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The Concept and Teaching of Place-Value
An analysis of representative literature concerning the widely recognized ineffective learning of "place-value" by American children arguably also demonstrates a widespread lack of understanding of the concept of place-value among elementary school arithmetic teachers and among researchers themselves. Just being able to use place-value to write numbers and perform calculations, and to describe the process is not sufficient understanding to be able to teach it to children in the most complete and efficient manner.
A conceptual analysis and explication of the concept of "place-value" points to a more effective method of teaching it. However, effectively teaching "place-value" (or any conceptual or logical subject) requires more than the mechanical application of a different method, different content, or the introduction of a different kind of "manipulative". First, it is necessary to distinguish among mathematical 1) conventions, 2) algorithmic manipulations, and 3) logical/conceptual relationships, and then it is necessary to understand each of these requires different methods for effective teaching. And it is necessary to understand those different methods. Place-value involves all three mathematical elements.
Practice versus Understanding
Almost everyone who has had difficulty with introductory algebra has had an algebra teacher say to them "Just work more problems, and it will become clear to you. You are just not working enough problems." And, of course, when you can't work any problems, it is difficult to work many of them. Meeting the complaint "I can't do any of these" with the response "Then do them all" seems absurd, when it is a matter of conceptual understanding. It is not absurd when it is simply a matter of practicing something one can do correctly, but just not as adroitly, smoothly, quickly, or automatically as more practice would allow. Hence, athletes practice various skills to make them become more automatic and reflexive; students practice reciting a poem until they can do it smoothly; and musicians practice a piece until they can play it with little effort or error. And practicing something one cannot do very well is not absurd where practice will allow for self-correction. Hence, a tennis player may be able to work out a faulty stroke himself by analyzing his own form to find flawed technique or by trying different things until he arrives at something that seems right, which he then practices. But practicing something that one cannot even begin to do or understand, and that trial and error does not improve, is not going to lead to perfection or --as in the case of certain conceptual aspects of algebra-- any understanding at all.
What is necessary to help a student learn various conceptual aspects of algebra is to find out exactly what he does not understand conceptually or logically about what he has been presented. There are any number of reasons a student may not be able to work a problem, and repeating to him things he does understand, or merely repeating (1) things he heard the first time but does not understand, is generally not going to help him. Until you find out the specific stumbling block, you are not likely to tailor an answer that addresses his needs, particularly if your general explanation did not work with him the first time or two or three anyway and nothing has occurred to make that explanation any more intelligible or meaningful to him in the meantime.
There are a number of places in mathematics instruction where students encounter conceptual or logical difficulties that require more than just practice. Algebra includes some of them, but I would like to address one of the earliest occurring ones -- place-value. From reading the research, and from talking with elementary school arithmetic teachers, I suspect (and will try to point out why I suspect it) that children have a difficult time learning place-value because most elementary school teachers (as most adults in general, including those who research the effectiveness of student understanding of place-value) do not understand it conceptually and do not present it in a way that children can understand it. (2) (3) Elementary school teachers can generally understand enough about place-value to teach most children enough to eventually be able to work with it; but they don't often understand place-value conceptually and logically sufficiently to help children understand it conceptually and logically very well. And they may even impede learning by confusing children in ways they need not have; e.g. trying to make arbitrary conventions seem matters of logic, so children squander much intellectual capital seeking to understand what has nothing to be understood.
And a further problem in teaching is that because teachers, such as the algebra teachers referred to above, tend not to ferret out of children what the children specifically don't understand, teachers, even when they do understand what they are teaching, don't always understand what students are learning -- and not learning. There are at least two aspects to good teaching: (1) knowing the subject sufficiently well, and (2) being able to find out what the students are thinking as they try to learn the subject, in order to be the most helpful in facilitating learning. It is difficult to know how to help when one doesn't know what, if anything, is wrong. The passages quoted below seem to indicate either a failure by researchers to know what teachers know about students or a failure by teachers to know what students know about place-value. If it is the latter, then it would seem there is teaching occurring without learning happening, an oxymoron that, I believe, means there is not "teaching" occurring, but merely presentations being made to students without sufficient successful effort to find out how students are receiving or interpreting or understanding that presentation, and often without sufficient successful effort to discover what actually needs to be presented to particular students. (4) Part of good teaching is making certain students are grasping and learning what one is trying to teach. That is not always easy to do, but at least the attempt needs to be made as one goes along. Teachers ought to have known for some time what researchers have apparently only relatively recently discovered about children's understanding of place-value: "The literature is replete with studies identifying children's difficulties learning place-value concepts. (Jones and Thornton, p.12)" "Mieko Kamii's (1980, 1982) pioneering investigations in this area revealed glaring misunderstandings that were surprisingly pervasive. His [sic; Her] investigation showed that despite several years of place-value learning, children were unable to interpret rudimentary place-value concepts. (Jones, p.12) (5) "
Since I have taught my own children place-value after seeing how teachers failed to teach it (6). and since I have taught classes of children some things about place-value they could understand but had never thought of or been exposed to before, I believe the failure to learn place-value concepts lies not with children's lack of potential for understanding, but with the way place-value is understood by teachers and with the ways it is generally taught. It should not be surprising that something which is not taught very well in general is not learned very well in general. The research literature on place-value also shows a lack of understanding of the principle conceptual and practical aspects of learning place-value, and of testing for the understanding of it. Researchers seem to be evaluating the results of conceptually faulty teaching and testing methods concerning place-value. And when they find cultural or community differences in the learning of place-value, they seem to focus on factors that seem, from a conceptual viewpoint, less likely causally relevant than other factors. I believe that there is a better way to teach place-value than it is usually taught, and that children would then have better understanding of it earlier. Further, I believe that this better way stems from an understanding of the logic of place-value itself, along with an understanding of what is easier for human beings (whether children or adults) to learn. (7)
And I believe teaching involves more than just letting students (re-)invent things for themselves. A teacher must at least lead or guide in some form or other. How math, or anything, is taught is normally crucial to how well and how efficiently it is learned. It has taken civilization thousands of years, much ingenious creativity, and not a little fortuitous insight to develop many of the concepts and much of the knowledge it has; and children can not be expected to discover or invent for themselves many of those concepts or much of that knowledge without adults teaching them correctly, in person or in books or other media. Intellectual and scientific discovery is not transmitted genetically, and it is unrealistic to expect 25 years of an individual's biological development to recapitulate 25 centuries of collective intellectual accomplishment without significant help. Though many people can discover many things for themselves, it is virtually impossible for anyone to re-invent by himself enough of the significant ideas from the past to be competent in a given field, math being no exception. Potential learning is generally severely impeded without teaching. And it is possibly impeded even more by bad teaching, since bad teaching tends to dampen curiosity and motivation, and since wrong information, just like bad habits, may be harder to build from than would be no information, and no habits at all. In this paper I will discuss the elements I will argue are crucial to the concept and to the teaching of place-value.
Understanding Place-value: Practical and Conceptual Aspects
There are at least five aspects to being able to understand place-value, only two or three of which are often taught or stressed. The other two or three aspects are ignored, and yet one of them is crucial for children's (or anyone's) understanding of place-value, and one is important for complete understanding, though not for merely useful understanding. I will first just name and briefly describe these aspects all at once, and then go on to more fully discuss each one individually.
1) Learning number names (and their serial order) and using numbers to count quantities, developing familiarity and facility with numbers, practicing with numbers --including, when appropriate not only saying numbers but writing and reading them (8). not in terms of rules involving place-value, etc. but in terms of just being shown how to write and read individual numbers (with comments, when appropriate, that point out things like "ten, eleven, twelve, and all the teens have a '1' in front of them; all the twenty-numbers have a '2' in front of them" etc.) without reasons about why that is (9).
2) "simple" addition and subtraction,
3) developing familiarity through practice with groupings, and counting physical quantities by groups (not just saying the "multiples" of groups -- e.g. counting things by fives, not just being able to recite "five, ten, fifteen. "), and, when appropriate, being able to read and write group numbers --not by place-value concepts, but simply by having learned how to write numbers before. Practice with grouping and counting by groups should, of course, include groupings by ten's,
4) representation (of groupings)
5) specifics about representations in terms of columns.
Aspects (1), (2), and (3) require demonstration and "drill" or repetitive practice. Aspects (4) and (5) involve understanding and reason with enough demonstration and practice to assimilate it and be able to remember the overall logic of it with some reflection, rather than the specific logical steps. (10)
1) Number Facility, Practice
The more familiar one is with numbers and what they represent, the easier it is, generally, to see relationships involving numbers. Hence, it is important that children learn to count and to be able to identify the number of things in a group either by counting or by patterns, etc. One way to see this is to take some slice of 10 letters out of the middle of the alphabet, say "k,l,m,n,o,p,q,r,s,t" and let them represent 0-9 in linear order. Even though most adults can say those letters in order, just as they and children can say the names of numbers in order, it is very difficult, unless one practices a lot, to be able to group things in sets of "n" or to multiply "mrk" times "pm" or to see that all multiples of "p" end in either a "p" or a "k". Yet, seeing the relationships between serially ordered items one can name in serial order, is much of what arithmetic is about. (Possibly really brilliant math prodigies and geniuses don't have to have number names in order to see number relationships, I don't know; but most of us would be lost in any sort of higher level arithmetic if we could not count by (the names of) numbers, recognize the number of things (by name), or use numbers (by name) in relatively simple ways to begin with.) Hence, children normally need to learn to count objects and to understand "how many" the number names represent. Parents and teachers tend to teach students how to count and to give them at least some practice in counting. That is important.
2) Simple Addition and Subtraction
By "simple addition and subtraction", I mean addition and subtraction with regard to quantities children can learn to add and subtract just by counting together at first and then, with practice, fairly quickly learn to recognize by memory. For example, children can learn to play with dominoes or with two dice and add up the quantities, at first by having to count all the dots, but after a while just from remembering the combinations. Children can play something like blackjack with cards and develop facility with adding the numbers on face cards. Or they can play "team war", where pairs of individuals each turn over a card, as do the individuals on the opposing team, and whichever team has the highest sum, gets all four cards for their pile. Adding and subtracting in this way (or in some cases, even multiplying or dividing) may involve quantities that would be regrouped if calculated by algorithm on paper, but they have nothing to do with regrouping when it is done in this "direct" or "simple" manner. For example, children who play various card games with full decks of regular playing cards tend to learn half of 52 is 26 and that a deck divided equally among four people gives them each 13 cards.
It is particularly important that children get sufficient practice to become facile with adding pairs of single digit numbers whose sums are not only as high as 10, but also as high as 18. And it is particularly important that they get sufficient practice to become facile with subtracting single digit numbers that yield single digit answers, not only from minuends as high as 10, but from minuends between 10 and 18. The reason for this is that whenever you regroup for subtraction, if you regroup "first" (11) you always END UP with a subtraction that requires taking away from a number between 10 and 18 a single digit number that is larger than the "ones" digit of the minuend (i.e. the number between 10 and 18). E.g. 15-7, 18-9, 11-4, etc. The reason you had to "regroup" or "borrow" in the first place was that the subtrahend digit in the column in question was larger than the minuend digit in that column; and when you regroup the minuend, those digits do not change, but the minuend digit simply gains a "ten" and becomes a number between 10 and 18. (The original minuend digit --at the time you are trying to subtract from it (12) -- had to have been between 0 and 8, inclusive, for you not to be able to subtract without regrouping. Had it been a nine, you would have been able to subtract any possible single digit number from it without having to regroup.) Another way of saying this is that whenever you regroup, you end up with a subtraction of the form:
where the digit after the 1 will be between 0 and 8 (inclusive) and will be smaller than the digit designated by the "x" (13).
Children often do not get sufficient practice in this sort of subtraction to make it comfortable and automatic for them. Many "educational" math games involving simple addition and subtraction tend to give practice up to sums or minuends of 10 or 12, but not up to 18. I believe lack of such practice and lack of "comfort" with regrouped subtractions tends to contribute toward a reluctance in children to properly regroup for subtraction because when they get to the part where they have to subtract a combination of the above form they think there must be something wrong because that is still not an "automatically" recognizable combination for them. Hence, they go to something else which they can subtract instead (e.g. by reversing the subtrahend and minuend digits in that column, so it will "come out" by allowing subtraction of a smaller digit from a bigger one) even though it ends up wrong. In a sense, doing what seems familar to them "makes sense" to them (14).
Memory can work very well after a bit of practice with "simple" additions and subtractions (sums or minuends to 18), since memory in general can work very well with regard to quantities. One of my daughters at the age of five or six learned how to get tremendously high scores on a computer game that required quickly and correctly identifying prime numbers. She had learned the numbers by trial and error playing the game over and over; she had no clue what being a prime number meant; she just knew which numbers (that were on the game) were primes. Similarly, if children play with adding many of the same combinations of numbers, even large numbers, they learn to remember what those combinations add or subtract to after a short while. This ability can be helpful when adding later by non-like groups (e.g. seven and eight, as opposed to adding by groups of all ten's). According to Fuson, many Asian children are given this kind of practice with pairs of quantities that sum to ten. But one can do other quantities as well; and single digit numbers summing up to and including 18, and single digit subtractions from minuends up to and including 18 that yield single digit answers, are important for children to practice. (One way to give such practice that children seem to enjoy would be for them to play a non-gambling version of blackjack or "21" with a deck of cards that has all the picture cards removed. The reason for removing the picture cards is to give more opportunity for practicing adding combinations that do not involve ten's, which are fairly easy.)
An analysis of the research in place-value seems to make quite clear that children incorrectly perform algorithmic operations in ways that they would themselves clearly recognize as mistakes if they had more familiarity with what quantities meant and with "simple" addition and subtraction. Fuson shows in a table (p. 376) fourteen different kinds of errors researchers have found children make in performing the adding and subtracting algorithms that require "regrouping" or "trading". But the errors I believe most significant are those involving children's getting an outrageous answer because they seem to have no idea what the algorithm is really an algorithm about. Two examples: children may write a sum for each column, so they add 375 to 466 and they get 71311. Or they "vanish the one" (i.e. just ignore and forget about it) so that they add 777 to 888 and get 555. Clearly, if children understood in the first case they were adding together two numbers somewhere around 400 each, they would know they should end up with an answer somewhere around 800, and that 71,000 is too far away. And they would understand in the second case that you cannot add two (positive) quantities together and get a smaller quantity than either. (15) It is not so bad for children to occasionally make simple calculating errors; anyone can have understanding and still make a mistake. And it is not so bad if children make algorithmic errors because they have not learned or practiced the algorithm enough to remember or to be able to follow the algorithmic rules well enough to work a problem correctly; that just takes more practice. But it should be of major significance that many children cannot recognize that the procedure, the way they are doing it, yields such a bad answer, that they must be doing something wrong! The answers Fuson details in her chart of errors of algorithmic calculation are less disturbing about children's use of algorithms than they are about children's understanding of number and quantity relationships and their understanding of what they are even trying to accomplish by using algorithms (in this case, for adding and subtracting).
Since counting large numbers of things one at a time gets to be tedious, counting by groups of two, three, five, ten, etc. is a helpful skill to facilitate. Students have to be taught and rehearsed to count this way, and generally they have to be told that it is a faster and easier way to count large quantities. (16) Also, it serves as a prelude to multiplication, since counting by groups (of, say, three) introduces one subconsciously to multiples of those groups (i.e. in this case, multiples of three). And, of course, grouping by 10's is a prelude to understanding those aspects of arithmetic based on 10's. Many teachers teach students to count by groups and to recognize quantities by the patterns a group can make (such as on numerical playing cards). This is important.
Aspects of elements 2) and 3) can be "taught" or learned at the same time. Though they are "logically" distinct; they need not be taught or learned in serial order or specifically in the order I mention them here. Many conceptually distinct ideas occur together naturally in practice.
4) Representations of Groups
This is what most elementary school teachers, since they are generally not math majors, do not understand, and can only teach with regard to columnar "place-value". But columnar place-value is (1) not the only way to represent groups, and (2) it is an extremely difficult way for children to understand representations of groups. There are more accessible ways for children to work with representations of groups. And I think it is easier for them to learn columnar place-value if one starts them out with more psychologically accessible group representations.
Once children have gained facility with counting, and with counting by groups, especially groups of 10's and perhaps 100's, and 1000's (i.e. knowing that when you group things by 100's and 1000's that the series go "100, 200, 300. 900, 1000; and 1000, 2000, 3000, etc.), I believe it is better to start them out learning about the kind of representational group values that children seem to have no trouble with -- such as colors, as in poker chips (or color tiles, if you feel that "poker" chips are inappropriate for school children; poker chips are just inexpensive, available, easy to manipulate, and able to be stacked) (17). Only one needs not, and should not, talk about "representation", but merely set up some principles like "We have these three different color poker chips, white ones, blue ones, and red ones. Whenever you have ten white ones, you can exchange them for one blue one; or any time you want to exchange a blue one for ten white ones you can do that. And any time you have ten BLUE ones, you can trade them in for one red one, or vice versa ." Then you can show them how to count ten blue ones (representing ten's), saying "10, 20, 30. 90, 100" so they can see, if they don't already, that a red one is worth 100. Then you do some demonstrations, such as putting down eleven white ones and saying something like "if we exchange 10 of these white ones
for a blue one, what will we have?" And the children will usually say something like "one blue one and one white one". And you can reinforce that they still make (i.e. represent) the same quantity "And that then is still eleven, right? [Pointing at the blue one] Ten [then pointing at the white one] and one is eleven." Do this until they catch on and can readily and easily represent numbers in poker chips, using mixtures of red, blue, and white ones. In this way, they come to understand group representation by means of colored poker chips, though you do not use the word representation, since they are unlikely to understand it.
Let the students get used to making (i.e. representing) numbers with their poker chips, and you can go around and quickly check to see who needs help and who does not, as you go. Ask them, for example, to show you how to make various numbers in (the fewest possible) poker chips -- say 30, 60, etc. then move into 12, 15, 31, 34, 39. 103, 135, etc. Keep checking each child's facility and comfort levels doing this.
Then, when they are readily able to do this, get into some simple poker chip addition or subtraction, starting with sums and differences that don't require regrouping, e.g. 2+3, 9-6, 4+5, etc. Then, when they are ready, get into some easy poker chip regroupings. "If you have seven white ones and add five white ones to them, how many do you have?" "Now let's exchange ten of them for a blue one, and what do you get? (18) " Add larger and larger numbers and also show them some easy subtractions -- like with the number 12 they just got before, with the blue one and the two white ones, "If we wanted to take 3 away from this 12, how could we do it?" [Someone will usually say, or the teacher could say the first time or two] "We need to change the blue one into 10 white ones, then we could take away 3 white ones from the 12 white ones we have." ETC. Keep practicing and changing the numbers so they sometimes need regrouping and sometimes don't; but so they get better and better at doing it. (They are now using the colors both representationally and quantitatively -- trading quantities for chips that represent them, and vice versa .) Then introduce double digit additions and subtractions that don't require regrouping the poker chips, e.g. 23 + 46, 32 + 43, 42 - 21, 56 - 35, etc. (The first of these, for example is adding 4 blues and 6 whites to 2 blues and 3 whites to end up with 6 blues and 9 whites, 69; the last takes 3 blues and 5 whites away from 5 blues and 6 whites to leave 2 blues and 1 white, 21.) When they are comfortable with these, introduce double digit addition and subtraction that requires regrouping poker chips, e.g. 25 + 25, 25 + 28, 23 - 5, 33 - 15, 82 - 57, etc.
As you do all these things it is important to walk around the room watching what students are doing, and asking those who seem to be having trouble to explain what they are doing and why. In some ways, seeing how they manipulate the chips gives you some insight into their understanding or lack of it. Usually when they explain their faulty manipulations you can see what sorts of, usually conceptual, problems they are having. And you can tell or show them something they need to know, or ask them leading questions to get them to self-correct. Sometimes they will simply make counting mistakes, however, e.g. counting out 8 white chips instead of 9. That kind of mistake is not as important for teaching purposes at this point as conceptual mistakes. They tend to make fewer careless mere counting errors once they see that gives them wrong answers.
After gradually taking them into problems involving greater and greater difficulty, at some point you will be able to give them something like just one red poker chip (100) and ask them to take away 37 from it, and they will be able to figure it out and do it, and give you the answer --not because they have been shown (since they will not have been shown), but because they understand.
Then, after they are comfortable and good at doing this, you can point out that when numbers are written numerically, the columns are like the different color poker chips. The first column is like white poker chips, telling you how many "ones" you have, and the second column is like blue poker chips, telling you how many 10's (or chips worth ten) you have. etc. This would be a good time to tell them that in fact the columns are even named like the poker chips -- the one's column, the ten's column, the hundred's column, etc. (Remember, they have learned to write numbers by rote and by practice; they should find it interesting that written numbers have these parts --i.e. the numerals and columns-- which "coincide" with how many one's, ten's, etc. there are in the quantity that the number names.) (19)
Then show them that adding and subtracting some double digit numbers (not requiring regrouping) on paper is like doing it with different color (i.e. group value) poker chips. Let them try some. Let them do additions and subtractions on paper, checking their answers and their manipulations with different color (group value) poker chips. E.g. let them subtract 43 from 67 and see that taking the 4 tens from the 6 tens and the 3 ones from the 7 ones is the same on paper as it is with blue and white poker chips -- taking 4 blue ones from 6 blue ones and 3 white ones from 7 white ones.
Then demonstrate how adding and subtracting numbers (that require regrouping) on paper is just like adding and subtracting numbers that their poker chips represent that require exchanging. This is a good time to introduce, somewhat casually, the algorithm for adding and subtracting numerals "on paper" using the "trading" or "borrowing/carrying" technique. You may want to stick representative poker chips above your columns on the chalk board, or have them use crayons to put the poker chip colors above their columns on their paper (using, say, yellow for white if they have white paper). Show them how they can "exchange" numerals in their various columns by crossing out and replacing those they are borrowing from, carrying to, adding to, or regrouping. (This is sometimes somewhat difficult for them at first because at first they have a difficult time keeping their substitutions straight and writing them where they can notice and read them and remember what they mean. They tend to start getting scratched-out numbers and "new" numbers in a mess that is difficult to deal with. But once they see the need to be more orderly, and once you show them some ways they can be more orderly, they tend to be able to do all right.) Let them do problems on paper and check their own answers with poker chips. Give them lots of practice, and, as time goes on, make certain they can all do the algorithmic calculation fairly formally and that they can also understand what they are doing if they were to stop and think about it.
Again, the whole time you can walk around and around the room seeing who might need extra help, or what you might have to do for everyone. Doing this in this way lets you almost see what they are individually thinking and it lets you know who might be having trouble, and where, and what you might need to do to ameliorate that trouble. You may find general difficulties or you may find each child has his own peculiar difficulties, if any. For a while my children tended to forget the "one's" they already had when they regrouped; they would forget to mix the "new" one's with the "old" one's. So, if they had 34 to start with and borrowed 10 from the thirty, they would forget about the 4 ones they already had, and subtract from 10 instead of from 14. Children in schools using small desk spaces sometimes get their different piles of poker chips confused, since they may not put their "subtracted" chips far enough away or they may not put their "regrouped" chips far enough away from a "working" pile of chips. There may be fairly unique or unusual difficulties that will test your own understanding of the concept and what possible misunderstanding the child could have about it, so that you can structure help that fits in with his/her thinking.
Columns (above one's) and colors ("above" white) are each representations of groups of numbers, but columns are a relational property representation, whereas colors are not. Colors are a simple or inherent or immediately obvious property. Columns are relational, more complex, and less obvious. Once color or columnar values are established, three blue chips are always thirty, but a written numeral three is not thirty unless it is in a column with only one (non-decimal) column to its right. Column representations of groups are more difficult to comprehend than color representations, and I suspect that is (1) because they depend on location relative to other numerals which have to be (remembered to be) looked for and then examined, rather than on just one inherent property, such as color (or shape), and (2) because children can physically exchange "higher value" color chips for the equivalent number of lower value ones, whereas doing that is not so easy or obvious in using columns. In regard to (1), as anyone knows who has ever put things together from a kit, any time objects are distinctly colored and referred to in the directions by those colors, they are made easier to distinguish than when they have to be identified by size or other relative properties, which requires finding other similar objects and examining them all together to make comparisons. In regard to (2), it is easy to physically change, say a blue chip, for ten white ones and then have, say, fourteen white ones altogether from which to subtract (if you already had four one's). But it is difficult to represent this trade with written numerals in columns, since you have to scratch stuff out and then place the new quantity in a slightly different place, and because you end up with new columns (as in putting the number "14" all in the one's column, when borrowing 10 from, say 30 in the number "34", in order to subtract 8). Further, (3) I suspect there is something more "real" or simply more meaningful to a child to say "a blue chip is worth 10 white ones" than there is to say "this '1' is worth 10 of this '1' because it is over here instead of over here"; value based on place seems stranger than value based on color, or it seems somehow more arbitrary. But regardless of WHY children can associate colors with numerical groupings more readily than they do with relative column positions, they do.
5) Specifics about Columnar Representations
Apart from the comments made in the last section about columnar representation, I would like to add the following, which is not important for students to understand while they are learning columnar representation (usually known as "place-values"), but may be helpful to teachers to understand. And it may be interesting to students at some later stage when they can absorb it. (I have taught this to third graders, but the presentation is extremely different from the way I will write it here; and that presentation is crucial to their following the ideas and understanding them. That presentation is detailed in the paper about a method of effectively teaching conceptual/logical material, "The Socratic Method -- Teaching by Asking Instead of by Telling .")
Columnar representation of groups is simply one way of designating groups. But it is important to understand why groups need to be designated at all, and what is actually going on in assigning what has come to be known as "place-value" designation. Groups make it easier to count large quantities; but apart from counting, it is only in writing numbers that group designations are important. Spoken numbers are the same no matter how they might be written or designated. They can even be designated in written word form, such as "four thousand three hundred sixty five" -- as when you spell out dollar amounts in word form in writing a check. And notice, that in spoken form there are no place-values mentioned though there may seem to be. That is we say "five thousand fifty four", not "five thousand no hundred and fifty four". "Two million six" is not "two million, no hundred thousands, no ten thousands, no thousands, no hundreds, no tens, and six." Even though we use names like "hundred," "thousand," "million," etc. which are the same as the names of the columns higher than the ten's column, we are not really representing groupings; we are merely giving the number name, when we pronounce it, just as when we say "ten" or "eleven". "Eleven" is just a word that names a particular quantity. Starting with "zero", it is the twelfth unique number name. Similarly "four thousand, three hundred, twenty nine" is just a unique name for a particular quantity. It could have been given a totally unique name (say "gumph") just like "eleven" was, but it would be difficult to remember totally unique names for all the numbers. It just makes it easier to remember all the names by making them fit certain patterns, and we start those patterns in English at the number "thirteen" (or some might consider it to be "twenty one", since the "teens" are different from the decades). We only use the concept of represented 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
Remember, all this could have been done differently. The abacus does it differently. Our poker chips did it differently. Roman numerals do it differently. And, in a sense, computers and calculators do it differently because they use only two representations (switches that are either "on" or "off") and they don't need columns of anything at all (unless they have to show a written number to a human who is used to numbers written a certain way -- in columns using 10 numerals). And though we can calculate with pencil and paper using this method of representation, we can also calculate with poker chips or the abacus; and we can do multiplication and division, and other things, much quicker with a slide rule, which does not use columns to designate numbers either, or with a calculator or computer.
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.
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Baroody, A.J. (1990). How and when should place-value concepts and skills be taught? Journal for Research in Mathematics Education, 21 (4), 281-286.
Cobb, Paul. (1992) Personal correspondence. October 9.
Kamii, C. (1989). Young children continue to reinvent arithmetic: 2nd grade. New York: Teachers College Press.