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The Teaching of Geometry for 11 to 16 Year Olds in Ontario and England, with Reference to Curriculum and Textbooks
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This essay is a comparative analysis of the education system of England with the
district of Ontario in Canada with particular reference to the teaching of geometry. I shall
begin by focussing on the assumptions made by both regions in how students learn to
reason geometrically. An emphasis is placed on the use of instructional tools which are
explored from the perspective of curriculum and a textbook analysis for each region.
Literature Review
This review will introduce the van Hiele levels used by educators as a framework for
developing instructional tools (O’Shea, 2003). The model is placed in a historical context by
introducing Piaget’s theory as it has heavily influenced the van Hieles’. This is followed by a
review of relevant instructional tools used in the teaching of geometry.
Piaget and Van Hiele. Piaget’s theory on conceptualisation of geometrical ideas is
constructivist and has two major themes. Firstly, experience is interpreted and is
dependent on the way a student internalises and thinks about geometry. It is not simply a
““reading off” of the spatial environment”” (Clements, 2001, p152) and is dependent on the
previous experiences of the student in relation to the specific geometric concept. Cognitive
strategies and knowledge acquired are therefore actively constructed by prior active
manipulation of that geometric environment (Bereiter, 1985, Scardamalia & Bereiter, 1983;
Wittrock, 1974). Secondly, geometrical knowledge builds in stages which are logical and not
historical. A student begins with scribbles and progresses to recognition of topological
properties of shapes (e.g. enclosure). The student then moves to a projective stage where a
differentiation is developed between these topological shapes and Euclidean shapes and
finally the student is able to draw Euclidean shapes like triangles (Piaget & Inhelder, 1956).
Piaget argued that students are not influenced by instruction, that their cognitive levels are
a function of the development of their mental structures (Piaget cited in Lehrer, Jenkins, &
Osana, 1998) and this progression is a biological one and not related to the learning process
(Choi-‐Koh, 1999). Four factors of development involved in this biological process are the
maturity of the student, his/her physical and social experiences, and finally the ability to be
able to integrate these three factors into a coherent whole, known as equilibration
(Gallagher, 2002).
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Dina van Hiele-‐Geldof and Pierre van Hiele were husband and wife and introduced their
theory in their doctoral dissertations in 1957 at Utrecht University. Van Hiele theory is
based on the notion that students pass through stages of development by progressing from
primitive levels of thought to more sophisticated forms in relation to geometrical reasoning
(Van Hiele, 1984). The model proposes that geometry learning is a discontinuous process
which requires students to display different levels of thinking and because of this, the five
levels of progression for a student are hierarchical, in sequence, and are disjoint. The model
implicitly integrates the previous concepts associated with level(s) of thinking as a student
progresses to more sophisticated thinking levels. Each level has its own specific language
which a teacher must be able to understand if they are to identify which level a student has
attained (ibid.). Building on Piaget’s constructivist model, the van Hieles differ in that
learning becomes a process by which to gain acquisition of new geometrical knowledge and
is not age-‐dependent (Jaime and Gutierrez, 1995). Instruction from a teacher (aided with
instructional tools) heavily influences the way students conceptualise and organise
geometry and the design of instruction must be specific to the respective developmental
level for each student (Crowley, 1987). The first level of recognition involves a purely visual
interpretation and figures are not differentiated by individual parts or their properties but
rather seen as a whole. For example a triangle is seen as not having three sides or its angles
adding up to 180 degrees but rather as a ‘raw’ figure. Geometric concepts therefore are
viewed as “total entities rather than having components or attributes” (ibid., p2). Students
at this level can even reproduce shapes but do not exhibit any real understanding of the
shape, for example a student could reproduce a square having seen it but would not be able
to distinguish that each angle is a right angle. Students can also use geometric vocabulary
such as recognise a square after seeing it. The second level of analysis involves the student
reasoning about basic concepts through looking at figures. They may colour the same
angles of an isosceles and, using an inductive process, be able to generalise characteristics
about this figure. In line with Piaget’s notion of equilibration, students at this level are
unable to make connections with neither multiple figures nor the relationship between
properties of figures. Definitions at this level are not understood. At the next level of
informal deduction, students start understanding the properties of concepts, build abstract
definitions, and start seeing the variance and invariance between shapes. For example, a
student can recognise that a square is a type of rectangle because it shares all its properties.
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Inductive and deductive techniques are mixed at this informal stage and formal proofs can
be followed but not constructed. It should be noted that this third level is mostly the stage
at which current secondary schools work to (ibid.). The penultimate level is that of formal
deduction and at this stage students can work formally within an axiomatic system including
proving and understanding the intra-‐relationships within this system. The final stage of
rigour involves being able to work in different axiomatic systems (i.e. non-‐Euclidean
geometries), which can be compared and geometries can be seen in the abstract.
Research on the van Hiele model of understanding has led to a confirmation in its
hierarchical nature (Fuys, Geddes, & Tischler, 1988; Usiskin, 1982). Usiskin investigated the
predictability of the levels for geometric achievement and found a significant correlation
while Fuys et. al. used a working model for the van Hiele levels and found a match with
student activity. The writing of formal proof related with the deductive process, as well as
development within geometry in general with respect to van Hiele levels, has further helped
in its corroboration (Bobango, 1988, De Villiers, 1999). The discrete nature of the levels and
their application to geometry as a whole has led to criticisms of the model. Scholars have
argued that students show signs of thinking at more than one level in the same or different
tasks and in different contexts, although one level of thinking may predominate (Gutiérrez,
Jaime, & Fortuny, 1991; Lehrer, Jenkins & Osana, 1998). They either used assessment in the
form of written tests ( Gutierrez & Jaime, 1987; Mayberry, 1983; Usiskin, 1982) or
interviews to ascertain the thinking level displayed (Burger & Shaughnessy, 1986; Fuys et al.,
1988). In both cases a large number of students displayed the presence of other levels in
their answers. Burger and Shaughnessy postulated that this was due to students being in
transition between the levels while Fuys et al. went further to indicate that students were at
two levels simultaneously and both levels were predominant in the student’s thinking.
Students were also found to be at different levels depending on the specific content area of
geometry being studied (Mayberry, 1981). Further, the prerequisite understanding of
previous levels has been called into question with Lawrie (1999) finding that high-‐level
students misinterpreted first and second level test items.
Instructional tools. Geometry teaching takes place through a combination of verbal
language, mathematical symbolism (formal or informal) and through visual forms (e.g.
diagrams). In the literature this is referred to as a multimodal discourse (Duval, 2000;
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Morgan, 1995; 1996; 2006; O’Halloran, 2003). In this review I will concentrate on the visual
forms of teaching through outlining the use of diagrams, manipulatives, and virtual
manipulatives in dynamic geometry software. According to Nesher and Kilpatrick geometry
learning begins when students understand the physical world around them and continues to
a high-‐level through inductive processes or within deductive systems (Nesher & Kilpatrick,
1990). Most students at school are unable to progress to the last two van Hiele levels of
understanding (Crowley, 1987) and so I will concentrate on these concrete tools used to
promote the lower levels of thinking.
Diagrams. Research has shown that considering spatial, visual and kinaesthetic
approaches to learning mathematics helps students to make links between these
representations and their underlying concepts (Bryant, 2009; Goldin, 1998). Moreover
Alshwaikh (2009) argues there is consensus that diagrams are important in teaching
mathematics mainly in “visualisation, mathematical thinking and problem solving” (ibid.,
p2). He has offered a simple framework for distinguishing different types of diagrams; offer-‐
labels type and demand-‐labels type. Offer-‐labels do not ask for any mathematical activity to
be carried out by the student and express either “geometrical relationships or specific
quantities” (ibid., p3).
Figure 1: Offer-‐labels expressing geometrical relationships (ibid.)
Figure 2: Offer-‐labels expressing specific quantities (ibid.)
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Demand-‐labels require mathematical action to be carried out by the student finding either
unknown quantities such as ‘c’ in figure 3 or “variable names” (ibid.) such as finding x, y and
z in figure 4.
Figure 3: Demand-‐labels: Unknown quantities Figure 4: Demand-‐labels: Variable
names (ibid.) Although diagrams are seen as an integral part of geometry teaching, they do have their
drawbacks. A theory of internal representation argues that students combine all mental
pictures and properties associated with the concept to create a ‘concept image’ (Vinner &
Hershkowitz, 1980). Further it is argued that concept images are the staple of student’s
reasoning rather than definitions of these concepts (Clements, 2001). In this context
diagrams can pose significant difficulty for a student, who may fail to understand that
drawings do not necessarily represent all the information about its representation (Parzysz,
1988). A simple example could be a drawing of a rectangle which does not give information
that the angles are equal or the sides are parallel. A student must be familiar with the
properties of the shape to be able to understand that a diagram is not a literal translation of
the object’s properties and implied concepts. In relation to test items on proof, students
were unable to differentiate essential and unessential characteristics of a diagram
(Clements & Battista, 1992). The concrete nature of diagrams is only useful therefore if a
student is able to understand the concepts associated with the diagram which may not be
specifically stated and recognise what information is relevant to the task. This is in line with
van Hiele in that students’ progress through the first three levels by developing their
understanding of concepts with the use of visual aids. The concrete nature of diagrams for
the van Hieles help students progress through levels of understanding in an inductive
manner, which eventually progresses to deductive thought and then formal rigour. O’Toole
highlights how “pure logic is abstract, but geometry makes logic concrete by applying the
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thinking to diagrams” (O’Toole, 1941, p. 319). It should be noted that more recent research
has argued there are several types of concrete knowledge and the more advanced of these
integrates concreteness and abstraction (Clements, 1999).
Manipulatives. Manipulatives can be defined as a tactile instructional tool which
gives the student a better opportunity to perceive a mathematical concept (Clements and
Battista, 1992). Examples of manipulatives include tangrams, cubes, spinners, etc. and they
are physically ‘manipulated’ to better grasp concepts associated with a specific task. An
example could be using a tangram and fitting smaller shapes into a bigger shape, thus
learning to appreciate that a shape can be split into parts and mathematical operations can
take place on these smaller parts. Manipulatives provide a hands-‐on activity for the student
to link what they are doing with abstract geometric concepts (Clements, 2001) by not only
seeing mathematics but by also providing a kinaesthetic element to the learning process.
The use of manipulatives intimately relates with real-‐world application by introducing
physical objects as a means to create more ‘meaning’ for a student. Manipulatives allow
students to experiment and discover relationships, construct their own knowledge of
geometry and be able to apply their understandings in real-‐world settings (Daniels, Hyde,
and Zemelman, 1993). Research has found manipulatives encourage students to participate
in mathematics, raise their self-‐confidence, and improve their understanding of geometry
(Ernest, 1994). Furthermore, manipulatives were found to help students make the jump
from concrete notions to more abstract ways of thinking (Hartshorn & Boren, 1990).
Students with large addition problems were found to perform better with manipulatives
than students without them (Carpenter & Moser, 1982; Steffe & Johnson, 1970). Increase in
mathematics achievement with the use of manipulatives was found by Suydam & Higgins
(1977) while Sowell’s meta-‐analysis into 60 studies found long-‐term use of manipulatives
was more beneficial than short-‐term use (1989). Further Sowell found student attitudes
improved with use of manipulatives (ibid.). Manipulatives were also found to have a
positive effect on student achievement compared to traditional instruction (Ruzic and
O’Connell, 2001). Criticisms of manipulatives stem from the notion that promoting ‘deep’
learning is not guaranteed by manipulatives (Baroody, 1989). Students may use
manipulatives in a rote manner to memorise procedures and then reproduce them without
understanding the underlying concepts (Clements, 2000).
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Dynamic Geometry Software. Dynamic geometry software (DGS) provides a
medium for the learning of Euclidean geometry in which students are able, through “direct
manipulation” (Laborde & Laborde, 2008, p31) to operate on theoretical geometric objects
in a “physical sense” (Laborde, 1995, p242) as diagrams on a computer screen. DGS in the
simplest sense is used as a means to create constructions which replace the pencil and
paper method of construction. However, a study into undergraduate mathematics
concluded that doing mathematics in the traditional manner leads to one understanding it
(Povey & Ransom, 2000) and decontextualizes acquired knowledge in geometry (Baki, Kosa,
& Guven, 2011). In contrast the notion of “situated abstractions” was introduced by Noss
and Hoyles (1996) where computer environments provide an acceptable model for learning
in which the original mathematical concepts are “preserved and extended by the learners”
(ibid., p125).
Users begin with creating a set of independent, freely existing objects-‐ usually points, and
then proceed by further constructing objects that are dependent on the former with respect
to geometric relationships (Gonzalez-‐Lopez, 2001). The power of DGS lies in the specific
context of its ‘dragging’ feature (Goldenberg & Cuoco, 1998) which allows students to
explore the dynamic behaviour of a construction by moving it (Kortenkamp, 1999). They
can observe variant and invariant aspects of a diagram under dragging and gain insight into
the particular construction through this feature (ibid.). Further dragging allows students to
observe and manipulate many examples and they obtain immediate feedback which cannot
be obtained by paper and pencil teaching (Marrades & Gutierrez, 2000). DGS can be used in
promoting explorations which aid in the conjecturing process (Baki, 2005). This view is
further supported by Hoyles (1998) who argues that DGS allows the student to make
justifications and explanations termed “proof by explanation” (Hanna, 1989). She argues
that DGS helps students make the transition from looking at particular cases to the general,
through manipulations which can be easily seen on the computer screen (DiSessa, Hoyles, &
Noss, 1995). The link therefore towards higher forms of van Hiele thinking levels which
require formal mathematics can be seen to be promoted by DGS. Baccaglini-‐Frank and
Mariotti (2010) argue that students can pass “from the phenomenological world to the
mathematical world” (ibid., p227) of formal rigour through DGS which can act as a
“potential bridge” (ibid.). A detailed study found a better grasp in the ability to visualise the
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geometrical character of a figure, a tangible understanding of the meaning of a theorem,
and clarity in what should be proved when students used DGS (Nomura & Nohda, 1999,
p93). The pitfalls of DGS have been highlighted however with the main criticism being that
DGS is used simply as a verifying tool, and not for problem-‐solving (Hölzl, 2001).
Corroborating the limited use of DGS, another study found “the software was mainly used
as an amplifier for visualizing properties” (Laborde, 2001, p92) and students didn’t use the
software as a tool for solving the task (ibid.). The highlighted need for teacher-‐input
therefore is a crucial factor in determining the effective use of DGS with teachers needing to
“carefully design tasks aimed at exploring and conjecturing” (Jones, 2000, p81).
Comparative Analysis
This analysis begins by providing a framework which is then used to assess
curriculum and textbooks for each region, which are considered separately in terms of
instructional tools.
Framework. The loose framework which I have chosen to work from is a cube (Bray,
Adamson, & Mason, 2007) with three faces representing three different aspects of
comparative education.
Figure 5: Bray and Thomas Cube (ibid., p9)
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The top face represents different demographic groups, for example age group, sex, etc. The
front face represents the geographical location such as countries, districts, etc. and the side
face represents aspects of education and society like curriculum, political change, etc. The
aim in using this cube is to focus on specific aspects of each face and as result a unit cube
within the whole cube is the area which is specific to a comparative study. The example
given in the diagram is the entire population of two states being compared in relation to
their respective curriculums. I have chosen to use this cube in conjunction with Bereday’s
model where the central theme is the idea of establishing a “tertium comparationis”
(Bereday, 1964, p9) which is a comparison with respect to stated criteria that is common in
the things compared and this is known as a juxtaposition.
Figure 6: Bereday’s Model for Comparative Studies (Bray et. al., 2007, p86)
I have chosen to compare the education system of Ontario, Canada with that of England.
Canada does not have a national curriculum and each state is responsible for managing its
own educational provision. Although with respect to the Bray and Thomas cube, it outlines
comparing geographically, like with like, I have chosen to avoid the “notion of one country,
one system” (ibid., p123). I chose England instead of the UK because the other countries
have “their own systems of education” (ibid., p138). I have decided to compare the two
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regions using one central theme, instructional tools, which will be my term of comparison
(Bereday, 1964) but from two different perspectives of curriculum and textbook content
(the side face of the cube). I believe that looking at the macro (curriculum) and the micro
(textbooks) may help to provide a better perspective in understanding how these systems
compare in terms of instructional tools. Ontario has two systems which run alongside each
other, a Francophone (French-‐speaking schools) and an Anglophone system (English-‐
speaking schools). Post-‐1997 curricula were developed separately for the two systems
(Ontario Ministry of Education (MoE), 1997)) so I have chosen to analyse the Anglophone
education provision. Furthermore, I have chosen to compare the secondary school age
group in both regions. The English curriculum for secondary school is split into Key Stage 3
which covers 11-‐14 year olds, and Key Stage 4 which covers 14-‐16 year olds (Department for
Education and Employment (DfEE), 1999(a)). The Ontario curriculum has separate
curriculums for the elementary years, which covers 5-‐13 year olds and then students move
to upper secondary school to complete their final four years (MoE, 2005).
For the textbook analysis I have chosen to compare year 9 from both regions as it covers the
same age group (13-‐14 year-‐olds). For the English system it covers the intermediate year of
secondary schooling and the first year of upper secondary school in Ontario. The textbook
analysis corresponds with the analysis of the English 1999 National Curriculum and so
accordingly Key Maths 93 Revised (KMR) published by Nelson Thornes was chosen on the
basis that it was the “best-‐selling” (Fujita & Jones, 2002, p80) textbook for 2001. For the
Ontario Curriculum, the MoE publishes a recommended list of textbooks known as the
“Trillium List” (Ontario Ministry of Education, 2013) and the textbook I have chosen,
Principles of Mathematics 9 (PoM), is included in this list. The two systems have different
policies when it comes to streaming, with it being prominent in England, and having been
abolished at grade 9 in 1995 in Ontario (Robertson, Cowell, & Olson, 1998). PoM for
Ontario covers students of all abilities, while KMR for the top strand was chosen for England
to allow a comparison to correspond to the opportunity for the ‘full’ curriculum to be
covered in each of the textbooks. The geometry sections of each textbook will be compared
with a focus on instructional tools, that of diagrams, manipulatives and DGS.
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Geometry Curriculum Comparison. Curriculum has different meanings in different
contexts (Beauchamp, 1982; Jackson, 1992; Walker, 2002) for which reason I have chosen to
specifically define the concept. Curriculum can be seen as providing a “specific set of
instructional materials that order content” (Clements, 2007, p36) and can be seen as an
“available curriculum” (ibid.). This view of curriculum is in contrast to the ideal, adopted,
implemented, achieved, or tested curriculum (ibid.) and my aim is to look at curriculum
purely in terms of potentially implemented. I have chosen to use the English 1999 National
Curriculum because it is more detailed than the 2007 curriculum. Further the 2007
curriculum contained additional elements of the “development and role of mathematics in
society” (Brown, 2011, p158) however due to the lack of teacher knowledge coupled with
no examining of this knowledge, “there are few signs of it yet in classrooms” (ibid.).
Van Hiele’s theory of progressive geometrical learning is evident in both England and
Ontario’s design of their curriculum. Ontario and generally North America has been
influenced by the National Council of Teachers in Mathematics (NCTM) (O’Shea, 2003)
which holds that students in secondary school make a transition from induction to
deduction, in line with van Hiele (NCTM, 2000). They further go on to recognise a
progression through a hierarchy of levels in relation to the development of geometric ideas
(NCTM, 1989). The English system at the primary level made use of van Hiele in the
National Numeracy Strategy (DfEE, 1999(b)) and is evident in secondary level mainly in how
the attainment levels are defined. The van Hiele levels are distributed over eight attainment
levels. The first van Hiele level is ignored as an assumption is made that this level has been
achieved at primary school and students at the outset are expected to “describe the
properties” (DfEE, 1999(a), p91) of 2D and 3D shapes. The final attainment level for
exceptional performance goes beyond analysis and informal deduction towards formal
deduction. Students are expected at this level to “use the conditions for congruent triangles
in formal geometric proofs” (ibid.) and this corresponds to the fourth van Hiele level. The
levels in between correspond to the second and then third level of van Hiele. Students are
expected to build their understanding of properties of shapes, for example “reflective
symmetry of 2D shapes” (ibid., p91) to understanding concepts like “congruence and
mathematical similarity” (ibid., p92).
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Instructional Tools. Both curricula in relation to instructional tools make specific
reference to diagrams. At both key stages, the English curriculum argues students should be
able to use “geometrical diagrams and related explanatory text” (Department for Education
and Skills ((DfES), 2004, p63) to communicate mathematically. Furthermore, students must
be able to “interpret, discuss and synthesise geometrical information” (ibid., p63) presented
to them in different mathematical forms such as diagrams. The Ontario curriculum places
diagrams (and other mathematical forms) in a context where they are used to “pose
questions about geometric relationships, investigate them, and present their findings”
(MoE, 2005, p37). Diagrams therefore are seen by both regions to promote learning, where
the English take a more generic view and see diagrams as a means of mathematical
communication; Ontarians see it to be used specifically in conjunction with investigation.
Through this way of analysing mathematics, students can “broaden their understanding of
the relationships among the properties” (ibid., p10). Ontario’s understanding of the use of
diagrams is related to the perception of how students learn effectively. The assumption is
that students need to be given “opportunities to investigate ideas and concepts” (ibid., p4)
and then through guidance gain insight into the “abstract mathematics involved” (ibid.).
The specific use of DGS is outlined in both curricula, but whereas the English mention it
largely in generic terms of ICT with a reference to manipulating geometrical
representations, the Ontarians refer to it more extensively in relation to geometry teaching
(see appendix). In the section where it highlights how students should be taught
knowledge, skills, and understanding, the National Curriculum outlines the use of
“appropriate ICT…. and knowing when it is not appropriate to use a particular form of
technology” (DfES, 2004, p69). In the brackets there is a mention of geometry packages;
however this is the only reference in relation to DGS in the whole document. The general
use of ICT is more detailed in this document, with it being seen as a medium “to overcome
difficulties in thinking and working in the abstract” (DfEE, 1999(a), p82). ICT is seen as
supporting students learning in all subjects at Key Stages 3 and 4 (ibid.) and should be used
to “refine their work and enhance its quality and accuracy” (ibid.). An attempt should also
be made to reflect “critically” (ibid.) as work progresses. The National Curriculum therefore
generally outlines the importance of the use of ICT and comments made can be extended to
be applicable to the use of DGS for geometry teaching. However within the specification for
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geometry teaching in Key stage 3 and Foundation Key stage 4 there are only two mentions
of the use of ICT, one in relation to problem-‐solving and the other related to loci (the higher
course has just one mention). So although the importance of ICT is stated, the instructional
way in which ICT and more specifically DGS can be used is highlighted in a minimal way. The
curriculum outlines the requirement of constructions by hand with instruments and there is
no mention of ICT to be used in conjunction with this way of constructing (ibid.). The
Ontario Curriculum seems to place a lot more importance on the use of technology and
more specifically DGS. Students at the outset are required to have the ability “to use
technology effectively” (MoE, 2005, p3) and highlights how with the advent of technology in
classrooms “previously time-‐consuming” (ibid.) problems can now be easily solved with
students being focused on the “underlying concepts” (ibid.). In fact the Ontario Curriculum
goes on to elaborate the use of DGS by students working with “pre-‐made sketches” (ibid.,
p14) meaning their work will be orientated towards manipulation of the sketch rather than
mindlessly “inputting of data or the designing of the sketch” (ibid.). Furthermore, within
the programme of study, I counted over ten references to the use of DGS in relation to sub-‐
topics in geometry for grades 9 and 10 written in the context of investigations being carried
out and pre-‐made sketches being used (see appendix).
Manipulatives are treated by the English Curriculum in a similar way to DGS, in that there is
no specific mention of how they can be used in the classroom. However there is a general
statement encouraging their application. The curriculum states that students should be
able to “do practical work with geometrical objects to develop their ability to visualise these
objects and work with them mentally” (DfES, 2004, p69). So manipulatives are seen as a
tool to promote the ‘inner eye’ of the student through practical work however a student
must “distinguish between practical demonstrations and proof” (ibid., p56). Within the
specification for both key stages, there is a reference to being able to “construct cubes,
regular tetrahedra, square-‐based pyramids and other 3-‐D shapes from given information”
(ibid., p65). However there is no mention of whether manipulatives could or should be
used, implying this is a requirement for accurately sketching these shapes onto paper. In
relation to finding the volume of cuboids, the curriculum outlines how students need to
appreciate a “connection to counting cubes” (ibid.) and although manipulatives are not
highlighted this requirement certainly allows the use of this instructional tool. The Ontario
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curriculum is much more concise and clear about the use of manipulatives and again there is
influence from NCTM. It defines them in clear terms as “concrete learning tools to make
models of mathematical ideas” (MoE, 2005, p15) and sees their use as promoting the
observation of relationships, making connections between the concrete and abstract and
evaluating and communicating reasoning (ibid.). The Ontario Curriculum highlights how
manipulatives are “necessary tools” (ibid., p23) for effective learning and how they provide
a medium for exploring abstract mathematical ideas through “varied, concrete, tactile, and
visually rich ways” (ibid.). The document further elaborates how teachers can understand
students’ thinking through “analysing students’ concrete representations of mathematical
concepts” (ibid.) and listening to their reasoning. Within the specification, the use of
manipulatives is highlighted in relation to the sub-‐topics of geometry with seven references
(see appendix). Further examples of manipulatives are given such as “paper-‐folding” (ibid.,
p37) for identifying the properties of polygons and using “toothpicks” (ibid., p36) for
determining the maximum area of a rectangle given a perimeter.
Textbook analysis. PoM is a larger book with dimensions 21cm by 26cm compared
to KMR (which has dimensions 17cm by 24cm) and a surface area which is 34% bigger on
each page. PoM also coupled with this, has over 50% more pages than KMR which suggests
there is significantly more mathematics coverage in this Canadian book. PoM and KMR have
similar proportions of the whole text dedicated to geometry, 39% and 37% respectively,
however because PoM is a larger book this suggests it either covers more geometry than
KMR or more thoroughly covers the same or less amount of geometry. Both books have
been published in colour with PoM dedicating the last four chapters out of nine to different
areas of geometry. KMR’s coverage is more fragmented with geometry being covered in
seven of sixteen chapters in an episodic manner throughout the book. The chapters are
significantly shorter in KMR with the longest geometry-‐related chapter being 28 pages; in
contrast PoM has 66 pages. PoM has specific learning aims outlined at the beginning of
each chapter (e.g. Dearling et. al., 2006, p360) while KMR does not. Both textbooks’
chapters are split into sub-‐sections exploring geometric concepts in a piecemeal fashion.
KMR provides mathematical concepts and theorems with worked examples at the beginning
of each chapter and then prior to when the students need to use the concept in application
to problems (Baker, Hogan, Job & Verity, 2001, p47 and p112). This approach is somewhat
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been criticised in that students are not encouraged to understand the fundamentals of
concepts, but rather they are just “merely stated rather than developed” (Fujita & Jones,
2002, p82). In line with the Ontarian curriculum aim that students learn best by
investigating, PoM introduces new topics in investigations needed to be carried out by the
student and introduces concepts in the context of an investigation (e.g. Dearling et. al.,
2006, p436). This is coupled with concepts introduced through worked examples similar to
KMR (ibid., p414). Mathematics encouraging Informal deduction, the third van Hiele level is
not easily identified in KMR as I was unable to find any questions or methods encouraging
the use of inductive or deductive informal reasoning. KMR corresponds to the first two van
Hiele levels, promoting memorisation and the application of mathematical properties and
concepts in a “narrative block, with examples and then exercises” (Fujita & Jones, 2003, p5-‐
p6). Mathematical concepts and demands made from the student are “always closed”
(Haggarty & Pepin, 2002, p579) and “help pupils achieve fluency in the use of routine skills
through repeated practice in exercises” (ibid., p586-‐p587). In contrast PoM couples the
KMR setup, usually early in the chapters, with encouraging students towards developing
inductive and deductive techniques. The main medium is through the idea of students’
conjecturing mathematical generalisations and then testing these conjectures by attempting
to find a counter-‐example (e.g. Dearling et. al., 2006, p397). Furthermore, through an
investigative approach promoting students to make hypothesis and then test them, PoM
encourages exploration of mathematical concepts and allow students to ‘discover’
mathematical relationships (e.g. ibid., p384).
Instructional tools. In line with the simple distinction for classifying different types
of diagrams, I counted the different types of offer-‐labels and demand-‐labels and worked
them out individually as a proportion of the total diagrams in the geometry sections in each
of the textbooks (Table 1). I found there were a larger proportion of offer-‐labels in PoM
(27%) compared to KMR (20%) suggesting there was more information given to students in
visual form as a total proportion of diagrams with respect to geometry. The proportion of
demand-‐labels shows that KMR makes more use of these types of diagrams with a large
proportion (80%) dominating the textbook. PoM in comparison has 73% which although still
a large proportion, is less than KMR. A breakdown of demand-‐labels reveals an interesting
17
finding that PoM has over a third of demand-‐labels asking for variable names while just
under two-‐thirds are asking for unknown quantities.
Figure 7: Example of a demand-‐label asking for variable-‐names in PoM (p382). KMR in comparison has less than a tenth of demand-‐labels asking for variable names with
the majority, over 90% being dedicated to unknown quantities.
Figure 8: Typical demand-‐label in KMR (p112). This corroborates the finding that “UK mathematics textbooks are designed around a set of
exercises” (Fujita and Jones, 2002, p82) and further elaborates that these exercises are
18
often in relation to finding an unknown, rather than looking for multiple variables which
requires a deeper understanding of geometrical concepts and theorems.
Diagrams Offer-‐labels Demand-‐labels Geometrical
Relationships Specific
Quantities Unknown Quantities
Variable Names
Principles of Maths 9 17% 10% 46% 27% Key Maths Revised 93 13% 7% 73% 7% Table 1: Distribution of different types of diagrams. Out of the demand-‐labels asking for variable names in PoM, a significant proportion (12%)
required the student (through investigation) to think inductively and deductively and in
generic terms through experimentation and conjecturing (e.g. Dearling et. al., 2006, p382).
Figure 9: Diagram related to conjecturing in PoM (p435). In contrast all of the questions requiring variable names were simply an extension of
unknown quantities and required finding multiple unknowns in KMR. The lack of analytic
thought required for geometry in KMR was surprising and corroborates Healy and Hoyles
(1999) in that UK textbooks focus on “calculation and the production of specific (usually
numerical) results” (cited in Fujita & Jones, 2002, p82).
In terms of manipulatives, both textbooks refer to the use of this tool, with PoM making five
references and KMS with just two. PoM however does use manipulatives in a more
effective way, for example by encouraging the use of geoboards (Dearling et. al., 2006,
p478), investigating rectangles with toothpicks (p481), constructing prisms from square
cubes (p491), and using tennis balls for understanding spheres (p462). KMR on the other
hand doesn’t actually promote the use of manipulatives in exploring mathematical
19
concepts, but instead demands the student to simply build them. Once the manipulative is
made, the exercise ends and it is questionable as to the usefulness of these manipulatives.
The two occasions they are mentioned requires the building of a presentation box with
specified features (Baker et. al., 2001, p119) and building a shape sorter (p353).
The use of DGS in each of the textbooks is very clear-‐cut in that KMR has no mention
including no reference to the use of ICT in general, while PoM has over 19 separate
occasions where this tool is encouraged to be used. PoM also features (p525-‐p531) a
‘basics’ guide to using Geometer’s Sketchpad (GS). The references in exercises made to
using GS covers the depth and breadth of its use from its basic use in exploring
constructions (p375-‐p376) and shapes such as polygons (p386-‐p388) or quadrilaterals
(p402) to investigating cones (p450), spheres (p461), area (p479-‐p480) and volume (p468-‐
p469). Exercises involving GS are also present in PoM with an emphasis on using this tool to
hypothesise and test conjectures to aid students in making the step to more formalised
thinking (e.g. p365-‐366). PoM also doesn’t compromise making constructions by hand with
over ten references (e.g. constructing polygons, p385). KMR on the other hand encourages
the use of constructions made by hand (in line with the curriculum) but “there is no
explanation of how to use a protractor” (Haggarty and Pepin, 2002, p582) or any other tools
in a geometry set.
Conclusion
The curriculum analysis shows that instructional tools used to promote the lower
levels of van Hiele levels are better utilised in the Ontario system in that they are clearly
defined and used in a more productive manner. Manipulatives and DGS are explained in
terms of how they facilitate the learning process and then are linked with the Programme of
Study through many examples of how they may be used (see appendix). The English
curriculum on the other hand although clearly defines diagrams and outlines how their use
promotes learning, are lacking in in defining and utilising manipulatives and DGS. The
National Curriculum fails to provide practical ways in which concrete instructional tools can
be used and this is reflected in the lack of DGS, manipulatives that explore mathematical
concepts, and diagrams which encourage abstract thinking in KMS. Concepts are simply
given as a fact which needs to be learned to be able to complete ‘closed’ questions. This
20
“approach that emphasizes computation, rules, and procedures, at the expense of depth of
understanding, is disadvantageous to students, primarily because it encourages learning
that is inflexible, school-‐bound, and of limited use” (Boaler, 1998, p60). PoM on the other
hand utilises the use of complex diagrams, not only building manipulatives but using them in
a lateral manner to investigate relationships, and DGS which encourages the exploration of
concepts via pre-‐made sketches. Further it promotes the third van Hiele level through
exercises which promote inductive and deductive arguments at an informal level. Overall,
although developing abstract mathematic concepts is an aim in the National Curriculum, the
English system is unable to move beyond the first two levels of van Hiele in application to
the textbook analysis. Ontario has a recommended reading list defined by MoE which
better correlates to teaching in line with the curricular aims that is better realised in PoM.
Lessons from both systems show that for concrete instructional tools to be better realised in
geometry teaching, curriculum should clearly define them and then instruct on why and
how they should be used in the classroom. Further providing a relevant context in their use
is advantageous to localising their use in the classroom. The Ontario system uses a context
of investigation which allows students to experience the exploration of mathematical ideas
and concepts in a fashion which suits their age-‐group. In relation to textbooks, there is a
case for the English Education Department providing a recommended reading list which
would allow a correlation of curricular goals with content taught.
21
Appendix
ENGLAND ONTARIO, CANADA
GENERAL COMMENTS
The English Curriculum clearly appreciates the need for students to be able to interpret and understand diagrams and places it as an essential tool in the teaching of geometry. Unlike Ontario however, there are no definitions or explanation of the different ways in which manipulatives and DGS can be used to promote learning. Further throughout the Programme of Study for both key stages there is minimal reference in providing specific examples of how these tools can be used in conjunction with what is being taught.
The Ontario Curriculum places importance on the use of these instructional tools. Diagrams are placed in the context of promoting an investigative approach, and are defined as a type of “mathematical model” (MoE, 2005, p62) which gives a description of a situation. Manipulatives are also clearly defined and are seen as “concrete learning tools” (p15) and in conjunction with DGS are emphasised with specific ways that they can promote learning. Manipulatives are seen to encourage the recognition of patterns and relationships, link the concrete with the abstract, be self-‐critical on their reasoning, improve memory and effectively communicate reasoning to others (p15). DGS in a similar fashion is seen as a problem-‐solving tool which can be used to reduce mechanical activities and speed up learning in comparison to the paper-‐and-‐pencil method (p14). Further Geometer’s Sketchpad is introduced as the sole DGS application which should be used in classrooms. Throughout the Programme of Study there are numerous references to how diagrams, manipulatives, and DGS can be used to promote specific geometrical concepts.
THE USE OF DIAGRAMS
Being able to draw diagrams on paper and using ICT (DfEE, 1999, p41). Presenting, organising and explaining diagrams (p49). Interpreting diagrams and drawing conclusions (p41). Effectively using diagrams as a medium for appreciating mathematics, communicating and gaining a perspective on a problem. (p65).
Using diagrams to express mathematical ideas (p16). Using diagrams to pose questions about geometric relationships, investigating them and presenting findings (p37). Using diagrams to promote problem-‐solving strategies (p13). Using diagrams to solve problems with respect to parallel lines with a transversal (p45).
THE USE OF MANIPULATIVES
Only reference in relation to doing practical work with geometrical objects to develop ability to visualise these objects mentally (DfES, 2004, p69).
Ability to be able to select appropriate manipulatives to perform particular mathematical tasks, to investigate mathematical ideas, and to solve problems (p14). Manipulatives are necessary tools for supporting effective learning and are a valuable aid to teachers (p23). Determining the maximum area of a rectangle with a given perimeter by using geoboards and toothpicks (p36). Using concrete materials to investigate the
22
effect of varying the dimensions on the surface area of square-‐based prisms and cylinders, given a fixed surface area (p36). Using concrete materials to investigate the formulas for the volume of a pyramid, a cone, and a sphere (p37). Using concrete materials to investigate and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons (p37). Using paper-‐folding to investigate the properties of polygons (p37).
THE USE OF DYNAMIC GEOMETRY SOFTWARE
Using dynamic geometry packages to manipulate geometrical configurations (DfEE, 1999, p8).
Use pre-‐made sketches when using DGS to manipulate data rather than simply inputting data or designing a sketch (p14). Representing mathematical ideas and modelling situations using DGS (p16). Using DGS to express and organise ideas and mathematical thinking using oral, visual, and written forms (p21). DGS can be used to support various methods of inquiry in mathematics (p28). Using DGS to collect data which can then be tabulated and graphed (p33). Verify parallel and perpendicular lines using DGS (p35). Verify through investigation with DGS, geometric properties and relationships involving two-‐dimensional shapes (p36). Determining the maximum area of a rectangle with a given perimeter by using DGS (p36). Using DGS to investigate and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons (p37). Using DGS to investigate the properties of polygons (p37). Using DGS to pose questions about geometric relationships, investigating them and presenting findings (p37). Using DGS to verify a geometric statement by scientific induction or falsify through a counter-‐example (P37). Using DGS to develop a formula for the midpoint of a line segment (p49). Using DGS to develop a formula for the length of a line segment (p49). Using DGS to investigate some characteristics and properties of geometric figures (p49). Using DGS to investigate the similarity of triangles (p51).
TABLE 2: Differences in curricula with respect to concrete instructional tools.
23
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