By the end of the course, students moved from being passive spectators of the past to enfranchised agents who could participate in the forms of thinking, reasoning, and engagement that are the hallmark of skilled historical cognition. For example, early in the school year, Ms.
Remember that your reader is basically ignorant, so you need to express your view as clearly as you can.
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Try to form your ideas from the beginning to a middle and then an end. In the middle, justify your view. What factors support your idea and will convince your reader? By January his responses to questions about the fall of the cotton-based economy in the South were linked to British trade policy and colonial ventures in Asia, as well as to the failure of Southern leaders to read public opinion accurately in Great Britain. Elizabeth Jensen prepares her group of eleventh graders to debate the following resolution:. Resolved: The British government possesses the legitimate authority to tax the American colonies.
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But today that voice is silent as her students take up the question of the legitimacy of British taxation in the American colonies. England says she keeps troops here for our own protection. On face value, this seems reasonable enough, but there is really no substance to their claims.
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First of all, who do they think they are protecting us from? The French?
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Quoting from our friend Mr. Maybe they need to protect us from the Spanish? Yet the same war also subdued the Spanish, so they are no real worry either. In fact, the only threat to our order is the Indians…but…we have a decent militia of our own…. So why are they putting troops here? The only possible reason is to keep us in line. With more and more troops coming over, soon every freedom we hold dear will be stripped away. The great irony is that Britain expects us to pay for these vicious troops, these British squelchers of colonial justice.
We moved here, we are paying less taxes than we did for two generations in England, and you complain? But did you know that over one-half of their war debt was caused by defending us in the French and Indian War…. Yet virtual representation makes this whining of yours an untruth. Every British citizen, whether he had a right to vote or not, is represented in Parliament. Why does this representation not extend to America?
Rebel: Okay, then what about the Intolerable Acts…denying us rights of British subjects. What about the rights we are denied?
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Loyalist: The Sons of Liberty tarred and feather people, pillaged homes— they were definitely deserving of some sort of punishment. For a moment, the room is a cacophony of charges and countercharges. The teacher, still in the corner, still with spiral notebook in lap, issues her only command of the day. Order is restored and the loyalists continue their opening argument from Wineburg and Wilson, She knows that her and year-olds cannot begin to grasp the complexities of the debates without first understanding that these disagreements were rooted in fundamentally different conceptions of human nature—a point glossed over in two paragraphs in her history textbook.
Rather than beginning the year with a unit on European discovery and exploration, as her text dictates, she begins with a conference on the nature of man. Students in her eleventh-grade history class read excerpts from the writings of philosophers Hume, Locke, Plato, and Aristotle , leaders of state and revolutionaries Jefferson, Lenin, Gandhi , and tyrants Hitler, Mussolini , presenting and advocating these views before their classmates. Six weeks later, when it is time to study the ratification of the Constitution, these now-familiar figures—Plato, Aristotle, and others—are reconvened to be courted by impassioned groups of Federalists and anti-Federalists.
These examples provide glimpses of outstanding teaching in the discipline of history. As we previously noted, this point sharply contradicts one of the popular—and dangerous—myths about teaching: teaching is a generic skill and a good teacher can teach any subject. The uniqueness of the content knowledge and pedagogical knowledge necessary to teach his-. As is the case in history, most people believe that they know what mathematics is about—computation. Most people are familiar with only the computational aspects of mathematics and so are likely to argue for its place in the school curriculum and for traditional methods of instructing children in computation.
In contrast, mathematicians see computation as merely a tool in the real stuff of mathematics, which includes problem solving, and characterizing and understanding structure and patterns. The current debate concerning what students should learn in mathematics seems to set proponents of teaching computational skills against the advocates of fostering conceptual understanding and reflects the wide range of beliefs about what aspects of mathematics are important to know.
A growing body of research provides convincing evidence that what teachers know and believe about mathematics is closely linked to their instructional decisions and actions Brown, ; National Council of Teachers of Mathematics, ; Wilson, a, b; Brophy, ; Thompson, Thus, as we examine mathematics instruction, we need to pay attention to the subject-matter knowledge of teachers, their pedagogical knowledge general and content specific , and their knowledge of children as learners of mathematics.
In this section, we examine three cases of mathematics instruction that are viewed as being close to the current vision of exemplary instruction and discuss the knowledge base on which the teacher is drawing, as well as the beliefs and goals which guide his or her instructional decisions.
For teaching multidigit multiplication, teacher-researcher Magdelene Lampert created a series of lessons in which she taught a heterogeneous group of 28 fourth-grade students. The students ranged in computational skill from beginning to learn the single-digit multiplication facts to being able to accurately solve n-digit by n-digit multiplications.
The lessons were intended to give children experiences in which the important mathematical principles of additive and multiplicative composition, associativity, commutativity, and the distributive property of multiplication over addition were all evident in the steps of the procedures used to arrive at an answer Lampert, It is clear from her description of her instruction that both her deep understanding of multiplicative structures and her knowledge of a wide range of representations and problem situations related to multiplication were brought to bear as she planned and taught these lessons.
Lampert described her role as follows:. I also taught new information in the form of symbolic structures and emphasized the connection between symbols and operations on quantities, but I made it a classroom requirement that students use their own ways of deciding whether something was mathematically reasonable in doing the work. On the part of the teacher, the principles might be known as a more formal abstract system, whereas on the part of the learners, they are known in relation to familiar experiential contexts.
But what seems most important is that teachers and students together are disposed toward a particular way of viewing and doing mathematics in the classroom. Magdelene Lampert set out to connect what students already knew about multidigit multiplication with principled conceptual knowledge. She did so in three sets of lessons.
Another set of lessons used simple stories and drawings to illustrate the ways in which large quantities could be grouped.
Finally, the third set of lessons used only numbers and arithmetic symbols to represent problems. Throughout the lessons, students were challenged to explain their answers and to rely on their arguments, rather than to rely on the teacher or book for verification of correctness. An example serves to highlight this approach; see Box 7.
They were able to talk meaningfully about place value and order of operations to give legitimacy to procedures and to reason about their outcomes, even though they did not use technical terms to do so. I took their experimentations and arguments as evidence that they had come to see mathematics as more than a set of procedures for finding answers. Clearly, her own deep understanding of mathematics comes into play as she teaches these lessons.
Helping third-grade students extend their understanding of numbers from the natural numbers to the integers is a challenge undertaken by another teacher-researcher. That is, she not only takes into account what the important mathematical ideas are, but also how children think about the particular area of mathematics on which she is focusing. She draws on both her understanding of the integers as mathematical entities subject-matter knowledge and her extensive pedagogical content knowledge specifically about integers. A wealth of possible models for negative numbers exists and she reviewed a number of them—magic peanuts, money, game scoring, a frog on a number line, buildings with floors above and below ground.
She decided to use the building model first and money later: she was acutely aware of the strengths and limitations of each.shop.reborn.mk/map37.php
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Teacher: And if I did this multiplication and found the answer, what would I know about those. Teacher: Okay, here are the jars. The stars in them will stand for butterflies. Now, it will be easier for us to count how many butterflies there are altogether, if we think of the jars in groups. Lampert then has the children explore other ways of grouping the jars, for example, into two groups of 6 jars. It is a sign that she needs to do many more activities involving different groupings.
Students continue to develop their understanding of the principles that govern multiplication and to invent computational procedures based on those principles.
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Students defend the reasonableness of their procedures by using drawings and stories. Eventually, students explore more traditional as well as alternative algorithms for two-digit multiplication, using only written symbols. She hoped that the positional aspects of the building model would help children recognize that negative numbers were not equivalent to zero, a common misconception. She was aware that the building model would be difficult to use for modeling subtraction of negative numbers.
Deborah Ball begins her work with the students, using the building model by labeling its floors. Students were presented with increasingly difficult problems. Ball then used a model of money as a second representational context for exploring negative numbers, noting that it, too, has limitations. Like Lampert, Ball wanted her students to accept the responsibility of deciding when a solution is reasonable and likely to be correct, rather than depending on text or teacher for confirmation of correctness.