Mathematical Proficiency

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Mathematical Proficiency

• What is it?

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Mathematical Proficiency

• Mathematics proficiency - what is it?
• It is an intertwining combination of the
  following five factors...
• Conceptual understanding - comprehension of
  mathematics concepts, operations and relations
• Procedural fluency -skill in carrying out
  procedures flexibly, accurately, efficiently, and
  appropriately
• Strategic competence - ability to formulate,
  represent, and solve mathematical problems
• Adaptive reasoning - capacity for logical
  thought, reflection, explanation, and
  justification
• Productive disposition - habitual inclination to
  see mathematics as sensible, useful, and
  worthwhile, coupled with a belief in diligence
  and one's own efficacy.
• Kilpatrick, James, Swafford, Jane, and Findell,
  Bradford, eds., Adding it Up - Helping Children
  Learn Mathematics, National research Council,
  National Academy Press, Washington, DC. 2001

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Intertwined Strands of Proficiency
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Conceptual Understanding

• An integrated and functional grasp of
  mathematical ideas
• Students with conceptual understanding
• Know more than isolated facts and methods
• Understand why a mathematical idea is important
  and the kinds of contexts in which it is useful
• Have organized knowledge into a coherent whole

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Significant Indicators of Conceptual Understanding

• being able to represent mathematical situations
  in different ways and knowing how different
  representations can be useful for different
  purposes
• seeing the connections among concepts and
  procedures and being able to give arguments to
  explain why some facts are consequences of others
• The degree of students conceptual understanding
  is related to the richness and extent of the
  connections they have made.

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Example

• Knowledge clusters help make learning easier
• Consider the development of whole number addition
• Children tend to learn doubles fairly easily
• Relationships extend their learning
• 6 6 12, 6 7 is just one more than 6 6
• Addition is commutative
• 5 3 3 5
• Learning of basic addition facts is reduced by
  almost half

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Example

• A student multiplies 9.83 and 7.65 and gets
  7519.95.

• Conceptual understanding allows this student to
  reason that 10 x 8 80.
• Therefore, multiplying two numbers less than 10
  and 8 will give a product less than 80.
• Suspect that the decimal point is incorrectly
  place and check that possibility.

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Examples in Higher Math

• True or False
• The graph of any equation in two variables is a
  straight line.
• The graph of a quadratic function f(x) ax2bxc
  (agt0) is never a straight line.
• If the selling price of a product is raised,
  total revenue from sales of the product must rise
  due to the increased revenue per item.

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Procedural Fluency

• Knowledge of procedures, knowledge of when and
  how to use them appropriately, and skill in
  performing them flexibly, accurately, and
  efficiently.
• Students with procedural fluency
• Know ways to estimate the results of a procedure
• Understand that procedures can be developed to
  solve entire classes of problems, not just
  individual problems
• Are able to use a variety of mental strategies
• Apply various methods of calculation
• Can modify or adapt procedures to make them
  easier to use

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Example

• Students with limited procedural understanding
  would need a paper and pencil to add
• 199 66

• Students with more understanding would see that
  199 is 1 less than 200, so they might add 200
  66, then subtract 1 from that sum

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Procedural Fluency

• "Procedural fluency" means "how many ways can we
  approach a given problem and how many solutions
  make sense based on a) what information was given
  and b)what was requested."

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Consider

• Learning a procedure with understanding vs.
  learning the procedure without understanding.
• Discuss
• What procedures do we teach that students may not
  or typically do not understand?
• What errors do they make in applying the
  procedures?
• What happens when they practice the procedures
  incorrectly?
• How does it impact their future learning?

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Consider

• This is not to say that one must learn every
  nuance behind the scenes of a procedure.
• Did you fully understand why the power rule for
  derivatives or antiderivatives worked?

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Strategic Competence

• The ability to formulate mathematical problems,
  represent them, and solve them.
• Similar to what has been called problem solving
  and problem formulation.
• Students with strategic competence
• Are able to represent problems using the
  essential elements or key features
• Know a variety of solution strategies as well as
  which strategies may be useful for solving a
  specific problem.

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Example

• At ARCO, gas sells for 1.13 per gallon.
• This is 5 cents less per gallon than gas at
  Chevron.
• How much does 5 gallons of gas cost at Chevron?

• Students with limited strategic competence focus
  on the numbers and so-called keywords to
  determine appropriate arithmetic operations.
• 1.13 less 5 1.08
• 1.08 x 5 gallons 5.40

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Example (Non-Routine Problems)

• A cycle shop has a total of 36 bicycles and
  tricycles in stock.
• Collectively there are 80 wheels.
• How many bikes and how many tricycles are there?

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Strategy 2

• Guess and Check
• If there were 20 bikes and 16 tricycles, there
  would be (20 x 2) (16 x 3) 88 wheels.
• Try 24 bikes and 12 tricycles.
• (24 x 2) (12 x 3) 84 wheels.
• Try 28 bikes and 8 tricycles.
• (28 x 2) (8 x 3) 80 wheels.

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Strategy 3

• Algebraic Solution
• Let b number of bikes, t number of tricycles
• b t 36
• 2b 3t 80
• The solution of the system of equations yields
  28 bikes and 8 tricycles.

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Alternative Strategies

• What alternative strategies did you use to
  represent and solve the problem?
• What strategy might a student who has not
  mastered multiplication use?

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Multiple Solution Problem

• Bookers Bakery
• Cookie prices (per cookie)
• Oatmeal 9
• Peanut Butter 12
• Chocolate Chip 15
• (Math Mountain, McRel 1998)

• What could you get in a 1.50 bag of cookies? Try
  to find several possibilities.
• What is the maximum number of cookies you could
  get in a 1.50 bag?
• Could you get a 1.50 bag of cookies with no
  chocolate chips?
• Could you get a 1.50 bag of cookies with an
  equal number of all three kinds of cookies?

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Adaptive Reasoning

• The capacity to think logically about the
  relationships among concepts and situations.
• Reasoning is correct and valid, and stems from
  careful consideration of alternatives.
• Includes knowledge of how to justify the
  conclusions.
• Adaptive reasoning is the glue that holds
  everything together.

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Adaptive Reasoning Broad View

• Includes not only
• formal proof and other forms of deductive
  reasoning
• But also
• informal explanation and justification
• As well as
• intuitive and inductive reasoning based on
  pattern, analogy, and metaphor.

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Example

• Estimate the sum
• 1, 2, 19, 21

• 55 of 13-year olds chose either 19 or 21 as the
  correct response
• 24 of the 13-year olds selected the correct
  response
• 37 of the 17-year olds selected the correct
  response
• (NAEP)

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Example Population Growth in Two Towns

• In 1980 the populations of Town A and Town B were
  5000 and 6000, respectively. The 1990 populations
  of Town A and Town B were 8000 and 9000,
  respectively.
• Phil claims that from 1980 to 1990 the
  populations of the two towns grew by the same
  amount. Use mathematics to explain how Brian
  might have justified his claim.
• Darlene claims that from 1980 to 1990 the
  population of Town A grew more. Use mathematics
  to explain how Darlene might have justified her
  claim.

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Meaningful Classroom Discourse

• Engaging students in conversations in the
  mathematics classroom provides important learning
  opportunities for students. NCTM, 2000
• When students are asked to orally justify their
  solution methods to their teachers and their
  peers, their own understanding deepens. NCTM,
  2000
• When teachers better understand the development
  of students mathematical thinking, students
  performance is enhanced. McREL, 2001

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Productive Disposition

• The tendency to see sense in mathematics, to
  perceive it as both useful and worthwhile, to
  believe that steady effort in learning
  mathematics pays off, and to see oneself as an
  effective learner and doer of mathematics.
• If students are to develop conceptual
  understanding, procedural fluency, strategic
  competence, and adaptive reasoning abilities,
  they must believe that mathematics is
  understandable, not arbitrary that, with
  diligent effort, it can be learned and used and
  that they are capable of figuring it out.
• Developing a productive disposition requires
  frequent opportunities to make sense of
  mathematics, to recognize the benefits of
  perseverance, and to experience the rewards of
  sense making in mathematics.

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Discuss

• What factors influence the development of
  productive disposition?
• What factors hinder the development of productive
  disposition?
• What can teachers do to help learners develop a
  healthy disposition towards mathematics?

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Properties of Mathematical Proficiency

• The strands of proficiency are interwoven
• Proficiency isnt all or nothing
• Proficiency develops over time

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Teaching for Mathematical Proficiency

• What is learned depends on what is taught and how
  it is presented.
• Consistently helping students learn worthwhile
  mathematical content.
• Being able to work effectively with a wide
  variety of students in different environments and
  across a range of mathematical concepts.

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References

• http//www.state.sd.us/deca/DDN4Learning/
• Principles and Standards for School Mathematics,
  NCTM