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Building New Course Structures
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6 Teaching Undergraduate History: A Problem-Based Approach
Robert K. Poch and Eskender Yousuf
problem solving, historical thinking skills, learning assessment
Introduction
Among the challenges that faculty encounter is facilitating active engagement with their discipline within classrooms of diverse undergraduate students (Calder, 2006; Rendon, 2009). We face this challenge regularly when teaching history. Within a mostly lecture-based format, it is easy to deny students opportunities to engage the discipline as historians.
While we concentrate primarily on the discipline of history, this approach can be applied within other disciplines where the aim is to provide diverse undergraduate students with the opportunity to “do” the work of those disciplines and find personal connection within them. Historians discover and use primary source documents, confront vexing contextual and interpretational problems, experience the diverse perspectives of peers, and so too can students. For many faculty, the temptation in undergraduate survey courses is to place full emphasis on content coverage and to ignore or minimize development of discipline-based skills (Calder, 2006; Sipress and Voelker, 2009). This produces poor results if our instructional goals include providing a more genuine experience with our discipline, developing analytic skills, and engaging students meaningfully (Weimer 2002). When students are passive recipients of disciplinary information with no apparent connection to themselves, they withdraw intellectually and emotionally (Freire 1970; Langer 1997).
However, it is possible for highly diverse students to experience the dynamic nature of disciplines such as history by “doing” – that is, by actively developing the skills and engaging the processes and problems involved in being a practitioner of the discipline (Sipress and Voelker, 2009; Weimer, 2002). This can happen in large and small classes and also in survey courses. We want students to encounter history as historians. We also want them to experience the excitement of historical discovery and personal meaning-making that first drew us into the work. In doing so, students also learn and retain substantive course content.
We focus below on a “problem-based” approach to teaching and learning history where students are active disciplinary practitioners engaged in addressing problem topics of relevance and connection to their diverse lives. In doing so, the following questions are addressed:
- What are some of the core elements of historical inquiry? What do skilled historians actually do?
- What actions encourage students from different cultural and disciplinary backgrounds to engage the core elements of historical inquiry as practitioners?
- How can course pedagogy, assignments, and assessments become consistent with what skilled historians do and foster student engagement?
- How can the results of these efforts be assessed? What are some of the assessment results from using a problem-based approach to learning history?
While we concentrate primarily on the discipline of history, this approach can be applied within other disciplines where the aim is to provide diverse undergraduate students with the opportunity to “do” the work of those disciplines and find personal connection within them (Gurung, Chick, and Haynie, 2009).
Identifying and Using Core Elements of Historical Inquiry
In creating learning environments where students become historians, it is necessary to consider what it is that historians do and what skills are necessary to be practitioners of the discipline. In a macro sense, many historians describe their work as problem solving guided by active questioning (Elton, 1967; Fischer, 1970; Marius and Page, 2005; Nevins, 1963). That is, questions are posed; sources and facts are collected, critically read, contextualized, and organized; and, an interpretation of the past is formed while recognizing that the complexities of history defy easy explanations (Ayers, 2006; Commager, 1965; Wineburg, 1991, 1999). Whenever possible, historians utilize primary sources to form their own interpretations rather than relying mostly on the interpretations of other historians.
Historians are regularly challenged to analyze, contextualize, and interpret the past from incomplete disparate sources. Such actions require a series of more discrete skills including the critical evaluation, interpretation, and communication of evidence; the detection of bias; and careful consideration of historical causation (Ayers, 2005; Barzun and Graff, 1977; Bloch, 1953; Carr, 1961; Commager, 1965; Elton, 1967; Evans, 1999; Fischer, 1970; Lerner, 1997; Nevins, 1963; Wood, 2008). These skills are expressed concisely as the “5Cs” of historical thinking: “change over time, causality, context, complexity, and contingency” (Andrews and Burke, 2007, 1).
Through using the 5Cs, students become increasingly aware of how much can change over time – such as political systems, landscapes, and social values – while, simultaneously, acknowledging retention of strong elements of the past such as holidays and the rituals surrounding them. Further, once developed through classroom engagement with historical sources, the elements of causality, context, complexity, and contingency enable students to identify and appreciate the incomplete nature of historical records and the intricate, simultaneous, and broad scale human interactions and competing interests in history (Ayers, 2006; Nevins, 1963; Wood, 2008). The awareness of complexity causes professional historians and students to probe more deeply into explanations of causality and to reject simplistic reasoning.
To have students engage history problems in a manner that is relevant and meaningful, it is necessary to also think carefully about how to invite students and their interests into the process of historical inquiry. While engineering, mathematics, physics and accounting are experienced as real, relevant, and practical, history is not experienced that way. Students often encounter it as an abstruse, fact-laden, memorization-based, irrelevant, impersonal discipline. We must, therefore, address how to engage students in being practitioners of historical inquiry and interpretation. These are some of the key components of the real work of historians and the historical reasoning used to create interpretations of the past. In working with undergraduate students – some of whom just graduated from high school – the 5Cs are a useful, understandable, and easy to remember toolset for engaging historical inquiry. With these parts of the work of historians and historical thinking in mind, it is possible to design classroom experiences that bring students into the dynamic nature of this work and its associated challenges. Students can then experience the discipline of history more fully and also learn how to create their own historical meaning from available sources (Sipress and Voelker, 2009). These components of historical thinking help to form the “history problems” that we utilize in our U.S. history classroom and which we describe further below.
However, to have students engage history problems in a manner that is relevant and meaningful, it is necessary to also think carefully about how to invite students and their interests into the process of historical inquiry. While engineering, mathematics, physics and accounting are experienced as real, relevant, and practical, history is not experienced that way. Students often encounter it as an abstruse, fact-laden, memorization-based, irrelevant, impersonal discipline. We must, therefore, address how to engage students in being practitioners of historical inquiry and interpretation.
Engaging Students in Core Elements of Historical Inquiry as Practitioners
The invitation to participate in our class, “America’s Past and Present: Multicultural Perspectives,” is underscored by bringing students into direct interaction with historical thinking skills, primary source materials that are reflective of multiple cultures on the American landscape, and with complex historical issues and problems that invite students to interpret history with their own voices rather than having a textbook or the instructor be the sole interpretive voices. We want students to gain more elegant and inclusive views of history that expose the complex dynamics between people over time and which stimulate curiosity about how life was experienced and interpreted by different diverse populations. This is enabled in part by the diversity of our students. The multiple complexities of persons in the past are reflected in our students. As observed by Lee, Poch, Shaw, and Williams (2012),
“…we have observed our institution’s student population become increasingly diverse in terms of racial and ethnic demographics. Historically, generalized categories of racial and ethnic identity have become more diffuse and complex. We are also more mindful of the often less visible forms of difference that are present in any learning environment, such as socioeconomic status, sexual orientation, religion, disability, and many others”
As students engage each other in class and discover how their classmates form different historical interpretations based in part on their different lived experiences, it stimulates and reinforces an understanding of the different perspectives and lived experiences of persons throughout history.
To better ensure the relevance of the history problems to diverse student interests, students are asked on the first day of the course what part of U.S. history between the Civil War and present time is of greatest interest to them. While some students do not know how to answer such a question at first, many others have some notion of their interests. Examples of these student responses from spring semester 2015 are as follows:
- World War II
- U.S. Civil Rights movements (including the role of youth in such movements)
- Vietnam War
- The Great Depression
- 9/11 and its effect on the world
- Other countries and perceptions of the U.S.
- Native Americans and Tribes
- Civil War & differing economies
When we, as instructors, are responsive to such interests, students more fully engage with the topics and are willing to invest the energy to do the challenging work of historical meaning-making using disciplinary thinking skills. These interests are invaluable in making the course ours – that is, a shared experience of historical investigation that reflects mutual interests rather than those of the instructor alone. It is a powerful opportunity to communicate to students at the beginning of the course that the instructors are engaged co-investigators of historical topics that the students suggest and that student interests are of great value. We use student historical interests to create substantive class discussion questions, short reading and writing assignments, and further engagement with historical thinking skills through lengthier history problems that come later in the semester. While assembling sources related to the interests, we spend the first three to four weeks of the course introducing and practicing the 5Cs of historical thinking. Initial reading and writing assignments selected before the course starts enable students to begin the process of understanding, recognizing, and using context, causality, complexity, change and continuity over time, and contingency. These historical thinking skills are then used to explore more deeply and intentionally the historical topics that students suggest. In doing so, student interests become integrated and useful parts of the course experience.
Students also tell us that they want to explore historical themes and issues that are not commonly approached in historical texts or rooted in fact memorization. For example, one student expressed in a topical interest survey that she wanted “…to learn the truth about history. The real original text. I want to find it and research it. Interest in causality and what caused all the events in history to happen? WHYYY! The reason things went down the way they did!” Students express that they want to explore the meaning and use of racism, the rise of feminism, and the perspectives of other nations whose histories intersect with those of the United States. They want to do so in a way that engages history through interesting questions full of encounters with ordinary people who experienced the past in powerful but mostly unknown ways. When we, as instructors, are responsive to such interests, students more fully engage with the topics and are willing to invest the energy to do the challenging work of historical meaning-making using disciplinary thinking skills.
Making Pedagogy, Assignments, and Assessments Consistent with what Historians Actually Do
Engaging students as historians takes careful thought and planning. Core elements of course design are important parts of this work. The course curriculum must provide space for the development of student evaluative and interpretive skills. This often comes with winnowing some course content as traditionally delivered through lengthy lectures (Calder, 2006). The process of winnowing involved using part of a summer break to critically review course materials to identify where unnecessary content was located that cluttered class time and reduced the capacity to develop student historical thinking skills. For example, a discussion of Civil War medicine and pro- and anti-U.S. imperialist arguments were removed given that they were peripheral to more important course themes. Further, those subjects tended to lead to more lecturing rather than active discussion. Rendon (1993) observed that “…many culturally diverse students do not learn best through lecture. Instead, we should focus on collaborative learning and dialogue that promote critical thinking, interpretation and diversity of opinion” (10).
Students can mine rich primary sources of the period as guided by research questions collaboratively developed by students and the instructor. Lectures are balanced with skill-building by doing – actively engaging students in learning how to develop researchable questions, engaging primary and secondary texts with critical lenses, forming interpretations from available evidence, and presenting results. Further, significant thought must be invested in designing course assignments and resources that enable skill development to occur and be assessed. Assessments must be constructed to evaluate student work in a manner consistent with skill development expectations. Class time invested in developing the foundational skills used within the discipline is necessary given that “…history teachers cannot simply present students with documents, tell them what to do, and then expect magical gains in the development of students’ historical sense. Much more elaborate and carefully thought out ‘scaffolding’ is needed to realize the potential of this approach” (Calder et al., 2002, 59).
This approach has significant student developmental implications. It may involve moving students and the course structure away from a dualistic form of learning history where questions are framed in terms of right and wrong response outcomes and there is strong dependency upon the instructor. Instead, there will be movement toward a course design wherein students are met with formulating questions within a course-related area of personal historical interest that has interpretive complexities associated within it (Donald, 2002, 3; Evans et al, 2010).
For example, rather than presenting students with a course design that asks them to identify within an exam three major outcomes of Reconstruction after the U.S. Civil War from lecture notes, students can experience the real problems of Reconstruction in depth by reading conflicting newspaper accounts in the North and the South regarding the political enfranchisement of African American men and the political balances of power that were at play (Langer, 1989). Rather than searching secondary and tertiary sources (such as many textbooks and lectures) alone for such information, students can mine rich primary sources of the period as guided by research questions collaboratively developed by students and the instructor. One research question that was developed in this manner focused on the tactics that some southern states utilized to stymie the voting capacity of black males following passage of the Fifteenth Amendment. In response, students were able to find and analyze different literacy tests for voting (the class even tried taking some of the tests which produced a high failure rate) and also details on the administration of poll taxes.
Such collaboration and student interpretive responsibilities can lead to movement from what psychologist Ellen Langer refers to as “mindlessness” wherein students are stuck with rote memorization and the search for the “right” answer rather than experiencing the rich contexts and possibilities that exist as part of the act of discovering and making meaning within disciplines (Langer, 1997). Langer notes that, “In math, teaching for understanding involves teaching students to think about what a problem means and to look for multiple solutions. Studies have confirmed that science is better taught through hands-on research and discovery than through memorization alone. In English, teaching for understanding means emphasizing the process of writing and exploring literature rather than memorizing grammar rules and doing drills. Understanding is encouraged in history by turning students into junior historians” (Langer, 1997, 71, 72). It is in that spirit that we developed history problems.
History Problems
“As an interpretive historian using the primary source readings that are provided in this problem, how do you define Jim Crow?” This question, which seems deceptively simple at first, quickly exposes the complexities of Jim Crow as a comprehensive system within American society that touched every aspect of life. Each history problem is comprised of three essential parts: an introduction to the problem with concise contextual information; open-ended problem questions designed to provide students with interpretive space to utilize their voice and perspectives (rather than the instructor’s voice or that of the textbook); and a set of primary source materials that reflect diverse authors and views. Students are given three history problems throughout the semester and they have approximately four weeks to complete them given the complexity of the readings. Although there is no required page length for the problem responses, students often write seven pages or more for each problem. The history problems used during the Spring 2015 semester were based on student interests expressed at the beginning of the semester and involved the following topics: “The challenges of Jim Crow and the dynamics found within it;” “’Equal protection under the law:’ The challenges of separate but equal – The struggle for Brown v. Board of Education ;” and, “September 11, 2001.”
The history problem questions provide students with the ability to create responses based on their own interpretation of the material. The questions replicate real challenges and problems for historians that are consistent with the 5Cs of historical thinking that we use in class. Some questions expose students to the complexities of powerful systems of racial oppression such as Jim Crow. Other questions focus on establishing context or examining change over time. For example, in the problem examining Jim Crow, students were asked the following: “As an interpretive historian using the primary source readings that are provided in this problem, how do you define Jim Crow?” This question, which seems deceptively simple at first, quickly exposes the complexities of Jim Crow as a comprehensive system within American society that touched every aspect of life.
To assist in developing a definition of Jim Crow, the history problem packet includes a variety of primary sources that include memoirs, excerpts from scholarly books and novels, and a 1949 travel guide for African American motorists. Within this particular problem packet, the sources included pieces from W.E.B. Dubois’ The Souls of Black Folk (1903); Ralph Ellison’s Invisible Man (1952); John Hope Franklin’s autobiography, Mirror to America (2005); Howard Thurman’s The Luminous Darkness: A Personal Interpretation of the Anatomy of Segregation and the Ground of Hope (1965); Richard Wright’s Uncle Tom’s Children (1940); and, The Negro Motorist Green Book (1949). The Green Book was published to provide African American travelers with “…information that will keep him from running into difficulties, embarrassments and to make his trips more enjoyable” (1). These sources provided different views of and experiences with Jim Crow and, unlike a textbook, did not provide the definition and interpretation of Jim Crow for the students. Instead, the students worked with the different texts, situated them contextually in their particular time and place and with consideration of who wrote them, and gradually developed their own definition of Jim Crow. Further, the sources spanned a number of decades so that some consideration could be given to change over time in addition to complexity. The sources worked well in providing multiple perspectives of how Jim Crow, as a system, affected different parts of life and also a sense of the varieties of materials that historians use.
The same history problem also asked students to consider how the sources in the packet related to any prior readings we had used in the course (such as Frederick Douglass’ 1865 speech, “What the Black Man Wants”), so that students could further analyze and gain familiarity with context, change over time, complexity, contingency, and causality. Douglass’ speech was useful not only as an earlier expression of the challenges and contradictions that Jim Crow created within a nation that professed freedom and democracy, but also served as a source to explore the challenging concept of contingency. By expressing how black men wanted political participation through receipt of the right to vote and full recognition for their intellectual capacity to be informed contributors to democracy, Douglass’ speech highlighted contingencies necessary for breaking explicit bonds of enslavement and more diffuse societal systems of oppression. These varied course and problem-based primary sources enabled students to make complex connections between forms of evidence and further solidified historical thinking skills as expressed within the 5Cs. Providing approximately four weeks to work with each history problem gave plenty of in-class time to further discuss the sources, the context of the sources, and to practice the skills necessary to utilize them effectively.
Assessing the Results of Using a Problem-Based Approach to Learning History
We utilized several forms of assessment to evaluate the problem-based approach to learning history: 1) the evaluation of student written responses to history problems, 2) individual interviews with students, and, 3) an independently conducted end-of-semester student survey. In each assessment approach, it was important that student voices were prominent and listened to attentively (Patton, 1980). Each of these assessment forms is discussed below.
Written responses to history problems
Student written responses to the four history problems were evaluated carefully for progressive use of the elements of historical thinking. Each of the papers went through two evaluative reviews – one by the course teaching assistant and the other by the primary instructor. The papers were scored on a standard A-F grading scale and were preceded by smaller writing assignments that practiced discrete elements of five identified historical thinking skills. For example, the Frederick Douglass speech, “What the Black Man Wants,” was used early in the semester in part so students could practice establishing and expressing context and recognizing some elements of contingency. This was done with multiple other pieces of short reading and writing exercises that involved the voices and writing of diverse speakers and authors. With this practice experience in place, students could move with greater confidence in addressing contextual issues in the first history problem and those that followed. The papers were also useful in assessing student command or struggle with certain components of historical thinking. We discovered in multiple early papers that students did not fully understand the idea of contingency and, in response, were able to spend more time discussing and practicing it in class.
Toward the end of the semester as students worked with perhaps the most challenging history problem involving the September 11, 2001 attacks, we could detect that students were engaging in far more sophisticated historical reasoning and explicit use of historical thinking skills. For example, one student, a freshman, having studied the presence and role of the United States in the Middle East since the early twentieth-century (using maps, documents, interviews, reports, and political cartoons provided in the fourth history problem), constructed a complex contextual background in her written response to one set of the problem questions (“Using information from our class sessions and the materials provided within this problem packet, describe the relationships that existed and some of the events that occurred between the United States and the ‘Middle East’ region prior to 9/11. With these sources in mind, what are some of the possible motives for the 9/11 attack?”). She responded in part in her introduction,
The events of September 11th, 2001 came as a shock to millions of American citizens; however, a complex history of rocky foreign relations combined with the struggles regarding religion and government in the ‘Middle East’ suggest that the attack was only one part of several interconnected issues. As we examine the context of the events surrounding the 9/11 attacks, the complexity of the United States’ position in world affairs, the major causes leading up to the attack, and contingency of other nations’ histories on our own, we can begin to analyze the affect that each of these has had on the aftermath of 9/11 over time… [the] ten to fifteen years before the attacks on the Twin Towers… show a deeply complex relationship between different nations. While the Saudi Arabian government aided the United States [in the Gulf War by providing a U.S. military staging ground along the border with Kuwait] there were other entities such as Iraq that were pitted against the United States, resulting in conflict between the nations in that area regarding involvement of the United States. To add to this complexity is the idea of a theocracy and questions on how to rule a nation when military forces within those nations do not share the political philosophy or religious beliefs of those nations.
In this brief excerpt which was supported by lengthier supporting text and examples, we could easily detect the use of historical thinking skills (some of which were explicitly mentioned), including greater recognition of the deep complexities that long preceded the events of 9/11.
The written responses to the history problem questions were returned to the students with evaluative comments that served, in part, to prepare students for individual meetings with the course instructor and the graduate research assistant. With highly diverse students from different nations, it was important to provide different opportunities for the students to express how they approached the problems that extended beyond their written responses.
Individual meetings with students about the history problems
Individual student interviews were conducted on two of the problem set essays (history problems one and three). Each student was given the opportunity to schedule a conversation with us to discuss their responses and to further sharpen their historical thinking skills based on the 5Cs. During these meetings, the students could gain additional points (but no subtraction of points) by further clarifying their written responses and the processes that they used to construct them. The meeting questions were provided to each student in advance and included: Where did you encounter the greatest challenges in responding to this history problem? How did you approach the challenges? What do you believe you learned through engaging in this history problem? An opening question was designed to further probe each student’s writing by asking: “We found some engaging interpretations within your paper [if this was truthful] and also some places where we would like to know more. Can you further describe [this was customized for each student paper]….” The questions presented opportunities for students to further explain their own interpretations and how they approached the problems over time. Through the questions, students reflected on and expressed their own historical interpretations in a manner that replicates much of the way that historians utilize peers within their professional communities.
Within the second set of individual meetings on the third history problem, students began to express more of the 5Cs of historical thinking. During the first set of interviews regarding the “Challenges of Jim Crow and the dynamics found within it,” we met individually with twenty-three students who expressed a wide array of historical thinking skills. For example, one student, in expanding upon her interpretation of Jim Crow, remarked that developing her own definition of Jim Crow enabled her to “…go far below the surface to see the complexity of Jim Crow and its relationship to definitions of racism.” Further, the student explained that examining the use of Jim Crow-related art revealed to her the complex strategies of Jim Crow systems of oppression. Another student observed that reading primary source accounts of African Americans who lived within Jim Crow brought forth “contradictions” within the imagery and terms used within Jim Crow such as using ideas of “light” and “visibility” to describe American society while those who lived under the weight of Jim Crow described darkness, shadows, and invisibility. Within these comments, and many others, we observed students expressing different elements of historical thinking including complexity (very explicitly), as well as change over time, and causality as students considered and described the structures of Jim Crow messaging and how the messaging was delivered in different ways during specific spans of time.
Within the second set of individual meetings on the third history problem, students began to express more of the 5Cs of historical thinking. We used a slightly modified set of questions that included: Did you feel that you were able to utilize any particular historical thinking skills in this problem? Students commented more extensively and easily about historical thinking skills and demonstrated within their papers greater complexity in historical thinking. For example, one student expressed complex differences in how African American’s responded to racial oppression over time. She interpreted primary sources in the first history problem as being “defensive in nature – how persons responded or protected themselves within the Jim Crow system” whereas in the third history problem on the legal strategies that African American attorneys used to eventually prevail in the Brown v. Board of Education decision, the strategy was “more offensive in nature in that it showed black persons taking back their rights and being more confident in doing so.” This student, and others, expressed greater awareness and mastery of complexity, causality, and change over time in their interpretations of primary historical source materials.
Independently conducted end-of-semester student survey
At the request of the instructor, a grant-supported survey was developed and given to students in the history course at the very end of spring semester 2015. Among those who responded to the survey (50% of a class of 24 first-year students where this problem-based approach was fully implemented), the following are representative of their responses:
Question: What parts of the course were particularly effective for you in developing historical thinking skills and the capacity to be an effective historian? What parts were particularly ineffective?
- “The parts of this course that were effective in developing historical thinking skills were definitely the history problems and also the 5Cs. I learned so much through the history problems that I would have never learned through a test and I will remember the information much better by writing about it in a history problem. I didn’t feel like any part was ineffective.”
- “The history problems were effective because they allowed us to give our own opinion on the matter and we got involved instead of just mindless memorization.”
- “I think that the most effective things that we did in class to develop my historical thinking skills were definitely the problem sets and our class discussions. Both of those two platforms pushed us to think for ourselves and contribute to a larger group discussion. I loved the problem sets because they forced me to think and form my own opinions using the historical thinking skills that we were given.”
- “The history problems really helped me see how contemporary historians actually applied their skills to modern problems.”
Question: Do you believe that you know and can apply the essential elements of historical thinking, as a result of this course? Please explain.
- “Yes, I have already applied it to other classes and feel very comfortable doing it.”
- “Yes, the history problems gave us that opportunity.”
- “I do believe that I could apply the elements of historical thinking into other classes and in my everyday life as a historian. I feel confident that I know the 5Cs of historical thinking and could use them in other situations. They were drilled into us, I won’t forget them.”
- “Yes. I believe that using the 5Cs from the beginning of this course helped me to gain further knowledge and also to help me dig deeper into historical problems and questions.”
- “I definitely feel that this course has aided my skills in critical analysis and historical thinking and I can see how to use these skills in different contexts and subjects besides history.”
Student survey responses found that 1) history thinking elements of the 5C’s gave them a grasp of historical thinking skills; 2) established a process whereby students could formulate their own interpretations of historical sources thereby moving from mindless memorization to mindfulness and, 3) students were able to utilize their own lived experiences and interests as historians engaged in investigating historical problems.
The use of history problems in our course stems from a twofold purpose. First, we want students to experience the discipline of history as actively engaged historians who use primary sources in addressing challenging questions of historical interpretation through use of well-defined historical thinking skills. Second, we want to facilitate personal interaction with the discipline by using sources and stories that are reflective of the diversity of our students and enabling their interpretive voices to emerge and be respected in our assessments of their learning. Through the use of history problems, unlike our past exams, we noticed the disappearance of instructor voice in student interpretations of historical source materials and an increase in deliberate use of the 5Cs of historical thinking: context, change over time, contingency, complexity, and causality. Continued exploration of the use of history problems in developing more focused development of particular historical thinking skills will occur in our future work as will the capacity to assess those skills effectively through combinations of written work and in-person conversational interactions with students.
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Acknowledgements
Facilitation and evaluation of this project was enabled by the invaluable support and guidance of Ilene Alexander, Jeff Lindgren, and J.D. Walker. We also acknowledge the work and influence of the following excellent undergraduate teaching assistants who, over the last seven years, contributed to the development, implementation, and the strengthening of the problem based approach to teaching and learning: Emily DePalma, Emily McCune, Julianna Ryburn, Jade Beauclair Sandstrom, and Chris Stewart
Innovative Learning and Teaching: Experiments Across the Disciplines Copyright © 2017 by Individual authors is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.
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How People Learn: Brain, Mind, Experience, and School: Expanded Edition (2000)
Chapter: 7 effective teaching: examples in history, mathematics, and science, 7 effective teaching: examples in history, mathematics, and science.
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: history, mathematics, and science. 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. To use Shulman’s (1987) language, effective teachers need pedagogical content knowledge (knowledge about how to teach in particular disciplines) rather than only knowledge of a particular subject matter.
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’ progress, 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 are sensitive to those aspects of the discipline that are especially hard or easy for new students to master.
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’ likely to hinder others” (McDonald and Naso, 1986:8). These conceptual barriers differ from discipline to discipline.
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. 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 the best of the best?
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 about the world?” The students begin enumerating their questions, “Can they be about silly, little things?” asks one student. “If they’re your questions that you really want answered, they’re neither silly nor little,” replies the teacher. 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.
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 class. 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 information 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 Johnson, “We decide what are the most compelling intellectual issues, devise ways to investigate those issues
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 learn more than we initially expected” (personal communication).
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. Says one student, “I just thought we were having fun. I didn’t realize we were learning, too!”
Barb Johnson’s teaching is extraordinary. It requires a wide range of disciplinary knowledge because she begins with students’ questions rather than with a fixed curriculum. Because of her extensive knowledge, she can map students’ questions 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 quickly 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 Johnson’s classes (Anderson and Smith, 1987).
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, 1987) that underlie expert teaching. They should help to clarify why effective teaching requires much more than a set of “general teaching skills.”
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 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, 1987). Unfortunately, many teachers do not present an exciting approach to history, perhaps because they, too, were taught in the dates-facts method.
Beyond Facts
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, 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 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 by Different Teachers
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 complete 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 question, “Evaluate the causes 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 .
Mr. Barnes’ comments on the actual content of the essays concentrated on the factual level. Ms. Kelsey’s comments addressed broader images of the nature of the domain, without neglecting important errors of fact. Overall, Mr. 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.
Studies of Outstanding History Teachers
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. This contributes to students’ beliefs that their textbooks are the history rather than a history.
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, then, is to put down on paper why they chose the items they did. In this way, the students explicitly articulate their underlying assumptions of what constitutes historical significance. Students’ responses are pooled, and he writes them on a large poster that he hangs on the classroom wall. This poster, which Bob Bain calls “Rules for Determining Historical Significance,” becomes a lightening rod for class discussions throughout the year, undergoing revisions and elaborations as students become better able to articulate their ideas.
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. In this instance, the students’ growing ability to understand the interpretative nature of history is aided by their teacher’s deep understanding of a fundamental principle of the discipline.
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 statement, “Every true history is contemporary history.” In the first week of the semester, Sterling thrust her students into the kinds of epistemological issues that one might find in a graduate seminar: “What is history?” “How do we know the past?” “What is the difference between someone who sits down to
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 colonies, but it was fought for English imperialism so you can’t blame them for not wanting to pay. The taxes were just the start of the slow turn toward 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 having 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. 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 a conspiracy in the royal government to extinguish America’s liberties. I think a lot of people also just were going with the flow, or were being pressured by the Sons of Liberty. 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. |
‘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 an evidentiary form of knowledge, not as clusters of fixed names and dates.
One might wonder about the advisability of spending 5 days “defining history” in a curriculum with so much to cover. But it is precisely Sterling’s framework of subject-matter knowledge—her overarching understanding of the discipline as a whole—that permits students entry into the advanced world of historical sense-making. 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. Sterling asked her students a question about the Constitutional Convention and “what were men able to do.” Paul took the question literally: “Uh, I think one of the biggest things that they did, that we talked about yesterday, was the establishment of the first settlements in the Northwest
—your topic sentence is weak —more factual detail would improve your essay —note spelling and grammar corrections
—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 a great place to start. —To make the essay however, you need to refine your organization strategies significantly. 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. In the beginning, tell what side you’re on: What made the colonists rebel— money, propaganda, conformity? In the middle, justify your view. What factors support your idea and will convince your reader? In the end, remind your reader again about your point of view. Go back and revise and hand this in again!
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area states.” But after 2 months of educating students into a way of thinking about history, Paul 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 understanding 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.
Debating the Evidence
Elizabeth Jensen prepares her group of eleventh graders to debate the following resolution:
Resolved: The British government possesses the legitimate authority to tax the American colonies.
As her students enter the classroom they arrange 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 booming voice. But today that voice is silent as her students take up the question of the legitimacy of British taxation in the American colonies.
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:
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? The French? Quoting from our friend Mr. Bailey on page 54, ‘By the settlement in Paris in 1763, French power was thrown completely off the continent of North America.’ Clearly not the French then. 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.
A loyalist responds:
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. …This sounds a little greedy, I mean what right do they have to take 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 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?
A rebel questions the loyalist about this:
Rebel: What benefits do we get out of paying taxes to the crown?
Loyalist: We benefit from the protection.
Rebel: (cutting in) Is that the only benefit you claim, protection?
Loyalist: Yes—and all the rights of an Englishman.
Rebel: Okay, then what about the Intolerable Acts…denying us rights of British subjects. What about the rights we are denied?
Loyalist: The Sons of Liberty tarred and feather people, pillaged homes— they were definitely deserving of some sort of punishment.
Rebel: So should all the colonies be punished for the acts of a few colonies?
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 bull.” Thirty-two students seem to be talking at once, while the presiding judge, a wiry student 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, 1991).
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 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. It is Elizabeth Jensen’s understanding of what she wants to teach and what adolescents already know that allows her to craft an activity that helps students get a feel for the domain that awaits them: decisions about rebellion, the Constitution, federalism, slavery, and the nature of a government.
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 demonstrate, 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 discipline 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 curriculum—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 necessary to teach his-
tory becomes clearer as one explores outstanding teaching in other disciplines.
MATHEMATICS
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, 1985; National Council of Teachers of Mathematics, 1989; Wilson, 1990a, b; Brophy, 1990; Thompson, 1992).
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 shows 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 teachers, 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 for instruction.
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 it 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 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.
Multiplication with Meaning
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, 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 bear 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 role as follows:
My role was to bring students’ 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 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. If one conceives of the teacher’s role 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 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. The first set used coin problems, 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 quantities could be grouped
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, rather than to rely on the teacher or book for verification of correctness. An example serves to highlight this approach; see Box 7.2 .
Lampert (1986:337) concludes:
…students used 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 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. It is worth noting that her goal of helping students see what is mathematically legitimate shapes the way in which she designs lessons to develop students’ understanding of two-digit multiplication.
Understanding Negative Numbers
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 another 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 discipline and of children as mathematical thinkers” (Ball, 1993). 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. Like 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.
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 reviewed a number of them—magic peanuts, money, game scoring, a frog on a number line, buildings with floors above and below ground. She decided to use the building model first and money later: she was acutely aware of the strengths and limitations of each
The teacher begins with a request for an example of a basic computation. Jessica: There were 12 jars, and each had 4 butterflies in it. Teacher: And if I did this multiplication and found the answer, what would I know about those Jessica: You’d know you had that many butterflies altogether. The teacher and students next illustrate Jessica’s story and construct a procedure for counting the butterflies. Sally: 10. 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 other ways of grouping the jars, for example, into two groups of 6 jars. The students are obviously surprised that 6×4 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 ). It is a sign that she needs to do many more activities involving different groupings. In subsequent lessons, 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 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. |
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 particular 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 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 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 another 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 second floor?” Working with the building model allowed students to generate a number of observations. For example, one student noticed that “any number below zero plus 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 then used a model of money as a second representational context for exploring negative numbers, noting that it, too, has limitations.
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 students’ 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 reasonable and likely to be correct, rather than depending on text or teacher for confirmation of correctness.
Guided Discussion
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 cognitively guided instruction helps illustrate 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-
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 decisions to assist students to construct their mathematical knowledge. In this approach, the idea of domains of knowledge for teaching (Shulman, 1986) is extended to include teachers’ knowledge of individual learners in their classrooms.
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’s thinking and uses that understanding to guide her teaching. A portrait of Ms. Keith’s classroom reveals also how her knowledge of mathematics and pedagogy influence her instructional decisions.
Word problems form the basis for almost 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 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-10 materials, such as base-10 blocks and counting frames, to solve word problems involving multidigit numbers.
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 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: she wants a variety of solutions presented so that students will have an opportunity 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 that is actually incorrect to be presented so that she can initiate a discussion of a common misconception. Or she
may select a solution that is more sophisticated than 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.
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 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 instructional decisions give her clear diagnoses of individual students’ current state 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 progress.
Model-Based Reasoning
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–12). 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, 1989; Hestenes, 1992; Lehrer and Romberg, 1996a, b; Schauble et al., 1995; see Box 7.3 ).
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-
Physical models, like models of solar systems or elbows, are 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 function 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 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, 1996a, b). |
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.
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
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 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, 1996b).
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 here 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 content knowledge, and knowledge of students in general.
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., 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., 1981). 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 7.1 . Furthermore, similar differences emerged in problem categorization: students who performed the hierarchical analyses considered principles (as opposed to surface features) more often in
deciding whether or not two problems would be solved similarly; see Figure 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 practice is an efficient way to promote expertise. 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., 1996). Undergraduate engineering students were instructed to write qualitative strategies for solving problems before 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 applying 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-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 7.3 .
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, 1984). 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, 1984). 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
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FIGURE 7.1 Effects of two methods of training on problem-solving, final answer, and principle understanding. SOURCE: Dufresne et al. (1992).
non-hierarchically. Similarly, students who received a hierarchical organization of problem-solving strategies performed much better than subjects who received the same strategies organized non-hierarchically. Thus, helping students to organize their knowledge is as important as the knowledge itself, since knowledge organization is likely to affect students’ intellectual performance.
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 to solve on their own (an instructional practice used in all the sciences), it is highly
![explain the utility of problem solving method in history teaching](https://nap.nationalacademies.org/openbook/0309070368/xhtml/images/p2000495fg175001.jpg)
FIGURE 7.2 Effects of two methods of training on considering principles for categorizing problems. SOURCE: Dufresne et al. (1992).
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 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
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problem solving, historical thinking skills, learning assessment . Introduction. Among the challenges that faculty encounter is facilitating active engagement with their discipline within classrooms of diverse undergraduate students (Calder, 2006; Rendon, 2009). We face this challenge regularly when teaching history.
There are several methods a teacher can use to make history more vibrant. Active learning techniques, films, library research, and historical fiction can all be used to make teaching and learning on history more invigorating. Regardless of what methods are used, however, it is important to apply a humanistic approach when teaching history.
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 list of physics terms or equations that were covered in the course, but the students are not able to ...
The problem solving may reveal as one of the most important pedagogical methods as it addresses the central question of pedagogy: by what means may we affect the representations of a human person ...
Problem-Solving Fellows Program Undergraduate students who are currently or plan to be peer educators (e.g., UTAs, lab TAs, peer mentors, etc.) are encouraged to take the course, UNIV 1110: The Theory and Teaching of Problem Solving. Within this course, we focus on developing effective problem solvers through students' teaching practices.
Teachers are far too busy teaching in class, and expertise with pedagogical problem solving is, we argue, largely tacit. Teacher educators make the tacit explicit for novices, so the Pedagogical Problem Typology may help this dialogue. We now explain each problem before illustrating them with a thick description of excerpts from the data.
Working on solutions. In the solution phase, one develops and then implements a coherent plan for solving the problem. As you help students with this phase, you might ask them to: identify the general model or procedure they have in mind for solving the problem. set sub-goals for solving the problem. identify necessary operations and steps.
The first teaching approach was based on individual problem-solving, the second involved problem-solving through discussion. Students in the discussion condition were prompted to discuss the value of studying history with the entire class, and were required to solve historical problems in groups.
Make students articulate their problem solving process . In a one-on-one tutoring session, ask the student to work his/her problem out loud. This slows down the thinking process, making it more accurate and allowing you to access understanding. When working with larger groups you can ask students to provide a written "two-column solution.".
In this study, 100 Italian eighth graders were divided into two groups to compare the effects of two instructional interventions - the first based on problem-solving through discussion, the ...
By "living" the decisions through problem-based simulations, our students would collectively be better prepared to engage in the larger questions that are debated in the discipline of history. Challenge Cycles. What did this look like in World History? We created challenge cycles based on each of the eras into which the course was divided.
PBL is a student-centered approach to learning that involves groups of students working to solve a real-world problem, quite different from the direct teaching method of a teacher presenting facts and concepts about a specific subject to a classroom of students. Through PBL, students not only strengthen their teamwork, communication, and ...
A last word, in which is stated what the writer believes to be a sensible, conservative attitude toward the use of problem solving in. teaching the social sciences. 1. Concepts must be present in the mind which are germane to problem on hand-. is the power to profit by experience with effort. The experience.
The problem-solving method of teaching is a student-centered approach to learning that focuses on developing students' problem-solving skills. In this method, students have to face real-world problems to solve. They are encouraged to use their knowledge and skills to provide solutions. The teacher acts as a facilitator, providing guidance and ...
In Dewey's (1916, p. 177) vision for teacher preparation, methods arise from a thorough understanding of one's disciplinary domain, but subject matter is always balanced by a deep understanding of the principles of learning and teaching: "In brief, the method of teaching is the method of an art, of action intelligently directed by ends ...
The problem-solving method is an effective teaching strategy that promotes critical thinking, creativity, and collaboration. It provides students with real-world problems that require them to apply their knowledge and skills to find solutions. By using the problem-solving method, teachers can help their students develop the skills they need to ...
Abstract. I describe the application of the problem method, sometimes known as the case study method, to the teaching of undergraduate research methods. Problems are assigned in advance, students use course material to solve the problem, and the solutions are discussed in class. This method is particularly applicable to courses in research ...
The teaching methods described in these papers involve two phases: first a generation (or invention) phase where students struggle to come up with a solution to the problem posed and second, a consolidation (or instruction) phase where they are given the canonical solution to the problem together with direct instruction in applying the canonical solution.
Advantages of Problem-Based Learning. 1. Development of Long-Term Knowledge Retention. Students who participate in problem-based learning activities can improve their abilities to retain and recall information, according to a literature review of studies about the pedagogy.. The literature review states "elaboration of knowledge at the time of learning" -- by sharing facts and ideas ...
Abstract. Problem-based learning is a recognized teaching method in which complex real-world problems are used as the vehicle to promote student learning of concepts and principles as opposed to ...
maps traces back centuries. These pictorial methods record knowledge and model systems, and have a long history in learning, brainstorming, memory, visual thinking, and problem solving by educators, engineers, psychologists, and others. Buzan (1993) argues that traditional outlines rely
The problem-solving method of teaching is the learning method that allows children to learn by doing. This is because they are given examples and real-world situations so that the theory behind it can be understood better, as well as practice with each new concept or skill taught on top of what was previously learned in class before moving onto another topic at hand.
What is the problem solving method of teaching? Luther A. Mahan, Luther A. Mahan. Stout State University, Menomonie, Wisconsin. Search for more papers by this author. Luther A. Mahan, Luther A. Mahan. Stout State University, Menomonie, Wisconsin. Search for more papers by this author.