Mathematical Literacy An idea to talk about..

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Mathematical Literacy An idea to talk
about..
The Mathematical Council of the Alberta Teachers
Association Red Deer, Alberta May 3, 2002
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A problem is not necessarily solved because the
correct answer has been made. A problem is not
truly solved unless the learner understands what
he has done and knows why his actions were
appropriate. -William A. Brownell, The
Measurement of Understanding (1946)
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Model of Mathematical Literacy

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Reasoning
Reasoning is essential to mathematics.
Developing ideas, exploring phenomena, justifying
results, and using mathematical conjectures help
students see and expect that mathematics makes
sense. (NCTM, 2000)
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Representing
Representing involves the learner in
constructing, and alternating between, various
mathematical models such as equations, matrices,
graphs, and other symbolic and graphical forms.
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Manipulating
Manipulating involves the learner in the
performance of calculations and the successful
use of algorithms and procedures. This process
involves the development of mental mathematics
skills including recognizing the reasonableness
of a result as well as the use of symbolic
manipulation software and calculators.
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Problem Solving
Problem solving is fundamental to mathematics.
It engages the learner in using prior knowledge
and skills toward resolution of a problem that
lacks an apparent solution.
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Technology
Technology should be used efficiently and
effectively in investigating mathematical ideas
and in finding solutions to mathematical
problems.
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Communication
Communication plays an important role in
supporting learners through clarifying, refining,
and consolidating their thinking. Mathematically
literate learners should be able to communicate
their mathematical ideas orally and in writing
while defending and offering justification for
such ideas.
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Values
Values includes learners emotions, beliefs, and
attitudes toward mathematics and learning. Such
affective processes interact with cognition and
are instrumental in empowering the learner to
take control of their own learning and express
confidence in their mathematical decisions.
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Robert Freeman (2002). A floor plan for
mathematical success. ENC Focus 9(2). (see
www.enc.org/focus)
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"Mathematics literacy is an individuals capacity
to identify and understand the role that
mathematics plays in the world, to make
well-founded mathematical judgements and to
engage in mathematics, in ways that meet the
needs of that individuals current and future
life as a constructive, concerned and reflective
citizen."
Organization for Economic Co-Operation and
Development Program for International Student
Assessment
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• PISA will assess mathematical literacy in three
  dimensions
• First, the content of mathematics, as defined
  mainly in terms of broad mathematical concepts
  underlying mathematical thinking (such as chance,
  change and growth, space and shape, reasoning,
  uncertainty and dependency relationships), and
  only secondarily in relation to "curricular
  strands" (such as numbers, algebra and geometry).
  The PISA 2000 assessment, in which mathematics is
  a minor domain, will focus on two concepts
  change and growth, and space and shape. These two
  domains will allow a wide representation of
  aspects of the curriculum without giving undue
  weight to number skills. Emphasis mine

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• Second, the process of mathematics as defined by
  general mathematical competencies. These include
  the use of mathematical language, modelling and
  problem-solving skills. The idea is not, however,
  to separate out such skills in different test
  items, since it is assumed that a range of
  competencies will be needed to perform any given
  mathematical task. Rather, questions are
  organised in terms of three "competency classes"
  defining the type of thinking skill needed
• The first class of mathematical competency
  consists of simple computations or definitions of
  the type most familiar in conventional
  mathematics assessments,
• The second class requires connections to be made
  to solve straightforward problems,
• The third competency class consists of
  mathematical thinking, generalisation and
  insight, and requires students to engage in
  analysis, to identify the mathematical elements
  in a situation and to pose their own problems.

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• Third, the situations in which mathematics is
  used, ranging from private contexts to those
  relating to wider scientific and public issues.

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• A new way of looking at student performance.
• PISA assessed young people's capacity to use
  their knowledge and skills in order to meet
  real-life challenges, rather than merely looking
  at how well they had mastered a specific school
  curriculum.
• PISA assessed literacy in reading, mathematics
  and science.
• Students had to understand key concepts, to
  master certain processes and to apply knowledge
  and skills in different situations.
• Information was also collected on student
  attitudes and approaches to learning.

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Question Design a graph (or graphs) that shows
the uneven distribution of the Indonesian
population.  Source de Lange et Verhage (1992).
Used with permission
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Charles saw advertisements for two cellular
telephone companies. Keep-in-Touch offers phone
service for a basic fee of 20.00 a month plus
0.10 for each minute used. ChitChat has no
monthly basic fee but charges 0.45 a minute.
Both companies use technology that allows them to
charge for the exact amount of time used they do
not "round up" the time to the nearest minute, as
many of their competitors do. Compare these two
companies' charges for the time used each month.
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José ate ½ of a pizza. Ella ate ½ of another
pizza. José said that he ate more pizza than
Ella, but Ella said they both ate the same
amount. Use words and pictures to show that José
could be right.
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Guidelines for Peer Assessment1. Read your
classmate's draft carefully so your response will
be helpful. 2. You are not to edit your
classmate's work rather you are to provide
feedback that will enable your classmate edit and
produce a response that is clear and informative.
3. Provide suggestions that will guide you and
your classmate in discussing important
mathematical ideas contained in the paper and
what needs to be improved. 4. Note what you
agree with in the mathematics, arguments, and
justification. 5. Check if there is evidence
to support important ideas, strategies, and
conclusions. 6. Note places where you as a
reader would like more information.
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Developing Metacognition
Moving beyond the what, when, and how of the
written protocols to the why and why not of
metacognition can begin as individual or
collaborative learning experiences. I use
open-ended questions to encourage students to
rethink and reflect on their problem solving.
Why did you do this? Why did you not do that?
What were you thinking when you...? What
would have happened if you had...? Is this way
better (or more effective, efficient, and so
forth) than that? Would you do anything
differently?
Martinex, J. G. R. (2001). Thinking and writing
mathematically 'Achilles and the Tortoise' as an
algebraic word problem. Mathematics Teacher,
94(4), 248-252.
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Pacific Regional Education Laboratory, 1996
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Experience with mathematical modes of thought
builds mathematical power a capacity of mind of
increasing value in this technological age that
enables one to read critically, to identify
fallacies, to detect bias, to assess risk, and to
suggest alternatives. Mathematics empowers us to
understand better the information-laden world in
which we live. (National Research Council, 1989)