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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
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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
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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
27 | P a g e
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
28 | P a g e
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.
29 | P a g e
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
30 | P a g e
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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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.
33 | P a g e
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.
34 | P a g e
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 (%)
35 | P a g e
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)
36 | P a g e
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 (%)
38 | P a g e
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).
39 | P a g e
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.
40 | P a g e
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).
41 | P a g e
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52 | P a g e
Sample coding for expectations
Expectation
Subtask 1: Non-application, moderate complexity
Subtask 2: Non-application, moderate complexity
Expectation: Application, Moderate complexity
53 | P a g e
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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