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Grades 6 - 8: Number and Operations on 2009-10-18
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In the middle grades, students
should expand their repertoire of meanings, representations, and uses
for nonnegative rational numbers
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Chapter 6: Standards for Grades 6 - 8 on 2009-10-18
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ee mathematics as an exciting, useful, and creative
field of study
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Middle-grades students should
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During this time, many students will
solidify conceptions about themselves as learners of mathematics
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Grades 3 - 5: Representation on 2009-10-18
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Students in grades 3–5 should continue to develop the habit of represent
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ng
problems and ideas to support and extend their reasoning
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In grades 3–5, students should create representations
that are
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more detailed and accurate than is expected in the primary grades
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representations should be portrayed as useful tools
» for building understanding, for
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communicating information,
and for demonstrating reasoning
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Learning to record
or represent thinking in an organized way, both in solving a probl
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m and
in sharing a solution, is an acquired skill for many students
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As students discuss their ideas and begin to develop conjectures based on representations
of the problem, the teacher might want to represent the stude
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nts' thinking
in other ways in order to support and extend their ideas
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Grades 3 - 5: Connections on 2009-10-18
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important that teachers plan ahead to integrate mathematics into
other subject areas and experiences that students will have during the
year
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connections between mathematics and
everyday experience, and connections between mathematics and other disciplines
can support learning.
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Standards for School Mathematics: Representation on 2009-10-18
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he ways in which mathematical ideas are represented is fundamental to how people
can understand and use those ideas.
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representation refers both to process and to product—in
other words, to the act of capturing a mathematical concept or relationship
in some form and to the form itself
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Some forms of representation—such as diagrams, graphical displays, and symbolic
expressions—have long been part of school mathematics
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often been taught and learned as
if they were ends in themselves. Representations should be treated as
essential elements in supporting students' understanding of mathematical
concepts and relationships
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Students should understand that written representations of mathematical
ideas are an essential part of learning and doing mathematics
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ir naive conceptions and the mathematical formalisms.
Research indicates, however, that students at all levels need to work
at developing their understandings of the complex ideas captured in conventional
representations
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record a solution method and
to describe the method to others. Teachers can gain valuable insights
into students' ways of interpreting and thinking about mathematics by
looking at their representations
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Students' use of representations can help make
mathematical ideas more concrete and available for reflection
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model is used to refer to physical materials
with which students work in school—manipulative models
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exemplification or simulation
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Standards for School Mathematics: Connections on 2009-10-18
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When students can connect mathematical ideas, their understanding is deeper and
more lasting
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mathematics
is an integrated field of study
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To emphasize the connections, teachers must know the needs
of their students as well as the mathematics that the students studied
in the preceding grades and what they will study in the following grades
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The notion that mathematical ideas are connected should permeate the
school mathematics experience at all levels
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hroughout the pre-K–12 span, students should routinely ask themselves,
"How is this problem or mathematical topic like things I have studied
before?
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As students progress through their school mathematics experience, their
ability to see the same mathematical structure in seemingly different
settings should increase
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mathematics
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p.
65 |
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School mathematics experiences at all levels should include opportunities
to learn about mathematics by working on problems arising in
» contexts outside of mathematics |
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opportunity for students to experience mathematics in a context is important
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Data analysis and statistics are useful in helping students clarify issues related
to their personal lives
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Standards for School Mathematics: Communication on 2009-10-18
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Communication
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Instructional programs from prekindergarten through grade
12 should enable all students to—
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Communication is a
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ommunication process also helps build meaning and permanence for
ideas and makes them public
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Students who are involved in discussions
in which they justify solutions—especially in the face of disagreement—will
gain better mathematical understanding as they work to convince their
peers about differing points of view
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Because mathematics is so often conveyed in symbols, oral and written communication
about mathematical ideas is not always recognized as an important part
of mathematics education.
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Students need to work with mathematical tasks that are worthwhile topics of discussion
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Students gain insights into their thinking when they present their methods
for solving problems, when they justify their reasoning to a
» classmate or teacher, or when they formulate a question
about something that is puzzling to them
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they share responsibility
with the teacher for the learning that occurs in the lesson
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Reflection and communication are intertwined processes
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Writing in mathematics can also help students consolidate their thinking because
it requires them to reflect on their work and clarify their thoughts about
the ideas developed in the lesson.
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Students need opportunities to test their ideas on the basis of shared
knowledge in the mathematical community of the classroom to see whether
they can be understood and if they are sufficiently convincing
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teachers must build a community
in which students will feel free to express their ideas
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students should
gradually take more responsibility for participating in whole-class discussions
and responding to one another directly
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By the end of the high school years, students should be able to write
well-constructed mathematical arguments using formal vocabulary.
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Examining and discussing both exemplary and problematic pieces of mathematical
writing can be beneficial at all levels.
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they should express themselves increasingly
clearly and coherently
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Often, a student who has one way of seeing a problem
can profit from another student's view, which may reveal a different aspect
of the problem
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t is difficult for students to learn to consider, evaluate, and build
on the thinking of others, especially when their peers are still developing
their own mathematical understandings. A good setting in which young students
can share and analyze one another's strategies is in solving arithmetic
problems, where students' invented strategies can become objects of discussion
and critique
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Teachers can help students see that some words that are used in everyday
language, such as similar, factor, area, or
function, are used in mathematics with different or more-precise
meanings
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However, it is important to
avoid a premature rush to impose formal mathematical language
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BrainPOP - Animated Educational Site for Kids - Science, Social Studies, English, Math, Arts & Music, Health, and Technology on 2009-10-14
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Apple Learning Interchange - What is Geometry on 2009-10-05
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Interactive Mathematics Miscellany and Puzzles, Index on 2009-10-01
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