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Marriott_Caroline_ 18448643

EDP243 Children as Mathematical Learners

Assessment 3 ePortfolio

Caroline Marriott

18448643


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.


3 Mathematics Content Knowledge TestOrdering Fractions and Equivalent Fractions

Question 1. Ordering fractions from largest to smallest using set models

15

12

16

13

14

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).

16

15

14

13

12

12

13

14

15

16

12

13

14

15

16

12

13

14

15

16


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).


Misconception: Students are unable to compare fractions using different fraction models


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


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.






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


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).


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.


ACMNA102: Compare and order common unit fractions, and locate and represent them on



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).



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


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.



their denominator (ACARA, 2015).


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


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).


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


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


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


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


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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