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Understanding learning

Understanding

mathematics learning

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

questionnaire.

1

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.

2

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

Better(US) Fall09.indb 12 14/10/09 13:05:03

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