Learning math is not as easy as 1, 2, 3, but is inﬂ uenced by
Despite some improvement between
students aged 12 years and students aged
some quite surprising factors. Celia Hoyles explains more
13 years, a substantial minority continued
to use “number pattern spotting” strategies
ALL STUDENTS LEARN MATHEMATICS in marks. One question asked annually in the which gave the incorrect solution of 180 grey
school, but what and how they actually second study was a number/algebra task tiles. Altogether, 35% of students aged 12
learn is inﬂ uenced in part by factors beyond involving a tile pattern, quite familiar to years gave such responses. Although this
the curriculum and even beyond the way in English students. The version for students fell to 21% for students aged 13 years, it
which lessons are delivered. Students may aged 14 years (see Figure 1) consisted of stayed at 21% for students aged 14 years.
be taught procedures to support calculation, two parts, A1a and A1b. The students were Interestingly, longitudinal data show that
yet these can be learned without a real grasp given one example of the relationship it was not the same students who always
of why they work as they do. Furthermore, showing 6 grey tiles and 18 white tiles; made pattern-spotting responses: rather,
progress in math can be subject to wider in part a, they had to generalize this to of those giving such a response in any
inﬂ uences. For example, although students another number (60) of white tiles and one year, only about half repeated that
prefer to tackle math with a common-sense explain their numerical calculation. In response in subsequent years. Students
approach, they often forego this for an part b, students were asked to write a in fact ﬂ ipped between pattern spotting
approach which they believe is more likely general relationship involving n white and structural reasoning, indicating that
to gain approval. Teachers need to be aware tiles. The version for students aged 12 mathematics learning is neither stable nor
of the fragility of students’ appreciation of years consisted only of part a, as most linear – a sobering reﬂ ection for teachers of
mathematical argument, and encourage students of that age have not experienced mathematics.
reasoning rather than “answer getting.” much algebra. The aim of the task was Face-to-face interviews and data analysis
to discover whether students would conﬁ rmed this fragility of students’
Seeing through numbers generalize on the basis of structure or use appreciation of mathematical argument.
Along with two colleagues, Lulu Healy spurious number patterns arguing, for It suggests that teachers need to work on
and Dietmar Küchemann, I conducted example, that 6 white tiles had 18 grey tiles scaffolding students’ structural perspective
two large-scale studies of students’ round them, so for 60 you multiply the 18 as well as on reinforcing recognition when
views of math in England between 1995 by 10 to give 180. students “get it.”
and 2003. These studies explored how
high-achieving students (in the top third) Figure 1: Number Patterns or by Mathematical Structure
justiﬁ ed mathematical conjectures, judged
mathematical arguments, and explained A1
their reasons. Both studies compared
ﬁ ndings at different levels (students/ Lisa has some white square tiles and some grey square tiles.
classes/schools) to interpret students’
conceptions and progress with reference to They are all the same size.
the landscape of school and teacher factors.
In the ﬁ rst study, 2,459 high-achieving She makes a row of
students aged 15 years from 94 classes in 90 white tiles.
schools completed two proof questionnaires
(one for algebra and one for geometry) while She surrounds the white
their teachers completed a teacher/school tiles by a single layer
The second study adopted of grey tiles.
a similar approach but added a longitudinal
dimension, analyzing the development of (a) How many grey tiles does she need to surround
mathematical reasoning for students from a row of 60 white tiles?
age 12 to 14 years.
Thus, 1,512 students
from 54 randomly selected schools Show how you obtained your answer.
completed a proof questionnaire.
We asked similar questions each year. In (b) Write an expression for the number of grey tiles needed to
some, students had to justify conjectures surround a row of n white tiles.
and present arguments to gain the best
12 Better: Evidence-based Education fall 2009
db 12 14/10/09 13:05:03
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