International Journal of Scientific and Research Publications, Volume 10, Issue 6, June 2020 110

ISSN 2250-3153

This publication is licensed under Creative Commons Attribution CC BY.

http://dx.doi.org/10.29322/IJSRP.10.06.2020.p10215 www.ijsrp.org

The use of Cuisenaire Rods on Learners’ Performance in

Fractions in Grade 9 in Public High Schools in Chris

Hani West District, South Africa.

George Adom, Prof E.O. Adu

Faculty of Education, University of Fort Hare, South Africa.

DOI: 10.29322/IJSRP.10.06.2020.p10215

http://dx.doi.org/10.29322/IJSRP.10.06.2020.p10215

Abstract- This study was aimed at examining the use of Cuisenaire

rods on grade 9 learners’ performance in fractions. Pre-test, Post-

test, and Control group quasi-experimental design was used for the

study. The study group was made up of 250 grade 9 learners’. One

hundred and twenty-five (125) learners were selected into the

experiment group whiles One hundred and twenty-five (125)

learners’ were selected into the control group through systematic

simple random method. The data collected were analysed using

Analysis of Covariance (ANCOVA) to find the Mean, Standard

Deviation and Sample T-test. The mean and standard deviation

were used to compare the pre-test and post-test between the

Experimental group and Control group. The analysed results of the

means, standard deviations and T-tests were used to reject the null

hypotheses. The analysed results of Cuisenaire rods showed that

the pre-test (mean = 8.372, SD=1.770) and post-test (mean =

12,428, SD=4.732), t=13,024 p< 0.05. The hypotheses were tested

at 0.05 level of significance.

Index Terms- Cuisenaire rods, Educators, Fractions, Grade 9,

Learners’.

I. INTRODUCTION

overnments and stakeholders all over the world continue to

devise means and strategies to make the learning of fractions

in mathematics to learners very easy and practical to learners’ in

this modern world (Govan Mbeki Mathematical Development

Unit (GMMDU), 2013). The American Mathematics Society

(AMS) in collaboration with other mathematical organizations

promoted mathematics, science, and research through funding to

create awareness of mathematics education and to project the

mathematics profession (AMS, 2019). Eurydice (2011) asserted

that most European countries have reviewed their mathematics

syllabi, embracing an outcome-based method which was aimed at

developing learners’ competencies and skills rather than on

theoretical approach. This integral approached was focused on an

all-inclusive and flexible in meeting the needs of different levels

of learners’ as well as to their ability to comprehend the tenacity

of mathematics applications in their daily lives.

An assessment conducted by researchers on U.S. Grade 8

learners’ on fraction addition, the results showed that out of the

closest whole number to 12 13

⁄ + 78

⁄ (The answer choices were

1; 2; 19; 21 and “I don’t know”) revealed that only 27% got the

correct answer to be (2) (Lortie-Forgues, Tian, & Siegler, 2015).

In a similar vein, the National Assessment of Educational Progress

(NAEP), conducted a test to a sample of U.S grade 8 learners. At

the end of the test, it was observed that only 50% of the

participants could correctly order 27

⁄, 59

⁄, and 112

⁄ from the

smallest to the largest (Martin, Strutchens, & Ellintt, 2007).

Siegler and Pyke (2013), observed that in addition and

subtraction of fraction problem, unequal denominator problems

elicited many more errors than equal denomitor ones. Learners’

in grade 6 correctly answered 41% of the problem and 57% of

grade 8 learners’ also correctly answered the problem. Siegler and

Pyke (2013), observed that 6th and 8th graders representing 68%

correctly answered decimal arithmetic problems. Performance

was high on addition and subtraction representing 90% and 93%

respectively but performance of learners’ was lower in terms of

multiplication and division representing 54% and 35%

respectively.

Fraction was introduced to Korean learners’ in grade three,

Japan in grade four to the elementary level whiles in Taiwan,

fraction was introduced to learners’ at grade three with much

emphasised on composition and decomposition of fractions (Son,

2011, Charalambous & Pitta-Pantazi, 2010, Watanabe, 2012). In

addition, East Asian countries used an amalgamation of carefully

selected mathematical materials that have prolonged existence in

terms of their application in demonstrating fractions and also

replicated the idea of fraction as a quantity. The focus was on the

linear model in connection with the bar model which was mostly

used in Japanese fractions instruction as well as Korean and

Taiwanese textbooks. This approached adopted by the East Asian

countries was in variance to the North American approached who

were preoccupied with the ‘pizza model’ or other circular area

models (Son, 2011, Charalambous & Pitta-Pantazi, 2010,

Watanabe, 2012).

Study conducted by (ICMI, 2009) showed that

Mathematics development was very low in Africa due to the few

numbers of secondary school educators teaching mathematics and

also the few number of graduates and post graduates pursuing

mathematics as a course at the masters and PhD levels and above

all the absence of innovations in most of our schools.

Van der Walt et al (2008) and Ndlovu (2011) were of the

view that the abysmal performance of learners in mathematics in

South African could be attributed to lack of inadequate learner

support materials, poor socio-economic background of learners,

G

International Journal of Scientific and Research Publications, Volume 10, Issue 6, June 2020 111

ISSN 2250-3153

This publication is licensed under Creative Commons Attribution CC BY.

http://dx.doi.org/10.29322/IJSRP.10.06.2020.p10215 www.ijsrp.org

medium of instructions, lacked of motivation, poor quality of

educators and inadequate study orientation.

Researchers asserted that, an overwhelming majority of

South Africa learners’ Mathematical knowledge was uncertain.

South African learners’ encountered serious problems related to

Mathematics methodological terminology (Van der Walt, Maree

and Ellis 2008:490). Trends in International Mathematics and

Science Study (TIMSS, 2015) conducted a test in Mathematics

and Science to evaluate the performance of learners in grade Eight

among some selected Africa countries (Tunisia, Egypt, Ghana,

South Africa, Morocco and Botswana). At the end of the

competition, South Africa recorded the lowest mark in Science

and Mathematics (TIMSS, 2015). South Africa learners’ were

rated 30.0% in numeracy and 48.1% in literacy (UNICEF, 2005).

Siegler and Fazio (2010) observed that learners all over the world

were faced with challenges in learning fractions and it was evident

that an average learner never gained an abstract knowledge of

fractions. Many learners and educators were faced with the

challenge of fraction learning in mathematics due to the complex

nature of it. Research showed that learners were confronted with

the tasked of understanding the concept of the subject especially

fractions (Charalambous & Pitta-Pantazi, 2010).

II. RESEARCH OBJECTIVES

The present study aimed to find out the use of Cuisenaire

Rods on Learners’ Performance in Fraction in Grade 9 in Public

High Schools in Chris Hani West District South Africa.

III. HYPOTHESES

The following null hypotheses were tested at 0.05 level of

significance:

H01: There is no significant difference between the Pre-test

and Post-test of the control and experimental group.

H02: There is no significant relationship between

Cuisenaire rods and grade nine learners’ performance in fraction.

IV. RESEARCH QUESTION

What was the effect of Cuisenaire rods on learners’ academic

performance in fraction in grade 9?

V. LITERATURE

The literature of this study was based on what scholars had

written about the use of Cuisenaire rods in the teaching of fractions

both locally and internationally.

5.1 Theoretical framework

The theoretical framework of this study was anchored on

two theories; Cognitive Development theory and Constructivism

theory. Cognitive development theory was the ability to make

intellectual judgement through the process of involving all the

mental faculties (De Witt, 2011) whiles constructivism keenly

involved the collaboration efforts of learners with the educator in

constructing new meaning (Atherthon, 2010).

5.2 Conceptualisation of fraction

Olanoff, Lo and Tobias (2014), opined that fraction was an

aspect of rational numbers which was expressed in the form 𝑎

𝑏

where “a” and “b” were both numerals, and “b” ≠ 0. Fractions was

an aspect of study of rational numbers. In a similar vein Lortie-

Forgues et. al. (2015 p.206) asserted that a fraction was made up

of three components, a numerator, a denominator, and a line

separating the two numbers eg. 12

⁄ . Studies showed that for one

to advance in the understanding of the concept of rational numbers

in general, one must undertake a study of different interpretations

of fractions (Lamon, 2007, 2012). Ball (cited in Olanoff et al.

2014 p. 272) asserted that fractions may be inferred to as; (a) in

part-whole terms, where the whole unit may vary; (b) as a number

on the number line; (c) as an operator (or scalar) that could shrunk

or stretched another quantity; (d) as a quotient of two integers; (e)

as a rate; and (f) as a ratio.

Fractions was an essential aspect of mathematics that

formed the bedrock of every learner’s success in the subject as

stipulated by the National Mathematics Advisory Panel (NMAP,

2008). Lortie-Forgues, Tian and Siegler (2015) argued that, the

prominence of fraction and decimal calculation for academic

accomplishment was not restricted to mathematics courses only.

Rational number arithmetic was also ever-present in physics,

chemistry, engineering, psychology, sociology, biology,

economics, and other spheres of studies. Gould, Outhred, and

Mitchelmore (2006), asserted that educators, learners and

academics have typically described fraction learning as a difficult

aspect of mathematics syllabus. Researchers underscored the fact

that learners found it problematic to comprehend the idea of “a

part as a whole” relationship in mathematics.

5.3 Application of manipulative concrete materials in

classroom teaching.

In 2013, the National Council of Supervisors of

Mathematics (NCSM) issued a position statement on the use of

manipulative concrete materials in classroom teaching to develop

learners’ accomplishment in mathematics. “In order to develop

every student’s mathematically proficiency, NCSM recommended

that learners and educators must systematically integrate the use

of concrete and virtual manipulative into classroom instruction at

all grade levels” (NCSM, 2013).

Understanding mathematical skills was very important in

today’s technological world (Burns & Hamm, 2011; Carbonneau,

Marley, & Selig, 2013). Johann Pestalozzi (1746 –1827)

influenced educators in the 19th century to use manipulative

concrete materials in teaching number sense at the basic level of

education including basic blocks (Saetter, 1990). Piaget’s

constructivism viewpoint of the 1970s, observed that theoretical

knowledge was established through discovering while using

physical materials rather than through auditory information via

person to person (Piaget, 1973). In this Morden world, there were

a variety of manipulative concrete materials stretching from

virtual computer software programmes to teacher-made materials

(Gaetano, 2014 p.5).

A manipulative concrete material was a physical object that

could be touched, felt, moved around by learners, appealed to the

faculties of the senses and also conveyed a mathematical

International Journal of Scientific and Research Publications, Volume 10, Issue 6, June 2020 112

ISSN 2250-3153

This publication is licensed under Creative Commons Attribution CC BY.

http://dx.doi.org/10.29322/IJSRP.10.06.2020.p10215 www.ijsrp.org

knowledge (Swan & Marshall, 2011). In addition, Cramer and

Henry (2013) asserted that manipulative materials were physical

objects which ranged in size, shape, and colour. They

encompassed physical prototypes such as fraction circles, paper

folding, pie pieces, Cuisenaire rods, fraction bars, dice, and chips

that enabled learners to establish cognitive images of fractions.

However, many scholars were of the view that the use of

manipulative concrete materials does not necessarily warrant the

understanding of mathematical ideas. Researchers were of the

notion that virtual manipulative concrete materials in reality do not

help learners in cultivating mathematical comprehension (Moyer-

Packenham & Westenskow, 2013).

5.4 Cuisenaire rods

Elia, Gagatsis, and Demetrico (2007), asserted that

Cuisenaire rods were hands-on and minds-on physical material

used for mathematical instruction of abstract concepts. It was a

significant mathematical material used for modelling

mathematical concepts of what was taught in the mathematics

classroom and what pertained at homes relating to classroom

experience to everyday life activities. Cuisenaire Rods were

invented over the past nine decades by George Cuisenaire a

Belgian mathematics educator. This distinctive mathematical tool

was to help learners understand abstract mathematical concepts by

manipulating painted wooden strips of different dimensions called

Cuisenaire rods. A package of Cuisenaire rods consisted of 74

rectangular rods in 10cm different dimensions and 10 varied

colours. Each colour related to a particular length. The content of

the pack was made up of 22 white rods of 1cm each, 12 red rods

of 2cm each 10 light green rods of 3cm each, 6 purple rods of 4cm

each, 4 yellow rods of 5cm each, 4 dark green rods of 6cm each, 4

black rods of 7cm each, 4 brown rods of 8cm each, 4 blue rods of

9cm each and 4 orange rods of 10cm each. These rods were used

as physical objects to teach any concept in mathematics

(Kurumeh, 2010).

Figure 5.1: Using Cuisenaire rods to do addition of fractions. (Kurumeh, 2010).

This harmonious interconnection of the senses acted as a

tool to enhance learning and memory. Cuisenaire rods enabled

learners to discover mathematical problems on their own

(Akarcay, 2012). Using Cuisenaire rod teaching approach

prepared learners to meet daily standards with daily critical

thinking activities. It prepared learners for school success in

mathematics and meets the needs of every age group. Cuisenaire

rods enabled every learner to work individually and in groups on

important mathematical contents such as fractions while the

educator offered individual assistance to learners (Akarcay, 2012;

Van de Walle, 2007).

VI. METHODOLOGY

6.1 Research Paradigm

The researcher adopted a positivist research paradigm for

this study. Positivism was often associated with quantitative

research method. Collins (2010), was of the view that positivism

hang on quantifiable interpretations that led themselves to

statistical analysis. The researcher used treatments on

experimental and control groups in the classroom as stipulated by

the positivism theory of laboratory experiment of study.

6.2 Research Design

For this study, the researcher adopted a Pre-test, Post-test,

and Control group quasi-experimental design to determine the

effectiveness of Cuisenaire rods on learners’ academic

performance in fraction in grade nine (9).

6.3 Sample and Sampling Techniques

Two hundred and fifty (250) grade 9 learners whose ages

ranged between 13-16 years and ten (10) educators teaching grade

9 mathematics were selected from 40 public high schools with the

use of stratified, systematic random sampling, convenience and

purposive sampling methods. One hundred and twenty-five (125)

learners were put into the experiment group and another One

International Journal of Scientific and Research Publications, Volume 10, Issue 6, June 2020 113

ISSN 2250-3153

This publication is licensed under Creative Commons Attribution CC BY.

http://dx.doi.org/10.29322/IJSRP.10.06.2020.p10215 www.ijsrp.org

hundred and twenty-five (125) learners’ were put into control

group through systematic random sampling method. One hundred

and two (102) learners were boys and One hundred and forty-eight

(148) learners were girls.

6.4 Data collection procedure

A Fraction Achievement Test (FAT) made up of multiple-

choice objective test of twenty (20) items were used. Each item

had one correct option (key) and three distractors, i.e. options A,

B, C, and D. The content area covered were; Proper fractions,

Improper fractions, Mixed fractions. A Pre-test was administered

to both the experimental group and the control group according to

their codes. The experimental group was then exposed to

Cuisenaire rods in solving different types of fractions with the help

of the researcher and the research assistant whiles the Control

group was not. On the third week, a Post-test was administered to

both experimental and control group. This was done according to

the codes assigned to them in the pre-test.

VII. DATA ANALYSIS

The data collected were analysed using Analysis of

Covariance (ANCOVA) to find the Mean, Standard Deviation and

Sample T-test. The mean and standard deviation were used to

compare the pre-test and post-test between the Experimental group

and Control group. The analysed results of the means, standard

deviations and T-tests were used to reject the null hypotheses. The

hypotheses were tested at 0.05 level of significance.

i). H01: There is no significant difference between the Pre-test and

Post-test of the control and experimental group.

Table 1.1 Summary result of Cuisenaire Rods Manipulative Tool Data Set.

Group

N

Trial

Mean

Standard

deviation

Min

Max

Controls

125

Pre-test

8.192

1.735

0

14

125

Post-test

7.992

1.406

1

11

Experiment

125

Pre-test

8.552

1.794

2

11

125

Post-test

16.864

1.820

12

20

Source: (Field study February, 2019).

Table 1.1, illustrated the descriptive statistics for

Cuisenaire Rods Manipulative Tool. The table showed the mean

scores and standard deviations of the experimental group and the

control group in the Pre-test (mean =8.552, SD=1.794) and

(mean =8.192, SD=1.735) respectively. The pre-test scores

showed that, there was no significant difference in the mean scores

and standard deviation between the experimental group and the

control group in the Pre-test. This could suggest that the initial

competencies of the two groups in fractions were equivalent prior

to the study. However, the mean scores and standard deviation of

the experimental group and control group in the Post-test were as

followed (mean =16.864, SD= 1.820) and (mean = 7.99,

SD=1.406) respectively. There was vast disparities in the mean

scores and standard deviation in the Post-test between the

Experimental Group and Control Group. (p < 0.05) therefor

hypothesis (H01) is rejected. The difference in the mean and

standard deviation scores could be attributed to the effects of

Cuisenaire Rods on the Experimental Group.

ii). H02: There is no significant relationship between Cuisenaire

rods and grade nine learners’ academic performance in fraction

Table 1.2: Shows the Paired Samples Statistics (N= 250)

Pair

Mean

Std. Deviation

Std. Error Mean

Cuisenaire rods manipulative

Post-test

12.428

4.732

.299

Pre-test

8.372

1.770

.112

Source: (Field work February, 2019).

International Journal of Scientific and Research Publications, Volume 10, Issue 6, June 2020 114

ISSN 2250-3153

This publication is licensed under Creative Commons Attribution CC BY.

http://dx.doi.org/10.29322/IJSRP.10.06.2020.p10215 www.ijsrp.org

Table 1.2: Shows a Paired Samples Test from

Pair

Paired Differences

t

df

Sig.

(2-

tailed)

Mean

Std.

Deviation

Std. Error

Mean

95% Confidence Interval of the

Difference

Lower

Upper

Cuisenaire rods

manipulative Post-test -

Pre-test

4.056

4.924

.311

3.443

4.669

13.024

249

0.000

Source:(Field work February, 2019).

Mean

Std.

Deviation

Effect size

4.056

4.924

0.82

Effect size

To prove that there was a significant relationship between

Cuisenaire rods and grade nine learners’ performance in fractions,

the pre-test and post-test mean and standard deviations scores of

Cuisenaire rods were compared using sample paired t-test. Table

19 showed the Pre-test and Post-test mean and standard deviation

of Pre-test scores (mean =8.372, SD=1.770) and Post-test

scores (mean =12,428, SD=4.732) respectively. The scores

indicated that, there were increased in the mean scores and

standard deviation of the Post-test. The t-test (t=13,024, p < 0.05)

indicated that there was a significant relationship between

Cuisenaire rods and grade nine learners performance in fractions

therefore the null hypothesis (H02) was rejected.

VIII. CONCLUSION

The study showed that Cuisenaire rods have a great effect

on learners’ academic performance in fraction. Cuisenaire rods as

a manipulative concrete material has aided learners’ to discover

solutions to mathematical problems involving fractions, motivate

learners’ to work independently and in groups and also enhanced

learners’ understanding of fractions. Elia, Gagatsis, and Demetrico

(2007), suggested that Cuisenaire rods are hands-on and minds-on

physical material used for mathematical instruction of abstract

concepts and made mathematics real to learners. In support

(Akarcay, 2012) alluded to the fact that Cuisenaire rods enabled

learners to discover mathematical problems on their own.

Cuisenaire rods motivate every learner to work individually and in

groups on important mathematical contents such as fractions while

the educator offered individual assistance to learners (Akarcay,

2012; Van de Walle, 2007). Cuisenaire rods made mathematics

real to learners since it was learner friendly, activity oriented, and

stimulated learners’ comprehension of the mathematical concepts

and facilitated higher understanding of mathematical concepts,

facts and principles Kurumeh (2010).

IX. RECOMMENDATIONS

1. Learners’ ought to use Cuisenaire rods frequently in their

mathematical lessons so that they would be abreast with

it and also increased their understanding in fractions.

2. Mathematics educators should ensure that they

incorporated Cuisenaire rods in their mathematical

instructions.

3. Educators ought to motivate and sustained the interest of

their learners’ during mathematics instruction by the use

of different teaching methods. Mathematics class must be

learner centred and not teacher centred form of

instruction.

4. Principals ought to ensure that there were adequate

manipulative concrete materials in their schools to

enhance the teaching and learning of mathematics.

5. It was obligatory for the Department of Education to

ensure that every school was well resourced with

manipulative concrete materials to enhance the

mathematical proficiency of the learners’.

REFERENCES

[1] Akarcay, S. (2012). "Cuisenaire Rods: Pedagogical and Relational

Instruments for Language Learning". Vermount, USA: Published Thesis of

the SIT Graduate Institute. Retrieved from https://digital

collections.site.edu/ipp-collection/521

[2] AMS. (2019). USA Science and Engineering festival 2016. Rhode Island

02904-2213. 201 Charles Street Providence: Unpblished

[3] Atherton, J. S. (2010). Learning and Teaching; Constructive in learning.

Retrieved from

http://www.learningandteaching.info/learning/constructivism.htm

[4] Bernard, H. R. (2011). Research methods in Anthropology: Qualitative and

quantitative approaches. UK: Altamira Press.

[5] Burns, B. A., & Hamm, E. M. (2011). A comparison of concrete and virtual

manipulative use in third- and fourth-grade mathematics. School Science And

Mathematics, 111(6), 256-261. Retrieved from Retrieved from:

http://ehis.ebscohost.com.dbsearch.fredonia.edu:2048/ehost/p

[6] Carbonneau, K. M., Marley, S., & Selig, J. (2013). A meta-analysis of the

efficacy of teaching mathematics with concrete manipulatives. Journal of

Educational Psychology, 105(2), 380-400.

[7] Charalambos, C., & Pitta-Pantazi, D. (2010). Drawing on a Theoretical

Model to Study Students’ Understanding of fractions . Educational Studies

in Mathematics, 64(3), 293-316.

International Journal of Scientific and Research Publications, Volume 10, Issue 6, June 2020 115

ISSN 2250-3153

This publication is licensed under Creative Commons Attribution CC BY.

http://dx.doi.org/10.29322/IJSRP.10.06.2020.p10215 www.ijsrp.org

[8] Collins, H. P. (2010). Creative Research: The Theory and Practice of

Research for the Creative Industries. AVA publishing. doi:2940411085,

9782940411085

[9] Cramer, K., & Henry, A. (2013). Using manipulative models to build number

sense for addition of fractions. Yearbook (National Council of Teachers of

Mathematics), 75, 365 - 371 .

[10] Creswell, J. W. (2014). Research Design: Qualitative, Quantitative and

Mixed Methods Approaches (4th ed.). Los Angelus: SAGE.

[11] De Witt, M. W. (2011). The young child in context: A thematic approach.

Perspectives from Educational Psychology and sociopedagogics (1st ed.).

Pretoria: Van Schaik.

[12] Druckman, D. (2005). Doing Research: Methods of Inquiry for Conflict

Analysis. Fairfax, USA: SAGE.

[13] Elia, L., Gagatsis, A., & Demetrico, A. (2007). The effects of different modes

of Representation on the solution of one-step additive problems.

[14] Eurydice. (2011). Mathematics Education in Europe: Common Challenges

and National Policies. Wim Varisteenkiste, Cummunication and Publication.

European Commission. Retrieved from

http://eacea.ec.europa.eu/education/eurydice/thematic-studies-en.php

[15] Gaetano, J. (2014). The effectiveness of using manipulatives to teach

fractions. A Thesis submitted to the department of Psychology College of

Science and Mathematics at Rowan University.

[16] Gould, P., Outhred, L. N., & Mitchelmore, M. C. (2006). One-third is three-

quarters of one-half. In R. Z. In P. Grootenboer, Identities, Cultures and

learning spaces (Proceeding of the 29th annual confernece of the

Mathematics Education Research Group of Australiasia (pp. 262 - 269 ).

Adelaide: MERGA.

[17] Govan Mbeki Mathematics Development Unit. (2015). Teaching the future

of mathematics and science education. Nelson Mandela Metropolitan

University. Retrieved from http://www.mbeki-maths-dev.nmmu.ac.za

[18] (ICMI), (2009). A commission of The International Mathematical Union

(IMU). (Adopted by the IMU Executive Committee by an electronic vote on

December 31, 2009).

[19] Kurumeh, M. S. (2010). Effect of Cuisenaire Rods' Approach on Students'

Interest in decimal fractions in Junior Secondary School Makurdi Metropolis.

Global Journal of Educational Research , 9(9), 25-31.

[20] Lamon, S. J. (2012). Teaching fractions and ratios for understanding

Essential knowledge and instructional strategies for teachers. New York:

Routledge/Taylor & Francis Group.

[21] Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward

a theoretical framework for research. In F.K. Lester, Jr. (Ed.), Second

handbook of research on mathematics teaching and learning. New York:

Routledge/Taylor & Francis Group.

[22] Lee, C., & Chen, M. (2010). Taiwanese Junior High School Students'

mathematics attitudes and perceptions towards virtual manipulatives. British

Journal of Educational Technology, 41(2), 17 - 21 . doi:10.1111/j.1467-

8535.2008.00877

[23] Lira , J., & Ezeife, A. N. (2008). Strengthening Intermediate-Level

Mathematics Teaching Using Manipulatives: A Theory-Backed Discourse. .

Ontario: Academic Exchange – EXTRA.

[24] Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction

and decimal arithmetic so difficult? Developmental Review, 38, 201-221.

doi:10.1016/j.dr.2015.07.008

[25] Martin, W. G., Strutchens, M. E., & Ellintt, P. C. (2007). The learning of

mathematics. Reston VA: National Council of Teachers of Mathematics.

[26] Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of virtual

manipulatives on student achievement and mathematics. (n.d.). International

Journal of Virtual and Personal Learning Environments, 4(3), 35–50.

doi:10.4018/jvple.2013070103

[27] Moyer - Packenham , P., & Suh, J. (2012). Learning mathematics with

technology: The influence of virtual manipulatives on different achievement

groups. Journal of computers in Mathematics and Science Teaching, 31(1),

39 - 59 .

[28] National Centre on Intensive Intervention (NCII). (2016). NCII User Guide.

National Centre on Intensive Intervention at American Institutes for

Research. Washington, DC: American Institutes for Research.

[29] Ndlovu, M. C. (2011). University-school partnerships for social justice in

mathematics and science education: The case of the SMILES project at

IMSTUS . South African Journal of Education, 31(1), 419-433.

[30] OECD. (2015). PISA 2012 results: Creative problem solving: Students’ skills

in tackling real-life problems . NY. Retrieved from http://www.oecd-

ilibrary.org/education/pisa-2012-results-skills-for-life-volume-v-

9789264208070-enq

[31] Olanoff, D., Lo, J. J., & Tobias, J. (2014). Mathematical Content Knowledge

for Teaching Elementary Mathematics: A Focus on Fractions;. The

Mathematics Enthusiast, 2(5). Retrieved from

http://scholarworks.unit.edu/tme/vol11/iss2/5

[32] Rahi, S. (2017). Research Design and Methods: A systematic Review of

Research Paradigms, Sampling Issues and Instruments Development. Int. J.

Econ Manag Sci, 6(403). doi:doi:10.4172/2162-6359.1000403

[33] Ross, C. J. (2008). The Effect of Mathematical Manipulative Materials on

Third Grade Students’ Participation, Engagement, and Academic

Performance. Doctoral Dissertation. . Florida: University of Central Florida.

[34] Saettler, L. P. (1990). The evolution of American educational technology.

Englewood Colo: Libraries Unlimited.

[35] Siegler, R. S., & Fazio, L. (2010). Developing effective fractions instruction:

A practice guide. Washington, DC: National Center for Education Evaluation

and Regional Assistance, Institute of Education Sciences, U.S. Department

of Education. doi:NCEE #2010 - 009

[36] Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual

differences in understanding fractions. Developmental Psychology, 49, 1994-

2004.

[37] Son, J. W. (2011). A global look at math instruction. Teaching Children

Mathematics, 17(6), Pg. 360 - 368.

[38] Special Connections . (2009, April 9). Special Connections From concrete to

representational to abstract. Retrieved April 9, 2009 . Retrieved from the

Special Connections:

http://www.specialconnections.ku.edu/cgibin/cgiwrap/specconn/main.php?c

at=instruction&subsection=math/cra.

[39] Swan, P., & Marshall, L. (2011). Revisiting mathematics manipulative

materials. Australian Primary Mathematics Classroom, 15(2), 13-19.

Retrieved from

http://ehis.ebscohost.com.dbsearch.fredonia.edu:2048/ehost/pdfviewer/pdfv

iewer?sid=7ee176b5-9b1c-47e2-a8d

[40] TIMSS. (2015). TIMSS 2015 International Results in Mathematics. Boston:

IEA TIMSS & PIRLS International Study Center, Boston College.

[41] Trespalacios, J. H. (2008). The Effects of Two Generative Activities on

Learner Comprehension of Part-Whole Meaning of Rational Numbers Using

Virtual Manipulatives. Blacksburg: Virginia: Virginia Polytechnic Institute

and State University.

[42] UNESCO/UNICEF. (2005). Monitoring Learning Achievement Project .

Retrieved from web log post:

http://www.literacyonline.org/explorer/un_back.htmlWestern Cape

Department of Education 2005

[43] Uribe Florez, L. J., & Wilkins, J. L. (2010). Elementary school teachers'

manipulative use. School Science and Mathematics, 110(7), 363-371.

[44] Van der Walt, M., Maree, K., & Ellis, S. (2008). A mathematics vocabulary

questionnaire for use in the intermediate phase. South African Journal of

Education, 28(1), 489-504

[45] Van de Walle, J. A., Karp, K. S., & Bay-Williams , J. M. (2008). Elementary

and Middle School Mathematics: Teaching developmentally. Boston, MA:

Allyn and Bacon.

[46] Watanabe, T. (2012). Thinking about Learning and Teaching sequences for

the Addition and Subtraction of Fractions. In C. Bruce (chair), Think Tank

on the Addition and Subtraction of Fractions. Ontario: Think Tank

Conducted in Barrie.

[47] Wijaya, S. (2017). Multi-level tensions in transport policy and planning: bus-

rapid transit (BRT) in Indonesia. Thesis to Massey university. New Zealand:

Unpublished.

AUTHORS

First Author – Adom George, University of Fort Hare, East

London Campus, South Africa, Email:

georgeadom90@gmail.com

International Journal of Scientific and Research Publications, Volume 10, Issue 6, June 2020 116

ISSN 2250-3153

This publication is licensed under Creative Commons Attribution CC BY.

http://dx.doi.org/10.29322/IJSRP.10.06.2020.p10215 www.ijsrp.org

Second Author – Second Author – Prof. E.O Adu, University of

Fort Hare, East London Campus, South Africa. Email: eadu@ufh.ac.za