# Teaching money to slow learners ppt

You're looking at the OpenBook, the full text of this publication, available for reading in full online. What would you like to do next? The preceding chapter explored implications of research on learning for general issues relevant to the design of effective learning environments. We now move to a more detailed exploration of teaching and learning in three disciplines: We chose these three areas in order to focus on the similarities and differences of disciplines that use different methods of inquiry and analysis.

A major goal of our discussion is to explore the knowledge required to teach effectively in a diversity of disciplines. We noted in Chapter 2 that expertise in particular areas involves more than a set of general problem-solving skills; it also requires well-organized knowledge of concepts and inquiry procedures. Different disciplines are organized differently and have different approaches to inquiry. For example, the evidence needed to support a set of historical claims is different from the evidence needed to prove a mathematical conjecture, and both of these differ from the evidence needed to test a scientific theory.

Discussion in Chapter 2 also differentiated between expertise in a discipline and the ability to help others learn about that discipline. Pedagogical content knowledge is different from knowledge of general teaching methods. In short, their knowledge of the discipline and their knowledge of pedagogy interact. But knowledge of the discipline structure does not in itself guide the teacher. For example, expert teachers are sensitive to those aspects of the discipline that are especially hard or easy for new students to master.

These conceptual barriers differ from discipline to discipline.

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An emphasis on interactions between disciplinary knowledge and pedagogical knowledge directly contradicts common misconceptions about what teachers need to know in order to design effective learning environments for their students. The misconceptions are that teaching consists only of a set of general methods, that a good teacher can teach any subject, or that content knowledge alone is sufficient.

Some teachers are able to teach in ways that involve a variety of disciplines. However, their ability to do so requires more than a set of general teaching skills. Consider the case of Barb Johnson, who has been a sixth-grade teacher for 12 years at Monroe Middle School.

By conventional standards Monroe is a good school. Standardized test scores are about average, class size is small, the building facilities are well maintained, the administrator is a strong instructional leader, and there is little faculty and staff turnover. What happens in her classroom that gives it the reputation of being the best of the best? During the first week of school Barb Johnson asks her sixth graders two questions: After the students list their individual questions, Barb organizes the students into small groups where they share lists and search for questions they have in common.

After much discussion each group comes up with a priority list of questions, rank-ordering the questions about themselves and those about the world. The students had the opportunity to seek out information from family members, friends, experts in various fields, on-line computer services, and books, as well as from the teacher.

Sometimes we fall short of our goal. At the end of an investigation, Barb Johnson works with the students to help them see how their investigations relate to conventional subject-matter areas.

They create a chart on which they tally experiences in language and literacy, mathematics, science, social studies and history, music, and art. Students often are surprised at how much and how varied their learning is.

It would not work to simply arm new teachers with general strategies that mirror how she teaches and encourage them to use this approach in their classrooms. Unless they have the relevant disciplinary knowledge, the teachers and the classes would quickly become lost. At the same time, disciplinary knowledge without knowledge about how students learn i. In the remainder of this chapter, we present illustrations and discussions of exemplary teaching in history, mathematics, and science. The three examples of history, mathematics, and science are designed to convey a sense of the pedagogical knowledge and content knowledge Shulman, that underlie expert teaching.

Most people have had quite similar experiences with history courses: This view of history is radically different from the way that historians see their work. Students who think that history is about facts and dates miss exciting opportunities to understand how history is a discipline that is guided by particular rules of evidence and how particular analytical skills can be relevant for understanding events in their lives see Ravitch and Finn, Unfortunately, many teachers do not present an exciting approach to history, perhaps because they, too, were taught in the dates-facts method.

In Chapter 2 , we discussed a study of experts in the field of history and learned that they regard the available evidence as more than lists of facts Wineburg, The study contrasted a group of gifted high school seniors with a group of working historians. Both groups were given a test of facts about the American Revolution taken from the chapter review section of a popular United States history textbook.

The historians who had backgrounds in American history knew most of the items, while historians whose specialties lay elsewhere knew only a third of the test facts.

Several students scored higher than some historians on the factual pretest. In addition to the test of facts, however, the historians and students were presented with a set of historical documents and asked to sort out competing claims and to formulate reasoned interpretations.

The historians excelled at this task. Most students, on the other hand, were stymied. Despite the volume of historical information the students possessed, they had little sense of how to use it productively for forming interpretations of events or for reaching conclusions.

Different views of history affect how teachers teach history. Consider the different types of feedback that Mr. Kelsey gave a student paper; see Box 7. Barnes saw the papers as an indication of the bell-shaped distribution of abilities; Ms. Kelsey saw them as representing the misconception that history is about memorizing a mass of information and recounting a series of facts. These two teachers had very different ideas about the nature of learning history.

Those ideas affected how they taught and what they wanted their students to achieve. For expert history teachers, their knowledge of the discipline and beliefs about its structure interact with their teaching strategies. Rather than simply introduce students to sets of facts to be learned, these teachers help people to understand the problematic nature of historical interpretation and analysis and to appreciate the relevance of history for their everyday lives.

One example of outstanding history teaching comes from the classroom of Bob Bain, a public school teacher in Beechwood, Ohio. Historians, he notes, are cursed with an abundance of data—the traces of the past threaten to overwhelm them unless they find some way of separating what is important from what is peripheral.

The assumptions that historians hold about significance shape how they write their histories, the data they select, and the narrative they compose, as well as the larger schemes they bring to organize and periodize the past.

Often these assumptions about historical significance remain unarticulated in the classroom. Bob Bain begins his ninth-grade high school class by having all the students create a time capsule of what they think are the most important artifacts from the past.

In this way, the students explicitly articulate their underlying assumptions of what constitutes historical significance. At first, students apply the rules rigidly and algorithmically, with little understanding that just as they made the rules, they can also change them.

But as students become more practiced in plying their judgments of significance, they come to see the rules as tools for assaying the arguments of different historians, which allows them to begin to understand why historians disagree. Leinhardt and Greeno , spent 2 years studying a highly accomplished teacher of advanced placement history in an urban high school in Pittsburgh. When the French and Indian war ended, British expected Americans to help them pay back there war debts.

If I had the choice between being loyal, or rebelling and having something to eat, I know what my choice would be. I think a lot of people also just were going with the flow, or were being pressured by the Sons of Liberty. 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. Try to form your ideas from the beginning to a middle and then an end.

What made the colonists rebel— money, propaganda, conformity? 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:.

The British government possesses the legitimate authority to tax the American colonies. 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. First of all, who do they think they are protecting us from?

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? Okay, then what about the Intolerable Acts…denying us rights of British subjects. What about the rights we are denied? 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: 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.

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: And if I did this multiplication and found the answer, what would I know about those.

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.

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.

The concept of cognitively guided instruction helps illustrate another important characteristic of effective mathematics instruction: Teachers, it is claimed, will use their knowledge to make appropriate instructional decisions to assist students to construct their mathematical knowledge.

Cognitively guided instruction is used by Annie Keith, who teaches a combination first- and second-grade class in an elementary school in Madison Wisconsin Hiebert et al. A portrait of Ms. Students spend a great deal of time discussing alternative strategies with each other, in groups, and as a whole class.

The teacher often participates in these discussions but almost never demonstrates the solution to problems. Important ideas in mathematics are developed as students explore solutions to problems, rather than being a focus of instruction per se. For example, place-value concepts are developed as students use base materials, such as base blocks and counting frames, to solve word problems involving multidigit numbers. Everyday first-grade and second-grade activities, such as sharing snacks, lunch count, and attendance, regularly serve as contexts for problem-solving tasks.

Mathematics lessons frequently make use of math centers in which the students do a variety of activities. On any given day, children at one center may solve word problems presented by the teacher while at another center children write word problems to present to the class later or play a math game.

She continually challenges her students to think and to try to make sense of what they are doing in math. She uses the activities as opportunities for her to learn what individual students know and understand about mathematics. As students work in groups to solve problems, she observes the various solutions and mentally makes notes about which students should present their work: Her knowledge of the important ideas in mathematics serves as one framework for the selection process, but her understanding of how children think about the mathematical ideas they are using also affects her decisions about who should present.

She might select a solution that is actually incorrect to be presented so that she can initiate a discussion of a common misconception. Both the presentations of solutions and the class discussions that follow provide her with information about what her students know and what problems she should use with them next.

She forms hypotheses about what her students understand and selects instructional activities based on these hypotheses. She modifies her instruction as she gathers additional information about her students and compares it with the mathematics she wants them to learn. Her approach is not a free-for-all without teacher guidance: Some attempts to revitalize mathematics instruction have emphasized the importance of modeling phenomena. Work on modeling can be done from kindergarten through twelth grade K— Modeling involves cycles of model construction, model evaluation, and model revision.

It is central to professional practice in many disciplines, such as mathematics and science, but it is largely missing from school instruction.

Modeling practices are ubiquitous and diverse, ranging from the construction of physical models, such as a planetarium or a model of the human vascular system, to the development of abstract symbol systems, exemplified by the mathematics of algebra, geometry, and calculus. The ubiquity and diversity of models in these disciplines suggest that modeling can help students develop understanding about a wide range of important ideas.

Modeling practices can and should be fostered at every age and grade level Clement, ; Hestenes, ; Lehrer and Romberg, a, b; Schauble et al.

Taking a model-based approach to a problem entails inventing or selecting a model, exploring the qualities of the model, and then applying the model to answer a question of interest.

For example, the geometry of triangles has an internal logic and also has predictive power for phenomena ranging from optics to wayfinding as in navigational systems to laying floor tile. Modeling emphasizes a need for forms of mathematics that are typically underrepresented in the standard curriculum, such as spatial visualization and geometry, data structure, measurement, and uncertainty. For example, the scientific study of animal behavior, like bird foraging, is se-.

Note, for instance, the rubber bands that mimic the connective function of ligaments and the wooden dowels that are arranged so that their translation in the vertical plane cannot exceed degrees. Though the search for function is supported by initial resemblance, what counts as resemblance typically changes as children revise their models.

For example, attempts to make models exemplify elbow motion often lead to an interest in the way muscles might be arranged from Lehrer and Schauble, a, b.

Increasingly, approaches to early mathematics teaching incorporate the premises that all learning involves extending understanding to new situations, that young children come to school with many ideas about mathematics, that knowledge relevant to a new setting is not always accessed spontaneously, and that learning can be enhanced by respecting and encouraging.

Rather than beginning mathematics instruction by focusing solely on computational algorithms, such as addition and subtraction, students are encouraged to invent their own strategies for solving problems and to discuss why those strategies work. Teachers may also explicitly prompt students to think about aspects of their everyday life that are potentially relevant for further learning. For example, everyday experiences of walking and related ideas about position and direction can serve as a springboard for developing corresponding mathematics about the structure of large-scale space, position, and direction Lehrer and Romberg, b.

Two recent examples in physics illustrate how research findings can be used to design instructional strategies that promote the sort of problem-solving behavior observed in experts. Undergraduates who had finished an introductory physics course were asked to spend a total of 10 hours, spread over several weeks, solving physics problems using a computer-based tool that constrained them to perform a conceptual analysis of the problems based on a hierarchy of principles and procedures that could be applied to solve them Dufresne et al.

This approach was motivated by research on expertise discussed in Chapter 2. The reader will recall that, when asked to state an approach to solving a problem, physicists generally discuss principles and procedures. Novices, in contrast, tend to discuss specific equations that could be used to manipulate variables given in the problem Chi et al. When compared with a group of students who solved the same problems on their own, the students who used the computer to carry out the hierarchical analyses performed noticeably better in subsequent measures of expertise.

For example, in problem solving, those who performed the hierarchical analyses outperformed those who did not, whether measured in terms of overall problem-solving performance, ability to arrive at the correct answer, or ability to apply appropriate principles to solve the problems; see Figure 7.

Furthermore, similar differences emerged in problem categorization: See Chapter 6 for an example of the type of item used in the categorization task of Figure 7. It is also worth noting that both Figures 7. In both cases, the control group made significant improvements simply as a result of practice time on task , but the experimental group showed more improvements for the same amount of training time deliberate practice.

Introductory physics courses have also been taught successfully with an approach for problem solving that begins with a qualitative hierarchical analysis of the problems Leonard et al. Undergraduate engineering students were instructed to write qualitative strategies for solving problems before attempting to solve them based on Chi et al. The strategies consisted of a coherent verbal description of how a problem could be solved and contained three components: That is, the what, why, and how of solving the problem were explicitly delineated; see Box 7.

Compared with students who took a traditional course, students in the strategy-based course performed significantly better in their ability to categorize problems according to the relevant principles that could be applied to solve them; see Figure 7. Hierarchical structures are useful strategies for helping novices both recall knowledge and solve problems.

For example, physics novices who had completed and received good grades in an introductory college physics course were trained to generate a problem analysis called a theoretical problem description Heller and Reif, The analysis consists of describing force problems in terms of concepts, principles, and heuristics. With such an approach, novices substantially improved in their ability to solve problems, even though the type of theoretical problem description used in the study was not a natural one for novices.

Novices untrained in the theoretical descriptions were generally unable to generate appropriate descriptions on their own—even given fairly routine problems. Skills, such as the ability to describe a problem in detail before attempting a solution, the ability to determine what relevant information should enter the analysis of a problem, and the ability to decide which procedures can be used to generate problem descriptions and analyses, are tacitly used by experts but rarely taught explicitly in physics courses.

Another approach helps students organize knowledge by imposing a hierarchical organization on the performance of different tasks in physics Eylon and Reif, Students who received a particular physics argument that was organized in hierarchical form performed various recall and problem-solving tasks better than subjects who received the same argument.

Similarly, students who received a hierarchical organization of problem-solving strategies performed much better than subjects who received the same strategies organized non-hierarchically. If students had simply been given problems to solve on their own an instructional practice used in all the sciences , it is highly.

Students might get stuck for minutes, or even hours, in attempting a solution to a problem and either give up or waste lots of time. In Chapter 3 , we discussed ways in which learners profit from errors and that making mistakes is not always time wasted. However, it is not efficient if a student spends most of the problem-solving time rehearsing procedures that are not optimal for promoting skilled performance, such as finding and manipulating equations to solve the problem, rather than identifying the underlying principle and procedures that apply to the problem and then constructing the specific equations needed.

In deliberate practice, a student works under a tutor human. Students enrolled in an introductory physics course were asked to write a strategy for an exam problem. Use the conservation of energy since the only nonconservative force in the system is the tension in the rope attached to the mass M and wound around the disk assuming there is no friction between the axle and the disk, and the mass M and the air , and the work done by the tension to the disk and the mass cancel each other out.

First, set up a coordinate system so the potential energy of the system at the start can be determined. There will be no kinetic energy at the start since it starts at rest. Therefore the potential energy is all the initial energy. Now set the initial energy equal to the final energy that is made up of the kinetic energy of the disk plus the mass M and any potential energy left in the system with respect to the chosen coordinate system. I would use conservation of mechanical energy to solve this problem.

The mass M has some potential energy while it is hanging there. When the block starts to accelerate downward the potential energy is transformed into rotational kinetic energy. Through deliberate practice, computer-based tutoring environments have been designed that reduce the time it takes individuals to reach real-world performance criteria from 4 years to 25 hours see Chapter 9!

Before students can really learn new scientific concepts, they often need to re-conceptualize deeply rooted misconceptions that interfere with the learning.

As reviewed above see Chapters 3 and 4 , people spend considerable time and effort constructing a view of the physical world through.

Mechanical energy is conserved even with the nonconservative tension force because the tension force is internal to the system pulley, mass, rope.

In trying to find the speed of the block I would try to find angular momentum kinetic energy, use gravity. I would also use rotational kinematics and moment of inertia around the center of mass for the disk.

There will be a torque about the center of mass due to the weight of the block, M.

The force pulling downward is mg. The moment of inertia multiplied by the angular acceleration. By plugging these values into a kinematic expression, the angular speed can be calculated. Then, the angular speed times the radius gives you the velocity of the block. The first two strategies display an excellent understanding of the principles, justification, and procedures that could be used to solve the problem the what, why, and how for solving the problem.

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The last two strategies are largely a shopping list of physics terms or equations that were covered in the course, but the students are not able to articulate why or how they apply to the problem under consideration.

Having students write strategies after modeling strategy writing for them and providing suitable scaffolding to ensure progress provides an excellent formative assessment tool for monitoring whether or not students are making the appropriate links between problem contexts, and the principles and procedures that could be applied to solve them see Leonard et al.

Starting with the anchoring intuition that a spring exerts an upward force on the book resting on it, the student might be asked if a book resting on the. The fact that the bent board looks as if it is serving the same function as the spring helps many students agree that both the spring and the board exert upward forces on the book. For a student who may not agree that the bent board exerts an upward force on the book, the instructor may ask a student to place her hand on top of a vertical spring.

She would then be asked if she experienced an upward force that resisted her push in both cases. Another effective strategy for helping students overcome persistent erroneous beliefs are interactive lecture demonstrations Sokoloff and Thornton, ; Thornton and Sokoloff, This strategy, which has been used very effectively in large introductory college physics classes, begins with an introduction to a demonstration that the instructor is about to perform, such as a collision between two air carts on an air track, one a stationary light cart, the other a heavy cart moving toward the stationary cart.

The teacher first asks the students to discuss the situation with their neighbors and then record a prediction as to whether one of the carts would exert a bigger force on the other during impact or whether the carts would exert equal forces. The vast majority of students incorrectly predict that the heavier, moving cart exerts a larger force on the lighter, stationary cart. Again, this prediction seems quite reasonable based on experience—students know that a moving Mack truck colliding with a stationary Volkswagen beetle will result in much more damage done to the Volkswagen, and this is interpreted to mean that the Mack truck must have exerted a larger force on the Volkswagen.

After the students make and record their predictions, the instructor performs the demonstration, and the students see on the screen that the force probes record forces of equal magnitude but oppositely directed during the collision. Several other situations are discussed in the same way: What if the two carts had been moving toward each other at the same speed?

What if the situation is reversed so that the heavy cart is stationary and the light cart is moving toward it? Students make predictions and then see the actual forces between the carts displayed as they collide. Both bridging and the interactive demonstration strategies have been shown to be effective at helping students permanently overcome misconceptions. This finding is a major breakthrough in teaching science, since so much research indicates that students often can parrot back correct answers on a test that might be erroneously interpreted as displaying the eradication of a misconception, but the same misconception often resurfaces when students are probed weeks or months later see Mestre, , for a review.

Minstrell uses many research-based instructional techniques e. He does this through classroom discussions in which students construct understanding by making sense of physics concepts, with Minstrell playing a coaching role.

The following quote exemplifies his innovative and effective instructional strategies Minstrell, The act of instruction can be viewed as helping the students unravel individual strands of belief, label them, and then weave them into a fabric of more complete understanding.

An important point is that later understanding can be constructed, to a considerable extent, from earlier beliefs. Sometimes new strands of belief are introduced, but rarely is an earlier belief pulled out and replaced. Rather than denying the relevancy of a belief, teachers might do better by helping students differentiate their present ideas from and integrate them into conceptual beliefs more like those of scientists.

Describing a lesson on force, Minstrell Today we are going to try to explain some rather ordinary events that you might see any day. You will find that you already have many good ideas that will help explain those events.

We will find that some of our ideas are similar to those of the scientist, but in other cases our ideas might be different. When we are finished with this unit, I expect that we will have a much clearer idea of how scientists explain those events, and I know that you will feel more comfortable about your explanations…A key idea we are going to use is the idea of force.

What does the idea of force mean to you? At some point Minstrell guides the discussion to a specific example: He asks students to individually formulate their ideas and to draw a diagram showing the major forces on the rock as arrows, with labels to denote the cause of each force. A lengthy discussion follows in which students present their views, views that contain many irrelevant e. With this approach, Minstrell has been able to identify many erroneous beliefs of students that stand in the way of conceptual understanding.

One example is the belief that only active agents e. Facets may relate to conceptual knowledge e. One of the obstacles to instructional innovation in large introductory science courses at the college level is the sheer number of students who are taught at one time. Classroom communication systems can help the instructor of a large class accomplish these objectives. One such system, called Classtalk, consists of both hardware and software that allows up to four students to share an input device e.

Answers can then be displayed anonymously in histogram. This technology has been used successfully at the University of Massachusetts-Amherst to teach physics to a range of students, from non-science majors to engineering and science majors Dufresne et al.

The technology creates an interactive learning environment in the lectures: The technology is also a natural mechanism to support formative assessment during instruction, providing both the teacher and students with feedback on how well the class is grasping the concepts under study.

The approach accommodates a wider variety of learning styles than is possible by lectures and helps to foster a community of learners focused on common objectives and goals. The examples above present some effective strategies for teaching and learning science for high school and college students. We drew some general principles of learning from these examples and stressed that the findings consistently point to the strong effect of knowledge structures on learning.

The approach stresses how discourse is a primary means for the search for knowledge and scientific sense-making. It also illustrates how scientific ideas are constructed. In this way it mirrors science, in the words of Nobel Laureate Sir Peter Medawar Like other exploratory processes, [the scientific method] can be resolved into a dialogue between fact and fancy, the actual and the possible; between what could be true and what is in fact the case.

The purpose of scientific enquiry is not to compile an inventory of factual information, nor to build up a totalitarian world picture of Natural Laws in which every event that is not compulsory is forbidden. We should think of it rather as a logically articulated structure of justifiable beliefs about a Possible World— a story which we invent and criticize and modify as we go along, so that it ends by being, as nearly as we can make it, a story about real life.

In addition, students design studies, collect information, analyze data and construct evidence, and they then debate the conclusions that they derive from their evidence. In effect, the students build and argue about theories; see Box 7. Students constructed scientific understandings through an iterative process of theory building, criticism, and refinement based on their own questions, hypotheses, and data analysis activities.

Within this structure, students explored the implications of the theories they held, examined underlying assumptions, formulated and tested hypotheses, developed evidence, negotiated conflicts in belief and evidence, argued alternative interpretations, provided warrants for conclusions, and so forth. The process as a whole provided a richer, more scientifically grounded experience than the conventional focus on textbooks or laboratory demonstrations.

The emphasis on establishing communities of scientific practice builds on the fact that robust knowledge and understandings are socially constructed through talk, activity, and interaction around meaningful problems and tools Vygotsky, The teacher guides and supports students as they explore problems and define questions that are of interest to them.

Students share the responsibility for thinking and doing: In addition, a community of practice can be a powerful context for constructing scientific meanings. Challenged by their teacher, the students set out to determine whether they actually preferred the water from the third floor or only thought they did. As a first step, the students designed and took a blind taste test of the water from fountains on all three floors of the building.

They found, to their surprise, that two-thirds of them chose the water from the first-floor fountain, even though they all said that they preferred drinking from the third-floor fountain.

The students did not believe the data. Their teacher was also suspicious of the results because she had expected no differences among the three water fountains. These beliefs and suspicions motivated students to conduct a second taste test with a larger sample drawn from the rest of the junior high. The students decided where, when, and how to run their experiment.

They discussed methodological issues: How to collect the water, how to hide the identity of the sources, and, crucially, how many fountains to include. They decided to include the same three fountains as before so that they could compare results. What do students learn from participating in a scientific sense-making community?

Individual interviews with students before and after the water taste test investigation see Box 7. In the interviews conducted in Haitian Creole , the students were asked to think aloud about two open-ended real-world problems—pollution in the Boston Harbor and a sudden illness in an elementary school.

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They worried about bias in the voting process. What if some students voted more than once? Each student in the class volunteered to organize a piece of the experiment.

About 40 students participated in the blind taste test. When they analyzed their data, they found support for their earlier results 88 percent of the junior high students thought they preferred water from the third-floor fountain, but 55 percent actually chose the water from the first floor a result of 33 percent would be chance.

Faced with this evidence, the students suspicions turned to curiosity. Why was the water from the first-floor fountain preferred? How can they determine the source of the preference?

They found that all the fountains had unacceptably high levels of bacteria.

**In praise of the slow learner**

In fact, the first-floor fountain the one most preferred had the highest bacterial count. They also found that the water from the first-floor fountain was 20 degrees Fahrenheit colder than the water from fountains on the other floors. Based on their findings, they concluded that temperature was probably a deciding factor in taste preference.

Not surprisingly, the students knew more about water pollution and aquatic ecosystems in June than they did in September. They were also able to use this knowledge generatively. One student explained how she would clean the water in Boston Harbor Rosebery et al. Chlorine and alum, you put in the water. Note that this explanation contains misconceptions. By confusing the cleaning of drinking water with the cleaning of sea water, the student suggests adding chemicals to take all microscopic life from the water good for drinking water, but bad for the ecosystem of Boston Harbor.

In September, there were three ways in which the students showed little familiarity with scientific forms of reasoning. First, the students did not understand the function of hypotheses or experiments in scientific inquiry. Ah, I could say a person, some person that gave them something…. Second, the students conceptualized evidence as information they already knew, either through personal experience or second-hand sources, rather than data produced through experimentation or observation.

In the June interviews, the students showed that they had become familiar with the function of hypotheses and experiments and with reasoning within larger explanatory frameworks. Elinor had developed a model of an integrated water system in which an action or event in one part of the system had consequences for other parts Rosebery et al. If you leave it on the ground, the water that, the earth has water underground, it will still spoil the water underground.

Or when it rains it will just take it and, when it rains, the water runs, it will take it and leave it in the river, in where the water goes in. In June, the students no longer invoked anonymous agents, but put forward chains of hypotheses to explain phenomena, such as why children were getting sick page The June interviews also showed that students had begun to develop a sense of the function and form of experimentation.

They no longer depended on personal experience as evidence, but proposed experiments to test specific hypotheses. In response to a question about sick fish, Laure clearly understands how to find a scientific answer page Teaching and learning in science have been influenced very directly by research studies on expertise see Chapter 2.

The examples discussed in this chapter focus on two areas of science teaching: Others illustrate ways to help students engage in deliberate practice see Chapter 3 and to monitor their progress. Learning the strategies for scientific thinking have another objective: Often, the barrier to achieving insights to new solutions is rooted in a fundamental misconception about the subject matter.

Another strategy involves the use of interactive lecture demonstrations to encourage students to make predictions, consider feedback, and then reconceptualize phenomena. Students learned to think, talk, and act scientifically, and their first and second languages mediated their learning in power-. Using Haitian Creole, they designed their studies, interpreted data, and argued theories; using English, they collected data from their mainstream peers, read standards to interpret their scientific test results, reported their findings, and consulted with experts at the local water treatment facility.

Outstanding teaching requires teachers to have a deep understanding of the subject matter and its structure, as well as an equally thorough understanding of the kinds of teaching activities that help students understand the subject matter in order to be capable of asking probing questions.

Numerous studies demonstrate that the curriculum and its tools, including textbooks, need to be dissected and discussed in the larger contexts and framework of a discipline. In order to be able to provide such guidance, teachers themselves need a thorough understanding of the subject domain and the epistemology that guides the discipline for history, see Wineburg and Wilson, ; for math and English, see Ball, ; Grossman et al.

The examples in this chapter illustrate the principles for the design of learning environments that were discussed in Chapter 6: They are learner centered in the sense that teachers build on the knowledge students bring to the learning situation.

They are knowledge centered in the sense that the teachers attempt to help students develop an organized understanding of important concepts in each discipline.

They are community centered in the sense that the teachers establish classroom norms that learning with understanding is valued and students feel free to explore what they do not understand. These examples illustrate the importance of pedagogical content knowledge to guide teachers. Expert teachers have a firm understanding of their respective disciplines, knowledge of the conceptual barriers that students face in learning about the discipline, and knowledge of effective strategies for working with students.

The teachers focus on understanding rather than memorization and routine procedures to follow, and they engage students in activities that help students reflect on their own learning and understanding.

The interplay between content knowledge and pedagogical knowledge illustrated in this chapter contradicts a commonly held misconception about teaching—that effective teaching consists of a set of general teaching strategies that apply to all content areas. This notion is erroneous, just as is the idea that expertise in a discipline is a general set of problem-solving skills that lack a content knowledge base to support them see Chapter 2.

The outcomes of new approaches to teaching as reflected in the results of summative assessments are encouraging. How these kinds of teaching strategies reveal themselves on typical standardized tests is another matter. In some cases there is evidence that teaching for understanding can increase scores on standardized measures e.

It is noteworthy that none of the teachers discussed in this chapter felt that he or she was finished learning. Many discussed their work as involving a lifelong and continuing struggle to understand and improve.

What opportunities do teachers have to improve their practice? First released in the Spring of , How People Learn has been expanded to show how the theories and insights from the original book can translate into actions and practice, now making a real connection between classroom activities and learning behavior. This edition includes far-reaching suggestions for research that could increase the impact that classroom teaching has on actual learning.

Like the original edition, this book offers exciting new research about the mind and the brain that provides answers to a number of compelling questions. When do infants begin to learn? How do experts learn and how is this different from non-experts? What can teachers and schools do-with curricula, classroom settings, and teaching methods--to help children learn most effectively? New evidence from many branches of science has significantly added to our understanding of what it means to know, from the neural processes that occur during learning to the influence of culture on what people see and absorb.

How People Learn examines these findings and their implications for what we teach, how we teach it, and how we assess what our children learn. The book uses exemplary teaching to illustrate how approaches based on what we now know result in in-depth learning. This new knowledge calls into question concepts and practices firmly entrenched in our current education system.

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Brain, Mind, Experience, and School: Expanded Edition Chapter: Examples in History, Mathematics, and Science. The National Academies Press. Page Share Cite. Different Views of History by Different Teachers. Studies of Outstanding History Teachers. In the end, remind your reader again about your point of view. Go back and revise and hand this in again! Wilson and Wineburg What benefits do we get out of paying taxes to the crown?

We benefit from the protection. Yes—and all the rights of an Englishman. So should all the colonies be punished for the acts of a few colonies? There were 12 jars, and each had 4 butterflies in it. And if I did this multiplication and found the answer, what would I know about those Jessica: Interactive Instruction in Large Classes. Science for All Children. Login or Register to save! How learning actually changes the physical structure of the brain.

How existing knowledge affects what people notice and how they learn. What the thought processes of experts tell us about how to teach. The amazing learning potential of infants. The relationship of classroom learning and everyday settings of community and workplace. Learning needs and opportunities for teachers. A realistic look at the role of technology in education. From Speculation to Science 3—28 II LEARNERS AND LEARNING 29—30 2 How Experts Differ from Novices 31—50 3 Learning and Transfer 51—78 4 How Children Learn 79— 5 Mind and Brain — III TEACHERS AND TEACHING — 6 The Design of Learning Environments — 7 Effective Teaching: The teacher begins with a request for an example of a basic computation.