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Developing
Children's Skills In Mathematical Explanation |
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School: N/a |
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Fellow: Maureen Loomes |
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Email: m.c.loomes@herts.ac.uk |
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Final report |
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My project for the Gatsby Teacher Fellowship
in 1999-2000 explored ways to support teachers
in developing children's skills in the area
of mathematical explanation. Since my Gatsby
project ended I have continued my research
into mathematical explanation, as a Primary
Mathematics lecturer in the Department of
Education at the University of Hertfordshire.
I have co-authored a paper on the use of ICT
(Teaching Explanation using the Audiograph
Presentation Tool) which was presented at
the 2001 Computer Aided Learning Conference
at the University of Warwick last April. I
am going to present an overview of this work
the Gatsby Teacher Fellowships Seminar, June
2001, University of Warwick. An extended version
of the paper (Teaching Mathematical Explanation
through Audiographic Technology) has been
submitted to the special conference edition
of the International Journal of Computers
and Education. |
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The paper discusses pilot
studies to explore the novel use of an existing
authoring system, the Audiograph, which is
designed for producing web-based presentations,
to support the teaching of mathematics at
primary school level. Providing opportunities
to practice and improve mathematical explanations
is quite a challenge for teachers, particularly
if both written and spoken forms are to be
used. The Audiograph provides an opportunity
for whole explanations to be developed incrementally,
stored for later reference, and discussed
as artifacts in their own right, thus providing
children with the opportunity to develop their
explanation skills in a supportive environment. |
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My project for the Gatsby
Teacher Fellowship in 1999-2000 explored ways
to support teachers in developing children’s
skills in the area of mathematical explanation.
Since my Gatsby project ended I have continued
my research into mathematical explanation,
as a Primary Mathematics lecturer in the Department
of Education at the University of Hertfordshire.
I co-authored the paper below on the use of
ICT (Teaching Explanation using the Audiograph
Presentation Tool) which was presented at
the 2001 Computer Aided Learning Conference
at the University of Warwick last April. An
extended version of the paper (Teaching Mathematical
Explanation through Audiographic Technology)
has also been submitted to the special conference
edition of the International Journal of Computers
and Education. |
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Teaching Explanation
using the Audiograph Presentation Tool |
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| Maureen Loomes |
Department of Education,
University of Hertfordshire |
| Martin Loomes |
Department of Education, University
of Hertfordshire |
| Alex Shafarenko |
Department of Education, University
of Hertfordshire |
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This paper discusses pilot
studies to explore the novel use of an existing
authoring system, the Audiograph, for web-based
presentations. It focuses on the teaching
of mathematics at primary school level, but
many of the ideas presented are applicable
to other disciplines and levels of education. |
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The Audiograph is a freely-available
authoring tool for producing presentations
based upon a lecturer’s conventional
presentation mode, namely the use of OHP slides,
accompanied by a spoken narrative which is
punctuated by hand-written annotations and
pointers references to various elements that
are present on screen. The Audiograph comprises
a tool for generating the presentation, which
imports the background slides (typically from
PowerPoint), then offers the author the opportunity
to add spoken dialogue, hand-written annotations
and mechanisms for highlighting text which
can be interleaved in real time. At any time
in recording a presentation, it can be edited
or converted to a Web site, containing the
slides, audio and graphics in a highly compressed
state, playable in real time even across slow
modem links. Playback requires the second
component of the Audiograph, a plug-in, which
enables standard web browsers to play back
the original content. The key point to note
here is the simplicity of the operation: pilot
studies suggest that a typical child, already
familiar with a home computer and mouse, can
learn to produce a simple presentation based
upon existing background slides, within a
few minutes. The hardest aspect seems to be
the use of a graphics tablet and stylus, rather
than a mouse, which is necessary to enable
the use of hand-written text and diagrams. |
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The National Curriculum
and National Numeracy Framework which currently
defines mathematics education in the UK has
within it the requirement for teachers to
“teach” pupils “how to explain
their answers to problems”. Whilst teachers
have always expected children to “show
their working” and provide “explanations”,
there is little evidence that this skill has
ever been taught explicitly, particularly
at the younger ages. Indeed, a review of published
schemes of work and associated teachers guides
reveal that the subject is rarely explicitly
addressed. One of the authors has been researching
this problem for several years, and has demonstrated
that it is possible to teach explanation skills,
but has highlighted a number of problems.
Of primary concern here are the difficulty
of explaining mathematics without a firm grasp
of notation, the confusions that often arise
between the “mathematics” and
the "meta-mathematical explanations",
how to provide non-threatening opportunities
for practising explanation skills in large
classes, how to motivate “explanation”
and the difficulty of providing constructive
feedback on the quality of explanations. It
is our hypothesis, supported by initial studies,
that Audiograph provides a good starting point
for addressing many of these problems. The
tool has been used to present questions to
pupils as if on an examination paper (the
background slides) but allowing them to answer
using hand-written text, spoken annotations
and pointing. Teachers can then “mark”
these attempts within Audiograph, interleaving
their own comments, annotations and pointers
as appropriate. |
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One of the major problems
encountered by children is that of referencing
the mathematical objects relevant to their
explanations. Techniques such a indexing by
subscript, denoting angles by three vertices
and referring to the semantic role of formula
elements (e.g. “coefficient” vs.
"variable") are often only mastered
post hoc. Phrases such as “the number
is added to the other number” or “the
two angles must add up to 180, so the answer
is 70” are commonly encountered. If
the pupils are able to point to items the
teacher can provide feedback on the reasoning,
and also judge when it is appropriate to encourage
the child to adopt a more formal notation.
Thus the explanation can be treated as a draft,
and refined rather like an English essay on
a word-processor. |
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Similar refinement can take
place in untangling the formal presentation
of mathematics from the meta-information of
what is being done and why. A common example
of this comes in algebra, when children write
things like “x + 2 = 5 = x = 3”,
but they say “x plus 2 equals 5, so
we get x equals 3”, overloading the
symbol "=" and introducing confusion
between the equality of the two sides of an
equation and the algebraic equivalence of
two equations. Correcting this sort of “error”
is difficult in a purely written form. The
teacher's ability to refer to the pupil's
spoken commentary, and to highlight elements
of the proposed written solution as feedback
is provided, overcomes some of these problems. |
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Providing opportunities
to practice explanations is quite a challenge
for teachers, particularly if both written
and spoken forms are to be used, whilst also
having a record to refer to when providing
feedback. Explanations to the whole class
can easily be dominated by presentation skills,
and detailed feedback relies upon the pupil
and teacher remembering accurately what was
said. The use of video for capturing presentations
overcomes this problem, but adds to the complications
of presentation skills. Explanations to peers
may provide good opportunities for practice,
but well-informed feedback is difficult to
achieve. Explanations given one-to-one to
teachers are probably the mechanism used most
commonly, but there the various roles of the
teacher confuse the situation. Pupils often
feel they are being tested, rather than genuinely
asked for an explanation, and teachers find
it difficult to avoid interaction during the
explanation, this influencing its development.
The Audiograph provides an opportunity for
whole explanations to be developed incrementally,
stored for later reference, and discussed
as artefacts in their own right. Because the
focus is on the artefact, rather than on the
pupil, the exercise is less threatening than
techniques such as group presentations and
videos. |
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In conclusion, we would
argue that Audiograph offers the opportunity
for children to develop their explanation
skills in a supportive environment. Teachers
can focus on particular aspects of the task,
provide constructive feedback and make “explanation”
a substantive area like “fractions”.
Records of progress can be kept, and pupils
can refer back to similar problems when they
forget how to tackle aspects of the task. |
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