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Master of Education

Essay 2

A comparison of Nigerian and English mathematics textbooks: The case of fractions

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Table of Contents

List of Tables .................................................................................................................................. 3

List of Figures ................................................................................................................................. 3

Introduction ..................................................................................................................................... 4

Literature Review ............................................................................................................................ 6

Understanding in mathematics .................................................................................................... 6

Understanding of fractions .......................................................................................................... 7

Sub-constructs of fractions .......................................................................................................... 9

Textbook analysis ...................................................................................................................... 12

Background ................................................................................................................................... 14

Nigeria ....................................................................................................................................... 14

England ...................................................................................................................................... 16

Methodology ................................................................................................................................. 20

Horizontal framework ............................................................................................................... 20

Vertical framework ................................................................................................................... 21

Comparison ................................................................................................................................... 24

Emphasis ................................................................................................................................... 24

Sub-constructs ........................................................................................................................... 29

Expectation ................................................................................................................................ 33

Conclusion .................................................................................................................................... 39

References ..................................................................................................................................... 41

Appendix ....................................................................................................................................... 48

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Keywords: Textbook, sub-constructs, instrumental understanding, relational understanding,

instrumentalisation, mathematical complexity, implemented curriculum, intended curriculum.

List of Tables

TABLE 1: DETAILS OF TEXTBOOKS 19

TABLE 2: FRACTION EMPHASIS 25

TABLE 3: FRACTION MODULES 26

TABLE 4 SUB-CONSTRUCTS 29

TABLE 5 TASK COMPLEXITY TABLE 34

TABLE 6: APPLICATION OF KNOWLEDGE 36

List of Figures

FIGURE 1: SUB-AREA SUB-CONSTRUCT 9 FIGURE 2: SUBSET SUBCONSTRUCT 10 FIGURE 3: MEASURE SUB-CONSTRUCT 10 FIGURE 4: RATIO SUB-CONSTRUCT 11 FIGURE 5: EXTRACT FROM THE NIGERIAN CURRICULUM 15 FIGURE 6: MULTIPLE SUB-CONSTRUCTS 21 FIGURE 7: FRACTIONS INTRODUCTION UM3 31 FIGURE 8: FRACTIONS INTRODUCTION IM1 31 FIGURE 9: CONTEXT OF WORD PROBLEMS 33 FIGURE 10: TASK COMPLEXITY CHART 34 FIGURE 11: UM TASK COMPLEX (LESSON) 35 FIGURE 12: IM TASK COMPLEXITY (LESSON) 35 FIGURE 13: APPLICATION OF KNOWLEDGE 37

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Introduction

The aim of most comparative inquiries is to reveal “taken-for-granted and hidden aspects of

teaching” (Hiebert et al., 2003, p.3), which could be unnoticed because they are inherent to the

system under investigation (Wilson, Andrew & Sourikova, 2001). Comparing different systems

promises to help researchers become more aware of their own implicit assumptions in

mathematics education (Knipping, 2003; Andrews, 2009; Kaiser, 1999a).

As a primary school pupil in Nigeria in the late 1980s, I found it challenging to understand the

method of adding and subtracting fractions. However, while teaching in England, I noticed that

students find it difficult understanding the whole fraction concept. The struggle with fractions, as

Charalambous, Delaney, Yu-Hsu and Mesa (2010) confirm, affects the development of other

mathematical ideas.

Researchers have suggested mathematics textbooks (books for instruction covering topics and

activities in a subject) have a profound influence on the learning opportunities presented to

students (Mesa, 2004) and might be responsible for some of the difficulties learners have with

fractions (Behr, Harel, Post & Lesh, 1993; Bezuk & Cramer, 1989).

My familiarity with both systems simplifies the identification of commonalities between the

systems, making the research process more straightforward (Phillips & Schweisfurth, 2012). For

instance, I can identify that in England there are likely to be differentiated textbooks, but in

Nigeria, there is one textbook per year group.

In this essay, I examined the position of theoretical standpoints on learning fractions and

textbook comparison, and then examined the background and intended curricula1 (laid down

policy document on the system’s expectation) of both countries. I qualitatively (since the

                                                                                                                         1This  will  be  referred  to  as  curriculum  or  curricula  in  this  study  

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frequency of learning opportunity presented to students influences the learning (Stigler, Fuson,

Han & Kim, 1986)) compared the textbooks, using empirical data to identify how fractions are

presented and developed in and across textbooks. This comparison should offer highlight and

explanations for the similarities and differences; diversity could be accounted for by educational

traditional and cultural differences while similarities would reflect the impact of international co-

operation (especially that of the English colonialism of mathematics education in Nigeria and

Nigerian importation of British expatriates in curriculum and examinational development) and

global trends in mathematics education. Since the countries differ significantly, this comparison

could contribute to the ongoing discussion regarding mathematics textbooks being cultural

(Haggarty & Pepin 2002). The results should enhance the development of textbooks and generate

questions for further investigation.

In this study, I do not claim that findings represent the entirety of educational systems; textbooks

are “almost certainly not the embodiment of the intended curriculum2” and are “not identical to

the implemented curriculum, as teachers make their own decisions” (Howson, 1995, p.6). It is

also important to highlight that not all printed materials in classrooms are textbooks because

materials like teacher support books, workbooks and worksheets cannot be classified as

textbooks.

                                                                                                                         2  Laid  down  policy  document  on  the  system’s  expectation  

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Literature Review

This review examines theoretical positions in four sections. In order to acquire a clearer view of

understanding fraction concepts and methods, it is important to evaluate the meaning of

understanding in mathematics as this could be subjective (Tall, 1978). Following the definition

of understanding, I will address understanding of fraction concepts and the challenges involved.

Thirdly, a closer examination of a highly discussed challenge (sub-constructs) of learning

fractions is evaluated, before reviewing theoretical positions on textbooks comparative research

to provide a basis for this comparison.

Understanding in mathematics

The difficult experience of teaching and learning mathematics well, as Cockcroft (1982) noticed,

might be grounded in the “many hierarchical characteristics” of the subject (Tanner & Jones,

2000, p.18), which require a cluster of lower skill fluency before advanced concepts are

understood (Cockcroft, 1982; Kilpatrick, Swafford & Findell, 2001). Understanding is

ubiquitous in discussions relating to mathematics education but the meaning, as Tall (1978)

noted, could be subjective. Skemp (1976) classified understanding into relational and

instrumental understanding. He explained that instrumental understanding is the knowing of

what and how to do, without considering why. Relational understanding is the knowing of why,

and depends on developed interconnected knowledge structures based on large ideas (Glaser,

1995).

Skemp’s categorisation of understanding has been a subject of critique because both types of

understanding “lie on a continuum and cannot always be separated” (Rittle-Johnson & Alibali,

1999, p.175). They argued that both forms of understanding mutually support themselves.

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Although forms of understanding are inseparable and exhibit mutual support, it has been

observed that some instructional practices promote instrumental understanding (Skemp, 1976).

This, as Skemp explained, is because: it is easier to understand in similar context; of curricular

pressures; and rewards are immediate. Focus on instrumental understanding should be to support

the development of concepts (Kilpatrick et al., 2001). For instance, better understanding of

addition of fractions depends on students’ efficiency at finding equivalent fractions. It also

supports understanding the parent concept. For instance, Gelman and Meck (1986) found that

children first learn how to count before understanding principles of counting. The critical point

in justifying instrumental understanding is the focus. This should be aimed at developing

efficiency (Kilpatrick et al., 2001) and knowledge of algorithm (Tall, 1978).

Relational understanding can be promoted by tasks that enhance learners’ abilities to see

mathematics as a dynamic and exploratory subject (Henningsen & Stein, 1997). These tasks, as

Henningsen and Stein argue, investigate patterns to understand mathematical structures and

relationships, formulate and solve mathematical problems, think and reason in flexible ways, and

justify and generalise ideas. Such activities place high cognitive demands on learners (National

Assessment Governing Board, 2008). Henningsen & Stein (1997) added that the set-up and

implement of such tasks could influence the cognitive demand of the process involved when

cognitive processing becomes more predictable and thinking more mechanical.

Understanding of fractions

Fractions have received substantial research focus because of the cognitive challenge they pose

to learners (Ball, 1993; Behr et al., 1993; Lamon, 1999). It is noted that when learners can

complete the necessary task, only few have the relational understanding (English & Halford,

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1995; Ochefu, 2013). Relational understanding of fractions is a prerequisite to other topics like

decimals, percentages and proportions (Charalambous & Pitta-Pantazi, 2007) and “many student

difficulties in algebra can be traced back to an incomplete understanding of fraction ideas” (Behr

et al., 1993, p.3). Cockcroft (1982) played down the importance of fractions due to technological

advancement; the difficulty encountered by learners in learning fractions and limited use of

fractions in real life. A historically example is the use of pounds, shillings and pence as monetary

units by England and Nigeria. Twenty pence is equivalent to a shilling, twelve shillings

equivalent to a pound. So when calculating money is easier is to use fractions because, for

instance, to add fifteen shillings to one pound and six shillings, it would be easier to convert to

fractions. In the 1970s, both countries converted to decimal monetary systems: Nigeria to Naira

and kobo (100 kobo equivalent to 1 naira), England to Pounds and pence (100 pence equivalent

to 1 pound). Fractions’ place in school mathematics is justified because of its right as a

mathematical object (Kilpatrick et al., 2001) and importance to further mathematical ideas (Behr

et al., 1993; Charalambous & Delaney, 2007).

Scholars attribute the difficulty in learning fractions to, firstly, the cognitive conflict that

fractions concept development poses with whole number concepts (Mack, 1995; Lamon, 1999),

and rules for calculating with whole numbers (Kilpatrick et al., 2001). Lamon argues that

learners are resistant to seeing fractions as numbers because they are not part of the counting

sequence. This resistance results in their not perceiving a fraction as an entity but two separate

whole numbers (Lamon, 1999). This poor conceptualisation is related to learners adding the

corresponding numerators and denominators when adding fractions (Mack, 1995).

Secondly, informal notions of fractions are not properly explored as a starting point for the

development of fraction concepts (Kilpatrick et al., 2001). Children are exposed to partitioning,

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sharing and comparing sizes actively within and outside school. This informal understanding

should be extensively explored and form the basis for the learning of fractions (Confrey, 1994;

Empson 1999; Mack 2005). Kilpatrick et al. (2001) argued that, for fractions, sharing would aid

the development of the fraction concept as counting does for whole numbers.

Sub-constructs of fractions

Studies have noted that learners are challenged by the multiple ways that a fraction can be

represented, referred to as sub-constructs of a fraction (Kieren, 1976; Dickson, Brown & Gibson,

1984). The sub-construct of fractions, in line with the classifications of Charalambous and Pitta-

Pantazi (2007), and Dickson et al. (1984), are as follows:

Sub-areas of a unit region: The fraction 3/43 could represent the shaded parts of this continuous

area.

Figure 14: Sub-area sub-construct

Dickson et al. (1984) claim that children’s earliest encounter with fractional concepts are of the

“spatial kind” (p.276). They further suggested that, within this part-whole sub-construct, a child

                                                                                                                         3  For  essay  fractions  like  ¾  will  be  given  as  3/4    4  (Benjamin,  D.  et  al.,  2000,  p.  154)  

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will more readily accept a three-dimensional representation than a two-dimensional one, as the

former appears to be easier to learn (Novillis, 1976), but such knowledge could be difficult to

relate to other sub-constructs (Dickson et al., 1984; Behr et al., 1993). Understanding that the

sub-areas are of equal sizes is critical in understanding this sub-construct.

Subset of a set of objects: This aspect is very similar to the area subset but these aspects of the

objects are discrete and, as Behr at al. highlighted, the cardinality of the discrete objects is

critical not the area covered. So the sizes of squares don’t need to be equal in Figure 2 for the

fraction of black squares to be 1/4.

Figure 2: Subset subconstruct

Measure: This sub-construct relates a fraction’s quantitative aspect and the measure assigned to

an interval (Moseley, 2005).

Figure 35: Measure Sub-construct

Despite the link between this representation and that of the sub-set of an area, understanding this

sub-construct is challenging for learners (Dickson et al., 1984). This challenge has been

                                                                                                                         5  Adaption  of  Benjamin  et  al.  (2000)  

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attributed to the abstract nature of the representation (Novillis, 1976) and learners’ focus on the

marks not the intervals (Baturo, 2004). This sub-construct has advantages as it readily relates to

improper fractions, addition of fractions (Behr et al., 1993), enables learners see natural

numbers6 as parts of fractions (Dickson et al., 1984) and fractions as numbers.

Result of a division: This is the aspect of fraction that defines a fraction as a result of a division.

For instance, 3÷5 could be expressed as 3/5.

Understanding this aspect is particularly significant, as Dickson et al. argued; “in order to change

a fraction to decimal or percentage … it is generally necessary to convert from, say, 2/7 to 2 ÷ 7”

(p.283).

Comparing two sets or Ratio: This aspect of fractions compares two different sizes or measures

(Dickson et al., 1984). It could be seen more as a comparative index (Carraher, 1996).

Exemplified in Figure 4, the relationship of black to white squares is 3/4.

Figure 4: Ratio sub-construct

Dickson and his colleagues highlighted that this sub-construct appears to be difficult to

understand and developed late. They submit that the sub-construct is an essential component in

solving many ratio problems.

Operator: This sub-construct relates a fraction to a function used to transform a number, set or

object (Behr et al., 1993). For instance 3/4 of 60 is 45. 3/4 is now seen as an operator on the

number 60.

                                                                                                                         6  Positive  whole  numbers  

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The first two sub-constructs have drawbacks in implementation when applied to improper

fractions (Novillis, 1976). Although Charalambous and Pitta-Pantazi (2007) grouped them

together, reviewing them separately, as suggested by Dickson et al., will provide a more

comprehensive view of fractions (Behr et al., 1993). Dickson et al. and Behr et al. agree that

these sub-constructs form bases for other sub-constructs. Learners learn fractions better when

they are exposed to multiple sub-constructs of fractions (Kieren, 1976; Baturo, 2004).

For instructional practice to adequately address the challenges associated with learning fractions,

it needs explore the learners’ informal knowledge, link as many as possible sub-constructs of

fractions and emphasise the understanding of the fraction concept. So the identified challenges

are mainly linked to the introduction of the fraction concept.

Textbook analysis

Mathematical textbooks have a profound influence on the implemented curriculum because they

are the ready intermediary between teaching intentions and realities (Ball & Cohen 1996); they

represent the curriculum in most classrooms (Valverde, Bianchi, Wolfe, Schmidt & Houang,

2002) and teachers rely heavily on textbooks when sourcing for teaching methods (Beaton et al.,

1996). Researchers have suggested mathematics textbooks might be responsible for some of the

difficulties learners have with fractions. For instance, Behr et al. (1993) criticised the nature of

tasks some textbooks provide for learners. Bezuk and Cramer (1989) argued that the textbooks’

rapid progression between fraction modules might be a source of misunderstanding.

There has been a growing focus on cross-systematic textbook research, perhaps due to its

comparative nature; it might expose the “textbook signatures” (Charalambous et al, 2010, p.143)

unique to a country’s textbooks but normally unnoticed (ibid). Hiebert and his colleagues argued

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that researchers better understand the strengths and weaknesses of their textbooks through cross

national comparison (Hiebert et al., 2003; Stigler & Hiebert, 2004). Textbook comparisons in

presenting opportunities of exploring across systems can, although there are critics like Freeman

and Porter (1989), offer explanations for the difference in students’ performance in international

comparative studies (Li, 2000). Its findings remain still probabilistic (Mesa, 2004) because it

depends on the teacher and/or student interaction with the textbooks and the level to which the

textbook reflects the intend curriculum.

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Background

I now present the background of the two countries whose textbooks are being compared, to

provide context for this comparison (Mason, 2007).

Nigeria

The Federal Republic of Nigeria is a country in Africa. Salman (2005) reports that fractions is

not perceived as a difficult topic by teachers in a region7 in Nigeria. In a later, paper she reports

that Nigerian primary teachers see equivalent fractions8 as the least challenging to teach among

15 topics (Salman, 2009). These findings tend to contradict the global trend about fractions

regarding it as challenging to teach (Kilpatrick et al., 2001; Cockcroft, 1982; Behr et al., 1993)

so warranting investigation.

Awofala (2012) suggested that between the 1930s and 1950s, teaching of mathematics in Nigeria

was characterised by the mastery of computational skills. During that period, arithmetic

textbooks contained topics that could be “described as arithmetic processes and nothing more”

(Badmus, 1977, p.15). Awofala opines that recent global influence, especially from England,

might have led to subsequent evolutions that promote the development of relational

understanding. The Nigerian curricular objectives emphasise the need to develop knowledge and

skills that can be applied, and be relevant for the future functioning of the learner (Awofala,

2012), and “solving mathematical problems encountered in one’s daily life becomes the

overriding concern in the curriculum” (Akinsola, 2012, p.8). This suggests that the curriculum

has a principle that focuses on the well-being of learners, and emphasis on their happiness. It

                                                                                                                         7  Kwara  central  8  The  only  review  fraction  module  

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follows that it is guided by utilitarian principles and its ideology is characterised by “social and

economic efficiency” (Mason, 2007, p.268).

The Nigerian curriculum is referred to as a “teaching curriculum” (Akinsola, 2012, p.7), because

it provides the teachers with when, what and how to teach. Figure 5 exemplifies how the

curriculum details the content and context of every lesson.

Figure 5: Extract from the Nigerian Curriculum

In 2010, the Nigerian government ordered mathematics textbooks for all school children (FMI,

2010). This suggests that textbooks are available to students in Nigeria. The series chosen for

this comparison is the Understanding Mathematics (UM) series, because: it is one of the

textbook series ordered for distribution by the Ministry of Education (FMI, 2010). The author

claims the “content of series meets the needs and goals of the 2006 UBE curriculum” (David–

Osuagwu, 2010c, p.i), and it is recommended by all Nigerian book lists available on the internet.

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From my experience, in common with other Nigerian textbooks, the book series has one book for

each year group. Despite the interest in cross systematic textbook research and the recent

availability of textbooks in Nigeria, I am yet to come across a Nigerian mathematics textbook

being compared in a published research.

The culture of having the same textbook for students in a year group (suggesting that all students

have access to the full curriculum) and the curriculum dictating what happens in the classrooms,

suggest that the culture of the Nigerian educational system could be seen as a form of

collectivism (Hofstede, 1991).

England

England is a country in the United Kingdom. Its curriculum aspires to promote the development

of pupils and prepare them for later life (Qualifications and Curriculum Authority, 2007). For

mathematics, the curriculum aims for learners to enjoy learning, make progress and achieve;

become confident individuals who are able to live safe, healthy and fulfilling lives and make a

positive contribution to society (Qualifications and Curriculum Authority, 2007).

The aspirations of the English curriculum and aims of mathematics in the curriculum call for the

development of skills for future individual functioning and societal benefit. These characteristics

suggest that the principle of the system is utilitarian and ideology is social and economic

efficiency (Mason, 2007).

The English curriculum only suggests to teachers what to teach, though there is a non-statutory

national framework that suggests how to teach (Qualifications and Curriculum Authority, 2007;

DfEE, 2001). So the teacher still decides how to teach.

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Teachers use pupils’ textbooks more than any other resources in presenting topics and exercises

in Key Stage 3 (Askew et al., 1993). If this trend remained constant, then textbook availability

and interaction would be good in England.

The Impact Maths (IM) series is chosen for this comparison because it is currently being used by

my school, and is recommended by teachers and the Cambridge International Examinations

(2011). The series follows the usual practice of differentiated textbooks per year group in

England9.

Dowling (1996) found that differentiated textbooks for the same year have differences in content,

abstraction, topics, expectations and aspiration. So to maintain balance of the difference, I chose

different ability textbooks for the three year groups10.

The variation in English textbooks, coupled with the allowance given by the curriculum in

allowing teachers to control how to teach, suggests the educational system has an individualist

view. Arguments about the system being individualist and utilitarian principled resonate with

other studies (Pepin, Haggarty & Keynes, 2001; Kaiser, 1999b).

The Nigerian and English backgrounds already highlight key similarities in the philosophy and

ideology of their educational systems. Their principles tend towards utilitarian and their

ideologies tend towards social and economic efficiency. However, there is a major disparity in

the educational tradition; while the Nigerian system tends to demonstrate a form of collectivism,

the English takes a more individualistic position. A closer look at the sequencing and order of

learning activities highlights another difference, the location of fractions in the curriculum

                                                                                                                         9  Three  textbooks  per  year  group  10  Details  in  Table  1  

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expectation. While a primary 411 student in Nigeria is expected to cover most fractional concepts

including computation with fractions (Sofolahan, 2005), this goal is not included in England

until late Key Stage 312(Tanner & Jones, 2000). This might have been motivated by Cockcroft’s

(1982) argument that computing fractions be removed from the primary curriculum because most

primary school children are still developing the cognitive ability to understand the concept and

notation involved. He argued that most fraction modules would be better understood in

secondary schools. As it would not be logical to compare textbooks of similar ages, I compared

textbooks from Nigeria covering primaries 3-613 and England years 7–914. Since the targeted age

range of the textbooks compared is different, this might affect the prior knowledge the textbooks

expect from the learners and so influence how learning opportunities are presented. Another

important disparity is that, in the English system, age influences year groups while in Nigeria,

attainment does (Omoifo, 2008). For instance, an 11-year-old student will be in year 6 but the

Nigerian promotion to primary 6 is consequent on performance during assessments in primary 5.

The table below describes the compared textbooks

                                                                                                                         11  Students’  age  is  about  9  years  old  12  Students’  age  is  about  13  years  old    13  Students’  age  range  8  –  11    14  Students’  age  range  12  –  14  15  (David-­‐Osuagwu,  2010a)  

Name Code Publishers Year

Understanding Mathematics For Nigeria

Primary 315 (Nigerian textbook) UM3 African First Publisher 2010

Understanding Mathematics For Nigeria UM4 African First Publisher 2010

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Table 1: Details of textbooks

I compared these series by answering these questions:

1. What is the emphasis placed on fractions evident in textbooks?

2. What sub-constructs are presented in textbooks’ learning opportunity?

3. What learning expectations do textbooks have of learners?

                                                                                                                         16  (David-­‐Osuagwu,  2010b)  17  (David-­‐Osuagwu,  2010c)  18  (David-­‐Osuagwu,  2009)  19  (Benjamin  et  al.,  2000)  20  (Benjamin  et  al.,  1999)  21  (Benjamin  et  al.,  2001)  

Primary 416 (Nigerian textbook)

Understanding Mathematics For Nigeria

Primary 517 (Nigerian textbook) UM5 African First Publisher 2010

Understanding Mathematics For Nigeria

Primary 618 (Nigerian textbook) UM6 African First Publisher 2009

Impact Maths 1R19 (English textbook: high

ability) IM1

Heinemann Educational

Publishers 2000

Impact Maths 2B20 (English textbook:

core/middle ability) IM2

Heinemann Educational

Publishers 1999

Impact Maths 3G 21 (English textbook: low

ability) IM3

Heinemann Educational

Publishers 2001

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Methodology

Studies have classified cross-national textbook analysis into three main categories: horizontal

(analyses the whole textbook), vertical (analyses specific areas) and contextual (Charalambous et

al., 2010). The contextual category compares interactions with textbooks, so unveils the

interaction between teachers and/or students, and textbooks (Mesa 2007) and accounts for

cultural aspects (Rezat, 2007). Establishing that both systems adequately use textbooks

motivated my comparing the textbooks using the vertical and horizontal frameworks.

Comparisons of this nature, as Charalambous et al. (2010) attests, provide a useful step towards

contextual analysis of textbooks.

Horizontal framework

The blocks of the curriculum “are chosen based on the belief that in everyday living one is often

faced with these elements in the order” (Akinsola, 2012, p.8) and textbook tends to reflect the

curriculum (Valverde et al., 2002). So the location of a topic could account for emphasis given to

it. My first criterion is the emphasis given to fraction modules in textbooks, reflecting the

importance. I found the percentage of preceding fractions22 and pages used for fraction23 (which

also reflects emphasis (Alajmi, 2012)). I then compared results from books of the same series to

provide information on shifting emphasis due to progression. Implementing this model

encountered a challenge - two of the English books had fractions and ratio in one chapter. I

therefore analysed pages on fractions. I could have considered comparing for location using the

chapter numbers but some chapters had more than one topic.

                                                                                                                         22  The  higher  the  percentage,  the  lower  the  emphasis  23  The  higher  the  percentage,  the  higher  the  emphasis  

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Vertical framework

Since relational understanding of fractions is enhanced by exposure to multiple sub-constructs of

fractions, then the second criterion for this comparison is to evaluate sub-constructs of fractions

presented to learners in examples and introductions to lessons (basic textbook units explaining a

concept). On discovering that there were situations with no or more than one sub-construct,

using a bottom-up approach I included the multiple sub–constructs category – where more than

one sub-construct is used in a context and the no sub-constructs category – where no sub-

construct is identified. This is in addition to the categories identified in previous studies. When

an opportunity is coded as multiple, coding will also be accorded to the individual sub-construct

it comprises so, individual sub-constructs in a task with multiple constructs are also accounted

for in the comparison. For example:

Figure 6: Multiple sub-constructs

This task is coded as Multiple and also sub-area and measure.

Since textbook tasks have been challenged for poor achievement in fractions (Behr et al., 1993),

the third criterion for this comparison is the expectation the textbook has of learners if they

complete all tasks in textbooks. The first framework used here is informed by the National

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Assessment Governing Board (2008) using a three-level model of mathematical complexity

(measuring the cognitive expectation of task). This framework classifies tasks according to their

level of mathematical complexity as follows: Low-complexity – which expects students to recall

or recognise concepts or procedures); Moderate-complexity – which expects students to involve

more flexibility of thinking, choice among alternatives on how to complete the task; High-

complexity (HC) – which expects students to engage in a more relational, reasoning, synthesis,

analysis, judgement, formulation and creative processes. The mathematical complexity addresses

what the students are asked to do without taking into account how the students might undertake it

(Neidorf, Binkley, Gattis & Nohara, 2006). Hence the choice of this model as this criterion

evaluates the textbook expectation rather than cognitive demand.

Since the education systems in England and Nigeria tend to have a utilitarian view and, as

Skovsmose (1994) noted, mathematics will be meaningful if children see its relevance to their

world, I compared the extent to which textbooks expect students to apply their learning in real

life situations. Following a framework used by Fan (1999), all problems are classified into two

categories – application problems and non-application problems. Their proportions were taken

and used for the comparison.

Since coding reliabilities are subject to individual and time (Mikhaylov, Laver & Benoit, 2012), I

re-rated all categories three weeks after the initial rating. Furthermore I sent scanned copies and

description of the categories to be rated by a teacher in both countries making four rating. Using

Cohen’s kappa due to its ability to evaluate inter-rater reliability with multi-variables

frameworks and not been subject to chance. The calculation was carried out using Online Kappa

Calculator (Randolph, 2008) and the reliability was sub-construct (k=92%), the application of

23  |  P a g e        

learning (k=98%) and the cognitive expectation (k=72%), all in excess of Landis and Koch’s

(1977) moderate inter-rater reliability benchmark.

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Comparison

The result of analysis and initial comparisons follows in three sections as described by the

methodology chapter.

Emphasis

The table below evaluates the emphasis on fractions.

 

Books Total pages

Pages before

fractions

Location Percentage

Fraction Pages

Pages Percentage

Impact Mathematics 1

(IM1) 336 154 45.8 12 3.6

Impact Mathematics 2

(IM2) 278 88 31.7 12 4.3

Impact Mathematics 3

(IM3) 198 132 66.6 19 9.6

Impact Maths series

average (IM) 48.1 6

Understanding

Mathematics 3 (UM3) 161 60 37.2 18 11.2

Understanding

Mathematics 4 (UM4) 200 84 42 20 10

Understanding

Mathematics 5 (UM5) 206 31 15 16 7.8

Understanding

Mathematics 6 (UM6) 236 18 8 7 3

Understanding

Mathematics series 25.5 8

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average (UM)

Table 224: Fraction Emphasis

From table 2, IM and UM columns present the mean values for the series. The location and page

percentages show that the UM series places more value on fractions than the IM series. The IM

series appears to have an increasing page proportion for fractions because the percentage of

pages addressing fractions increases from 3.6 to 9.6. For the location, in both IM1 and IM2,

fractions appeared in the second quarter of the book, while in IM3, fractions appeared in the third

quarter of the book showing a stable but declining emphasis. For the Nigerian books, the page

proportion decreases from 11.2 to 3 showing decreasing emphasis. But regarding location, in

UM3 and UM4, fractions appeared in the second quarter of the books but progressively moved

towards the front part of the books (UM5 – 15%, UM6 – 8%) showing increasing emphasis. A

contradiction could easily be spotted with the Nigerian series (since location percentage suggests

progressive emphasis and page percentage suggests retrogressive emphasis) so comparing based

on this data could be could over-simplify the situation. Therefore, I considered the topics

progress (an identified textbook issue (Bezuk & Cramer, 1989)) which could influence the page

allocation. Below is the table of topics addressed in each of textbook.

Textbook Modules covered

Impact Maths 1R

Using numbers to represent fractions; Mixed numbers and

improper fractions; Finding a fraction of a quantity; Finding more

than one part; Equivalent fractions; Adding and subtracting

fractions

Impact Maths 2B Finding fractions of a quantity; Equivalent fractions; Putting

Fractions in order of size; Type of fractions; Adding and

                                                                                                                         24  Values  on  all  tables  are  rounded  to  one  decimal  place  

26  |  P a g e        

subtraction fractions

Impact Maths 3G

Mixed numbers and improper fractions; Equivalent fractions;

Putting Fractions in orders of size; Adding and subtracting

fractions; Multiplying and dividing fractions;

Understanding Mathematics 3

Meaning of a fraction; Numerator and denominator; halves,

quarters and eighths; Thirds, sixths and twelfths of a whole

object; Fifths and tenths of a whole object; Equivalent fractions;

Ordering of fractions; Whole numbers and fractions; Addition

and subtraction of like fractions

Understanding Mathematics 4

Proper and improper fractions; Mixed numbers; Equivalent

fractions; Fractions in their lowest terms; Ordering fractions;

Division of a whole number by another to get a fractions;

Addition, Subtraction and Multiplication of fractions; Fractions of

a group of objects; Word problems; Quantitative reasoning

Understanding Mathematics 5

Addition, Subtraction, multiplication of fractions; Division of

fractions; Division of mixed numbers by other mixed numbers;

Expressing one quantity as a fraction of another quantity; Order

of Operations; Word problems

Table 3: Fraction Modules

Table 3 reveals a repetition of topics in the English books. For instance, adding and subtracting

fractions, comparing fractions, equivalent fractions appeared in all three books. But in the

Nigerian books, there is less repetition of topics. This might account for the falling emphasis on

fractions in the Nigerian books as evidenced by Table 2. Bearing in mind the repetition in topics

seen in the English books might be because the sampled books are differentiated, the assessment

influencing year groups seen in the Nigerian curriculum could be responsible for this disparity.

This is because, if students demonstrate good understanding of the concept, it is irrelevant

repeating it. However, the critical question to consider is how learners would demonstrate the

required understanding of concepts taught during assessments. Another possible and related

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cause could be linked to curricular dominance of lesson content since textbooks reflect the

curriculum directly with no room for variations.

Repetition of topics is not new to the English curriculum as Kaiser’s (1999b) characterises it as

spiral-typed involving “frequent repetitions of mathematical terms and methods which have

already been taught” (p.143). A further research examine if progression might influence the

content of similar topic for instance addition of fractions in year 7 and year 9.

Despite the falling emphasis seen with the UM series, it still emphasises fractions more than the

IM series25 which agrees with Hodgen, Küchemann, Brown and Coe’s (2009) claim regarding

English fraction emphasis. The Cockcroft Report might have influenced the lower emphasis on

fractions in the English curriculum (Cockroft, 1982). His argument about primary school

children’s ability with fractions could be challenged by evidence from Hodgen et al., that reveal

that 14-year-old students performed better in fractions in 1977 (15%, when calculating with

fractions was taught in primary schools) than 2008 (6%, when calculating with fractions was

taught in secondary school).

It is noted that word problems appear in the content of the Nigerian textbooks, but not in the

content of the English textbooks. Does this suggest that the English books do not address word

problems or is it a case of a different approach? This discussion is carried out in more detail

under the expectation subsection.

There are limitations to the finding on this criterion because in analysing pages, I did not taken

into consideration the variation in page distribution due to topic nature. Some topics might

                                                                                                                         25  Average  of  5.83  

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require more pages due the nature of the topics, for instance, shapes might require more pages to

demonstrate a concept due to the representations involved.

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Sub-constructs

Table 4 shows the result of the textbook analysis for the learning opportunities presented to the

students with focus on the sub-constructs.

Table 4 Sub-constructs

Table 4 evidences the sub-area domination (England 43%, Nigeria 30.7%) of other sub-

constructs in the examined series. This confirms the global emphasis placed on this sub-construct

as noted by Charalambous and Pitta-Pantazi (2007). This dominance, as they argue, could be

responsible for children finding the area sub-construct easiest to understand because learners can

easily relate their understanding using this sub-construct. It is easy to justify this emphasis based

Books Sub-

Area sub-set Ratio

Operato

r Quotient Measure Multiple None Total

# % # % # % # % # % # % # % # % #

IM1 15 44.1 0 0 1 2.9 7 20.6 0 0 0 0 1 2.9 11 31.4 34

IM2 11 50 0 0 0 0 2 9.1 1 4.5 0 0 1 4.5 8 34.7 22

IM3 5 33.3 0 0 0 0 0 0 0 0 1 6.7 0 0 9 60 15

IM 31 43.7 0 0 1 1.4 9 12.7 1 1.4 1 1.4 2 2.8 28 42.1 71

UM3 17 45.9 3 8.1 0 0 6 16.2 11 29.7 0 0 12 32.4 0 0 37

UM4 17 32.7 8 15.4 0 0 5 9.6 1 1.9 4 7.7 9 17.3 17 27.9 52

UM5 4 16.7 0 0 2 8.3 2 8.3 0 0 1 4.2 0 2 15 65.4 24

UM6 1 7.1 0 0 1 7.1 1 7.1 0 0 0 0 1 7.1 11 78.6 14

UM 39 30.7 11 8.7 3 2.4 14 11 12 9.4 5 3.9 22 17.3 43 43 127

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on the presumption that this sub-construct provides basis for learning others, but caution should

be exercised in promoting this sub-construct. This is because learning dominated by the area sub-

construct could foster learners to relate fractions understanding to mostly this sub-construct

(Behr et al., 1993) and also how something is taught impacts on how it is learnt (Streefland,

1991). Such domination makes understanding with other sub-constructs difficult (Behr et al.,

1993).

The ratio sub-construct is the least presented in learning opportunities across both countries, and

such was identified as a global trend (Charalambous & Pitta-Pantazi, 2007). This limited

learning opportunity could explain the difficulty, Dickson et al. (1984) noted, that learners

experience with this sub-construct.

The Nigerian textbooks presented a higher proportion of multiple sub-constructs than the English

books, suggesting that the Nigerian books might better promote better relational understanding

than the English, since exposing students to multiple fraction sub-constructs help the students’

understanding to be relational as each representation relates to particular cognitive structures

(Kieren 1976).

Since the identified challenges in fractions found in the introduction, I now looked at how

fractions are introduced in textbooks.

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Figure 7: Fractions introduction UM3

Figure 8: Fractions introduction IM1

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The UM3 introduces fractions using an orange cut in half and a set of pencils. This already

accounts for two sub-constructs (sub-area and sub-set). It also introduces the operator sub-

construct in ‘½ group of 8’ and explores dividing. Orange and pencils are very familiar objects to

students and are three dimensional so this introduction should elicit proper conceptualisation of

fractions since it involves the informal knowledge of sharing (Kilpatrick et al., 2001), objects

familiar to the learners (Confrey, 1994), using multiplicity of sub-constructs (Baturo, 2004), and

three dimensional representations (Dickson et al, 1984). With IM1, although it has used

representations, all appear two dimensional. It includes area sub-constructs and introduces (but

doesn’t explore) the operator and hardly explores the informal knowledge of the students. The

introduction to key words just appeared without connection to any concept, informal or prior

knowledge. Given the individualist view in the English system, the teacher would be expected to

give individual attention to every student. This reduces the contact time between each student

and the teacher. I would expect textbooks to involve detailed discussion to promote learning

when the teacher is not available. While being cautious that the books target students who of

different age ranges and prior understanding of fractions but motivated by similar findings (like

Bierhoff, 1996; Haggarty & Pepin, 2002), this approach seen in the English book seems to

promote instrumental understanding. That said, it will be interesting to see how other textbooks

in both systems introduce fractions and other topics.

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Expectation

It should be noted that cognitive expectation of a task is influenced by the context within which

the task is undertaken, for example:

Figure 9: Context of word problems

Despite the complexity of these tasks, I rate them as low complexity because, although the tasks

appear different, they are mostly following the procedure shown by the example so there is no

room for flexibility of thinking (National Assessment Governing Board, 2008). Although that

learners need to formulate mathematical in the task, but the lesson focuses on fractions so the

formulation needs to relate more to fraction than to computation as seen in the tasks.

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Table 5 shows the result of the cognitive expectation of textbook tasks.

Books Low

(#)

Complexity

(%)

Moderate

(#)

Complexity

(%)

High

(#)

Complexity

(%) Total

IM1 96 77.4 18 14.5 10 8.1 124

IM2 55 76.3 12 16.7 5 7 72

IM3 101 71.6 28 19.9 12 8.5 141

IM 252 74.8 58 17.2 27 8 337

UM3 171 96.6 6 3.4 0 0 177

UM4 318 94.3 19 5.7 0 0 337

UM5 176 78.6 47 21 1 0.5 224

UM6 106 84.1 16 12.7 4 3.2 126

UM 771 89.2 88 10.2 5 0.6 864

Table 5 Task complexity table

Figure 10: Task complexity chart

Evidence from Table 5 suggests that English books have a higher proportion of moderate to

higher complexity tasks than the Nigerian books. A critical evaluation reveals that the context

reduces the complexity of tasks because examples are very similar to problems, so there is

limited room for flexible thinking thus it becomes mainly a case of understanding what and how.

0  

10  

20  

30  

40  

50  

60  

70  

80  

90  

100  

IM1   IM2   IM3   UM3   UM4   UM5   UM6  

High  (%)  

Moderate  (%)  

Low  (%)  

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Also task sequence in each lesson in the English series appears to become increasingly more

complex but the complexity is mostly constant in the Nigerian series, as exemplified below:

Figure 11: UM Task complex (Lesson)

Figure 12: IM Task complexity (Lesson)

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The Nigerian textbook might be aiming for fluency using this approach but one might wonder

how the English textbooks promote fluency. It is important to note that exercises should deepen

understanding (Kaiser, 1999b).

Table 6 and Figure 13 present findings about learning application in real life:

Books Application Task

(#) (%)

Non Application Task

(#) (%) Total

Lesson

#

Ave

Task/Lesson

IM1 38 30.6 86 69.4 124 8 15.5

IM2 19 26.4 53 73.6 72 5 14.4

IM3 37 26.2 104 73.7 141 6 23.5

IM

27.7

72.3 337 19 17.7

UM3 1 0.6 176 99.4 177 17 10.4

UM4 39 11.6 298 88.4 337 19 17.7

UM5 50 22.3 174 77.7 224 11 20.4

UM6 28 22.2 98 77.8 126 6 21

UM 118 14.2 746 85.8 864 53 16.3

Table 6: Application of knowledge

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Figure 13: Application of knowledge

Within the series, there is a slight decline with the English series but the Nigerian series tends to

increase in the proportion of task appearing from fraction learning to real life. Despite the

utilitarian view demonstrated by both curricula, it is surprising that the proportion of task-

applying knowledge falls below the Fan (1990) benchmark of 40-50% of tasks related to real

life.

Tasks applying knowledge to real-life situations are part of lessons with the parent concept in

English textbooks, as exemplified in Figure 12. But the Nigerian series have the tasks applying

knowledge are found in standalone lessons as exemplified in Figure 9 where examples are given

and tasks are of similar nature. Since application of knowledge could imply relation of

knowledge to a different situation so it should enhance relational understanding, then word

problems should be part of lessons with parent concepts. As in the case of task complexity in the

Nigerian series, the resultant of understanding expected from isolated application tasks, due to

their context, could be instrumental.

0  

10  

20  

30  

40  

50  

60  

70  

80  

90  

100  

IM1   IM2   IM3   UM3   UM4   UM5   UM6  

Applica]on  (%)  

Non  Applica]on  (%)  

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Mindful that the inference from other textbooks might contradict these findings, since

opportunities for relational understanding are lost through their context and the presentation of

word problems causes cognitive processes of solving them become “channeled into more

predictable and … mechanic forms of thinking” (Henningsten & Stein, 1997, p.535). So in the

case of fractions, the UM series tends to instrumentalises (reduces to instrumental)

understanding. This finding agrees with the position of some scholars that learning expectation in

Nigeria is dominated by rote learning (Awolofa, 2012). This resonates with Ochefu’s (2013)

findings that out of the 70% of students who correctly multiply fractions, only 20% could

verbalise what they have done, and none of the students appeared to have conceptual

understanding of the multiplication of fractions. I am interested to see if instrumentalisation

(reduction of tasks’ cognitive expectation so they target instrumental understanding) is common

to other topics and textbooks in Nigeria. Instrumentalisation might account for how easy the

teachers in Salman’s study found teaching fractions (Salman 2005, 2009) since instrumental

understanding flourishes in similar contexts and its rewards are immediate (Skemp, 1976). It is

also important to remember that understanding is not black and white; my conjectures could be

further evaluated.

The Nigerian culture may be responsible for instrumentalisation because it “is so dictatorial and

conservative that it does not encourage inquisitiveness” (Georgewill, 1990, p.380). The

curriculum dictation of what happens in textbooks might be grounded in the dictatorial culture of

the country. The curricular dictation makes it challenging for instructional practice to develop

relational understanding, because concepts need to be taught and learnt within a fixed time

frame, so practice might resort to targeting instrumental understanding (Skemp, 1976).

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Conclusion

This essay has compared textbooks from England and Nigeria with focus on fractions. Although

it has not conclusively shown consistency or textbook signatures in the series examined, on all of

the criteria, it has shown that there appear to be overarching factors influencing the educational

system, which appear culturally and contextually motivated. Therefore, it situates itself in the

body of research that suggests that mathematics and textbooks can be cultural.

It identified that, although the Nigerian textbooks demonstrated good opportunities to learn

fractions, they had lost opportunities of the students developing relational understanding due to

the context of tasks and suggested that the underlying cause could be cultural. Bezuk and Cramer

(1989) blamed textbook for learners’ fraction misunderstanding but this study shows that in the

case of Nigerian textbooks, the rapid progression is mainly consequent on the curricular

structure. Similarly, Behr et al. (1993) criticism of textbooks’ task nature appeared to be caused

by the curricular expectation. The Nigerian curriculum design needs to be more flexible and the

textbook writers mindful of the context of tasks so the cognitive expectation is maintained

(mitigating instrumentalisation).

The English textbooks provided a weaker learning opportunity but demanded more from the

students. English books should aim to include a conglomerate picture of fractions by including

more sub-constructs in developing instructions to mitigate lack of understanding and relate

concepts to the informal knowledge of the learners. Textbooks from both countries need to

increase the proportion of tasks applying knowledge to real life situations and those of higher

cognitive demand.

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Findings concerning English textbooks resonate with Haggarty and Pepin’s (2002) position on

limited learning opportunities, although it appears to contrast their findings in relation to the

expectation of task. The context of the comparison might provide reason for this disparity. This

essay compares England to Nigeria, and there being more evidence of higher complexity tasks in

English books does not necessarily imply that English books adequately address higher

complexity tasks.

The English government recently announced that most fraction modules are to return to the

primary school curriculum (DfE, 2012). Does this imply an increased emphasis?

Further research areas could investigate the effect of using different frameworks to answer the

research questions. Since comparative research reveals deep assumptions, what would be the

effect of using different countries or textbooks for this essay? It would be useful to see the

difference in the findings if other researcher(s) were to code these elements in the books. Also, it

will be interest to confirm the findings of this essay using a qualitative method.

This study has demonstrated the importance of stepping out of one’s culture in an attempt to

unveil hidden commonality inherent in one’s practice (Stigler, Gallimore, & Hiebert, 2000).

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References

Akinsola, K. M. (2012). Primary Mathematics Curriculum and Methods. Lagos: National Open University of Nigeria.

Alajmi, A. H. (2012). How do elementary textbooks address fractions? A review of mathematics textbooks in the USA, Japan and Kuwait. Educational Studies in Mathematics, 79, 239-261. doi: 10.1007/s10649-011-9342-1.

Andrews, P. (2009). Comparative studies of mathematics teachers' observable learning objectives: validating low inference codes. Education Studies in Mathematics, 71, 97-122. doi:10.1007/s10649-008-9165-x.

Askew, M., Brown, M., Johnson, D., Millet, A., Prestage, S., & Walsh, A. (1993). Evaluation of the Implementation of National Curriculum Mathematics at Key Stages 1, 2 and 3. Volume 1: Report. London: SCAA.

Awofala, A. (2012). An Analysis of the New 9-Year Basic Education Mathematics Curriculum in Nigeria. Acta Didactica Napocensia, 5(1), 17-29.

Badmus, G. (1977, August 1-3). An Evaluation of the Mathematics Education component of Nigeria Primary School Curricula between 1930 and 1960. A paper presented to the 15th Annual Conference of Mathematical Association of Nigeria. Zaria: Ahmadu Bello University.

Ball, D. L. (1993). Halves, pieces and twoths: Constructing and using representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157-195). Hillsdale: Erlbaum.

Ball, D. L., & Cohen, D. L. (1996). Reform by the book: What is - or might be - the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6-8, 14.

Baturo, A. (2004). Empowering Andrea to help year-5 students construct fraction understanding. In M. Hoines, & A. Fuglestad (Eds.), Proceedings of the 28th Conference for Psychology of Mathematics Education (pp. 95-102). Bergen: University College.

Beaton, A. E., Mullis, I. V., Martin, M. O., Gonzalez, E. J., Kelly, D. L., & Smith, T. A. (1996). Mathematics Achievement in the Middle School Years: IEA's Third International Mathematics and Science Study (TIMSS). Chestnut Hill, MA: Boston College, Center for the Study of Testing, Evaluation, and Educational Policy.

42  |  P a g e        

Behr, M., Harel, G., Post, T., & Lesh, R. (1993). Rational Number: Towards a semantic analysis. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: An integration of research (pp. 13-47). Hillsdale, NJ: Erlbaum.

Benjamin, D., Bright, S., Clough, T., Cole, G., Debrida, D., Fraser, R., Grantham, B., et al. (2000). Impact Maths 1R. Oxford: Heinemann Educational Publisher.

Benjamin, D., Bright, S., Clough, T., Cole, G., Debrida, D., Fraser, R., Jolly, P., et al. (2001). Impact Maths 3G. Oxford: Heinemann Educational Publishers.

Benjamin, D., Clough, T., Cole, G., Debrida, D., Fraser, R., Jolly, P., Kent, D., et al. (1999). Impact Maths 2B. Oxford: Heinemann Educational Publishers.

Bezuk, N., & Cramer, K. (1989). Teaching fractions: What, when and how? In P. Trafton (Eds.), New directions for elementary school mathematics: 1989 Yearbook (pp. 156-167). Reston: National Council of Teachers of Mathematics.

Bierhoff, H. (1996). Laying the foundations of Numeracy : a Comparison of Primary School Textbooks in Britain, Germany and Switzerland (Discussion Paper no. 90). London: National Institute of Economics and Social Research.

Cambridge International Examinations. (2011, December). Cambridge Secondary 1 Reseource List. Retrieved from University of Cambridge International Examinations: http://www.cie.org.uk/docs/qualifications/checkpoint/resource%20lists/

Carraher, D. W. (1996). Learning about fractions. In L. P. Steffe, P. Nesher, G. Goldin, P. Cobb, & B. Greer (Eds.), Theories of Mathematical Learning (pp. 241-266). Hillsdale, NJ: Erlbaum.

Charalambous, C., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students' understanding of fractions. Educational studies in Mathematics, 64, 293-316 doi 10.1007/s10649-006-9036-2.

Charalambous, C., Delaney, S., Yu-Hsu, H., & Mesa, V. (2010). A Comparative Analysis of the Addition and Subtraction of Fractions in the Textbooks from Three Countries. Mathematical Thinking and Learning: An International Journal, 12(2), 117-151 doi 10.1080/10986060903460070.

Cockcroft, W. H. (1982). Mathematics Counts. London: HMSO.

Confrey, J. (1994). Splitting, similarity and rate of change: A new approach to multiplication and exponential functions. In G. Harel, & J. Confrey (Eds.), The development of multiplicative reasonning in the learning of mathematics (pp. 291-330). Albany: State University New York Press.

43  |  P a g e        

David-Osuagwu, M. N. (2009). Understanding Mathematics for Nigeria 6. Onitsha: Africana first Publishers.

David-Osuagwu, M. N. (2010a). Understanding Mathematics for Nigeria 3. Onitsha: Africana first Publishers.

David-Osuagwu, M. N. (2010b). Understanding Mathematics for Nigeria 4. Onitsha: Africana first Publishers.

David-Osuagwu, M. N. (2010c). Understanding Mathematics for Nigeria 5. Onitsha: Africana first Publishers.

DfE. (2012, June 11). New primary curriculum to bring higher standards in English, maths and science. Retrieved from GOV.UK: https://www.gov.uk/government/news/new-primary-curriculum-to-bring-higher-standards-in-english-maths-and-science

DfEE. (1999). The National Numeracy Strategy: Framework for teaching mathematics from Reception to Year 6. Retrieved August 27, 2013, from www.edu.dudley.gov.uk: http://www.edu.dudley.gov.uk/primary/Strategymaterials/NNS/NNS%20Framework/Introduction.pdf

DfEE. (2001). Key Stage 3 National Strategy for Teaching Mathematics: Year 7, 8 and 9. London: Department for Education and Employment.

Dickson, L., Brown, M., & Gibson, O. (1984). Children Learning Mathematics: A Teacher's guide to Recent Research. London: Holf, Rinehart and Winston.

Dowling, P. (1996). A Sociological analysis of school mathematics texts. Educational studies in Mathematics, 31, 389-415.

Empson, S. (1999). Equal Sharing and Shared Meaning: The Developement of Faction Concepts in a First-Grade Classroom. Cognitive and Instruction, 17(3), 283-342, doi: 10.1207/S1532690XCI1703 3.

English, L. D., & Halford, G. S. (1995). Mathematics education: models and processes. Mahwah, NY: Lawrence Erlbaum Associates.

Fan, L. (1990). Some comparisons and thoughts on the reforms of mathematics curriculum. Zhejiang Educational Science, 23, 22-28.

Fan, L. (1999). Application of Arithmetic in US and Chinese Textbooks: A Comparative Study. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International Comparisons in Mathematics Education (pp. 151-164). London: Falmer Press.

44  |  P a g e        

FMI. (2010, September 1). Federal Ministry of Information. Retrieved from fmi.gov.ng: http://fmi.gov.ng/federal-executive-council-resolution-for-wednesday-1st-september-2010/

Foxman, D. (1999). Mathematics across the world some evidence from the third international mathematics and science study (TIMSS). Slough: National Foundation for Education Research.

Freeman, D. J., & Porter, A. C. (1989). Do textbooks dictate the content of mathematics instruction in elementary schools? American Educational Research Journal, 26, 403-421.

Gelman, R., & Meck, E. (1986). The notion of principle: The case of counting. In J. Hiebert (Ed.), The relationship between procedural and conceptual competence (pp. 29-57). Hillsdale, NJ: Erlbaum.

Georgewill, J. W. (1990). Causes of poor achievement in West African school certificate mathematics examinations in Rivers State secondary schools, Nigeria. International Journal of Mathematical Education in Science and Technology, 21(3), 379-385.

Glaser, R. (1995). Expect knowledge and the process of thinking. In P. Murphy, M. Selinger, J. Bourne, & M. Bridges, Subject Learning in the Primary Curriculum (pp. 274-288). London: Routledge/OUP.

Haggarty, L., & Pepin, B. (2002). An investigation of mathematics textbooks and their use in English, French and German classroom: who gets an opportunity to learn what. British Educational Research Journal, 28(4), 567-590. doi:10.1080/0141192022000005832.

Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28,524-549.

Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., et al. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: U.S. Department of Education, National Center for Education Statistics.

Hodgen, J., Küchemann, D., Brown, M., & Coe, R. (2009). Lower secondary school students' knowledge of fractions. British Society for Research into Learning Mathematics 29(1), 55-60.

Hofstede, G. (1991). Cultures and organizations: Software of the mind. London: McGraw-Hill.

Howson, G. (1995). Mathematics textbooks: A comparative study of grade-8 texts. Vancouver, Canada: Pacific Education Press.

45  |  P a g e        

Kaiser, G. (1999a). International Comparisons in mathematics Education under the Perspective of Comparative Education. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International Comparisons in Mathematics Education (pp. 3-15). London: Falmer Press.

Kaiser, G. (1999b). Comparative Studies on Teaching Mathematics in England and Germany. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International Comparisons in Mathematics Education (pp. 140-150). London: Falmer Press.

Kieren, T. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh, Number and measurement: Papers from a research workshop (pp. 101-144). Columbus, OH: ERIC/SMEAC.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: helping children learn mathematics. Washington, DC: The National Academies.

Knipping, C. (2003). Learning from comparing: a review and reflection on qualitative oriented comparisons of teaching and learningmathematics in different countries. ZDM. Zentralblatt fur Didaktik der Mathematik, 35(6), 282-293. doi:10.1007/BF02656692.

Lamon, S. (1999). Teaching fractions and ratios for understanding. Mahwah, NJ: Erlbaum.

Landis, J. R., & Koch, G. G. (1977). The measurement of oberver agreement for categorical data. Biometrics, 33, 159-174.

Li, Y. (2000). A comparison of problems that follow selected content presentations in American and Chinese mathematics textbooks. Journal for Research in Mathematics Education, 31, 234-241.

Mack, N. (1995). CONFOUNDING WHOLE-NUMBER AND FRACTION CONCEPTS WHEN BUILDING ON INFORMAL KNOWLEDGE. Journal for Reseach in Mathematics Education, 422-441.

Mason, M. (2007). Comparing Cultures. In M. Bray, B. Adamson, & Mason M., Comparative Education Research Approaches and Methods (pp. 165-196). Hong Kong: Comparative Education Research Centre.

Mesa, V. (2004). Characterizing pratices assiociated with functions in middle school textbooks: An empirical approach. Educational Studies in Mathematics, 56, 255-286.

Mesa, V. (2007). Instructors and textbooks. Paper presented at the 10th RUME conference. San Diego, CA.

Mikhaylov, S., Laver, M., & Benoit, R. (2012). Coder reliability and misclassification in the human coding of party manifestos. Political Analysis, 78-91.

46  |  P a g e        

Moseley, B. (2005). Students' Early Mathematical Representation Knowledge: The effects of Emphasizing Single or Multiple Perspectives of Rational Number Domain Problem Solving. Educational studies in Mathematics, 60, 37-69.

National Assessment Governing Board. (2008). Mathematics framework for the 2009 National Assessment of Educational progress. Washington, DC: U.S. Department of Education.

Neidorf, T. S., Binkley, M., Gattis, K., & Nohara. (2006). Comparing Mathematics Content in the National Assessment of Educational Progress (NAEP), Trends in International Mathematics and Science Study (TIMSS), and Program for International Student Assessment (PISA) 2003 Assessments. Techinal Report. Washington, DC: National Center for Education Statistics.

Novillis, C. (1976). An analysis of the fraction concept into a hierarchy of selected subconcepts and the testing of the hierarchy dependencies. Journal for research in Mathematics Education, 7, 131-144.

Ochefu, A. (2013). Development of the concept of multiplication with fractions. Universal journal of Education and General Studies, 2(3), 079-083.

Omoifo, C. N. (2008). An Overview of the New 9-Year Basic Education Curriculum. Benin City: UBE.

Pepin, B., Haggarty, L. & Keynes, M. (2001). Mathematics textbooks and their use in English, French and German classrooms: a way to understand culture. ZDM. Zentralblatt fur Didaktik der Mathematik, 33(5) 158-175.

Phillips, D., & Schweisfurth, M. (2012). Comparative and International Education An introduction to theory , methods and pratice. London: Continuum.

Qualifications and Curriculum Authority. (2007). Mathematics 2007 Programme of Study for Key Stage 3 and attainment targets. Retrieved 8 2013, 8, from Department for Education: http://www.education.gov.uk/schools/teachingandlearning/curriculum/secondary/b00199003/mathematics/ks3/programme

Randolph, J. (2013, August 25). Online Calculator. Retrieved from Online Kappa Calculator: http://justus.randolph.name/kappa

Rezat, S. (2007). The use of mathematics textbooks by students. Paper presented at 31st Conference of the Psychology of Mathematics Education. Korea.

Rittle-Johnson, B., & Alibali, M. (1999). Conceptual and Procedural Knowledge of Mathematics: Does One Lead to the Other. Journal of Educational Psychology, 175-189.

47  |  P a g e        

Salman, M. (2005). Teachers identification of the difficulty levels of topics in the primary school mathematics curriculum in Kwara state abacus. The Journal of MAthematical Association of Nigeria, 30(1), 20-29.

Salman, M. (2009). ACTIVE LEARNING TECHNIQUES (ALT) IN A MATHE,ATHEMATICS WORKSHOP; NIGERIAN PRIMARY SCHOOL TEACHERS' ASSESSMENT. International Electronic Journal of Mathematics Education, 23-25 ISSN: 1306-3030.

Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.

Skovsmose, O. (1994). Towards a Philosophy of Critical Mathematics Education. Dodrecht: Kluwer.

Sofolahan, J. A. (2005). NEW PRIMARY SCHOOL CURRICULUM MODULES. Ibadan: Evans.

Stigler, J. W., & Hiebert, J. (2004). Improving mathematics teaching. Educational leadership, 61(5), 12-17.

Stigler, J. W., Fuson, K. C., Han, M., & Kim, M. S. (1986). An analysis of addition and subtraction word problems in American and Soviet elementary mathematics textbooks. Cognitive and Instruction, 3(3) 153-171.

Stigler, J., Gallimore, R., & Hiebert, J. (2000). Using video surveys to compare classrooms and teaching across cultures. Educational Psychologist, 35(2), 87-100.

Streefland, L. (1991). Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research. The Netherlands: Kluwer Academic Publishers.

Tall, D. (1978). The Dynamics of Understanding Mathematics. Mathematics Teaching, 81, 50-52.

Tanner, H., & Jones, S. (2000). BECOMING A SUCCESSFUL TEACHER OF MATHEMATICS. Oxon: RoutledgeFalmer.

Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H. & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of books. Boston: Kluwer.

Wilson, L., Andrews, C., & Sourikova, S. . (2001). Shape and structure in primary mathematics lessons: a comparative study in North-east of England and St Petersburg, Russia - some implications for thedaily mathematics lesson. British Educational Research Journal, 27(1), 30-58. doi:10.1080/01411920125004.

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Appendix

Sample coding for learning opportunities

Sub-construct: Measure

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Sub-construct: sub-area

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Sub-­‐construct:  sub-­‐set      

Sub-construct – None

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Sub-construct: Multiple, sub-area, measure

Sub-­‐construct:    Ratio  

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Sample coding for expectations

Expectation

Subtask 1: Non-application, moderate complexity

Subtask 2: Non-application, moderate complexity

Expectation: Application, Moderate complexity

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Expectation

Subtask 1: Application, Moderate complexity

Subtask 2: Application, Moderate complexity

Subtask 3: Application, Moderate complexity

Expectation: Application, High complexity

Expectation: Non-application, Moderate complexity

Expectation: Non-application, Low-complexity

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Expectation: Application, High complexity

Task and sub task were evaluated as tasks.