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EDP243 Children as Mathematical Learners
Assessment 3 ePortfolio
Caroline Marriott
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Table of Contents
1 Introduction....................................................................................................................................1
2 Rationale.......................................................................................................................................1
3 Mathematics Content Knowledge Test.........................................................................................2
4 Diagnostic Interview......................................................................................................................6
4.1 Rationale................................................................................................................................6
5 Conceptual Development of Multiplicative Thinking....................................................................10
5.1 Early Number.......................................................................................................................10
5.2 Mental strategies for Addition and Subtraction....................................................................11
5.3 Conceptual strategies for multiplication and division...........................................................11
5.4 Mental strategies for multiplication and division...................................................................12
5.5 Fractions and Decimals........................................................................................................13
5.6 Ratio, proportion and percent..............................................................................................14
6 Conclusion...................................................................................................................................15
References.........................................................................................................................................17
1 Introduction One of the main intended outcomes of school mathematics is to develop the big ideas of
number in students so mathematical concepts can be understood and used flexibly, and a key
element in a student's learning is the quality of teacher explanations (Siemon as cited by
Victorian Department of Education [VDOE], 2007). In order to teach key concepts of
mathematics, a teacher requires both mathematical content knowledge [MCK] and pedagogical
content knowledge [PCK] to help students understand specific concepts (Schulman, as cited by
Livy & Vale, 2011). For example, a teacher with sound MCK might have a sound understanding
of the scope and sequence of teaching a strand and sub-strand of mathematics in a particular
year level, while a teacher with sound PCK might demonstrate a range of effective strategies for
teaching mathematics, identify individual student needs, and make adjustments to how a
concept is being taught. In this report, the MCK and PCK needed to effectively teach
multiplicative thinking, fractions, and decimals will be demonstrated and explained in terms of
the underpinning concepts, how each concept is related, and how concepts are best developed
in the classroom, together with a National Assessment Program – Literacy and Numeracy
(Australian Curriculum, Assessment and Reporting Authority [ACARA], 2015) style fraction test
and a diagnostic interview to determine fraction misconceptions.
2 RationaleMultiplicative thinking can be described as the ability to work flexibility with multiplicative
concepts, using a variety of strategies, resources and representations to solve problems
involving whole numbers, fractions, decimals, and percent, and has been described as the ' ‘big
idea’ of number (Siemon, 2011). Multiplicative thinking is necessary if students are to
participate successfully in mathematics in later years or access post-compulsory training options
as important mathematical concepts such as measurement, proportional reasoning, and
algebraic reasoning are dependent on multiplicative thinking (Siemon, as cited by Hurst &
Hurrell, 2015). Multiplicative thinking also connects 'big number' ideas. For example,
multiplicative thinking underpins and informs place value, and in turn, place value underpins
multiplicative thinking (Siemon, as cited by Hurst & Hurrell, 2015). This reciprocal relationship
between mathematical concepts highlights the importance of a teacher's sound mathematical
content knowledge because when the link is understood, a teacher can use a range of
scaffolding practices to support students to make the same important connections.
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3 Mathematics Content Knowledge TestOrdering Fractions and Equivalent Fractions
Question 1. Ordering fractions from largest to smallest using set models
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A)
B)
C)
D)
Answer: B
Rationale: This questions will determine if the student understands the rule of ordering fractions;
when factions have the same numerator the fractions are ordered using their denominator.
However, although rules are effective, students need time to consider the relative size of
fractions in order to develop a number sense about fractions (Van de Walle, 2013), so this
question would also suit a diagnostic interview so that a teacher could probe how the problem
was tackled and the thinking behind the strategy used. This question also provides the context
of sets and can help determine if students understand that a set of objects is also a single unit
(Reys et al., 2012).
Misconception: The student understands the algorithmic rule only when comparing fractions
(Van de Walle, 2013). When using the algorithmic method, it is possible for the student to
misapply the rule and understand that the largest denominator represents the largest fraction
when the numerator is the same. This error occurs when working with fractions when students
apply familiar whole number strategies whereby the biggest number is greatest in size (Bezuk &
Cramer, 1989).
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12
13
14
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16
12
13
14
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ACMNA102: Compare and order common unit fractions and locate and represent them on a
number line Elaboration: Recognising the connection between the order of unit fractions and
their denominators (ACARA, 2015).
Question 2. Finding equivalent fractions using different part-whole models.
1. 2.
3.
4.
A) 1 and 3
B) 2 and 3
C) 1 and 4
D) 1 and 2
Answer: D
Rationale: Without the fractional notation provided, a student must first determine the numerator
and denominator of each area model before comparing and determining which ones are
equivalent. A deep understanding of fractions can be attained using different representations of
fraction models (Van de Walle, 2013) to chunk quantities into parts and name them. Having a
sound conceptual knowledge of equivalent fraction representations will prepare students for
many mathematical tasks including fraction computation (Van de Walle, 2013).
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Misconception: Students are unable to compare fractions using different fraction models
(Reys et al., 2012).
ACMNA102: Compare and order common unit fractions and locate and represent them on
a number line. Elaboration: Recognising the connection between the order of unit fractions and
their denominators (ACARA, 2015).
Question 3. Comparing and Ordering Fractions on a number line.
A) 34
B) 48
C) 12
D) 28
Answer: D, C & B, A
Rationale: Using a number line, and other length models, help students to determine ordered
fractions and help recognise fractions as numbers (Van de Walle, 2013). Research indicates
that number lines are an essential model for teaching fractions (Van de Walle, 2013). This
question will help determine if the student has had the necessary experience to calibrate the
number line, plot simple fractions according to their magnitude, and determine if the student is
able to recognise fractions as more than parts of an area model (Reys et al., 2012). The ability
to recognise equivalent fractions will also be able to be determined.
Misconception: When comparing points on a number line the student may experience difficulty if
a student has a restricted definition of fractions or experiences have been limited to region
models (Reys et al., 2012), only be able to calibrate the number line and plot fractions with the
same denominator, demonstrating an inability to convert all fractions make them equivalent.
ACMNA102: Compare and order common unit fractions and locate and represent them on
a number line. Elaboration: Recognising the connection between the order of unit fractions and
their denominators (ACARA, 2015).
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Question 4. Ordering fractions using a variety of fraction models.
A B C D E
Order the fractions from smallest to largest
A) A, B, C, D, E
B) C, D, B, A, E
C) B, E, A, C, D
D) B, E, C, D, A
Answer: C
Rationale: Without the fractional notation provided, a student must first determine the
numerator and denominator of each area model before comparing and ordering them. When
comparing fraction models it is sometimes difficult to tell which parts are larger (Reys et al.,
2012), especially is the shaded parts are not adjacent.
Misconception: Students are unable to compare fractions using different fraction models
(Reys et al., 2012).
ACMNA102: Compare and order common unit fractions and locate and represent them on a
number line Elaboration: recognising the connection between the order of unit fractions and their
denominators (ACARA, 2015).
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4 Diagnostic Interview
4.1 RationaleThis diagnostic interview is designed to determine the strategies used by the students to make
sense of fractions on a number line using comparing, ordering, addition and subtraction. A
significant number of year 5 students demonstrate a difficulty with fractions, with a number of
students misunderstanding the meaning of the denominator (VDEO, 2015). Reys et al. (2012)
explains that students need to develop a conceptual understanding of fractions before fluency in
the computation of a fraction can be achieved, and partitioning and equivalence are two key
conceptual understandings that can bind fraction concepts together. Misconceptions can be
identified using this method as it provides additional insight into a student's thinking (Burns,
2010). Representing common fractions on a number line are typically demonstrated by
students between the ages of 11 and 13 which coincides with the end of the operating phase of
the First Steps Mathematics diagnostic map for number (Department of Education, 2013), and
is the focus of the following diagnostic interview.
Task 1. Key Understanding 5: ordering fractions on a number line.
Explain that fractions can be compared and ordered using a number line (Department of
Education, 2013). Using an 8 point blank number line from 0 to 1, ask the student to estimate a
selection of fractions with a common denominator and mark the number line at the correct point.
48 , 28 ,
58 ,
78
Ask the students to explain their thinking. What fraction would be equal to the 1 on the number line?
What do you notice about the position of 48 ?
Misconception: The student may not understand that when fractions have the same
denominator, fractions are ordered using their numerator (Bezuk & Cramer, 1989). Students
may demonstrate difficulty making mental representations of mathematical concepts and/or
make the shift to a continuous line instead of counting a number of objects (Van de Walle,
2013). Students may also disregard the distance between fractions on a number line
(McNamara & Shaughnessy, as cited by Van de Walle, 2013) demonstrating a difficulty with
proportional reasoning.
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ACMNA102: Compare and order common unit fractions, and locate and represent them on
a number line. Elaboration: Recognising the connection between the order of unit fractions and
their denominators (ACARA, 2015).
Task 2. Key Understanding 5: ordering fractions on a number line.
Explain that fractions can be compared and ordered using a number line (DOE, 2013). Using
blank number lines from 0 to 1, ask the student to represent a selection of common use
fractions with different denominators at the correct point.
12 ,
58 ,
14 ,
38
Ask the student to explain their thinking. How did you decide how many parts to divide the number line into? How did you find the mid-point?
Misconception: The student may not understand that when factions have the a different denominator, fractions are ordered using both numerator and denominator (Bezuk & Cramer, 1989) and difficulties may be due to limited experience working with number lines (Van de Walle, 2013).
ACMNA102: Compare and order common unit fractions and locate and represent them on
a number line. Elaboration: Recognising the connection between the order of unit fractions and
their denominator (ACARA, 2015).
Task 3. Key Understanding 5: solving fractions problems on a number line.
Explain that addition and subtraction problems can be solved using a number line (DOE, 2013).
Using an 8 point number line from 0 - 1 with fractions marked in eighths, ask the student to
solve addition and subtraction problems involving jumps on the number line. Ask the student to
explain their thinking.
68 -
18 Answer: 58
28 + 28 Answer: 48
88 -
88 Answer: 0
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Misconception: The student is unable to recognise fractions as points on a number line for
performing addition and subtraction problems, demonstrating a limitation to other types of
fraction models. A student may also fail to recognise a fraction as a whole number, instead
seeing a fraction as two numbers with the numerator and denominator as separate whole
number (Reys et al., 2012).
ACMNA103: Investigate strategies to solve problems involving addition and subtraction of
fractions with the same denominator. Elaboration: Modelling and solving addition and
subtraction problems involving fractions by using jumps on a number line, or making diagrams
of fractions as parts of shapes (ACARA, 2015).
Task 4. Key Understanding 5: interpreting fractions greater than 1 using a number line
Explain that number lines can be used to show fractions greater than 1 (Van de Walle, 2013).
Ask the student to mark the point on the number line marked from 0 to 2 where each fraction
would belong.
1210 ,
210 ,
610 ,
1910
Why did you place the fractions in that place?
What fraction would each whole number be?
Pointing to 112 on the number line, ask the student to say what fraction would go there?
Misconception: the students is unable to recognise the whole and additional parts in a fraction
greater than 1 (Van de Walle, 2013). The student is unable to determine an accurate
benchmark on the number line and the density properties the fractions provided (Reys et al.,
2012), as well an convert an improper fraction to mixed numbers for plotting purposes.
ACMNA102: Compare and order common unit fractions and locate and represent them on
a number line. Elaboration: Recognising the connection between the order of unit fractions and
their denominator (ACARA, 2015).
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5 Conceptual Development of Multiplicative ThinkingMultiplicative thinking is underpinned by a number of key ideas and strategies (Siemon, as cited
by the [VDOE], 2013) that each provide conceptual pre-requisites for the next idea that follows.
Multiplicative thinking requires students to work with three aspects of a multiplicative situation;
groups of equal size, the number of groups, and the total amount, and for fractions and
decimals this idea is extended to parts of equal size, number of parts, and the total amount
(Reys et al., 2012).
5.1 Early NumberIn the early years, teaching is focussed on early number skills; counting, subitising, part-part-
whole, trusting the count, place-value, and composite units (Reys et al., 2012). For example, a
teacher could provide opportunities for students to visualise the whole number 20 to be 4
groups of 5 and 5 groups of 3 by using concrete manipulative such as beans or counters to
understand composite units. Trusting the count is deemed by Hurst and Hurrell, (2015) as one
of the big idea of early number skills as it requires two understandings; recognising a collection
without needing to count each individual object, and developing flexible mental images for
numbers 0 to 10 (Siemon, 2007), and this can be developed by teaching students to 'count on'
from a given number without needing to start from 1 by stating, but hiding, the first collection
(Siemon, 2007). The understanding of 'groups of' help reinforce counting strategies, make-all
and count-all strategies, skip counting, and repeated addition, and the models used most
commonly when illustrating these aspects are number lines, sets of objects, and arrays
(Reys et al., 2012). Working with dot arrays provides a connection to equal sharing and
grouping as well as part-part-whole partitioning which underpin the later ideas of mental
multiplication and division (Hurst & Hurrell, 2015) and part to whole ratios are in turn linked to
percentages and extended to proportion concepts (Van de Walle, 2013). Furthermore, early
estimation language of 'more than, less than' and 'about' can provide a foundation to
computational estimation for all other key ideas of multiplicative thinking and help students to
develop flexible thinking about numbers (Van de Walle, 2013). Place value is deemed by Hurst
and Hurrell, (2015) as another of the big idea of early number. Working with place value
requires students to be able to assign values to numbers based on their position which is
represented by successive powers of ten (Siemon, 2011) and failing to recognise the basis for
recording multi-digit numbers and the structure of the base 10 number can cause difficulty for
students when working with mental and written computation at a later date as place value is
fundamental to multiplicative thinking (Siemon, as cited by VDOE, 2013).
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5.2 Mental strategies for Addition and SubtractionMental strategies for addition and subtraction; the next key idea of multiplicative thinking
(Siemon, as cited by VDOE, 2013), includes counting on from larger numbers, thinking about
doubles and near doubles, and combinations-to-ten (Reys et al., 2012). Using mental imagery
is a skill that students will use to add and subtract single digit numbers when working on Key
Understanding 1 in the First Steps Mathematics: Number outcomes and a teacher can help
develop this skill by asking students to imagine adding or subtracting from a group of objects
without touching them (DOE, 2013). A variety of models can be used to illustrate this operation
to students including Unifix cubes, a bead abacus, or Numicon shapes. As students learn fact
strategies such as combinations-to-ten; 3 + 7, 2 + 8, students can be extended with 10 frames
to 20 to help with bridge-to-10 strategies and place value (Reys et al., 2012). As addition and
subtraction both require an understanding of place value, a teacher could plan experiences
involving regrouping and trading; using bundling sticks or a 3 prong abacus to model and
represent numbers in a variety of ways. Like addition, children often develop different strategies
when moving from concrete to written equations with subtraction (Reys et al., 2012).
Introducing students to working with fact families may help them see the inverse relationship
between addition and subtraction (VDEO, 2014). Early estimation skills are further developed
when working with mental addition and subtraction, and strategies include the front end method,
whereby students focus on the leading digit, the rounding method; substituting one number with
a compatible number to make the computation easier, compatible numbers; using compatible
whole number substitutes, and using tens and hundreds (Van de Walle, 2013). Estimation
activities that include finding answers within a range may help students who are anxious about
finding the exact answer. For example, a teacher might plan activities that involve estimating a
quantity of object in a container. By providing a number range, the students can use their
estimation skills and strategies to find an answer within the range provided.
5.3 Conceptual strategies for multiplication and divisionConceptual strategies for multiplication and division is the next key idea of multiplicative thinking
(Siemon, as cited by VDOE, 2013), and is underpinned by concepts such as 'groups of',
arrays/regions, area, Cartesian product, rate, and factor x factor = product. The Cartesian
product strategy, explores the combinations possible with 3 or more items, while arrays, have a
multiplicative structure and can help students visualise multiplication by seeing an array of dots
as a grid and as one entity, and turning it 90 degrees then demonstrates commutative
properties (Van de Walle, 2013). A teacher might plan lessons whereby students use tiles,
geoboards or graph paper (Reys et al., 2012) to model problems such as 4 x 2 = 8, and 2 x 4 =
8 to help with the transition from additive thinking to multiplicative thinking. Arrays are a very
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powerful and useful tool to show fractions, multiplication, division, area, ratio and commutative
properties which provides a conceptual way of learning which can be used in a concrete and
semi-concrete manipulatives. Students working in First Steps Key Understanding 1 could use
ten frames as an introduction to arrays to see visualise how numbers can be broken up to help
with calculation. Distributive property of operations is also important to students learning as it
allows numbers to be separated and helps make problems easier to solve (Van de Walle,
2013). For example, in the problem 3 x 99, students can change the problem to 3 x 100 minus
1 removing the requirement of a calculator pen and paper and experiences using the
fundamental properties of the operations can be a valuable scaffolding strategy to the next key
idea.
5.4 Mental strategies for multiplication and divisionThe multiplication and divisional mental strategies idea works with strategies of doubles,
doubles and 1 more, relate to 10, and commutative property (Siemon, as cited by VDOE, 2013).
A teacher might demonstrate that two numbers can be multiplied in any order and the answer
will be the same using Cuisenaire rods and a ruler, or dominos and the game Switcheroo. When
working with division, students should also recognise that division is the inverse of multiplication
as this is fundamental to computational fluency and algebraic thinking in later years (Siemon,
2011). Links can also be made between division and fractions by using the terms such as
'halving' as this strategy also simplifies division problems. Many of the computational skills
required for multiplication and division stem directly from other key ideas and strategies such as
partitioning, place value, addition, and subtraction (Van de Walle, 2013). For example, when
partitioning mentally a student may use existing number facts such as 10 x 8 = 80 so 9 x 9 = 80
and Hurst and Hurrell (2015) deem partitioning as big idea of number. Multiplication is often
viewed as repeated addition, and while it is possible to solve multiplication problems using this
strategy, the process remains additive (Siemon, as cited by Australian Association of
Mathematics Teachers [AAMT], 2013) which is limiting. For example, when the student begins
to multiply fractions and decimals, or multiplication is used to determine the area in geometry,
the repeated addition strategy fails to work. Hurst (2015), suggests the use of the term 'times
as many' when the scaling relationship is introduced . For example, when working with the
factor of 18, the number of groups could be 6 and the number in each group would be 3, so the
3 is 'scaled up' by a factor of 6. A teacher could also model specific language to help prepare
students for working with ratios and proportions in the future.
5.5 Fractions and DecimalsThe related processes of partitioning and working with composite units in the previous key ideas
of multiplicative thinking form a strong foundation for the key idea of working with fractions and
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decimals. For example, computation with multiplication and division helps students when
working with decimals as the same concepts are used and only have to work on estimation of
decimal point placement (Van de Walle, 2013). Key strategies associated with fractions and
decimals include making, naming, recording, renaming, comparing, and ordering equal parts of
a whole (Siemon, as cited by VDOE, 2013). Reys et al. (2012) explains the conceptual
understandings of fractions as six key ideas including the 3 meanings of fractions; part-whole,
quotient and ratio, models of part-whole meaning; region, length and set, and making sense of
fractions; partitioning, the vocabulary of fractions, counting the parts, the meaning of symbols,
drawing and extending models, ordering and equivalent fractions, benchmarks, mixed and
proper fractions, and operations with fractions. A teacher could model part-whole relationships
with folded pieces of string to show 2/3, model the part-whole fractions of items in a student's
everyday life such as pizza, or folded origami squares of paper (Reys et al., 2012). Fraction
ordering tasks could include concrete and pictorial models such as fraction tiles and diagrams in
the early years, before moving to symbolic representations of fractions in later years, Similarly
when working with benchmarks, to help student determine if a fraction is near 0 or 1, number
lines can help students visualise position, compare fractions with different denominators, and
help explain the density property of fractions (Reys et al., 2012). Fractions are dependent on
identifying measurement, particularly when working with number lines (Van de Walle, 2013). For
example, when determining the fraction points on a number line, determining the position of 1/2
and 1/4 assists students to use a measure to estimate placement of other fractions according to
the total unit. Benchmarks also build on the concept of estimation which, although a complex
mental procedure involving different choices and methods, is an effective way of promoting
procedural fluency when working with fractions (Van de Walle, 2013). By Level 5 of the First
Steps Mathematics Number outcomes; Key Understanding 7, students should be able to
mentally calculate unit fractions using addition and subtraction and recognise well-known
equivalences (DOE, 2013).
Decimals, proportional reasoning; ratio, proportion, and percent, all form the final key idea of
multiplicative thinking (Siemon, as cited by VDOE, 2013). The decimal concept becomes a
focus in years 4 and 5, whereas the ratio and percentage concept features in the content
descriptions in the Australian Curriculum for Years 6 to 8 (Reys et al., 2012). The development
of key understanding related to decimals might include activities partitioning decimals in
different ways (DOE, 2013). For example, using whole number partitioning skills, students
could partition decimal numbers such as 5.25 into 5 as the whole number and .25 as the
decimal or use number expanders with a decimal point. Interestingly, it is not necessary to
complete the study of fractions before introducing decimals (Reys et al., 2012), in fact, it is
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important to relate decimals to what a student already knows about fractions as the
understanding of fractions underpins the understanding of decimals. Ryes et al. (2012)
recommends relating a fraction such as a 4/10 area model to remind students that a tenth has
ten equal parts so that students can transfer their understanding of 4/10 to be 0.4 in a decimal
notation and that both equal four tenths. By using the place value names of tenths, hundredths,
and thousandths, students can be introduced to the placement of the decimal point on an
extended place value chart (Reys et al., 2012). A teacher could return to activities with concrete
materials for groupings, but this time discuss grouping by tenths instead of tens, or use a
decimat to play games to represent the size of decimals and their place value (Roche, 2010).
Decimats may also provide a clearer explanation of why tenths are bigger than hundreds when
the opposite is true for places to the left of the decimal point (Van de Walle, 2013). Ryes et al.
(2012) cautions, however, that students need to have made the connections in tenths to model,
symbol, and word before moving on to hundredths and thousandths places. Careful
explanation from a teacher should see students be able to write fractions and decimals notation
and switch between the two with confidence. An although operations with decimals create little
difficulty for students who are already working well with addition and subtraction, however,
some students may need to be reminded to line-up the decimal point, especially when working
with word problems (Reys et al., 2012).
5.6 Ratio, proportion and percent
Ratio, proportion, and percentage demonstrate quantitative thinking; counting, and relational
thinking; how one mathematical concept or understanding might relate to another (Reys et al.,
2012), and together, form the last key idea of multiplicative thinking. There are three types of
ratios; part-whole, quotients, and rates, and these all require the cognitive tasks of multiplicative
comparison and thinking of ratios as a composed unit (Van de Walle, 2013). A teacher could,
for example, provide everyday examples of each thinking task by presenting problems such as
a 5:4 ration of men and women on a train where there are 5 men to every 4 women, or
comparing a 10 cm length of wire to a 25cm length of wire and determining that the second
piece of wire is 2.5 times as long as the first. Part-part and part-to-whole ideas, patterns, and
fractions all underpin working with ratios, proportion and percentage (Siemon, as cited by
VDOE, 2013). For example, a ratio of 1:3 red and blue smarties are the same as the ratio 2:6
so the ratios are proportional, and 25% is a ratio of 25:100. Similarly, percent can be seen is a
ratio with 100 as the denominator (Van de Walle, 2013) and students who can demonstrate the
relative magnitude of ratios will have achieved level 5 of the First Steps in Mathematics Number
outcomes (DOE, 2013). Linking symbols to models will also help to minimise confusion when
students first begin working with ratio notation (Reys et al., 2012) so using fraction tiles with the
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fractional notation on one side and percentages on the side may be helpful to understand
equivalences, relative magnitude, as well as the relationship to decimals. It is the flexible
understanding of these concepts that help with everyday estimation and mental calculation in
contexts of shopping, cooking, reading maps, or home renovation. A teacher can help students
to make links between ratio, proportion, and percentage, and apply this knowledge to real-world
situations such as making play dough. Two batches of play dough could be made at the same
time by students for example; one batch with half the quantity of ingredients to demonstrate
direct proportion. The flour to salt could be calculated to demonstrate ratio, and quantity of
dough could be compared at the end to demonstrate proportion before being divided to
calculate the percentage each student receives. By year 7 students should be able to connect
fractions, decimals, and percentages as well as carry out simple conversions and solve
problems involving simple ratios (ACARA, 2015).
6 Conclusion
It is clear that students need significant teacher support to transition from additive thinking to
multiplicative thinking. It is also clear that teaching should be targeted to the key ideas and
strategies outlined in this report to increase the degree to which connections are made between
them to help students build a capacity to work flexibly with a variety of numbers and ensure
future mathematical success. The mathematical content knowledge that has been synthesised
in this report has outlined a range of assessment techniques to identify student thinking so that
a teacher can help students master new multiplicative understandings that can be applied to a
variety of problems, as well as the depth and breadth of mathematical content and pedagogical
knowledge required to help students understand the key ideas of multiplicative thinking. By
using mathematic reasoning, a student can choose which skill to use in each context and
communicate their thinking in a variety of ways. The pedagogical content knowledge outlined in
this report has highlighted the importance for a teacher to determine what a student already
knows in order to decide what needs to be learned next, how the content can be divided into
manageable chunks, which resources and activities are the most appropriate, and the various
representations of a key idea that students need to know. Most importantly, students need to
learn through play and mathematical games and identify real life possibilities of applying
knowledge to situations that students find interesting.
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References
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Bezuk, N., & Cramer, K. (1989). Teaching About Fractions: What, When, and How? Retrieved from http://www.cehd.umn.edu/ci/rationalnumberproject/89_1.html
Burns, M. (2010).Snapshot of student. Retrieved from https://lms.curtin.edu.au/bbcswebdav/pid-4155240-dt-content-rid-23568886_1/xid-23568886_1
Department of Education. (2013). First Steps in Mathematics: Number Book 1. Retrieved from http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/
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