Preparing Teachers Building Evidence for Sound Policy (2010) / Chapter Skim
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6 Preparing Mathematics Teachers
6 Preparing Mathematics Teachers
Pages 103-124
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From page 103... ...
Preparation for middle grades mathematics teachers varies from place to place, and certification requirements reflect the ambiguous status of middle school. For example, many states offer grade K-8 certification to teachers prepared as generalists, as well as grade 7-12 certification to those specifically prepared to teach mathematics.
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From page 104... ...
The time allotted for mathematics content in the preparation of many elementary and middle school teachers is unlikely to be adequate, and many secondary school mathematics teachers (including those in the middle grades who are prepared as specialists) may also be receiving training that does not prepare them to teach advanced-level mathematics (e.g., algebra, geometry, and trigonometry)
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From page 105... ...
, the Conference Board of the Mathematical Sciences, and, most recently, the National Mathematics Advisory Panel have published some of the most widely known documents. Each of those documents is the product of extensive efforts to collaborate, develop consensus, and distill practical guidance from theoretical models as well as research, and we have relied heavily on them.
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From page 106... ...
A series of more recent publications have also been important. Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000)
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From page 107... ...
. The report describes mathematical proficiency as having five intertwined strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
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From page 108... ...
All of these documents -- Principles and Standards for School Mathematics, Curriculum Focal Points, Adding It Up, and Foundations for Success -- attempt to answer questions about what successful mathematics students know. But none of the documents could rely on empirical research that demonstrates that students who have mastered these domains of mathematical knowledge and skill are more productive or successful at their schoolwork or in life.
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From page 109... ...
Yet there is consensus that for students to be successful in high school mathematics courses, they need preparation in the basic topic areas -- including number, operations, and fractions -- and there is little disagreement that students also need to develop conceptual understanding, procedural fluency, and confidence in their capacity to learn mathematics. In general, successful mathematics learning entails the cumulative development of increasingly sophisticated conceptual understanding, procedural fluency, and capacity for reasoning and problem solving.4 Moreover, there is broad general agreement about the topics to be included in the curriculum for grades K-8, though the relative emphasis they should receive and their exact placement by grade is not settled.
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From page 110... ...
, and both have been influential in mathematics education. How Students Learn begins by applying three broad principles about learning to the teaching of mathematics: that engaging students' prior conceptions is critical to successful learning, that learning with understanding entails an integration of factual knowledge and conceptual frameworks, and that students need to learn how to monitor their learning.
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From page 111... ...
The panel did, however, find some evidence to support the effectiveness of cooperative learning practices and the regular use of formative assessment in elementary mathematics instruction as a tool for tailoring instruction to students' needs. The panel called attention to the limited amount of rigorous empirical research available to answer questions about mathematics teaching and learning, and it recommended a variety of research to test hypotheses about the most effective approaches.
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From page 112... ...
These studies supported the development of more nuanced descriptions of teachers' knowledge and skills by illuminating the ways that the mathematical knowledge needed for teaching differs from the mathematical knowledge needed to succeed in advanced courses. The concept of pedagogical content knowledge gave a name to the knowledge of content as it applies to and can be used in teaching (see, e.g., Shulman, 1986; Ball, Lubienski, and Mewborn, 2001)
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From page 113... ...
An additional obvious difference between the requirements for elementary teachers and high school teachers is that teaching older students carries a greater demand for both technical skills and abstract reasoning. These developments in the understanding of mathematics teaching provide a critical framework for teacher education, although they do not provide empirical support for a concrete description of precisely how to teach mathematics (Ball, Lubienski, and Mewborn, 2001)
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From page 114... ...
Moreover, the report observes that the challenges of teaching each level are distinct and require different preparation, as current certification requirements reflect. It notes in particular that teachers of middle grades mathematics often "have been prepared to teach elementary school mathematics and lack the broader background needed to teach the more advanced mathematics of the middle grades" (Conference Board of the Mathematical Sciences, 2001, see http://www.cbmsweb.org/MET_Document/chapter_4.htm [November 2009]
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From page 115... ...
at least 12 semester-hours on fundamental ideas of school mathematics appropriate for middle grades teachers. 5 The National Council on Teacher Quality (Greenberg and Walsh, 2008)
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From page 116... ...
a 6-hour capstone course connecting their college mathematics courses with high school mathematics. David Monk (1994)
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From page 117... ...
The relevant body of work on what instructional opportunities are most valuable for mathematics teachers is growing but thus far is largely descriptive, and it has not identified causal relationships between specific aspects of preparation programs and measures of prospective teachers' subsequent effectiveness. Nevertheless, the field of mathematics education has established a firm consensus that to prepare effective K-12 mathematics teachers, a program should provide prospective teachers with the knowledge and skills described by the Conference Board of the Mathematical Sciences: • a deep understanding of the mathematics they will teach, • ourses that focus on a thorough development of basic mathemati c cal ideas, and • ourses that develop careful reasoning and mathematical "common c sense" in analyzing conceptual relationships and solving problems, and courses that develop the habits of mind of a mathematical thinker.
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From page 118... ...
We had two major sources of information about how teacher preparation in mathematics is currently being conducted -- state requirements and coursework -- although much of the information is somewhat indirect. State Requirements We begin with the requirements states have established for licensing mathematics teachers, which influence the goals teacher preparation programs set for themselves.
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From page 119... ...
With the analyses of Florida commissioned for our study, we were able to look in more detail at the average number of mathematics credits earned by Florida teachers by certification area: see Table 6-1. Although these data do not provide information about the content or nature of the coursework, they do suggest significant overall exposure to mathematics, corresponding roughly to 4 three-credit courses for elementary teachers, 15 courses for teachers certified for middle school mathematics, and 19 courses for teachers certified for high school mathematics.8 7 These results correspond to McCrory's emerging results as well as the self-reports of new teachers surveyed as part of the Teacher Pathways Project being conducted by the University at Albany and Stanford University (see http://www.teacherpolicyresearch.org/ TeacherPathwaysProject/tabid/81/Default.aspx [February 2010]
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From page 120... ...
. Current elementary teachers and middle and secondary level mathematics teachers were asked about the extent to which their teacher preparation program gave them the opportunity to do and learn a variety of things, such as learning about the typical difficulties students have with aspects of mathematics.
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From page 121... ...
teachers scored in the middle or close to the bottom in comparison with teachers in the countries whose students performed well on the TIMSS study. The results suggested possible differences in preparation across countries, and the study's authors concluded that the TABLE 6-2 New York City Teachers' Reported Exposure to Mathematics Preparation Opportunity to Learn Mathematics Education Mean Response on 1-5 Scalea Approach or Strategy Elementary Teachers Learn typical difficulties students have with 2.71 place value.
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From page 122... ...
conducted an analysis of career pathways for that state's mathematics and science teachers. They found that 10 percent of middle school and 12 percent of high school mathematics teachers were teaching out of field, and that 40 percent of novice high school mathematics teachers were not well prepared (defined as lacking a preliminary credential)
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From page 123... ...
Regarding what students need to know, mathematicians and mathematics educators are in accord that successful mathematics learning is most likely when core topics in school mathematics and the five strands of mathematical proficiency identified in Adding It Up (conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition) are interwoven at each level of schooling, and students are provided with a coherent curriculum in which clear objectives, based on a logical conception of the mathematics learning trajectory, guide each year of mathematics study.
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From page 124... ...
The data regarding what is currently happening in teacher preparation for mathematics is extremely limited, but the information that is available clearly indicates that such preparation is not sufficient. That is, because it appears that many preparation programs fall short of guidelines such as The Mathematical Education of Teachers recommendations, it is likely that: Conclusion 6-2: Many, perhaps most, mathematics teachers lack the level of preparation in mathematics and teaching that the professional community deems adequate to teach mathematics.
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