groupings when we write numbers using numerals.
What happens in writing numbers numerically is that if we are going to use ten numerals, as we do in our everyday base-ten "normal" arithmetic, and if we are going to start with 0 as the lowest single numeral, then when we get to the number "ten", we have to do something else, because we have used up all the representing symbols (i.e., the numerals) we have chosen -- 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Now we are stuck when it comes to writing the next number, which is "ten". To write a ten we need to do something else like make a different size numeral or a different color numeral or a different angled numeral, or something. On the abacus, you move all the beads on the one's row back and move forward a bead on the ten's row. What is chosen for written numbers is to start a new column. And since the first number that needs that column in order to be written numerically is the number ten, we simply say "we will use this column to designate a ten" -- and so that you more easily recognize it is a different column, we will include something to show where the old column is that has all the numbers from zero to nine; we will put a zero in the original column. And, to be economical, instead of using other different columns for different numbers of tens, we can just use this one column and different numerals in it to designate how many tens we are talking about, in writing any given number. Then it turns out that by changing out the numerals in the original column and the numerals in the "ten" column, we can make combinations of our ten numerals that represent each of the numbers from 0 to 99. Now we are stuck again for a way to write one hundred. We add another column.(20) And we can get by with that column until we pass nine hundred ninety nine. Etc.
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 lMaths Problems Mathamatics Mathlearndirect Szh Modified Railway Math Learn Direct The Concept and Teaching of Place-Value in Mathn Math Learn Direct l Math Math Learn Direct rMaths Problems Mathamatics Mathlearndirect Szh Modified Railway Math Learn Direct The Concept and Teaching of Place-Value in Mathl n p p Direct Kenya's%20math%20worksheets u Math Math Learn Direct