The national mathematics curriculum as a stimulus to better teaching and learning of mathematics

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The national mathematics curriculum as a stimulus
to better teaching and learning of mathematics

• Peter Sullivan
• Monash University

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Overview

• Goals of the mathematics curriculum
• Key decisions
• Definitions
• Assessment
• Using the teaching of fractions to exemplify the
  points about 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 curriculum will be clear and succinct, and
  this is about pedagogy
• currently teachers feel they have to rush from
  one topic to the next, and this is bad for
  teaching
• all students can be challenged within basic
  topics, including the advanced students

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Numeracy

• There are some who see numeracy as a separate
  subject
• There are some who think numeracy does not exist
• There will be a numeracy continuum 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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Three content strands (nouns)

• Number and algebra
• Measurement and geometry
• Statistics and probability

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Expectations for proficiency (verbs)

• Understanding
• Fluency
• Problem solving
• Reasoning

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To give an example

• Why choose fractions?
• How can it be described simply?
• What is the nature of the learning?
• What sorts of activities can we consider?

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Why would I talk about fractions to a group of
secondary teachers?

• In the 2007 Victorian Year 9 AIM assessment, the
  following question was posed
• Carol eats 1/3 of the pizza. Theresa eats 1/4 of
  the pizza. How much is left for Peter? (it was a
  write in answer)
• What percentage of SA Year 9 students would get
  that correct?
• Less than one quarter of Year 8/9 Students could
  work out a/5 a/10

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What else do we know about teaching fractions

• Takes up a lot of time
• Students dont seem to improve much (about 10
  improvement in year 8, for example)
• The topic serves to alienate some students
• We use confusing models (shape and number), and
  sometimes we dont connect them well

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The fractions curriculum (my view)

• Definitions and language
• Rational numbers (Counting in fractions)
• Equivalence
• Addition and subtraction
• Converting and comparing
• As an operator
• Multiplication and division

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The Victorian upper primary curriculum

• Students use decimals, ratios and percentages to
  find equivalent representations of common
  fractions (for example, 3/4 9/12 0.75 75
  3 4 6 8). They explain and use mental and
  written algorithms for the addition, subtraction,
  multiplication and division of natural numbers
  (positive whole numbers). They add, subtract, and
  multiply fractions and decimals (to two decimal
  places) and apply these operations in practical
  contexts, including the use of money. They use
  estimates for computations and apply criteria to
  determine if estimates are reasonable or not.
• Progression point 4.25
• knowledge of decimal and percentage equivalents
  for1/2, 1/4, 3/4, 1/3, 2/3 expression of single
  digit decimals as fractions in simplest form and
  conversion between ratio, fraction, decimal and
  percentage forms division of fractions using
  multiplication by the inverse

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Teaching ideas
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We count before we operate

• Race to 10, start at 0, counting by 1 or 2
• Race to 3, start at 0, counting by ½ or ¼
• Race to 3, start at 0, counting by 1/6 or 1/3

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Different ways to write the one fraction

• In a group, ½ of the people were Crows
  supporters. How many people might be in the room
  and how many are Crows supporters. Please write
  it like this

Crows
People
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Different ways to write the one fraction

• In a group, 1/3 of the people were Port
  supporters. How many people might be in the room
  and how many are Port supporters. Please write it
  like this

Port
People
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• What is the smallest group size common to both?

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Addition

• Why cant we add like we used to?
• What types of addition are there?

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What cant we add like this


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Types of addition
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To illustrate how easy questions can be
interesting
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The following is a description of an idea that
might be used as the basis of a lesson

• Which is bigger

or
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A question for you

• Please work out the answer

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Some solution methods

• Finding common denominators
• Using a calculator to divide numerator by
  denominator to convert to decimals
• the residual needed to build 200/300 and 201/301
  to the whole
• the first fraction requires 100/300
• the second fraction requires 100/301,
• because the 301ths are smaller parts, the
  residual needed is smaller.

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The sugar rule
gt
So long as b ? 0, a lt b
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• A teacher in the United States used a basketball
  analogy to give a convincing alternative
  strategy. He argued that we could think of
  200/300 as a basketball players free throw
  success rate. In moving to 201/301, the
  basketball player has had one more throw, which
  was successful. His average must therefore have
  improved, and so 201/301 must be larger.

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• It is possible to engage with simple topics in
  sophisticated ways

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Conclusion

• The process of writing will allow input
• Let us think about the big ideas and how to
  express them in ways that will inform teaching