Geometry Resource portfolio- Contents
2-4
Using posters as a pedagogical tool: Phil Loweth
5-7
Using textbooks as a pedagogical tool : Celia Trevan
8-9 ICT which teachers use :Áine Corrigan
10-11 ICT which teachers use: Yi Wang
12-14 ICT used by teachers : Rachel Murgatroyd
15-17 ICT used by teachers : Joe Watson
18-21 Physical resources: Joe Stuart
1
Using Posters as a pedagogical tool
Phil Loweth
‘While there is great demand for such posters by classroom teachers, too frequently their
educational value is unrealized and many are simply forgotten “pretty pictures” on the wall.’
(Hubenthal, 2009)
This quote from Hubenthal suggests that although posters are omnipresent throughout our
educational system, little credence is given to their power as a pedagogical tool. It would
seem as though many posters serve a purely aesthetic purpose, to brighten up an otherwise
empty wall. Glorified wallpaper, if you will, that is purchased, put on the wall and then
forever ignored. However, one could argue that the process of creating posters allows
learners to exercise their own creativity within a given task and make links that they may
otherwise not have made.
The creation of posters by learners within given parameters allows them to explore a topic
for themselves and Bonwell et al. (1991) make a case for the promotion of active learning. In
their article ‘Active learning: Creating excitement in the classroom’, they give ‘some of the
major characteristics associated with active learning strategies’. They say that ‘students are
involved in more than passive listening’ which can help learners retain information. In a
lecture environment, much of the information can be lost and actively engaging in a task can
assist with the integration of knowledge. If a learner lacks interest then it is highly unlikely
that they will remember the information. They also argue that ‘ student motivation is
increased’ and ‘students can receive immediate feedback from their instructor’. These are
incredibly important within a classroom environment. Getting learners motivated, often
through feedback, is essential in getting them to progress and want to progress.
So in a general context, the creation of posters can boost student motivation and increase
the amount of feedback that learners receive. Looking at the advantages of posters in a
more mathematics specific context. Morgen et al. (2004) claim that ‘Mathematics is
sometimes taught and learnt as a set of discrete facts and procedures, each of which is
encountered and memorized separately.’ Of course mathematics is far from a ‘set of
discrete facts’ and the interconnectedness, not only between different mathematical topics
but also with other subjects, is vast. Posters can be used to collate information and although
the poster may be on a specific topic, there is still the scope to include other related
information and draw knowledge from other areas of mathematics, or outside the subject,
which the learners are aware of. Creating links between information that you are learning
and knowledge that you already have can increase retention.
Jones (2002) argues that, ‘geometry appeals to our visual, aesthetic and intuitive senses.’
Much of mathematics, geometry in particular, is incredibly visual and often one requires a
gradual escalation of key visual concepts to fully understand the entire concept. The main
positives of posters being that they are entirely visual and designed to be aesthetically
2
pleasing would suggest that utilising the creation of posters for the teaching of geometry
seems entirely appropriate.
During a research lesson at a comprehensive secondary school in South London. Year 7
learners in the top set had been doing a unit on polygons and tessellations. They were asked
to create posters using a polygon of their choice that will tessellate. The posters allowed the
learners to include all the information about their chosen polygon that they had learnt in
the unit. So it was a collation of everything that they had been taught and it also gave them
the opportunity to expand the topic. Many of the learners were practicing Muslims and had
noticed that there were a lot of tessellating patterns in mosques. This led to the information
that they use these patterns to decorate, as they are not allowed to create graven images
like they do in Christianity and many other religions. Some of the learners decided to put
this information on their posters and it provided a cross curricular link and, possibly more
importantly, a very personal link between the subject content and their lives.
Another research lesson in the same South London comprehensive, foundation set learners
in year 10 were doing a unit on circle theorems. The lesson involved creating posters to
show the ‘Double Angle Theorem’, which says that the angle at the centre of a circle is
double the angle at the circumference. The learners were all given three pieces of coloured
paper with a representation of the same double angle theorem on them. Two of the angles
at the circumference could be fitted into the angle in the middle. All the learners got
different sized angles to do it with and this created a number of very visual examples
showing that the theorem worked. Even without a rigorous proof the learners could now
see that the theorem held no matter the angle. Many learners went on to expand the
information on their individual posters including relating it to the language of circles. Both
lessons allowed valuable time for feedback for the learners. Not only from the class teacher
but also from their peers to say in which areas the work could be improved.
Jones (2002) stressed the importance of displays and creativity within the classroom whilst
teaching geometry. ‘Use your imagination and tap into that of your pupils. Create striking
classroom displays, suspend geometrical models from the ceiling of your classroom, involve
your pupils in making things and imagining things, get them to decide on definitions and
then explore the logical consequences.’
In conclusion, the use of posters as a pedagogical tool for teaching mathematics, and in
particular geometry, has many benefits. It can be used to collate all the information that the
learners know on a particular topic in a creative way. It can also be used to expand the topic
into other areas of mathematics and even other subjects whilst boosting learner
participation and motivation. So perhaps in future posters will be seen as a legitimate
pedagogic tool and more than just ‘forgotten “pretty pictures” on the wall.’ (Hubenthal,
2009)
3
References:
Hubenthal, M. (2009). Revisiting your classroom's walls: The pedagogical power of posters.
Available:
http://www.iris.edu/hq/files/programs/education_and_outreach/poster_pilot/Poster_Guid
e_v2a.pdf. Last accessed 30th Nov 2013.
Bonwell, Charles C., and James A. Eison. Active learning: Creating excitement in the
classroom. Washington, DC: School of Education and Human Development, George
Washington University, 1991.
Jones, K. (2002), Issues in the Teaching and Learning of Geometry. In: Linda Haggarty (Ed),
Aspects of Teaching Secondary Mathematics: perspectives on practice. London:
RoutledgeFalmer. Chapter 8, pp 121-139.
Morgan, C, Watson, A & Tickly, C. (2004). In: Teaching School Subjects 11-19: Mathematics.
London: Routledge/Falmer. Chapter 4, pp 48-68.
4
Using textbooks to set up, support and sustain student’s engagement with geometry
Celia Trevan
Textbooks are regular resources in Secondary schools, particular in Mathematics lessons.
Due to the amount that they are relied on, it is extremely important to critically analyse
them and question whether or not they give pupils opportunities to learn rich mathematics.
It is the responsibility of the teacher to judge the textbooks they choose, with the pupils
learning as the only priority. In English schools it is increasingly difficult for teachers to find
time to prepare lessons and will use textbooks to define the boundaries of their lesson
sequences (Haggarty & Pepin, 2002). It is therefore essential to question their advantages in
the classroom.
The exercises that textbooks can offer for pupils to practice mathematical methods are one
of the main features that make them an ideal resource. The teacher should choose exercises
that will fulfil the purpose of the lesson and sufficiently challenge students. As well as being
stimulating for pupils, it is important to have a variety of work in the textbooks. Textbooks in
England include investigations, but were not integral and would be used marginally [H&P].
The exploratory nature of investigations is essential to many topics in Mathematics, and
therefore teachers in the UK should be conscious to try to include investigative thinking into
their lessons.
Another feature of textbooks that is important to consider is the use language and of
diagrams. Textbooks used in England were found to have ‘no language curriculum reflected
in the textbooks’ (Haggarty & Pepin, 2002). To progress in studying mathematics, language
is used for communicating ideas and explanations. There are variances in the use of
language, diagrams and practices between different textbooks. Paul Dowling found
contrasting practice for different students, stating that ‘high ability students get
mathematics whilst low ability pupils get ad hoc principles that lay claim to, but clearly do
not constitute official school knowledge’ (Dowling, 2001). Here he clearly highlights that a
different language and style of practice are present in many textbooks, which does not
advantage the ‘low ability’ students.
Textbooks can be useful to aid lessons by testing pupils through individual exercises. in
French school were found to give students the opportunity to engage in the process of
doing mathematics by being given challenging mathematics with stimulating exercises,
because the intention of the textbooks was to help pupils understand mathematical
concepts, ‘through guided discovery’ and ‘through practice to achieve fluency in the use of
these concepts’ (Haggarty & Pepin, 2002). Real life examples are often offered to pupils in
textbooks in an attempt to engage students, for example by examining objects that are
triangular in shape. Dowling argues that it may mean that students ‘may not fully recognise
that a game is being played,…,let alone recognise the rules by which they have to play’
(Dowling, 2001). While teachers are in the knowledge that mathematics is not ‘all around
5
us’ as some textbooks lay claim, they teach students that it is, but ‘a world of physical
shapes and manual activity is constructed as the public domain of school geometry’
(Dowling, 2001).
Textbooks are often set out in a linear way, with chapters in no particular order, indicating
that the topics stand alone and the links between them seem coincidental. Textbooks can
hold the risk of being a static body of information, consisting of symbols and rules that were
‘immutable and true’ (Haggarty & Pepin, 2002). Practicing routine skills and the use of
Mathematics as a ‘bag of tools’ (Haggarty & Pepin, 2002) can be seen as key features of a
Mathematics textbook, but we should focus on the textbooks (particularly in France) which
emphasise the developmental aspects of mathematical knowledge. Every textbook that I
have come across has a contents page at the start indicating the chapters that topics belong
to, which give the impression of a body of knowledge which is linearly organised.
I taught a lesson to a year 8 class and used a textbook for practice. The objective for my
lesson was to know the sum of interior and exterior angles of a polygon. We looked at a
triangle, quadrilateral, pentagon and hexagon on the board and using dynamic software, the
angles were measured and we could calculate the sum of the interior angles. The students
noted down their results and discussed whether they could come up with a general formula
for the number of triangles you could inscribe in an n sided shape, and what the interior
angles summed to. The textbook offered this in the ‘Discussion’ point of the chapter. There
was also a practical activity and written exercises offered to students.
One of the issues with the textbook was the order of the questions. The first question asked
students how many triangles a shape can be divided into and the questions that followed
were unrelated. It wasn’t until question 7 that they were asked to put this into use, and
prove the sum of interior angles of polygons. This is a good example of where the teacher
would have to pick these questions out to be done consecutively, in order not to run the risk
of some pupils doing question 1 with no outcome.
Figure 3: Extract from Framework Mathematics 8+ p.270-271
6
The textbook was the favoured option of orientation for this task, because the students
could practice the formula they had just found in question 2. Questions 4 and 5 gave them
the opportunity to test their knowledge further.
One of the disadvantages of textbooks is that they all have their own narrative, which is set
out in a linear way. Textbooks can guide pupils through discovery to arrive at concepts,
particularly in Geometry where students have ‘known facts’ which they can deduce facts
from. However this depends on the textbook being used. It is evident from the study by
Haggarty and Pepin that the opportunities to study rich mathematics depends on the view
of Mathematics held by a countries department of education. It is up to the schools and
teachers to choose the textbooks they use wisely. With the main aim to discourage the
limits that textbooks can put on pupils, whether it is a limit on the content of the
mathematics in the book, or a limit on how it is being explained.
Bibliography
Dowling, P., 2001. Reading mathematics texts. In: Issues in Mathematics Teaching.
s.l.:Routledge.
Haggarty, L. & Pepin, B., 2002. An Investigation of Mathematics Textbooks and their Use in
English, French and German Classrooms: who gets an opportunity to learn what?. British
Educational Research Journal, 28(4), pp. 567-590.
7
ICT used by teachers
Áine Corrigan
This report will analyse a number of different ICT resources used in the classroom to support
learning. The report will consider interactive whiteboards, PowerPoint and the Internet,
used within year 7, 9 and 10 class groups.
The year 7 group during the proceeding class had worked on probability and a number of
experiments, including rolling dice to find its relative frequency. The teacher intended to
use ICT in the form of the internet to demonstrate that the relative frequency can be
different every time you redo an experiment, however the more trials run, the closer the
relative frequency gets to the theoretical probability. The teacher used a dice rolling
simulation from the Internet, which was projected on the board. The simulation enabled the
teacher to choose between a biased or unbiased dice, manage the number of times the dice
is rolled and graphically display the probability of each event occurring. The teacher 1 st
chose a biased dice and allowed the diced role 6 times, the output was displayed on the
board for the class to see. The class was then asked whether they thought the dice was
biased or unbiased, and the number of students that chose each option was recorded on
the board. The experiment was then continued to 20 roles. The teacher again asked the
class their opinion and recorded the results. Finally the dice rolled so that 300 outcomes
were recorded and the class made their final decision. The process was then repeated with
an un-biased dice.
The use of the resource in this way appeared to be very effective. Resource was used to
build upon prior knowledge and generate large amounts of data to demonstrate the more
trials run the closer the relative frequency gets to the theoretical probability. Had this
resource not been available this five-minute task would have taken much longer. The
students enjoyed watching the dice roll, trying to guess the correct answer and debating the
correct answer after the 1st 6 roles. The Internet enabled the teacher display large amounts
of data being gathered in real time in an easy, fun and interesting way, which is a very
important factor for teachers decision to use ICT in the classroom (Cox, 2000).
The Internet was also used with the year 9 group. The group had previously been working
on indices and during this lesson were introduced to standard form. The ICT resource/the
Internet was used to demonstrate why indices and standard form are important and where
they are used today. The teacher displayed on the board an image of the Milky Way as 10
million light years from the Earth, she was then able to decrease the view magnitude
zooming in on the Milky Way galaxy, our solar system, the Earth, the United States, Florida,
and oak leaf, the cells on a leaf surface, all the way down to strands of DNA, carbon atoms
and quarks. The students loved the visual demonstration, they could also see the uses for
indices and standard form not just in maths but also in their other subjects like science and
astronomy.
8
Again this use of ICT was well received by the students. The demonstration enabled
students to see the links between maths science and astronomy. There was a number of
gasps and impressed sounds as the demonstration went from positive indices showing a leaf
to negative indices showing cell structures. Following the demonstration students also
asked for the URL of the website, this perhaps indicating an interest in the demonstration
motivating students to investigate the topic further.
The ICT resource used during the year 10 class was PowerPoint and the interactive
whiteboard. The teacher had planned to present the lesson using a PowerPoint
presentation, within which there was a number of 2-D and 3-D diagrams as well as tables.
The teacher had intended to annotate and fill in the tables on the interactive whiteboard.
Unfortunately 10 minutes before the lesson there was a network outage in the school. As a
result there was no PowerPoint and the teacher needed to draw the diagrams and tables on
the board. This took more time than the teacher had planned, which meant less material
was covered in the class. Also the teacher had forgot to label one of the sides of a shape,
which caused some confusion for the students. The teacher did spot this mistake and
corrected it but 10 minutes had elapsed in the meantime. This was not an effective use of
resource. The teacher was reliant on the network and ICT, and these are never 100%
guaranteed. The teacher should have had a backup plan, and been more observant when
drawing diagrams. As ICT is not a “unproblematic innovation that will somehow lead to
enhanced learning” (Sutherland et al., 2004), back up plans are essential should the ICT fail.
The teachers use of ICT during the year 7 and year 9 classes was successful because teacher
used the technology to support the teaching and learning within the class, the ICT was “not
a catalyst for change but rather it’s tool” (Watson, 2001). Both of these lessons considered
the lesson objectives and how ICT could be used to enhance their students learning.
Cox, M. J. ;Cox (2000) What factors support or prevent teachers from using ICT in their
classrooms? [online]. Available from:
http://www.leeds.ac.uk/educol/documents/00001304.htm (Accessed 24 November 2013).
Sutherland, R. et al. (2004) Transforming teaching and learning: embedding ICT into
everyday classroom practices: Transforming teaching and learning. Journal of Computer
Assisted Learning. [Online] 20 (6), 413–425.
Watson, D. M. (2001) Pedagogy before Technology: Re-thinking the Relationship between
ICT and Teaching. Education and Information Technologies. [Online] 6 (4), 251–266.
9
Use of teacher’s ICT in school classroom
Yi Wang
The use of information and communication technology (ICT) in teaching mathematics is
commonly seen these days. Introducing interactive white boards (IWBs) into the classroom
aims to achieve interactive learning by bringing technology and pedagogy together. It has
been shown in the literature that IWBs can best be used in the secondary school (Moss et
al., 2007). In this article, I will focus on the use of interactive whiteboards (IWBs) as a
pedagogical tool in educational settings.
The school I am currently teaching at is a technology specialized school. Every teacher is
given a laptop and there are IWBs in each of the maths classrooms. Maths teachers are
expected to use IWBs in every single lesson. There are two main pedagogic uses of IWBs:
text design-displaying, and interactive activities.
Text design remains a key activity for the majority (78%) of teachers using IWBs. (C. Jewitt et
al., 2007). Here are some of the positive effects that IWB text designs bring in. Firstly,
introducing IWBs into classrooms widely extends the potential for text design. It brings
colorful images, sounds and videos into the lesson. For instance, in a lesson on deriving
formulae, I showed a video clip of the story about Carl Friedrich Gauss solving the problem
of adding 1 to 100, and asked the students to derive the formula for calculating the sum.
The students all concentrated fully while watching the video and engaging in the discussions
afterwards. This could not have been done if we were in a traditional classroom with only a
blackboard in the front. In contrast, some teachers use IWBs in much of the same ways as
they would use a traditional blackboard. They write notes on the board from memory and
teach from the front. This is more common for older and more experienced teachers. They
are more confident in using IWBs this way, as they do not need to spend hours preparing
the presentation before hand. Secondly, using IWB display also provides consistency for the
lessons. For example, I create a PowerPoint slideshow for all my lessons. The first slide
always contains the dates, learning objectives and all the key words clearly laid out. It is
shown on the board as students are coming in, so that they can sit down quietly and start to
copy these into their book while I take the register and wait for the students to be ready for
the lesson to start. The color and font of the words can be easily adjusted to meet the needs
of the class, especially where there are EAL and SEN students in the class. This is especially
beneficial to the students if the teacher’s handwriting is unclear. Thirdly, IWBs allow the
teacher to move between screens and applications. In geometry lessons, dynamic software
is very helpful for the students’ learning. When I taught my students the sum of exterior
angles of polygons I switched between my slideshows and GeoGebra constantly. This
requires the teacher to practice before the lesson so that he/she can move between
applications and linked materials easily. Moreover, having the slideshow ready for the
lesson gives the teacher confidence in delivering the lesson as well as tracking the pace. This
is especially true for new teachers, when they struggle with time management. The number
of the slide shows can give them a rough guide of which stage they are at in the lesson.
Furthermore, everything that has been written on the board can be saved as a file at the
end of the lesson. This allows me to evaluate the lesson effectively afterwards and shape my
future lesson planning. If I want to refer to anything that has been taught in the previous
lesson, I can just open the file and show it to my students.
10
IWBs are also intended to be used for interactive activities . There are a lot of interactive
games, which are designed for this purpose. For example, in the fraction lesson, I asked
students to come up to the front to drag the shape with shaded areas into the correct
fraction boxes. There are also card-pairing games for practicing decimals and fractions.
Students love this way of learning, as everyone is actively involved in the lesson. However,
behavior management needs to be taken into account while planning these kinds of
activities. The teacher also needs to be fair to the students by giving everyone a chance to
take apart in the activity. Moreover, since this kind of activities is heavily dependent on the
technology, if one particular interactive activity is planned as an important aspect of the
lesson and a technical failure occurs such as the laptop stops working or the IWB loses
connection, the whole lesson plan is instantly disrupted. The gaps and delays it causes can
be quite costly. Therefore, back up plans are always needed for interactive activities.
In conclusion, IWBs can support a teacher’s interactive teaching in the classroom. However,
the teacher must decide how IWBs should be used to meet the needs for a particular
learning objective and for a particular group of students. For instance, in the algebra lesson
of expanding bracelets, I would list all the expressions in my slideshow., copy them onto the
normal whiteboard which is next to the IWB and show the working to the students, rather
than preparing all the answers in my slides. This method makes the steps of multiplying out
and collecting the like terms together clearer to the students. The best teachers know how
to use a variety of resources along with the IWB to stimulate pupils’ discussion and generate
their individual thinking.(D. Glover et al.,2007) Otherwise the IWB will just be seen as a
modern version of the traditional blackboard.
In other words, IWBs ‘should be used in unique and creative ways otherwise it will not make
any difference to using the normal whiteboards or other projection methods.’ (H. J. Smith et
al., 2008).
References
Derek Glover, David Miller, Douglas Averis and Victoria Door (2007) ‘The evolution of an
effective pedagogy for teachers using the interactive whiteboard in mathematics and
modern languages: an empirical analysis from the secondary sector?’, Learning, Media and
Technology 32(1): 5-20
Heather J. Smith, Steve Higgins, Kate Wall & Jen Miller (2005) ‘Interactive whiteboards:
boon or bandwagon? A critical review of the literature’, Journal of Computer Assisted
Learning 21(2):91-101
Jewitt, C., Moss, G. and Cardini, A. (2007) 'Pace, interactivity and multimodality in teachers;
design of texts for interactive whiteboards in the secondary school classroom', Learning,
Media and Technology 32(3): 303-317
Moss, G., Jewitt, C., Levacic, R., Armstrong, V., Cardini, A. & Castle, F. (2007) The interactive
whiteboards, pedagogy and pupil performance evaluation. Research report 816 (London,
DfES).
11
ICT used by students
Rachel Murgatroyd
Pupils’ use of ICT as a resource to aid the teaching of geometry has been developing since
the 1980s. I’ve explored this resource as a pedagogical orientation for geometry and its
purpose in the classroom. These range from being used as an accurate way of construction
or improving the skill of using dynamic software, to being used for exploration, allowing
pupils to spot geometrical patterns and come up with conjectures regarding geometrical
properties.
I used the dynamic software package, Geogebra, in a lesson I taught on introducing the
trigonometric ratios. I introduced the concepts of trigonometry by using the unit circle, so
that students would have a better understanding of trigonometry for use in further study. I
then related this to the ratio definitions for students to use when finding missing lengths
and angles in right angled triangles. This was set out as follows; There is a unit circle centred
on (0,0), with a line from the centre of the circle to a point lying on the circle. An angle, , is
subtended between this line and the x-axis. The value of sin is equal to the y-coordinate of
the point. So if  is between 0° and 90°, a right angled triangle can be created with a vertical
line from the point to the x- axis and a line horizontally from here to (0,0). So sin measures
the side of the triangle, opposite . Moving away from dynamic software, when the
hypotenuse is h rather than 1, through similar triangles the side opposite  is hsin. This can
be rearranged to make sin the subject. Cos is explored in a similar way using the unit
circle. Cos gives the x coordinate of the point on the circle. When  is between 0° and 90°,
a right angled triangle can be created. Cos then measures the side of the triangle adjacent
to the right angle and . For any hypotenuse h, hcos gives the adjacent side. This can be
rearranged to make cos the subject.
The software used an already constructed unit circle, showing the angle being subtended as
the point was dragged around the circle. Tick boxes allowed the user to choose whether
sin or cos should be displayed. When sin was selected, I circulated the class to ask pupils
what they noticed about the value of sin for any given , as they dragged the point around.
I prompted them if they were struggling, asking if they noticed a relationship between sin 
and the position of the point. As was pointed out in an experiment by Choi-Koh (2003,
p.366) whilst using a graphic calculator to explore trigonometric graphs, “Students should
not… get a quick answer to a question by pushing a button. Instead, teachers should ask
why and how when trying to stimulate students to think mathematically and meaningfully.”
I therefore felt that prompting pupils with open questions such as these was necessary.
The dynamic geometry software allowed pupils to spot a pattern quickly through the speed
of the software. Prichard (1993) suggests an approach using a pencil and paper method,
however I felt the use of ICT was advantageous due to its speed. This is often an advantage
of dynamic software, as pupils can explore a topic with ease and once they have formed a
12
conjecture, can test it without having to redraw and recalculate. However, as is pointed out
by Goldstein (2002), whilst speed enables students to spot patterns, this is not enough.
Often students are under the impression that their conjectures are correct, simply because
they have tested it and it has not yet been proved false. “The computer has allowed the
students to explore… it is through exploration with the computer that the results may have
been suggested in the first place. But when a student is able to argue and convince
someone else, then some new mathematics has been understood.” (Goldstein, 2002,
pp.154-155) So work away from the computer is necessary to fully understand the
mathematics. ICT shouldn’t be relied on as the only resource to use and should be used
appropriately, for the purpose of exploration.
Due to rounding errors in the software, misleading results sometimes appeared, such as
displaying that sin4° = 0. Pupils did query this as some had already formed a conjecture,
which this result would have falsified. I asked pupils why they thought this error might have
occurred, showing the importance of checking whether a result is what they would have
expected. As is mentioned by Ruthven, Hennessy and Deaney (2008), some teachers
purposefully design activities that avoid rounding errors, but rounding errors can actually be
used to an advantage, as a lesson to pupils that they should always question what is given to
them.
Whilst “Computers should be used in mathematics lessons to serve the
curriculum…”(Goldstein, 2002, p.157), this focus on the contents of the curriculum often
means that teachers use software only to establish ideas, which “may differ markedly from
the exploratory orientation advocated” (Ruthven, Hennessy and Deaney, 2008, p.300) by its
pioneers. Admittedly, using ICT as an orientation is often more time consuming than other
alternatives, but it gives further opportunity for pupils’ own discovery, enforcing that “…the
teacher’s role is not to inform the students but to introduce activities that help students to
use their awareness in coming to know what is necessary.” (Hewitt, 1999, p.52) The pupil’s
use of ICT means that pupils actually develop their own understanding of the mathematics,
instead of it being treated by the teacher as arbitrary.
The pupils’ use of ICT as an orientation for geometry can be a powerful resource, if used
appropriately by the teacher. Enabling a pupil to discover geometric properties and
relationships for themselves where possible, rather than being informed of it by a teacher,
allows them to develop a deeper understanding of mathematics. Using ICT can motivate this
process, making spotting patterns and forming conjectures easier. However, this is only
effective if the teacher creates an appropriate activity and probes students to give reasoning
behind what they have noticed. The teacher should not rely on ICT orientations alone, but
use them alongside pencil-and-paper approaches and physical activities, ensuring that they
are all related.
13
References
Choi-Koh, S., 2003. Effect of a Graphing Caclulator on a 10th-Grade Student’s Study of
Trigonometry. The Journal of Educational Research [e-journal] 96(6) Available at:
<http://dx.doi.org/10.1080/00220670309596619> [Accessed 12 November 2013].
Goldstein, G., 2002. Integrating Computers into the Teaching of Mathematics, In: L.Haggarty
ed. 2002. Teaching Mathematics in Secondary School. London: Routledge Famler. Ch.10.
Hewitt, D., 1999. Arbitrary and Necessary: a way of viewing the mathematics curriculu, In: L.
Haggarty ed. 2002. Teaching Mathematics in Secondary School. London: Routledge Falmer.
Ch. 4.
Prichard, V., 1993, Introducing and Extending Trigonometry Using the 'Trig-Wheel'.
Mathematics in School 22(2) Available at: <http://www.jstor.org/stable/30214973>
[Accessed 1 November 2013]
Ruthven, K., Hennessy, S., Deaney, R., Constructions of dynamic geometry: A study of the
interpretative flexibility of educational software in classroom practice. 2008 Computers and
Education [e-journal] 51 Available at:
<http://www.sciencedirect.com/science/article/pii/S0360131507000553> [Accessed 9
November 2013].
14
ICT used by students
Joe Watson
Students’ use of ICT in Mathematics is something which can potentially enhance learning
significantly, however, there are many barriers that have been, and still are, stopping this
happening. ICT for students’ use has become far more accessible as technology has become
cheaper, and as there are more dedicated learning programmes and software being
created. It is now often the case that state schools have multiple ICT rooms, as well as
laptops available for students’ use in standard classrooms. However, preliminary
observations found that teachers are not fully utilising these facilities in their teaching
(Chong Chee Keong et al, 2005). Six major barriers were identified: lack of time in the school
schedule for projects involving ICT, insufficient teacher training opportunities for ICT
projects, inadequate technical support for these projects, lack of knowledge about ways to
integrate ICT to enhance the curriculum, difficulty in integrating and using different ICT tools
in a single lesson and unavailability of resources at home for the students to access the
necessary educational materials.
When discussing lesson plans with teachers, the most common phrase you’re likely to hear
as an excuse for a poor lesson is “I would have done it like this if I had more the time”. This
concern of time, coupled with the pressure teachers feel to “teach to the exam” could mean
that experimenting with new methods and technology could be seen as a risk compared to
the tried and tested existing methods. Software such as Geogebra can be used to help
students learn through discovery in a wide range of topics, however, from conversations
with teachers it can be the case that this sort of learning can be considered to be sacrificing
effective teaching time at the risk that knowledge may not be discovered in ICT based
lessons.
Considering the lack of time dedicated to projects involving ICT, it is no wonder that other
such barriers should arise, such as a lack of training, lack of confidence in using ICT, and lack
of knowledge on how to integrate ICT with the curriculum. There is also one other barrier
that I would add: the availability of ICT at school. Despite technology being much more
accessible than it was in the past, it is still often the case that there will only be one
computer room to share between multiple classes. For example, in my placement school I
have yet to see students using ICT despite being here for nearly two months. The teachers
often use ICT themselves but there are only a couple of dedicated ICT rooms in a school of
over one thousand students, and hence the students themselves rarely get a chance to use
such technology.
It also seems to be the case that when teachers teach lessons where students don’t use ICT,
they get into a routine of doing so. If we want students using ICT more often, we need
teachers getting in the habit of using it. Students using ICT to learn Maths should be the
norm, as opposed to an infrequent occurrence, because as long as ICT lessons are in the
minority, it can still be seen as more of a hassle for teachers than a benefit.
15
As previously mentioned there are more and more computer packages being made available
for students to use when learning Mathematics. Many of these, if used well, can significantly
aid learning. However, there are other forms of ICT that can also help. For example, there
are virtual learning environments such as Edmodo which can work very well. Although
networks like this can often be seen as a tool for teachers, they can be extremely useful for
students as well. These networks can give students the ability to receive and respond to
teacher assessment instantly, without the wait of handing books in and writing responses to
feedback, as well as a platform to share materials and work with other students. Again
though, this sort of resource is at its most useful when more people use it. There would be
little point in a student using it if none of their peers used it, for example, as there would be
no one else to share content with.
Geometry is a prime example of a topic in Mathematics where students could really benefit
from the visual representation that ICT can offer. For example, a teacher can use ICT to
demonstrate circle theorems to the class; however, this approach does not allow students
to work through the problem themselves. This ostensive method could therefore be
improved by creating a situation composition whereby the students used ICT to discover
these theorems for their selves.
Through discussions with teachers I have come across applications of ICT for students in a
variety of topics, such as statistics and calculus. Understanding that the when you
differentiate e^x you get e^x becomes much easier when a student can construct the graph
with a moving gradient for themselves, and then observe what happens, for example.
Another interesting, and not immediately obvious use is conditional probability:
representing the famous breast cancer problem with a Venn diagram, you can then use
sliders to explore how the sizes of the sets influence the conditional probability. Hence
students would be exploring the relationship of the conditional probability and the
probability of each event.
The more teachers get their students to use ICT when learning maths, the more interesting
and useful ways can be found of using it. At the moment it seems there is a risk of a vicious
circle where the lack of use leads to a lack of ideas for teaching, a lack of confidence and
time allocated to it, which in turn would deter teachers from using it further.
To conclude, I ICT can be an extremely powerful tool to aiding students’ learning but there is
still a long way to go if we want to get anywhere near the full potential of its use. There are
some underlying problems in the education system which will mean that the previously
mentioned barriers to its use are likely to re-occur. However, if we can get teachers to
become more confident with getting their students to use ICT, then this will set the ball
rolling for more use, and hopefully better use in the future.
References
16
Chong Chee Keong, Sharaf Horani & Jacob Daniel. (2005) A Study on the Use of ICT in
Mathematics Teaching
17
Physical resources
Joe Stuart
In this report I will be evaluating the use of cards (especially the matching of pairs/groups of
cards) and other related tasks, as mathematical pedagogic resources. I will conduct a brief
review of relevant literature, reflect upon my experience teaching with the resource, as well
as my observations of associated activities being employed in other mathematics classes.
Rather than a ‘transmission’ orientation (which primarily uses explanations, examples and
exercises), Swan (2008, p. 2) advocates a more collaborative orientation in which students
work in groups on ‘interconnected’ tasks. The teacher has various roles in this model of
learning, including:
assessing students and making constructive use of prior knowledge; making the purposes of
activities clear; challenging students through effective, probing questions; managing small
group and whole class discussions; encouraging the discussion of alternative viewpoints;
drawing out the important ideas in each lesson; and helping students to make connections
between their ideas (Swan, 2008, p. 2).
Swan (2008, p. 3) developed “five task ‘types’ that encourage concept development”, they
include: ‘Interpreting multiple representations’; a card matching activity which creates links
between different representations of the same mathematical idea and ‘Evaluating
mathematical statements’; in which pupils decide whether the statements are true or not
(always, sometimes or never) and are encouraged to give their reasoning, giving
examples/counterexamples as further justification. Though the latter task does not specify
that cards are used, such activities can be found in the Standards Unit; which aims “to
exemplify effective and enjoyable ways of teaching and learning mathematics” (Swan, 2005,
p. 1).
Before starting a card sorting activity, Swan (2008, p. 4) would get the students to answer a
few questions relating the task; which would be put aside until the end of the session.
During the main task: pupils take turns matching cards and filling out additional or missing
information on blank cards (or spaces); students may find some cards are equivalent or that
there are not enough cards for one-to-one matching; the teacher can encourage groups to
test out their ideas; then “arranging the matching groups of cards onto posters and writing
explanations alongside” (Swan, 2008, p. 6). During the plenary, further questioning is used
to discuss what has been discovered and students correct their initial work, identifying the
causes of any mistakes; raising awareness of potential changes in their thinking (Swan, 2008,
p. 7). If conducted as described, then a card sorting task satisfies Swan’s (2001, p. 160)
principles for being an effective classroom activity for use in ‘diagnostic teaching’; of which
“the essential distinctive feature is that pupils’ own ideas and methods are acknowledged
and discussed before any teaching input is given” (Swan, 2001, p. 158).
18
I designed a card matching activity focusing on significant figures (sf) to serve as a starter for
a year 8 class. I had previously taught the same class a starter on rounding, where students
would, for example, guess the circumference of the earth to the nearest ten thousand.
Continuing along the same theme, I put one of the following on each card: ‘Length of the
Great Wall of China’; ‘Length of the Great Wall of China (4 sf)’; ‘Rounding values to 4 sf,
estimate: 211.862×100.049 km’; ‘21196.2 km’; ‘21200 km’; ‘21190 km’. Each ‘word’ card can
be uniquely matched to a ‘number’ card giving three correct pairs. There were three other
sets (containing six cards each). I deliberately chose to ask for the length to be rounded to 4
sf, as the answer (21200 km) has only 3 non-zero significant figures; a common mistake
when rounding to 𝑥 significant figures, stems from not knowing whether a zero can be
significant or not. Similarly I chose the estimate question so it would give an answer (21190
km) that could appear to be to 4sf; as it contains four non-zero digits. The estimation
question (once the values have been rounded to 4 sf) becomes: 211.9×100.0, a trivial
calculation which should highlight if any mistakes had been made. The students received the
cards all mixed up, one whole set (24 cards, 12 pairs) between two. I found it hard to make
sure that pupils were not just guessing or randomly sorting the cards, I walked around the
class prompting double check (or make a first attempt) at the rounding or estimations. I
then got students to compare their results with other groups and discuss any differences.
Very little conflict arose and answers were easily agreed upon.
Although card resources are readily available (there are free online sources) or can be easily
created and produced in a relatively quickly; I found card sorts are frequently overlooked as
a potential class activity. After several weeks of observing various mathematics classes, I
saw one instance of cards being deployed during a lesson; it was with a year 12 class. I
should note that I was only present for the first half of the period and so missed the
arrangement of cards into a domino chain. Firstly the students were given ‘sheet 1’ on
manipulating surds (N11) from the Standards Unit; they were asked to determine (using
calculators) whether statements were true or false and to decide if certain expressions were
equal or not. The statements consisted of equations which were easily verified as true or
false (a real number could be calculated for each side of the equation), the expressions
however were all given in terms of positive integers 𝑎, 𝑏, 𝑥 and 𝑦. Since these expressions
could not be directly evaluated on ordinary scientific calculators, some students needed the
teacher to reinforce (as per the suggested approach in the Standards Unit) that the
generalised expressions could be linked the [true or false] numerical examples. Without
having to explicitly tell pupils they could substitute in arbitrary values (in place of the
variables) to test if two expressions were equal, they used their previous answers (from
whether a numerical equation was valid or not) to inform their decision. Listening to a few
brief discussions (the small class was reluctant to interact with each other or talk at length
with the teacher, even with gentle encouragement; afterwards I was told many of the
students were quite shy and were slowly building confidence) I witnessed individuals
challenge and then confirm/refute their own or each other’s rules regarding surds. From the
19
short time I spent watching this class, I gained the impression that all the students had
deepened their understanding of surds; via a method which was significantly more enriching
than if the lesson had been given in ‘the didactic approach’ (Swan, 2001, p. 149).
I think card matching as a starter has its limitations, mainly due to the time constraints
involved; it would be greatly advantageous to incorporate the task as an orientation activity
for a whole lesson, in which there would be ample time for sufficient reasoning, discussion
and further questioning. Although activities such as card sorts and evaluating statements
may be difficult for teachers to manage, I think the potential gain in students’ learning
outcomes is worth plentiful effort.
Bibliography
Swan, M., 2001. Dealing with misconceptions in mathematics, in: Gates, P. ed. Issues in
Mathematics Teaching. Hoboken: Taylor and Francis, p. 147 to 165.
Swan, M., 2005. Improving learning in mathematics: challenges and strategies. Sheffield:
Teaching and Learning Division, Department for Education and Skills Standards Unit.
Swan, M., 2008. A Designer Speaks. Educ. Des. [Online] 1. Available at:
http://www.educationaldesigner.org/ed/volume1/issue1/article3/pdf/ed_1_1_swa_08.pdf
[Accessed: 1 December 2013].
20
21