The Concept and Teaching of Place-Value in Math

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.

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.

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⁾,

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,

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⁾

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).

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.

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.

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.

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.

Cobb, Paul. (1992) Personal correspondence. October 9.

Fuson, K.C. (1990). Conceptual structures for multiunit numbers: implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7(4), 343-403.

Jones, G.A., & Thornton, C.A. (1993). Children's understanding of place value: a framework for curriculum development and assessment. Young Children, 48(5), 12-18.

Kamii, C. (1989). Young children continue to reinvent arithmetic: 2nd grade. New York: Teachers College Press.

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.

(7) In second semester calculus, there were three chapters full of formulas that could all be derived from the first formula in the chapter, but neither the book nor any of the teachers pointed out that all but the first formula was derivative. There appeared to be much memorization needed to learn each of these individual formulas. I happened to notice the relationship the night before the midterm exam, purely by luck and some coincidental reasoning about something else. I figured I was the last to see it of the 1500 students in the course and that, as usual, I had been very naive about the material. It turned out I was the only one to see it. I did extremely well but everyone else did miserably on the test because memory under exam conditions was no match for reasoning. Had the teachers or the book simply specifically said the first formula was a general principle from which you could derive all the others, most of the other students would have done well on the test also.

There could be millions of examples. Most people have known teachers who just could not explain things very well, or who could only explain something one precise way, so that if a student did not follow that particular explanation, he had no chance of learning that thing from that teacher. The structure of the presentation to a particular student is important to learning. (Return to text.)

Footnote 4. In a small town not terribly far from Birmingham, there is a recently opened McDonald's that serves chocolate shakes which are off-white in color and which taste like not very good vanilla shakes. They are not like other McDonald's chocolate shakes. When I told the manager how the shakes tasted, her response was that the shake machine was brand new, was installed by experts, and had been certified by them the previous week --the shake machine met McDonald's exacting standards, so the shakes were the way they were supposed to be; there was nothing wrong with them. There was no convincing her. After she returned to her office I realized, and mentioned to the sales staff, that I should have asked her to take a taste test to try to distinguish her chocolate shakes from her vanilla ones. That would show her there was no difference. The staff told me that would not work since there was a clear difference: "Our vanilla shakes taste like chalk." They understood there was a problem.

Unfortunately, too many teachers teach like that manager manages. They think if they do well what the manuals and the college courses and the curriculum guides tell them to do, then they have taught well and have done their job. What the children get out of it is irrelevant to how good a teacher they are. It is the presentation, not the reaction to the presentation, that they are concerned about. To them "teaching" is the presentation (or the setting up of the classroom for discovery or work). If they "teach" well what children already know, they are good teachers. If they make dynamic well-prepared presentations with much enthusiasm, or if they assign particular projects, they are good teachers, even if no child understands the material, discovers anything, or cares about it. If they train their students to be able to do, for example, fractions on a test, they have done a good job teaching arithmetic whether those children understand fractions outside of a test situation or not. And if by whatever means necessary they train children to do those fractions well, it is irrelevant if they forever poison the child's interest in mathematics. Teaching, for teachers like these, is just a matter of the proper technique, not a matter of the results.

Well, that is not any more true than that those shakes meet McDonald's standards just because the technique by which they are made is "certified". I am not saying that classroom teachers ought to be able to teach so that every child learns. There are variables outside of even the best teachers' control. But teachers ought to be able to tell what their reasonably capable students already know, so they do not waste their time or bore them. Teachers ought to be able to tell whether reasonably capable students understand new material, or whether it needs to be presented again in a different way or at a different time. And teachers ought to be able to tell whether they are stimulating those students' minds about the material or whether they are poisoning any interest the child might have.

All the techniques in all the instructional manuals and curriculum guides in all the world only aim at those ends. Techniques are not ends in themselves; they are only means to ends. Those teachers who perfect their instructional techniques by merely polishing their presentations, rearranging the classroom environment, or conscientiously designing new projects, without any understanding of, or regard for, what they are actually doing to children may as well be co-managing that McDonald's. (Return to text.)

Footnote 5. Some of these studies interpreted to show that children do not understand place-value, are, I believe, mistaken. Jones and Thornton explain the following "place-value task": Children are asked to count 26 candies and then to place them into 6 cups of 4 candies each, with two candies remaining. When the "2" of "26" was circled and the children were asked to show it with candies, the children typically pointed to the two candies. When the "6" in "26" was circled and asked to be pointed out with candies, the children typically pointed to the 6 cups of candy. This is taken to demonstrate children do not understand place value. I believe this demonstrates the kind of tricks similar to the following problems, which do not show lack of understanding, but show that one can be deceived into ignoring or forgetting one's understanding.

(1) There is a ship in the harbor with a very long rope-ladder hanging overboard whose rungs are 8 inches apart. At the beginning of the tide's coming in, three rungs are under water. If the tide comes in for four hours at the rate of 1 foot per hour, at the end of this period, how many rungs will be submerged?

The answer is not nine, but "still just three, because the ship will rise with the tide." This does not demonstrate respondents do not understand buoyancy, only that one can be tricked into forgetting about it or ignoring it.

(2) Three men went into a hotel in 1927 and got a suite of rooms for $30 total, which they paid in advance in cash, each man contributing $10. After they went up to the room, the desk clerk realized he made a mistake and that the suite was only $25. He gave the bellhop $5 to take back to the men. The bellhop did not know how to divide the money evenly among the men, so he merely returned $1 to each of them and kept two for himself. That meant the men paid $9 each for a total of $27. The bellhop kept $2, so that is $29. But there was $30 to begin with, so what happened to the other dollar?

This tends to be an extremely difficult problem --psychologically-- though it has an extremely simple answer. The money paid out must simply equal the money taken in. $27 was paid out (ultimately); $2 of it went to the bellhop and $25 went to the desk. You must subtract the $2 the bellhop kept, not add it back to the amount the men paid out. There is no reason to add the $2 to the $27 other than to get a number close enough to the original $30 to confuse the listener into thinking something is wrong and that $1 is unaccounted for. People who cannot solve this problem, generally have no trouble accounting for money, however; they do only when working on this problem.

(3) The following problem is difficult the more calculus you know. If you know no calculus, the problem is not especially difficult. It is a favorite problem to trick unsuspecting math professors with.

Two trains start out simultaneously, 750 miles apart on the same track, heading toward each other. The train in the west is traveling 70 mph and the train in the east is traveling 55 mph. At the time the trains begin, a bee that flies 300 mph starts at one train and flies until it reaches the other, at which time it reverses (without losing any speed) and immediately flies back to the first train, which, of course, is now closer. The bee keeps going back and forth between the two ever-closer trains until it is squashed between them when they crash into each other. What is the total distance the bee flies?

The computationally extremely difficult, but psychologically logically apparent, solution is to "sum an infinite series". Mathematicians tend to lock into that method. The easy solution, however, is that the trains are approaching each other at a combined rate of 125 mph, so they will cover the 750 miles, and crash, in 6 hours. The bee is constantly flying 300 mph; so in that 6 hours he will fly 1800 miles. (One mathematician is supposed to have given the answer immediately, astonishing a questioner who responded how incredible that was "since most mathematicians try to sum an infinite series." The mathematician responded with astonishment of his own, "but that is what I did.")

It is not that mathematicians do not know how to solve this problem the easy way; it is that it is constructed in a way to make them not think about the easy way.

I believe that the problem Jones and Thornton describe acts similarly on the minds of children. Though I believe there is ample evidence children, and adults, do not really understand place-value, I do not think problems of this sort demonstrate that, any more than problems like those given here demonstrate lack of understanding about the principles involved.

It is easy to see children do not understand place-value when they cannot correctly add or subtract written numbers using increasingly more difficult problems than they have been shown and drilled or substantially rehearsed "how" to do (by specific steps; i.e., by algorithm). By increasingly difficult, I mean, for example, going from subtracting or summing relatively smaller quantities to relatively larger ones (with more and more digits), going to problems that require (call it what you like) regrouping, carrying, borrowing, or trading; going to subtraction problems with zeroes in the number from which you are subtracting; to consecutive zeroes in the number from which you are subtracting; and subtracting such problems that are particularly psychologically difficult in written form, such as "10,101 - 9,999". Asking students to (demonstrate how they) solve (the kinds of) problems they have been "taught" and rehearsed on merely tests their attention and memory, but asking students to (demonstrate how they) solve new kinds of problems (that use the concepts and methods you have been demonstrating, but "go just a bit further" from them) helps to show whether they have developed understanding. However, the kinds of problems at the beginning of this endnote do not do that because they have been contrived specifically to psychologically mislead, or they are constructed accidentally in such a way as to actually mislead. They go beyond what the students have been specifically taught, but do it in a tricky way rather than a merely "logically natural" way. I cannot categorize in what ways "going beyond in a tricky way" differs from "going beyond in a 'naturally logical' way" in order to test for understanding, but the examples should make clear what it is I mean.

Further, it is often difficult to know what someone else is asking or saying when they do it in a way that is different from anything you are thinking about at the time. If you ask about a spatial design of some sort and someone draws a cutaway view from an angle that makes sense to him, it may make no sense to you at all until you can "re-orient" your thinking or your perspective. Or if someone is demonstrating a proof or rationale, he may proceed in a step you don't follow at all, and may have to ask him to explain that step. What was obvious to him was not obvious to you at the moment.

The fact that a child, or any subject, points to two candies when you circle the "2" in "26" and ask him to show you what that means, may be simply because he is not thinking about what you are asking in the way that you are asking it or thinking about it yourself. There is no deception involved; you both are simply thinking about different things -- but using the same words (or symbols) to describe what you are thinking about. This is similar to someone's quoting a price of "nineteen ninety five" when you mistakenly think you are looking at costume jewelry, and you think he means $19.95, while he is meaning $1995. Or, ask someone to look at the face of a person about ten feet away from them and describe what they see. They will describe that person's face, but they will actually be seeing much more than that person's face. So, their answer is wrong, though understandably so. Now, in a sense, this is a trivial and trick misunderstanding, but in photography, amateurs all the time "see" only a face in their viewer, when actually they are too far away to have that face show up very well in the photograph. They really do not know all they are seeing through the viewer, and all that the camera is "seeing" to take. The difference is that if one makes this mistake with a camera, it really is a mistake; if one makes the mistake verbally in answer to the question I stated, it may not be a real mistake but only taking an ambiguous question the way it deceptively was not intended. Asking a child what a circled "2" means, no matter where it comes from, may give the child no reason to think you are asking about the "twenty" part of "26" --especially when there are two objects you have intentionally had him put before him, and no readily obvious set of twenty objects. He may understand place-value perfectly well, but not see that is what you are asking about -- especially under the circumstances you have constructed and in which you ask the question. (Return to text.)

Footnote 6. If you understand the concept of place-value, if you understand how children (or anyone) tend to think about new information of any sort (and how easy misunderstanding is, particularly about conceptual matters), and if you watch most teachers teach about the things that involve place-value, or any other logical-conceptual aspects of math, it is not surprising that children do not understand place-value or other mathematical concepts very well and that they cannot generally do math very well. Place-value, like many concepts, is often taught as though it were some sort of natural phenomena --as if being in the 10's column was a simple, naturally occurring, observable property, like being tall or loud or round-- instead of a logically and psychologically complex concept. What may be astonishing is that most adults can do math as well as they do it at all with as little in-depth understanding as they have. Research on what children understand about place-value should be recognized as what children understand about place-value given how it has been taught to them, not as the limits of their possible understanding about place-value. (Return to text.)

Footnote 7. Baroody (1990) categorizes what he calls "increasingly abstract models of multidigit numbers using objects or pictures" and includes mention of the model I think most appropriate --different color poker chips --which he points out to be conceptually similar to Egyptian hieroglyphics-- in which a different looking "marker" is used to represent tens. And he says "Using a different-looking ten marker may help some children --particularly those of low ability-- to bridge the gap between highly concrete size embodiments and the [next/last] relatively abstract model [involving relative position of markers]."

I do not believe that his categories are categories of increasingly abstract models of multidigit numbers. He has four categories; I believe the first two are merely concrete groupings of objects (interlocking blocks and tally marks in the first category, and Dienes blocks and drawings of Dienes blocks in the second category). And the second two --different marker type and different relative-position-value-- are both equally abstract representations of grouping, the difference between them being that relative-positional-value is a more difficult concept to assimilate at first than is different marker type. It is not more abstract; it is just abstract in a way that is more difficult to recognize and deal with.

Further, Baroody labels all his categories as kinds of "trading", but he does not seem to recognize there is sometimes a difference between "trading" and "representing", and that trading is not abstract at all in the way that representing is. I can trade you my Mickey Mantle card for your Ted Kluzewski card or my tuna sandwich for your soft drink, but that does not mean Mickey Mantle cards represent Klu cards or that sandwiches represent soft drinks. Children in general, not just children with low ability, can understand trading without necessarily understanding representing. And they can go on from there to understand the kind of representing that does happen to be similar to trading, which is the kind of representing that place-value is. But with regard to trading, as opposed to representing, it is easier first to apprehend or appreciate (or remember, or pretend) there being a value difference between objects that are physically different, regardless of where they are, than it is to apprehend or appreciate a difference between two identical looking objects that are simply in different places. It makes sense to say that something can be of more or less value if it is physically changed, not just physically moved. Painting your car, bumping out the dents, or re-building the carburetor makes it worth more in some obvious way; parking it further up in your driveway does not. It makes sense to a child to say that two blue poker chips are worth 20 white ones; it makes less apparent sense to say a "2" over here is worth ten "2's" over here. Color poker chips teach the important abstract representational parts of columns in a way children can grasp far more readily. So why not use them and make it easier for all children to learn? And poker chips are relatively inexpensive classroom materials. By thinking of using different marker types (to represent different group values) primarily as an aid for students of "low ability", Baroody misses their potential for helping all children, including quite "bright" children, learn place-value earlier, more easily, and more effectively. (Return to text.)

Footnote 8. Remember, written versions of numbers are not the same thing as spoken versions. Written versions have to be learned as well as spoken versions; knowing spoken numbers does not teach written numbers. For example, numbers written in Roman numerals are pronounced the same as numbers in Arabic numerals. And numbers written in binary form are pronounced the same as the numbers they represent; they just are written differently, and look like different numbers. In binary math "110" is "six", not "one hundred ten". When children learn to read numbers, they sometimes make some mistakes like calling "11" "one-one", etc. Even adults, when faced with a large multi-column number, often have difficulty naming the number, though they might have no trouble manipulating the number for calculations; number names beyond the single digit numbers are not necessarily a help for thinking about or manipulating numbers.

Karen C. Fuson explains how the names of numbers from 10 through 99 in the Chinese language include what are essentially the column names (as do our whole-number multiples of 100), and she thinks that makes Chinese-speaking students able to learn place-value concepts more readily. But I believe that does not follow, since however the names of numbers are pronounced, the numeric designation of them is still a totally different thing from the written word designation; e.g., "1000" versus "one thousand". It should be just as difficult for a Chinese-speaking child to learn to identify the number "11" as it is for an English-speaking child, because both, having learned the number "1" as "one", will see the number "11" as simply two "ones" together. It should not be any easier for a Chinese child to learn to read or pronounce "11" as (the Chinese translation of) "one-ten, one" than it is for English-speaking children to see it as "eleven". And Fuson does note the detection of three problems Chinese children have: (1) learning to write a "0" when there is no mention of a particular "column" in the saying of a number (e.g., knowing that "three thousand six" is "3006" not just "36"); (2) knowing that in certain cases when you get more than nine of a given place-value, you have to convert the "extra" into a higher place-value in order to write it (e.g., you can say "five one hundred's and twelve ten's" but you have to write it as "620" because you [sort of] cannot write it as "5120". [I say, "sort of" because we do teach children to write "concatenated" columns --columns that contain multi-digit numbers-- when we teach them the borrowing algorithm of subtraction; we do write a "12" in the ten's column when we had two ten's and borrow 10 more.] (3) Writing numbers normally without "concatenating" them (e.g., learning to write "five hundred twelve" as "512" instead of "50012", where the child writes down the "500" and puts the "12" on the end of it).

But there is, or should be, more involved. Even after Chinese-speaking children have learned to read numeric numbers, such as "215" as (the Chinese translation of) "2-one hundred, one-ten, five", that alone should not help them be able to subtract "56" from it any more easily than an English-speaking child can do it, because (1) one still has to translate the concepts of trading into columnar numeric notations, which is not especially easy, and because (2) one still has to understand how ones, tens, hundreds, etc. relate to each other so that one can trade between higher and lower column-name-designations; e.g., between thousands and hundreds or between millions and hundred thousands, etc. And although it may seem easy to subtract "five-ten" (50) from "six-ten" (60) to get "one-ten" (10), it is not generally difficult for people who have learned to count by tens to subtract "fifty" from "sixty" to get "ten". Nor is it difficult for English-speaking students who have practiced much with quantities and number names to subtract "forty-two" from "fifty-six" to get "fourteen". Surely it is not easier for a Chinese-speaking child to get "one-ten four" by subtracting "four-ten two" from "five-ten six". Algebra students often have a difficult time adding and subtracting mixed variables [e.g., "(10x + 3y) - (4x + y)"]; is it going to be easier for Chinese-speaking children to do something virtually identical? I suspect that if Chinese-speaking children understand place-value better than English-speaking children, there is more reason than the name designation of their numbers. And Fuson points out a number of things that Asian children learn to do that American children are generally not taught, from various methods of finger counting to practicing with pairs of numbers that add to ten or to whole number multiples of ten.

From a conceptual standpoint of the sort I am describing in this paper, it would seem that sort of practice is far more important for learning about relationships between numbers and between quantities than the way spoken numbers are named. There are all kinds of ways to practice using numbers and quantities; if few or none of them are used, children are not likely to learn math very well, regardless of how number words are constructed or pronounced or how numbers are written. (Return to text.)

Footnote 9. Because children can learn to read numbers simply by repetition and practice, I maintain that reading and writing numbers has nothing necessarily to do with understanding place-value. I take "place-value" to be about how and why columns represent what they do and how they relate to each other, not just knowing what they are named. Some teachers and researchers, however (and Fuson may be one of them) seem to use the term "place-value" to include or be about the naming of written numbers, or the writing of named numbers. In this usage then, Fuson would be correct that --once children learn that written numbers have column names, and what the order of those column names is -- Chinese-speaking children would have an advantage in reading and writing numbers (that include any ten's and one's) that English-speaking children do not have. But as I pointed out earlier, I do not believe that advantage carries over into doing numerically written or numerically represented arithmetical manipulations, which is where place-value understanding comes in.

And I do not believe it is any sort of real advantage at all, since I believe that children can learn to read and write numbers from 1 to 100 fairly easily by rote, with practice, and they can do it more readily that way than they can do it by learning column names and numbers and how to put different digits together by columns in order to form the number.

When my children were learning to "count" out loud (i.e., merely recite number names in order) two things were difficult for them, one of which would be difficult for Chinese-speaking children also, I assume. They would forget to go to the next ten group after getting to nine in the previous group (and I assume that, if Chinese children learn to count to ten before they go on to "one-ten one", they probably sometimes will inadvertently count from, say, "six-ten nine to six-ten ten"). And, probably unlike Chinese children, for the reasons Fuson gives, my children had trouble remembering the names of the subsequent sets of tens or "decades". When they did remember that they had to change the decade name after a something-ty nine, they would forget what came next. But this was not that difficult to remedy by brief rehearsal periods of saying the decades (while driving in the car, during errands or commuting, usually) and then practicing going from twenty-nine to thirty, thirty-nine to forty, etc. separately.

Actually a third thing would also sometimes happen, and theoretically, it seems to me, it would probably happen more frequently to children learning to count in Chinese. When counting to 100 my children would occasionally skip a number without noticing or they would lose their concentration and forget where they were and maybe go from sixty six to seventy seven, or some such. I would think that if you were learning to count with the Chinese naming system, it would be fairly easy to go from something like six-ten three to four-ten seven if you have any lapse in concentration at all. It would be easy to confuse which "ten" and which "one" you had just said. If you try to count simple mixtures of two different kinds of objects at one time --in your head-- you will easily confuse which number is next for which object. Put different small numbers of blue and red poker chips in ten or fifteen piles, and then by going from one pile to the next just one time through, try to simultaneously count up all the blue ones and all the red ones (keeping the two sums distinguished). It is extremely difficult to do this without getting confused which sum you just had last for the blue ones and which you just had last for the red ones. In short, you lose track of which number goes with which name. I assume Chinese children would have this same difficulty learning to say the numbers in order. (Return to text.)

Footnote 10. There is a difference between things that require sheer repetitive practice to "learn" and things that require understanding. The point of practice is to become better at avoiding mistakes, not better at recognizing or understanding them each time you make them. The point of repetitive practice is simply to get more adroit at doing something correctly. It does not necessarily have anything to do with understanding it better. It is about being able to do something faster, more smoothly, more automatically, more naturally, more skillfully, more perfectly, well or perfectly more often, etc. Some team fundamentals in sports may have obvious rationales; teams repetitively practice and drill on those fundamentals then, not in order to understand them better but to be able to do them better.

In math and science (and many other areas), understanding and practical application are sometimes separate things in the sense that one may understand multiplication, but that is different from being able to multiply smoothly and quickly. Many people can multiply without understanding multiplication very well because they have been taught an algorithm for multiplication that they have practiced repetitively. Others have learned to understand multiplication conceptually but have not practiced multiplying actual numbers enough to be able to effectively multiply without a calculator. Both understanding and practice are important in many aspects of math, but the practice and understanding are two different things, and often need to be "taught" or worked on separately.

Similarly, physicists or mathematicians may work with formulas they know by heart from practice and use, but they may have to think a bit and reconstruct a proof or rationale for those formulas if asked. Having understanding, or being able to have understanding, are often different from being able to state a proof or rationale from memory instantaneously. In some cases it may be important for someone not only to understand a subject but to memorize the steps of that understanding, or to practice or rehearse the "proof" or rationale or derivation also, so that he can recall the full, specific rationale at will. But not all cases are like that. (Return to text.)

Footnote 11. In a discussion of this point on Internet's AERA-C list, Tad Watanabe pointed out correctly that one does not need to regroup first to do subtractions that require "borrowing" or exchanging ten's into one's. One could subtract the subtrahend digit from the "borrowed" ten, and add the difference to the original minuend one's digit. For example, in subtracting 26 from 53, one can change 53 into, not just 40 plus 18, but 40 plus a ten and 3 one's, subtract the 6 from the ten, and then add the diffence, 4, back to the 3 you "already had", in order to get the 7 one's. Then, of course, subtract the two ten's from the four ten's and end up with 27. This prevents one from having to do subtractions involving minuends from 11 through 18.

That in turn reminded me of two other ways to do such subtraction, avoiding subtracting from 11 through 18: (1) akin to the way you would do it with an abacus, you subtract as many one's as you can from the one's in the "existing" minuend; and then you subtract the rest of the one's you need to subtract after you convert a ten to 10 one's. (In the case of 53-26, you subtract all three one's from the 53, which leaves three more one's that you need to subtract once you have converted the ten from fifty into 10 one's. Then, of course, you subtract the 20.)

(2) You can go into negative numbers, so in the same problem, when you subtract the 6 from the 3, you get -3, and combine that -3 with the 10 one's after converting the ten, and then subtract the 20 from the 47, i.e., the 4 ten's and 7 one's.

If you don't teach children (or help them figure out how) to adroitly do subtractions with minuends from 11 through 18, you will essentially force them into options (1) or (2) above or something similar. Whereas if you do teach subtractions from 11 through 18, you give them the option of using any or all three methods. Plus, if you are going to want children to be able to see 53 as some other combination of groups besides 5 ten's and 3 one's, although 4 ten's plus 1 ten plus 3 one's will serve, 4 ten's and 13 one's seems a spontaneous or psychologically ready consequence of that, and it would be unnecessarily limiting children not to make it easy for them to see this combination as useful in subtraction. (Return to text.)

Footnote 12. I say at the time you are trying to subtract from it because you may have already regrouped that number and borrowed from it. Hence, it may have been a different number originally. If you subtract 99 from 1001, the 0's in the minuend will be 9's when you "get to them" in the usual subtraction algorithm that involves proceeding from the right (one's column) to the left, regrouping, borrowing, and subtracting by columns as you proceed. (Return to text.)

Footnote 13. When I explained about the need to practice these kinds of subtractions to one teacher who teaches elementary gifted education, who likes math and mathematical/logical puzzles and problems, and who is very knowledgeable and bright herself, she said "Oh, you mean they need practice regrouping in order to subtract these amounts." That was a natural conceptual mistake on her part, since you do NOT regroup to do these subtractions. These subtractions are what you always end up with AFTER you regroup to subtract. If you try to regroup to subtract them, you end up with the same thing, since changing the "ten" into 10 ones still gives you 1_ as the minuend. For example, when subtracting 9 from 18, if you regroup the 18 into no tens and 18 ones, you still must subtract 9 from those 18 ones. Nothing has been gained. (Return to text.)

Footnote 14. In a third grade class where I was demonstrating some aspects of addition and subtraction to students, if you asked the class how much, say, 13 - 5 was (or any such subtraction with a larger subtrahend digit than the minuend digit), you got a range of answers until they finally settled on two or three possibilities. I am told by teachers that this is not unusual for students who have not had much practice with this kind of subtraction. (Return to text.)

Footnote 15. There is nothing wrong with teaching algorithms, even complex ones that are difficult to learn. But they need to be taught at the appropriate time if they are going to have much usefulness. They cannot be taught as a series of steps whose outcome has no meaning other than that it is the outcome of the steps. Algorithms taught and used that way are like any other merely formal system -- the result is a formal result with no real meaning outside of the form. And the only thing that makes the answer incorrect is that the procedure was incorrectly followed, not that the answer may be outlandish or unreasonable. In a sense, the means become the ends.

Arithmetic algorithms are not the only areas of life where means become ends, so the kinds of arithmetic errors children make in this regard are not unique to math education. (A formal justice system based on formal "rules of evidence" sometimes makes outlandish decisions because of loopholes or "technicalities"; particular scientific "methods" sometimes cause evidence to be missed, ignored, or considered merely aberrations; business policies often lead to business failures when assiduously followed; and many traditions that began as ways of enhancing human and social life become fossilized burdensome rituals as the conditions under which they had merit disappear.)

Unfortunately, when formal systems are learned incorrectly or when mistakes are made inadvertently, there is no reason to suspect error merely by looking at the result of following the rules. Any result, just from its appearance, is as good as any other result.

Arithmetic algorithms, then, should not be taught as merely formal systems. They need to be taught as short-hand methods for getting meaningful results, and that one can often tell from reflection about the results, that something must have gone awry. Children need to reflect about the results, but they can only do that if they have had significant practice working and playing with numbers and quantities in various ways and forms before they are introduced to algorithms which are simply supposed to make their calculating easier, and not merely simply formal. Children do not always need to understand the rationale for the algorithm's steps, because that is sometimes too complicated for them, but they need to understand the purpose and point of the algorithm if they are going to be able to (learn to) apply it reasonably. Learning an algorithm is a matter of memorization and practice, but learning the purpose or rationale of an algorithm is not a matter of memorization or practice; it is a matter of understanding. Teaching an algorithm's steps effectively involves merely devising means of effective demonstration and practice. But teaching an algorithm's point or rationale effectively involves the more difficult task of cultivating students' understanding and reasoning. Cultivating understanding is as much art as it is science because it involves both being clear and being able to understand when, why, and how you have not been clear to a particular student or group of students. Since misunderstanding can occur in all kinds of unanticipated and unpredictable ways, teaching for understanding requires insight and flexibility that is difficult or impossible for prepared texts, or limited computer programs, alone to accomplish.

Finally, many (math) algorithms are fairly complex, with many different "rules", so they are difficult to learn just as formal systems, even with practice. The addition and subtraction algorithms (how to line up columns, when and how to borrow or carry, how to note that you have done so, how to treat zeroes, etc., etc.) are fairly complex and difficult to learn just by rote alone. I think the research clearly shows that children do not learn these algorithms very well when they are taught as formal systems and when children have insufficient background to understand their point. And it is easy to see that in cases involving "simple addition and subtraction", the algorithm is far more complicated than just "figuring out" the answer in any logical way one might; and that it is easier for children to figure out a way to get the answer than it is for them to learn the algorithm. Rule-based derivations are helpful in cases too complex to do by memory, logic, or imagination alone; but they are a hindrance in cases where learning or using them is more difficult than using memory, logic, or imagination directly on the problem or task at hand. (This is not dissimilar to the fact that learning to read and write numbers --at least up to 100-- is easier to do by rote and by practice than it is to do by being told about column names and the rules for their use.) There is simply no reason to introduce algorithms before students can understand their purpose and before students get to the kinds of (usually higher) number problems for which algorithms are helpful or necessary to solve. This can be at a young age, if children are given useful kinds of number and quantity experiences. Age alone is not the factor. (Return to text.)

Footnote 16. Thinking or remembering to count large quantities by groups, instead of tediously one at a time, is generally a learned skill, though a quickly learned one if one is told about it. Similarly, manipulating groups for arithmetical operations such as addition, subtraction, multiplication and division, instead of manipulating single objects. The fact that English-speaking children often count even large quantities by individual items rather than by groups (Kamii), or that they have difficulty adding and subtracting by multi-unit groups (Fuson) may be more a lack of simply having been told about its efficacies and given practice in it, than a lack of "understanding" or reasoning ability. I do not think this is a reflection on children's understanding, or their ability to understand.

There are many subject areas where simple insights are elusive until one is told them, and given a little practice to "bind" the idea into memory or reflex. Sometimes one only needs to be told once, sees it immediately, and feels foolish for not having realized it oneself. Many people who take pictures with a rectangular format camera never think on their own to turn the camera vertically in order to better frame and be able to get much closer to a vertical subject. Most children try to balance a bicycle by shifting their shoulders though most of their weight (and balance then) is in their hips, and the hips tend to go the opposite direction of the shoulders; so that correcting a lean by a shoulder lean in the opposite direction usually actually hastens the fall. The idea of contour plowing in order to prevent erosion, once it is pointed out, seems obvious, yet it was never obvious to people who did not do it. Counting back "change" by "counting forward" from the amount charged to the amount given, is a simple, effective way to figure change, but it is a way most students are not taught to "subtract", so store managers need to teach it to student employees. It is not because students do not know how to subtract or cannot understand subtraction, but because they may have not been shown this simple device or thought of it themselves. I believe that counting or calculating by groups, rather than by one's or units, is one of these simple kinds of things one generally needs to be told about when one is young (and given practice in, to make it automatic) or one will not think about it.

I do not believe having to be told these simple things necessarily shows one did not have any understanding of the principles they involve. As in the trick problems given earlier, sometimes our "understanding" simply gets a kind of blind spot or a focus in a different direction that blocks a particular piece of knowledge. Since understanding is so immediate upon simply being told the insight, it seems a different kind of thing from teaching someone a whole new idea they did not understand before, were not ready to understand, or could not understand. I suspect that often even when children are taught to recognize groups by patterns or are taught to recite successive numbers by groups (i.e., recite the multiples of groups -- e.g., 5, 10, 15, 20...) they are not told that is a quicker way to count large quantities of things -- i.e., first grouping things and then counting the groups. And they are not given practice counting objects that way. So they don't make the connection; and when asked to count large quantities, do it one at a time. (Return to text.)

Footnote 17. Different color poker chips alone, as Fuson notes (p. 384) will not generate understanding about quantities or about place-value. Children can be confused about the representational aspects of poker chip colors if they are not introduced to them correctly. And if not wisely guided into using them effectively, children can learn "face-value (superficial grouping)" facility with poker chips that are not dissimilar to the face value, superficial ability to read and write numbers numerically. The point, however, is not to let them just use poker chips to represent "face-values" alone, but to guide them into using them for both (face-value) representation and as grouped physical quantities. What I wrote here about the use of poker chips to teach place-value involves introducing them in a particular (but flexible) way at a particular time, for a particular reason. I give examples of the way they need to be used to teach place-value in the text. The time they need to be introduced this way is after children understand about grouping quantities and counting quantities "by groups". And I explain in this article precisely why different color poker chips, when used correctly, can better teach children about place-value than can base-ten blocks alone. Poker chips, used and demonstrated correctly, can serve as an effective practical and conceptual bridge between physical groups and columnar representation, because they are both physical and representational in ways that make sense to children --with minimal demonstration and with monitored, guided, practice. And since poker chips stack fairly conveniently, they can be used at earlier stages for children to count individually and by groups, and to manipulate by groups. (Columns of poker chips can also be used effectively to teach understanding about many of the more difficult conceptual and representational aspects of fractions, which is another matter about teaching that I only mention here to point out the usefulness of having a large supply of poker chips in classrooms for a number of different mathematics educational purposes.) (Return to text.)

Footnote 18. There is a difference between regrouping poker chips between 10 and 18, and regrouping written numbers between 10 and 18, since when you regroup with poker chips, you change ten of the white ones into a blue one, (or vice versa) but when you regroup 18 in written form you merely end up with a number that looks like what you started with. "One ten and 8 ones" in numerically written form looks just like "18 ones". (When you regroup and borrow in order to subtract, say in the problem 35 - 9, you regroup the 35 into "20 and 15" or, as I say pointedly to students "twentyfifteen". Then you write the "15" in the one's column where the digit "5" was and you have a "2" in the column where the "3" was, so it even kind of looks like "twentyfifteen". However, in numerical written form, when you start with a number from 10 through 18, if you "scratch out" the "1" and then add ten to the "8" in the one's column, you end up with "18" in the one's column, which is essentially the same in appearance as what you started with. There is a perceptual point in changing 35 into 2[15]; there is not a perceptual point in changing 18 into [18]. With poker chips there is a perceptual difference between "one (blue) ten and eight (white) ones" and "18 (white) ones". That is part of how poker chips help children conceptually understand representational regrouping. (Return to text.)

Footnote 19. Instead of teaching them to construct numbers by numerals and columns, you have previously taught them simply to write numbers. By using the poker chips, you have helped them group quantities representationally in terms of ten's and one's where the ten's are different from the one's in some characteristic. Then you show them that written numbers actually also group quantities that way --that written numbers are not just indivisible monadic symbols but that they have a logical structure and rationale to them. That gives them a feeling of discovery and it makes more sense to them than does trying to start out teaching them to write numbers in terms of numerals and columns, which will mean nothing to them, or seem of no special significance. (Return to text.)

Footnote 20. In any base math, you simply add another column whenever you "get stuck" because you have run out of numeric symbols and combinations of them. And you call that column by the name of the first number you need to have a new column in order to write the number. Hence, in binary arithmetic, you have "one's", "two's", "four's", "eight's", "sixteen's", "thirty-two's" columns, etc., since after you write "0" and "1", you need a new column to write "two", since you don't have any more numerals. Then you can write "10" for "two" and "11" for "three", and you again run out of numerals and combinations. To write "four" you need a new column (hence, it is the "four's" column) and you then can make four different combinations ("100" for "four", "101" [one four, no two's, and one one] for "five", "110" [one four, one two, and no one's] for a "six", and "111" [a four, a two, and a one] for a "seven"). (Return to text.)

Footnote 21. 35 times 43 for example is the following, if you remember from algebra (30 + 5)(40 +3), which ends up being [(30)(40) + (30)(3) + (5)(40) + (5)(3)]. And this is how we actually do the calculation (though in a different order) when we multiply, since you multiply five times three and then five times forty and then add it together (in the same number) and add that to the sum of thirty times three and thirty times forty. But, of course, we don't think of it this way; and many people who can perfectly well multiply would be unable to think of it this way on their own.

Further, seeing why "a(b+c)" is the same as "(ab + ac)" is not easy in terms of written numbers, though it is easy to see if you lay out poker chips in rows and columns. You can see that five rows of seven, for example is the same as five rows of four plus five rows of three, because the two sets of five rows are lying right beside each other. And by doing this in poker chips with a few sets of numbers, it is fairly easy for the imagination to see that "a" rows of "(b+c)" is the same as "a" rows of "b" plus "a" rows of "c" and vice versa.

And seeing why "(a+b)(c+d)" is (ac + ad + bc + bd) is possible also (though somewhat more difficult) by putting poker chips in rows and columns, e.g., 12 by 23 [(10+2) by (20+3)] and marking them off in portions that match ac, ad, bc, and bd, and seeing these are all mutually exclusive segments that combine to make the total number of chips. (See figure.)

At any rate, the manipulations we learn using pencil and paper, have a rationale, but the rationale is not something we generally learn, and not something that in a sense is as easy as the manipulations.

Further, for large numbers, conceptualization and physical representation are difficult or impossible. So once one learns the rationale or is able to understand or see it, one does not necessarily employ the conceptualization of it for every application. (Return to text.)

Footnote 22. I believe there is a certain irony in calling actual physical quantities of things manipulatives, while considering "pure" numbers not to be manipulatives. In a sense it seems to me that is just the reverse of the truth. "Pure" numbers allow us to represent quantities apart from what they are quantities of (so that if we know that five sets of five are 25, we don't have to calculate separately what five sets of five tires and five sets of five candies and five sets of nickels are); but pure numbers are more often simply what we can manipulate "mathematically" rather than sets of objects. It is much more feasible to figure amounts of things on paper (or in a calculator) than to assemble the requisite number of things we are talking about in order to add, subtract, multiply, or divide them, especially when we are talking about large numbers of things. And this is true whether we are talking about billions of dollars of money or thousands of gallons of gasoline. In liquid measures we often calculate volumes by multiplying dimensions, not by individually scooping out and transferring unit volumes. In all these cases, we manipulate numbers, not things.

Unfortunately in real life, quantities do not conform to simple arithmetic, and so science is empirical rather than a priori. Velocities do not combine with each other by simple addition (although at relatively low velocities they seem to); forces do not combine with each other by simple addition; nor are forces three times the distance acting at one third the strength; working twice as fast may not get you done in half the time (because you may wear out before you finish if you work harder than your capacity); and 10,000 T-shirts purchased at one time probably won't cost 10,000 times the price of one T-shirt. Mixing equal volumes of things that dissolve in one another won't give you twice the volume of either of them. Figuring out the way in which various quantities of things relate to each other is part of what science is about; and it is not always a very easy endeavor that conforms to the arithmetical manipulation of numbers.

In other words, real objects do not always manipulate in the same way that numbers do; and manipulating objects is not the same thing as manipulating numbers. And, it seems to me, the child who is manipulating objects in rows and columns in order to demonstrate or understand multiplication is doing something quite different from the person who is manipulating numbers on paper or in his head. Multiplication is easily seen to be commutative (i.e., six sets of eight will equal eight sets of six) when manipulating objects in rows and columns (because if you change your vantage point 90 degrees, your rows and columns simply reverse, but the total quantities remain the same); whereas it is not so obvious if you do it merely in your head or merely with pure numbers why, or that, six bags each containing eight candies will be the same number of candies as eight bags each containing six candies. (Return to text.)

Footnote 23. In correspondence with me from Peabody, Paul Cobb has said he "would argue that mathematics at the elementary level should not involve mechanical skills even though it is currently often taught this way. The ideal would be that conceptually based problem solving would be infused in all children's mathematical activity in school." I disagree.

Although children should not be taught arithmetic only mechanically, there are some mechanical skills that can be relatively easily learned by children, and that are important or necessary for seeing more "interesting" number relationships. For example, memorizing multiplication tables is not (and should neither be seen nor used as) just an exercise to enable one to multiply like a very slow calculator. It gives facility with multiples that can help one more readily understand the concept of division, and more readily understand fractions and relationships between fractions --such as when seeking common denominators or converting between "mixed" numbers and fractions. It gives increased ability to understand and use factoring in algebra or in calculus.

I am not saying that all the things children learn mechanically in elementary math are necessary to learn or are best learned mechanically. But some are. And I would consider learning to recite number names in order and number names by groups ("counting" and "counting by groups") and learning to do what I have referred to as simple addition and subtraction as examples of crucially important mechanical skills. Some mechanically learned skills simply allow you to make intellectual leaps you might not have been able to make at all if you were not able to quickly and somewhat automatically perceive relationships you had not become extremely familiar with or "primed for" previously by memorization, repetition, drill, and practice. (Return to text.)

Footnote 24. I used to play an imagination "bag game" with my children that asked them things like "I have a bag and you have a bag; my bag has three less than your bag; and you have five things in your bag. How many things do I have in mine?" As they got better at doing this, I made the problems harder. "I have a bag and you have a bag, and together we have eight things; but you have four more things than I do. How many do we each have?" We are now up to "I have a bag and you have a bag. You have five more than I do in your bag, but if we triple what I have, I will have five more than you. How much do we each have?" Children can work out these things by thinking. They don't have to go through particular steps they are trained in. We also do number progressions where they have to reason out what the next number would be. You can do these in really weird, tricky, but actually simple progressions and they often love it; e.g., 2, 10, 4, 20, 8, 30, 16, ?. [One correct answer would be "40", since these are two different progressions that are interspersed: 2, 4, 8, 16, ..., and 10, 20, 30, .... We did most of these math games in the car while commuting places. Professor Richard Feynman's father used to do color tile patterns with him when Richard was a still in a high chair. There are all kinds of mathematical things you can do with very young children that they can successfully figure out and learn from, and that they can enjoy.

Math learning does not have to go in some particular arithmetical order only, at some particular age. There are all kinds of mathematical types of things that children can do at various ages. There is more to math than just algorithmic arithmetic; and children can do the "more" even in some cases where they cannot yet do the algorithmic arithmetic. Children can reason; they just sometimes need some help or practice or feedback, or they sometimes need a reasonable or reasonably channeled challenge, in order to hone their reasoning skills. (Return to text.)

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