17108124 Nikola Kurtz EDP245 Assessment 2

Introduction

The purpose of this report is to determine the level of understanding students have in relation

to fraction sense and the mathematical elements associated with this understanding through

the use of diagnostic assessment. This report will cover the initial planning of diagnostic

assessment for two grade five students, give insight into what took place during the activities

and discuss the results of the activities to give a diagnosis of the students level of

understanding in accordance with the Australian Curriculum and other sources. The report

will outline what steps to take after the students’ level of understanding has been confirmed

through a discussion of effective teaching strategies, which is then applied to a lesson plan

designed to move the student on to the next level of understanding.

Background

The key idea behind the development of fraction sense lies within ones ability to acquire an

internal feel for the mathematical elements of fraction knowledge (Reys, Lindquist, Lambdin

and Smith, 2012). With this internal feel students have a sense of the magnitude of the

notion of fractions, and a basis of understanding to assist in development of more difficult

and sophisticated concepts (AAMT, 2013). One with fraction sense has developed further

from simply memorising procedures, but is able to explain their mathematical thinking and

understand not only the how but why behind processes (AAMT, 2013). Having an aptitude

for the intuitive knowledge of fractions, allows students to express their development through

working confidently and flexibly across a range of contexts, representations and problems

(McNamara & Shaugnessy, 2010).

Fraction sense is a required skill within the Australian Curriculum as students begins to learn

about partitioning and the part-whole model (ACARA, 2014). A feel for fractions is needed as

students progress to modelling fractions, comparing, ordering on a number line and solving

problems (ACARA, 2014). Diagnostic assessment allows teachers to find out what their

students know, but also to identify how students use their mathematical understandings and

probe the depth of understandings to gain insight into their level of fraction sense (Booker,

Bond, Sparrow & Swan, 2010). Diagnostic assessment allows teachers to assess a range of

factors related to fraction sense such as misconceptions, use of language, confidence,

application of knowledge, use of strategies and mastery of concepts (AAMT, 2013).

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Diagnostic Assessment

Two year five students Sam* and Jessica* were assessed on their fractional understandings.

The two students vary in their mathematical ability with Sam* being in the highest-level math

class of three. Sam and his parents have expressed Sam’s enthusiasm towards mathematics

and his intrinsic motivation towards number and fractions as well as other topics. Sam is also

very confident in his mathematical skills and understandings. This is reflected through Sam’s

results in his 2014 half yearly report. The reports show Sam’s level of mathematics to be in

the high levels of proficiency and indicate Sam is a highly competent and capable student.

Jessica in comparison struggles to maintain a place in the middle level class. Jessica’s

parents have stated that she is not a strong maths student and lacks understanding in

strands such as number and algebra and geometry. This was made clear by Jessica’s

negative and anxious attitude towards mathematics, when asked if she enjoyed maths

lessons. Jessica expressed she did not enjoy maths lessons or doing maths and found it

stressful and lacked motivation. However, Jessica performed better than usual in her most

recent maths test. Furthermore Jessica’s report shows a sound level of achievement in most

areas of mathematics with some areas needing more development.

Planning for both students’ assessment tasks began with reviewing the background

information of the students’ capabilities and then looking at the Australian Curriculum to find

a suitable content descriptor for their year level. The year five content descriptor “Compare

and order common unit fractions and represent and locate them on a number line”

(ACMNA102) linked well with Key Understanding Five in the First steps in Mathematics:

Number learning resource, which explores ordering and comparing fractions and using

number lines (ACARA, 2014). Both students’ activities were planned from Key

Understanding Five, however the activities varied in difficulty and sophistication based on

their pre judged level of understanding.

The planning of Sam’s diagnostic assessment was based on his high level of proficiency and

confidence towards mathematics. Activities from the later stages of key understanding 5

were chosen as they were more sophisticated and involved ordering and comparing fractions

as well as showing understanding of number lines. The activity “Places on a number line”

was chosen but changed slightly to using a 60cm number line from 0 to 1. In the case that

Sam found the task too easy another number line from 1 to 2 was created with mixed

fractions to assess the extensiveness of his knowledge. This activity was chosen as it would

clearly elicit the students understanding of partitioning, size of fractions and equivalence

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whilst giving insight into the development of fraction sense (Reys et al., 2012). “Comparing

fractions” and “Fraction Cards” were also chosen in case Sam was unable to complete or

found the first task to simple.

The background information on Jessica implied she was not a confident mathematics student

but had performed well in a recent assessment. Tasks from the middle level of key

understanding 5 were chosen so they were not overly complex but still challenging enough to

show her level of understanding. The activity “Cheesecake” was chosen as it assesses the

student’s ability to order and compare two commonly used fractions. The fractions in the

problem have different denominators, which makes the task more complex and in line with

Jessica’s mathematics level. The task also asks the student to explain thinking through

diagrams or a number line to ensure that the answer was not guessed or from memory.

“What number am I” and “Comparing lengths” were also chosen in case “Cheesecake” was

not at the correct instructional level (see appendix C).

Sam completed the first task very quickly however majority of the fractions placed on the

number line were incorrect. The student was asked questions to elicit the reasoning behind

the answers such as “what mathematical knowledge did you use to help place your fractions

on the number line”? Sam then attempted the comparing fractions task, which asked him to

order six fractions from smallest to largest. Sam found this task difficult and changed the

order of the fractions several times. Sam was given two number lines, which indicated thirds

on one and fourths on the other to help him (see Appendix C) Sam also attempted to use

paper folding to help order the fractions but was not sure how to use this properly.

The main outcomes of Sam’s diagnostic assessment mainly revealed areas of difficulty and

misconceptions. However, Sam was able to locate one half and one quarter on a number

line. Sam was able to use the number line manipulatives to assist slightly in the second task

as well as attempt paper folding but was not sure how to use the strategy correctly. Sam also

has some basic understanding that the value of the denominator does not mean the quantity

of the fraction as he placed one half as the largest fraction in the second task. Issues of

difficulty included locating only two out of fourteen fractions correctly on the number line and

then ordering the fractions incorrectly in the second task. The fractions were ordered by

value of the numerator and when the numerator was the same the student ordered by the

value of the denominator.

Jessica completed all three activities and was able to do two out of the three correctly. The

first task “cheesecake” was completed incorrectly however Jessica attempted to draw a

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number line to help solve the problem but located the fractions incorrectly. The simpler task

was then given to Jessica, which she solved correctly however, Jessica needed to re read

the question several times, re write the question and use a manipulative to reach the answer.

Jessica was asked how she reached the answer and her reply indicated she had a good

sense of partitioning. Jessica was then given the more challenging task of the three using the

number line in which she was highly competent.

Jessica demonstrated many understandings and learned concepts in her activities as well as

some areas of confusion. Jessica was able to draw a number line and diagrams to help solve

the first task. She was able to determine the fraction in between one half and one as two

thirds and explain her answer. Jessica also located ten out of eleven fractions correctly on a

number line showing an excellent understanding of thirds, fourths and fifths. However there

were difficulties that were noted such as stating that one-quarter is bigger than one third

because four is bigger than three. This shows that Jessica is confused about the role of the

denominator, however, was able to label the same fractions correctly on the number line.

Sam’s results show that he has attempted the tasks with some informal knowledge of

fractions and a very basic understanding of partitioning and the relationship between the

numerator and denominator. Looking at the results it is apparent that Sam’s level of

understanding of fractions is currently at a year two to three level. Sam has shown that he

understands what halves and quarters are through labelling them on the number line and

paper folding activities, however this may be due to informal knowledge and frequent use of

such terms. This idea is reinforced, as Sam was unable to demonstrate year three skills such

as represent or model fractions such as thirds, and fifths, represent them on a number line or

relate the parts to the size of a fraction, only halves and fourths (ACARA, 2014). The NCTM

standards also state that year two students should have an understanding of these

commonly used fractions before moving on to years three to five where students should

recognise equivalent forms of fractions and recognise fractions as parts of wholes, skills in

which Sam is not competent with (NCTM, n.d).

Jessica’s results indicate that she is at a year four to five level of understanding in relation to

the Australian Curriculum. Jessica has demonstrated that she is past a year three and four

level of understanding as she is able to model and represent halves, thirds, quarters and

fifths correctly on a number line and model fractions to a whole (ACARA, 2014). Jessica was

able to locate thirds and quarters correctly on the number line however failed to complete the

first task successfully as she stated one quarter was larger than one third because four was

larger than three. This may be due to difficulty in understanding the relationship to fractions

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on a number line to quantities in real life contexts. Jessica’s competency in the number line

task shows that she is able to demonstrate skills at a year five level by locating almost every

fraction correctly on the number line, which relates to the first content descriptor ACMNA102

(ACARA, 2014). According to the NCTM standards Jessica is developing skills within the

grades three to five expectations and has mastered the year two expectations of

representing and modelling commonly used fractions (NCTM, n.d).

Teaching Fractions

Both student’s demonstrated difficulties with the part whole concept and the role of the

numerator and denominator in determining the size of fractions. Thus the next steps to

increase the students understanding will be focused on mathematical elements such as

partitioning, the part whole concept, the relationship of the numerator and denominator and

representing fractions through models, words, symbols and on a number line. To assist the

students in developing these basic fractional concepts there are important instructional

strategies that will make learning more effective and engaging such as the use of concrete

and representational manipulatives, fractional language and using active, student centred

learning activities (Booker et al., 2010).

The use of manipulatives such as fraction strips, pattern blocks and fraction bars are

important to help express the abstract idea that fractions are a part of something, a skill

which both students demonstrated difficulties with (Forbringer & Fuchs, 2014). Using

instructional strategies such as creating and shading or modelling parts of a region model

allows students to experience a wide range of fractions, partitioning, equivalence and the

relationship of the numerator and denominator (Reys et al., 2012). This strategy is seen in

the lesson plan for Sam (Appendix A) as the student is introduced to partitioning through

fraction strips and then labels and discusses the relationships of the parts to the whole.

Including instructional strategies that encourage the students to reflect on their mental

processes is crucial and also supported by the use of manipulatives as they allow students to

represent their thinking, also seen in the lesson as questioning of students work is included

within the formative assessment (Naiser, Wright & Capraro, 2003). Encouraging Sam to

represent his ideas through manipulatives can also assist him in clarifying and deepening his

conceptual understandings (Forbringer & Fuchs, 2014).

Making learning active and student centred is also an important instructional strategy to

ensure students are motivated and interested. Using manipulatives in collaboration with

problem solving activities and real life contexts that students can find meaning in and can

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help engage students and interest them classroom activities (Naiser et al., 2003). This is

important for students such as Jessica who have low motivation towards maths and is also

represented in the lesson plan through the use of hands on activities, group discussion,

student questioning and range of manipulatives. Additionally the use of fractional language is

key to developing an understanding of the partitioning and part whole concept (Booker et al.,

2010). As both students showed confusion towards the numerator and denominator, using

language that clearly and explicitly states that fractions are parts out of equal parts and using

the correct terms for parts such as one fifth and three fifths is essential for both students to

understand fractions are a single quantity and not two separate numbers (Booker et al.,

2010).

Conclusion

It is clear, therefore, that the use of diagnostic assessment to find out what understandings,

difficulties and misconceptions students hold before beginning new mathematical concepts is

critical to ensuring instruction is aimed at the right developmental level. The report displayed

the discrepancies that may be revealed between a students actual understanding of

concepts compared to what a students attitude, previous performances and reputation in a

subject area may expose. The use of diagnostic assessment with both students assisted in

eliciting understandings and misconceptions that differed greatly from what was expected

based on the students background information. Most importantly, it is clear that diagnostic

assessment is essential to ensure instruction is planned at the correct instructional level and

that it is tailored to suit the distinct needs of the individual learner.

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References

ACARA. (2014). Mathematics F-10 Curriculum. Retrieved from

http://www.australiancurriculum.edu.au/mathematics/Curriculum/F-10?layout=1

Australian Association of Mathematics Teachers. (2013). Resources for teachers of Mathematics:

Fractions. Retrieved from http://topdrawer.aamt.edu.au/Fractions

Booker, G., Bond, D., Sparrow, L., & Swan, P. (2010). Teaching Primary Mathematics. Frenchs

Forest: NSW, Pearson.

Department of Education WA. (2013). First steps in Mathematics: Number. Retrieved from

http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/

Forbringer, L. & Fuchs, W. (2014). Rtl in Math: Evidence-Based Interventions for Struggling

Students. Retrieved from http://link.library.curtin.edu.au/p?

pid=CUR_ALMA51118573990001951

McNamara, J., & Shaughnessy, M, M. (2010). Beyond Pizzas & Pies: 10 Essential Strategies for

Supporting Fraction Sense, Grades 3-5. Retrieved from http://books.google.com.au/books?

id=iHwVuR4qndMC&dq=fraction+sense&lr=&source= gbs_navlinks_s

Naiser, E, A., Wright, W, E., & Capraro, R, M. (2003) Teaching Fractions: Strategies Used for

Teaching Fractions to Middle Grades Students, Journal of Research in Childhood Education,

18:3, 193-198, DOI: 10.1080/02568540409595034

NCTM. (n.d). Standards and Focal Points – Number and Operations. Retrieved from

http://www.nctm.org/standards/content.aspx?id=7564

Reys, R., Lambdin, D., Lindquist, M., & Smith, N. (2012). Helping Children Learn Mathematics.

Danvers, MA: Wiley

AppendixA: Lesson Plan (Sam)

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Lesson Title: Looking back at Fractions Learning area: Mathematics/Number

Year: 5 Time: 60 MinutesLearning objectives

By the end of the lesson the students will be able explain and demonstrate that whole objects can be partitioned into equal sized parts

By the end of the lesson students will be able to create, represent and name common fractions

By the end of the lesson students will be able to explain that a fraction is a quantity in between two wholes

By the end of the lesson students will be able to understand the role of the numerator and denominator

Curriculum links Recognise and interpret common uses of halves, quarters and eighths of shapes and collections (ACMNA033) (ACARA, 2014).(Lesson will include thirds, fifths, sixths and sevenths also)

Prior knowledge Students can explain, represent and model halves and quarters Students are able to partition wholes into halves and fourths Students have an understanding of whole numbers

Resources Pens, paper, scissors, glueFraction StripsChocolate BarCardboard CircleWhite board

Introduction (15 Minutes)

Whole class setting

Engage students in topic by asking: “what is a fraction?” and “what have we learnt about fractions so far?”

Write ideas on the whiteboard Introduce concept by labelling a block of chocolate as a “whole” and

breaking into “halves” and then “fourths” and “eighths” Ask students what the parts are called and represent and write the word and

symbol for the fraction on the board Explain the relationship of the numerator and denominator and how they

represent the size of the parts and the number of parts

Body of the lesson (30 Minutes)

Students are in pairs

Explain and discuss with students the action of partitioning the chocolate into equal parts that make up one whole and the part whole model

Form students into pairs and give out fraction strips and piece of cardboard to each pair

Model and demonstrate folding a fraction strip and cutting in half to represent halves

Paste strip to board and write halves next to it and the symbol for one half Explain to students they will be folding and cutting their fraction strips to

make halves, thirds, fourths, fifths, sixths, sevenths and eighths and then representing them on their cardboard and labelling the fractional parts with the word and symbol

Students work together to partition fraction strips, label and represent on cardboard, which shows the model, symbol, and word

Concluding the lesson (15 Minutes)

Teacher selects a poster and engages whole class in discussion about findings using key vocabulary and terms

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Whole class setting

Assessment

Key Questions

Adjustment of lesson for high and low achievers

Questioning throughout entire lesson to ensure students are involved in the discussion, understand learning material and can explain how they completed activities

Teacher moves around the room observing students to ensure they are completing activity and providing scaffolding for any students who are having difficulty

“How many parts make up this whole?”

“What are the names for these parts?”

“What does the numerator tell us?”

“What does the denominator tell us?”

“If I had two, three etc. parts of this whole how would I say/write it?”

“Which fractional part is larger?”

“How many parts of this fraction would I need to equal this fraction?”

High achievers: Students are given fraction cards and asked to locate fractions on a number

line in addition to creating a poster Students are asked to partition fraction strips larger than eighths Students are given extra materials such as pattern blocks, Cuisenaire rods

and circles instead of fraction strips

Low achievers: Provide extra scaffolding Ask student to work mainly with halves, thirds and quarters Pair student with a high achieving student Give student word bank to assist with new vocabulary

B: Consent Letters

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C: Student Tasks

Jessica (Year 5)

Task: Cheesecake

The word problem say’s “ Dad told louise and Matthew there were two pieces of cheesecake

left in the fridge. One piece was one third of the cheesecake the other piece was one quarter

of the cheesecake. Dad said the older child should get the bigger piece. Louise got one third

and Matthew got one quarter” Jessica must figure out which child was the oldest by solving

the problem of which fraction quantity of the cake is biggest. Jessica must also explain her

answer verbally and through the use of manipulatives, diagrams and/or a number line.

Task: What Number am I

I am less than one but more than zero. I am bigger than one half. What number am I? Draw diagrams or pictures to show how you figured this out.

Task: Comparing lengths

Mark the given lengths on the provided number line. The number line is from 0 to 10.

Sam (Year 5)

Task: Places on a number line

Sam is given a number line starting with 0 at one end and 1 at the other end with half way

marked. Sam is given a variety of fraction cards and asked to determine where they would sit

on the line and why. Sam is asked further questions and prompts if the task proves to be too

easy or too difficult.

Task 2: Fraction cards

Order sets of fraction cards and justify your reasons for ordering the cards as you have.

Task 3: Comparing Fractions

Which number is larger, two thirds or four fifths? How do you know? Use a number line to

prove that your answer is correct.

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Manipulative

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D: Activities completed by teacher Cheesecake:

One piece is one third and one piece is one quarter

A third is when a whole has been partitioned into three equal parts

A quarter is when a whole has been partitioned into four equal parts

Three equal parts of a cake would be bigger than four as the parts have come from the same

whole therefore Louise is the older child

Places on a number line:

Fractions between 0 and 1 are placed on a number line

The fractions that have the numerator closer to the denominator are closer to the 1 and the fractions that have the numerator further away from the denominator are closer to the 0

With each fraction I looked at the denominator and imagined partitioning the number line into that many equal parts and then counted where the numerator would go and marked the fraction there

For fractions that had the same numerator and denominator I marked them at 1

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