What Is the Definition of a Mathematical Average

In mathematics and statistics, average refers to the sum of a group of values divided by n, where n is the number of values in the group. An average is also known as a mean. Like the median and the mode, the average is a measure of central tendency, meaning it reflects a typical value in a given set. Averages are used quite regularly to determine final grades over a term or semester. Averages are also used as measures of performance. For example, batting averages express how frequently a baseball player hits when they are up to bat. Gas mileage expresses how far a vehicle will typically travel on a gallon of fuel. In its most colloquial sense, average refers to whatever is considered common or typical. Mathematical Average A mathematical average is calculated by taking the sum of a group of values and dividing it by the number of values in the group. It is also known as an arithmetic mean. (Other means, such as geometric and harmonic means, are calculated using the product and reciprocals of the values rather than the sum.) With a small set of values, calculating the average takes only a few simple steps. For example, let us imagine we want to find the average age among a group of five people. Their respective ages are 12, 22, 24, 27, and 35. First, we add up these values to find their sum: 12 22 24 27 35 120 Then we take this sum and divide it by the number of values (5): 120 Ã · 5 24 The result, 24, is the average age of the five individuals. Mean, Median, and Mode The average, or mean, is not the only measure of central tendency, though it is one of the most common. The other common measures are the median and the mode. The median is the middle value in a given set, or the value that separates the higher half from the lower half. In the example above, the median age among the five individuals is 24, the value that falls between the higher half (27, 35) and the lower half (12, 22). In the case of this data set, the median and the mean are the same, but that is not always the case. For example, if the youngest individual in the group were 7 instead of the 12, the average age would be 23. However, the median would still be 24. For statisticians, the median can be a very useful measure, especially when a data set contains outliers, or values that greatly differ from the other values in the set. In the example above, all of the individuals are within 25 years of each other. But what if that were not the case? What if the oldest person were 85 instead of 35? That outlier would bring the average age up to 34, a value greater than 80 percent of the values in the set. Because of this outlier, the mathematical average is no longer a good representation of the ages in the group. The median of 24 is a much better measure. The mode is the most frequent value in a data set, or the one that is most likely to appear in a statistical sample. In the example above, there is no mode since each individual value is unique. In a larger sample of people, though, there would likely be multiple individuals of the same age, and the most common age would be the mode. Weighted Average In an ordinary average, each value in a given data set is treated equally. In other words, each value contributes as much as the others to the final average. In a weighted average, however, some values have a greater effect on the final average than others. For example, imagine a stock portfolio made up of three different stocks: Stock A, Stock B, and Stock C. Over the last year, Stock As value grew 10 percent, Stock Bs value grew 15 percent, and Stock Cs value grew 25 percent. We can calculate the average percent growth by adding up these values and dividing them by three. But that would only tell us the overall growth of the portfolio if the owner held equal amounts of Stock A, Stock B, and Stock C. Most portfolios, of course, contain a mix of different stocks, some making up a larger percentages of the portfolio than others. To find the overall growth of the portfolio, then, we need to calculate a weighted average based on how much of each stock is held in the portfolio. For the sake of example, well say that Stock A makes up 20 percent of the portfolio, Stock B makes up 10 percent, and Stock C makes up 70 percent. We weight each growth value by multiplying it by its percentage of the portfolio: Stock A 10 percent growth x 20 percent of portfolio 200Stock B 15 percent growth x 10 percent of portfolio 150Stock C 25 percent growth x 70 percent of portfolio 1750 Then we add up these weighted values and divide them by the sum of the portfolio percentage values: (200 150 1750) Ã · (20 10 70) 21 The result, 21 percent, represents the overall growth of the portfolio. Note that it is higher than the average of the three growth values alone—16.67—which makes sense given that the highest performing stock also makes up the lions share of the portfolio.

Economics and Production Possibilities Curve - 2007 Words

Principles of Miroeconomics - Final Exam Instructions Please put your name and the course information on the scantron (ECON 2302 Professor J. Bikis) Please put your DBU ID number on the scantron in the area market ID NUMBER Please write your webadvisonr id on the back of the scantron in the area marked WEBADVISOR ID. (This ID is the same ID that you use to get into your DBU email or into BlackBoard) Please select the best answer for each question given and fill in the respective answer option on the scantron for questions 1-40. Good luck! 1. Which of the following is not a positive economics statement? a. Prices of essential goods and services should be set by the government. b. Interest rates†¦show more content†¦What quantity of output will the company produce in order to maximize profit? a. 2 units b. 3 units c. 4 units d. 5 units 11. Take a look at the exhibit above. At what price will firm sell its output to maximize profit? a. $2 b. $4 c. $8 d. Any price 12. Take a look at the exhibit above. If the firm maximizes profit, what will be the firm s total profit? a. $24 b. $12 c. $8 d. $4 13. The process of negotiation between union and management to arrive at a labor contract is called a. arbitration b. mediation c. reconciliation d. collective bargaining e. impasse 14. You are stranded on an island and can either catch four fish in one hour or pick eight bananas in one hour. From this information, the opportunity cost of one fish is: a. 2 bananas. b. 1/2 bananas. c. 5 bananas. d. 2 1/2 bananas. e. zero, since you can t compare bananas and fish. 15. Whatever else you learned about profit-maximization, you should have learned this: Maximum profit is obtained at the production level where a. P = AC b. TR = TC c. MR =Show MoreRelatedProduction Possibility Curve Essay1259 Words   |  6 Pageseconomies have a production possibility curve and there any many different things that effect it. 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It has big economies such as France, England and Germany but it does not have much raw materialsRead MoreThe Production Possibilities Frontier1083 Words   |  4 Pageseconomy. The production possibilities frontier will move outward, and the economy will move closer to the production possibilities frontier. Both outcomes are related. Freer trade creates more markets for the country for its goods and services. This allows companies to produce more, but it does not directly change the production capacity of the nation. Thus, the economy moves closer to the production possibilities frontier (Rittenberg Tregarthen, 2009). The reason that the production possibilitiesRead MoreMacroeconomics : A Nation s Levels Of Unemployment1376 Words   |  6 PagesCongregado et al, 2012). The curve for production possibilities indicate that outcomes or else combinations of the productions that can be produced at levels which are available in terms of the resources. 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A PPF is show the consumers goods andRead MoreReview for Midterm I1634 Words   |  7 PagesShawn can produce more donuts in one day than Sue can produce in one day, then a. | Shawn has a comparative advantage in the production of donuts. | b. | Sue has a comparative advantage in the production of donuts. | c. | Shawn has an absolute advantage in the production of donuts. | d. | Sue has an absolute advantage in the production of donuts. | ____3.Which of the following statements is not correct? a. | Trade allows for specialization. | bRead MoreMacro Eco1739 Words   |  7 PagesEconolands Production Possibility Frontier (PPF) we first need to understand what is a PPF. A production possibilities frontier (PPF) is a graph showing the different quantities of two goods that an economy can efficiently produce with limited productive resources. Points along the curve describe the trade-off between the two goods, that is, the opportunity cost. Opportunity cost here measures how much an additional unit of one good costs in units forgone of the other good. The curve illustrates

The Amber Spyglass Chapter 1 The Enchanted Sleeper Free Essays

string(123) " take refuge in the cave, but it was rumored that she had a companion with her who was in some way dangerous and powerful\." The morning comes, the night decays, the watchmen leave their stations; The grave is burst, the spices shed, the linen wrapped up; The bones of death, the cov’ring clay, the sinews shrunk dry’d Reviving shake, inspiring move, breathing, awakening, Spring like redeemed captives when their bonds bars are burst. Let the slave grinding at the mill run out into the field, Let him look up into the heavens laugh in the bright air; Let the inchained soul, shut up in darkness and in sighing, Whose face has never seen a smile in thirty weary years, Rise and look out; his chains are loose, his dungeon doors are open; And let his wife and children return from the oppressor’s scourge. They look behind at every step believe it is a dream, Singing: â€Å"The Sun has left his blackness has found a fresher morning, And the fair Moon rejoices in the clear cloudless night; For Empire is no more, and now the Lion Wolf shall cease. We will write a custom essay sample on The Amber Spyglass Chapter 1 The Enchanted Sleeper or any similar topic only for you Order Now † from â€Å"America: A Prophecy† by William Blake O stars, isn’t it from you that the lover’s desire for the face of his beloved arises? Doesn’t his secret insight into her pure features come from the pure constellations? from â€Å"The Third Elegy† by Rainer Maria Rilke Fine vapors escape from whatever is doing the living. The night is cold and delicate and full of angels Pounding down the living. The factories are all lit up, The chime goes unheard. We are together at last, though far apart. from â€Å"The Ecclesiast† by John Ashbery Chapter 1. The Enchanted Sleeper In a valley shaded with rhododendrons, close to the snow line, where a stream milky with meltwater splashed and where doves and linnets flew among the immense pines, lay a cave, half-hidden by the crag above and the stiff heavy leaves that clustered below. The woods were full of sound: the stream between the rocks, the wind among the needles of the pine branches, the chitter of insects and the cries of small arboreal mammals, as well as the birdsong; and from time to time a stronger gust of wind would make one of the branches of a cedar or a fir move against another and groan like a cello. It was a place of brilliant sunlight, never undappled. Shafts of lemon-gold brilliance lanced down to the forest floor between bars and pools of brown-green shade; and the light was never still, never constant, because drifting mist would often float among the treetops, filtering all the sunlight to a pearly sheen and brushing every pine cone with moisture that glistened when the mist lifted. Sometimes the wetness in the clouds condensed into tiny drops half mist and half rain, which floated downward rather than fell, making a soft rustling patter among the millions of needles. There was a narrow path beside the stream, which led from a village – little more than a cluster of herdsmen’s dwellings – at the foot of the valley to a half-ruined shrine near the glacier at its head, a place where faded silken flags streamed out in the perpetual winds from the high mountains, and offerings of barley cakes and dried tea were placed by pious villagers. An odd effect of the light, the ice, and the vapor enveloped the head of the valley in perpetual rainbows. The cave lay some way above the path. Many years before, a holy man had lived there, meditating and fasting and praying, and the place was venerated for the sake of his memory. It was thirty feet or so deep, with a dry floor: an ideal den for a bear or a wolf, but the only creatures living in it for years had been birds and bats. But the form that was crouching inside the entrance, his black eyes watching this way and that, his sharp ears pricked, was neither bird nor bat. The sunlight lay heavy and rich on his lustrous golden fur, and his monkey hands turned a pine cone this way and that, snapping off the scales with sharp fingers and scratching out the sweet nuts. Behind him, just beyond the point where the sunlight reached, Mrs. Coulter was heating some water in a small pan over a naphtha stove. Her daemon uttered a warning murmur and Mrs. Coulter looked up. Coming along the forest path was a young village girl. Mrs. Coulter knew who she was: Ama had been bringing her food for some days now. Mrs. Coulter had let it be known when she first arrived that she was a holy woman engaged in meditation and prayer, and under a vow never to speak to a man. Ama was the only person whose visits she accepted. This time, though, the girl wasn’t alone. Her father was with her, and while Ama climbed up to the cave, he waited a little way off. Ama came to the cave entrance and bowed. â€Å"My father sends me with prayers for your goodwill,† she said. â€Å"Greetings, child,† said Mrs. Coulter. The girl was carrying a bundle wrapped in faded cotton, which she laid at Mrs. Coulter’s feet. Then she held out a little hunch of flowers, a dozen or so anemones bound with a cotton thread, and began to speak in a rapid, nervous voice. Mrs. Coulter understood some of the language of these mountain people, but it would never do to let them know how much. So she smiled and motioned to the girl to close her lips and to watch their two daemons. The golden monkey was holding out his little black hand, and Ama’s butterfly daemon was fluttering closer and closer until he settled on a horny forefinger. The monkey brought him slowly to his ear, and Mrs. Coulter felt a tiny stream of understanding flow into her mind, clarifying the girl’s words. The villagers were happy for a holy woman, such as herself, to take refuge in the cave, but it was rumored that she had a companion with her who was in some way dangerous and powerful. You read "The Amber Spyglass Chapter 1 The Enchanted Sleeper" in category "Essay examples" It was that which made the villagers afraid. Was this other being Mrs. Coulter’s master, or her servant? Did she mean harm? Why was she there in the first place? Were they going to stay long? Ama conveyed these questions with a thousand misgivings. A novel answer occurred to Mrs. Coulter as the daemon’s understanding filtered into hers. She could tell the truth. Not all of it, naturally, but some. She felt a little quiver of laughter at the idea, but kept it out of her voice as she explained: â€Å"Yes, there is someone else with me. But there is nothing to be afraid of. She is my daughter, and she is under a spell that made her fall asleep. We have come here to hide from the enchanter who put the spell on her, while I try to cure her and keep her from harm. Come and see her, if you like.† Ama was half-soothed by Mrs. Coulter’s soft voice, and half-afraid still; and the talk of enchanters and spells added to the awe she felt. But the golden monkey was holding her daemon so gently, and she was curious, besides, so she followed Mrs. Coulter into the cave. Her father, on the path below, took a step forward, and his crow daemon raised her wings once or twice, but he stayed where he was. Mrs. Coulter lit a candle, because the light was fading rapidly, and led Ama to the back of the cave. Ama’s eyes glittered widely in the gloom, and her hands were moving together in a repetitive gesture of finger on thumb, finger on thumb, to ward off danger by confusing the evil spirits. â€Å"You see?† said Mrs. Coulter. â€Å"She can do no harm. There’s nothing to be afraid of.† Ama looked at the figure in the sleeping bag. It was a girl older than she was, by three or four years, perhaps; and she had hair of a color Ama had never seen before – a tawny fairness like a lion’s. Her lips were pressed tightly together, and she was deeply asleep, there was no doubt about that, for her daemon lay coiled and unconscious at her throat. He had the form of some creature like a mongoose, but red-gold in color and smaller. The golden monkey was tenderly smoothing the fur between the sleeping daemon’s ears, and as Ama looked, the mongoose creature stirred uneasily and uttered a hoarse little mew. Ama’s daemon, mouse-formed, pressed himself close to Ama’s neck and peered fearfully through her hair. â€Å"So you can tell your father what you’ve seen,† Mrs. Coulter went on. â€Å"No evil spirit. Just my daughter, asleep under a spell, and in my care. But, please, Ama, tell your father that this must be a secret. No one but you two must know Lyra is here. If the enchanter knew where she was, he would seek her out and destroy her, and me, and everything nearby. So hush! Tell your father, and no one else.† She knelt beside Lyra and smoothed the damp hair back from the sleeping face before bending low to kiss her daughter’s cheek. Then she looked up with sad and loving eyes, and smiled at Ama with such brave, wise compassion that the little girl felt tears fill her gaze. Mrs. Coulter took Ama’s hand as they went back to the cave entrance, and saw the girl’s father watching anxiously from below. The woman put her hands together and bowed to him, and he responded with relief as his daughter, having bowed both to Mrs. Coulter and to the enchanted sleeper, turned and scampered down the slope in the twilight. Father and daughter bowed once more to the cave and then set off, to vanish among the gloom of the heavy rhododendrons. Mrs. Coulter turned back to the water on her stove, which was nearly at the boil. Crouching down, she crumbled some dried leaves into it, two pinches from this bag, one from that, and added three drops of a pale yellow oil. She stirred it briskly, counting in her head till five minutes had gone by. Then she took the pan off the stove and sat down to wait for the liquid to cool. Around her there lay some of the equipment from the camp by the blue lake where Sir Charles Latrom had died: a sleeping bag, a rucksack with changes of clothes and washing equipment, and so on. There was also a case of canvas with a tough wooden frame, lined with kapok, containing various instruments; and there was a pistol in a holster. The decoction cooled rapidly in the thin air, and as soon as it was at blood heat, she poured it carefully into a metal beaker and carried it to the rear of the cave. The monkey daemon dropped his pine cone and came with her. Mrs. Coulter placed the beaker carefully on a low rock and knelt beside the sleeping Lyra. The golden monkey crouched on her other side, ready to seize Pantalaimon if he woke up. Lyra’s hair was damp, and her eyes moved behind their closed lids. She was beginning to stir: Mrs. Coulter had felt her eyelashes flutter when she’d kissed her, and knew she didn’t have long before Lyra woke up altogether. She slipped a hand under the girl’s head, and with the other lifted the damp strands of hair off her forehead. Lyra’s lips parted and she moaned softly; Pantalaimon moved a little closer to her breast. The golden monkey’s eyes never left Lyra’s daemon, and his little black fingers twitched at the edge of the sleeping bag. A look from Mrs. Coulter, and he let go and moved back a hand’s breadth. The woman gently lifted her daughter so that her shoulders were off the ground and her head lolled, and then Lyra caught her breath and her eyes half-opened, fluttering, heavy. â€Å"Roger,† she murmured. â€Å"Roger†¦ where are you†¦ I can’t see†¦Ã¢â‚¬  â€Å"Shh,† her mother whispered, â€Å"shh, my darling, drink this.† Holding the beaker in Lyra’s mouth, she tilted it to let a drop moisten the girl’s lips. Lyra’s tongue sensed it and moved to lick them, and then Mrs. Coulter let a little more of the liquid trickle into Lyra’s mouth, very carefully, letting her swallow each sip before allowing her more. It took several minutes, but eventually the beaker was empty, and Mrs. Coulter laid her daughter down again. As soon as Lyra’s head lay on the ground, Pantalaimon moved back around her throat. His red-gold fur was as damp as her hair. They were deeply asleep again. The golden monkey picked his way lightly to the mouth of the cave and sat once more watching the path. Mrs. Coulter dipped a flannel in a basin of cold water and mopped Lyra’s face, and then unfastened the sleeping bag and washed Lyra’s arms and neck and shoulders, for Lyra was hot. Then her mother took a comb and gently teased out the tangles in Lyra’s hair, smoothing it back from her forehead, parting it neatly. She left the sleeping hag open so the girl could cool down, and unfolded the bundle that Ama had brought: some flat loaves of bread, a cake of compressed tea, some sticky rice wrapped in a large leaf. It was time to build the fire. The chill of the mountains was fierce at night. Working methodically, she shaved some dry tinder, set the fire, and struck a match. That was something else to think of: the matches were running out, and so was the naphtha for the stove; she must keep the fire alight day and night from now on. Her daemon was discontented. He didn’t like what she was doing here in the cave, and when he tried to express his concern, she brushed him away. He turned his back, contempt in every line of his body as he flicked the scales from his pine cone out into the dark. She took no notice, but worked steadily and skillfully to build up the fire and set the pan to heat some water for tea. Nevertheless, his skepticism affected her, and as she crumbled the dark gray tea brick into the water, she wondered what in the world she thought she was doing, and whether she had gone mad, and, over and over again, what would happen when the Church found out. The golden monkey was right. She wasn’t only hiding Lyra: she was hiding her own eyes. Out of the dark the little boy came, hopeful and frightened, whispering over and over: â€Å"Lyra, Lyra, Lyra†¦Ã¢â‚¬  Behind him there were other figures, even more shadowy than he was, even more silent. They seemed to be of the same company and of the same kind, but they had no faces that were visible and no voices that spoke; and his voice never rose above a whisper, and his face was shaded and blurred like something half-forgotten. â€Å"Lyra†¦ Lyra†¦Ã¢â‚¬  Where were they? On a great plain, where no light shone from the iron-dark sky, and where a mist obscured the horizon on every side. The ground was bare earth, beaten flat by the pressure of millions of feet, even though those feet had less weight than feathers; so it must have been time that pressed it flat, even though time had been stilled in this place; so it must have been the way things were. This was the end of all places and the last of all worlds. â€Å"Lyra†¦Ã¢â‚¬  Why were they there? They were imprisoned. Someone had committed a crime, though no one knew what it was, or who had done it, or what authority sat in judgment. Why did the little boy keep calling Lyra’s name? Hope. Who were they? Ghosts. And Lyra couldn’t touch them, no matter how she tried. Her baffled hands moved through and through, and still the little boy stood there pleading. â€Å"Roger,† she said, but her voice came out in a whisper. â€Å"Oh, Roger, where are you? What is this place?† He said, â€Å"It’s the world of the dead, Lyra, I dunno what to do, I dunno if I’m here forever, and I dunno if I done bad things or what, because I tried to be good, but I hate it, I’m scared of it all, I hate it†¦Ã¢â‚¬  And Lyra said, â€Å"I’ll get us out of here, Roger, I promise. And Will’s coming, I’m sure he is!† He didn’t understand. He spread his pale hands and shook his head. â€Å"I dunno who that is, and he won’t come here,† he said, â€Å"and if he does, he won’t know me.† â€Å"He’s coming to me,† she said â€Å"and me and Will, oh, I don’t know how Roger but I swear we’ll help. And don’t forget there’s others on our side. There’s Serafina and there’s Iorek, and they will come, the will!† â€Å"But where are you Lyra?† How to cite The Amber Spyglass Chapter 1 The Enchanted Sleeper, Essay examples

Classroom Field Study Report free essay sample

General Description Disston Elementary School, located at Knorr and Cottage Streets, in the Tacony/Wissinoming neighborhood of Philadelphia, is a four-story brick building. The school, K through 8, has a gym, auditorium, and library. There is also a computer lab, consumer education classroom, and a music room. The floors are organized by grade levels, starting with the lower grades on the first floor and working their way up the building. The ground floor, or basement, is reserved for specialty classes, such as music, etc. The school, built in the early 1900’s, has very few modern improvements, mainly because it is a historical landmark. There are no elevators or ramps making it inaccessible to those with walking disabilities or are wheelchair bound. The school, which has approximately 800 students and 40 teachers on staff, is maintained well and kept clean. The staff members I encountered at the school were well informed and professional. The neighborhood is predominately middle class, but does include some low-income families. We will write a custom essay sample on Classroom Field Study Report or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page In addition, about 15% of the students are bussed to Disston from neighborhoods that are at or below the poverty level. The cooperating teacher, Ms. Bledy, was happy to share her classroom as well as her experiences. She provided a pleasant atmosphere and gave me the opportunity to observe her seventh grade science, mathematics, and social studies classes, and fifth grade reading and English classes, which she also instructed. Her seventh grade classes consist of 29 students and her fifth grade RELA (Reading/Language Arts) class consists of 25 students. Learner Differences In the 7th grade class, there are 26 students with average or near-average ability and three students who have been identified with reading and math disabilities. These 3 students attend special classes for those subjects and return to Ms. Bledy for science and social studies. Ms. Bledy adapts the content in science and social studies according to each of the special education students’ I. E. P. (Individual Education Plan). Since all three students are at, or below a second grade reading level, Ms. Bledy uses the school library to find appropriate stories, books, or articles at each students grade level. She looks for material that is parallel, or as close as possible, to the content that is being covered in science and social studies. While lower grade level books have less detail, the teacher tries to provide the special education children with some reading material on the same or a related topic. For example, in a science class on metamorphosis, Ms. Bledy found a story called â€Å"The Very Hungry Caterpillar† by Eric Carle. This grade 1 story, through pictures and words, shows the four life stages from an egg to a butterfly (Surprisingly, the students of average ability also enjoyed this 1st grade book). Ms. Bledy also provides daily hands-on and visual learning experiences for all her students; therefore she finds it easy to adapt lessons for the special education students mainstreamed into her classroom. With the exception of the 3 special educations students, the remaining 26 students in Ms. Bledy’s math class are instructed on a 7th grade level. The teacher uses various visual, auditory, and hands-on learning techniques to adjust to each students learning style. Disston School provides a â€Å"reading cycle† for 90 minutes each day from 10:15 to 11:45 a. m. During this time, students throughout the school change classes to attend reading and language arts at their ability levels. Ms. Bledy teaches level 5 reading, writing, and English. In this class, there is a mixture of 28 students from grades four through eight. Within the groups of students I observed, there were no students with physical handicaps, nor did I observe anyone with severe emotional behavior differences. Ms. Bledy did inform me that 2 students receive the prescription drug Ritalin daily, administered either by the school nurse or a parent. There were very few behavior difficulties in Ms. Bledys classroom. She has created a structured, firm and fair atmosphere that the students find comfortable. Motivation Techniques Ms. Bledy facilitates safety, belonging, and positive self-worth to each of her students on a daily basis. The atmosphere in the class is one of calm, encouragement, respect, and unity. During the entire time I attended her classes, I saw students actively engaged in learning, any disruptions were addressed quickly, quietly and effectively. The children seemed to have little difficulty in dividing work in cooperative learning groups and they readily helped each other during class activities. I observed Ms. Bledy’s classes in the last few weeks of school, but it was apparent that the students had become accustomed to routines and had a sense of trust among them. The special education students mainstreamed into the regular education classes were not afraid or embarrassed to show their group members any lower level materials they were using. Although the students were grouped heterogeneously, Ms. Bledy is careful to place the special education students in groups with students that are more patient and have a kinder disposition. These easy-going students are more compassionate toward a special education group member who might take longer to complete an assignment. Some examples of motivation techniques I observed are: Safety: While Ms. Bledy was giving directions for a map activity in social studies class, one of her students, Brandon, rolled up a few small pieces of paper and was proceeding to put them into an empty pen tube. Just as he was about to put the pen to his lips, Ms. Bledy made eye contact with him as she continued giving directions for the activity. She walked over to Brandon and put out her hand so that he could hand over the â€Å"spitball† tube. She never stopped giving directions, but did stop this young man from throwing spitballs around the class causing a disruption. She then pointed to the behavior book and motioned for Brandon to sign it. He shrugged, but quietly walked over, found the page in the book with his name on it and wrote the date and what he did. Brandon wrote, â€Å"trying to make spit ball tube, but Ms. Bledy took it from me. This is a warning, if I do something disruptive again today I will receive a penalty. † At the beginning of the school year Ms. Bledy taught and practiced with her students the class rules and what was expected of them. She feels that if the students write down what they did and make some notation of either a consequence or some other thing that helps them improve their behavior, that there are fewer disruptions in class. Effort and Improvement: Eric is having difficulty creating a circle graph from information on a complicated bar graph. Ms. Bledy gives Eric some individual attention. She goes over to his desk and helps him create some simple circle graphs from simple bar graphs. She then has him practice a few more circle/bar graphs escalating the difficulty each time. She gives encouragement and approval when he is correct and guidance as needed. In a short time, Eric is on his own, confident that he can probably handle the more difficult work. Ms. Bledy moves on, but keeps a watchful eye out for Eric. By the end of class, Eric is successful in completing his graphs. Rewards: Ms. Bledy makes positive comments to her students regularly. She also gives students simple, but effective, rewards. One student, Melissa, not only was helpful to the teacher, but she went out of her way to help Antoine (a special education student) locate the Allied countries of WWI on a map. At the end of class, Ms. Bledy used her cell phone to call Melissa’s mother and told her what a wonderful person Melissa is. Multiple Intelligence: Like most classrooms Ms. Bledy’s class has a variety of intelligences. One of the techniques she uses to address this is by having students divide up work in cooperative learning groups according to their interests. In science class where the students were working on a project on acid rain, the students in each co-op group were required to provide research, complete an experiment, keep a daily journal of results and conclusions, make drawings and graphs of the results, and complete a cover page. Each group is permitted to divide the work according to their talents, although all group members must share in the experiment. Behavioral Learning Principles Ms. Bledy’s classroom atmosphere stresses learning, from the various information provided on classroom bulletin boards, posters, and at workstations to the organization and structure of daily activities she creates for her students. Here are some examples I observed: Fact Learning: Ms. Bledy begins each math class with a math warm-up. The students complete a few simple questions (no more than 5) in two or three minutes. These warm-ups are designed to practice various skills. Students practice basic multiplication, addition, subtraction, and division skills by playing the math game â€Å"24†. Students prepare study cards, usually for homework. They are similar to a flash card, where they put a definition on one side and the term on the other side. This is done on 35 index cards. In class, the students play a memory game either in pairs or small groups, using these cards. Reinforcement: Ms. Bledy uses positive reinforcement in her classroom. When she praises a student she uses concrete words that describe the achievement made. One student, Regina, who struggles with math but excels in social studies, was trying to figure out the average winter temperature in Celsius in the country of Germany. She knew the temperature in Fahrenheit. Regina recalled that during a math and science class, Ms. Bledy had shown them how to convert Fahrenheit into Celsius. During this social studies class, Regina applied those skills and was successful in finding the answer. When she told Ms. Bledy the answer, the teacher complimented her on her math skills. Regina beamed the rest of the afternoon and took pride in showing her classmates how to convert the temperatures. Punishment: I observed Ms. Bledy use both presentation and removal punishment techniques. She regularly removes items from the students that can cause them to be disruptive, such as when she removed Brandon’s spitball tube. But, I only saw her need to remove one student to a time-out area. Patrick refused to cooperate with his group and Ms. Bledy had him complete a writing activity (presentation punishment) on cooperation in a time-out area (removal punishment). When he completed this activity he was eager to get back to his group. Ms. Bledy allowed him to rejoin the group where he participated mannerly. Schedules of Reinforcement: Ms. Bledy uses a variable ratio as the schedule of reinforcement in her classroom. She calls on students randomly who offer answers. Although Ms. Bledy schedules particular educational trips with specific dates, she randomly assigns points that students need to earn toward that trip. At the beginning of an activity on statistics in math class, Ms. Bledy told the class that each student could earn 10 points toward Spirit Day, by completing the activity in a timely and thorough manner. One of the requirements she listed was that each group member helps each other complete the assignment. Behavior Modification: Ms. Bledy uses a daily report for students to help them improve behavior. On this report the students identify the behavior they are trying to improve, whether it is disciplinary or academic. The student presents the daily report to all teachers throughout the day. Each teacher writes whether or not the child improved that targeted behavior for that class and lists any suggestions the student can follow for further improvement. At the end of the day, the child takes the daily report home for a parental signature (This is not always a requirement. It is done on a case-by-case situation). Ms. Bledy and the student review the daily report each morning and discuss other ways the child can help him/herself. Modeling: Ms. Bledy uses modeling as part of her direct instruction at the beginning of most classes. In one instance, the students were to create a time line of their future. Ms. Bledy created a time line for herself and shared it with the class. She showed them how she started out by making a list of 10 goals for herself on paper. Then she explained how long she thought (realistically) it would take her to achieve each goal. Finally, she picked 2001 as a starting point and systematically set up her time line. The students grasped the idea and went quickly to work. Cognitive Learning Principles Ms. Bledy increases her students learning by using a number of cognitive learning principles. Concepts are learned through practice and examples, information is processed by using chunking, categorizing, etc. and her students regularly participate in meaningful activities linking new information to existing knowledge. Concept Learning: Ms. Bledy writes a list (or row) of terms that are related and adds one that is unrelated. In a science lesson on solutions she listed the terms â€Å"dissolving, hard water, soft water, bacteria, solute, solvent. † The students identified the term â€Å"bacteria† as the one that did not fit into this list. She continued by listing 3 or more groups. Information Processing: Ms. Bledy makes constant use of charts and graphs. The students created bar graphs for a math class I observed based on statistical information they collected. Students surveyed Ms. Bledy’s class as well as seven other classrooms on how they would like, or wish, to spend their summer vacations. They combined their information and converted it into percentages. Then they created bar graphs and later in the week made circle graphs. Constructivism: Ms. Bledy’s students went on to accumulate information on how they actually spent their summer vacations (meaningful activity) and created graphs using this information. As a conclusion to this assignment, Ms. Bledy had the students compare the way they actually spend vacation to how they wish their vacations would be like. Classroom Management Ms. Bledy’s classroom is well organized and well managed. She has three osters above the side blackboard that lists rules, penalties, and rewards (see attached copy of Ms. Bledy’s Discipline Plan).