Literacy and Numeracy · 2011. 4. 12. · Literacy and Numeracy Pilots Final Report 7 Chart 2 Student confidence post-project 2010 Note that the lowest level of confidence was halved

Literacy and Numeracy

Pilots

FINAL REPORT

Literacy and Numeracy Pilots Final Report

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Section 1 – Executive Summary

The Middle Years Mental Computation Project (MYMCP) targets teachers and their students in the middle

years of schooling (years 4-8). Its focus is the number strand with particular emphasis on the sub-strand

rational number. The major aims of the project are to build teachers‟ pedagogical content knowledge in

mental computation and to support the project teachers as they embed the mental computation program

throughout their own school setting.

The MYMCP builds teachers‟ conceptual knowledge and provides them with a repertoire of appropriate

strategies for teaching mental computation meaningfully.

The project emphasises the use of effective concrete representations of concepts. These representations

provide students with a platform on which to construct mental models. Mental images allow students to

carry out mental calculations flexibly and meaningfully. As students continue to develop, the models are

gradually replaced by efficient symbolic representation alone.

Research by Alistair McIntosh, Shelley Dole and Dianne Siemon has been used in the development of the

MYMCP. The 2004 resource, Mental Computation: A Strategies Approach is a required resource for project

teachers. In response to the many requests from teachers explicit links have been developed between

MYMCP and the NSW Department of Education and Training Count Me In Too (CMIT) professional

development program.

Figure 1: Mental Computation Development Cycle

The project was implemented strategically to ensure sustainability by the ACT Department of Education

and Training (ACT DET). The project provided participating teachers with 5 days of professional learning

throughout the year of their involvement. Two project teachers from each school were nominated for this

professional learning. Participants received weekly support from project officers, including, timetabled

collaborative planning sessions (the planning document in appendix 3 was used to scaffold these

sessions), in-class modelling, coaching and teacher experience reporting was undertaken by each

participant. A gradual release of responsibility model1 was adopted by project officers to ensure that the

new professional understanding was consolidated into project teachers‟ teaching practice. Participating

1 Gradual Release of Responsibility Model, (Pearson and Gallagher, 1983)

2. Students are encouraged to

develop MENTAL IMAGES

3. Students refine images for use in

EFFICIENT MENTAL COMPUTATION

4. Students derive WRITTEN METHODS

from their understanding

5. Students GENERALISE understandings to

engage in more complex mathematics

1. Students create/utilise

MANIPULATIVE/ concrete materials


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school principals were advised to nominate emerging leaders in their school for the program to maximise

the chance that the professional learning undertaken would be adopted school wide. To foster this, regular

communication between project officers‟ and school leadership was established.

Achievement of the project‟s outcomes for both teachers and students is evident in the analysis of the

responses of pre and post testing. The tools used were surveys; an ACT DET developed mental

computation test (MYMCT); and the National Assessment Program for Literacy and Numeracy (NAPLAN),

(NAPLAN analysis has been completed for students in 2009 only).

Project teachers grew in confidence to teach mental computation (12 percent pre-program to 76 percent

post-program in 2010) and mathematics generally (62 percent pre-program to 90 percent post-program).

They were able to demonstrate an increased understanding of effective teaching pedagogy. This was

evident through teachers in post testing having gained the knowledge to describe an increased range of

mental computation strategies as well as having learned an increased number of effective tools to assist

students in developing a deep understanding of number.

Students demonstrated improved facility with mental computation; particularly in the sub-strand of rational

number. The following table highlights the improvement for students in four of the substrands tested (the

test schedule is in appendix 2).

Mental Computation Skill 2010 pre-

test

2010

post-test

2010

growth

Fraction sense (+ -) 32.56% 61.15% 28.59%

Fraction (x ÷) 26.86% 50.69% 23.83%

Decimals (+ -) 41.96% 67.25% 25.29%

Decimals (x ÷) 24.44% 44.53% 20.09%

NAPLAN student progress scores (evaluated as a residual score based on expected growth using the

students 2008 score) were compared using a class with an MYMC project teacher and a class within the

same school with a teacher who did not receive any MYMC professional learning. Care must be taken in

using these results as the data set is small. There are a number of factors that contribute to student

progress such as transitions and supportive pastoral care. A methodology for using NAPLAN as an

evidence based longitudinal measure for the MYMC program is currently being considered.

In 2010 project officers noticed project teachers‟ willingness to adopt a leading role in the planning and

implementing of the mental computation sessions resulting in the project officers‟ role more quickly

becoming one of mentor rather than instructor. Providing this intensive level of scaffolding to two teachers

only results in a greater likelihood that there will be quality outcomes for them and their classes. With two

teachers skilled within a school and with the support of the school executive, the resulting uptake of the

program is more likely to be successful the following year.

Future directions identified for the MYMC program are to develop facilitator workshops to build the capacity

of teachers in the system and ensure sustainability of the professional learning. It is anticipated that each

facilitator will receive two days professional learning and specialise in, whole number, fractions, decimals or

percentage/ratio. In addition to this the School Improvement Division of the Department employs two

numeracy executive officers. They are able to have a coaching role in schools when this level of support is

requested.


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A connected learning community (cLc)2 has been established to enable past project teachers to remain

connected with new ideas and research. The learning community is in early development, however the

concept has been well received by past project teachers. MYMCP‟s existing resources are currently being

mapped to the Australian Curriculum and will be uploaded on the website for teachers to access. This

includes both print and Interactive Whiteboard (IWB) resources; instructions; adaptations; and, where

appropriate, a cultural inclusion activity to engage students either learning a language other than English or

those who have English as an additional language or dialect (EAL/D).

The MYMCT is currently being reviewed and there are plans to develop additional mental computation

assessment DVDs that would support schools in adopting consistent practices for mental computation and

further enable effective communication between High Schools and their respective feeder Primary Schools.

The MYMCP is endorsed as a strategic professional learning program for ACT schools by the Department.

The program has been positively received in ACT schools. Teachers and the Literacy and Numeracy

section will continue to foster the program‟s growth to meet the needs of the system, its educators and its

students.

2 The implementation of the connected Learning community (cLc) is a key activity for supporting schools through

innovative technologies within the Department‟s Operational Plan 2011, Everyone matters. It is provided by Learnology©2010.


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Section 2 – Framework Data

Students

The MYMCP was conducted as two separate programs in the 2009 and 2010 calendar years. Whilst the

intended student outcomes did not differ, the way in which data was collected did. The intended outcomes

for students were:

an increased confidence in mathematics

the transferability of mental computation to other learning areas

an increase in student attitude towards mathematics

an increase in student achievement, both in mental computation facility and NAPLAN numeracy

results.

Outcome 1: Student confidence

2009

A student survey was conducted at the beginning and end of the in-class support. Students were read a

number of statements and asked to circle those they agreed with. The following statements surveyed

student confidence, categorised as low confidence, confident, and highly confident.

Low confidence Confident Highly confident

I feel like I never get the right

answers in maths.

I give up when the problem

seems too hard to solve.

Mathematics time makes me

nervous.

I don‟t like to give my answer

to a maths problem in class.

I feel worried when I have to

do a mathematics problem.

I can solve the maths

problems without too much

difficulty.

I never ask for help in maths

classes.

I understand what is being

asked in mathematics

questions.

If I don‟t understand the

maths being taught to me I

ask my teacher to explain.

I keep trying to find the

answer to a maths problem

even when it is very hard for

me.

I think maths is easy.

I am sure the answers I get

are correct.

I like to share my answers

with the class.

Table 1: Student confidence survey questions 2009

Student confidence levels are shown in the table below. Note: students could select any number of

statements they agreed with. This means that student responses could fall into more than one category

and the percentages of student responses will not add to 100.

CONFIDENCE LEVEL

Low Confident High n

Pre-project 36.44% 36.87% 62.21% 404

Post-project 31.59% 37.44% 52.18% 390

Table 2: Confidence level of students 2009


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The table shows the number of times each category of question was selected. Results show a decrease in

the number of high confidence questions selected. Reasons for this may include:

Students are now more aware of misconceptions and critically reflect on answers

Students may be more likely to seek help from peers rather than their teacher

Students‟ individual needs are catered for through differentiation so maths is no longer easy, but

challenging

A slight decrease in the response to low confidence questions supports these ideas and may be seen as

closing the gap in confidence for students of lower ability. It is proposed that these students now feel more

engaged in their learning.

2010

In 2009 the format of the survey made it very difficult to ascertain a student‟s feelings towards mathematics

so the survey was simplified. Again the 2010 student survey (appendix 1) was completed at the beginning

and end of in-class support. However, the response option was modified to a sliding scale to make it easier

for students to commit to a response. There were three questions as detailed below:

Confidence:

Circle the place on the scale (strongly disagree, disagree, neutral, agree, strongly agree) that best

describes how much you agree with the following statements:

a. I feel like I always get the wrong answers in Maths

b. I keep trying to find the answer to a maths problem even when it is very hard for me

c. Mathematics time makes me nervous

Chart 1 Student confidence pre-project 2010

0.7 4.59.7

28.7

32.7

22.6

Not answered

Lowest confidence

Low confidence

Neutral

Confident

Highly confident


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Chart 2 Student confidence post-project 2010

Note that the lowest level of confidence was halved with one year of exposure to the program. With a whole

school approach to mental computation it may be inferred that the number of students in this category

would continue to decline. Student confidence in 2010 increased overall with low confidence dropping from

14.2 percent to 11.2 percent. The increase in confidence at the confident and highly confident measured

(55.3% to 61.7%) shows an effect on the students‟ confidence attributed to their year of MYMC learning.

Outcome 2 – Mental Computation Transferability (Use of mental computation in other learning

areas)

2009

Teachers in 2009 stated the following areas for transference of mental computation strategies:

consumer arithmetic

sustainability assignment

measurement activities

mapping scales.

2010

Teachers in 2010 stated the following areas for transference of mental computation strategies:

distance on a map

in literacy plotting a leaders life span and achievements on a timeline

students used mental computation strategies at the school canteen

in Studies of Society and the Environment (SOSE) to calculate how many people in the army

using timelines and finding the difference in years

0.6 2.68.6

25.6

36.8

24.9

Not answered

Lowest confidence

Low confidence

Neutral

Confident

Highly confident


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in art, finding out how long an artist lived for

measurement in physical education

in music

in Science when writing and timing their observations

relationship to probability in mathematics

an inquiry unit of running a business

working out how much time before lunch

integrated units of earn and learn

in Science during physics

in art when working out area

in writing exposition and assigning a numerical value to pros and cons based on their worth.

Outcome 3 – Student Attitude

2009

Students were asked to rate how much they enjoyed mathematics lessons in the 2009 survey.

Chart 3: Students’ enjoyment of mathematics lessons 2009

The graphs for pre and post project show an increase in students‟ enjoyment of mathematics lessons.

Fewer students are feeling unhappy in mathematics lessons.

The survey questions relating to student attitude are summarised in the table below:

37%

50%

13%

Happy

Neutral

Unhappy

Pre-project

41%

49%

10%

Post-project


9

ATTITUDE

Negative Neutral Positive n

Pre-project 29.58% 53.22% 37.87% 404

Post-project 29.74% 51.28% 46.79% 390

Table 3: Attitude towards mathematics 2009

Student attitude levels show an improvement in positive responses, this includes students enjoying finding

new ways to solve problems and engaging in extra tasks when available. Again, the students were able to

choose questions in more than one category which made it difficult to ascertain from the data exactly how

the students were feeling. The question surveying students‟ enjoyment of mathematics shows that the

number of students indicating a dislike of mathematics during the pre survey has not shifted significantly in

the post survey. 13% of students indicated a dislike of mathematics pre-project and 10% post-project.

Dispositions towards mathematics both positive and negative are built up over years and the result of a

complex mix of experiences. These include the home experiences and classroom experiences over years

and once formed are resistant to change. However, it should be noted that the structure of the question

was ambiguous and therefore was modified in 2010 to ask questions and have students give a response on

a sliding scale.

2010

Once again in 2009 the format of the survey made it very difficult to ascertain a student‟s attitude towards

mathematics so the survey was simplified. In 2010 the attitude section of the survey response option was

modified to a sliding scale to make it easier for students to respond. There were three questions as

detailed below:

Attitude:

Circle the place on the scale (never, rarely, sometimes, mostly, always) that best describes how often you:

a. Enjoy maths lessons

b. Find maths lessons interesting

c. Try in maths because you feel it is important


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Chart 4: Student attitude towards mathematics in 2010 pre-project

Chart 5: Student attitude towards mathematics in 2010 post-project

The results show that student attitude whilst decreasing in the never category also decreased in the always

category. This indicates that it is difficult to measure student attitude and the results are impacted on by

how students are feeling at the particular time of the survey. As previously mentioned a student‟s attitude to

3.79.5

31.1

32.3

23.1

0.3

Never

Rarely

Sometimes

Mostly

Always

Not Answered

1.812.4

30.5

33.8

20.90.6

Never

Rarely

Sometimes

Mostly

Always

Not Answered


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mathematics often has many more contributing factors such as previous years of schooling, real-world

experiences and parental influence.

Quantitative

Outcome 1 – Student Achievement

Student Assessment – Middle Years Mental Computation Test

The Middle Years Mental Computation Test (MYMCT) measures achievement in mental computation. The

test assesses mental computation across all sub-strands of number. The test is administered via a DVD

that uses both visual and audio delivery, see appendix 2 for questions and the timing schedule. There are

three versions of the test; the first is administered at the beginning of the program, a second, as a mid-test

at the teachers discretion and the third at the end of the program.

Teachers are provided with a spreadsheet for the collection and entry of the MYMCT data.

The initial data entry screen for each test provides teachers with an immediate visual display of successes

and gaps for each question, for each student and cumulatively for the class. See Figure 2.

Figure 2: Example of MYMCT entry screen

Cumulative class percentages at the bottom of the screen are colour coded indicating class achievement.

o Green >75% correct

o Amber >50% correct

o Red >25% correct

o Black 0-24% correct

Individual student achievement is highlighted horizontally by the use of colour. Gaps (no colour) show

incorrect answers providing a powerful visual representation of error patterns. Each colour is representative


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of a set of mental computation concepts or skills, (tens facts; 2 and 1 digit; groups of ten; harder addition

and subtraction; basic facts multiplication and division; doubles and multiply by 10; extended basic facts

multiplication and division; harder multiplication and division; fraction sense addition and subtraction;

fraction multiplication and division; decimal addition and subtraction; decimal multiplication and division;

and percentages). The Scaffolding Mental Computation planning document (appendix 3) uses this colour

scheme to support teachers in linking the assessment to explicit teaching and learning opportunities.

Each question may be analysed by reading the vertical columns, highlighting particular error patterns for

the class.

The above spreadsheet information can also be represented using a range of options through an

interactive chart (spider graph). Options include:

class data, with tests 1, 2 or 3. An overview of class achievement

cohort data, with tests 1, 2 or 3. An average of classes from that year level

student data, with tests 1, 2 or 3. An overview of individual achievement

any combination of the above.

Figure 3: Example of MYMCT interactive chart showing pre and post test results

The spider graph shows the sub-strands tested within the MYMCT with the percentage correct represented

on the radii beginning with basic 10s facts through to percentages, showing the hierarchy of difficulty for

mental computation.

The graphs provide powerful visual information about class or an individual student‟s conceptual

development. Teachers in collaboration with the project officer analyse the error patterns of the MYMCT.


13

The analysis determines the areas to be targeted in the mental computation sessions and provides project

teachers with vital data to help with the selection of their experience report topics.

An example of two individual students achievement can be seen in the charts below, (one is a lower ability

student and the other a higher ability student, both in year 7).

Chart 6: Individual student pre and post MYMCT spider chart

The chart shows this student had correct answers only in the 2 and 1 digit addition and subtraction section

of the assessment in pre-testing. In post testing the student was able to answer questions correctly in tens

facts; 2 and 1 digit addition and subtraction; groups of tens; basic facts multiplication and division; double

and multiply by 10; fraction sense addition and subtraction; decimal addition and subtraction; and

percentages. The chart highlights a great improvement in understanding for a student of lower ability in a

mixed ability year 7 class.

Chart 7: Individual student pre and post MYMCT spider chart


14

The chart shows the student improved in all aspects of the assessment, (except in double and multiply by

10 where they maintained 100% correct), in particular their basic facts for multiplication and division and

fractions, decimals and percentages. The data is from a student in the same mixed ability year 7 class as

the previous student‟s data. By broadening the use of the assessment spreadsheets throughout a school,

the whole school would benefit from the display of growth for individual students using this evidence-based

framework.

The data collected is a key component for planning sessions between the project officer and teachers. The

students‟ areas of weakness are identified and the mental computation sessions focus on explicit

instruction to improve students‟ facility with number. The Scaffolding Mental Computation planning

document (appendix 3) has been developed for teachers to facilitate them finding explicit teaching

strategies and activities to consolidate the identified problems. This resource is based on the same colour

scheme as the mental computation assessment spreadsheet.

2009

Students in all participating classes completed at least two versions of the MYMCT in the year. The results

of the pre-test informed the focus for the mental computation sessions that were delivered each week.

Data is used for class level, team cohort, northside/southside cohort and as a total project cohort. The total

project cohort results in the pre and post testing are shown in the following graph.

Chart 8: 2009 Growth chart MYMCT

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

10

fa

cts

(+,-

)

2 a

nd

1 d

igit

(+

,-)

gro

up

s o

f te

n (

+,-

)

mix

ed

(+

,-)

Ba

sic

Fact

s (x

,/)

Do

ub

le a

nd

X1

0 (

x,/)

ext

nd

ba

sic

fact

s (x

,/)

mix

ed

(x,

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fra

ctio

n s

en

se

fra

ctio

n (

x,/)

de

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als

(+

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als

(x,

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be

nch

ma

rk p

erc

en

tag

es

& m

ult

iple

s o

f 1

0%

Post-test

Pre-test


15

Mental Computation Skill 2009 pre-project 2009 post-project 2009 growth

10s facts (+ -) 81.05% 86.84% 5.79%

2 and 1 digit (+ -) 70.98% 79.64% 8.66%

Groups of ten (+ -) 63.36% 69.53% 6.17%

Mixed (+ -) 36.04% 42.66% 6.61%

Basic facts (x ÷) 40.49% 52.26% 11.77%

Doubles and x 10 (x ÷) 65.25% 77.73% 12.48%

Extended basic facts (x ÷) 31.04% 43.90% 12.86%

Mixed harder (x ÷) 19.22% 27.49% 8.26%

Fraction sense (+ -) 34.47% 50.49% 16.02%

Fraction (x ÷) 25.05% 40.03% 14.98%

Decimals (+ -) 39.28% 54.28% 15.00%

Decimals (x ÷) 23.23% 38.29% 15.06%

Percentage 21.38% 34.94% 13.57%

Table 4: Student growth in MYMCT 2009

The data collected shows a significant improvement across all substrands tested. Of particular interest is

the growth in fraction sense, the major focus of the project. Pre-testing results for fraction sense questions

were 34.47% correct and post-testing was 50.49% correct. This is an improvement of 16.02%. Other

growth is evident in questions assessing operating with multiplication and division of fractions, an

improvement of 14.98% overall. There was an improvement of 15% for questions assessing decimal

operations. Results for the substrand percentages, whilst not a focus for all classes shows an improvement

overall of 13.57%. This suggests that students‟ conceptual understanding of fractions and decimals has

had a transfer effect on percentage operations.

2010

Again students were given pre and post testing for the MYMCT. In reading these results it is worth noting

the differences made to the project in 2010 as follows:

the adoption of only two teachers from each participating school for the intensive in-class support

was implemented (schools were still eligible to enrol more teachers in the professional learning

workshops)

the professional learning modules were re-developed to include whole number addition and

subtraction

teachers were given electronic and paper based resources at the completion of each professional

learning module

the format for delivery was adjusted to allow more time in between each workshop so teachers were

not overwhelmed by the course content

in-class support was increased to weekly and a greater emphasis on a gradual release of

responsibility model (Pearson and Gallagher, 1983) was used by project officers


16

a more explicit focus on teacher experience reporting was adopted by project officers and project

teachers

a timetabled collaborative planning session was scheduled for each project teacher and project

officer; the Scaffolding Mental Computation planning document was used to engage project

teachers in a framework for planning.

Chart 9: MYMCT pre and post testing 2010

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

10

fac

ts (

+,-)

2 a

nd

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igit

(+,

-)

gro

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Bas

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(x,

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Do

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ben

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ark

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of

10

%

Post-test

Pre-test


17

Mental Computation Skill 2009 pre-

project

2010 pre-

project

2009

post-

project

2010

post-

project

2009

growth

2010

growth

10s facts (+ -) 81.05% 83.73% 86.84% 91.54% 5.79% 7.81%

2 and 1 digit (+ -) 70.98% 73.61% 79.64% 86.25% 8.66% 12.64%

Groups of ten (+ -) 63.36% 67.18% 69.53% 78.96% 6.17% 11.78%

Mixed (+ -) 36.04% 39.06% 42.66% 50.63% 6.61% 11.58%

Basic facts (x ÷) 40.49% 41.26% 52.26% 61.11% 11.77% 19.84%

Doubles and x 10 (x ÷) 65.25% 68.22% 77.73% 83.45% 12.48% 15.23%

Extended basic facts (x ÷) 31.04% 31.96% 43.90% 52.40% 12.86% 20.44%

Mixed harder (x ÷) 19.22% 18.16% 27.49% 38.09% 8.26% 19.93%

Fraction sense (+ -) 34.47% 32.56% 50.49% 61.15% 16.02% 28.59%

Fraction (x ÷) 25.05% 26.86% 40.03% 50.69% 14.98% 23.83%

Decimals (+ -) 39.28% 41.96% 54.28% 67.25% 15.00% 25.29%

Decimals (x ÷) 23.23% 24.44% 38.29% 44.53% 15.06% 20.09%

Percentage 21.38% 23.46% 34.94% 41.64% 13.57% 18.18%

Table 5: Student growth in MYMCT 2009 and 2010

The increased improvement can be attributed to the adaptations to the program and the consistency of

project officers (in 2009 one project officer resigned after term 2). The data obtained through the

assessment process is useful to guide teaching as well as being beneficial reflectively to evaluate the

student‟s learning based on their initial understanding. The spreadsheet for collecting data has been used

by project teachers for analysing student results as well as students‟ identifying personal learning goals and

teachers communicating with parents on student progress. Future directions are to develop an assessment

schedule that aligns with the excel spreadsheet for data entry and can be used at multiple year levels. The

eventual aim is to have a DVD assessment for year 1/2, 3/4, 5/6, 7/8 and 9/10 that uses the same reporting

process. Each test will incorporate a number of linking items that will be used to track students

longitudinally. This will support schools to have a common approach to assessment and reporting for

mental computation and enable assessment for learning in a wider range of year levels. In particular the

DVD assessment will ensure students who require further assessment such as Schedule for Early Number

Assessment (SENA) 1 or 2 (CMIT) will be identified early and intervention or differentiated learning

experiences can be applied.

In 2010 a stronger emphasis was placed on project teachers undertaking experience reporting. Topics

were chosen based on their class assessments, which ensured there was strategic vision for developing

pedagogy and building capacity for the teacher. Some examples of the experience reporting undertaken by

project teachers are provided in appendix 4.

Student achievement NAPLAN

Using NAPLAN data as a comparative measure of student achievement has been difficult due to the ACT

system‟s practice of keeping these results confidential. Level of access to the data is in the control of


18

school principals. Due to the sensitivity about these results it was difficult to access these records for

research purposes.

Year 6 students in 2009 were deemed to be the most suitable cohort for NAPLAN measures. The student

results for year 5 numeracy in 2008 and their year 7 2010 growth was analysed.

There were complications in finding a comparison group of students in order to compare and analyse

results. The intention was to use a project class in a school and compare them to a non-project class in the

same school. However one of the objectives of the project was to build the capacity of the project teacher

and have them share collaboratively as facilitators within their school. Many schools selected all teachers to

participate in the project in 2009, which left only a small number of schools to choose from. North Canberra

schools had an interruption in project officer support due to a project officer resigning at the end of term 2.

Some schools streamed classes so that the comparison class was clearly different in structure and culture

of learning. Taking into account all of the difficulties in obtaining the data and finding a comparison group of

students it was decided to use one school as a case study.

School 30963

School 30963 is situated in south Canberra. It has a diverse population of approximately 275 students with

approximately 7 percent Indigenous and 16 percent with English as an additional language or dialect

(EAL/D). It is an identified National Partnership school and in 2010 an additional school leader was placed

in the school by the Department to support and build the capacity of Literacy and Numeracy teaching and

learning. In 2009 two teachers were nominated to participate as project teachers, a teacher of year 6 and a

teacher of year 5. The year 6 class was used in this case study (MYMC class); there was a second year 6

class in the school that was used as the comparison class (non-MYMC class).

The 2008 and 2010 student mean scores, growth and student residual are shown for the two classes in the

table below. The student residual is the actual growth minus the student‟s expected growth, (which varies

according to the student‟s 2008 score).

Class 2008 mean score

(year 5 numeracy)

2010 mean score

(year 7 numeracy) Mean growth

Mean student

residual n

MYMC

class 482.59 554 71.41 0.4 17

Non-MYMC

class 464.94 526.71 61.76 -11.4 17

ACT 485.2 557.9 69.5 unknown 1559

Table 6: NAPLAN comparison for school 30963

Care must be taken in reading the results, n=17 is not a statistically sound sample. Other factors should

also be considered such as the individual students, teachers, parent and leadership relationships; the

dynamics of the different classes; the impact of transitions on the students moving from year 6 into year 7,

(5 different high schools were attended in 2010); and the students‟ relative literacy levels.

Looking at the data in quartiles for each group of students some inferences can be made. The non-MYMC

class were similar in ability, there was however an outlier in the data for the MYMC class in 2008 and two in

2010; these were all high achieving outliers and have been removed from the graphs. The additional outlier

in 2010 indicates the learning opportunities in the MYMC class were such that higher ability students were

extended.


19

Chart 10: NAPLAN comparison 2008 to 2010

The median has shifted more in the MYMC class (77 mean points compared to 57 in the non-MYMC class);

so has the inter-quartile range. This indicates the learning in the MYMC class catered for the 50% of

students in the inter-quartile range. The lowest score in the 2010 MYMC class indicates a need to provide

continued support for these students; however if you put names to the data, the lowest student in the

MYMC class in 2008 actually moved above the lower quartile in 2010. There was more movement of

individual students in the 25th percentile in the MYMC class than that of the non-MYMC class, which means

in the non-MYMC class if the students‟ started in the bottom 25 percent they tended to stay there.

Using a sample group of 17 students makes the interpretation of these data unreliable and the results

should not be generalised to the wider ACT school population. However the small sample results are

encouraging for further research to be undertaken. In planning for 2011 and beyond further work will be

undertaken using NAPLAN data in a longitudinal study.

0

100

200

300

400

500

600

700

800

non-MYMC 2008 non-MYMC 2010 MYMC 2008 MYMC 2010


20


Teachers

The intended teacher outcomes for the MYMC project were to increase:

confidence; to teach mental computation and mathematics in general

motivation to teach mental computation explicitly

capacity and pedagogical content knowledge for teaching mental computation.

Again, because the project was delivered as two separate programs the results have been displayed

according to the year of involvement.

Outcome 1: Teacher confidence

2009

The pre and post survey (appendix 5) teachers‟ responses to the confidence they have in teaching mental

computation and mathematics are shown in the graphs below:

Chart 11: 2009 Teacher confidence – mental computation

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Not Confident Highly Confident

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Confidence Level

Confidence in Teaching Mental Computation

Pre Survey

Post survey


21

Chart 12: 2009 Teacher confidence – mathematics

Results show growth in teacher confidence to teach mental computation, with the two highest confidence

levels increasing from 29 percent to 77 percent. There was also growth in teacher confidence to teach

mathematics more generally (54 percent to 70 percent). This indicates that increasing teachers‟

pedagogical strategies for mental computation also has an effect on confidence to teach mathematics.

2010

The pre and post survey results for teaching mental computation and mathematics generally are displayed

in the following charts.

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Confidence in Teaching Mathematics

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Post survey


22

Chart 13: 2010 Teacher confidence – mental computation

Chart 14: 2010 Teacher confidence – mathematics generally

The impact of the project is clearly shown with a shift in confidence for teaching mental computation and

mathematics generally. Teachers‟ confidence to teach mental computation increased from 12 percent to 76

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Confidence in Teaching Mental Computation

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23

percent and to teach mathematics generally from 62 percent to 90 percent. There are mechanisms for

providing continued support into 2011 for project teachers. This includes the cLc, email contact, a

continued relationship by the project officer with the school and the regular professional learning support

provided by the literacy and numeracy section.

Outcome 2: Teacher motivation

2009 and 2010

Selected responses teachers made about their perception of the importance of mental computation and

mathematical understanding follow.

Teacher 1 “After seeing the improved results of my students I am definitely going to change my teaching

practice to further implement mental computations in my teaching practice.”

Teacher 2 “As far as the future of mental computations in the school it is my/our intent that it should

permeate the whole school with all staff becoming accomplished users of the strategies and implementing

them in all classes.”

Teacher 3 “I will definitely use resources and strategies used from the MYMC project next year and I look

forward to sharing my knowledge with other members of the school. It has been great how my team have

all embraced the project and how our enthusiasm has influenced others within the school and they are

becoming more involved in mental computations.”

Teacher 4 “My attitude towards teaching fractions changed as well. In the past I had seen fractions as dry

and boring, and did not look forward to teaching fraction concepts. Seeing how engaged the students were,

and how much they improved really changed the way I think about it. I was motivated to continue, and so

were the students.”

Teacher 5 “Mental computation is valuable because it is the type of maths that we mostly use in our day to

day lives. I think that a lot of students that I work with need maths that they will and can use in their daily

lives, now and in the future.”

Teacher 6 “Mental computation is a fundamental part of a maths curriculum; it is the base on which

students build their understanding of all maths concepts.”

Teacher 7 “My belief is that this is invaluable! The positive outcomes of this as a maths program support

(sic) has been instrumental in my class making bold positive advancements in their skills.”

Teacher 8 “Mental computation plays a crucial role in a child‟s mathematical development and in enhancing

their understanding of more complex concepts.”

Teacher 9 “It (mental computation) forms the basis of what the children do with number. It arms them with

explicit strategies to use when working with number. It frees up their working memory.”

Teacher 10 “MYMC complements any maths program by helping students develop an ever-growing

understanding and language they can use to become more independent, creative and confident in maths

(all areas).”


24

Outcome 3: Teacher capacity

2009

Analysis of teachers‟ knowledge of mental computation and number sense strategies

Strategy/Tool Number of times mentioned Percentage of response

PRE POST PRE POST

Concrete materials 37 38 72.5 86.4

Pictorial representations 13 22 25.5 50

Mental images 2 20 3.9 45.5

Interactive whiteboards 16 9 31.4 20.5

No strategies mentioned 21 5 80.4 11.4

Class/peer discussion 31 38 60.8 86.4

Specific strategies identified 5 35 9.8 79.5

Quick mentals 11 0 21.6 0

Maths journals 5 14 9.8 31.8

Table 7: 2009 Teacher references to current pedagogy and tools

During MYMCP, participants have been introduced to a wide range of appropriate representations and

resources to use for developing particular mental computation concepts and skills. The post survey

highlights this.

There was a significant increase in teachers stating the use of pictorial representations and mental images

for teaching mental computation skills (highlighted in yellow). This indicates teachers are now familiar with

and using effective strategies that were introduced during the project to develop number sense and mental

computation.

The stated use of Interactive Whiteboards (IWBs) decreased, (see green highlight). This may be due to

teachers now having a repertoire of well targeted hands-on resources for students. Another factor is

teachers becoming more self-aware of effective ways to use IWB technology for explicit teaching of number

concepts, such as multiplying by ten using number sliders.

The number of teachers explicitly identifying strategies for mental computation grew from 9.8% to 79.5%,

(highlighted in blue). This result shows significant growth in the project aim of building teachers‟

pedagogical content knowledge in mental computation.

Teachers reported the adoption of maths journals as reflection tools for students and they are being valued

as a quality part of a mathematics program. Teachers are now utilising students‟ individual explanations of

understandings and encouraging thinking and reflecting on strategies. Teachers now see a value for

explicitly focussing both on the strategies used, as well as the answer given.

Specific strategies mentioned were:

PRE SURVEY POST SURVEY

Bridging to 10

Splitting tens - partitioning

Jumping on number line

Compensation

Doubling

Halving

Conceptual x 10, ÷ 10

Reconstructing times tables (strategies approach)

Using 10s facts

Skip counting

Using 10%

Using arrays


25

PRE SURVEY POST SURVEY

Doubles

Skip counting

Split method

Jump – forward and backward

Compensation

Linking decimals to tenths as a fraction

Language focus

Visualising

Relating % to fractions

Double 10%

Bridging to nearest whole number/10/100

Near doubles

Partitioning numbers

Sharing

Number tracking

Using fraction families

Count in 10s

Part-part-whole

Open-ended questioning

Counting on/back

Estimating

Counting on/off decade

Benchmarking

Table 8: 2009 Teachers known mental computation strategies pre and post professional learning

The table above demonstrates that the project has provided teachers with a comprehensive repertoire of

strategies that effectively target mental computation. Growth in student achievement confirms the

successful implementation of the strategies by the project teachers.

2010

The following table shows the strategies and tools mentioned in the pre-project survey.

Strategies Mentioned Tools Mentioned

Tricks

Simplifying fractions

Folding paper

Applying fraction strategies to %

Estimation to nearest ten

Place value

Rounding

Add-on

Subitising

Rote Learning

Knowledge of money

Using 10%

Relationships between operations

Cents/dollars for decimals

% is a decimal without the decimal point

Number bonds

MAB

Glossaries

Games

Posters

SCOOTLE/Internet

Counters

Number lines/grids

Matchsticks

Fraction cakes

Dartboard

Calculators

100 chart

Dice

Cards

Mathletics

IWB games (none specifically mentioned though)


26


Bridging

Splitting (partitioning) numbers

Using number lines

Skip counting

Counting on/back

Doubling

Pictorial representations

Understanding the language of operations

Group 10s

Multiplicative thinking (more efficient than additive)

Part/Whole

Divide bottom number into top number

Go back to food/money for motivation

Convert between fractions/decimals/percentages

Equivalent fractions

Trading

Near doubles

Partitioning and combining

Equal groups

100 is magic number (unclear what this

means)

Factors/multiples

Key words

Visual manipulation

Understanding of base 10

What to do with the decimal dot when

performing + - x ÷

Numerical relationships

Arrays

Ben Dunbar resources

Modulo art

Blocks

Number frames

Cuisenaire rods

Number expanders

Fraction wall

Empty number line

Place value mats

Own materials from project based student learning

Table 9: Strategies and tools mentioned pre survey 2010

Initial survey responses indicate that the teachers in 2010 are starting from a different foundation to 2009

teachers. Whilst not all strategies mentioned are ideal (those that are not are highlighted in bold), the

number of strategies being named and the resources listed indicate that there is growing awareness in the

ACT system for teaching mental computation strategies. This is also evident in the attendance of 50

teachers in a supplementary course offered in 2010. The growing number of ACT teachers participating in

the MYMC professional learning has the benefit of schools using common language and implementing a

strategies approach to number sense and mental computation.


27

In the post-project survey the following strategies and tools were mentioned:


Bridging to 10, 100, 1000


Counting on and off the decade

Visualising equivalent fractions

Fraction representations for + and –

Visualising using a decimal number line

Magnifying number line

Skip counting for decimal x and ÷

Doubling and halving

Number movement

Linking decimals and percentage to fractions

Binary to 10, 20, 50 and 100

Compensation

Strategies approach for multiplication

Array method for multiplication

Sharing for division

Skip counting

Visualisation of fractions

Benchmarking fractions

Converting % and decimals to fractions

Place value understanding

Using 10%

Jump method for + and –

Part-part-whole knowledge

Add-on/jump back/compensation for subtraction

Fraction families

Complements for fraction addition

Making 100%

Fractions are part of a whole

Recognising there is no 1 way to work a problem

Groups of ten

Decimal operation strategies are the same as

whole number

Finding 10%

Fraction calculators

Fraction wall

Pattern blocks (attribute blocks)

Number line

Empty number lines

Fraction rulers

Empty rectangle

Sharing mats

Decimal rulers

Percentage rulers

Number cards

Decimats

MAB

100 charts

Grid paper

NLVM.com

Balance scales

Dice

Consolidation activities (hex board, connect four,

bingo, circuits, complements boards, noughts and

crosses, serve, another way, spy hunter, boxes,

snakes and ladders)

Thinkboards

Dominoes

Mathletics

Targeting maths

Concrete materials

Drawing and visualisation

Ten frames

MYMC IWB resources

Unifix blocks

Discussion cards

Multiple representations sheets


28


Finding % by applying strategies in fractions and

decimals

Subitising

Relationship of fractions to ratio – for young

students this needs to be visual/tactile

Table 10: Strategies and tools mentioned post survey 2010

The quality of the responses has increased from pre to post project. The teacher responses were more

focused on effective use of teaching tools to complement sound pedagogy. Teachers were also asked to

provide additional comments at the end of the survey; all of the responses from 2010 are given in appendix

6.

At the completion of each calendar year project teachers were asked how effective they found various

aspects of the project. The results were used to inform directions for the project. The results show that

despite teachers starting from an increased awareness of mental computation strategies in 2010 the project

was still highly effective.

Chart 15: Teachers response to effectiveness of professional learning workshops

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29

Chart 16: Teachers response to effectiveness of the length of the project

Chart 17: Teachers response to effectiveness of in-class support

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Chart 18: Teachers response to effectiveness of mentoring by project officer

Chart 19: Teachers response to effectiveness of planning time

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31

Chart 20: Teachers response to effectiveness of resources supplied

Chart 21: Teachers response to effectiveness of collegial support

The graphs show an overall improvement in satisfaction for the program and demonstrate the effectiveness

of the adaptations in 2010.

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School Leaders

The MYMCP intended outcomes for school leadership was to ensure an explicit focus on a whole school

approach to mental computation was adopted. The measures of this were how many schools requested

further support and what type of support was requested.

The measures were collected through conversations and observations from project officers and the

increased professional learning requests into the literacy and numeracy section of the Department for

MYMC.

12 out of 14 schools in 2010 have requested or initiated further consolidation of the MYMCP in varying

ways:

workshops requested for all other staff

whole school workshops for mental computation and Interactive Whiteboard (IWB) resources

additional support with planning for regular mathematics lessons

scoping and sequencing mental computation across the school

school leaders attending MYMC professional learning modules and shared strategies at staff meetings

school leaders observing mental computation lessons and then coaching year 3/4 teachers

school leaders and numeracy coordinators attending MYMC extension workshops

planning for 2011 support in team meetings and using the same model of gradual release of

responsibility with 3 additional teachers

the whole maths faculty attending additional mental computation workshops

release time given to staff to create resources for use across the school

additional mental computation workshops delivered in 2010 and a middle school workshop delivered in

2011 to consolidate.

It is also worth noting that the MYMCP has become an endorsed numeracy program (along with NSW DET

CMIT) for numeracy by the Department. This means that the stature of the program has been affecting in-

school leadership as well as system leadership. It also ensures sustainability and continual improvement,

with primary and secondary numeracy executive officers employed within the literacy and numeracy section

to drive the project further.

The strategic direction for the literacy and numeracy section is to develop facilitators courses in various

modules; assessment and planning; whole number operations; fractions; decimals; and percentage and

ratio. This will be offered to selected teachers within the Department and they will specialise in one module,

(all facilitators will be required to complete the assessment and planning module). This approach will

enable teachers to deeply understand the research and teaching implications for one area and not burden

them with having to know too much, it is anticipated this will ensure key information is passed on effectively

and the capacity of the system to support explicit teaching of mental computation strategies will continue to

grow and improve.


33


Parent and Community

The parent and community involvement was measured by the number of schools publishing information in

the school newsletter and the requests for project officers to share at school open nights.

8 out of 14 schools had articles in the school newsletter with one particular school giving an academic

achievement award for mental computation lessons.

Further examples of parent and community involvement were:

parent information evening that project officers attended – parents communicated an interest and value

of mental computation – the Principal stated that the teachers participating commented it is the best

professional learning they have ever had

student learning journeys involved mental computation strategies and parents were able to take

resources home

parent numeracy evening to inform parents of mental computation strategies and ideas for home.


34

Section 3 – Delivery Summary The administrative and management efforts required to deliver the pilot were considered to be aspects of

the numeracy executive officer roles already employed by ACT DET. The MYMC program has been able to

evolve and develop into a professional learning program widely valued by students, teachers and leaders.

The major cost of the project was employee expenses, however the way in which the money was applied

the Australian Government funding provided one school leader position and ACT DET employed another.

This left each officer free to contribute to other initiatives across the system and ensured a collaborative

approach to the project was utilised. Other connections within the School Improvement Division were able

to be made, particularly in the development of IWB resources with a joint project with the learning

technologies section of the Department in 2009 providing a structure for MYMC project officers to develop

resources to align with the program. The resources are now being mapped to the Australian Curriculum for

mathematics and will be available on a connected learning community for teachers to access in 2011.

The implementation and ongoing management of similar pilots would be improved if some aspects were

completed prior to the commencement of the project; an example is the evidence framework. This was

finalised after the project had begun and some of the requirements were difficult to deliver because the

original intentions of the project did not align with the priorities set and some data was collected for the

sake of collecting data rather than for strategic purposes. The parent and community involvement is an

example of this.

The strength of the MYMC pilot project is the flexibility to adapt to the needs of the school requesting

support. In 2011 15 schools have submitted requests and are being supported by numeracy executive

officers. The type and frequency is negotiated between the literacy and numeracy section and school

leadership, it is a requirement that the professional learning links with the school priorities set out in the

annual operational plan. 11 out of the 15 schools have adopted a whole school approach to mental

computation and professional learning has incorporated tools to be used in the earlier years of schooling

with feedback from P-2 teachers being that they were surprised and delighted the mental computation

resources applied to their classroom practice as well. School leadership have been able to foster the

development of common language across the school and there is a greater awareness that conceptual

understanding in number is essential for developing fluency in mental computation; this begins much earlier

than the middle years the project originally targeted.

Provide details on any variations that occurred to the Pilot from the delivery arrangements as

specified in your Funding Agreement at Schedule 1 Item C, including any changes to the timeline.

No variations were made to the delivery arrangement.


35

Please provide a detailed itemised Income and Expenditure statement against your Budget as

specified at Attachment B of your Funding Agreement.

Budget Actual Variance

2008-09 2009-10 2010-11 Total

Income

Receipts from

DEEWR 286,920 143,460 57,384 57,384 258,228

10% co-investment 28,692 28,692

Total Income 315,612 286,920 -28,692

Variance Comment DEEWR will provide remaining balance after final report

Expenditure

Salaries and wages

(includes

superannuation)

37,942 103,097 54,858 195,897

Administration (7% by

financial services) 10,042 4,017 14,059

CRS (from 41320) 41,280 41,280

CRS (from 44065) 38,000 38,000

Professional

Consultation 2,900 2,250 5,150

Supplies and Services 4,123 9,143 1,179 14,445

Total Expenses 308,831 -6,781

Variance Comment There will be further funds spent on developing the facilitator courses once the

final funds from DEEWR are received.

Table 11: Project income and expenditure


36

Section 4 – Sustainability As previously mentioned the MYMCP will continue to develop and improve in the ACT. This includes the

content as well as the strategic implementation. Through this process avenues will be explored to publish

parts of the program using NEALS to ensure the most effective sharing across the ACT and other

jurisdictions. A priority for the ACT will be to ensure the schools identified in National Partnerships are

included in this development process.

A key component of the program is the professional learning modules. They develop a shared

understanding and commitment to researching the way students learn number concepts. They ensure the

conversations about planning for teaching and learning between the project officer and teacher are focused

and built upon assessment for learning. The development of facilitator courses will enable others to deliver

the MYMCP and once established in the ACT effectively the Literacy and Numeracy Section will investigate

options for distributing the program wider than the ACT jurisdiction.


37

Section 5 – Other Information

Key contacts:

Katie King

Numeracy Executive Officer

ACT Department of Education and Training

[email protected]

Greg Taylor

Numeracy Executive Officer

ACT Department of Education and Training

[email protected]

mailto:[email protected]:[email protected]


38

Appendix 1

2010 Student survey

Name:

School:

Class:

1. What is your favourite subject in school?

2. Circle the place on the scale that best describes how often you:

a. Enjoy maths lessons

b. Find maths lessons interesting

c. Try in maths lessons because you feel it is important

3. Circle how you feel today…

NEVER RARELY SOMETIMES MOSTLY ALWAYS




39

4. Circle the place on the scale that best describes how much you agree with the following

statements:

a. I feel like I always get the wrong answers in Maths

b. I keep trying to find the answer to a maths problem even when it is very hard for me

c. Mathematics time makes me nervous

5. Please finish these sentences...

a. When I am in maths class I feel...

b. My favourite part of maths class is...

c. What I hate about maths class is...

STRONGLY DISAGREE

DISAGREE NEUTRAL AGREE STRONGLY

AGREE

STRONGLY DISAGREE

DISAGREE NEUTRAL AGREE STRONGLY

AGREE

STRONGLY DISAGREE

DISAGREE NEUTRAL AGREE STRONGLY AGREE


40

Appendix 2

Test One - Archimedes

Category Questions Timing 10s facts 4 + 7

9 + 5 10 – 8 8 – 3 17 – 7 11 – 5 6 + 6

1 second each – verbal only, no visual displayed

2 and 1 digit 6 + 13 36 – 5 21 + 4 58 – 3 3 +48 57 + 9 42 – 6 31 - 4

3 seconds each – verbal and visual display of question

Groups of ten 50 + 70 140 – 60 60 – 13 30 + 22 76 + 40 54 – 20 65 – 35 15 + 25 43 – 12 33 + 15


Mixed 92 – 34 27 + 25 105 – 26 264 – 99 111 – 67


Basic Facts (multiplication/division)

6 x 9 21 ÷ 3 20 ÷ 4 8 x 3 72 ÷ 9 7 x 8 24 ÷ 6

1 second each – verbal only, no visual displayed

Double and x 10 2 x 40 Halve 18 60 x 2 Halve 46 17 x 2 10 x 19 28 x 10


Extended Basic Facts 30 x 5 80 ÷ 4 200 ÷ 5 7 x 200 13 x 20 40 x 70


Fraction Sense ¼ + ½ 10 seconds each – verbal and


41

½ + 5 tenths 1 ¼ - ½ 7/8 + ½ 1 – 1 fifth 3 – ½ ¾ - ½ 6 sevenths – 2 sevenths 4 fifths + 2 fifths

visual display of question

Further fractions 1/3 of 12 ¾ of 20 1/10 of 40 5 x one third Double ¾ 2 ÷ ½


Decimal + - 0.5 + 0.5 0.25 + 0.25 1 – 0.2 4.5 – 3 2 – 0.6 0.7 + 1.8 0.19 + 0.01


Decimal x ÷ 3 x 0.2 5 x 0.3 0.1 x 40 3 ÷ 0.5 2 ÷ 0.1 0.1 ÷ 0.1


Benchmark percentages and

multiples of 10%

100% of 35 50% of 28 25% of 80 20% of 30 10% of 40 33 1/3% of 12 70% of 80 40% of 40 10% of $5.50


Appendix 3

Scaffolding Mental Computation Planning tool linking Count Me In Too and MYMC assessment spreadsheet

Link to useful Count Me In Too learning objects http://www.curriculumsupport.education.nsw.gov.au/countmein/children.html

Link to Scaffolding Numeracy in the Middle Years http://www.education.vic.gov.au/studentlearning/teachingresources/maths/snmy/default.ht

m

Link to Every Chance to Learn http://activated.act.edu.au/ectl/framework.htm

http://www.curriculumsupport.education.nsw.gov.au/countmein/children.htmlhttp://www.education.vic.gov.au/studentlearning/teachingresources/maths/snmy/default.htmhttp://www.education.vic.gov.au/studentlearning/teachingresources/maths/snmy/default.htmhttp://activated.act.edu.au/ectl/framework.htm

http://www.curriculumsupport.education.nsw.gov.au/countmein/learning_framework_in_number.html

Mixed



Compensation strategy

Building addition and

subtraction through

grouping (facile strategies)

CMIT Learning

framework: Number

Building addition and

subtraction through

counting by ones

CMIT Learning

framework: Number

Place Value Level 2 in CMIT

Counting by 10s and 100s Level 2

Combining and Partitioning

Level 2

Place Value Level 1

Facile Strategies

Combining and Partitioning Level 1 and 2

Place Value Level

2 and 3

Groups of ten

Counting by 10s

Extending 10s facts

Bridging to 100

Combining and partitioning

2 and 1 digit

Count on and back

Bridging to multiples of 10

Tens Facts

Counting on and back by 1, 2, 3

Using known tens facts

Doubles/Near doubles

Add/subtract 10

Bridge to 10


44


Mixed

Distributive property

Division by chunking

Building multiplication and

division through equal

grouping and counting

CMIT Learning

framework: Number

Focus is on moving students from

skip counting (level 4 CMIT) to using

a strategies approach for

application to beyond basic facts

Extended Basic Facts

Extending use of strategies

Skip counting where

appropriate

Basic facts

x5, halve x10

x3, double plus one lot

x9, x10 subtract one lot

x6, x5 plus one lot or double x3

x7, x5 plus double

Double, x10

Partitioning to double

x2, x4, x8 link

Conceptual understanding of

x10

Multiplication and Division

Level 5


45


Fraction division

Division understanding

Division of a fraction by a

fraction

Building fractions through

equal sharing

CMIT Learning

framework: Number

Fraction multiplication

Fractions of quantities

Whole number multiplied by

fractions

Skip counting Fraction sense

Adding and subtracting –

common denominators

Adding and subtracting –

readily identifiable

denominators Fraction sense

Fraction representations –

developing part of whole

Equivalence

Symbolic representations

Benchmarks

Fraction operations:

Simple addition and subtraction of halves and quarter separately, e.g. half + half (Early

Childhood (K-2))

Half and quarter together e.g. a half plus a quarter is three quarters (Later Childhood (3-5))

Fractions with readily identifiable common denominators (Early Adolescence (6-8))

Initial fraction understandings:

half and quarter (Early Childhood (K-2)) (CMIT scope)

denominators to tenths (Later Childhood (3-5))

any denomination (Early Adolescence (6-8))


46


Multiplication and division

Skip counting (whole number

multiplied by a decimal)


Division of a whole number by

a decimal

Building place value through

grouping

CMIT Learning

framework: Number

Level 4: Decimal place value in CMIT develops the positional

value of decimals, however, operations with decimals is

beyond CMIT

Performing operations with decimals begins in later

childhood band of development (addition and subtraction to

hundredths)

Addition and subtraction

Extend to larger numbers

Bridging to nearest whole

number

Ordering decimals (to

hundredths and beyond)

Addition and subtraction

Developing tenths as a fraction

Symbolic representation

Addition and subtraction of

tenths


47

Percentages

Benchmarks

Percent-Fraction-Decimal

equivalence

Developing 10%

Percentages

Using 10%

Developing 1%

% increase and decrease

Greater than 100%

Percentage is beyond CMIT and is incorporated into Every

Chance to Learn from Early Adolescence

Initial benchmarks (year 6) are 100% and 50% following

this include 25% and 10%.

Other benchmarks needed are 75%, 333

1% and 20%.

Using 10% to find 5% and 1% and all other percentages

should come after the development of benchmarks

48

Mental Computation – Sequence of Strategies

Addition and Subtraction Focus (From assessment

schedule)

Strategies Modules (McIntosh

resource)

Activities

Tens Facts

Counting On and Back by 1, 2,

3

10s facts

Doubles

Add / Subtract 10

Bridge to 10 (single digit +

single digit)

Near Doubles

2.4, 2.12

2.5, 2.11

2.6, 2.10

2.7,2.13

2.5, 2.11,

2.8, 2.10

10 frames - subitising

Make 10 snap

Dice doubles

Complements boards

Number cards (0-11)

Card concentration/memory (combinations to 10)

Spy Hunter

Empty Number Lines

Mine or Yours games – include tracking

Dominoes

Dice games

Flash cards

Card Bingo – Subitising

Place value charts

100 chart games

Number of the day

Target Number

Turn around Dominos Facts

CMIT Activities

DENS 1 – Diffy Towers p.118, Apple Turnovers p.

121

Figurative – DENS 1 p.161-187

DENS 2 p.20-60

49

Focus (From assessment

schedule)


resource)

Activities

2 and 1 digit

(+ -)

Count on and Back

Bridging to Multiples of 10 +

Partitioning Numbers

4.1

4.7

Number line tracking

Mine vs yours

Up and down the river

Bundling – straws, paddlepop sticks

Spy Hunter

Empty Number Lines

Mine or Yours games – include tracking

Dominoes

Dice games

Flash cards

Card Bingo – Subitising

Place value charts

100 chart games

Ten Frames

Number of the day

Target Number

Ten Frame Race

Dice Doubles

Race to 0

Addition Wheel

CMIT

DENS 1 p. 149- 156

DENS p.161-187

DENS 233-267

DENS 2 –p.20-60

50


schedule)


resource)

Activities

Groups of

ten

Counting by 10s

Extending 10s facts

4.2, 4.3, 4.4

4.5, 4.6, 4.8

Complements boards

Hundreds charts

Place value charts

100 chart games

Number Cards

2 Different Ways

CMIT

DENS 1 p. 149- 156

DENS 233-267

Bucket Count

DENS 2 p.62-90

p.180-196

Mixed


Partitioning

Compensation Strategy

4.2, 4.3, 4.4,

4.5, 4.6, 4.8

Complements boards

Hundreds charts

Place value charts

100 chart games

Number cards

DENS 2 p.62-90

p.180-196

p.284-292

51


Multiplication and Division Focus (From assessment

schedule)


resource)

Activities

Basic Facts

2x – Double

3x Adding one lot,

e.g 3 x6 double 6 and add 6.

1 x and 0 x

5 x - half of 10 x

Double, Double (x4)

6 x , 5 x + 1 lot

9 x - 10 x take 1 lot

Double, Double, Double (8x)

7 x - 5 x + 2 x

*Commutatively, Developing -

Skip Counting

(Embedded across strategies)

3.1, 3.8

3.2, 3.9

3.3, 3.4, 3.10, 3.11

3.6, 3.13

3.7, 3.14

Multiplication Toss

1-12 cards (0-11)

Dice games e.g. area dice

Sharing Mats/Grouping Mats

Multiple Madness (100 grid)

Empty Rectangle

Reconstructing times tables

Array book (finger painting, square counters,

dot page etc)

Turn around facts

Stories

Chatterboxes

Target numbers

Clumps (groups of)

Fluency practise

- Beat the clock

- Mathletics

Around the world/Buzz

Eliminator

Think boards

Times circuits

Trisaw/squaresaw

Product Pairs

Today‟s Target

Connection Charts

Hex

Serve

CMIT

DENS 1 p.189-203

52


schedule)


resource)

Activities

DENS 1 p.269-277

DENS 2 p.92-102

DENS 2 p.198-206

DENS 2 p.274-283

Double and x

10

Doubling Strategies-

partitioning, known facts

10 times – Conceptual

Understanding

3.5, 3.13 Cherry Strategy (partitioning) Number Sliders

Dice Doubles

Sharing Mats / Grouping Mats

Arrays

Spy Hunter

Empty Rectangle

Reconstructing times tables

Turn around facts

Think boards

Trisaw/squaresaw

Connection Charts

Dice Games

Connect 4

Bingo

Hex

Noughts and Crosses

53


schedule)


resource)

Activities

Extended

Basic Facts

Extending 1 digit facts – 2

digit by single digit

Skip counting (where

appropriate)

4.10, 4.11

Spy Hunter

Connect 4

Bingo

Hex

Noughts and Crosses

Empty rectangle

Three Throw

Dice Games


Mixed Facts

Distributive property x and ÷

Division 2 and 3 digit divided

by 1 digit

Division by chunking

(partitioning numbers)

4.12

4.13, 4.14


Rectangular arrays

Noughts and Crosses

Hex

Division Dash

Division Box Game

54


Fractions Focus (From assessment

schedule)


resource)

Activities

Fraction

Sense

Fraction representations –

developing part of a whole

Equivalence

Symbolic Representation

Benchmarks

Adding and Subtracting –

common denominators

Adding and Subtracting –

different denominators

5.1

5.2

Fraction calculators

Paper Folding

Fraction Rulers

Fraction Wall game

Benchmarking

Number lines

From Here to There

Pattern Blocks

Everything about my fraction

Fraction Snap

Fractions, Pikelets and Lamingtons

Hex Addition Games

Bingo

Noughts and Crosses

Complements Boards

Equivalence Boards

Fractions

Multiplication

and Division

Fractions of Quantities

Whole number multiplied by

fractions

Skip counting

Division Understanding

Division of fraction by

fraction

5.3, 5.4

5.5

5.6

Grouping and sharing mats

Rectangular arrays

Pattern Blocks

Paper Folding

Number lines

Adapting recipes

Bingo

Noughts and Crosses

Spy Hunter

55


Decimals Focus (From assessment

schedule)


resource)

Activities

Addition and

Subtraction

Developing tenths as a

fraction

Symbolic representation

Addition and subtraction of

tenths

Extend to larger numbers

e.g. 3.7 + 1.6

Bridging to 1s

Developing hundredths

Ordering decimals

5.7 Decimal rulers Complements boards (make 1, Make 5 , Make 10)

Think boards

Hex

Decimal calculators

Name a Decimal

Decimal discussion and ordering cards

Target number

Decimal jigsaw

Determining the decimal

Closest to 10

Decimal difference

Target Practice

Number expanders

Number line tracking

Decimat

Spy Hunter

Nought and Crosses

Connect Four

Multiplication

and Division

Skip counting e.g. 4 lots of

0.5


5.8

5.9

Hex

Mm square grids

Number lines

Noughts and Crosses

Spy Hunter

Connect Four

56


Percentages Focus (From assessment

schedule)


resource)

Activities

Percentages

Benchmarks 50%, 100%,

25%, 33 1/3 %, 75%, 20%

Percent – Fraction – Decimal

Equivalence

Developing 10%

Using 10%

Developing 1%



greater than 100%

6.5, 6.7

6.6

6.8

6.9

6.10

6.11, 6.13

6.12

Complements Boards

What Percent Are…?

Percentage / fraction / decimal snap

Percentage Match

Graphic Conversions

Percentage Grids

Benchmarking

Percentage Play

Money for Context

Discounts

Percentage Box Game

Hex

Noughts and Crosses

57

References

NSW Department of Education and Training, 2001, Count Me In Too – Learning Framework in number, Curriculum Support Directorate

McIntosh, A, Dole, S, 2005, Mental Computation: A Strategies Approach, Department of Education, Tasmania

Siemon, D, 2006, Scaffolding Numeracy in the Middle Years, RMIT University, Victoria


Appendix 4

Experience Report - Fractions

School and Class Information

My class has 23 students in years 4 and 5. There are 7 year 4 students and 16 year 5 students. The

school has 10 mainstream classes and two Autism classes. The area is mainly middle class with a few

lower socio-economic families attending the school.

The first MYMC testing identified the strengths of the Year 4 students. They consistently achieved

higher results than the Year 5 students throughout all testing. The Year 5 students, on average,

were performing at a low level. A number of teaching points were easily identified.

Focus

The focus for the experience report was fractions. My aim was to build an increased understanding

of fractions and to build the students‟ ability and confidence. I focussed on fraction sense and the

addition and subtraction of fractions. Fractions of collections was another area of teaching and

learning that was addressed.

I chose fractions as the focus because in the first test of the program I was able to identify large

gaps of missed learning. Fractions can be a difficult topic to teach and learn. My aim was to teach

fractions with a variety of tools to ensure deep understanding that can be transferred to other

situations.

Activities

The first activity to introduce fractions was to identify a half. The students discussed their

Documents

Literacy and Numeracy · 2011. 4. 12. · Literacy and Numeracy Pilots Final Report 7 Chart 2 Student confidence post-project 2010 Note that the lowest level of confidence was halved