the concept and teaching of place

8/6/2019 The Concept and Teaching of Place

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first time but does not understand, is generally not going to help him. Until you find outthe specific stumbling block, you are not likely to tailor an answer that addresses hisneeds, particularly if your general explanation did not work with him the first time or twoor three anyway and nothing has occurred to make that explanation any more intelligibleor 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 includessome 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 arithmeticteachers, I suspect (and will try to point out why I suspect it) that children have a difficulttime learning place-value because most elementary school teachers (as most adults ingeneral, 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 canunderstand 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 childrenunderstand 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 arbitraryconventions seem matters of logic, so children squander much intellectual capital seekingto understand what has nothing to be understood.

And a further problem in teaching is that because teachers, such as the algebra teachersreferred to above, tend not to ferret out of children what the children specifically don'tunderstand, teachers, even when they do understand what they are teaching, don't alwaysunderstand what students are learning -- and not learning. There are at least two aspectsto 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 mosthelpful in facilitating learning. It is difficult to know how to help when one doesn't knowwhat, 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 toknow what students know about place-value. If it is the latter, then it would seem there isteaching occurring without learning happening, an oxymoron that, I believe, means thereis not "teaching" occurring, but merely presentations being made to students withoutsufficient 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 ismaking certain students are grasping and learning what one is trying to teach. That is notalways easy to do, but at least the attempt needs to be made as one goes along. Teachersought to have known for some time what researchers have apparently only relativelyrecently discovered about children's understanding of place-value: "The literature isreplete with studies identifying children's difficulties learning place-value concepts.(Jones and Thornton, p.12)" "Mieko Kamii's (1980, 1982) pioneering investigations inthis area revealed glaring misunderstandings that were surprisingly pervasive. His [sic;Her] investigation showed that despite several years of place-value learning, childrenwere unable to interpret rudimentary place-value concepts. (Jones, p.12)(5)"

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Since I have taught my own children place-value after seeing how teachers failed to teachit(6), and since I have taught classes of children some things about place-value they couldunderstand but had never thought of or been exposed to before, I believe the failure tolearn place-value concepts lies not with children's lack of potential for understanding, butwith 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 generalis not learned very well in general. The research literature on place-value also shows alack 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 theresults of conceptually faulty teaching and testing methods concerning place-value. Andwhen they find cultural or community differences in the learning of place-value, theyseem to focus on factors that seem, from a conceptual viewpoint, less likely causallyrelevant than other factors. I believe that there is a better way to teach place-value than itis 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 hastaken civilization thousands of years, much ingenious creativity, and not a little fortuitousinsight to develop many of the concepts and much of the knowledge it has; and childrencan not be expected to discover or invent for themselves many of those concepts or muchof 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 isunrealistic to expect 25 years of an individual's biological development to recapitulate 25

centuries of collective intellectual accomplishment without significant help. Thoughmany 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 becompetent in a given field, math being no exception. Potential learning is generallyseverely impeded without teaching. And it is possibly impeded even more by badteaching, since bad teaching tends to dampen curiosity and motivation, and since wronginformation, just like bad habits, may be harder to build from than would be noinformation, and no habits at all. In this paper I will discuss the elements I will argue arecrucial to the concept and to the teaching of place-value.

Understanding Place-value: Practical and Conceptual Aspects

There are at least five aspects to being able to understand place-value, only two or threeof which are often taught or stressed. The other two or three aspects are ignored, and yetone of them is crucial for children's (or anyone's) understanding of place-value, and oneis important for complete understanding, though not for merely useful understanding. I







will first just name and briefly describe these aspects all at once, and then go on to morefully discuss each one individually.

1) Learning number names (and their serial order) and using numbers to count quantities,developing familiarity and facility with numbers, practicing with numbers --including,

when appropriate not only saying numbers but writing and reading them(8)

, not in terms of rules involving place-value, etc., but in terms of just being shown how to write and readindividual numbers (with comments, when appropriate, that point out things like "ten,eleven, twelve, and all the teens have a '1' in front of them; all the twenty-numbers have a'2' in front of them" etc.) without reasons about why that is(9),

2) "simple" addition and subtraction,

3) developing familiarity through practice with groupings, and counting physicalquantities by groups (not just saying the "multiples" of groups -- e.g., counting things byfives, 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 havinglearned how to write numbers before. Practice with grouping and counting by groupsshould, of course, include groupings by ten's,

4) representation (of groupings)

5) specifics about representations in terms of columns.

Aspects (1), (2), and (3) require demonstration and "drill" or repetitive practice. Aspects(4) and (5) involve understanding and reason with enough demonstration and practice toassimilate it and be able to remember the overall logic of it with some reflection, rather

than the specific logical steps.

(10)

1) Number Facility, Practice

The more familiar one is with numbers and what they represent, the easier it is, generally,to see relationships involving numbers. Hence, it is important that children learn to countand to be able to identify the number of things in a group either by counting or by patterns, etc. One way to see this is to take some slice of 10 letters out of the middle of the alphabet, say "k,l,m,n,o,p,q,r,s,t" and let them represent 0-9 in linear order. Eventhough most adults can say those letters in order, just as they and children can say thenames of numbers in order, it is very difficult, unless one practices a lot, to be able to

group things in sets of "n" or to multiply "mrk" times "pm" or to see that all multiples of "p" end in either a "p" or a "k". Yet, seeing the relationships between serially ordereditems one can name in serial order, is much of what arithmetic is about. (Possibly really brilliant math prodigies and geniuses don't have to have number names in order to seenumber relationships, I don't know; but most of us would be lost in any sort of higher level arithmetic if we could not count by (the names of) numbers, recognize the number of things (by name), or use numbers (by name) in relatively simple ways to begin with.)Hence, children normally need to learn to count objects and to understand "how many"









the number names represent. Parents and teachers tend to teach students how to count andto give them at least some practice in counting. That is important.

2) Simple Addition and Subtraction

By "simple addition and subtraction", I mean addition and subtraction with regard toquantities children can learn to add and subtract just by counting together at first andthen, with practice, fairly quickly learn to recognize by memory. For example, childrencan learn to play with dominoes or with two dice and add up the quantities, at first byhaving to count all the dots, but after a while just from remembering the combinations.Children can play something like blackjack with cards and develop facility with addingthe numbers on face cards. Or they can play "team war", where pairs of individuals eachturn over a card, as do the individuals on the opposing team, and whichever team has thehighest sum, gets all four cards for their pile. Adding and subtracting in this way (or insome cases, even multiplying or dividing) may involve quantities that would beregrouped if calculated by algorithm on paper, but they have nothing to do with

regrouping when it is done in this "direct" or "simple" manner. For example, childrenwho play various card games with full decks of regular playing cards tend to learn half of 52 is 26 and that a deck divided equally among four people gives them each 13 cards.

It is particularly important that children get sufficient practice to become facile with

adding pairs of single digit numbers whose sums are not only as high as 10, but also as

high as 18. And it is particularly important that they get sufficient practice to become facile with subtracting single digit numbers that yield single digit answers, not only from

minuends as high as 10, but from minuends between 10 and 18. The reason for this is thatwhenever you regroup for subtraction, if you regroup "first"(11) you always END UP witha 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 10and 18). E.g., 15-7, 18-9, 11-4, etc. The reason you had to "regroup" or "borrow" in thefirst place was that the subtrahend digit in the column in question was larger than theminuend digit in that column; and when you regroup the minuend, those digits do notchange, but the minuend digit simply gains a "ten" and becomes a number between 10and 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 withoutregrouping. Had it been a nine, you would have been able to subtract any possible singledigit number from it without having to regroup.) Another way of saying this is thatwhenever you regroup, you end up with a subtraction of the form:

1?- x

where the digit after the 1 will be between 0 and 8 (inclusive) and will be smaller than thedigit designated by the "x"(13).

Children often do not get sufficient practice in this sort of subtraction to make itcomfortable and automatic for them. Many "educational" math games involving simple









simple calculating errors; anyone can have understanding and still make a mistake. And itis 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 ruleswell enough to work a problem correctly; that just takes more practice. But it should beof 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 lessdisturbing about children's use of algorithms than they are about children's understandingof number and quantity relationships and their understanding of what they are even tryingto accomplish by using algorithms (in this case, for adding and subtracting).

3) Groups

Since counting large numbers of things one at a time gets to be tedious, counting bygroups of two, three, five, ten, etc. is a helpful skill to facilitate. Students have to betaught and rehearsed to count this way, and generally they have to be told that it is a

faster and easier way to count large quantities.

(16)

Also, it serves as a prelude tomultiplication, since counting by groups (of, say, three) introduces one subconsciously tomultiples of those groups (i.e., in this case, multiples of three). And, of course, grouping by 10's is a prelude to understanding those aspects of arithmetic based on 10's. Manyteachers teach students to count by groups and to recognize quantities by the patterns agroup can make (such as on numerical playing cards). This is important.

Aspects of elements 2) and 3) can be "taught" or learned at the same time. Though theyare "logically" distinct; they need not be taught or learned in serial order or specifically inthe order I mention them here. Many conceptually distinct ideas occur together naturallyin practice.

4) Representations of Groups

This is what most elementary school teachers, since they are generally not math majors,do not understand, and can only teach with regard to columnar "place-value". Butcolumnar place-value is (1) not the only way to represent groups, and (2) it is anextremely difficult way for children to understand representations of groups. There aremore accessible ways for children to work with representations of groups. And I think itis easier for them to learn columnar place-value if one starts them out with more psychologically accessible group representations.

Once children have gained facility with counting, and with counting by groups, especiallygroups 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 representationalgroup 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 blueone; or any time you want to exchange a blue one for ten white ones you can do that. Andany 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 youdo some demonstrations, such as putting down eleven white ones and saying somethinglike "if we exchange 10 of these white ones for a blue one, what will we have?" And thechildren will usually say something like "one blue one and one white one". And you canreinforce that they still make (i.e., represent) the same quantity "And that then is stilleleven, right? [Pointing at the blue one] Ten [then pointing at the white one] and one iseleven." 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 understandgroup representation by means of colored poker chips, though you do not use the wordrepresentation, 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, asyou 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 12they just got before, with the blue one and the two white ones, "If we wanted to take 3away from this 12, how could we do it?" [Someone will usually say, or the teacher couldsay the first time or two] "We need to change the blue one into 10 white ones, then wecould take away 3 white ones from the 12 white ones we have." ETC. Keep practicingand changing the numbers so they sometimes need regrouping and sometimes don't; butso they get better and better at doing it. (They are now using the colors bothrepresentationally and quantitatively -- trading quantities for chips that represent them,and vice versa.) Then introduce double digit additions and subtractions that don't requireregrouping 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 whitesto leave 2 blues and 1 white, 21.) When they are comfortable with these, introduce doubledigit 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 studentsare doing, and asking those who seem to be having trouble to explain what they are doingand why. In some ways, seeing how they manipulate the chips gives you some insight





because it is over here instead of over here"; value based on place seems stranger thanvalue based on color, or it seems somehow more arbitrary. But regardless of WHYchildren can associate colors with numerical groupings more readily than they do withrelative column positions, they do.

5) Specifics about Columnar Representations

Apart from the comments made in the last section about columnar representation, I wouldlike to add the following, which is not important for students to understand while they arelearning columnar representation (usually known as "place-values"), but may be helpfulto teachers to understand. And it may be interesting to students at some later stage whenthey can absorb it. (I have taught this to third graders, but the presentation is extremelydifferent from the way I will write it here; and that presentation is crucial to their following the ideas and understanding them. That presentation is detailed in the paper about a method of effectively teaching conceptual/logical material, "The Socratic Method-- Teaching by Asking Instead of by Telling.")

Columnar representation of groups is simply one way of designating groups. But it isimportant to understand why groups need to be designated at all, and what is actuallygoing on in assigning what has come to be known as "place-value" designation. Groupsmake 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 wordform, 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 fiftyfour", 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, andsix." Even though we use names like "hundred," "thousand," "million," etc., which are thesame as the names of the columns higher than the ten's column, we are not reallyrepresenting 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. Itcould have been given a totally unique name (say "gumph") just like "eleven" was, but itwould be difficult to remember totally unique names for all the numbers. It just makes iteasier 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 "twentyone", 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 0as the lowest single numeral, then when we get to the number "ten", we have to dosomething 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

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the next number, which is "ten". To write a ten we need to do something else like make adifferent 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 moveforward a bead on the ten's row. What is chosen for written numbers is to start a newcolumn. 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 includesomething 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 anddifferent numerals in it to designate how many tens we are talking about, in writing anygiven number. Then it turns out that by changing out the numerals in the original columnand the numerals in the "ten" column, we can make combinations of our ten numeralsthat represent each of the numbers from 0 to 99. Now we are stuck again for a way towrite one hundred. We add another column.(20) And we can get by with that column untilwe pass nine hundred ninety nine. Etc.

Representations, Conventions, Algorithmic Manipulations, and Logic

Remember, all this could have been done differently. The abacus does it differently. Our poker chips did it differently. Roman numerals do it differently. And, in a sense,computers and calculators do it differently because they use only two representations(switches that are either "on" or "off") and they don't need columns of anything at all(unless they have to show a written number to a human who is used to numbers written acertain way -- in columns using 10 numerals). And though we can calculate with penciland paper using this method of representation, we can also calculate with poker chips or the abacus; and we can do multiplication and division, and other things, much quicker

with a slide rule, which does not use columns to designate numbers either, or with acalculator or computer.

The written numbering system we use is merely conventional and totally arbitrary and,though it is in a sense logically structured, it could be very different and still be logicallystructured. Although it is useful to many people for representing numbers and calculatingwith numbers, it is necessary for neither. We could represent numbers differently and docalculations quite differently. For, although the relationships between quantities is "fixed"or "determined" by logic, and although the way we manipulate various designations inorder to calculate quickly and accurately is determined by logic, the way we designatethose 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 analgorithm for multiplying and dividing Roman numerals. But following algorithms isneither understanding the principles the algorithms are based on, nor is it a sign of understanding what one is doing mathematically. Developing algorithms requiresunderstanding; using them does not.





conventions or learning algorithms (whose logic is far more complicated than being ableto 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 algorithmicmanipulations are matters of logic they need to understand, is doing them a severedisservice.

On the other hand, children do need to work on the logical aspects of mathematics, someof which follow from given conventions or representations and some of which havenothing to do with any particular conventions but have to do merely with the wayquantities relate to each other. But developing children's mathematical insight andintuition requires something other than repetition, drill, or practice.

Many of these things can be done simultaneously though they may not be in any wayrelated to each other. Students can be helped to get logical insights that will stand them ingood stead when they eventually get to algebra and calculus(24), even though at a differenttime 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 variousways, in some cases through playing miniature golf on all kinds of strange surfaces,through origami, through making periscopes or kaleidoscopes, through doing somesurveying, 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 chipcolors and the column representations of groups. What is important is that teachers canunderstand which elements are conventional or conventionally representational, whichelements are logical, and which elements are (complexly) algorithmic so that they teachthese different kinds of elements, each in its own appropriate way, giving practice inthose things which benefit from practice, and guiding understanding in those thingswhich require understanding. And teachers need to understand which elements of

mathematics are conventional or conventionally representational, which elements arelogical, and which elements are (complexly) algorithmic so that they can teach thosedistinctions themselves when students are ready to be able to understand and assimilatethem.

Footnote 1. Mere repetition about conceptual matters can work in cases whereintervening experiences or information have taken a student to a new level of awarenessso that what is repeated to him will have "new meaning" or relevance to him that it didnot before. Repetition about conceptual points without new levels of awareness willgenerally not be helpful. And mere repetition concerning non-conceptual matters may behelpful, as in interminably reminding a young baseball player to keep his swing level, ayoung 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 thenames they do. That is, why is the tens column the tens column or the hundreds columnthe hundreds column? And, could there have been some method other than columns thatwould 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


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particular columns that we use? In informal questioning, I have not met any primarygrade teachers who can answer these questions or who have ever even thought aboutthem 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 extremelyimportant 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) sayinga 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, triplefive". It is even difficult for an American to grasp a phone number if you pause after thefourth 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 thesame 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-sectionalcomparisons after studying each region in entirety, but I could not construct a wholeregion 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 onetraining 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 childwas justifiably riding at a 30 degree angle to the bike. When I took off the other trainingwheel to teach her to ride, it took about ten minutes just to get her back to a normalnovice'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 tostudents, though it does not "sink" in to students fully at the end of that time. ("Sinkingin" or ready facility requires practice along with understanding.) Many people I havetaught have taken whole courses in photography that were not structured very well, andmy perspective enlightens their understanding in a way they may not have achieved in thedirection they were going.

(5) I studied European history for the first time when I was in college. My lecturer did notstructure the material for us, and to me the whole thing was an endless, indistinguishablecollection of popes and kings and wars. I tried to memorize it all and it was virtuallyimpossible. I found out at the end of the term that the other professor who taught thecourse (to all my friends) spent each of his lectures simply structuring a framework inorder to give a perspective for the students to place the details they were reading. Theylearned it.

(6) The year I took organic chemistry, one professor tested out a new textbook thatstructured the material in a new way, and he lectured in the same structure as the book.

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He admitted at the end of the year that was a big mistake; students did not learn as wellusing 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 bemuch memorization needed to learn each of these individual formulas. I happened tonotice the relationship the night before the midterm exam, purely by luck and somecoincidental reasoning about something else. I figured I was the last to see it of the 1500students in the course and that, as usual, I had been very naive about the material. Itturned out I was the only one to see it. I did extremely well but everyone else didmiserably 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 ageneral principle from which you could derive all the others, most of the other studentswould have done well on the test also.

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

Footnote 4. In a small town not terribly far from Birmingham, there is a recently openedMcDonald's that serves chocolate shakes which are off-white in color and which tastelike 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 machinewas 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 theway they were supposed to be; there was nothing wrong with them. There was noconvincing her. After she returned to her office I realized, and mentioned to the salesstaff, that I should have asked her to take a taste test to try to distinguish her chocolateshakes from her vanilla ones. That would show her there was no difference. The staff toldme that would not work since there was a clear difference: "Our vanilla shakes taste likechalk." They understood there was a problem.

Unfortunately, too many teachers teach like that manager manages. They think if they dowell 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 isirrelevant 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 thesetting up of the classroom for discovery or work). If they "teach" well what childrenalready know, they are good teachers. If they make dynamic well-prepared presentationswith much enthusiasm, or if they assign particular projects, they are good teachers, evenif no child understands the material, discovers anything, or cares about it. If they traintheir students to be able to do, for example, fractions on a test, they have done a good jobteaching 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 thatclassroom teachers ought to be able to teach so that every child learns. There arevariables outside of even the best teachers' control. But teachers ought to be able to tellwhat 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 studentsunderstand 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 stimulatingthose students' minds about the material or whether they are poisoning any interest thechild 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 toends. 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 tochildren 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 thechildren 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, thechildren typically pointed to the 6 cups of candy. This is taken to demonstrate children donot understand place value. I believe this demonstrates the kind of tricks similar to thefollowing problems, which do not show lack of understanding, but show that one can bedeceived into ignoring or forgetting one's understanding.

(1) There is a ship in the harbor with a very long rope-ladder hanging overboard whoserungs 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." Thisdoes not demonstrate respondents do not understand buoyancy, only that one can betricked 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, thedesk 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





algorithm). By increasingly difficult, I mean, for example, going from subtracting or summing relatively smaller quantities to relatively larger ones (with more and moredigits), going to problems that require (call it what you like) regrouping, carrying, borrowing, or trading; going to subtraction problems with zeroes in the number fromwhich you are subtracting; to consecutive zeroes in the number from which you are

subtracting; and subtracting such problems that are particularly psychologically difficultin 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 becausethey have been contrived specifically to psychologically mislead, or they are constructedaccidentally 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 theexamples should make clear what it is I mean.

Further, it is often difficult to know what someone else is asking or saying when they doit 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 thatmakes 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 thinkingabout 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 -- butusing the same words (or symbols) to describe what you are thinking about. This issimilar 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 anddescribe what they see. They will describe that person's face, but they will actually beseeing much more than that person's face. So, their answer is wrong, thoughunderstandably 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 theyare too far away to have that face show up very well in the photograph. They really donot know all they are seeing through the viewer, and all that the camera is "seeing" totake. 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 realmistake 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 thechild 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 readilyobvious set of twenty objects. He may understand place-value perfectly well, but not seethat is what you are asking about -- especially under the circumstances you haveconstructed 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 easymisunderstanding is, particularly about conceptual matters), and if you watch mostteachers teach about the things that involve place-value, or any other logical-conceptualaspects 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 psychologicallycomplex concept. What may be astonishing is that most adults can do math as well asthey 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 understandabout place-value given how it has been taught to them, not as the limits of their possibleunderstanding 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 conceptuallysimilar to Egyptian hieroglyphics-- in which a different looking "marker" is used torepresent tens. And he says "Using a different-looking ten marker may help somechildren --particularly those of low ability-- to bridge the gap between highly concretesize 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 concretegroupings 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 abstractrepresentations 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 isnot more abstract; it is just abstract in a way that is more difficult to recognize and dealwith.

Further, Baroody labels all his categories as kinds of "trading", but he does not seem torecognize there is sometimes a difference between "trading" and "representing", and thattrading is not abstract at all in the way that representing is. I can trade you my MickeyMantle card for your Ted Kluzewski card or my tuna sandwich for your soft drink, butthat does not mean Mickey Mantle cards represent Klu cards or that sandwiches representsoft drinks. Children in general, not just children with low ability, can understand tradingwithout 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 thekind of representing that place-value is. But with regard to trading, as opposed torepresenting, 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 wherethey 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 someobvious way; parking it further up in your driveway does not. It makes sense to a child tosay that two blue poker chips are worth 20 white ones; it makes less apparent sense to saya "2" over here is worth ten "2's" over here. Color poker chips teach the importantabstract 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 arerelatively 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 spokenversions. Written versions have to be learned as well as spoken versions; knowing spokennumbers does not teach written numbers. For example, numbers written in Romannumerals are pronounced the same as numbers in Arabic numerals. And numbers writtenin binary form are pronounced the same as the numbers they represent; they just arewritten 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 somemistakes 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 digitnumbers 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 Chineselanguage 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 thenames of numbers are pronounced, the numeric designation of them is still a totallydifferent thing from the written word designation; e.g., "1000" versus "one thousand". Itshould 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 anyeasier for a Chinese child to learn to read or pronounce "11" as (the Chinese translationof) "one-ten, one" than it is for English-speaking children to see it as "eleven". AndFuson does note the detection of three problems Chinese children have: (1) learning towrite 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 incertain 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


http://www.garlikov.com/PlaceValue.html#fuson


http://www.garlikov.com/PlaceValue.html#fuson



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 borrowingalgorithm of subtraction; we do write a "12" in the ten's column when we had two ten'sand 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 childwrites 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 havelearned to read numeric numbers, such as "215" as (the Chinese translation of) "2-onehundred, one-ten, five", that alone should not help them be able to subtract "56" from itany more easily than an English-speaking child can do it, because (1) one still has totranslate the concepts of trading into columnar numeric notations, which is not especiallyeasy, and because (2) one still has to understand how ones, tens, hundreds, etc. relate toeach 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 tosubtract "fifty" from "sixty" to get "ten". Nor is it difficult for English-speaking studentswho 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 oftenhave 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 virtuallyidentical? I suspect that if Chinese-speaking children understand place-value better thanEnglish-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 thatsort of practice is far more important for learning about relationships between numbersand 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, childrenare 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 withunderstanding place-value. I take "place-value" to be about how and why columnsrepresent what they do and how they relate to each other , not just knowing what they arenamed. Some teachers and researchers, however (and Fuson may be one of them) seem touse the term "place-value" to include or be about the naming of written numbers, or thewriting of named numbers. In this usage then, Fuson would be correct that --oncechildren learn that written numbers have column names, and what the order of thosecolumn 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 nothave. But as I pointed out earlier, I do not believe that advantage carries over into doingnumerically written or numerically represented arithmetical manipulations, which iswhere place-value understanding comes in.

And I do not believe it is any sort of real advantage at all, since I believe that children canlearn to read and write numbers from 1 to 100 fairly easily by rote, with practice, andthey can do it more readily that way than they can do it by learning column names andnumbers 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 inorder) 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 countto ten before they go on to "one-ten one", they probably sometimes will inadvertentlycount 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 thesubsequent sets of tens or "decades". When they did remember that they had to changethe decade name after a something-ty nine, they would forget what came next. But thiswas not that difficult to remedy by brief rehearsal periods of saying the decades (whiledriving in the car, during errands or commuting, usually) and then practicing going fromtwenty-nine to thirty, thirty-nine to forty, etc. separately.

Actually a third thing would also sometimes happen, and theoretically, it seems to me, itwould probably happen more frequently to children learning to count in Chinese. Whencounting to 100 my children would occasionally skip a number without noticing or theywould 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 theChinese naming system, it would be fairly easy to go from something like six-ten three tofour-ten seven if you have any lapse in concentration at all. It would be easy to confusewhich "ten" and which "one" you had just said. If you try to count simple mixtures of twodifferent kinds of objects at one time --in your head-- you will easily confuse whichnumber is next for which object. Put different small numbers of blue and red poker chipsin 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 twosums distinguished). It is extremely difficult to do this without getting confused whichsum you just had last for the blue ones and which you just had last for the red ones. Inshort, you lose track of which number goes with which name. I assume Chinese childrenwould 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 atavoiding mistakes, not better at recognizing or understanding them each time you makethem. The point of repetitive practice is simply to get more adroit at doing somethingcorrectly. It does not necessarily have anything to do with understanding it better. It isabout being able to do something faster, more smoothly, more automatically, more





naturally, more skillfully, more perfectly, well or perfectly more often, etc. Some teamfundamentals in sports may have obvious rationales; teams repetitively practice and drillon those fundamentals then, not in order to understand them better but to be able to dothem better.

In math and science (and many other areas), understanding and practical application aresometimes separate things in the sense that one may understand multiplication, but that isdifferent from being able to multiply smoothly and quickly. Many people can multiplywithout understanding multiplication very well because they have been taught analgorithm for multiplication that they have practiced repetitively. Others have learned tounderstand multiplication conceptually but have not practiced multiplying actual numbersenough 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 aretwo 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 haveunderstanding, are often different from being able to state a proof or rationale frommemory instantaneously. In some cases it may be important for someone not only tounderstand 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, specificrationale 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 tothe 3 you "already had", in order to get the 7 one's. Then, of course, subtract the two ten'sfrom the four ten's and end up with 27. This prevents one from having to do subtractionsinvolving minuends from 11 through 18.

That in turn reminded me of two other ways to do such subtraction, avoiding subtractingfrom 11 through 18: (1) akin to the way you would do it with an abacus, you subtract asmany one's as you can from the one's in the "existing" minuend; and then you subtract therest 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 youneed 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 6from the 3, you get -3, and combine that -3 with the 10 one's after converting the ten, andthen 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 withminuends 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 wantchildren 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 aspontaneous or psychologically ready consequence of that, and it would be unnecessarilylimiting children not to make it easy for them to see this combination as useful insubtraction. (Return to text.)

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

Footnote 13. When I explained about the need to practice these kinds of subtractions toone teacher who teaches elementary gifted education, who likes math andmathematical/logical puzzles and problems, and who is very knowledgeable and brightherself, she said "Oh, you mean they need practice regrouping in order to subtract theseamounts." That was a natural conceptual mistake on her part, since you do NOT regroupto do these subtractions. These subtractions are what you always end up with AFTER youregroup to subtract. If you try to regroup to subtract them, you end up with the samething, 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, youstill 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 additionand subtraction to students, if you asked the class how much, say, 13 - 5 was (or any suchsubtraction 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 thatthis 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 thatare difficult to learn. But they need to be taught at the appropriate time if they are goingto have much usefulness. They cannot be taught as a series of steps whose outcome hasno meaning other than that it is the outcome of the steps. Algorithms taught and used thatway are like any other merely formal system -- the result is a formal result with no realmeaning outside of the form. And the only thing that makes the answer incorrect is thatthe 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 thekinds 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 makesoutlandish decisions because of loopholes or "technicalities"; particular scientific"methods" sometimes cause evidence to be missed, ignored, or considered merelyaberrations; 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 madeinadvertently, 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 tellfrom reflection about the results, that something must have gone awry. Children need toreflect about the results, but they can only do that if they have had significant practiceworking 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 rationalefor the algorithm's steps, because that is sometimes too complicated for them, but theyneed 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 stepseffectively involves merely devising means of effective demonstration and practice. Butteaching an algorithm's point or rationale effectively involves the more difficult task of cultivating students' understanding and reasoning. Cultivating understanding is as muchart 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 impossiblefor prepared texts, or limited computer programs, alone to accomplish.

Finally, many (math) algorithms are fairly complex, with many different "rules", so theyare difficult to learn just as formal systems, even with practice. The addition andsubtraction algorithms (how to line up columns, when and how to borrow or carry, howto note that you have done so, how to treat zeroes, etc., etc.) are fairly complex anddifficult to learn just by rote alone. I think the research clearly shows that children do notlearn these algorithms very well when they are taught as formal systems and whenchildren have insufficient background to understand their point. And it is easy to see thatin cases involving "simple addition and subtraction", the algorithm is far morecomplicated than just "figuring out" the answer in any logical way one might; and that itis easier for children to figure out a way to get the answer than it is for them to learn thealgorithm. 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 usingthem is more difficult than using memory, logic, or imagination directly on the problemor 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 toldabout column names and the rules for their use.) There is simply no reason to introducealgorithms before students can understand their purpose and before students get to thekinds of (usually higher) number problems for which algorithms are helpful or necessaryto 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 istold about it. Similarly, manipulating groups for arithmetical operations such as addition,subtraction, multiplication and division, instead of manipulating single objects. The factthat English-speaking children often count even large quantities by individual itemsrather than by groups (Kamii), or that they have difficulty adding and subtracting bymulti-unit groups (Fuson) may be more a lack of simply having been told about itsefficacies and given practice in it, than a lack of "understanding" or reasoning ability. I donot 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, andgiven a little practice to "bind" the idea into memory or reflex. Sometimes one only needsto 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 shouldersthough most of their weight (and balance then) is in their hips, and the hips tend to go theopposite direction of the shoulders; so that correcting a lean by a shoulder lean in theopposite 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 theamount charged to the amount given, is a simple, effective way to figure change, but it isa way most students are not taught to "subtract", so store managers need to teach it tostudent employees. It is not because students do not know how to subtract or cannotunderstand 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 byone's or units, is one of these simple kinds of things one generally needs to be told aboutwhen 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 haveany 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 differentdirection that blocks a particular piece of knowledge. Since understanding is soimmediate upon simply being told the insight, it seems a different kind of thing fromteaching someone a whole new idea they did not understand before, were not ready tounderstand, or could not understand. I suspect that often even when children are taught torecognize 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 thegroups. And they are not given practice counting objects that way. So they don't make theconnection; and when asked to count large quantities, do it one at a time. (Return totext.)

Footnote 17. Different color poker chips alone, as Fuson notes (p. 384) will not generateunderstanding about quantities or about place-value. Children can be confused about therepresentational 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 tolet them just use poker chips to represent "face-values" alone, but to guide them intousing them for both (face-value) representation and as grouped physical quantities. WhatI wrote here about the use of poker chips to teach place-value involves introducing themin a particular (but flexible) way at a particular time, for a particular reason. I giveexamples 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 andcounting 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 aneffective practical and conceptual bridge between physical groups and columnar representation, because they are both physical and representational in ways that makesense 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 moredifficult 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 largesupply 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 youregroup 18 in written form you merely end up with a number that looks like what youstarted 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, youregroup the 35 into "20 and 15" or, as I say pointedly to students "twentyfifteen". Thenyou write the "15" in the one's column where the digit "5" was and you have a "2" in thecolumn where the "3" was, so it even kind of looks like "twentyfifteen". However, innumerical 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 startedwith. There is a perceptual point in changing 35 into 2[15]; there is not a perceptual pointin 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 chipshelp 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 theten's are different from the one's in some characteristic. Then you show them that writtennumbers actually also group quantities that way --that written numbers are not justindivisible 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 tryingto start out teaching them to write numbers in terms of numerals and columns, which willmean 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 callthat 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", youneed a new column to write "two", since you don't have any more numerals. Then youcan write "10" for "two" and "11" for "three", and you again run out of numerals andcombinations. 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, notwo'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, sinceyou multiply five times three and then five times forty and then add it together (in thesame 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 multiplywould 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 writtennumbers, though it is easy to see if you lay out poker chips in rows and columns. You cansee 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 doingthis in poker chips with a few sets of numbers, it is fairly easy for the imagination to seethat "a" rows of "(b+c)" is the same as "a" rows of "b" plus "a" rows of "c" and viceversa.

And seeing why "(a+b)(c+d)" is (ac + ad + bc + bd) is possible also (though somewhatmore 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 theseare all mutually exclusive segments that combine to make the total number of chips. (Seefigure.)




http://www.garlikov.com/pics/rowcol.htm









At any rate, the manipulations we learn using pencil and paper, have a rationale, but therationale is not something we generally learn, and not something that in a sense is as easyas 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 doesnot 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 asense it seems to me that is just the reverse of the truth. "Pure" numbers allow us torepresent quantities apart from what they are quantities of (so that if we know that fivesets of five are 25, we don't have to calculate separately what five sets of five tires andfive sets of five candies and five sets of nickels are); but pure numbers are more oftensimply what we can manipulate "mathematically" rather than sets of objects. It is muchmore 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 istrue 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, wemanipulate numbers, not things.

Unfortunately in real life, quantities do not conform to simple arithmetic, and so scienceis empirical rather than a priori. Velocities do not combine with each other by simpleaddition (although at relatively low velocities they seem to); forces do not combine witheach 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 youmay 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 thevolume of either of them. Figuring out the way in which various quantities of thingsrelate to each other is part of what science is about; and it is not always a very easyendeavor 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 tome, 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 ismanipulating numbers on paper or in his head. Multiplication is easily seen to becommutative (i.e., six sets of eight will equal eight sets of six) when manipulating objectsin rows and columns (because if you change your vantage point 90 degrees, your rowsand columns simply reverse, but the total quantities remain the same); whereas it is not soobvious 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 eachcontaining six candies. (Return to text.)







Footnote 23. In correspondence with me from Peabody, Paul Cobb has said he "wouldargue that mathematics at the elementary level should not involve mechanical skills eventhough 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." Idisagree.

Although children should not be taught arithmetic only mechanically, there are somemechanical skills that can be relatively easily learned by children, and that are importantor necessary for seeing more "interesting" number relationships. For example,memorizing multiplication tables is not (and should neither be seen nor used as) just anexercise to enable one to multiply like a very slow calculator. It gives facility withmultiples that can help one more readily understand the concept of division, and morereadily understand fractions and relationships between fractions --such as when seekingcommon denominators or converting between "mixed" numbers and fractions. It givesincreased 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 arenecessary 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 andsubtraction as examples of crucially important mechanical skills. Some mechanicallylearned skills simply allow you to make intellectual leaps you might not have been ableto make at all if you were not able to quickly and somewhat automatically perceiverelationships you had not become extremely familiar with or "primed for" previously bymemorization, 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; andyou 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, andtogether we have eight things; but you have four more things than I do. How many do weeach have?" We are now up to "I have a bag and you have a bag. You have five morethan I do in your bag, but if we triple what I have, I will have five more than you. Howmuch do we each have?" Children can work out these things by thinking. They don't haveto go through particular steps they are trained in. We also do number progressions wherethey 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 areinterspersed: 2, 4, 8, 16, ..., and 10, 20, 30, .... We did most of these math games in thecar 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 successfullyfigure 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.)

References

Baroody, A.J. (1990). How and when should place-value concepts and skills be taught? Journal for Research in Mathematics Education, 21(4), 281-286.

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

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: aframework 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.





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