# How much does single parents get paid

The ATUS, which began in , is a nationally representative telephone survey that measures the amount of time people spend doing various activities throughout the day. It is sponsored by the U. Bureau of Labor Statistics and is conducted by the U. Data collected from through include interviews with more than , respondents. Comparable time diary data are available going back as far as , allowing for an analysis of trends over a nearly year period. There is no significant gap in attitudes between mothers and fathers: With so many demands on their time, many parents wonder whether they are spending the right amount of time with their children.

Fathers are much more likely than mothers to feel this way. Analysis of time use data shows that fathers devote significantly less time than mothers to child care an average of seven hours per week for fathers, compared with 14 hours per week for mothers. Only half of fathers say the same.

## The Parenting Payment

But with these changes have come the added pressures of balancing work and family life, for mothers and fathers alike. Trends in time use going back to clearly show how the increased participation of women in the workforce has affected the amount of time mothers devote to paid work.

In , mothers spent, on average, 21 hours per week on paid work, up from eight hours in Over the same period, the total amount of time mothers spend in non-paid work has gone down somewhat. For their part, fathers now spend more time engaged in housework and child care than they did half a century ago. And the amount of time they devote to paid work has decreased slightly over that period. Fathers have by no means caught up to mothers in terms of time spent caring for children and doing household chores, but there has been some gender convergence in the way they divide their time between work and home.

In those households, on average, fathers spend more time than mothers in paid work, while mothers spend more time on child care and household chores. However, when their paid work is combined with the work they do at home, fathers and mothers are carrying an almost equal workload. Mothers give themselves somewhat higher ratings than do fathers: Working mothers give themselves slightly higher ratings than non-working mothers for the job they are doing as parents. The report is divided into two main sections. Chapter 2 looks at the challenges mothers and fathers face in attempting to balance work and family life.

Chapter 3 explores how these challenges are affecting parents—both in terms of their overall happiness and in how they evaluate the job they are doing raising their children. Section II of the report, Time Use Findings Chapters , primarily draws from time use surveys and includes public opinion questions related to time use when available. Chapter 4 provides an overview of how mothers and fathers spend their time in the workplace and at home and how they feel about their time. Chapter 5 goes into detail about the long-term trend in time use among men and women—and fathers and mothers—over the past five decades.

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. 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: 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". 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.

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. 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.

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.

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. 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 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. 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: 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. 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. 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 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. People who cannot solve this problem, generally have no trouble accounting for money, however; they do only when working on this problem.

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, 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 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 mph, so they will cover the miles, and crash, in 6 hours. The bee is constantly flying mph; so in that 6 hours he will fly 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. 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. 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, - 9,".

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.

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. 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.

Baroody 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. 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.

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 "" 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. 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 , 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. 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: But there is, or should be, more involved. Even after Chinese-speaking children have learned to read numeric numbers, such as "" 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.

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. 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. 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 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.

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. 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 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. 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. 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 This prevents one from having to do subtractions involving minuends from 11 through That in turn reminded me of two other ways to do such subtraction, avoiding subtracting from 11 through In the case of , 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 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. 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 , 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. 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. 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. 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. 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 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. 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. 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. Different color poker chips alone, as Fuson notes p. 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. 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.

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.

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