Mathematics is both a doorway and a hurdle

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Mathematics is both a doorway and a hurdle

• Peter Sullivan
• Monash University

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Abstract

• Successful mathematics study opens many doors.
  Mathematics also acts as a hurdle.
• The challenge for mathematics curriculum and
  teaching is to help all students to jump the
  hurdles, so that the doors are open when they
  reach them.

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Overview

• Goals of the mathematics curriculum
• Key decisions
• Definitions
• Assessment
• Presentation of the curriculum

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The curriculum needs to foster

• development of expert mathematicians
• expert users of mathematics in the professions
• a workforce capable of meeting all numeracy
  requirements
• citizens able to use the mathematics they need

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Some of the key decisions

• mathematics success creates opportunities and all
  should have access to those opportunities
• inclusive for all to end of year 9 (and
  compulsory in year 10)

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The senior secondary curriculum

• Fundamental ( Mathematics A)
• Rigorous, substantial, worth studying
• Business, statistics, measurement, applied
  geometry
• Intermediate (Mathematics B)
• Also worth studying
• Leading to university mathematics subjects
• Calculus, statistical distributions, graphs
• Advanced (Mathematics C)
• For mathematics specialists (a little over 10)
• Vectors, mechanics, more calculus
• Vocational (pre vocational)

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• the curriculum will be clear and succinct, and
  this is about pedagogy
• clarity is needed for interactivity
• not about setting low expectations for teachers
• demands for additional detail should be resisted
• currently teachers feel they have to rush from
  one topic to the next, and this is bad for
  teaching
• studying combined topics in more depth
• some topics are more important than others

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• all students can be challenged within basic
  topics, including the advanced students
• extension by moving to advanced topics is
  ultimately counterproductive

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Numeracy

• There are some who see numeracy as a separate
  subject
• There are some who think numeracy does not exist
• Decision is to write a numeracy curriculum within
  mathematics
• as well as including references to numeracy
  aspects of other subjects in mathematics, and to
  numeracy in history, English and science

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MATHEMATICS ASSESSMENT

• Assessing Maths In 1970
• A logger sells a lorry load of timber for
  1000.His cost of production is 4/5 of the
  selling price. What is his profit?
• Assessing Maths In 1980
• A logger sells a lorry load of timber for 1000.
  His cost of production is 4/5 of the selling
  price, or 800. What is his profit?
• Assessing Maths In 1990
• A logger sells a lorry load of timber for 1000.
  His cost of production is 800. Did he make a
  profit?

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• Assessing Maths in 2000
• A logger sells a lorry load of timber for 1000.
  His cost of production is 800 and his profit is
  200. Underline the number 200.
• Assessing Maths in 2008
• A logger cuts down a beautiful forest because he
  is selfish and inconsiderate and cares nothing
  for the habitat of animals or the preservation of
  our woodlands. He does this so he can make a
  profit of 200. What do you think of this way of
  making a living?

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The curriculum driving any assessment

• The curriculum will outline expectations
• But it is cheaper to assess fluency than
  reasoning
• Assessing only one type of mathematical action
  will distort the curriculum
• In any case, if national assessments are to
  inform us on achievement of goals, then all
  aspects must be included

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Definitions
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Three content strands

• Number and algebra
• Measurement and geometry
• Statistics and probability
• Features
• 3 not 5
• traditional terms (but not intent)
• maximise interconnections
• not all aspects equally important at every level

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Expectations for proficiency

• Understanding
• Fluency
• Problem solving
• Reasoning
• Features
• More than working mathematically
• All are needed

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• Numeracy expectations as well

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The presentation of the curriculum
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Number and Algebra ( K 2)

• Counting
• Adfsdfsd asdfasdfasd asfdsdfsad asdfsdfas
  asdfsdfas asdfasd
• Place Value
• Dfsadfsd sdfsdfasd sadfsdfsd asdfsdfsd safdasdf
  adfsdfasd asdfsd fdsa
• Addition
• Adfsdfsd asdfsdfsd safdasdfsd asdfsdfsd asdfsdfsd
  asdfsdfs
• Subtraction
• Backwards counting modelling situations in which
  one part of the whole is unknown number
  strategies that are useful for subtraction
  solving subtraction word problems
• Multiplication
• Sdfsdfsd sfdasdfsd asdfsdfsd asdfsdfsd asdfsd
  asfsdfds asdfsdfsd asdfasd asdfsd
• Division
• Adfsdfs asdfsdfsad sdfasfasd asdfasdfsd sadfsdfsd
  sadfsdfasd sadfsd asfdsdfsadfasdfsd

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Number and Algebra ( K 2)(initial description)

• Counting
• Adfsdfsd asdfasdfasd asfdsdfsad asdfsdfas
  asdfsdfas asdfasd
• Place Value
• Dfsadfsd sdfsdfasd sadfsdfsd asdfsdfsd safdasdf
  adfsdfasd asdfsd fdsa
• Addition
• Adfsdfsd asdfsdfsd safdasdfsd asdfsdfsd asdfsdfsd
  asdfsdfs
• Subtraction
• Backwards counting modelling situations in which
  one part of the whole is unknown number
  strategies that are useful for subtraction
  solving subtraction word problems
• Multiplication
• Sdfsdfsd sfdasdfsd asdfsdfsd asdfsdfsd asdfsd
  asfsdfds asdfsdfsd asdfasd asdfsd
• Division
• Adfsdfs asdfsdfsad sdfasfasd asdfasdfsd sadfsdfsd
  sadfsdfasd sadfsd asfdsdfsadfasdfsd

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Number and Algebra ( K 2)(elaboration)

• Subtraction
• Backwards counting
• Adsfasd sdafjsadljfsda sdafasdfsda sdfasddsaf
  asdfsadfsad sadfasdsdasdffds
• modelling situations in which one part of the
  whole is unknown
• Asdfasdfsad sdaafasdfsad sdafasdfsdaa
  sdffasdfjasdl dsaff asdfasd sdffafsdfasd
• number strategies that are useful for
  subtraction
• Key facts that form the basis of strategies for
  subtraction are
• taking away 1 and taking away 2
• what do we need to build to 10 (6 ? 10,
  etc.)
• doubles (the idea that if 3 3 is known then 6
  3 can be developed)
• subtracting 10.
• solving subtraction word problems

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Number and Algebra ( K 2)

• Counting
• Adfsdfsd asdfasdfasd asfdsdfsad asdfsdfas
  asdfsdfas asdfasd
• Place Value
• Dfsadfsd sdfsdfasd sadfsdfsd asdfsdfsd safdasdf
  adfsdfasd asdfsd fdsa
• Addition
• Adfsdfsd asdfsdfsd safdasdfsd asdfsdfsd asdfsdfsd
  asdfsdfs
• Subtraction
• Backwards counting modelling situations in which
  one part of the whole is unknown number
  strategies that are useful for subtraction
  solving subtraction word problems
• Multiplication
• Sdfsdfsd sfdasdfsd asdfsdfsd asdfsdfsd asdfsd
  asfsdfds asdfsdfsd asdfasd asdfsd
• Division
• Adfsdfs asdfsdfsad sdfasfasd asdfasdfsd sadfsdfsd
  sadfsdfasd sadfsd asfdsdfsadfasdfsd

Understanding
Fluency
Problem solving
Reasoning
Numeracy
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Number and Algebra ( K 2)

• Counting
• Adfsdfsd asdfasdfasd asfdsdfsad asdfsdfas
  asdfsdfas asdfasd
• Place Value
• Dfsadfsd sdfsdfasd sadfsdfsd asdfsdfsd safdasdf
  adfsdfasd asdfsd fdsa
• Addition
• Adfsdfsd asdfsdfsd safdasdfsd asdfsdfsd asdfsdfsd
  asdfsdfs
• Subtraction
• Backwards counting modelling situations in which
  one part of the whole is unknown number
  strategies that are useful for subtraction
  solving subtraction word problems
• Multiplication
• Sdfsdfsd sfdasdfsd asdfsdfsd asdfsdfsd asdfsd
  asfsdfds asdfsdfsd asdfasd asdfsd
• Division
• Adfsdfs asdfsdfsad sdfasfasd asdfasdfsd sadfsdfsd
  sadfsdfasd sadfsd asfdsdfsadfasdfsd

Understanding
Fluency
Problem solving
Reasoning
Numeracy
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Subtraction - Understanding

• Students see connections between take away, less
  than, difference between
• Students connect with other forms of
  representations, such as empty number lines
• Student see connections between addition and
  subtraction. They understand that if they know 7
  8 15, they also know
• 15 7 15 8
• 15 ? 7 15 ? 8
• ? 7 8 ? 8 7

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12 - 4
2
2
12
8