modeling and using tools: mathematical practices of the common core
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Modeling and using tools: Mathematical Practices of the Common Core. Statewide Instructional Technology Project. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
MODELING AND
USING TOOLS: MATHEMATICAL
PRACTICES OF THE COMMON CORE
Statewide Instructional Technology Project
Introduction
This is the fourth in a series of five webinars on the Standards of Math Practice. Links to resources, including recordings of previous webinars can be found at (http://tinyurl.com/StandardsMathPractice)
Official ADE documents
2010 Arizona Mathematics Standards Overview of the 2010 Mathematical
Standards PDF Standards for Mathematical Practices
PDF Mathematics Introduction (Coming
Soon) Mathematics Glossary PDF Summary of Updates to Explanations
and Examples PDF
Mathematical Practices
Standards of Math Practice
Teacher Centered
Student Centered
Teachers Fountains of Knowledge
Create an environment that supports
learningStudents Receptive
Learners Active
Learners Teacher/Student
Relationship
Adversarial Collaborative
Reflect Changes in Teaching and Learning
MP. 4 Model with mathematics
MP. 5 Use appropriate tools strategically
Model and Use Tools
MP. 4 Model with Mathematics
“__________________ is seen as the process towards a solution where sometimes an exact answer does not exist”
MP. 4 Model with Mathematics
“___________________ activities are inherently social experiences, where students work in small teams to develop a product that is explicitly sharable.”
MP. 4 Model with Mathematics
“_____________________________ is a means to develop competencies … that are important in highly technological societies, and for making decisions that are imperative for the maintaining and further development of democracy.”
MP. 4 Model with MathematicsMathematically proficient students can… Choose an appropriate and efficient
model Model, represent and solve real-world
problems. Simplify by making assumptions and
approximations. Interpret results and revise the model if
necessary.
Mathematics is the language with which God has writ the universe.” Galileo
MP. 4 Model with Mathematics
Choose an appropriate and efficient modelModels are systems of elements, operations, relationships, and rules that can be used to describe, explain, or predict the behavior of some other experienced system.
MP. 4 Model with Mathematics
Why model? Reframe math from a set of abstract formal structures
into a useful tool for making sense of the world. Apply math to problems and situations of real interest to
children Give students experiences that lead them to see and
interpret our world using the language and lens of mathematics.
Demonstrate the necessity and usefulness of simplifying assumptions and approximations. About how many children are in our school? $1.23 x 0.78 = ?
Help children develop spatial understanding. (Think maps, diagrams, and physical models with cross-curricular applications)
MP. 4 Model with Mathematics
Modeling can look like: Counters to solve early number problems
involving grouping and separation are direct modeling
An array A number line Model-drawing method to solve word
problems Mapping the structure of a problem
situation and the corresponding symbolic expression
A physical structure
MP. 4 Model with Mathematics
Modeling can also look like: A Problem or Situation in which students –
are confronted with the need to develop a model
clearly recognize the need to revise or refine their current ways of thinking about the given problem
are challenged to express their understandings in ways they can test themselves and revise as often as necessary
develop models that can be shared with others and applied in other situations
(Lesh & Yoon, 2004)
Real World Problem
Problem Representat
ion
Mathematical
Representation
Real World Solution
MP. 4 Model with MathematicsModeling can also look like a Mathematical Modeling Process
Interpreting Problem Situation
Reevaluating
Application
Deducing
Reevaluating
ReevaluatingAbstracting Reevaluating
Great Model-Eliciting Problems include: Real world contexts and real world solutions Model exploration and application where students can
build, consolidate and refine their conceptual systems Multiple approaches, multiple solutions - which provides
opportunities to study, reflect, interpret, and re-interpret the problem information
Shared responsibility and 21st Century Skills A variety of end products so students can explore
creative approaches and representational fluency High level cognitive engagement
MP. 4 Model with Mathematics
MP. 4 Model with Mathematics
• to unlearn
• relearn
• enhance skills that make students’ thinking visible
• relinquish some control
• fully understand content
Teachers Need….
MP. 4 Model with Mathematics
• Coach students
• Persevere in solving problems
• Communicate with precision
• Reason and makearguments
Teachers should….
MP. 4 Model with Mathematics
• Talk about thinking
• Draw connections
• Start with simple situations
• Facilitate
Teachers should….
MP. 4 Model with Mathematics
Research Shows…
"subjects who worked cooperatively spent more time working on practice exercises and reported greater satisfaction than those who worked individually.“
Effects of cooperative learning James D. Klein and Doris R. Pridemore
"Studies have shown that groups out perform individuals on learning tasks, and further that individuals who work in groups do better on later individuals assignments as well”
(Barron, 2000b, 2003; O'Donnell & Danserau, 1992)." Powerful Learning by Linda Darling-Hammond, page 19.
MP. 4 Model with Mathematics
Mathematically proficient students can… Identify mathematical tools and recognize their
strengths and weaknesses. Select and use appropriate tools. Use estimation to predict reasonable solutions and/or
detect errors. Identify and successfully use external mathematical
resources. Use a variety of technologies, including digital content,
to explore, confirm, and deepen conceptual understanding.
MP. 4 Model with Mathematics
Strategic tool use Make tools available Introduce tools Allow exploration Demonstrate how the tool can be
used Keep tools continuously available Know your students, different
developmental levels need different tools
MP. 4 Model with mathematics
MP. 5 Use appropriate tools strategically
Model and Use Tools
MP. 5 Use appropriate tools strategically. Resources to support
(Password = SITP)
Mathematical Practices
Expertise that we each seek to develop in our students What does it mean to do mathematics? What does it mean to understand mathematics?
As teachers, our goal is to provide regular and consistent opportunities to develop and build these habits of mathematical thinking.
Mathematical Practices
How to transition Focus on the Mathematical Practices
• How do your students model the Mathematical Practices?
• In what ways do your classroom strategies foster development of the Mathematical Practices?
Implement the Critical Ideas Look at the ADE website for standards,
crosswalk, and summary of changes.
Resources for Further Exploration The Illustrative Mathematics Project
http://illustrativemathematics.org/ Math Common Core Coalition
http://www.nctm.org/standards/mathcommoncore/ Achieve the Core http://www.achievethecore.org/ National Council Teacher of Mathematics
http://www.nctm.org/standards/content.aspx?id=23273 Guiding Principles for Mathematics Curriculum and Assessment
Mathematics Problem Solving http://jwilson.coe.uga.edu/emt725/PSsyn/Pssyn.html
Mathematics Through Problem Solving http://www.mathgoodies.com/articles/problem_solving.html
Resources for Further Exploration Inside Mathematics
http://www.insidemathematics.org/index.php/standard-1
Curriculum Exemplars from EngageNY http://engageny.org/resource/curriculum-exemplars/
Indiana Dept of Ed Implementing the Standards for Mathematical Practice
Indiana Dept of Ed - Implementing the Standards for Mathematical Practice http://media.doe.in.gov/commoncore/2011-05-10-StandforMath.html
Tools for the Common Core Standards http://commoncoretools.me/
Resources for Further Exploration New Jersey Center for Teaching & Learning
Progressive: Mathematics Initiative http://njctl.org/programs/ Free digital course content for over twenty courses, these initiatives span K-12 mathematics and high school science.
Wiki on Standards of Practice http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice
To view the official ADE documents 2010 Arizona Mathematics Standards
Overview of the 2010 Mathematical Standards PDF
Standards for Mathematical Practices PDF
Mathematics Introduction (Coming Soon)
Mathematics Glossary PDF Summary of Updates to Explanations
and Examples PDF
