In grades , children should "understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and. The amortisation emerges from an accounting principle that implies that securities purchased at prices below face value have to be revalued upwards over. First, it is necessary to distinguish among mathematical 1) conventions, 2) algorithmic manipulations, and 3) logical/conceptual relationships, and then it is. BITCOIN ARTICLE
And they would understand in the second case that you cannot add two positive quantities together and get a smaller quantity than either. And it is not so bad if children make algorithmic errors because they have not learned or practiced the algorithm enough to remember or to be able to follow the algorithmic rules well enough to work a problem correctly; that just takes more practice. But it should be of major significance that many children cannot recognize that the procedure, the way they are doing it, yields such a bad answer, that they must be doing something wrong!
The answers Fuson details in her chart of errors of algorithmic calculation are less disturbing about children's use of algorithms than they are about children's understanding of number and quantity relationships and their understanding of what they are even trying to accomplish by using algorithms in this case, for adding and subtracting.
Students have to be taught and rehearsed to count this way, and generally they have to be told that it is a faster and easier way to count large quantities. And, of course, grouping by 10's is a prelude to understanding those aspects of arithmetic based on 10's. Many teachers teach students to count by groups and to recognize quantities by the patterns a group can make such as on numerical playing cards.
This is important. Aspects of elements 2 and 3 can be "taught" or learned at the same time. Though they are "logically" distinct; they need not be taught or learned in serial order or specifically in the order I mention them here. Many conceptually distinct ideas occur together naturally in practice. But columnar place-value is 1 not the only way to represent groups, and 2 it is an extremely difficult way for children to understand representations of groups. There are more accessible ways for children to work with representations of groups.
And I think it is easier for them to learn columnar place-value if one starts them out with more psychologically accessible group representations. Once children have gained facility with counting, and with counting by groups, especially groups of 10's and perhaps 's, and 's i. Only one needs not, and should not, talk about "representation", but merely set up some principles like "We have these three different color poker chips, white ones, blue ones, and red ones.
Whenever you have ten white ones, you can exchange them for one blue one; or any time you want to exchange a blue one for ten white ones you can do that. And any time you have ten BLUE ones, you can trade them in for one red one, or vice versa. Then you do some demonstrations, such as putting down eleven white ones and saying something like "if we exchange 10 of these white ones for a blue one, what will we have? And you can reinforce that they still make i.
In this way, they come to understand group representation by means of colored poker chips, though you do not use the word representation, since they are unlikely to understand it. Let the students get used to making i.
Ask them, for example, to show you how to make various numbers in the fewest possible poker chips -- say 30, 60, 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.
Then, when they are ready, get into some easy poker chip regroupings. Keep practicing and changing the numbers so they sometimes need regrouping and sometimes don't; but so they get better and better at doing it. They are now using the colors both representationally and quantitatively -- trading quantities for chips that represent them, and vice versa. Then introduce double digit additions and subtractions that don't require regrouping the poker chips, e. The first of these, for example is adding 4 blues and 6 whites to 2 blues and 3 whites to end up with 6 blues and 9 whites, 69; the last takes 3 blues and 5 whites away from 5 blues and 6 whites to leave 2 blues and 1 white, When they are comfortable with these, introduce double digit addition and subtraction that requires regrouping poker chips, e.
As you do all these things it is important to walk around the room watching what students are doing, and asking those who seem to be having trouble to explain what they are doing and why. In some ways, seeing how they manipulate the chips gives you some insight into their understanding or lack of it. Usually when they explain their faulty manipulations you can see what sorts of, usually conceptual, problems they are having.
And you can tell or show them something they need to know, or ask them leading questions to get them to self-correct. Sometimes they will simply make counting mistakes, however, e. That kind of mistake is not as important for teaching purposes at this point as conceptual mistakes. They tend to make fewer careless mere counting errors once they see that gives them wrong answers. After gradually taking them into problems involving greater and greater difficulty, at some point you will be able to give them something like just one red poker chip and ask them to take away 37 from it, and they will be able to figure it out and do it, and give you the answer --not because they have been shown since they will not have been shown , but because they understand.
Then, after they are comfortable and good at doing this, you can point out that when numbers are written numerically, the columns are like the different color poker chips. The first column is like white poker chips, telling you how many "ones" you have, and the second column is like blue poker chips, telling you how many 10's or chips worth ten you have This would be a good time to tell them that in fact the columns are even named like the poker chips -- the one's column, the ten's column, the hundred's column, etc.
Remember, they have learned to write numbers by rote and by practice; they should find it interesting that written numbers have these parts --i. Let them try some. Let them do additions and subtractions on paper, checking their answers and their manipulations with different color group value poker chips. Then demonstrate how adding and subtracting numbers that require regrouping on paper is just like adding and subtracting numbers that their poker chips represent that require exchanging.
You may want to stick representative poker chips above your columns on the chalk board, or have them use crayons to put the poker chip colors above their columns on their paper using, say, yellow for white if they have white paper. Show them how they can "exchange" numerals in their various columns by crossing out and replacing those they are borrowing from, carrying to, adding to, or regrouping.
This is sometimes somewhat difficult for them at first because at first they have a difficult time keeping their substitutions straight and writing them where they can notice and read them and remember what they mean. They tend to start getting scratched-out numbers and "new" numbers in a mess that is difficult to deal with.
But once they see the need to be more orderly, and once you show them some ways they can be more orderly, they tend to be able to do all right. Let them do problems on paper and check their own answers with poker chips. Give them lots of practice, and, as time goes on, make certain they can all do the algorithmic calculation fairly formally and that they can also understand what they are doing if they were to stop and think about it.
Again, the whole time you can walk around and around the room seeing who might need extra help, or what you might have to do for everyone. Doing this in this way lets you almost see what they are individually thinking and it lets you know who might be having trouble, and where, and what you might need to do to ameliorate that trouble. You may find general difficulties or you may find each child has his own peculiar difficulties, if any. For a while my children tended to forget the "one's" they already had when they regrouped; they would forget to mix the "new" one's with the "old" one's.
So, if they had 34 to start with and borrowed 10 from the thirty, they would forget about the 4 ones they already had, and subtract from 10 instead of from Children in schools using small desk spaces sometimes get their different piles of poker chips confused, since they may not put their "subtracted" chips far enough away or they may not put their "regrouped" chips far enough away from a "working" pile of chips.
Columns above one's and colors "above" white are each representations of groups of numbers, but columns are a relational property representation, whereas colors are not. Colors are a simple or inherent or immediately obvious property. Columns are relational, more complex, and less obvious.
Once color or columnar values are established, three blue chips are always thirty, but a written numeral three is not thirty unless it is in a column with only one non-decimal column to its right. Column representations of groups are more difficult to comprehend than color representations, and I suspect that is 1 because they depend on location relative to other numerals which have to be remembered to be looked for and then examined, rather than on just one inherent property, such as color or shape , and 2 because children can physically exchange "higher value" color chips for the equivalent number of lower value ones, whereas doing that is not so easy or obvious in using columns.
In regard to 1 , as anyone knows who has ever put things together from a kit, any time objects are distinctly colored and referred to in the directions by those colors, they are made easier to distinguish than when they have to be identified by size or other relative properties, which requires finding other similar objects and examining them all together to make comparisons. In regard to 2 , it is easy to physically change, say a blue chip, for ten white ones and then have, say, fourteen white ones altogether from which to subtract if you already had four one's.
But it is difficult to represent this trade with written numerals in columns, since you have to scratch stuff out and then place the new quantity in a slightly different place, and because you end up with new columns as in putting the number "14" all in the one's column, when borrowing 10 from, say 30 in the number "34", in order to subtract 8. Further, 3 I suspect there is something more "real" or simply more meaningful to a child to say "a blue chip is worth 10 white ones" than there is to say "this '1' is worth 10 of this '1' because it is over here instead of over here"; value based on place seems stranger than value based on color, or it seems somehow more arbitrary.
But regardless of WHY children can associate colors with numerical groupings more readily than they do with relative column positions, they do. I have made a narrated Power Point slide show of the use of poker chips to teach what is described above. It downloads when you click the link, and will play automatically when you open the download.
And it may be interesting to students at some later stage when they can absorb it. I have taught this to third graders, but the presentation is extremely different from the way I will write it here; and that presentation is crucial to their following the ideas and understanding them. But it is important to understand why groups need to be designated at all, and what is actually going on in assigning what has come to be known as "place-value" designation.
Groups make it easier to count large quantities; but apart from counting, it is only in writing numbers that group designations are important. Spoken numbers are the same no matter how they might be written or designated.
They can even be designated in written word form, such as "four thousand three hundred sixty five" -- as when you spell out dollar amounts in word form in writing a check. And notice, that in spoken form there are no place-values mentioned though there may seem to be. That is we say "five thousand fifty four", not "five thousand no hundred and fifty four". Starting with "zero", it is the twelfth unique number name.
Similarly "four thousand, three hundred, twenty nine" is just a unique name for a particular quantity. It could have been given a totally unique name say "gumph" just like "eleven" was, but it would be difficult to remember totally unique names for all the numbers. It just makes it easier to remember all the names by making them fit certain patterns, and we start those patterns in English at the number "thirteen" or some might consider it to be "twenty one", since the "teens" are different from the decades.
We only use the concept of represented groupings when we write numbers using numerals. What happens in writing numbers numerically is that if we are going to use ten numerals, as we do in our everyday base-ten "normal" arithmetic, and if we are going to start with 0 as the lowest single numeral, then when we get to the number "ten", we have to do something else, because we have used up all the representing symbols i.
Now we are stuck when it comes to writing the next number, which is "ten". To write a ten we need to do something else like make a different size numeral or a different color numeral or a different angled numeral, or something. On the abacus, you move all the beads on the one's row back and move forward a bead on the ten's row.
What is chosen for written numbers is to start a new column. And since the first number that needs that column in order to be written numerically is the number ten, we simply say "we will use this column to designate a ten" -- and so that you more easily recognize it is a different column, we will include something to show where the old column is that has all the numbers from zero to nine; we will put a zero in the original column.
And, to be economical, instead of using other different columns for different numbers of tens, we can just use this one column and different numerals in it to designate how many tens we are talking about, in writing any given number. Then it turns out that by changing out the numerals in the original column and the numerals in the "ten" column, we can make combinations of our ten numerals that represent each of the numbers from 0 to Now we are stuck again for a way to write one hundred.
We add another column. Representations, Conventions, Algorithmic Manipulations, and Logic Remember, all this could have been done differently. The abacus does it differently. Our poker chips did it differently. Roman numerals do it differently.
And, in a sense, computers and calculators do it differently because they use only two representations switches that are either "on" or "off" and they don't need columns of anything at all unless they have to show a written number to a human who is used to numbers written a certain way -- in columns using 10 numerals. And though we can calculate with pencil and paper using this method of representation, we can also calculate with poker chips or the abacus; and we can do multiplication and division, and other things, much quicker with a slide rule, which does not use columns to designate numbers either, or with a calculator or computer.
The written numbering system we use is merely conventional and totally arbitrary and, though it is in a sense logically structured, it could be very different and still be logically structured. Although it is useful to many people for representing numbers and calculating with numbers, it is necessary for neither.
We could represent numbers differently and do calculations quite differently. For, although the relationships between quantities is "fixed" or "determined" by logic, and although the way we manipulate various designations in order to calculate quickly and accurately is determined by logic, the way we designate those quantities in the first place is not "fixed" by logic or by reasoning alone, but is merely a matter of invented symbolism, designed in a way to be as useful as possible.
There are algorithms for multiplying and dividing on an abacus, and you can develop an algorithm for multiplying and dividing Roman numerals. But following algorithms is neither understanding the principles the algorithms are based on, nor is it a sign of understanding what one is doing mathematically.
Developing algorithms requires understanding; using them does not. But what is somewhat useful once you learn it, is not necessarily easy to learn. It is not easy for an adult to learn a new language, though most children learn their first language fairly well by a very tender age and can fairly easily use it as adults.
The use of columnar representation for groups i. And further, it is not easy to learn to manipulate written numbers in multi-step ways because often the manipulations or algorithms we are taught, though they have a complex or "deep" logical rationale, have no readily apparent basis, and it is more difficult to remember unrelated sequences the longer they are.
Most adults who can multiply using paper and pencil have no clue why you do it the way you do or why it works. Now arithmetic teachers and parents tend to confuse the teaching and learning of logical, conventional or representational, and algorithmic manipulative computational aspects of math. And sometimes they neglect to teach one aspect because they think they have taught it when they teach other aspects. That is not necessarily true. The "new math" instruction, in those cases where it failed, was an attempt to teach math logically in many cases by people who did not understand its logic while not teaching and giving sufficient practice in, many of the representational or algorithmic computational aspects of math.
The traditional approach tends to neglect logic or to assume that teaching algorithmic computations is teaching the logic of math. There are some new methods out that use certain kinds of manipulatives 22 to teach groupings, but those manipulatives aren't usually merely representational. Instead they simply present groups of, say 10's, by proportionally longer segments than things that present one's or five's; or like rolls of pennies, they actually hold things or ten things or two things, or whatever.
Students need to learn three different aspects of math; and what effectively teaches one aspect may not teach the other aspects. The three aspects are 1 mathematical conventions, 2 the logic s of mathematical ideas, and 3 mathematical algorithmic manipulations for calculating.
There is no a priori order to teaching these different aspects; whatever order is most effective with a given student or group of students is the best order. Students need to be taught the "normal", everyday conventional representations of arithmetic, and they need to be taught how to manipulate and calculate with written numbers by a variety of different means -- by calculators, by computer, by abacus, and by the society's "normal" algorithmic manipulations 23 , which in western countries are the methods of "regrouping" in addition and subtraction, multiplying multi-digit numbers in precise steps, and doing long division, etc.
Learning to use these things takes lots of repetition and practice, using games or whatever to make it as interesting as possible. But these things are generally matters of simply drill or practice on the part of children. But students should not be forced to try to make sense of these things by teachers who think that these things are matters of obvious or simple logic. These are not matters of obvious or simple logic, as I have tried to demonstrate in this paper.
Children will be swimming upstream if they are looking for logic when they are merely learning conventions or learning algorithms whose logic is far more complicated than being able to remember the steps of the algorithms, which itself is difficult enough for the children.
And any teacher who makes it look to children like conventions and algorithmic manipulations are matters of logic they need to understand, is doing them a severe disservice. On the other hand, children do need to work on the logical aspects of mathematics, some of which follow from given conventions or representations and some of which have nothing to do with any particular conventions but have to do merely with the way quantities relate to each other.
But developing children's mathematical insight and intuition requires something other than repetition, drill, or practice. Many of these things can be done simultaneously though they may not be in any way related to each other. Students can be helped to get logical insights that will stand them in good stead when they eventually get to algebra and calculus 24 , even though at a different time of the day or week they are only learning how to "borrow" and "carry" currently called "regrouping" two-column numbers.
They can learn geometrical insights in various ways, in some cases through playing miniature golf on all kinds of strange surfaces, through origami, through making periscopes or kaleidoscopes, through doing some surveying, through studying the buoyancy of different shaped objects, or however. Or they can be taught different things that might be related to each other, as the poker chip colors and the column representations of groups.
What is important is that teachers can understand which elements are conventional or conventionally representational, which elements are logical, and which elements are complexly algorithmic so that they teach these different kinds of elements, each in its own appropriate way, giving practice in those things which benefit from practice, and guiding understanding in those things which require understanding. And teachers need to understand which elements of mathematics are conventional or conventionally representational, which elements are logical, and which elements are complexly algorithmic so that they can teach those distinctions themselves when students are ready to be able to understand and assimilate them.
But if you find it meaningful and helpful and would like to contribute whatever easily affordable amount you feel it is worth, I will appreciate it. References Baroody, A. How and when should place-value concepts and skills be taught? Journal for Research in Mathematics Education, 21 4 , Cobb, Paul.
October 9. Fuson, K. Conceptual structures for multiunit numbers: implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7 4 , Jones, G. Children's understanding of place value: a framework for curriculum development and assessment. Young Children, 48 5 , Kamii, C. Young children continue to reinvent arithmetic: 2nd grade.
New York: Teachers College Press. Footnote 1. Mere repetition about conceptual matters can work in cases where intervening experiences or information have taken a student to a new level of awareness so that what is repeated to him will have "new meaning" or relevance to him that it did not before. Repetition about conceptual points without new levels of awareness will generally not be helpful. And mere repetition concerning non-conceptual matters may be helpful, as in interminably reminding a young baseball player to keep his swing level, a young boxer to keep his guard up and his feet moving, or a child learning to ride a bicycle to "keep peddling; keep peddling; PEDDLE!
Footnote 2. If you think you understand place value, then answer why columns have the names they do. That is, why is the tens column the tens column or the hundreds column the hundreds column? And, could there have been some method other than columns that would have done the same things columns do, as effectively?
If so, what, how, and why? If not, why not? In other words, why do we write numbers using columns, and why the particular columns that we use? In informal questioning, I have not met any primary grade teachers who can answer these questions or who have ever even thought about them before. Return to text. Footnote 3. How something is taught, or how the teaching or material is structured, to a particular individual and sometimes to similar groups of individuals is extremely important for how effectively or efficiently someone or everyone can learn it.
Sometimes the structure is crucial to learning it at all. A simple example first: 1 saying a phone number such as to an American as "three, two, three pause , two, five, five, five" allows him to grasp it much more readily than saying "double thirty two, triple five". It is even difficult for an American to grasp a phone number if you pause after the fourth digit instead of the third "three, two, three, two pause , five, five, five".
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.
They learned it. 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.
Footnote 4. In a small town not terribly far from Birmingham, there is a recently opened McDonald's that serves chocolate shakes which are off-white in color and which taste like not very good vanilla shakes. They are not like other McDonald's chocolate shakes. When I told the manager how the shakes tasted, her response was that the shake machine was brand new, was installed by experts, and had been certified by them the previous week --the shake machine met McDonald's exacting standards, so the shakes were the way they were supposed to be; there was nothing wrong with them.
There was no convincing her. After she returned to her office I realized, and mentioned to the sales staff, that I should have asked her to take a taste test to try to distinguish her chocolate shakes from her vanilla ones. That would show her there was no difference. The staff told me that would not work since there was a clear difference: "Our vanilla shakes taste like chalk. 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. Footnote 5. Some of these studies interpreted to show that children do not understand place-value, are, I believe, mistaken.
Jones and Thornton explain the following "place-value task": Children are asked to count 26 candies and then to place them into 6 cups of 4 candies each, with two candies remaining. When the "2" of "26" was circled and the children were asked to show it with candies, the children typically pointed to the two candies. When the "6" in "26" was circled and asked to be pointed out with candies, the children typically pointed to the 6 cups of candy. This is taken to demonstrate children do not understand place value.
I believe this demonstrates the kind of tricks similar to the following problems, which do not show lack of understanding, but show that one can be deceived into ignoring or forgetting one's understanding. 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. 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. Footnote 6. If you understand the concept of place-value, if you understand how children or anyone tend to think about new information of any sort and how easy misunderstanding is, particularly about conceptual matters , and if you watch most teachers teach about the things that involve place-value, or any other logical-conceptual aspects of math, it is not surprising that children do not understand place-value or other mathematical concepts very well and that they cannot generally do math very well.
Place-value, like many concepts, is often taught as though it were some sort of natural phenomena --as if being in the 10's column was a simple, naturally occurring, observable property, like being tall or loud or round-- instead of a logically and psychologically complex concept. What may be astonishing is that most adults can do math as well as they do it at all with as little in-depth understanding as they have.
Research on what children understand about place-value should be recognized as what children understand about place-value given how it has been taught to them, not as the limits of their possible understanding about place-value. Footnote 7. 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.
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.
Footnote 8. This system is purely a place-value system where the digit zero has its own importance. Representing a number: To write a number, the digits are placed from left to right under the places labelled as Units, Tens, Hundreds, Thousands… etc. For example, consider number The following is the way in which it is represented or written and read.
This number can be represented or written as, This number can be read as,Fifty nine crores thirty lakh seventy one thousand four hundred and twenty three. In international numbering system, millions are written after thousands while in Indian system, lakhs are written after thousands. Indian system is followed in India, Pakistan and Bangladesh. It is still called as Indian number system owing to it Union Indian sub-continent during the British rule.
Million and Billion are international units of number system. This system is based on ten to the power three It has a three digit split in the numbers. This system is commonly used in western countries. Whereas, Lakh and Crores are Indian units of number system. This system starts with ten to the power three and then is followed by two digit split after the hundredth place.
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