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Examining the role and nature of mediation
that 'steps' pupils into problem solving and considering its value
to an online mathematics enrichment environment
Project Team: Jennifer Piggott,
Jennifer Green, Liz Pumfrey
Final Report, January 2006
Research Context and Aims
The problems featured on the NRICH website are designed to be more
challenging than those typically found in textbooks. Mathematical
ideas are often presented in new or unusual contexts so that a
problem-solving strategy for a particular problem may not be
immediately obvious. In a classroom setting, there is often a lot
of mediation by the teacher at the start of lessons in which they
prepare students for what they are about to do and on-going support
that offers clarification and prompts. This feature of classroom
life is not available to students tackling problems on the NRICH
website. We were interested to know if it is possible to support
pupils "stepping into" problems which appear unfamiliar on a
website like NRICH without diminishing the challenge for users to
think for themselves. Concept mapping is a heuristic device
developed by Novak and Gowin (1984), which involves the pictorial
representation of concepts as individual nodes, linked together
within a hierarchical structure. The aim of this study was to
explore whether this technique might be an effective means of
'stepping' children into the kinds of problems found on the NRICH
website, which may at first seem difficult or which they may be
unsure how to approach.
Three classes of Year 6 children were recruited from three local
schools. Each class consisted of between 18 and 30 pupils, and
there were 75 pupils in all. A single morning was spent collecting
data in each school, which was divided into 3 one-hour sessions
spent with each class as a whole. Children were informed that we
were from the NRICH website, and that we were going to ask them to
try out some of our problems and then let us know what they thought
of them in brief questionnaires. They were told that there would be
three problems they would work on individually (for 25 minutes
each) and one we would do together as a group. We did not mention
concept maps (CM) in this introduction.
Pupils were asked to work on the first problem individually and
requested to resist the temptation to confer with their classmates.
Next children were given a 30-minute instruction session on concept
mapping during which a sample problem was used to demonstrate how a
CM might be used to work through a problem. For the purposes of
this study, a 'starter concept map' was created, which contained
two starting nodes ('what I know' and 'what I need to know') as
well as four prompts (e.g. 'I could try....', 'It reminds me
of...') to help children think about what information they might
put in their maps. After the instruction session, the second and
third problems were distributed along with the blank starter maps
which pupils were encouraged to use to help them work through the
problems. After spending 10 minutes on the problems, children were
reminded that the maps were there to help them, to be sure they did
not simply forget to use them. The position of each test problem in
the order was rotated so that if problem A was the first (or
'pre-test') problem in one school, it was the second problem in the
next school, and the third in the next. The same sample problem was
used in all three schools.
After each of the three test problems, a brief, one-page
questionnaire was distributed which was designed to assess what
pupils thought of each problem - how difficult they found it,
whether they knew how to solve it, understood directions,
instructions, etc. The questionnaires for the third and fourth
problems (those presented after the CM instruction session)
included a question about whether they felt the concept map to have
been helpful. Additionally a two-page questionnaire was presented
at the end of the last session, which aimed to assess pupils'
attitudes toward concept mapping generally, and the extent to which
they had found them to be helpful. In two of the three schools, a
brief question session was conducted in which children were asked
for their opinions on the problems and on concept mapping.
Results and Discussion
It was originally planned that children's performance on the
pre-test problem (without CM) would be compared with their
performance on the post-test problems (with CM), to see whether
using the maps affected their performance. However in the event, so
few children actually created maps that such an analysis was deemed
to be impracticable. Each of the 75 children had two chances to
make a map as there were two post-test problems. Of these 150
possible maps, 111 were returned completely blank. Of the 39 maps
that were actually utilised, 25 had only one or two marks.
It was not surprising, therefore, to learn from the questionnaire
data that children did not find the CMs to be particularly useful.
Table 1 summarises pupils' response to questionnaire items in which
they were asked to rate how helpful or useful they found concept
mapping to be. Only a very few reported finding the maps useful.
Interestingly, however, response was more favourable when pupils
were asked whether they felt CMs to be useful for solving maths
Table 1 -Perceived usefulness of concept mappings a) with respect
to individual problems and b) more generally.
In addition to saying whether they found the CM to be useful with
each post-test problem, pupils were provided with space to explain
the reasons for their answer in their own words. Table 2 shows the
kinds of reasons provided for not finding the concept maps to be
helpful. The most common response was 'I didn't need it', or 'I
already knew what to do'. Less common but related responses were
variations of 'it was easier to do in my head' and 'it was easier
to just to try things out'. The second most common response was 'I
didn't understand how to do concept mapping'.
Table 2 -Children's reasons why concept maps were not found to be
It may be that we were not successful in selecting problems that
lent themselves well to concept mapping, despite our efforts so to
do. It is also possible that the thirty minutes of instruction and
one example problem were not adequate to achieve the level of
understanding necessary for children to make effective use of the
maps. Response to a questionnaire item asking how well children
felt they understood concept mapping almost half of the children
(48%) reported understanding concept mapping 'well' or 'very well'.
About the same proportion of children (47%) said they felt they had
learned enough to make a CM on their own. The feedback received
from the participating pupils, along with the fact that almost no
maps were actually produced, would suggest that concept mapping
would not be an appropriate mediation tool for the NRICH
Novak, J. & Gowin, D.B. (1984) Learning How lo Learn. Cambridge
University press, Cambridge.