To what extent are students expected to participate in specialised mathematical discourse? Change over time in school mathematics in England

Figures & data

Table 1. Extract of the EDSM analytic scheme: specialisation and objectification.

Property of the discourse	Subsidiary research questions	Textual indicators
Specialisation	To what extent is specialised vocabulary used?	lexical items used in accordance with mathematical definitions or in ways distinctive to school mathematics, considered at the level of: vocabulary sentence text unit (question) extra-mathematical context depth of engagement with context
Reification of the discourse	To what extent does the discourse speak of properties of objects and relations between them rather than of processes?	nominalisation: use of a ‘grammatical metaphor,’ converting a process (verb, e.g. rotate) into an object (noun, e.g. rotation) the use of specialised mathematical nouns such as function, sequence which encapsulate processes into an object

Figure 1. Coding tree for specialisation.

Figure 2. Proportions of specialised school mathematics and non-specialised objects and processes: (a) Proportions of object words specialised to school mathematics and non-specialised objects; (b) Proportions of process words specialised to school mathematics and non-specialised processes.

Table 2. Proportions of specialised vocabulary – objects and processes.

	1980	1987	1991	1995	1999	2004	2010	2011
proportion of specialised objects	0.63	0.51	0.44	0.47	0.53	0.52	0.48	0.48
Z-score comparing each year to 1980	…	5.48	8.43	7.58	4.35	4.71	6.40	6.54
Z-score comparing each year to 1991	-8.43	-3.52	…	-1.39	-4.31	-3.93	-1.88	-1.73
proportion of specialised processes	0.55	0.45	0.39	0.39	0.43	0.43	0.40	0.39
Z-score comparing each year to 1980	…	2.76	4.15	4.40	2.99	3.04	3.90	4.05

Table 3. Specialised objects – symbolic and verbal.

	1980	1991	2011
All SM objects (% of all objects)	553 (63%)	570 (44%)	507 (48%)
Objects in symbols (% of SM objects)	246 (44.5%)	179 (31.4%)	193 (38%)
Distinct SM non-symbolic object words (rate per question)	115 (3.19)	143 (3.11)	118 (1.30)

Table 4. Frequency of the most common school mathematics processes found in 1980, 1991 and 2011.

	1980	1991	2011
Calculate	30	55	12
Find	31	36	14
Show	10	7	7
Simplify	5	1	8
Give	23	19	3
Prove	8	1	1
Solve	3	6	13
Work out	1	1	30

Figure 3. Edexcel GCSE Mathematics (Linear) – 1380, June 2011, Paper 4 Higher Tier, question 1.

Figure 4. Variation in instructions to students to calculate, find or work out.

Figure 5. Implicit expectation of algebraic manipulation. University of London GCE Ordinary Level Mathematics Syllabus B Paper 2, June 1980.

Table 5. Occurrence of prove and show that/show how.

	1980	1987	1991	1995	1999	2004	2010	2011
prove	8	0	1	4	2	7	0	1
show that/how	8	6	5	4	2	5	3	2
total	16	6	6	8	4	12	3	3
occurrences per question of prove and show that/how	0.44	0.11	0.13	0.09	0.06	0.16	0.03	0.03

Figure 6. Degree of contextualisation (all questions).

Figure 7. Degree of contextualisation (contextualised questions only).

Figure 8. Specialised reified objects.

Figure 9. Density of specialised reified objects per question.

Figure 10. University of London, GCE Examination Ordinary Level, Mathematics Syllabus B (361), Paper 2, June 1980.

Figure 11. OCR GCSE Mathematics Syllabus A, Paper 4 (Higher Tier), June 2011.