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While no additional big ideas, essential
questions, or enduring understandings are listed for this standard,
the mathematical processes are imbedded in the content-specific
ideas, questions, and understandings delineated for the first four
standards. References to the relevant processes can be found above. |
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4.5 A.
Problem Solving |
|
Descriptive Statement: Problem posing and problem solving
involve examining situations that arise in mathematics and other
disciplines and in common experiences, describing these situations
mathematically, formulating appropriate mathematical questions, and
using a variety of strategies to find solutions. Through problem
solving, students experience the power and usefulness of
mathematics. Problem solving is interwoven throughout the grades to
provide a context for learning and applying mathematical ideas. |
|
Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
|
1.
Learn mathematics through problem solving, inquiry, and
discovery. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
2.
Solve problems that arise in mathematics and in other contexts
(cf. workplace readiness standard 8.3).
·
Open-ended problems
·
Non-routine problems
·
Problems with multiple solutions
·
Problems that can be solved in several ways
|
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
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3.
Select and apply a variety of
appropriate problem-solving strategies (e.g., “try a simpler problem” or “make
a diagram”) to solve problems. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
4.
Pose problems of various types and levels of difficulty. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
5.
Monitor their progress and reflect on the process of their problem
solving activity. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
6.
Distinguish relevant from irrelevant information, and identify missing
information. |
Instructional Focus:
• This CPI was added by the State Board of Education on January 9,
2008..
• Assessment of this CPI is within the context of one or more of the
content CPIs 4.1 through 4.4. |
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4.5 B. Communication |
|
Descriptive Statement: Communication of mathematical ideas
involves students’ sharing their mathematical understandings in oral
and written form with their classmates, teachers, and parents. Such
communication helps students clarify and solidify their
understanding of mathematics and develop confidence in themselves as
mathematics learners. It also enables teachers to better monitor
student progress. |
|
Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
|
1.
Use communication to organize and clarify their mathematical
thinking.
·
Reading and writing
·
Discussion, listening, and questioning |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
2.
Communicate their mathematical
thinking coherently and clearly to peers, teachers, and others, both orally and
in writing. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
3.
Analyze and evaluate the mathematical thinking and strategies of
others. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
4.
Use the language of mathematics to express mathematical ideas
precisely. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
4.5 C. Connections |
|
Descriptive Statement: Making connections involves seeing
relationships between different topics, and drawing on those
relationships in future study. This applies within mathematics, so
that students can translate readily between fractions and decimals,
or between algebra and geometry; to other content areas, so that
students understand how mathematics is used in the sciences, the
social sciences, and the arts; and to the everyday world, so that
students can connect school mathematics to daily life. |
|
Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
|
1.
Recognize recurring themes across mathematical domains (e.g.,
patterns in number, algebra, and geometry). |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
2.
Use connections among mathematical ideas to explain concepts
(e.g., two linear equations have a unique solution because the lines they
represent intersect at a single point). |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
3.
Recognize that mathematics is used in a variety of contexts
outside of mathematics. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
4.
Apply mathematics in practical situations and in other
disciplines. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
5.
Trace the development of
mathematical concepts over time and across cultures (cf. world languages and
social studies standards). |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
6.
Understand how mathematical ideas interconnect and build on one
another to produce a coherent whole. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
4.5 D. Reasoning |
|
Descriptive Statement: Mathematical reasoning is the
critical skill that enables a student to make use of all other
mathematical skills. With the development of mathematical reasoning,
students recognize that mathematics makes sense and can be
understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and
describe solutions, and recognize how those solutions can be
applied. |
|
Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
|
1.
Recognize that mathematical facts, procedures, and claims must be
justified. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
2.
Use reasoning to support their
mathematical conclusions and problem solutions. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
3.
Select and use various types of reasoning and methods of proof. |
This may be included in classroom enrichment
activities at this grade level, but is more of a focus at secondary
grade levels. |
|
4.
Rely on reasoning, rather than answer keys, teachers, or peers, to
check the correctness of their problem solutions. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
5.
Make and investigate mathematical conjectures.
·
Counterexamples as a means of disproving conjectures
·
Verifying conjectures using informal reasoning or proofs.
|
This may be included in classroom enrichment
activities at this grade level, but is more of a focus at higher
grade levels. |
|
6.
Evaluate examples of mathematical reasoning and determine whether
they are valid. |
This is more of a focus at secondary grade
levels. |
|
4.5 E. Representations |
|
Descriptive Statement: Representations refers to the use
of physical objects, drawings, charts, graphs, and symbols to
represent mathematical concepts and problem situations. By using
various representations, students will be better able to communicate
their thinking and solve problems. Using multiple representations
will enrich the problem solver with alternative perspectives on the
problem. Historically, people have developed and successfully used
manipulatives (concrete representations such as fingers, base ten
blocks, geoboards, and algebra tiles) and other representations
(such as coordinate systems) to help them understand and develop
mathematics. |
|
Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
|
1.
Create and use representations
to organize, record, and communicate mathematical ideas.
·
Concrete representations (e.g.,
base-ten blocks or algebra tiles)
·
Pictorial representations
(e.g., diagrams, charts, or tables)
·
Symbolic representations (e.g.,
a formula)
·
Graphical representations
(e.g., a line graph)
|
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
2.
Select, apply, and translate among mathematical representations to
solve problems. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
3.
Use representations to model and interpret physical, social, and
mathematical phenomena. |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
4.5 F. Technology |
|
Descriptive Statement: Calculators and computers need to
be used along with other mathematical tools by students in both
instructional and assessment activities. These tools should be used,
not to replace mental math and paper-and-pencil computational
skills, but to enhance understanding of mathematics and the power to
use mathematics. Students should explore both new and familiar
concepts with calculators and computers and should also become
proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems). |
|
Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
|
1.
Use technology to gather, analyze, and communicate mathematical
information. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
2.
Use computer spreadsheets, software, and graphing utilities to
organize and display quantitative information. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
3.
Use graphing calculators and computer software to investigate
properties of functions and their graphs. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
4.
Use calculators as problem-solving tools (e.g., to explore
patterns, to validate solutions). |
Assessment of this CPI is within the context of
one or more of the content CPIs 4.1 through 4.4. |
|
5.
Use computer software to make
and verify conjectures about geometric objects. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
|
6.
Use computer-based laboratory technology for mathematical
applications in the sciences. |
Instructional Focus:
• This CPI is largely an instructional CPI and is assessed within
the context of one or more of the content CPIs 4.1 through 4.4. |
