footnote 3. How something is taught, or how the teaching or material is structured, to a particular individual (and sometimes to similar groups of individuals) is extremely important for how effectively or efficiently someone (or everyone) can learn it. Sometimes the structure is crucial to learning it at all. A simple example first: (1) saying a phone number such as 323-2555 to an American as "three, two, three (pause), two, five, five, five" allows him to grasp it much more readily than saying "double thirty two, triple five". It is even difficult for an American to grasp a phone number if you pause after the fourth digit instead of the third ("three, two, three, two (pause), five, five, five").
(2) I was able to learn history of art from a book that structured it by taking the reader through one kind of art in one kind of region for a long period of time, and then doing the same for another region. I had a difficult time learning from a book that did many regions simultaneously in different cross-sections of time. I could make my own cross-sectional comparisons after studying each region in entirety, but I could not construct a whole region from what, to me, were a jumble of cross-sectional parts.
(3) I saw a child trying to learn to ride a bicycle by her father's having removed one training wheel and left the other fully extended to the ground. The only way to keep the bike from tipping over was to lean far out over the remaining training wheel. The child was justifiably riding at a 30 degree angle to the bike. When I took off the other training wheel to teach her to ride, it took about ten minutes just to get her back to a normal novice's initial upright riding position. I don't believe she could have ever learned to ride by the father's method.
(4) I explain the elements of photography in three hours in a way that makes sense to students, though it does not "sink" in to students fully at the end of that time. ("Sinking in" or ready facility requires practice along with understanding.) Many people I have taught have taken whole courses in photography that were not structured very well, and my perspective enlightens their understanding in a way they may not have achieved in the direction they were going.
(5) I studied European history for the first time when I was in college. My lecturer did not structure the material for us, and to me the whole thing was an endless, indistinguishable collection of popes and kings and wars. I tried to memorize it all and it was virtually impossible. I found out at the end of the term that the other professor who taught the course (to all my friends) spent each of his lectures simply structuring a framework in order to give a perspective for the students to place the details they were reading. They learned it.
(6) The year I took organic chemistry, one professor tested out a new textbook that structured the material in a new way, and he lectured in the same structure as the book. He admitted at the end of the year that was a big mistake; students did not learn as well using this structure. I did not become good at organic chemistry. oMath Learn Direct Save Mathlearndirect Glossary Definitions Mvp Term Semiconductor Gpk 1003 Math Learn Direct The Concept and Teaching of Place-Value in Mathv Real gMath Learn Direct Save Mathlearndirect Glossary Definitions Mvp Term Semiconductor Gpk 1003 Math Learn Direct The Concept and Teaching of Place-Value in Mathf n n Learn h h Direct Math Learn Direct