Process Standards: Teaching and Learning Mathematics

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Process Standards Teaching and Learning
Mathematics

• Presented by
• David S. Allen, Ed.D.
• Melisa J. Hancock, TIR
• For
• The Infinite Mathematics Project (Year 3)

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Process Standards Teaching and Learning
Mathematics

• About Teaching Mathematics (Concepts)
• About Learning Mathematics (Concepts)
• Pedagogy (Act of Teaching)
• Content (N,A,G,M,DA)
• Pedagogical Content Knowledge (How do I teach
  this content while focusing on the big ideas of
  mathematics)
• Mathematics of Today for Tomorrow

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Math Standards

• Content Standards
• Number and Operations
• Algebra
• Geometry
• Measurement
• Data Analysis and Probability

• Process Standards
• Problem Solving
• Reasoning and Proof
• Communication
• Connections
• Representation

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Problem Solving

• Problem Solving Promotes understanding of
  concepts and allows application of knowledge. A
  problem solving disposition includes the
  confidence and willingness to take on new and
  difficult tasks. Problem solvers are
  resourceful, seek out information to help solve
  problems and make effective use of what they
  know. Problem solving means engaging in a task
  for which the solution method is NOT known in
  advance.

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Pedagogical Approach to Problem Solving

• Four Step
• Process
• Understanding the problem
• Choosing a strategy
• 3. Implementing a strategy
• 4. Finding and reporting a solution

• New Blooms
• Taxonomy
• Remembering
• Understanding
• Applying
• Analyzing
• Evaluating
• Creating

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Problem Solving Purpose
To meet new challenges in work, school, and
life, students will have to adapt and extend
whatever mathematics they know. Doing so
effectively lies at the heart of problem
solving. (NCTM, 2001)
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Skills are to mathematics what scales are to
music or spelling is to writing. The objective
of learning is to write, to play music, or to
solve problemsnot just to master skills.
Everybody Counts (1989)
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Problem Solving Goals and Objectives

• Problem Solving Requires
• Mathematical Content Knowledge
• Knowledge of problem-solving strategies
• Hueristics (Polya, 1954)
• Effective self-monitoring
• Productive disposition to pose and solve problems

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Race to 20!!!!

• Game played with a partner.
• Partners take turns counting 1 or 2 numbers
  starting at 1.
• Goal is to be the person to say 20.
• How can you guarantee a win every time?
• Pine Cones Game

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Problem Solving Sample
How many rectangles are there on a standard
8 X 8 checkerboard? Count only those rectangles
(including squares) whose sides lie on grid
lines. For example, there are nine rectangles on
a 2 X 2 board.
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Problem Solving Sample

• The Value of this problem
• What determines a rectangle?
• Examine a simpler problem
• Adding sequence of consecutive integers
• Problem lends itself to discovering mathematical
  applications based on number concepts
• Multiple correct solution strategies can be
  applied to arrive at a solution
• Variety of solutions are important

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Problem Solving
Problem solving is at the core of any
mathematics curriculum it is integral to all
mathematical activity. As such, it should
permeate the entire mathematics program. Students
who are consistently presented with challenging
problems learn to develop and apply new
strategies. When they are also given
opportunities to communicate their strategies
with others and reflect on their thinking, their
problem solving abilities are further enhanced.
(Fennell et al. 2000)
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Problem Solving Samples

• Problem Solving Tools General Ideas from NCTM
• Cooperative Problem Solving Cards
• Work with the cards
• Make and take for your grade level content
• Sample NCTM Context Starters
• Welcome to the Fair-Grade Level Activity
• Evaluating Problem Solving Task 3-5

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Reasoning Proof

• Define Reasoning
• Is this different in mathematics than in other
  areas of the real world?
• Define Proof
• Does proof always have to be done in a specific
  manner? (ie. Euclidean two column proofs)
• What do other forms of proof look like?

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Reasoning and Proof

• Reasoning Should be a natural, ongoing part of
  ALL classroom discussions. Students should
  constantly be asked to justify their conclusions.
  Students need opportunities to develop
  compelling arguments with enough evidence to
  convince someone who is not part of their own
  learning community. Reasoning and proof is a
  habit of mind, and like all habits, it must be
  developed through consistent use in many
  contexts.

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Reasoning and Proof

• Reasoning is a state of mind that causes students
  to explore, to justify, and to validate.
• Students are reasoning when they interpret data,
  when they solve problems, and when they view
  geometric patterns and shapes.
• When students are presented new problems, they
  use reasoning skills to apply previously acquired
  information and to test the validity of their
  solutions.
• Reasoning is the process by which students make
  sense of mathematics.

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Reasoning and Proof

• 5. Reasoning begins with intuition.
• 6. Intuition is used by even the youngest
  children as they begin to make sense of
  mathematics and should be encouraged at all
  levels.
• 7. Reasoning and proof can be incorporated in
  the mathematics classroom through questioning.
• How did you get your answer? Tell me how you
  thought about that. Why does your solution work?
  Do you think that strategy will always work?

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Reasoning Proof

• What factors inhibit the application of reasoning
  to solve problems?
• Activity Analysis
• 64 65 Proof. Can you Debunk this Proof?

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Communication

• What is Communication?
• Forms
• Purpose
• Ability

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Communication

• Communication In classrooms where students are
  challenged to think and reason, communication is
  an essential feature as students express the
  results of their thinking orally and in writing.
  To be prepared for the future, students must be
  able to collaborate and communicate their ideas
  clearly and effectively with others, both orally
  and in writing.

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Communication

• The communication skills of reading, writing,
  listening, and speaking provide the means for
  sharing ideas and promoting mathematical
  understanding.
• Provides students the opportunity to clarify
  their thinking and reinforce their comprehension
  of the concepts they are working with.
• By listening to their peers students are exposed
  to ideas they may not have thought of.
• Vygotsky and Piaget believed that to develop
  their reasoning students must engage in social
  interaction.

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Communication

• In middle school, mathematics begins to become
  more abstract. New concepts need to be introduced
  conceptually, but students need to move from
  concrete representations to symbolic notation
  more quickly than in elementary school. Effective
  communication of ideas becomes even more
  important.
• Putting ideas down on paper is another means of
  helping students organize their thinking.
• Written reflection can be an important tool for
  teachers in assessing their students
  mathematical understanding.

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Problem Solving Sample
A string is stretched corner to corner on a
floor tiled with square tiles. If the floor is 28
tiles long and 35 tiles wide, over how many tiles
does the string pass?
Remember the task involves communicating
your problem solving process and the reasoning
that drove that process.
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Communication

• Tools
• Activity Analysis
• Grade Level Application

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Connections

• ConnectionsThinking involves making connections
  and making connections builds understanding.
  Students should develop an increased capacity to
  link ideas and a deeper understanding of how more
  than one approach to the same problem can lead to
  equivalent results, even though the approaches
  might look quite different. When students can
  connect mathematical ideas, their understanding
  is deeper and more lasting.

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Connections

• There are three aspects to making connections in
  mathematics.
• Connections are made when one mathematical idea
  is used to build another.
• How is counting related to addition, addition to
  subtraction, addition to multiplication,
  multiplication to area?
• Connections are made among different mathematical
  ideas.
• Teachers need to know what mathematics students
  learned previously in order to build on that
  knowledge. Teachers should also be aware of what
  their students will be studying in subsequent
  grades.

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Connections

• c) Connections are made between mathematics
  and contexts outside the field of mathematics.
• Mathematics permeates other curriculum areas
  and is found in the everyday experience outside
  of school. The use of shapes and patterns is
  prevalent in art and architecture measurement
  skills and classification skills are important in
  science measurement skills and knowledge of
  fractions are utilized in cooking and in building
  models and measurement skills, data gathering,
  and statistics are applied in the social
  sciences.
• (Fennell et al. 2000)

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Connections

• NCTM Middle School Task
• Activity Handouts

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Representation

• RepresentationStudents deepen their
  understanding of concepts when given
  opportunities to create, compare, and use various
  representations. Representations such as
  physical objects, drawings, charts, graphs and
  symbols also help students communicate their
  thinking.

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Representation

• Representations provide vehicles for expressing
  and internalizing mathematical thought.
• Include physical objects, pictures, symbols
• Mental images, words, and ideas
• Formal/Informal representations-
• Conventional symbols, graphs, diagrams
• Informal forms are often invented by students as
  a way of making sense of mathematical ideas and
  communicating those ideas to classmates or the
  teacher.

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Representation

• How can you communicate the idea that
  adding more salt to the popcorn at the movie
  theatre increases drink sales?
• Students do not always see the
  mathematics the way we see it. Our formal
  education has caused us to see mathematics in a
  traditionally abstract or symbolic manner. We
  need to help students access mathematics as a
  product of their environment.

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The value of a tool is in its usefulness.
Being able to do pencil-paper computation will
not serve students without the ability to
interpret a problem, analyze what needs to be
done, and evaluate the solution.