Abstract
Teachers whose lessons make a significant difference to students’ understanding of mathematical ideas appear to adapt mathematical modes to the restricted frames of school mathematics. We explored one of these frames, the preparation of teaching resources, to investigate our hypothesis about the central role of mathematical modes of enquiry. We set up an artificial resource preparation exercise amongst a group of knowledgeable mathematics educators and recorded their collaboration. We found that our personal mathematical modes were transforming, and the results of this process were embedded into our planning. We argue that teachers’ fluency with mathematical modes is the basis of their unique contribution in providing something that a textbook or annotated website cannot provide.
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- 1.
In this Group, ‘Disciplinary Mathematics’ was intended to mean, and was taken to mean, mathematics as a research discipline.
- 2.
We acknowledge the help of Pedro Palhares and John Mason in preparing this chapter.
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Appendices
Appendix
From Rayner, D. (2000). Higher GCSE mathematics: Revision and practice (p. 188). Oxford, UK: Oxford University Press. Reproduced by kind permission of Oxford University Press.
Exercise 12
-
1.
Rewrite the statements connecting the variables using a constant of variation, k.
(a) \(x \propto \frac{1}{y}\)(b) \(s \propto \frac{1}{{t^2 }}\)(c) \(t \propto \frac{1}{{\sqrt q }}\)
(d) m varies inversely as w (e) z is inversely proportional to t 2
-
2.
T is inversely proportional to m. If T = 12 when m = 1, find:
(a) T when m = 2 (b) T when m = 24.
-
3.
L is inversely proportional to x. If L = 24 when x = 2, find:
(a) L when x = 8 (b) L when x = 32.
-
4.
b varies inversely as e. If b = 6 when e = 2, calculate:
(a) the value of b when e = 12 (b) the value of e when b = 3.
-
5.
x is inversely proportional to y 2. If x = 4 when y = 3, calculate:
(a) the value of x when y = 1 (b) the value of y when x = 2\(\frac{1}{4}\).
-
6.
p is inversely proportional to \(\sqrt y\). If p = 1.2 when y = 100, calculate:
(a) the value of p when y = 4 (b) the value of y when p = 3.
-
7.
Given that \(z \propto \frac{1}{y}\), copy and complete the table:
y
2
4
\(\frac{1}{4}\)
z
8
16
-
8.
Given that \(v \propto \frac{1}{{t^2 }}\), copy and complete the table:
t
2
5
10
z
25
\(\frac{1}{4}\)
-
9.
e varies inversely as (y – 2). If e = 12 when y = 4, find:
(a) e when y = 6 (b) y when e = 1.
-
10.
The volume V of a given mass of gas varies inversely as the pressure P. When V = 2 m3, P = 500 N/m2. Find the volume when the pressure is 400 N/m2. Find the pressure when the volume is 5 m3.
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Watson, A., Barton, B. (2011). Teaching Mathematics as the Contextual Application of Mathematical Modes of Enquiry. In: Rowland, T., Ruthven, K. (eds) Mathematical Knowledge in Teaching. Mathematics Education Library, vol 50. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9766-8_5
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