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.
The course, however, turned out to be a surprise in fact, three surprises. First, despite his pre-course assumption that his greatest challenge would be staying awake, he found the instructor, Sonya Reynolds, a gallery owner and artist of regional renown, riveting. She was not conventionally beautiful, not in the archetypal Northern European Catherine Deneuve mode.
Her mouth was too pouty, her cheekbones overly prominent, her nose too strong. But somehow the imperfect parts were unied into a uniquely striking whole by large eyes of a deep smoky green and by a manner that was completely relaxed and naturally sensual.
There were not many men in the class, just six of the twenty-six attendees, but she had the absolute attention of all six. The second surprise was his positive reaction to the subject matter. Because it was a special interest of hers, Sonya devoted considerable time to art derived from photography photography that had been manipulated to create images that were more powerful or communicative than the originals.
The third surprise came three weeks into the twelve-week course, on the night that she was commenting enthusiastically on a contemporary artists silk-screen prints derived from solarized photographic portraits. As Gurney gazed at the prints, the idea came to him that he could take advantage of an unusual resource to which he had special access and to which he could bring a special perspective. The notion. The last thing hed expected from an artappreciation course was excitement.
Once this occurred to himthe concept of enhancing, clarifying, intensifying criminal mug shots, particularly the mug shots of murderers, in ways that would capture and convey the nature of the beast he had spent his career studying, pursuing, and outwitting it took hold, and he thought about it more often than he would have been comfortable admitting.
He was, after all, a cautious man who could see both sides of every question, the aw in every conviction, the navet in every enthusiasm.
Im leaving these here, said Madeleine Gurney in a voice that to anyone else might have sounded casual but to her husband was fraught.
He looked over his shoulder, his eyes narrowing at the sight of the small burlap sack leaning against the door. Leaving what? Tulips, said Madeleine in the same even tone. You mean bulbs? It was a silly correction, and they both knew it. It was just a way of expressing his irritation at Madeleines wanting him to do something he didnt feel like doing.
What do you want me to do with them in here? Bring them out to the garden. Help me plant them. He considered pointing out the illogic of her bringing into the den something for him to bring back out to the garden but thought better of it. As soon as I nish with this, he said a little resentfully. He realized that planting tulip bulbs on a glorious Indian-summer day in a hilltop garden overlooking a rolling panorama of crimson autumn.
He just hated being interrupted. And this reaction to interruption, he told himself, was a by-product of his greatest strength: Madeleine peered over his shoulder at the computer screen.
How can you work on something so ugly on a day like this? The ten-acre pasture was surrounded by woods of cherry, maple, and oak. The house retained its original architectural simplicity. During the year theyd owned it, the Gurneys had restored to a more appropriate appearance the previous owners unfortunate updates replacing, for example, bleak aluminum windows with wood-framed versions that possessed the divided-light style of an earlier century. They did it not out of a mania for historical authenticity but in recognition that the original aesthetics had somehow been right.
This matter of how ones home should look and feel was one of the subjects on which Madeleine and David were in complete harmony a list that, it seemed to him, had lately been shrinking. This thought had been eating like acid at his mood most of the day, activated by his wifes comment about the ugliness of the portrait he was working on. It was still at the edge of his consciousness that afternoon when, dozing in his favorite Adirondack chair after the tulip-planting activity, he became aware of Madeleines footsteps brushing toward him through the ankle-high grass.
When the footsteps stopped in front of his chair, he opened one eye. Do you think, she said in her calm, light way, its too late to. Her voice positioned the words deftly between a question and a challenge.
Madeleine was a slim, athletic forty-ve-year-old who could easily be mistaken for thirty-ve.
Her eyes were frank, steady, appraising. Her long brown hair, with the exception of a few errant strands, was pulled up under her broad-brimmed straw gardening hat.
He responded with a question from his own train of thought. Do you really think its ugly? Of course its ugly, she said without hesitation. Isnt it supposed to be? He frowned as he considered her comment. You mean the subject matter?
What else would I mean? I dont know. He shrugged. You sounded a bit contemptuous of the whole thing the execution as well as the subject matter. She didnt seem sorry. As he teetered on the edge of saying so, she changed the subject. Are you looking forward to seeing your old classmate?
Not exactly, he said, adjusting the reclining back of his chair a notch lower. Im not big on recollections of times past. Maybe hes got a murder for you to solve. Gurney looked at his wife, studied the ambiguity of her expression. You think thats what he wants? Isnt that what youre famous for?
Anger was beginning to stiffen her voice. It was something hed witnessed in her often enough in recent months that he thought he understood what it was about. They had different notions of what his retirement from the job was supposed to mean, what kind of changes it was supposed to make in their lives, and, more specically, how it was supposed to change him. Recently, too, ill feeling had been growing around his new avocation the portraits-of-murderers project that was absorbing his time.
He suspected that Madeleines negativity in this area might be partly related to Sonyas enthusiasm. Did you know hes famous, too? Your classmate. Not really. He said something on the phone about writing a book, and I checked on it briey. I wouldnt have thought he was well known. Two books, said Madeleine. Hes the director of some sort of institute in Peony, and he did a series of lectures that ran on PBS. I printed out copies of the book jackets from the Internet.
You might want to take a look at them. I assume hell tell me all there is to know about himself and his books. He doesnt sound shy. Have it your way.
I left the copies on your desk, if you change your mind. By the way, Kyle phoned earlier. He stared at her silently. I said youd get back to him. Why didnt you call me? His son didnt call often. I asked him if I should get you. He said he didnt want to disturb you, it wasnt really urgent.
Did he say anything else? She turned and walked across the thick, moist grass toward the house. When she reached the side door and put her hand on the knob, she seemed to remember something else, looked back at him, and spoke with exaggerated bafement.
According to the book jacket, your old classmate seems to be a saint, perfect in every way. A guru of good behavior.
Its hard to imagine why hed need to consult a homicide detective. A retired homicide detective, corrected Gurney. But shed already gone in and neglected to cushion the slam of the door.
It was the picture of October in a New England calendar. Gurney rose at 7: The patio and the French doors leading to it were additions hed made to the house at Madeleines urging. She was good at that sort of thing, had a sensitive eye for what was possible, what was appropriate.
It revealed a lot about her her positive instincts, her practical imagination, her unfailing taste. But when he got tangled in their areas of contention the mires and brambles of the expectations each privately cultivated he found it difcult to focus on her remarkable strengths.
He must remember to return Kyles call. He would have to wait three hours because of the time difference between Walnut Crossing and Seattle. He settled deeper into his chair, cradling his warm coffee mug in both hands.
Think of a Number
He glanced at the slim folder hed brought out with his coffee and tried to imagine the appearance of the college classmate he hadnt seen for twenty-ve years.
The photo that appeared on the book jackets that Madeleine printed out from a bookstore website refreshed his recollection not only of the face but of the personality complete. When they were undergraduates at Fordhams Rose Hill campus in the Bronx, Mark Mellery was a wild character whose spurts of humor and truth, energy and ambition were colored by something darker.
He had a tendency to walk close to the edge a sort of careening genius, simultaneously reckless and calculating, always on the brink of a downward spiral. According to his website bio, the direction of the spiral, which had taken him down rapidly in his twenties, had been reversed in his thirties by some sort of dramatic spiritual transformation. Balancing his coffee mug on the narrow wooden arm of the chair, Gurney opened the folder on his lap, extracted the e-mail hed received from Mellery a week earlier, and went over it again, line by line.
Hello, Dave: I hope you dont nd it inappropriate to be contacted by an old classmate after so much time has elapsed. One can never be sure what may be brought to mind by a voice from the past. Ive remained in touch with our shared academic past through our alumni association and have been fascinated by the news items published over the years concerning the members of our graduating class.
I was happy to note on more than one occasion your own stellar achievements and the recognition you were receiving. Then, about a year ago, I saw that youd retired from the police department and that youd moved to up here to Delaware County.
It got my attention because I happen to be located in the town of Peony just down the road apiece, as they say.
Have you seen our maths skills in science posters?
I doubt that youve heard of it, but I now run a kind of retreat house here, called the Institute for Spiritual Renewal pretty fancy-sounding, I know, but in reality quite down to earth. Although it has occurred to me many times over the years that I would enjoy seeing you again, a disturbing situation has nally given me the nudge I needed to stop thinking about it and get in touch with you.
Its a situation in which I believe that your advice would be most helpful. What Id love to do is pay you a brief visit. If you could nd it possible to spare me half an hour, Ill come to your home in Walnut Crossing or to any other location that might better suit your convenience. My recollections of our conversations in the campus center and even longer conversations in the Shamrock Bar not to mention your remarkable professional experience tell me that youre the right person to talk to about the perplexing matter before me.
Its a weird puzzle that I suspect will interest you. Your ability to put two and two together in ways that elude everyone else was always your great strength. Whenever I think of you, I always think of your perfect logic and crystal clarity qualities that I dearly need more of right now.
Ill call you within the next few days at the number that appears in the alumni directory in the hope that its correct and current. With many good memories, Mark Mellery P.
Even if you end up as mystied by my problem as I am, and have no advice to offer, it will still be a delight to see you again. The promised call had come two days later. Gurney had immediately recognized the voice, eerily unchanged except for a distinct tremor of anxiety. After some self-deprecating remarks about his failure to stay in touch, Mellery got to the point. Could he see Gurney within the next few days? The sooner the better, since the situation was urgent.
Another development had occurred. It really was impossible to discuss over the phone, as Gurney would understand when they met. There were things Mellery had to show him. No, it wasnt a matter.
No, it wasnt a legal matter, not yet, anyway. No crime had been committed, nor was one being specically threatened not that he could prove. Lord, it was so difcult to talk about it this way; it would be so much easier in person. Yes, he realized that Gurney was not in the private-investigation business. But just half an hour could he have half an hour? With the mixed feelings hed had from the beginning, Gurney agreed. His curiosity often got the better of his reticence; in this instance he was curious about the hint of hysteria lurking in the undertone of Mellerys melliuous voice.
And, of course, a puzzle to be deciphered attracted him more powerfully than he cared to admit. 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. He was a man who depended on an audience a man who was drawn up to his full height only in the nourishing light of admiration. Gurney opened the folder and glanced through the e-mail yet again. He was bothered by Mellerys depiction of their relationship.
The contact between them had been less frequent, less signicant, less friendly than Mellerys words suggested. But he got the impression that Mellery had chosen his words carefully that despite its simplicity, the note had been written and rewritten, pondered and edited and that the attery, like everything else in the letter, was purposeful.
But what was the purpose? The obvious one was to ensure Gurneys agreement to a face-to-face meeting and to engage him in the solution of whatever mystery had arisen. Beyond that, it was hard to say. The problem was clearly important to Mellery. There was also the small matter of the P. In addition to subtly challenging him with the suggestion that he might be defeated by the puzzle, whatever it was, it also appeared to obstruct an easy exit route, to vitiate any claim Gurney might be tempted to make that he was not in the private-investigation business or would not be likely to be helpful.
The thrust of its wording was to characterize any reluctance to meet as a rude dismissal of an old friend. Oh, yes, it was carefully crafted. That was something new, wasnt it? Denitely not a cornerstone quality of the old Mark Mellery. This apparent change interested Gurney. On cue, Madeleine came out through the back door and walked about two-thirds of the way to where Gurney was sitting.
Your guest has arrived, she announced atly. Where is he? In the house. He looked down. An ant was zigzagging along the arm of his chair. He sent it ying with a sharp ick of his ngernail. Ask him to come out here, he said. Its too nice to be indoors.
It is, isnt it? By the way, he looks exactly like his picture on the book jacket even more so. Even more so? Whats that supposed to mean? She was already returning to the house and did not answer.
He approached Gurney as if planning to embrace him, but something made him reconsider. My God! Mellery went on. You look the same! God, its good to see you! Great to see you looking the way you do! Davey Gurney! Still do havent changed a bit! If I didnt know you were forty-seven like me, Id say you were thirty! He clasped Gurneys hand with both of his as though it were a precious object. Driving over today, from Peony to Walnut Crossing, I was remembering how calm and collected you always were.
An emotional oasis thats what you were, an emotional oasis! And you still have that look. Davey Gurney calm, cool, and collected plus the sharpest mind in town. How have you been? Ive been fortunate, said Gurney, extricating his hand and speaking in a voice as devoid of excitement as Mellerys was full of it. 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. Speed and memorization are two directions that we urgently need to move away from, not towards.
I'm thinking of a number...
What research tells us is that students understand more complex functions when they have num- ber sense and deep understanding of numerical principles, not blind memorization or fast recall Boaler, This idea starts early in classrooms and is one we need to eradicate. The highest achievers in the world are those who focus on big ideas in mathematics, and connections between ideas.
Students develop a connected view of mathematics when they work on mathematics conceptually and blind memorization is replaced by sense making. In the UK directives have similar potential for harm. As students work on meaningful number activities they will commit math facts to heart at the same time as understanding numbers and math. They will enjoy and learn important mathematics rather than memorize, dread and fear mathematics.
This is an ideal short teaching activity that teachers can start lessons with or parents can do at home. It involves posing an abstract math problem such as 18 x 5 and asking students to solve the problem mentally. The teacher then collects the different methods and looks at why they work. For example a teacher may pose 18 x 5 and find that students solve the problem in these different ways: Students love to give their different strategies and are usually completely engaged and fascinated by the different methods that emerge.
Students learn mental math, they have opportunities to memorize math facts and they also develop conceptual understanding of numbers and of the arithmetic properties that are critical to success in algebra and beyond. Two books, one by Cathy Humphreys and Ruth Parker in press and another by Sherry Parish illustrate many different number talks to work on with secondary and elementary students, respectively.
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.
This short paper illustrates both the damage that is caused by the practices that often accompany the teaching of math facts — speed pressure, timed testing and blind memorization — as well as summarizes the research evidence of something very different — number sense.
High achieving students use number sense and it is critical that lower achieving students, instead of working on drill and memori- zation, also learn to use numbers flexibly and conceptually. Memorization and timed testing stand in the way of number sense, giving students the impression that sense making is not important. We need to ur- gently reorient our teaching of early number and number sense in our mathematics teaching in the UK and the US.
When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics.With many good memories, Mark Mellery P. The eyes were always the hardest the eyes and the mouth but they were the key.
However, they may give you some clue to whats happening, or why its happening, once I explain my problem. Not exactly, he said, adjusting the reclining back of his chair a notch lower. Theres no scientic way to do that. Even more disturbing, the scale of his ambition seems to widen as events unfold. She was already returning to the house and did not answer. The whole business has me a bit confused.