MATH
Understanding learning
The infl uence of audience: Seeking
approval Figure 2
All our student proof questionnaires
Arthur, Bonnie, Ceri, Duncan, Eric and Yvonne were trying to prove whether the following
included multiple-choice questions
statement is true or false:
presenting mathematical conjectures with
several supporting arguments. Students
When you add any 2 even numbers, your answer is always even.
were asked to make two selections from
these arguments: the argument nearest to
Arthur’s answer Bonnie’s answer
their own approach and the argument which
a is any whole number 2 + 2 = 4 4 + 2 = 6
they believed would receive the best mark
b is any whole number 2 + 4 = 6 4 + 4 = 8
from their teacher (see Figure 2).
2a and 2b are any two even numbers 2 + 6 = 8 4 + 6 = 10
Fascinatingly, both surveys (like fi ndings
2a + 2b = 2 (a + b)
So Bonnie says it’s true.
elsewhere) found that students held two
So Arthur says it’s true.
different conceptions of a mathematical
argument simultaneously: those
arguments which they considered would
Ceri’s answer Duncan’s answer
receive the best mark and those which they
Even numbers are numbers that can be Even numbers end in 0, 2, 4, 6, or 8.
would adopt themselves. When they were
divided by 2. When you add numbers When you add any two of these the
trying to please the teacher, arguments
with a common factor, 2 in this case, answer will still end in 0, 2, 4, 6, or 8.
using letters (to students, a strong signal
the answer will have the same common
So Duncan says it’s true.
for algebra) were popular – despite their
factor.
sometimes making no sense (41% of
So Ceri says it’s true.
students thought Eric’s argument would
receive the best mark). For themselves,
students preferred arguments that they Eric’s answer Yvonne’s answer
could evaluate using common sense. This
Let x = any whole number,
+
preference tended to exclude algebra
y = any whole number
(the most popular answer, which 29% of
x + y = z =
students chose, was Duncan’s).
z – x = y
These choices provide startling evidence
z – y = x
of the disconnect between the justifi cations
z + z – (x + y) = x + y = 2z
chosen by students for themselves and
So Yvonne says it’s true
So Eric says it’s true.
those they considered suitable for the
teacher’s approval. We listened to what the
students themselves said. We found that
From the above answers, choose one that would be closest to
algebra was deemed appropriate in proofs
what you would do if you were asked to answer this question.
for the teacher, but largely because it makes
From the above answers, choose the one to which your
the answer seem so complicated, while
teacher would give the best mark.
other forms were judged perfectly adequate
for the students themselves. detach the language of algebra from “letters
References
to please the teacher” and help students
Challenging classes grasp it as a tool for expressing their own
1. Healy L & Hoyles C (2000) A Study of
Student responses to proof questionnaires ideas and arguments in increasingly rigorous
Proof Conceptions in Algebra, Journal for
were susceptible to the wider infl uences and unambiguous ways?
Research in Mathematics Education, 31(4),
framing effective teaching. For students Since students fi nd it diffi cult to link
396–428.
aged 14-15 years, the percentage of their own informal and narrative arguments
2. Küchemann D & Hoyles C (2009) From
children in the class entered for higher-tier with deductive arguments in algebraic
Empirical to Structural Reasoning in
GCSE (General Certifi cate of Secondary symbolism, can teachers fi nd ways to
Mathematics: Tracking Changes Over Time,
Education: an academic qualifi cation scaffold these connections?
in Stylianou DA, Blanton ML & Knuth EJ
awarded in UK schools) had a statistically How can teachers build on learners’ pre-
(Eds.) Teaching and Learning Proof Across
signifi cant effect on student proof scores: existing knowledge to encourage reasoning
the Grades K–16 Perspective, Lawrence
the higher the percentage of higher-tier rather than “answer getting”?
Erlbaum Associates 171– 191
students in the class, the better the class
proof score. Reasons may include the About the author
Further reading
motivating effect of “real” mathematics: Celia Hoyles OBE is Professor of
more curriculum time, higher expectations Mathematics Education at the Institute
On-line panel discussion on proof
of the students, and more challenging of Education, University of London. She
www.ncetm.org.uk/proofdiscussion.
questioning. has directed more than 30 research and
Online community for teachers
consultancy projects concerned with
www.ncetm.org.uk/proofcommunity.
Questions for teachers mathematics, and published widely in
If students see algebraic proof as a ticket to articles and books. From 2004–2007
Principles for effective teaching of
the best marks, they have not understood its Professor Hoyles was the UK Government’s
mathematics www.ncetm.org.uk/
mathematical signifi cance. How can teachers Chief Adviser for Mathematics.
mathematicsmatters.
fall 2009 Better: Evidence-based Education 13
Better(US) Fall09.indb 13 14/10/09 13:05:08
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