In This Issue...
Ken Bain is founding director of the Center for Teaching Excellence at New York University and the author of What the Best College Teachers Do (Harvard University Press, 2004), a book in which he describes and analyzes the findings of a study on the characteristics of highly effective teachers. Excerpts from this book can be downloaded free of charge from http://www.hup.harvard.edu/pdf/BAIBES_excerpt.pdf Some of the findings follow below (numbers added):
1. Without exception, outstanding teachers know their subjects extremely well. They are all active and accomplished scholars, artists, or scientists...
They have an unusually keen sense of the histories of their disciplines, including the controversies that have swirled within them, and that understanding seems to help them reflect deeply on the nature of thinking within their fields. They can then use that ability to think about their own thinking--what we call "metacognition"--and their understanding of the discipline qua discipline to grasp how other people might learn. They know what has to come first, and they can distinguish between foundational concepts and elaborations or illustrations of those ideas. They realize where people are likely to face difficulties developing their own comprehension, and they can use that understanding to simplify and clarify complex topics for others, tell the right story, or raise a powerfully provocative question... The people we analyzed have generally cobbled together from their own experiences working with students conceptions of human learning that are remarkably similar to some ideas that have emerged in the research and theoretical literature on cognition, motivation, and human development...
2. The teachers we encountered believe everybody constructs knowledge and that we all use existing constructions to understand any new sensory input. When these highly effective educators try to teach the basic facts in their disciplines, they want students to see a portion of reality the way the latest research and scholarship in the discipline has come to see it. They don't think of it as just getting students to "absorb some knowledge," as many other people put it. Because they believe that students must use their existing mental models to interpret what they encounter, they think about what they do as stimulating construction, not "transmitting knowledge." Furthermore, because they recognize that the higher-order concepts of their disciplines often run counter to the models of reality that everyday experience has encouraged most people to construct, they often want students to do something that human beings don't do very well: build new mental models of reality...
3. They conduct class and craft assignments in a way that allows students to try their own thinking, come up short, receive feedback, and try again. They give students a safe space in which to construct ideas, and they often spend a great deal of time creating a kind of scaffolding to help students engage in that construction (which is different from the popular notion of "covering" the material, but in ways that are sometimes difficult to grasp). Because they attempt to place students in situations in which some of their mental models will not work, they try to understand those models and the emotional baggage attached to them. They listen to student conceptions before challenging them. Rather than telling students they are wrong and then providing the "correct" answers, they often ask questions to help students see their own mistakes...
4. They believe that students must learn the facts while learning to use them to make decisions about what they understand or what they should do. To them, "learning" makes little sense unless it has some sustained influence on the way the learner subsequently thinks, acts, or feels. So they teach the "facts" in a rich context of problems, issues, and questions...
(2) How Students Learn: History, Mathematics, and Science in the Classroom (Edited by M. Suzanne Donovan and John D. Bransford)Source: National Research Council (publisher: National Academies Press)
How do you get a fourth-grader excited about history? How do you even begin to persuade high school students that mathematical functions are relevant to their everyday lives? In this volume--How Students Learn: History, Mathematics, and Science in the Classroom--practical questions that confront every classroom teacher are addressed using the latest research on cognition, teaching, and learning.
This volume builds on the discoveries detailed in the best-selling book, How People Learn. The findings are presented in a way that teachers can use immediately, to revitalize their work in the classroom for even greater effectiveness.
The book explores how the principles of learning can be applied in teaching history, science, and math topics at three levels: elementary, middle, and high school. Leading educators explain in detail how they developed successful curricula and teaching approaches, presenting strategies that serve as models for curriculum development and classroom instruction.
The book explores the importance of balancing students' knowledge of historical fact against their understanding of concepts, such as change and cause, and their skills in assessing historical accounts. It discusses how to build straightforward science experiments into true understanding of scientific principles. And it shows how to overcome the difficulties in teaching math to generate real insight and reasoning in math students. It also features illustrated suggestions for classroom activities.
How Students Learn offers a highly useful blend of principle and practice. It will be important not only to teachers, administrators, curriculum designers, and teacher educators, but also to parents and the larger community concerned about children's education. It can be browsed free of charge at http://www.nap.edu/books/0309074339/html/
(3) How Students Learn: Mathematics in the Classroom (Edited by M. Suzanne Donovan and John D. Bransford)Source: National Research Council (publisher: National Academies Press)
URL (online book): http://www.nap.edu/books/0309089492/html/
How Students Learn: Mathematics in the Classroom is comprised of chapters from How Students Learn: History, Mathematics, and Science in the Classroom that are relevant to the teaching of mathematics at the elementary, middle, and high school levels. This volume is available at the above Web sites for free browsing or for purchase.
[Book Excerpt] "Chapter 5: Mathematical Understanding: An Introduction" (by Karen C. Fuson, Mindy Kalchman, and John D. Bransford, pp. 217-219)
For many people, free association with the word "mathematics" would produce strong, negative images. Gary Larson published a cartoon entitled "Hell’s Library" that consisted of nothing but book after book of math word problems. Many students--and teachers--resonate strongly with this cartoon’s message. It is not just funny to them; it is true.
Why are associations with mathematics so negative for so many people? If we look through the lens of How People Learn, we see a subject that is rarely taught in a way that makes use of the three principles that are the focus of this volume. Instead of connecting with, building on, and refining the mathematical understandings, intuitions, and resourcefulness that students bring to the classroom (Principle 1), mathematics instruction often overrides students' reasoning processes, replacing them with a set of rules and procedures that disconnects problem solving from meaning making. Instead of organizing the skills and competences required to do mathematics fluently around a set of core mathematical concepts (Principle 2), those skills and competencies are often themselves the center, and sometimes the whole, of instruction. And precisely because the acquisition of procedural knowledge is often divorced from meaning making, students do not use metacognitive strategies (Principle 3) when they engage in solving mathematics problems...
A recent report of the National Research Council, Adding It Up, reviews a broad research base on the teaching and learning of elementary school mathematics. The report argues for an instructional goal of "mathematical proficiency," a much broader outcome than mastery of procedures. The report argues that five intertwining strands constitute mathematical proficiency:
1. Conceptual understanding--comprehension of mathematical concepts, operations, and relations
2. Procedural fluency--skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
3. Strategic competence--ability to formulate, represent, and solve mathematical problems
4. Adaptive reasoning--capacity for logical thought, reflection, explanation, and justification
5. Productive disposition--habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy
These strands map directly to the principles of How People Learn. Principle 2 argues for a foundation of factual knowledge (procedural fluency), tied to a conceptual framework (conceptual understanding), and organized in a way to facilitate retrieval and problem solving (strategic competence). Metacognition and adaptive reasoning both describe the phenomenon of ongoing sense making, reflection, and explanation to oneself and others. And...the preconceptions students bring to the study of mathematics affect more than their understanding and problem solving; those preconceptions also play a major role in whether students have a productive disposition toward mathematics, as do, of course, their experiences in learning mathematics.
The chapters...on whole number, rational number, and functions look at the principles of How People Learn as they apply to those specific domains. In this introduction, we explore how those principles apply to the subject of mathematics more generally...
[Letter from Patrick Gonzales]
I am pleased to announce that the International Study Center at Boston College has just released a report on the scaling of the TIMSS 2003 cognitive domains in mathematics. The report [entitled, IEA's TIMSS 2003 International Report on Achievement in the Mathematics Cognitive Domains: Findings from a Developmental Project] documents the process undertaken to produce scales in three cognitive domains: knowing, applying, and reasoning. Included are the final scales showing differences among countries, as well as within countries.
The report can be accessed at http://isc.bc.edu/timss2003i/mcgdm.html [free download] and is also available for purchase.
In addition, the new TIMSS 2007 Frameworks have also been posted. A new Frameworks document was undertaken to better define the mathematics and science included in the curricula of the now almost 60 countries participating in TIMSS. As a result, some content categories have been streamlined to better account for the ways in which mathematics and science is organized around the world, and reflecting changes in emphases that have occurred since the publication of the 2003 Frameworks. Moreover, the new Frameworks provide a more concise discussion of the cognitive dimensions of mathematics and science learning, based on the developmental work conducted for the mathematics cognitive domain scaling report. The new Frameworks can be accessed at http://isc.bc.edu/TIMSS2007/frameworks.html.
It is the intention of the IEA to produce content and cognitive domain scales for TIMSS 2007, in mathematics and science, at both grades 4 and 8.
The addition of the cognitive domain scales provides educators and policymakers with critical information without requiring increased burden from study respondents.
[Note: A few excerpts from IEA's TIMSS 2003 International Report on Achievement in the Mathematics Cognitive Domains appear below.]
"Based on this process and final confirmatory rounds of classifying the TIMSS 2003 fourth- and eighth-grade items, the experts felt comfortable with three cognitive domains:
• Knowing Facts, Procedures, and Concepts,
• Applying Knowledge and Understanding,
"The first domain, knowing facts, procedures, and concepts, covers what the student needs to know, while the second, applying knowledge and conceptual understanding, focuses on the ability of the student to apply what he or she knows to solve routine problems or answer questions. The third domain, reasoning, goes beyond the solution of routine problems to encompass unfamiliar situations, complex contexts, and multi-step problems" (p. 7).
"There was a substantial number of items in each domain: 65 in knowing, 93 in applying, and 36 in reasoning at eighth grade; and 58 in knowing, 63 in applying, and 38 in reasoning at fourth grade. Within each domain, there was a good spread of item type (constructed-response or multiple-choice) at both grades, although as might be expected, relatively more of the knowing items were multiple choice and relatively more reasoning items constructed response" (p. 8).
[Note: Gender differences were reported in Chapter 3 of this report.]
"At the eighth grade, girls had the advantage in more countries in the knowing domain of mathematics and, even more so in the reasoning domain. Internationally across the TIMSS 2003 participants, girls had significantly higher achievement, on average, than boys in both these domains. Boys had the advantage in more countries in the applying domain.
"At the fourth grade, while performance was about the same internationally for boys and girls in the knowing domain, there was a significant difference, on average, favoring boys in the applying domain. Also, boys had significantly higher achievement in considerably more countries than did girls. In the reasoning domain, there was essentially no difference internationally between [fourth-grade] boys and girls, but in the few countries where significant differences were found, girls had higher performance" (pp. 41-42).
COMET is sponsored in part by a grant from the California Mathematics Project.
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