447 matches found for How People Learn Brain,Mind,Experience,and School Expanded Edition. in 7 Effective Teaching: Examples in History, Mathematics, and Science
Select a page to see where your word(s) or phrase(s) are located in the OpenBook. Excerpts from the chapter provide context.
In the middle of
page 155...
... Effective Teaching: Examples in History, Mathematics, and Science...
At the bottom of
page 155...
... 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: history, mathematics, and science. We chose these three areas in order to focus on the similarities and differences of ... 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....
At the bottom of
page 155...
... 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 ... 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 ... and the ability to help others learn about that discipline. To use Shulman’s (1987) language, effective teachers need pedagogical content knowledge ...
At the bottom of
page 155...
... Pedagogical content knowledge is different from knowledge of general teaching methods. Expert teachers know the structure of their disciplines, and this knowledge provides them with cognitive roadmaps that guide the assignments they give students, the assessments they use to gauge students’ ... , and the questions they ask in the give and take of classroom life. 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 ...
In the middle of
page 156...
... This means that new teachers must develop the ability to “understand in a pedagogically reflective way; they must not only know their own way around a discipline, but must know the ‘conceptual barriers’ ... to hinder others” (McDonald and Naso, 1986:8). These conceptual barriers differ from discipline to discipline....
In the middle of
page 156...
... 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 ...
At the bottom of
page 156...
... 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 ... leader, and there is little faculty and staff turnover. However, every year parents sending their fifth-grade students from the local elementary schools to Monroe jockey to get their children assigned to Barb Johnson’s classes. What happens in her classroom that gives it the reputation of being ...
At the bottom of
page 156...
... During the first week of school Barb Johnson asks her sixth graders two questions: “What questions do you have about yourself?” and “What questions do you have ...
At the bottom of
page 156...
... Back together in a whole group session, Barb Johnson solicits the groups’ priorities and works toward consensus for the class’s combined lists of questions. These questions become the basis for guiding the curriculum in Barb’s ... . One question, “Will I live to be 100 years old?” spawned educational investigations into genetics, family and oral history, actuarial science, statistics and probability, heart disease, cancer, and hypertension. The students had the opportunity to seek out ... from family members, friends, experts in various fields, on-line computer services, and books, as well as from the teacher. She describes what they had to do as becoming part of a “learning community.” According to Barb ...
In the middle of
page 157...
... and start off on a learning journey. Sometimes we fall short of our goal. Sometimes we reach our goal, but most times we exceed these goals—we ...
In the middle of
page 157...
... 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....
In the middle of
page 157...
... onto important principles of relevant disciplines. 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 ... become lost. At the same time, disciplinary knowledge without knowledge about how students learn (i.e., principles consistent with developmental and learning psychology) and how to lead the processes of learning (i.e., pedagogical knowledge) would not yield the kind of learning seen in Barb ... ’s classes (Anderson and Smith, 1987)....
At the bottom of
page 157...
... 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 (...
At the bottom of
page 157...
... Most people have had quite similar experiences with history courses: they learned the facts and dates that the teacher and the text deemed relevant. This view of history is radically different from the way that historians see their work. Students who think that history is ... 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, 1987). Unfortunately, many teachers ...
In the middle of
page 158...
... history and learned that they regard the available evidence as more than lists of facts (Wineburg, 1991). 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 ...
At the bottom of
page 158...
... Different views of history affect how teachers teach history. For example, Wilson and Wineburg (1993) asked two teachers of American history to read a set of student essays on the causes of the American Revolution not as an unbiased or ... and definitive accounts of people and events, but to develop plans for the students’ “remediation or enrichment.” Teachers were provided with a set of essays on the ... of the American Revolution,” written by eleventh-graders for a timed, 45-minute test. Consider the different types of feedback that Mr. Barnes and Ms. Kelsey gave a student paper; see Box 7.1....
At the bottom of
page 158...
... 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 ...
In the middle of
page 159...
... Studies of Outstanding History Teachers...
In the middle of
page 159...
... 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 ... help people to understand the problematic nature of historical interpretation and analysis and to appreciate the relevance of history for their everyday lives....
At the bottom of
page 159...
... 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 ...
At the bottom of
page 159...
... 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. The students’ task, ...
At the bottom of
page 159...
... 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 ... to see the rules as tools for assaying the arguments of different historians, which allows them to begin to understand why historians disagree. In this instance, the students’ growing ability to understand the interpretative nature of history is aided by their ... ’s deep understanding of a fundamental principle of the discipline....
At the bottom of
page 159...
... Leinhardt and Greeno (1991, 1994) spent 2 years studying a highly accomplished teacher of advanced placement history in an urban high school in Pittsburgh. The teacher, Ms. Sterling, a veteran of over 20 years, began her school year by having her students ponder the meaning of the ...
In the middle of
page 160...
... When the French and Indian war ended, British expected Americans to help them pay back there war debts. That would be a reasonable request if the war was fought for the ... rebellion another factor was when parliament decided to forbid the colonial government to make any more money, Specie became scarcer than ever, and a lot of merchants were pushed into a “two way squeeze” and faced bankruptcy. If I had the choice between being loyal, or rebelling and ... something to eat, I know what my choice would be. The colonist who were really loyal never did rebel, and 1/3 support the revolution....
At the bottom of
page 160...
... The main thing that turned most people was the amount of propaganda, speeches from people like Patrick Henry, and organizations like the “Association.” After the Boston Massacre and the issuing of the Intolerable acts, people were convinced there was ... . Merchants who didn’t go along with boycotts often became the victims of mob violence. Overall though, people were sick of getting overtaxed and walked on and decided let’s do something about it....
At the bottom of
page 160...
... ‘write history’ and the artifacts that are produced as part of ordinary experience?” The goal of this extended exercise is to help students understand history as ... evidentiary form of knowledge, not as clusters of fixed names and dates....
At the bottom of
page 160...
... 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. Sterling asked her students a question about the Constitutional Convention and “what were men able to do.” Paul took the ...
In the middle of
page 161...
... —note spelling and grammar corrections...
In the middle of
page 161...
... —The greatest strength of this essay is its outstanding effort to grapple thoughtfully with the question, why did the colonists rebel? Keep thinking personally, “What if I were here?” It is ...
In the middle of
page 161...
... 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....
In the middle of
page 161...
... In the beginning, tell what side you’re on: What made the colonists rebel— money, propaganda, conformity?...
In the middle of
page 161...
... In the middle, justify your view. What factors support your idea and will convince your reader?...
In the middle of
page 161...
... In the end, remind your reader again about your point of view....
In the middle of
page 161...
... Go back and revise and hand this in again!...
At the bottom of
page 161...
... SOURCE: Wilson and Wineburg (1993:Fig. 1). Reprinted by permission....
At the bottom of
page 161...
... began to catch on. 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. Ms. Sterling’s own ... of history allowed her to create a classroom in which students not only mastered concepts and facts, but also used them in authentic ways to craft historical explanations....
At the top of
page 162...
... their desks into three groups—on the left of the room a group of “rebels,” on the right, a group of “loyalists,” and in the front, a group of “judges.” Off to the side with a spiral notebook on her lap sits Jensen, a short woman in her late 30s with a ...
At the top of
page 162...
... The rebels’ first speaker, a 16-year-old girl with a Grateful Dead T-shirt and one dangling earring, takes a paper from her notebook and begins:...
In the middle of
page 162...
... 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. ... …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 ...
In the middle of
page 162...
... We moved here, we are paying less taxes than we did for two generations in England, and you complain? Let’s look at why we are being taxed— the main reason is probably because England has a debt of £140,000,000. … ... our money simply because they have the power over us. But did you know that over one-half of their war debt was caused by defending us in the French and Indian War…. Taxation without representation isn’t fair. Indeed, it’s tyranny. Yet virtual representation makes this whining of ...
At the bottom of
page 162...
... Loyalist: Yes—and all the rights of an Englishman....
At the bottom of
page 162...
... Loyalist: The Sons of Liberty tarred and feather people, pillaged homes— they were definitely deserving of some sort of punishment....
In the middle of
page 163...
... For a moment, the room is a cacophony of charges and countercharges. “It’s the same as in Birmingham,” shouts a loyalist. A rebel snorts disparagingly, “Virtual representation is ... with horn-rimmed glasses, bangs his gavel to no avail. The teacher, still in the corner, still with spiral notebook in lap, issues her only command of the day. “Hold still!” she thunders. Order is restored and the loyalists continue their opening argument (from Wineburg and Wilson, ...
In the middle of
page 163...
... Another example of Elizabeth Jensen’s teaching involves her efforts to help her high school students understand the debates between Federalists and anti-Federalists. She knows that her 15- and 16-year-olds cannot begin to grasp the ...
At the bottom of
page 163...
... These examples provide glimpses of outstanding teaching in the discipline of history. The examples do not come from “gifted teachers” who know how to teach anything: they ... , instead, that expert teachers have a deep understanding of the structure and epistemologies of their disciplines, combined with knowledge of the kinds of teaching activities that will help students come to understand the ... for themselves. 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. Numerous studies demonstrate that any ... —including a textbook—is mediated by a teacher’s understanding of the subject domain (for history, see Wineburg and Wilson, 1988; for math, see Ball, 1993; for English, see Grossman et al., 1989). The uniqueness of the content knowledge and pedagogical knowledge ...
In the middle of
page 164...
... tory becomes clearer as one explores outstanding teaching in other disciplines....
At the bottom of
page 164...
... 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 ...
At the bottom of
page 164...
... Teachers’ ideas about mathematics, mathematics teaching, and mathematics learning directly influence their notions about what to teach and how to teach it—an interdependence of beliefs and knowledge about pedagogy and subject matter (e.g., Gamoran, 1994; Stein et al., 1990). It ... that teachers’ goals for instruction are, to a large extent, a reflection of what they think is important in mathematics and how they think students best learn it. Thus, as we examine mathematics instruction, we need to pay attention to the subject-matter knowledge of ... , their pedagogical knowledge (general and content specific), and their knowledge of children as learners of mathematics. Paying attention to these domains of knowledge also leads us to examine teachers’ goals ...
At the bottom of
page 164...
... If students in mathematics classes are to learn mathematics with understanding—a goal that is accepted by almost everyone in the current debate over the role of computational skills in mathematics classrooms—then ... is important to examine examples of teaching for understanding and to analyze the roles of the teacher and the knowledge that underlies the teacher’s enactments of those roles. In this section, we examine three cases of mathematics instruction that ... 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....
In the middle of
page 165...
... 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 ... the procedures used to arrive at an answer (Lampert, 1986:316). 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 ... as she planned and taught these lessons. It is also clear that her goals for the lessons included not only those related to students’ understanding of mathematics, but also those related to students’ development as independent, thoughtful problem solvers. Lampert (1986:339) described her ...
At the bottom of
page 165...
...; ideas about how to solve or analyze problems into the public forum of the classroom, to referee arguments about whether those ideas were reasonable, and to sanction students’ intuitive use of mathematical principles as legitimate. I also taught new information in the form of symbolic structures ... 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 ... in this way, it is difficult to separate instruction in mathematics content from building a culture of sense-making in the classroom, wherein teacher and students have a view of themselves as responsible for ascertaining the legitimacy of procedures by reference to known mathematical principles. On the ... , 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....
At the bottom of
page 165...
..., such as “Using only two kinds of coins, make $1.00 using 19 coins,” which encouraged children to draw on their familiarity with coins and mathematical principles that coin trading requires. Another set of lessons used simple stories and drawings to illustrate the ways in which large ...
In the middle of
page 166...
... for easier counting. 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, ...
In the middle of
page 166...
... principled knowledge that was tied to the language of groups to explain what they were seeing. 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 ... their experimentations and arguments as evidence that they had come to see mathematics as more than a set of procedures for finding answers....
In the middle of
page 166...
... Clearly, her own deep understanding of mathematics comes into play as she teaches these lessons. It is worth noting that her goal of helping students see what is mathematically ... shapes the way in which she designs lessons to develop students’ understanding of two-digit multiplication....
At the bottom of
page 166...
... Understanding Negative Numbers...
At the bottom of
page 166...
... Helping third-grade students extend their understanding of numbers from the natural numbers to the integers is a challenge undertaken by another teacher-researcher. Deborah Ball’s work provides ... snapshot of teaching that draws on extensive subject content and pedagogical content knowledge. Her goals in instruction include “developing a practice that respects the integrity both of mathematics as a ... and of children as mathematical thinkers” (Ball, 1993). That is, she not only takes into account what the important mathematical ideas are, but ... 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. ... Lampert, Ball’s goals go beyond the boundaries of what is typically considered mathematics and include developing a culture in which students conjecture, experiment, build arguments, and frame and solve problems—the work of mathematicians....
At the bottom of
page 166...
... Deborah Ball’s description of work highlights the importance and difficulty of figuring out powerful and effective ways to represent key mathematical ideas to children (see Ball, 1993). A wealth of possible models for negative numbers exists and she ... 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...
In the middle of
page 167...
... BOX 7.2 How Many Altogether?...
In the middle of
page 167...
... Jessica: There were 12 jars, and each had 4 butterflies in it....
In the middle of
page 167...
... Teacher: And if I did this multiplication and found the answer, what would I know about those...
In the middle of
page 167...
... The teacher and students next illustrate Jessica’s story and construct a procedure for counting the butterflies....
At the bottom of
page 167...
... 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. And as usual, ...
At the bottom of
page 167...
... The lesson progresses as the teacher and students construct a pictorial representation of grouping 10 sets of four butterflies and having 2 jars not in the group; they recognize that 12×4 can be thought of as 10×4 plus 2×4. Lampert then has the children explore ...
At the bottom of
page 167...
...;4 plus 6×4 produces the same number as 10×4 plus 2×4. For Lampert, this is important information about the students’ understanding (formative assessment—see Chapter 6). It is a sign that she needs to do many more activities involving different groupings. In subsequent ... , students are challenged with problems in which the two-digit number in the multiplication is much bigger and, ultimately, in which both numbers are quite large—28×65. Students continue to develop their understanding of the principles that govern ... 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....
In the middle of
page 168...
... model as a way for representing the key properties of numbers, particularly those of magnitude and direction. Reading Deborah Ball’s description of her deliberations, one is struck by the complexity of selecting appropriate models for ... mathematical ideas and processes. She hoped that the positional aspects of the building model would help children recognize that negative numbers were not equivalent to ...
In the middle of
page 168...
... Deborah Ball begins her work with the students, using the building model by labeling its floors. Students readily labeled the underground floors and accepted them as “below zero.” They then explored what happened as little paper people entered an elevator at some floor and rode to ... floor. This was used to introduce the conventions of writing addition and subtraction problems involving integers 4−6=−2 and −2+5=3. Students were presented with increasingly difficult problems. For example, “How many ways are there for a person to get to the ... that same number above zero equals zero” (Ball, 1993:381). However, the model failed to allow for explorations for such problems 5+(−6) and Ball was concerned that students were not developing a sense that −5 was less than −2—it was lower, but not necessarily less. Ball ...
In the middle of
page 168...
... Clearly, Deborah Ball’s knowledge of the possible representations of integers (pedagogical content knowledge) and her understanding of the important mathematical properties of integers were foundational to her planning and her instruction. Again, her goals related to developing ... ’ mathematical authority, and a sense of community also came into play. Like Lampert, Ball wanted her students to accept the responsibility of deciding when a solution is ... and likely to be correct, rather than depending on text or teacher for confirmation of correctness....
At the bottom of
page 168...
... The work of Lampert and Ball highlights the role of a teacher’s knowledge of content and pedagogical content knowledge in planning and teaching mathematics lessons. It also suggests the importance of the teacher’s understanding of children as learners. The concept of ... another important characteristic of effective mathematics instruction: that teachers not only need knowledge of a particular topic within mathematics and knowledge of how learners think about the particular topic, but also need to develop knowledge about how the indi-...
In the middle of
page 169...
... vidual children in their classrooms think about the topic (Carpenter and Fennema, 1992; Carpenter et al., 1996; Fennema et al., 1996). Teachers, it is claimed, will use their knowledge to make appropriate instructional ...
In the middle of
page 169...
... 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., 1997). Her instructional practices are an example of what is possible when a teacher understands children’...
At the bottom of
page 169...
... all instruction in Annie Keith’s classroom. 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 ... than being a focus of instruction per se. For example, place-value concepts are developed as students use base-10 materials, such as base-10 blocks and counting frames, to solve word problems involving multidigit numbers....
At the bottom of
page 169...
... Mathematics instruction in Annie Keith’s class takes place in a number of different settings. Everyday first-grade and second-grade activities, such as sharing snacks, lunch count, and attendance, regularly serve as contexts for problem-solving tasks. Mathematics ...
At the bottom of
page 169...
... 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 ... 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: she wants a variety of solutions presented so that students will have an ... to learn from each other. 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 ...
In the middle of
page 170...
... most students have used in order to provide an opportunity for students to see the benefits of such a strategy. 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....
In the middle of
page 170...
... Annie Keith’s strong belief that children need to construct their understanding of mathematical ideas by building on what they already know guides her instructional decisions. She forms hypotheses about what her students ... 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 instructional decisions give her clear diagnoses of individual students’ current ... of understanding. Her approach is not a free-for-all without teacher guidance: rather, it is instruction that builds on students’ understandings and is carefully orchestrated by the teacher, who is aware of what is mathematically important and also what is important to the learner’s ...
At the bottom of
page 170...
..., 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 ...
At the bottom of
page 170...
... 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 ... (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 ...
In the middle of
page 171...
... microcosms of systems that draw heavily on children’s intuitions about resemblance to sustain the relationship between the world being modeled and the model itself. The photograph below displays a child’s model of the elbow. Note, for instance, the rubber bands that mimic the connective ... of ligaments and the wooden dowels that are arranged so that their translation in the vertical plane cannot exceed 180 degrees. Though the search for function is ... 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, 1996a, b)....
At the bottom of
page 171...
... verely limited unless one also has access to such mathematical concepts as variability and uncertainty. Hence, the practice of modeling introduces the further explorations of important “big ideas” in disciplines....
At the bottom of
page 171...
... 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 ...
In the middle of
page 172...
... children to try out the ideas and strategies that they bring to school-based learning in classrooms. Rather than beginning mathematics instruction by focusing solely on computational algorithms, such as addition and ...
In the middle of
page 172...
... As research continues to provide good examples of instruction that help children learn important mathematics, there will be better understanding of the roles that teachers’ knowledge, beliefs, and goals play in their instructional thinking and actions. The examples we have provided ... make it clear that the selection of tasks and the guidance of students’ thinking as they work through tasks is highly dependent on teachers’ knowledge of mathematics, pedagogical ... knowledge, and knowledge of students in general....
At the bottom of
page 172...
... 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., 1992). 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., ...
In the middle of
page 173...
... 7.2. (See Chapter 6 for an example of the type of item used in the categorization task of Figure 7.2.) It is also worth noting that both Figures 7.1 and 7.2 illustrate two other issues that we have discussed in this volume, namely that time on task is a major indicator for learning and that deliberate ...
At the bottom of
page 173...
... attempting to solve them (based on Chi et al., 1981). The strategies consisted of a coherent verbal description of how a problem could be solved and contained three components: the major principle to be applied; the justification for why the principle was applicable; and the procedures for ... the principle. That is, the what, why, and how of solving the problem were explicitly delineated; see Box 7.4. Compared with students who took a traditional course, students in the strategy-...
At the bottom of
page 173...
... 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 ... a problem analysis called a theoretical problem description (Heller and Reif, 1984). The analysis consists of describing force problems in terms of concepts, principles, and heuristics. With such an approach, novices ... 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 ...
At the bottom of
page 173...
... Another approach helps students organize knowledge by imposing a hierarchical organization on the performance of different tasks in physics (Eylon and Reif, 1984). Students who received a particular physics argument that was organized in hierarchical form performed various recall and problem-solving ...
In the middle of
page 174...
... FIGURE 7.1 Effects of two methods of training on problem-solving, final answer, and principle understanding. SOURCE: Dufresne et al. (1992)....
At the bottom of
page 174...
... These examples demonstrate the importance of deliberate practice and of having a “coach” who provides feedback for ways of optimizing performance (see Chapter 3). If students had simply been given problems ...
At the bottom of
page 175...
... unlikely that they would have spent time efficiently. 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 ... 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 ...
In the middle of
page 176...
... A disk of mass, M=2 kg, and radius, R=0.4 m, has string wound around it and is free to rotate about an axle through its center. A block of mass, M=1 kg, is attached to the end of the string, and the system is released from ... with no slack in the string. What is the speed of the block after it has fallen a distance, d=0.5 m. Don’t forget to provide both a strategy and a solution....
At the bottom of
page 176...
... Strategy 1: 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 ... 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 ... 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....
At the bottom of
page 176...
... 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...
In the middle of
page 177...
... of the disk and kinetic energy of the falling mass. Equating the initial and final states and using the relationship between v and ω the speed of M can be found. Mechanical energy is conserved even with the nonconservative tension force because the tension force is internal ...
In the middle of
page 177...
... 3: 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....
In the middle of
page 177...
... 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). The last two strategies are largely a shopping ...
At the bottom of
page 177...
... 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 ... appropriate links between problem contexts, and the principles and procedures that could be applied to solve them (see Leonard et al., 1996)....
At the bottom of
page 177...
... experiences and observations, and they may cling tenaciously to those views— however much they conflict with scientific concepts—because they help them explain phenomena ...
At the bottom of
page 177...
... instructional strategy, termed “bridging,” has been successful in helping students overcome persistent misconceptions (Brown, 1992; Brown and Clement, 1989; Clement, 1993). The bridging strategy attempts to bridge from students’ correct beliefs (called anchoring conceptions) to their ...
At the bottom of
page 178...
... FIGURE 7.3 Percent correct choices under strategy-based and traditional teaching conditions by problem number in a categorization, multiple-choice task. SOURCE: Leonard et al. (1996)....
At the bottom of
page 178...
... from the board. 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 ... a student to place her hand on top of a vertical spring...
In the middle of
page 179...
... and push down and to place her hand on the middle of the springy board and push down. She would then be asked if she experienced an upward force that resisted her push in both cases. Through this type of dynamic probing of ... ’ beliefs, and by helping them come up with ways to resolve conflicting views, students can be guided into constructing a coherent view that is applicable across a ...
In the middle of
page 179...
... Another effective strategy for helping students overcome persistent erroneous beliefs are interactive lecture demonstrations (Sokoloff and Thornton, 1997; Thornton and Sokoloff, 1997). This strategy, which has been used very effectively in large introductory college physics classes, ... heavy cart moving toward the stationary cart. Each cart has an electronic “force probe” connected to it which displays on a large screen and in real-time the force acting on it during the collision. The teacher first asks the students to discuss the situation with their neighbors and then ...
At the bottom of
page 179...
...—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. Yet, notwithstanding the major damage to the ... , Newton’s Third Law states that two interacting bodies exert equal and opposite forces on each other....
At the bottom of
page 179...
... 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 ... : 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. In all ... , students see that the carts exert equal and opposite forces on each other, and with the help of a discussion moderated by the instructor, the students begin to build a consistent view of Newton’s Third Law that ... their observations and experiences....
In the middle of
page 180...
... of an event occurring in real-time significantly inhibits the learning of the underlying concept (Brasell, 1987)....
In the middle of
page 180...
... Both bridging and the interactive demonstration strategies have been shown to be effective at helping students permanently overcome misconceptions. This finding is a ...
In the middle of
page 180...
... One of the best examples of translating research into practice is Minstrell’s (1982, 1989, 1992) work with high school physics students. Minstrell uses many research-based instructional techniques (e.g., bridging, making students’ thinking visible, facilitating ...
At the bottom of
page 180...
... Students’ initial ideas about mechanics are like strands of yarn, some unconnected, some loosely interwoven. The act of instruction can be viewed as helping the students unravel individual strands of ... , 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 ... 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 ...
At the bottom of
page 180...
... 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 ...
In the middle of
page 181...
... ergy and momentum. At some point Minstrell guides the discussion to a specific example: he drops a rock and asks students how the event can be explained using their ideas about force. He asks students to individually formulate their ideas and to draw a ...
In the middle of
page 181...
... 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.g., people) can exert forces, that passive agents (e.g., a table) cannot. Minstrell (1992) ... developed a framework that helps both to make sense of students’ reasoning and to design instructional strategies. (For a related theoretical framework for classifying and explaining student reasoning, see the discussion of & ... knowledge (e.g., passive objects do not exert force), to strategic knowledge (e.g., average velocity can be determined by adding the initial and final velocities and dividing by two), or generic reasoning (e.g., the more the X, the more the Y). Identifying students’ facets, what cues ... in different contexts, and how students use them in reasoning are all helpful in devising instructional strategies....
At the bottom of
page 181...
... an instructor provide an active learning experience, provide feedback, accommodate different learning styles, make students’ thinking visible, and provide scaffolding and tailored instruction to meet specific student needs when facing more than 100 students at a time? Classroom communication ... 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.g., a fairly inexpensive graphing calculator) to “sign on” to a ... communication network that permits the instructor to send questions for students to work on and permits students to enter answers through their input device. Answers can then be displayed anonymously in histogram...
In the middle of
page 182...
... form to the class, and a permanent record of each student’s response is recorded to help evaluate progress as well as the effectiveness of instruction....
In the middle of
page 182...
... 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., 1996; Wenk et al., 1997; Mestre et al., 1997). The technology creates an interactive learning environment in the ... : students work collaboratively on conceptual questions, and the histogram of students’ answers is used as a visual springboard for classwide discussions when students defend the reasoning they used to ... at their answers. This technology makes students’ thinking visible and promotes critical listening, evaluation, and argumentation in the class. The teacher is a coach, providing scaffolding where needed, tailoring “mini-lectures” to clear up points of ... , or, if things are going well, simply moderating the discussion and allowing students to figure out things and reach consensus on their own. The technology is also a natural mechanism to support formative assessment during instruction, providing both the ... 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 ... possible by lectures and helps to foster a community of learners focused on common objectives and goals....
At the bottom of
page 182...
... 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 ...
At the bottom of
page 182...
... be educationally “at risk.” One approach that has been especially useful in science teaching was developed with language-minority grade-school children: Chèche Konnen, which in Haitian Creole means search for knowledge (Rosebery et al., 1992). The approach stresses how discourse is a ...
In the middle of
page 183...
... 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 ... . 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....
At the bottom of
page 183...
... teaching began by creating “communities of scientific practice” in language-minority classrooms in a few Boston and Cambridge, MA public schools. “Curriculum” emerges in these classrooms from the students’ questions and beliefs and is shaped in ongoing interactions that ...
At the bottom of
page 183...
... Students constructed scientific understandings through an iterative process of theory building, criticism, and refinement based on their own questions, hypotheses, and data analysis activities. Question posing, theorizing, and argumentation formed the ... scientific activity. 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 ... , and so forth. The process as a whole provided a richer, more scientifically grounded experience than the conventional focus on textbooks or laboratory ...
At the bottom of
page 183...
... 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, 1978). The teacher guides and ... students as they explore problems and define questions that are of interest to them. A community of practice also provides direct cognitive and social support for the efforts of the group’s individual members. Students share the responsibility for thinking and doing: they distribute ... . In addition, a community of practice can be a powerful context for constructing scientific meanings. In challenging one another’s thoughts and beliefs, students must be explicit about their meanings; they must negotiate conflicts in belief or evidence; and they must share and synthesize ...
In the middle of
page 184...
... drinking water from the fountain on the third floor, where the junior high was located, was superior to the water from the other fountains in their school. Challenged by their teacher, the students set out to determine whether they actually preferred the water from the third floor or only thought they ...
In the middle of
page 184...
... 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 ... -floor fountain was the worst because “all the little kids slobber in it.” (The first-floor fountain is located near the kindergarten and first grade classrooms.) Their teacher was also suspicious of the results because she had expected no differences among the three water fountains. ... beliefs and suspicions motivated students to conduct a second taste test with a larger sample drawn from the rest of the junior high....
At the bottom of
page 184...
... 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, ...
At the bottom of
page 184...
... achieve understanding (Brown and Palincsar, 1989; Inagaki and Hatano, 1987)....
At the bottom of
page 184...
... 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. The researchers were interested in changes in students’ conceptual knowledge about aquatic ecosystems and in students’ uses of ...
In the middle of
page 185...
.... Why was the water from the first-floor fountain preferred? How can they determine the source of the preference? They decided to analyze the school’s water along several dimensions, among them acidity, salinity, temperature and bacteria. They found that all the fountains had unacceptably ...
At the bottom of
page 185...
... 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 ...
At the bottom of
page 185...
... Like you look for the things, take the garbage out of the water, you put a screen to block all the paper and stuff, then you clean the water; you put chemical products in it to clean the water, and you’d take all the microscopic life out. Chlorine and ... , you put in the water. They’d gather the little stuff, the little stuff would stick to the chemical products, and they would clean the water....
In the middle of
page 186...
...; type of explanation. It is also clear that by making the student’s thinking visible, the teacher is in an excellent position to refine her (and perhaps the class’s) understanding....
In the middle of
page 186...
..., 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. When asked for their ideas about what could be making the children sick, the ...
In the middle of
page 186...
... 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. When asked to generate an experiment to justify an hypothesis—“...
At the bottom of
page 186...
... text comprehension question for which there was a “right” answer. They frequently responded with an explanation or assertion of knowledge and consistently marked their responses as explanatory (“because”): “Because fish don’t eat garbage. They eat plants under the ...
At the bottom of
page 186...
... 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 ...
At the bottom of
page 186...
... 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. Those things, poison things, you aren’t ...
In the middle of
page 187...
... Like, you could test what the kids ate and, like, test the water, too; it could be the water that isn’t good, that has microbes, that might have microscopic animals in it to make them ...
In the middle of
page 187...
... 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 ... to a question about sick fish, Laure clearly understands how to find a scientific answer (page 91):...
In the middle of
page 187...
... I’d put a fish in fresh water and one fish in a water full of garbage. I’d give the fresh water fish food to eat and the other one in the nasty water, I’d give it food to eat to see if the fresh water, if the one in the fresh water would die with the food I ... , if the one in the dirty water would die with the food I gave it…. I would give them the same food to see if the things they eat in the water and the things I give them now, which will make them healthy and which wouldn’t make them healthy....
At the bottom of
page 187...
... research studies on expertise (see Chapter 2). The examples discussed in this chapter focus on two areas of science teaching: physics and junior high school biology. Several of the teaching strategies illustrated ways to help students think about the general principles or “big” ideas in ...
At the bottom of
page 187...
... about the subject matter. One strategy for helping students in physics begins with an “anchoring intuition” about a phenomenon and then gradually bridging it to related phenomena that are less intuitive to the student but involve the same physics principles. Another strategy ... the use of interactive lecture demonstrations to encourage students to make predictions, consider feedback, and then reconceptualize phenomena....
At the bottom of
page 187...
...èche Konnen demonstrates the power of a sense-making approach to science learning that builds on the knowledge that students bring with them to school from their home cultures, including their familiar discourse practices. Students learned to think, talk, and act scientifically, and their first and ...
In the middle of
page 188...
... ful ways. 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 ... findings, and consulted with experts at the local water treatment facility....
At the bottom of
page 188...
... 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....
At the bottom of
page 188...
... 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 ... epistemology that guides the discipline (for history, see Wineburg and Wilson, 1988; for math and English, see Ball, 1993; Grossman et al., 1989; for science, see Rosebery et al., 1992)....
At the bottom of
page 188...
... chapter illustrate the principles for the design of learning environments that were discussed in Chapter 6: they are learner, knowledge, assessment, and community centered. They are learner centered in the sense that teachers build on the knowledge students bring to the learning situation. They are ... centered in the sense that the teachers attempt to help students develop an organized understanding of important concepts in each discipline. They are assessment centered in the sense that the teachers attempt to make students’ thinking ... so that ideas can be discussed and clarified, such as having students (1) present their arguments in debates, (2) discuss their solutions to problems at a qualitative level, and (3) ... predictions about various phenomena. 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....
At the bottom of
page 188...
... 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 ... . Teachers’ knowledge of their disciplines provides a cognitive roadmap to guide their assignments to students, to gauge student progress, and to support the questions students ask. The teachers focus on understanding rather than memorization and routine procedures to follow, and they engage ... in activities that help students reflect on their own learning and understanding....
In the middle of
page 189...
... The interplay between content knowledge and pedagogical knowledge illustrated in this chapter contradicts a commonly held misconception about teaching—that effective teaching consists of ...
At the bottom of
page 189...
... of students’ discussions in classrooms indicate that they learn to use the tools of systematic inquiry to think historically, mathematically, and scientifically. How these kinds of teaching strategies reveal themselves on typical standardized tests is another matter. In some cases there is ... that teaching for understanding can increase scores on standardized measures (e.g., Resnick et al., 1991); in other cases, scores on standardized tests are unaffected, but the students show sizable advantages on assessments that are sensitive to their comprehension and understanding ...
At the bottom of
page 189...
... 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? The next chapter explores teachers& ... ; chances to improve and advance their knowledge in order to function as effective professionals....
A total of pages of uncorrected, machine-read text were searched in this chapter. Please note that the searchable text may be scanned, uncorrected text, and should be presumed inaccurate. Page images should be used as the authoritative version.