King’s College London
Mathematics PGCE
Subject Studies Assignment
Extending lower
secondary students’
concept of area to
incorporate
parallelograms,
triangles and
trapezia.
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Contents
1.
Introduction.................................................................................................................................... 3
2.
Literature Review ......................................................................................................................... 3
2.1 Misconceptions and possible approaches ............................................................................. 7
2.2 Conclusion ................................................................................................................................ 12
3.
Sequence Planning .................................................................................................................... 12
3.1 Data Collection and Anonymity ............................................................................................. 13
3.2 Structure.................................................................................................................................... 14
3.21 Lesson One ........................................................................................................................ 14
3.22 Lesson Two ........................................................................................................................ 15
3.23 Lesson Three ..................................................................................................................... 15
3.24 Lesson Four ....................................................................................................................... 15
3.25 Lesson Five ........................................................................................................................ 16
3.26 Lesson Six .......................................................................................................................... 16
4.
Discussion and Analysis............................................................................................................ 16
4.1 Conservation of Area .............................................................................................................. 19
4.2 Area vs. Perimeter ................................................................................................................... 23
4.3 Application of the Formula for the Area of a Rectangle ..................................................... 24
4.5 From Parallelogram to Triangle to Trapezium .................................................................... 27
4.6
5.
Success of the Sequence ............................................................................................. 34
Conclusion ................................................................................................................................... 36
References .......................................................................................................................................... 37
Appendix ............................................................................................................................................... 42
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1. Introduction
This paper aims to assess the merits and pitfalls of a sequence of five lessons given
to an all male year 8 class entitled Area. The lessons will attempt to extend their
concept of area to incorporate parallelograms, triangles and trapezia, as in the aims
of the National Curriculum (DfE 2013). It will critically analyse - in terms of the
difficulties students experience - the role that the sequencing of ideas has on the
class’ understanding, and how well this sequence scaffolds their journey into their
zone of proximal development (Vygotsky 1980; Bruner 2009). The sequence of
progression of areas of different shapes will be based on what is considered best
practice from literature.
A review of literature will highlight the importance of area as a topic, inform the
sequencing of ideas in the lessons, and anticipate possible problems whilst
suggesting strategies. The sequence of lessons is then described in the context of
the literature review, and the following discussion will focus on what the students
learnt, and didn’t, and how this relates to the literature.
2. Literature Review
Area can be defined as “an invariable attribute, a definite measureable size of the
plane surfaces enclosed by figures which may be conserved while the shape of its
figure is altered” (Piaget et al. 1960; Kordaki & Balomenou 2006). Its importance is
highlighted by numerous authors, for a plethora of reasons: it has a variety of direct
practical applications in the public domain as well as in science and technology; it is
related to multiplication structures, for example imbuing understanding of the
distributive law, and furthermore is a key conceptual idea at a more abstract level –
area under a curve underlies integral calculus, which in turn opens up a range of
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more sophisticated applications (French 2004; Huang & Witz 2009; Kordaki & Potari
1998; Hirstein 1978; Huang & Witz 2012; Malkevitch 2009). Area can be classified
under the heading of measurement or of geometry, and as such it represents a
bridge between the physical world of real objects and the more abstract world of
mathematical structures (Hiebert 1981). This suggests it could be a topic in which
mathematical modelling is explicitly discussed, which has been given increased
emphasis in the new National Curriculum (DfE 2013). Further to this, it is a topic that
offers the opportunity to develop understanding of other important ideas, such as
that of invariance and proof (Piaget et al. 1960; Waring 2000; Johnstone-Wilder &
Mason 2005); and important skills, such as deductive reasoning, conjecturing and
proving (Jones 2012; Watson et al. 2013).
Despite its common usage and obvious utility, numerous studies have shown the
concept of area to be poorly understood by secondary school students, mathematics
undergraduates and trainee teachers alike (Woodward 1983; Kospentaris et al.
2011; Beattys & Maher 1985). This lack of comprehension is surprising given its
importance and leads to two obvious questions: Why? And what can be done to
change this? These questions will be explored in the context of a lower secondary
school male class undergoing a teaching sequence with an aim to developing an
understanding of the area of parallelograms, triangles, trapezia and other 2D
polygons.
A constructivist approach would say that students must actively construct their own
knowledge, and must necessarily build on their prior knowledge (Ryan & Cooper
2010). There are two important ramifications from this that will be discussed – firstly
that we must build on students’ prior knowledge, and secondly we must consider
effective ways in which to build one’s own knowledge, which will be discussed later.
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If one of the most critical factors that influences learning is a persons’ prior
knowledge, a teacher must try and ascertain what this is and teach accordingly
(Wiliam 2011). Indeed, harnessing children’s implicit knowledge about spatial
relations is an important challenge in mathematical education (Bryant 2007).
However, it is difficult to know exactly what experiences and concept images
students arrive with (Confrey 1990). To assess diagnostically for learning, there are
two things that we must be able to do – firstly gain enough understanding of what
mathematical experiences a learner may have had with the topic in order to use an
appropriate diagnostic test, and secondly use an appropriate method to analyse their
responses to the diagnostic test to probe understanding in a way that allows the
planning of effective teaching.
A common method that is used to describe geometrical thinking is to assign them
hierarchical levels described by the van Hiele model (Fuys et al. 1988a). Relating to
area, and renumbered 1-5 (as advocated by Clements et al. 1997 for consistency)
these are:
Level 1: Students use the concept of coverage to find the area of a figure
Level 2: They discover procedures for finding the area of certain types of shapes and
know when they do and don’t apply
Level 3: Students can interrelate knowledge and make informal arguments, and can
for example justify the formula for the area of a triangle
Level 4: Students can make deductive arguments to prove conjectures
Level 5: Students can argue proofs from an axiomatic basis, and can compare
different axiomatic systems
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Research largely agrees that these levels have some use in describing the
progression of geometrical thought. As Piaget et al. (1960) described, the learning
of geometry is now almost entirely in the reverse of its chronological emergence in
history, and the teaching of geometry tends to coincide with the van Hiele
progression of ideas - from topological attributes in Level 1, to formal deductive and
Euclidean arguments operating in levels 4 and 5. Although this is generally viewed
as having lead to an improvement on previous systems with a strong Euclidean
focus; Waring (2000) debates whether this is an ideal teaching model, and whether it
might be better to have a curriculum whereby geometric ideas of all types develop
over time in a coherent way, which might raise the currently low status proof
occupies in schools. The debate stems from Piaget’s conclusions that children are
egocentric and unable to decentre themselves and therefore reason, but these
experiments have been criticised for lacking validity due to their unconventional and
de-contextualised nature (Donaldson 1978). This alternative viewpoint is shared by
some who dispute that the levels of thinking are discrete and sequential as van Hiele
suggests (Walcott et al. 2009). For example, many researchers have reported that in
tasks where students compare areas of different shapes they may be able to
confidently explain which of two rectangles has the larger area and why, i.e.
operating within level 3, but regress to a level 1 when confronted with a similar task
involving triangles, or irregular shapes, even when the exact same reasoning that
they have used previously is appropriate (Kordaki & Balomenou 2006; Steele 2012;
Fuys et al. 1988).
Students are typically taught the concept of area firstly as coverage or direct
comparison of area via overlapping (Zacharos 2006), and by counting squares. This
is followed by noticing that when finding the area of a rectangle one can simply
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multiply the number of rows by the number of columns, and eventually leads into the
formula for the area of a rectangle (Kamii & Kysh 2006; Huang & Witz 2012). After
investigating this with non integer lengths, the rectangle is then sheared to form a
parallelogram and the concept of area conservation is highlighted, to derive the
formula for the area of a parallelogram, and from that a triangle as half its area. The
area of a trapezium can then be derived by noticing it is half the area of a
parallelogram formed by two congruent parallelograms (Waring 2000; Fuys et al.
1988a; Spitler 1982; Pereira-Mendoza 1984). From there more complex polygons’
areas can be derived by partitioning or surrounding with rectangles and other known
shapes (Woodward 1983; Rickard 1996).
2.1 Misconceptions and possible approaches
The sequencing of ideas in area stated above is the most commonly espoused one
in literature as good practice, and can be summarised by Fig 1.
Figure 1. Route 1: The sequencing of ideas that different authors claim should be taught to
students, in time from left to right. The arrows indicate from what previous shape or idea the
new set of shapes should be introduced.
However, Pereira-Mendoza (1984) also discusses two other routes, summarised in
Fig 2.
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Figure 2. Sequence of ideas in time from left to right. The arrows indicate from what previous
shape or idea the new set of shapes should be introduced. Route 2 (upper): Here, triangles
are introduced as halves of rectangles. The idea of shearing is explored and by putting two
congruent triangles together parallelograms’ areas are found. Route 3 (lower): Here ,
triangles and parallelograms’ areas are found via comparisons to rectangles. Note that all
three routes have in common going from arrays to rectangles, and from parallelograms to
trapezia.
Although these sequences are often mentioned, there appears to be a dearth of
literature that comments critically on what problems arise from the sequencing of
ideas itself, and compares the effectiveness of learning outcomes from different
sequences.
Nevertheless, if it is the case that the original (route 1) sequencing of ideas is good
practice, but outcomes of learning about area are poor, then execution of this
sequence must be poor. To test this and to help understand why this may be, we
must consider common misconceptions that arise. A wealth of literature has
highlighted the struggles students often encounter in moving from finding the area of
a shape by counting squares or by covering it with squares, to using a formula to
calculate the area of a rectangle (Clements et al. 1997; Kospentaris et al. 2011;
Huang & Witz 2012). Huang & Witz (2009) explain that emphasising the
memorisation of a formula rather than a conceptual understanding of area
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contributes to children’s difficulties. In their study, they show that the best learning
outcomes for children results when they are taught geometric motions – such as
recognising congruent shapes in different orientations - in conjunction with
numerical calculations of area measurement. The group taught in this way performed
better than groups that were taught only geometric motions or only area
measurement calculations. This shows that whilst it is important that the formula for
the area of a rectangle is learnt, mechanical calculations of area whereby two sides
are given by themselves do not lead to the best learning outcomes. Furthermore,
they point out that the procedures used in tiling or area coverage do not directly
demonstrate the properties of multiplication, and that the structure of arrays must be
made explicit. A 2-D array cannot simply be understood by multiplying length and
width (Nunes et al. 1993): the links between coverage, arrays, their multiplicative
structure and the formula must be made explicit. Area conservation is a key concept
that must be understood by children as a prerequisite for understanding the formula
for the area of a parallelogram. That cutting and pasting sections of a shape does
not change the area is not immediately obvious to many children (Beattys & Maher
1985; Kordaki & Balomenou 2006). This suggests a reason why Huang & Witz
(2009) found better outcomes for children who learnt geometrical motions alongside
area measurement calculations.
A common misconception that arises in lessons about area is the mixing up of area
and perimeter (Rickard 1996). French (2004) argues that this is more than just
mixing up the names but in fact that:
“There is a deeper problem to do with understanding the subtle relationship between
the two ideas which requires an appreciation that changing one does not necessarily
change the other”. (French 2004:69)
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This implicitly suggests that a strategy to overcome this misconception is to include
activities where one of the properties is invariant whilst the other changes.
A specific problem with deriving and understanding the formulae for the area of
parallelograms and triangles is drawing line segments as a perpendicular height. For
obtuse triangles and some parallelograms, this line segment is not necessarily
contained within the shape, and so constructing this segment often represents a
major difficulty for students (Spitler 1982). A different problem is recognising that
“base” does not necessarily mean the “bottom” of the shape (Waring 2000), which
she suggests can be helped by students creating parallelograms from two congruent
triangles, to reinforce the notion that we can define the base ourselves.
Many of the misconceptions and difficulties children have suggest obvious courses
of action which have already been expressed. For example, to help bridge the link
between the concept of area and the formula for the area of a rectangle we have a
clear sequence of ideas whereby the area formula is discovered by the multiplicative
structure of arrays. However, there are some more general teaching strategies which
are also pertinent to this topic. At the heart of geometry is reasoning and proof, and
there are two particularly interesting avenues through which these skills can be
encouraged.
Beattys & Maher (1985) show that investigating area concepts using geoboards is
more effective than a traditional paper and pencil based approach, both in a post test
and even more so in a retention test taken 6 weeks after the teaching sequence.
These findings are supported elsewhere (Cass et al. 2003; Sowell 1989), although
the latter study implies that we should be somewhat sceptical of any effects if the
geoboards are just used over the course of a few lessons. Students are able to
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quickly generate examples and explore areas for themselves by counting squares,
without the use of formulae (Spitler 1982). Although it could be claimed there is
some potential loss of generality, this is gained in accuracy of construction in
comparison to paper and pencil approaches (Anderson & Arcidiacono 2014).
Another is by utilising the power of computer based dynamic geometry systems
(DGS). As well as providing a pool of visual representations (Koyuncu et al. 2014),
they serve ‘‘to create experimental environments where collaborative learning and
student exploration are encouraged’’ (Ruthven et al. 2008; Erez & Yerushalmy
2007). In particular, the drag function allows a student to test a hypothesis with
thousands of examples in a short period of time, and therefore show a quite dramatic
demonstration of invariance, for example. There is evidence that using these DGS
systems can lead to improved learning outcomes for students (Reisa 2010).
However, despite the potential benefits offered by these two options from the
literature, there is some evidence to suggest that in the practical classroom setting
they are difficult to implement. In open ended novel problems of any type a teacher
has competing tensions that serve to truncate the potential of the activity (Herbst
2003), a major one being that despite the apparent open nature of a task, the
teacher has a desired outcome in mind that acts to limit the activity’s openness. For
computer based systems there are even more practical problems to be overcome,
and it appears that it is only exceptional cases whereby the software is used in a
maximal way, which itself depends on the context the teacher is in (Ruthven et al.
2008).
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2.2 Conclusion
Research consistently agrees that area is an important topic in the curriculum that is
nevertheless often poorly understood. There is consensus that a large amount of the
problems experienced by students correspond to an underdeveloped concept of
area and an over reliance on mechanistic uses of formulae, without being able to
reason why or when to use them. Some researchers argue that this is a problem not
localised to area in mathematics but is endemic to our teaching systems (Boaler
2009). Regardless of the truth of this, there appears to be some consensus on
strategies to create a more coherent mathematical journey through the concept of
area that we can implement from research, and further that using manipulatives and
dynamic geometry systems can powerfully increase the efficacy of this.
3. Sequence Planning
The sequence will consist of six 50 minute lessons focused on the aims of the
scheme of work based on the National Curriculum:
“Pupils should be taught to be able to derive and apply formulae to calculate and
solve problems involving the area of triangles, parallelograms and trapezia.” (DfE
2013:8)
There will be a particular focus on developing students’ conceptual understanding of
area, which has been highlighted as critical to achieving the wider aims of the
teaching for the students – being able to retain knowledge of how to find the areas of
various shapes; being able to apply this knowledge appropriately and in different
contexts, and to gain experience of mathematical reasoning (Jones 2012). The
lessons aim to address various misconceptions highlighted in the literature, and will
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focus on understanding area concepts rather than applying formulae without
understanding why they work.
3.1 Data Collection and Anonymity
Before the first and last lesson of the sequence, a test will be administered to the
students consisting of 10 questions. Each question is designed to address or
highlight a particular misconception; to give insight into the original state of the
students’ understanding; and to allow them and the teacher to see what has been
learned and what is still to be learned. Additionally, it will provide data on to what
extent particular concepts have been understood, and so aim to help improve the
commentary of how students react to the lessons. During each lesson, conversations
with students will be recorded and notes taken on class discussions. Classwork will
be marked and assessed to gather more data.
The misconceptions children experience will be commented on in reference to the
sequencing of ideas itself – referred to in Fig 1 – as well as commenting on what
about the sequence led to particularly easy transitions to new ideas for the students.
As well as gathering data to analyse the students’ learning, the data will be used
formatively: The questions asked during conversations and class discussions will
help guide the students’ thinking, their marked classwork and tests will give formative
feedback, and will inform the teaching of starters of following lessons and the final
lesson respectively. Indeed, this is essential in order to appropriately scaffold their
learning into their zones of proximal development (ZPD) (Vygotsky 1980; Bruner
2009). As Black & Wiliam (1998) say:
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“Firm evidence shows that formative assessment is an essential component of
classroom work and its development can raise standards of achievement.” (Black &
Wiliam, 1998:1)
A substantial amount of data will be taken in these lessons that will be used
formatively, and the quality of the teaching the students experience will not be
compromised by the study. Names of children have been changed to anonymise
them.
3.2 Structure
Lessons were planned in light of the responses to the first test, and follow the
general sequence of ideas shown in Fig 1. As well as numerical questions, there
were three sections that asked students to explain their reasoning. These indicated
that most students were operating in level 2 of Clements et al. (1997) modified
version of van Hiele’s levels of geometric reasoning. Many sought to mechanistically
apply formulae. To be able to progress to level 3 whereby students could make
informal arguments for how to find the area of a triangle say, it was important that
each step leading to that justification must be emphasised.
3.21 Lesson One
The first lesson will focus on understanding area as space inside of a shape, without
reference to formulae, advocated by Zacharos (2006). It will also address the idea of
conservation of area, and the difference between area and perimeter. The main
activity to do this will be comparing the areas of different isoperimetric shapes by
cutting and overlapping. By showing that rotations and other rigid body
transformations do not affect the area, geometric motions are being taught to
increase the effectiveness of their learning, as found by Huang & Witz (2009).
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3.22 Lesson Two
The second lesson will lead into the formula for the area of rectangles, by making
explicit the multiplicative nature of arrays and how this relates to area as Nunes et
al., (1993), Kamii & Kysh (2006) encourage. Huang & Witz (2009) showed that
students must still perform calculations of areas of shapes using formulae, even if
there are associated problems. Furthermore, it is crucial that this formula is
understood in order to find the areas of the more complex shapes further in the
sequence. This will be extended into finding the area of rectilinear compound
shapes, which follows directly from understanding the formula developed in this
lesson.
3.23 Lesson Three
The third lesson will draw on the ideas of conservation of area promulgated in lesson
one, to find the area of a parallelogram, whereby they cut off and move the corner of
a rectangle to form a parallelogram. It will focus on trying to get students to
distinguish between slant and perpendicular height, by using questions with
parallelograms on grids whereby they have to draw the height in themselves (see
Appendix 4). Additionally, it will introduce the idea that the word “base” does not
necessarily mean the side at the bottom of the shape.
3.24 Lesson Four
In lesson four, the area of triangles will be found by constructing parallelograms on
geoboards from two congruent triangles. By finding the area of the parallelograms,
and recognising that a triangle must have half the area, the children are lead into
discovering the formula for the area of the triangle themselves. Discovery learning
was advocated as a more powerful tool for learning by Piaget (1952), but it should be
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noted that this is guided discovery, which is agreed by many authors to be more
effective than pure discovery learning (Mayer 2004).
Beattys & Maher (1985) show that the use of geoboards can lead to improved
learning outcomes. Additionally, they give two other important benefits for this task –
the perpendicular height has to be found by the students, rather than being given by
the question. Furthermore, by using geoboards in pairs, students are naturally
prompted towards speaking mathematically about what they are doing. This social
element of learning is in line with social constructivism as being crucial in helping
students to formulate and refine their ideas (Powell & Cody 2009), as well as
promoting feedback from their peers.
3.25 Lesson Five
After consolidation of previous concepts, lesson five will introduce the area of a
trapezium via an almost identical activity to that above – on a geoboard, constructing
a parallelogram from two congruent trapezia, and thereby finding the area of each
trapezium.
3.26 Lesson Six
This lesson is based on what they are found to struggle with in the post-test that they
will complete between lessons five and six (in a lesson otherwise outside of the
sequence).
4. Discussion and Analysis
The presentation of the global responses from the tests will be presented and
analysed. Different topics and themes will be discussed in more detail using
individual answers in the test, work done in class and classroom discussions. They
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will approximately be addressed in the order at which they were relevant in the
lessons.
The pre-test administered is shown in Fig 3:
Figure 3. Each question tries to elicit responses pertaining to different misconceptions or
different applications of knowledge. Q1: Whether they can see area as counting squares.
Q2: Straightforward application of formula for a rectangle. Q3: Whether they can correctly
see that one of the pieces of information given is superfluous. Q4: Whether they can apply
the formula of a rectangle appropriately, or whether they will mechanistically try and multiply
just two of the lengths together. Q5: Prior knowledge: Whether they can find the area of a
triangle with all sides given. Q6: Whether they can find the perpendicular height for an
obtuse triangle. Q7: Whether they can recognise that base isn’t necessarily the bottom side
of a shape. Q8: Prior knowledge: Whether they can find the area of a trapezium. Q9:
Conservation of area: Whether they can see that areas can be rotated without changing in
magnitude, and added. Q10: Whether they can distinguish between the concepts of
perimeter and area.
The post test was almost identical to this but with different examples used, and is
shown in Appendix 7. Fig 4 shows the correct response rate per question.
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Mean correct responses per question
Correct response rate
1.2
Post test
1
Pre test
Increase
0.8
0.6
0.4
0.2
0
-0.2
1
2
3
4
5
6
7
8
Question Number
9
10
11
Figure 4. The blue line shows the mean mark for each question on the pre test. The red line
shows the mean mark for each question after 5 lessons, on the post test. The green line
shows the increase in marks from the pre test to the post test. A correct response gave a
mark of 1 and an incorrect response gave 0*. The mean mark is shown, for example if every
student got a question correct then the mean score for that question would be 1. The graph
shows that on most questions more students gave correct responses. However, for question
8 there was no improvement and on questions 9 and 10 the students’ performance
decreased.
*Except for 6 and 7. Here, 0.5 marks were awarded for an incorrect answer that still correctly
found the perpendicular height, and use the correct base and height, respectively.
The graph shows that most questions were answered correctly by more students
after the teaching sequence. There was a sharp increase in correct response rate for
finding the area of a compound rectilinear shape. There was a significant increase in
the number of correct responses for the areas of the triangles. Interestingly, for
question 8 on finding the area of a trapezium there was no improvement. On
questions 9 and 10 there was a decrease in students’ performance, although this is
likely to be due to harder questions in the post test, which is discussed further below.
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As well as the answers given, the workings out and explanations often helped to
illuminate the students’ thinking. The themes will be explored in the context of the
tests and conversations in lessons.
4.1 Conservation of Area
The first and third lessons dealt with ideas of invariance and conservation of area.
Lesson one began with questions on the board leading into a class discussion
concluding that you could say that a shape had less area than another if it could fit
completely inside it (See Appendix 8 for details). They were given scissors and
isoperimetric rectangles – and one triangle as an extension – and in groups were to
sort the shapes from largest to smallest area.
I listened to one group of 3, all of whom had originally not attempted question 9 or
answered it incorrectly. They initially tried to just estimate which shape was largest,
but with a little prompting decided that they needed to show it by cutting shapes such
that they could be completely enclosed by another.
Me: If we take this piece you’ve cut out, and put it back like this (so that it makes an
L shape), is the area the same?
Daniel: No… wait… yeah yeah it is.
(Agreement from the other two)
Me: Why?
Riley: It’s just the… it’s ‘cause you’ve not like… done anything to it…
Daniel: Yeah ‘cause before it was there and now it’s there but it’s still the same size.
The students have expressed the idea of conservation of area in their own words
and demonstrated a relational understanding of area conservation. They then went
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on to explain that if the cut up shape overlapped with itself, the total area would no
longer be the same (See Appendix 1).
After a class discussion consolidating these ideas, students attempted a plenary
question – which marking revealed they mostly managed correctly - whereby they
had to match shapes that had the same area, shown in Fig 5.
However, in hindsight with respect to the test results, it may be that students
answered the questions correctly due to cues in the question that made the question
possible without having an understanding of the concept that it is trying to get at.
Figure 5. H and K are simply rotations of each other. Other pairings involve shapes that have
been cut and pasted. G has no pairing, and C and F requires a slightly more taxing mental
reformation, or more simply can be paired by reasoning with fractions.
One particularly interesting comment made during this task was:
“You can’t know though. It’s impossible. Because from my perspective they (A and I)
look the same but you don’t know.”
This suggests that the student had taken on the previous challenge of trying to prove
the order of the area of the shapes, and was now unhappy that he was being asked
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to do something without proof. Highlighting this insight turned into a useful class
discussion about the nature of geometry and proof.
In the third lesson when introducing the area of a trapezium, the area of a rectangle
they were given was found. After cutting the corner off and moving it to the other side
to form a parallelogram, they were asked:
“What is the area of the parallelogram?”
Immediately, hands went up, with a variety of similar answers explaining that the
area was the same. Again, this suggests that students felt confident that area was
conserved under this operation.
Question 9 in the tests pertains specifically to area conservation, and given the
understanding expressed in the conversations throughout the lessons I was
surprised that the performance in this question appeared to decrease. Only one of
the three I’d observed closely in the lesson got this right, despite appearing to be
confident with the concept. It may be because in the pre test, the question requires
only mentally rotating and translating shapes to see that the areas add to one whole
little square. The post-test question is arguably more challenging (see Fig 6), as it
cannot be done in this way. Instead, a student must realise that each of the two
shaded areas in D are half of a whole square.
Additionally, there was a substantial difference in what the explanation section
revealed about their understanding. For the pre test, 9 of 22 students simply did not
attempt the question, and of attempted wrong answers explanations are summed up
by one of the answers given:
“They look about the same.”
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In the post test, only 3 of the 22 did not attempt it, and this time most of the wrong
answers were similar to the true and insightful comment,
“The area of the shaded bits in (T) add up to one square, just like C.”
Here, the wrong answers now demonstrate that they understand that area is
invariant under translation.
This highlights an important point, raised by Black et al. (2004):
“The assessment methods that teachers use are not effective in promoting good
learning.” (Black et al. 2009:9)
Had this test not have included a section whereby the students explain their
answers, and had I not conversed with students I would have believed that they did
not understand conservation of area. I may therefore have spent significant time –
which could be used more usefully on other things – trying to reinforce a concept
most students were confident with.
Figure 6. Q9. Left: Pre-test diagram. Middle: Post-test diagram. Right: Pre-test question. In
the pre-test, translations and rotations are needed to answer the question correctly. In the
post-test, one must also either use fractional knowledge to answer the question.
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4.2 Area vs. Perimeter
Rickard (1996) and French (2004) both highlighted difficulties students have in
mixing up concepts of area and perimeter. Q10 is an adapted version of a test used
by Woodward (1983) which he claimed could distinguish between students who “do
and don’t functionally understand area.” Whilst this is a bold claim without evidence
or reasoning, Rickard (1996) also advocates teaching to distinguish between the two
concepts by holding one parameter constant and varying the other. This question
shows rectangular gardens with the same perimeter but different areas. Unlike the
students in Woodward’s sample, the majority of students in this class got this
question correct and furthermore of those that didn’t, only one indicated that the
gardens were the same size. This suggests that this misconception was mostly not
evident in this particular class. The one student who did mix them up had answered
every single question in the pre-test as though the word area meant perimeter, and
was consistent in his definition. In the post-test, he consistently found areas. This
suggests that it was not a “Deeper problem to do with understanding the subtle
relationship between the two ideas,” but in fact was just mixing up the words.
In response to the class mostly answering or attempting the question correctly, the
post-test was changed to make the calculations more difficult. This was to test
whether this extra difficulty would cause any students to revert to trying to use the
easier to calculate perimeter instead. In fact, all but one student still attempted to
calculate areas, even though few correctly calculated 12.5 x 12.5. The one response
which did confuse the area and perimeter was interesting – they’d calculated the
area for all the shapes with integer lengths, but for the square with sides of 12.5
they’d calculated the perimeter. This could be attributed to different possible
reasons:
23 | P a g e
-
A weakness with number caused his concepts of area and perimeter to falter.
-
Non-integer lengths had not been dealt with in the lesson sequence, so he
thought that the formula for area didn’t apply.
-
He thought that it was unlikely that he’d be expected to carry out that
calculation, and so the question must require something different.
Unfortunately it wasn’t possible to question him further. Nevertheless, despite the
drop in actual correct answers, the explanations revealed that the vast majority still
approached the question in the right way.
4.3 Application of the Formula for the Area of a Rectangle
Only one student had got question 3 wrong on the pre-test by multiplying the three
given lengths together. However, question 4 with an L-compound shape caused
significant problems, with many students just multiplying the two longest lengths.
This is a prime example of where a formula is being applied without understanding
(Huang & Witz 2009). After the sequence, this question showed the largest
improvement in responses.
Fig 7 shows slides that formed part of a quiz that sought to make explicit the links
between arrays’ multiplicative nature and area, which is an important step in
understanding the formula for the area of rectangles (Nunes et al. 1993; Huang &
Witz 2009).
24 | P a g e
Figure 7. Four slides that were shown to illustrate the progression from arrays to the area of
a rectangle. The actual slides included intermediate steps between these. The purpose was
to get the students to naturally use the multiplicative nature of the arrays and to see how that
relates to rectangles without being told. They had to answer how many objects there were,
but after the first they were not given enough time to count them all one by one. They would
therefore have to use a multiplicative method in order to answer the questions in the allotted
time.
After this quiz, which was marked, the student who had multiplied the three lengths
together for question 3 was asked if he could explain what his mistake had been:
Jacob: Yes…. I... I multiplied all 3 together but you have to just multiply these two
(pointing at the perpendicular lengths).
Me: Why do we have to multiply these two?
Jacob: Because… those two (pointing at the opposite parallel sides) are the same.
Me: Ah I see. What if I drew this rectangle? (I draw a square and label three of the
sides). How would I find the area now?
Jacob: Erm. I don’t know.
Me: …
Jacob: Multiply those sides’ together (points at the perpendicular lines).
Me: Why?
Jacob: Erm... Er... I don’t know.
This extract indicates that he still does not have an understanding of why the formula
for the area of a rectangle is length x width. However, in the post test, not only did he
get this question correct, he also now correctly found the area of the compound
25 | P a g e
shape, which requires more than just mechanistic use of a formula. It is possible that
he understood why the formula works, but is stuck in level 2 of Clements’ levels of
reasoning – he cannot even informally argue why it works, even though he knows,
and may or may not understand why - it does.
The next activity in the second lesson consisted of a worksheet extending and
consolidating this knowledge, shown in Appendix 4. The plenary of this second
lesson showed an imagined student’s attempt to find the area of an L shaped
compound shape by multiplying the lengths of the two longest sides together.
“Ohhhh… I did that,” commented one student. (Then shortly after), “But that just
finds the area of the squares for the whole rectangle.” (Referring to the enclosing
rectangle).
Again this comment indicates a more relational understanding of the concept of why
we multiply the lengths of the sides of the rectangle.
Although this lesson was successful in that 64% of the whole class changed from
answering this question incorrectly to correctly between the two tests, conversations
throughout the room such as the two examples above suggest different outcomes:
some students have reconstructed their existing schema of ideas to incorporate
meaning into the formula for the area of a rectangle, whereas others may have
merely been reminded of the procedure of how to find the area of compound
rectilinear shapes. Again, this shows that raw marks in a test are insufficient to gain
a good understanding of a students’ ZPD. Nevertheless, the results provide evidence
that teaching the formula for the area of a rectangle through the multiplicative
structure of arrays is an effective scaffold and good route for learning.
26 | P a g e
4.5 From Parallelogram to Triangle to Trapezium
The formula for the area of a triangle was found via the area of a parallelogram, as in
the sequence that was shown in Fig 1. The rationale for this is fairly simple: to
reason that the area of an obtuse triangle is half the area of a rectangle requires a
good understanding of the shear transformation. The area of an obtuse triangle as
half the area of a parallelogram does not. However, understanding this does require
secure knowledge of the area of a parallelogram. This is plausible for a group of year
8 students, but it is unlikely that this knowledge will be as secure as that of the area
of a rectangle. It may be that learning the area of a triangle via rectangles may lead
to a more secure understanding for non-obtuse triangles, but that the route via the
parallelogram s leads to a more secure understanding of the formula for triangles
that are. This is leant evidence by one students’ answer to Q6 in the post test where
they had drawn a rectangle rather than a parallelogram to justify how they’d found
the area (see Fig 8). Even though he had shown no such inclination on the pre-test
to suggest this was from prior teaching, and it had never been represented as such
in the lessons, he’d still found it easier to think of the triangle as half of a rectangle
rather than half of a parallelogram.
Figure 8. A student’s response to Q6 in the post test. He has drawn a rectangle to justify his
answer to the question. He had found the area to be 16 cm2. (The dotted lines forming a
parallelogram were drawn on by me whilst marking).
27 | P a g e
With practice students were able to find and use the perpendicular height, which was
aided by having had to use the perpendicular height of parallelograms in the
previous lesson, which is a positive feature of this sequencing. Constructing the
parallelogram with congruent triangles proved difficult for many students, with many
pairs struggling to get opposite sides completely parallel. However, this didn’t seem
to be a barrier to their understanding that the area of the triangle was half of that of
the parallelogram, and a few students were even trying to use a co-ordinate
system/vectors type approach to finding parallel lines, a unique positive gifted by
using geoboards. The teacher drew the class together and using a labelled picture of
a parallelogram split into triangles as shown in Fig 8, asked the students to find the
area and write it on their whiteboards. They were asked to share their answers with
adjacent students and change their answers if they wished. When the class showed
their answers, all but five students got the correct answer, which someone in the
class was then asked to explain:
“I timesed the 6 and the 4 together and then divided by 2.”
This was clarified and with prompting the student explained why they had ignored
the given slant height of 5 cm. They were then asked to write down the worded
formula for a triangle using the visual prompt shown in Fig 9, and finally repeated
with b and h representing base and perpendicular height respectively. The second
answer was written correctly – although sometimes in esoteric forms – by all but a
couple of students, although the third was slightly less well answered without about a
third of the class unable to form an algebraic formula. Nevertheless, it was
concluded that most of the class were able to find the area of a triangle.
Unfortunately, the homework giving practice and consolidating this learning –
especially when the triangles were drawn without helpful parallelograms enclosing
28 | P a g e
them - was not due in until the day following the test. This may explain why there
were a significant number of students who were still unable to correctly find the area
of a triangle in the post test. Despite this, for question 5 very few students used the
slant height at all in their calculations in comparison to the pre-test. In question 6,
many now found the perpendicular height correctly. However, it became clear that a
number of students tried to count the squares inside, and in the pre-test had done
this correctly, but in the post test (with a slightly taller triangle) did so incorrectly.
Therefore, the original mean mark for Q6 in the pre test may be slightly inflated.
Figure 9. The image on the left was the first shown on the interactive whiteboard. After
students had found the area of one of the triangles, the second was shown to help them
write the formula in words. The final image on the right was shown after this. The
parallelograms were tlted slightly to aid generality.
29 | P a g e
There was a further problem with this sequencing of parallelograms to triangles.
After having written out the formula in their books for the area of a triangle, including
the words:
“The area of a triangle is half the area of a parallelogram”
They were presented with what was intended to be an easy example, shown in Fig
10.
Figure 10. An “easy” example of a triangle to find the area of.
Controversy erupted.
“That’s not half a parallelogram! That’s half a rectangle!” (Accompanying jeers).
In fact, Samuel’s objection is valid for some definitions of parallelograms (PereiraMendoza 1984). This raises a significant issue in the teaching of the area of a
triangle via the area of a parallelogram. Students’ must have a thorough
understanding of classes of quadrilaterals, specifically that a rectangle is a special
case of a parallelogram. Otherwise, this can lead to problems with inconsistent – in
the students’ eyes – definitions of the area of a triangle. It may be that being able to
confidently work out the area of an obtuse triangle has been gotten at the expense of
not being able to work out the area of a right angled triangle! There appears to be
little research studying the learning outcomes of teaching the area of a triangle via a
30 | P a g e
parallelogram in comparison to via a rectangle. Without firm evidence suggesting
that one route to the area of a triangle is better than another, this might be an
appropriate question that could be answered by a large scale randomised trial, as
recently advocated by Goldacre (2013).
In this sequence, the area of a trapezium is also found via the area of a
parallelogram. Students create parallelograms on geoboards using two congruent
trapezia. Through a similar process to that for deriving the area of a triangle, the area
of a trapezium is derived. However, in this case students were much less confident.
Few students could form parallelograms from appropriate trapezia:
Me: You’ve tried to turn the trapezium into a parallelogram, but we need to make
both trapezia the same.
Charlie: I thought that was a parallelogram.
Me: It is but we need to make the two trapezia congruent, equal.
Charlie: Oh. Ummm. Like this..?
Me: Almost. Just a bit more (correcting it)...
Charlie: Oh okay.
Me: There. So what’s the area of this parallelogram?
Charlie: (Answers, correctly, after some time).
Me: Okay, so what have we done to find this length (that has length a +b).
Charlie: Ermmm… counted it?
Me: Yes, but what is that made up from? Look at the two trapezia.
Charlie: Errrr… (Baffled expression).
This is an example of one of many similar conversations during this lesson. As well
as being difficult to form the parallelogram, it was not obvious to them that the
trapezium was half its. Additionally, they couldn’t find the area of the parallelogram
algebraically when scaffolded in the same way as for the triangles on mini
whiteboards. After some time of this unsuccessful activity, the teacher gave the
formula to them and showed examples by turning trapezia into parallelograms by
31 | P a g e
drawing another trapezium. Despite making it explicit, students were unhappy that
the lengths of the parallel sides of the parallelograms formed by two trapezia were
given by a + b in the formula. Students were quiet and unwilling to contribute during
this class discussion, in hindsight because they did not understand. In the examples
following the teacher’s exposition, a few students could correctly substitute numbers
into the formula to produce correct answers.
The lack of success of this lesson can be seen by the post-test results whereby there
were no more correct responses than in the pre-test.
It could be that the task that had worked for the area of the triangle was not
appropriate for this, or that the teachers’ exposition was less clear. Clearly, more
scaffolding was needed for students to make the cognitive jump.
The final lesson was spent consolidating the area of a triangle – students had now
had more practice having done the homework - and teaching the area of a trapezium
in a different way.
The way taught was via the area of a rectangle – different to all three routes
mentioned by Pereira-Mendoza (1984). Students were asked to find the average of
12 and 16, which appeared to be easy:
“It’s 14.” (After prompting): “I added the two numbers together and then divided by 2.”
The teacher drew two horizontal parallel lines, the lower of which was slightly longer,
and labeled them 12 cm and 16 cm respectively. Students agreed that the average
length of the two lines was 14 cm. The teacher now drew a point midway between
the left edge of the two lines (see Fig 11):
“This line will be 14 cm long. Where will it end?”
32 | P a g e
A student duly pointed out that it would end midway between the ends of the other
two. By drawing the rectangle with the average length of the two parallel sides, a
rectangle has been formed with exactly the same area as the trapezium, which can
be understood by using the concept of conservation of area. There were a few
exclamations of “Ohhhhh I get it!” when this was done, and in the subsequent
worksheet questions students by and large got the correct answers after marking,
which was judged visually after using the simple instruction:
“Of the first 5 questions, put up the number of fingers on your hand that you got
right.”
Figure 11. The diagram on the left was drawn and at this point a student was asked when
the (middle) line would stop if it was to be 14 cm. This was turned into the middle diagram, to
show how a rectangle formed with length (1/2)(a+b) could be used to find the area of the
trapezium. The diagram on the right shows that the area included in the rectangle is equal to
the area it doesn’t include of the trapezium.
One cannot conclude from this that the second method of teaching the area of a
parallelogram was better than the first – this was the second lesson devoted to it.
The issue it does raise is that the route by which the area of a trapezium is best
taught – just like for a triangle – is not sufficiently well supported in the literature. To
paraphrase the description of Tall (1990):
33 | P a g e
‘Professional mathematicians (often believe) that the best way to help students is to
present the materials in (the most) logical and coherent manner… but my
investigations reveal fundamental inadequacies into students’ conceptions. The
implicit and explicit agreements between mathematicians of what is most
mathematically accurate and coherent can cause further difficulties to students, and
may be presented in a sequence that is inappropriate for cognitive development.’
(Tall, 1990:49-50)
I believe that understanding the area of a trapezium as half of the parallelogram
formed by two congruent trapezia falls into this category – mathematically coherent
and accurate, but conceptually more difficult. The best way to test this belief would
be – as stated above – via controlled randomised trials.
4.6 Success of the Sequence
The success of the sequence must be measured in terms of students’ outcomes.
This has already been discussed explicitly for many of the questions and concepts
asked in the test. It is useful to look at students’ individual test results, summarised
by Fig 12.
34 | P a g e
Change in Correct Responses to
test
Change in Score for each Student
10
8
6
4
2
0
-2
-4
1
3
5
7
9
11
13
Student
15
17
19 21
Increase
Pre test
Post test
Figure 12. The red line shows students’ results on the pre-test, the green line shows the
results on the post-test and the blue line shows the increase in marks between the two tests.
Most students improved their results, especially considering the extra difficulties of questions
6, 9 and 10. (Q6: squares inside could no longer be counted, Q9: much improved written
explanations but more difficult question, Q10: understanding of issue present but actual
calculations proving too difficult). These changes explain 5 of the 6 students who showed no
improvement in performance: analysis of their scripts showed that all had actually improved
on 1 or more questions, but had also lost marks on Q6, 9 and 10 for reasons stated above.
Student 16 got all questions other than Q8 on the area of a trapezium correct on both tests.
However, this does not mean he made no progress: in fact, throughout the sequence of
lessons he was given extension work on compound shapes, and this learning is not picked
up in this test.
The graph shows that most students moved forward in their learning over this
sequence. I would suggest that the learning moved forward more than is suggested
by this graph, for reasons relating to specific questions mentioned earlier, as well as
35 | P a g e
that they each completed and marked homework on the area of a triangle, and had
an additional lesson on the area of a trapezium.
5. Conclusion
This sequence of lessons on area was successful in improving students’
understanding of the concept of conservation of area. Additionally, it improved
understanding of and ability to use appropriately the formula for the area of a
rectangle, parallelogram and triangle. This can be justified by the improvement in
results on the test, conversations with students in lessons, and answers to questions
– verbal or on mini whiteboards – during classroom discussions. Classroom work in
the final lesson following the post-test suggested that many students also
progressed in their ability to find the area of trapezia.
Ideally a retention test could be carried out after a number of weeks to see what was
retained, although this was not possible on this occasion.
Data gathered from students suggests that teaching the formula for the area of a
rectangle through making explicit the multiplicative nature of arrays and their link with
area is a good strategy. Similarly, that the idea of conservation of area is a good
basis for the area of a trapezium. It shows support that the area of a triangle can be
taught successfully via the area of a parallelogram, but does not give evidence that
the area of a trapezium can be successfully taught via the area of a parallelogram
(although it does not refute it either). Crucially, this one study and evaluation cannot
meaningfully comment on how effective this progression of ideas is for student
outcomes in comparison to other routes. To do this, many routes – such as those
suggested by Pereira-Mendoza (1984) - as well as others such as that taught in
lesson six of this sequence must be tried by many different teachers. Only once this
36 | P a g e
is done in a controlled fashion can meaningful data be produced to show which
sequence of ideas is the most effective way to present the area of these shapes to
the subset of the population that these tests are representative of.
References
Anderson, D.R. & Arcidiacono, M.J., 2014. Area Ratios of Quadrilaterals.
, 82(3), pp.176–184.
Beattys, C.B. & Maher, C.A., 1985. Approaches to Learning Area
Measurement and its Relation to Spatial Skill.
Black, P. et al., 2004. Working Inside the Black Box: Assessment for
Learning in the Classroom. , 86(September), pp.8–21.
Black, P. & Wiliam, D., 1998. Inside the Black Box: Raising Standards
Through Classroom Assessment. Phi Delta Kappan.
Boaler, J., 2009. The Elephant in the Classroom: Helping Children Learn
and Love Maths,
Bruner, J.S., 2009. Actual minds, possible worlds., Harvard University
Press.
Bryant, B.P., 2007. Paper 5: Understanding space and its representation
in mathematics.
Cass, M. et al., 2003. Effects of Manipulative Instruction on Solving Area
and Perimeter Problems by Students with Learning Disabilities.
Learning Disabilities Research and Practice, 18(2), pp.112–120.
Available at: http://doi.wiley.com/10.1111/1540-5826.00067.
Clements, D.H. et al., 1997. Development of Students ’ Spatial Thinking
in a Unit on Geometric Motions and Area. , 98(2), pp.171–186.
Confrey, J., 1990. What Constructivism Implies for Teaching. Journal for
Research in Mathematics Educatin, 4(1990), pp.107–122.
37 | P a g e
DfE, 2013. Mathematics programmes of study : key stage 3 National
curriculum in England,
Donaldson, M., 1978. Children’s Minds, London: Fontana.
Erez, M.M. & Yerushalmy, M., 2007. “If You Can Turn a Rectangle into a
Square, You Can Turn a Square into a Rectangle ...” Young
Students Experience the Dragging Tool. International Journal of
Computers for Mathematical Learning, 11(3), pp.271–299. Available
at: http://link.springer.com/10.1007/s10758-006-9106-7 [Accessed
December 13, 2014].
French, D., 2004. Teaching and learning geometry,
Fuys, D., Geddes, D. & Tischler, R., 1988a. The Van Hiele Model of
Thinking in Geometry among Adolescents. Journal for Research in
Mathematics Education. Monograph, 3, pp.i–196 CR – Copyright
© 1988 National Council. Available at:
http://www.jstor.org/stable/749957.
Fuys, D., Geddes, D. & Tischler, R., 1988b. The van Hiele Model of
Thinking in Geometry Among Adolescents. Journal for Research in
Mathematics Education, 3, pp.44–68.
Goldacre, B., 2013. Building Evidence Into Education, Available at:
https://www.gov.uk/government/news/building-evidence-intoeducation.
Herbst, P.G., 2003. Using Novel Tasks in Teaching Mathematics: Three
Tensions Affecting the Work of the Teacher. American Educational
Research Journal, 40(1), pp.197–238. Available at:
http://www.jstor.org/stable/3699431.
Hiebert, J., 1981. Units of Measure : Results and from Implications
National Assessment. The Arithmetic Teacher, 28(6), pp.38–43.
Hirstein, J., 1978. Student Misconceptions About Area Measure.
Arithmetic Teacher, 25.6(10-16).
Huang, H.-M.E. & Witz, K.G., 2012. Children’s Conceptions of Area
Measurement and Their Strategies for Solving Area Measurement
Problems. Journal of Curriculum and Teaching, 2(1), pp.10–26.
Available at:
38 | P a g e
http://www.sciedu.ca/journal/index.php/jct/article/view/2041
[Accessed December 19, 2014].
Huang, H.-M.E. & Witz, K.G., 2009. Developing children’s conceptual
understanding of area measurement: A curriculum and teaching
experiment. Learning and Instruction, 21(1), pp.1–13. Available at:
http://linkinghub.elsevier.com/retrieve/pii/S0959475209000930
[Accessed December 3, 2014].
Johnstone-Wilder, S. & Mason, J., 2005. Developing Thinking in
Geometry,
Jones, K., 2012. Promoting productive reasoning in the teaching of
geometry in lower secondary school: Towards a future research
agenda.
Kamii, C. & Kysh, J., 2006. The difficulty of “length×width”: Is a square
the unit of measurement? The Journal of Mathematical Behavior,
25(2), pp.105–115. Available at:
http://linkinghub.elsevier.com/retrieve/pii/S0732312306000125
[Accessed December 12, 2014].
Kordaki, M. & Balomenou, A., 2006. Challenging Students to View the
Concept of Area in Triangles in a Broad Context: Exploiting the
Features of Cabri-II. International Journal of Computers for
Mathematical Learning, 11(1), pp.99–135. Available at:
http://link.springer.com/10.1007/s10758-005-5380-z [Accessed
December 12, 2014].
Kordaki, M. & Potari, D., 1998. A learning environment for the
conservation of area and its measurement: a computer microworld.
Computers & Education, 31(4), pp.405–422. Available at:
http://linkinghub.elsevier.com/retrieve/pii/S0360131598000505.
Kospentaris, G., Spyrou, P. & Lappas, D., 2011. Exploring students’
strategies in area conservation geometrical tasks. Educational
Studies in Mathematics, 77(1), pp.105–127. Available at:
http://link.springer.com/10.1007/s10649-011-9303-8 [Accessed
December 13, 2014].
Koyuncu, I., Akyuz, D. & Cakiroglu, E., 2014. Investigating plane
geometry problem-solving strategies of prospective mathematics
teachers in technology and paper-and-pencil environments. ,
(January).
39 | P a g e
Malkevitch, J., 2009. What is geometry? In Understanding geometry for
a changing world. pp. 3–15.
Mayer, R.E., 2004. Should there be a three-strikes rule against pure
discovery learning? American Psychologist, 59(14).
Nunes, T., Light, P. & Mason, J., 1993. Tools for thought: the
measurement of length and area. Learning and Instruction, 3(1),
pp.39–54. Available at:
http://linkinghub.elsevier.com/retrieve/pii/S0959475209800042
[Accessed December 12, 2014].
Pereira-Mendoza, L., 1984. Developing Area Formulas. , 84(April 1984),
pp.297–307.
Piaget, J., 1952. The origins of intelligence in children, New York:
International Universities Press.
Piaget, J., Inhelder, B. & Szeminska, A., 1960. The child’s conception of
geometry E. a. Translated from French by Lunza, ed., Routledge and
Kegan Paul.
Powell, K. & Cody, K., 2009. Cognitive and social constructivism:
Developing tools for an effective classroom. Education, 130(241).
Reisa, Z.A., 2010. Computer supported mathematics with Geogebra.
Procedia - Social and Behavioral Sciences, 9, pp.1449–1455.
Available at:
http://linkinghub.elsevier.com/retrieve/pii/S1877042810024535
[Accessed December 8, 2014].
Rickard, A., 1996. Connections and Confusion : Teaching Perimeter and
Area With a Problem-Solving Oriented Unit. , 327, pp.303–327.
Ruthven, K., Hennessy, S. & Deaney, R., 2008. Constructions of
dynamic geometry: A study of the interpretative flexibility of
educational software in classroom practice. Computers & Education,
51(1), pp.297–317. Available at:
http://linkinghub.elsevier.com/retrieve/pii/S0360131507000553
[Accessed October 6, 2014].
Ryan, K. & Cooper, J.M., 2010. Those Who Can, Teach,
40 | P a g e
Sowell, E.J., 1989. EFFECTS OF MANIPULATIVE MATERIALS IN
MATHEMATICS INSTRUCTION. Journal for Research in
Mathematical Education, 20(5), pp.498–505.
Spitler, B.G., 1982. The Shear Joy of Area. , 29(8), pp.36–38.
Steele, M.D., 2012. Exploring the mathematical knowledge for teaching
geometry and measurement through the design and use of rich
assessment tasks. Journal of Mathematics Teacher Education,
16(4), pp.245–268. Available at:
http://link.springer.com/10.1007/s10857-012-9230-3 [Accessed
December 13, 2014].
Tall, D., 1990. Inconsistencies in the Learning of Calculus and Analysis.
, 63, pp.49–63.
Vygotsky, L., 1980. Mind in society: The development of higher
psychological processes, Harvard University Press.
Walcott, C., Mohr, D. & Kastberg, S.E., 2009. Making sense of shape:
An analysis of children’s written responses. The Journal of
Mathematical Behavior, 28(1), pp.30–40. Available at:
http://linkinghub.elsevier.com/retrieve/pii/S0732312309000194
[Accessed December 12, 2014].
Waring, S., 2000. Can you prove it? Developing concepts of proof in
primary and secondary schools, The Mathematical Association.
Watson, A., Jones, K. & Pratt, D., 2013. Key Ideas in Teaching
Mathematics,
Wiliam, D., 2011. What is assessment for learning? Studies in
Educational Evaluation, 37(1), pp.3–14. Available at:
http://linkinghub.elsevier.com/retrieve/pii/S0191491X11000149
[Accessed July 9, 2014].
Woodward, E., 1983. Area : Included Topic , Neglected Concept. ,
83(April), pp.343–347.
Zacharos, K., 2006. Prevailing educational practices for area
measurement and students’ failure in measuring areas. The Journal
of Mathematical Behavior, 25(3), pp.224–239. Available at:
http://linkinghub.elsevier.com/retrieve/pii/S0732312306000332
[Accessed December 12, 2014].
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Appendix
1. Conversation during lesson one task comparing the area of rectangles by
overlapping:
Riley: Which do you predict?
Samuel: Which has got the most surface area?
Daniel: I dunno… Which do you think..?
(Me): Could we find out?
Samuel: (Putting one on top of the other, and gesticulating) Because these bits are sticking
out… we can slice that and put it there…
(They put rectangles on top of each other and compare the bits sticking out by eye).
Riley: Cut this (A) and try it…
(Me): Have you decided any?
Daniel: We think D is first… (to the others) Let’s just cut it… It’s E it’s E!
Samuel: This then (putting D top which he’d just shown A to fit inside of it)
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(Me): (Taking cut out piece of A and putting it alongside A to make an L shape). Does this
shape I’ve made have the same area as it had originally?
Daniel: No… wait… yeah yeah it does
(Agreement from the other two)
Me: Why?
Riley: It’s just the… it’s ‘cause you’ve not like… done anything to it…
Daniel: Yeah ‘cause before it was there and now it’s there but it’s still the same size.
(Me): (taking piece and putting it alongside it so that it now obviously overlaps a little bit)
What about now?
Daniel: No no it can’t overlap
(Me): Why?
Daniel: Cause then… cause you’ve got to count this bit (points at the overlapped region)
cause otherwise you… you need to count that bit too.
(After they’ve come to a consensus on the order of some of the rectangles)
(Me): So… even though they have the same perimeter, you think that these two rectangles
(pointing at A and D) have different area?
Samuel: Yeah…
(Me): But they have the same perimeters. Doesn’t that make a difference?
Samuel:… no it doesn’t make any difference!
2. Worksheets showing rectangles for overlapping task in lesson one:
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3. Worksheet for area of a rectangle used in lesson two.
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4. Worksheet for area of a parallelogram used in lesson three.
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5. Worksheet to encourage seeing the base of a triangle as any side:
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6. Homework on the area of a triangle:
7. Post test given between the fifth and sixth lessons
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8. Class discussion of how to compare two areas.
A question on the board showed two squares. It said:
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‘You have no ruler and no formula. How could you show that the (smaller) square on the
right has a smaller area than the square on the left?’
Students were initially bemused at this question, with responses including:
““It’s obvious,” “You could just look at it!” “You just know!”
The question was too easy, and as it’s on a computer screen the idea of moving
the shapes is not obvious. Undeterred, the teacher showed another example, this
time with a square and a long rectangle of smaller area, but that was less
obviously smaller. Students looked at it with slightly more interest.
“It’s still smaller… I think both sides are still smaller…”
At this point the teacher took the opportunity afforded by this response to show
that the rectangles could be moved. The rectangle was moved on top of the
square. Some of the rectangle could not fit in the square, but the amount
protruding was small.
“It’s still obvious... that bit sticking out is less… less than the area left of the square.”
The teacher then asked whether if we were to cut that part of the rectangle out
and place it within the square, it would show the area of the rectangle was less
than the square. There was general agreement.
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