© ATM 2008 • No reproduction (including Internet) except for legitimate academic purposes • copyright@atm.org.uk for permissions
A CREATIVITY TAXONOMY
George Hardy describes his categorisation of practical mathematical tasks
using drama, body maths and a host of outlandish props.
Over the last year or so I have been experimenting with my approach to teaching maths
students according to their preferred learning
styles, in particular, the visual and the kinæsthetic. The motivation to work with kinæsthetic
styles came from my rediscovery of People
Maths, Hidden Depths (Bloomfield and Vertes,
2005). They define ‘people maths’ as ‘using
people to form the moving pieces of a mathematical activity, be it a puzzle, a sum, a diagram
or a demonstration ... with the emphasis
heavily on discussion within the group taking
part’ (Bloomfield and Vertes, 2005: 7). These
discussions within participant groups then lead
to the discovery of the hidden mathematical
depths to which their title refers.
I liked this idea but also wanted to involve the
students in a kinæsthetic way that was entertaining
and therefore a visual stimulus to those students
who elected to watch instead of participate. This
way the kinæsthetic learners could engage and
benefit from various levels of discussion and the
visual learners could observe and be brought into
discussion by the teacher at appropriate times.
I decided to try some ideas with my Y10 class
(set 2 of 5, 30 students) and I issued them with
diaries in which they could record their experiences
and thoughts after a ‘people maths’ lesson. In my
own reflections, I initially began to categorise the
lessons under the following three headings, partly
as a means of furthering the invention of more
activities within each classification.
Fig 1
Fig 2
Fig 3
1 Demonstration
In this category, students are used as props or
dynamic parts of a topic. For example, students
form ‘living’ straight-line graphs as a directed role
play, with x-y axes taped on the floor (figures 1-2).
Another example would be using individual whiteboards to expand two brackets using the ‘box
method’, with students constructing a ‘living’ box
(figure 3).
26
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2 Teamwork
Here, students also role play, but design the
dramatic input and choreography themselves. One
striking example of this is the manipulation of
equations. Students use individual whiteboards to
write out the initial equation; then, as a chorus line,
they step forward for each stage of the equation’s
development, amending their whiteboards as
necessary and reciting the developed equation as
they proceed (figure 4).
Fig 5
Fig 6
Starting with the end in mind
3 Creative display
Fig 4
With students in their established people maths
teams, they were asked to display a concept, with
an ‘anything goes’ attitude to the items employed.
For example, students were asked to construct a
3D toy for a child and calculate its volume. Possible
items that students could utilise included carrots,
parsnips, ice-cream cornets, traffic cones and
hoops (from the PE department), beach balls and
polystyrene spheres. Using these, and sugar paper, a
wide range of high quality 2D and 3D displays was
generated.
I have found it quite helpful to focus on a prop to
use in a classroom situation without initially
knowing what the application will be. (I refer to
the prop in question as being ‘classroom-able’ in
some way.) One example was a plastic foldaway
skip that holds sand or waste. I knew I wanted to
put students into one of these! (This, I think, can
be traced back to watching Tiswas on Saturday
morning children’s TV in the 1970s!) After mulling
the idea over, I decided to sit several students
under the skip and demonstrate selection without
replacement, using a Tiswas-ian selection device of a
boxing glove on a metre rule to tap the chosen
student gently to indicate that they had been
selected (figure 7).
Having created several lessons that could be
categorised in the manner described above, my
thoughts turned to finding a more specific classification that would highlight the creative process
underlying each invented lesson, and this is the
creativity taxonomy that I came up with.
Fig 7
Fig 8
Roleplay / choreography / song
The essential idea here is that students become the
mathematics under consideration. Often the adjective ‘living’ can precede the mathematical topic
being studied; in demonstration or team mode,
students have enacted living lines, living graphs,
living equations, living circles and living cones in
the classroom (figures 5-6). Writing and
performing songs as an aid to memory also falls
into this category.
Another idea came as the result of office banter,
when one colleague commented on another’s
patterned shirt, referring to it as his ‘graph-paper’
shirt. My initial reaction was to want to draw a
graph on the shirt in question, which quickly developed into a ‘eureka’ moment in which graph paper
T-shirts were created. These T-shirts (figure 8) turn
drawing graphs, solving simultaneous equations and
shading inequalities into physical activities, and are
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© ATM 2008 • No reproduction (including Internet) except for legitimate academic purposes • copyright@atm.org.uk for permissions
made from transfer paper ironed on to inexpensive
T-shirts. (Boys have transfers on front and back,
and for girls just the back has a transfer.)
Starting with a lively student in mind
When planning for a lesson in which I had to
manage students full of energy, I considered the
lesson’s learning objectives and then mentally
picked my ‘cast of players’. I found myself thinking
that two of my students would enjoy physically
demonstrating two supplementary angles using
their outstretched arms, for example. With a
photograph of this fed straight into my electronic
whiteboard, I could then superimpose straight lines
and angles over these students and engage the class
in finding missing angles. The fact that students
known to the class appeared on the board made
the others focus all the more.
Always on the job
I firmly believe that everywhere we go, whatever
we do, there are ideas that can feature in a creative
lesson. For instance, the ingredients on a box of
muesli can lead to a practical, with ingredients
being combined in the correct ratio to make a
muesli mix. On Brighton pier I noticed one of
those displays that you pop your head into and have
a photo taken, with the caption ‘just married’ or
something similar. I took this idea into the classroom with the help of the design and art departments, and created a ‘Pythagoras plenary’ board for
students to use when feeding back to the class on
what they had learned (figure 9). The crawlthrough tunnel from my own children’s old tent
served well as part of a number machine exercise,
in which crawling through the tunnel adds 4 to
each participant’s starting number.
Fig 9
Using pupils as props
As we saw before, students can use their arms to
show that approximately three diameters form a
circumference (see figure 5). In a ‘living’ equation,
students can form the equation by each holding a
part of it on an individual white-board. On occasion,
if I want to focus on the operations carried out to
each side of an equation, I choose two tall students
to be margin lines, and have other students holding
whiteboards either side of the margins, indicating
the given step. Students can be props in finding
how many cubic centimetres are in a cubic metre
(figure 10) and can each become two units when
demonstrating Pythagorean Triples. (Figures 11 and
12 show a 3-4-5 and a 7-24-25 triple being
modelled using six and 28 volunteers respectively,
each holding two rulers.)
Choose to use props
I have used a ‘hop, jump’ investigation in which
students walk over hoops laid flat on the floor,
choosing either to hop one hoop or to jump two
hoops, and investigate the various distances walked.
The decision to use actual hoops in the classroom,
as opposed to an artist’s illustration on a page,
completely transforms the lesson and focuses the
students on the task.
Raid the kitchen
Fig 10
28
Under the original broad heading of ‘creative display’,
items such as carrots, parsnips and ice cream
cornets were borrowed from the kitchen. Loaves of
bread can be used to demonstrate the cross-section
MATHEMATICS TEACHING INCORPORATING MICROMATH 211 / NOVEMBER 2008
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Fig 11
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Fig 12
of a rectangular prism (and other prisms). In my
experience, students’ interest increases when any
type of food is brought in to the lesson.
Raid the toybox
Like the kitchen items, toys attract students’ attention. I extended a discussion on co-ordinates in all
four quadrants of my classroom by placing a toy
doll on a light fitting and asking students to supply
x and y co-ordinates and estimate the z coordinate, which was the height of the doll above
the floor. If I had just asked for an estimate of the
height of the light fitting, the response would have
been slender compared to the enthusiasm students
exhibited towards the doll figure. I have also used
dolls to ‘walk’ the parts of a vector sum.
Conclusions
Comments from the students’ diaries included the
following:
• “It’s good for all the students because it applies
to all learning styles; this makes you feel
involved and it helps you learn.”
• “As an auditory learner, I feel that watching the
people maths helps me to be a visual and active
learner also.”
• “They say that if a lesson is fun, kids learn
better. And I like people maths because it is
fun.”
• “I like the Pythagoras plenary because it is
funny to watch the people talk about the
lesson.”
Students have found these lessons stimulating
and memorable; this latter quality is especially
important when revisiting topics for the purpose of
revision. A lesson that has a memorable ‘hook’ to it
is likely to be retained by the student. My ‘creativity
taxonomy’ set out here is an invitation to the classroom teacher to design their own kinæsthetic and
visual lessons by dipping, in an inventive way, into
the categories listed above.
Reference
Bloomfield, A. and
Vertes, R. (2005) People
Maths, Hidden Depths,
ATM
www.atm.org.uk/buy
online/products/act059
George Hardy is an advanced skills teacher and
lead practitioner at Northampton Academy.
g.hardy@northampton-academy.org
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29
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