Studying the role of human agency in school mathematics

Figures & data

Table 1. Summary characteristics of the form of mathematical discourse.

Discourse properties	Subsidiary research questions to guide the analysis
Vocabulary
a. specialisation	To what extent is specialised mathematical language used?
b. objectification of the discourse	To what extent does the discourse speak of properties of objects and relations between them rather than of processes?
Syntax
c. grammatical complexity	How much ‘unpacking’ is needed to identify the mathematical information involved?
d. logical complexity	What kinds of logical relationships are present and how explicit are they?
Visual mediators
e. the presence of multiple modes of representation	To what extent does the discourse make use of specialised mathematical modes?
f. transitions between modes of representation	What transformations need to be made between different modes and are these provided or to be produced by the student?
Routines
g. the types of action demanded of students	What areas of mathematics are involved and what is the form of student engagement in these areas?
Endorsed narratives
h. the origin of mathematical knowledge	What is the degree of alienation of the discourse? To what extent is mathematics presented as a human activity?
i. the status of mathematical knowledge as absolute or contingent	To what extent does the text indicate that decisions or choices are possible during mathematical activity?

Figure 1. East Anglian Examinations Board, London Regional Examining Board, University of London Examinations Board, Joint O Level/ CSE Examination, Mathematics Paper 4, June 1987.

Table 2. Discursive forms found in Figure 1 that contribute to alienation.

discursive form	how it contributes to alienation
Figure 2 shows	The figure itself is an agent, independent of any human creator. The presentation of representational objects such as figures, tables or graphs as agents in such communicative processes is a common phenomenon in the examination questions.
a tangent AT touching the circle The bisector of angle BAT meets the circle BP produced meets AT	The mathematical objects tangent, bisector, and BP are agents in material processes (touching, meets). Not only the objects themselves but also the relationships between them occur without human intervention.
BP produced meets AT	The non-finite form produced obscures the agency in the process. It is not apparent who or what decided to produce BP or carried out the material act.
Given also that AP = AQ	Again a non-finite form Given obscures agency, this time in the formation of the argument. Who or what ‘gives’ this information?

Figure 2. Edexcel, GCSE Mathematics (Linear) 1380/3H, June 2011.

Figure 3. OCR, GCSE Mathematics Syllabus A, Paper 4 (Higher Tier), June 2010.

Figure 4. London Examinations, GCSE Mathematics A/B – 1385/1386, Paper 6 (Higher Tier), June 1999.

Figure 5. Midland Examining Group, GCSE Mathematics (without coursework), Paper 3 (Further Tier), June 1995.

Table 3. Complete analytic tool for characterising how a text construes the origin of mathematical knowledge.

Discourse property	Subsidiary research questions guiding the analysis	Textual indicators
the origin of mathematical knowledge	What is the degree of alienation of the discourse?	mathematical objects as agents in processes agency obscured by: non-finite verb forms passive voice
	To what extent is mathematics construed as involving material action or as atemporal objects and their properties?	mathematical objects involved in: material processes relational processes
	To what extent is mathematics presented as a human activity?	human agents in mathematical processes thinking scribbling

Figure 6. Types of agency in processes found in examination papers.

Figure 7. Types of agency in specialised school mathematics processes.

Table 4. Proportions of all processes having human agency.

	% of all processes having human agency (%)	Z-scores
		1987	1991	1995	1999	2004	2010	2011
1980	27	-1.4147	-3.7544**	-2.2672*	-3.3675**	-4.1262**	-3.7836**	-5.0546**
1987	34		-2.8640**	-1.0009	-2.4236*	-3.3263**	-2.9209**	-4.4273**
1991	46			2.0306*	0.1946	-0.7252	-0.3276	-1.7653
1995	38				-1.6324	-2.5828**	-2.1601*	-3.7243**
1999	45					-0.8510	-0.4822	-1.8136
2004	49						0.3645	-0.9391
2010	47							-1.3087
2011	54

Two-tailed Z-test, *p < 0.01, **p < 0.05.

Table 5. Proportions of specialised mathematical processes having human agency.

	% of specialised processes having human agency (%)	Z-scores
		1987	1991	1995	1999	2004	2010	2011
1980	17	-0.3299	-1.7620	-3.1851**	-2.7638**	-3.7671**	-2.8943**	-3.6973**
1987	19		-1.6652	-3.3834**	-2.8380**	-3.9708**	-2.9207**	-3.8567**
1991	30			-1.3770	-1.0357	-2.1699*	-1.3748	-1.3748
1995	41				0.2563	-1.0926	-0.3427	-1.1400
1999	39					-1.2211	-0.5137	-1.2628
2004	50						0.5340	-0.1168
2010	44							-0.6128
2011	51

Two-tailed Z-test, *p < 0.01, **p < 0.05.