Thinkofa Number Malcolm E Lines Ideas, concepts and problems which challenge the mind and baffle the experts Think o. think of a number (johnny ball) Discover the world's research. The Ivy Press Limited , 10cr, 11cl, 12bl, 13bl, 13bcl, 42bl, 42bc, 42br, 43bl, 43bc, 43br. Think of a Number by John Verdon - Excerpt - Free download as PDF File .pdf), Text File .txt) or read online for free. An extraordinary fiction debut, Think of a.
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Think of a number. Add 5. Double the answer. Subtract 4. Add on your original number. Divide by 3. Add 6. Subtract your original number. What is your final. The first book in the Dave Gurney series, Think of a Number is a heart-pounding game of cat and mouse that grows relentlessly darker and more frightening. Teaching ideas. Use as a 10 minute mental starter. It will generate lots of discussion and your learners will want to know why the answer is.
At the same time, many children find it difficult to learn the written algorithms when carrying and borrowing are involved. A search for research on multi-digit addition and subtraction yields a vast number of papers that cannot be reviewed in detail. However, clear lessons have been learned from this work, which we summarize here.
These are common examples and certainly well known to primary school teachers. Many others can be described. The reason Brown and VanLehn termed these errors bugs in the algorithm is that they are not simply a result of lack of attention or a whim; they seem to be systematic rules used by the same children across examples and by quite a few children too for a different view, see Hennessy Second, the quantitative and qualitative results from research on multi-digit addition and subtraction, which suggested that children were learning rules in a meaningless fashion, reinforced the concerns about teaching for meaning in mathematics education.
Thus, much subsequent research focused on teaching children about place value more explicitly, using different sorts of concrete materials to help children understand the connection between place value and the quantities represented by digits e. The interpretation of meaning here is restricted to the analytical meaning of number. Children were taught to manipulate the materials used to represent place value in restricted ways so that they could learn how to manipulate the numbers.
Bruner is known for having asserted that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. His theory proposed that children first learn things using enactive i. The materials used in studies included base blocks in which ones are represented by small cubes, tens are represented by bars made of 10 small cubes, and hundreds by squares with 10 by 10 cubes , unifix blocks single cubes that can be attached to form blocks of tens or sticks tied in bundles of tens, which can be tied into 10 bundles to form These items provide iconic representations for children to learn about place value.
Some researchers concluded that this approach showed some degree of success Fuson and Briars ; Fuson et al. Cobb and Gravemeijer et al. At any rate, the use of manipulatives to represent ones, tens, and hundreds was an attempt to give meaning to the written digits by focusing on the analytic meaning of numbers, not to take advantage of the representational meaning of numbers by connecting them to quantities.
The relation between the numbers and the quantities they represented was not the focus of this research endeavor. In fact, the results of many subtractions where bugs in the algorithm existed were numbers which could not possibly be correct if the students thought about the quantities: how could one take 8 from and end up with ? Research on the teaching of written algorithms as well as on the loss of meaning after their correct implementation led many researchers started to query when the standard algorithms should be taught Baroody or actually whether to teach them at all Carroll Thus new themes appear in research on arithmetic in mathematics education after the research on teaching written algorithms: mental arithmetic [which is not new in the curriculum; according to Shulmann it appeared in the California State Board examination for elementary school teachers from March ] and flexibility.
While standard written procedures focus on calculation with single digits the ones, the tens, the hundreds, etc. Mental calculation strategies for multi-digit addition and subtraction have been categorized in various ways see e.
Criteria for classification have been defined as splitting up numbers into tens and ones both numbers or only one number and as rounding and compensating.
The six categories just described represent idealized types of strategies and do not reflect individual variations for solving a problem Threlfall Rathgeb-Schnierer proposed a model for describing the process of mental calculation in detail, with distinct but interrelated dimensions that have different functions and different degrees of explication.
Methods of calculation can sometimes be directly observed but do not provide information about how an answer is determined. Although the minuend and the subtrahend are written correctly, it is not obvious how he figured out the difference in each column separately, because this was not part of the observable behavior but was carried out mentally.
Regarding the column of the ones, there were many possible tools for solution that could be have led to the difference between 16 and 9, such as counting up or back, drawing on basic facts, or using adaptive strategic means e. The method of calculation by itself does not reveal the mental processes that lead to the solution of a computation. There are several possibilities: he could have estimated that the answer had to be less than 30 40 minus almost 20 has to be less than 30 , he could have checked the answer by adding and estimating 33 plus almost 20 should be about 50 , or he could have counted up from 33 and found that he got to 41 more quickly than expected.
A second dimension in mental calculation refers to cognitive elements. The tools she used for solving seem to rely on a combination of basic facts and strategic means, composing, and decomposing.
However, one cannot tell whether her solution derives from a learned procedure, from the recognition and use of number patterns and relations, or from a combination of both. In the Rathgeb-Schnierer and Green model of process of calculation, cognitive elements are defined as specific mental experiences that sustain the learning process. These can be learned procedures such as computing algorithms or recognition of number characteristics such as number patterns and relations. In reality, it is difficult to reconstruct the basic cognitive elements that lie behind an overt solution process.
In order to describe her mental calculation abilities more precisely, it would be critical to know which cognitive elements her solution entails. Cognitive elements are not sufficient to find the answer to a computation; they rely on additional tools that are used and combined in the given context to solve the problem. The specific tools for solution may involve counting, referring to basic facts, or employing adaptive strategic means Rathgeb-Schnierer and Green Strategic means are devices that modify complex problems to make them easier.
Whenever an arithmetic problem is solved, all three dimensions are involved. Michael: Because then I have [points at the number 36] — 35, and 35 is half of 70, and then here [points at the number 71] I have Michael has solved the problem mentally method for calculation by transforming the actual problem into a new one that preserves the difference between the numbers.
The transformed problem seems to be almost trivial for him because he found the answer immediately. He has used and combined tools for solution according to this specific subtraction problem; his approach to solve the computation emerged during the process and was unique for that specific situation. Michael recognized the numerical proximity of 71 to 70 and adapted this knowledge to the situation by transforming the problem.
Although the term mental arithmetic is widely adopted in research and teaching, some researchers have suggested an alternative description. Nunes et al. Each of these forms of arithmetic uses a different medium: written or oral number systems. In either form, mental manipulations of numbers are implemented based on mentally stored information about relations between numbers. The most striking findings of the studies on oral arithmetic are that 1 this practice often emerges among groups with limited or no schooling in the context of their occupations; 2 calculation is very flexible, using different moves that are appropriate for the context; 3 oral arithmetic refers to quantities, not to written symbols; and 4 oral arithmetic is accurate in the absence of higher levels of written calculation skill Gay and Cole ; Reed and Lave ; Moll et al.
Oral addition and subtraction rely on the associative property of these operations; multiplication and division rely on the distributive property of multiplication with respect to addition and of division with respect to subtraction. Multiplication often uses doubling and division uses halving when these facilitate the process. All four operations often make the use of the inverse operation during calculation.
We transcribe below two examples of calculation using oral arithmetic that illustrate the references to quantities rather than digits and the use of maneuvers which are not part of written arithmetic. Both children were in third grade.
R: I have cruzeiros the Brazilian currency at the time in my pocket. I want to download this bag of marbles.
Fluency Without Fear
You are selling the bag for 75 cruzeiros. How much money will I have left? C: You just give me the two hundred [he seems to have meant or changed his mind]. How many marbles will each one get? There are 25 left over. Fifteen each Nunes et al.
Number systems are necessary because without a number system that defines relations between numbers, arithmetic is not possible. When one uses an oral number system, we say the hundreds, then the tens, and then the ones. The direction of calculation in oral arithmetic is invariably in line with this way of enunciating numbers: hundreds, tens, and ones.
In contrast, when people write down a number, they can operate in the opposite direction and separate by ones, tens, and hundreds. In this process, number meaning is often lost, and errors such as those described by Brown and VanLehn are not necessarily recognized by the person doing the calculation. An aspect that has not been explored systematically in the research described in this and in the subsequent section is the form of presentation of the calculation tasks; this might turn out to be more important than it appears and could be a new aspect for investigation in the future.
This has led to multiple and sometimes inconsistent perspectives on the concept Star and Newton These varying definitions have resulted in different ways of operationalizing the concept see, for example, Verschaffel et al. There has been some consensus among all the definitions on the idea that flexibility in mental calculation includes two central features: the knowledge of different solution methods and the ability to adapt them appropriately when solving a problem.
However, it is exactly in this respect that crucial differences in the definitions emerge. Rechtsteiner-Merz systematically analyzed the various notions of flexibility in the literature and identified three different approaches regarding what is meant by the adaptive use of strategies and how this can be identified: 1 appropriateness of solution path and task characteristics, 2 accuracy and speed, and 3 appropriateness of cognitive elements that sustain the solution process.
Each of these is explained in turn. It is based on the assumption that there is one most suitable solution method for each specific task, and that this method is chosen consciously or unconsciously see e.
Rather than the choice of the most suitable solution or the quickest way of obtaining a solution, appropriateness is conceived as a match between the combination of strategic means and the recognition of number patterns and relations of a given problem during the computation process. All three approaches to mental flexibility can be linked to the model proposed for the process of calculation presented in Fig. The first two approaches focus predominantly on a single domain of the calculation process: either the domain methods of calculation or the domain tools for solution.
Consequently, evidence of flexibility in mental calculation can exist only if the tools for solution are linked in a dynamic way to problem characteristics, number patterns, and relationships. The paradigm is simple: in the choice condition, the participants are allowed to solve the task in whichever way they desire.
In the no-choice condition, participants are shown a demonstration of the method they should use in the task. I have no complaints. Mellery enunciated the syllables as if trying to recall the meaning of a foreign word. Its a nice place you have here.
Very nice. Madeleine has a good eye for these things. Shall we have a. Gurney motioned toward a pair of weathered Adirondack chairs facing each other between the apple tree and a birdbath. Mellery started in the direction indicated, then stopped. I had something. Could this be it? Madeleine was walking toward them from the house, holding in front of her an elegant briefcase.
Understated and expensive, it was like everything else in Mellerys appearance from the handmade but comfortably broken in and not too highly polished English shoes to the beautifully tailored but gently rumpled cashmere sport jacket a look seemingly calculated to say that here stood a man who knew how to use money without letting money use him, a man who had achieved success without worshipping it, a man to whom good fortune came naturally.
A harried look about his eyes, however, conveyed a different message. Ah, yes, thank you, said Mellery, accepting the briefcase from Madeleine with obvious relief. But where. You laid it on the coffee table.
Yes, of course. My brain is kind of scattered today. Thank you! Would you like something to drink? We have some iced tea already made. Or, if youd prefer something else. No, no, iced tea would be ne. Thank you. As Gurney observed his old classmate, it suddenly occurred to him what Madeleine had meant when she said that Mellery looked exactly like his book jacket photograph, only more so. The quality most evident in the photograph was a kind of informal perfection the illusion of a casual, amateur snapshot without the unattering shadows or awkward composition of an actual amateur snapshot.
It was exactly that sense of carefully crafted carelessness the ego-driven desire to appear ego-free that Mellery exemplied in person. As usual, Madeleines perception had been on target.
In your e-mail you mentioned a problem, said Gurney with a get-to-the-point abruptness verging on rudeness. Yes, Mellery answered, but instead of addressing it, he offered a reminiscence that seemed designed to weave another little thread of obligation into the old school tie, recounting a silly debate a classmate of theirs had gotten into with a philosophy professor. During the telling of this tale, Mellery referred to himself, Gurney, and the protagonist as the Three Musketeers of the Rose Hill campus, striving to make something sophomoric sound heroic.
Gurney found the effort embarrassing and offered his guest no response beyond an expectant stare. Well, said Mellery, turning uncomfortably to the matter at hand, Im not sure where to begin.
If you dont know where to begin your own story, thought Gurney, why the hell are you here? Mellery nally opened his briefcase, withdrew two slim softcover books, and handed them, with care, as if they were fragile, to Gurney.
They were the books described in the website printouts he had looked at earlier.
The other was called Honestly! You may not have heard of these books. They were moderately successful, but not exactly blockbusters. Mellery smiled with what looked like a well-practiced imitation of humility.
Im not suggesting you need to read them right now. He smiled again, as though this were amusing.
However, they may give you some clue to whats happening, or why its happening, once I explain my problem. The whole business has me a bit confused. And more than a bit frightened, mused Gurney. Mellery took a long breath, paused, then began his story like a man walking with fragile determination into a cold surf. I should tell you rst about the notes Ive received. He reached into his briefcase, withdrew two envelopes, opened one, took from it a sheet of white paper with handwriting on one side and a smaller envelope of the size that might be used for an RSVP.
He handed the paper to Gurney.
Think of a Number
This was the rst communication I received, about three weeks ago. Gurney took the paper and settled back in his chair to examine it, noting at once the neatness of the handwriting. The words were precisely, elegantly formed stirring a sudden recollection of Sister Mary Josephs script moving gracefully across a grammar-school blackboard.
But even stranger than the painstaking penmanship was the fact that the note had been written with a fountain pen, and in red ink. Red ink? Gurneys grandfather had had red ink. He had little round bottles of blue, green, and red ink. He remembered so little of his grandfather, but he remembered the ink.
Could one still download red ink for a fountain pen? Gurney read the note with a deepening frown, then read it again.
There was neither a salutation nor a signature. Do you believe in Fate?
I do, because I thought Id never see you again and then one day, there you were. It all came back: If someone told you to think of a number, I know what number youd think of. You dont believe me? Ill prove it to you. Think of any number up to a thousandthe rst number that comes to your mind. Picture it. Now see how well I know your secrets.
Open the little envelope. Gurney uttered a noncommittal grunt and looked inquiringly at Mellery, who had been staring at him intently as he read. Do you have any idea who sent you this? None whatever. Any suspicions? Did you play the game? The game? Clearly Mellery had not thought of it that way.
If what you mean is, did I think of a number, yes, I did.
Teaching and Learning About Whole Numbers in Primary School
Under the circumstances it would have been difcult not to. So you thought of a number? Mellery cleared his throat. The number I thought of was sixve-eight. He repeated it, articulating the digitssix, ve, eight as though they might mean something to Gurney. When he saw that they didnt, he took a nervous breath and went on. The number six fty-eight has no particular signicance to me. It just happened to be the rst number that came to mind. Ive racked my brains, trying to remember anything I might associate it with, any reason I might have picked it, but I couldnt come up with a single thing.
Its just the rst number that came to mind, he insisted with panicky earnestness. Gurney gazed at him with growing interest.
And in the smaller envelope.
Mellery handed him the other envelope that was enclosed with the note and watched closely as he opened it, extracted a piece of notepaper half the size of the rst, and read the message written in the same delicate style, the same red ink: Does it shock you that I knew you would pick ?
Who knows you that well? Send that exact amount to P. Box , Wycherly, CT Make it out to X. That was not always my name. After reading the note again, Gurney asked Mellery whether he had responded to it. I sent a check for the amount mentioned. What do you mean? Its a lot of money.
Why did you decide to send it? Because it was driving me crazy. The number how could he know? Has the check cleared? No, as a matter of fact, it hasnt, said Mellery. Ive been monitoring my account daily.
Thats why I sent a check instead of cash. I thought it might be a good idea to know something about this Arybdis person at least know where he deposited his checks. I mean, the whole tone of the thing was so unsettling. What exactly unsettled you?
The number, obviously! How could he possibly know such a thing? Good question, said Gurney. Why do you say he?
Oh, I see what you mean. I just thought. I dont know, its just what came to mind.
I suppose X. Arybdis sounded masculine for some reason. Odd sort of name, said Gurney. Does it mean anything to you? Ring any bell at all? The name meant nothing to Gurney, but it did not seem completely unfamiliar, either. Whatever it was, it was buried in a subbasement mental ling cabinet. After you sent the check, were you contacted again?
Oh, yes! I received this one about ten days ago. And this one the day after I sent you my e-mail asking if we could get together. He thrust them toward Gurney like a little boy showing his father two new bruises.
They appeared to be written by the same meticulous hand with the same pen as the pair of notes in the earlier communication, but the tone had changed. The rst was composed of eight short lines: How many bright angels can dance on a pin? How many hopes drown in a bottle of gin? Did the thought ever come. The eight lines of the second were similarly cryptic and menacing: What you took you will give when you get what you gave.
I know what you think, when you blink, where youve been, where youll be. You and I have a date, Mr. Over the next ten minutes, during which he read each note half a dozen times, Gurneys expression grew darker and Mellerys angst more obvious. What do you think? Mellery nally asked. You have a clever enemy. I mean, what do you think about the number business?
What about it? How could he know what number would come to my mind? Offhand, I would say he couldnt know. He couldnt know, but he did! I mean, thats the whole thing isnt? No one could possibly know that the number six fty-eight would be the number I would think of, but not only did he know it he knew it at least two days before I did, when he put the damn letter in the mail!
Mellery suddenly heaved himself up from his chair, pacing across the grass toward the house, then back again, running his hands through his hair. Research tells us that the best mathematics classrooms are those in which students learn number facts and number sense through engaging activities that focus on mathematical understanding rather than rote memorization.
The following five activities have been chosen to illustrate this principle; the appendix to this document provides a greater range of activities and links to other useful resources that will help stu- dents develop number sense. Each child makes a train of connecting cubes of a specified number.
Children take turns going around the circle showing their re- maining cubes. The other children work out the full number combination. For example, if I have 8 cubes in my number train I could snap it and put 3 behind my back. I would show my group the remaining 5 cubes and they should be able to say that three are missing and that 5 and 3 make 8.
How Many Are Hiding? In this activity each child has the same number of cubes and a cup. They take turns hiding some of their cubes in the cup and showing the leftovers. Example: I have 10 cubes and I decide to hide 4 in my cup. My group can see that I only have 6 cubes. Multiplication Fact Activities How Close to ?
This game is played in partners. Two children share a blank grid. The first partner rolls two number dice. The numbers that come up are the numbers the child uses to make an array on the grid. They can put the array anywhere on the grid, but the goal is to fill up the grid to get it as full as possible. After the player draws the array on the grid, she writes in the number sentence that describes the grid. The game ends when both players have rolled the dice and cannot put any more arrays on the grid.
How close to can you get? Pepperoni Pizza: In this game, children roll a dice twice. The first roll tells them how many pizzas to draw. The second roll tells them how many pepperonis to put on EACH pizza. I roll again and I get 3 so I put three pepperonis on each pizza. These usually include 2 unhelpful practices — memorization without understanding and time pressure. In our Math Cards activity we have used the structure of cards, which children like, but we have moved the emphasis to number sense and the understanding of multiplication.
The aim of the activity is to match cards with the same numerical answer, shown through different representations.
Lay all the cards down on a table and ask children to take turns picking them; pick as many as they find with the same answer shown through any representation. For example 9 and 4 can be shown with an area model, sets of objects such as dominoes, and the number sentence. When student match the cards they should explain how they know that the different cards are equivalent.
This activity encourages an understanding of multiplication as well as rehearsal of math facts. A full set of cards is given in Appendix A. Conclusion: Knowledge is Power The activities given above are illustrations of games and tasks in which students learn math facts at the same time as working on something they enjoy, rather than something they fear. The different activities also focus on the understanding of addition and multiplication, rather than blind memorization and this is critically important.
Appendix A presents other suggested activities and references. As educators we all share the goal of encouraging powerful mathematics learners who think carefully about mathematics as well as use numbers with fluency. But teachers and curriculum writers are often unable to access important research and this has meant that unproductive and counter-productive classroom practices continue.He was a natural actor, undisputed star of the college dramatic society a young man who, however full of life he might be at the Shamrock Bar, was doubly alive on the stage.
As students work on meaningful number activities they will commit math facts to heart at the same time as understanding numbers and math. He couldnt know, but he did!
For example, if I have 8 cubes in my number train I could snap it and put 3 behind my back. This book is a work of ction. Some researchers concluded that this approach showed some degree of success Fuson and Briars ; Fuson et al.
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