Level-spanning proof-production strategies to enhance students’ understanding of the proof structure in school mathematics

Figures & data

Figure 1. Framework of understanding of the structure of deductive proofs (source: Miyazaki, Fujita, & Jones, 2017)

Figure 2. Proof with a logical circularity

Table 1. Coding for proof-production strategies

Level of understanding		SKC	SKRc
Holistic-structural		C1: Find the premise and conclusion	Rc1: Review/modify the premise and conclusion
		C2: Formulate the proof (conclusion to premise/premise to conclusion)	Rc2: Reorder the proof (conclusion to premise/premise to conclusion)
Partial-structural	Relational	C3: Find the sequence for applying the theorem (e.g. congruence condition → property of congruent shape)	Rc3: Change the sequence for applying the theorem (e.g. congruence condition → property of congruent shape)
		C4: Identify the potentially applicable theorem (e.g. congruence condition)	Rc4: Modify the potentially applicable theorem (e.g. congruence condition)
	Elemental	C5: Find the sequence of the problem’s properties	Rc5: Change the sequence of the problem’s properties
		C6: Identify the problem-specific properties (e.g. AB = AC)	Rc6: Modify the problem’s properties (e.g. AB = AC)
Pre-structural		None	None

Figure 3. Percentage of proof-production strategies featuring in each class

Table 2. Proof-production strategies featuring in each class

Class	Teacher A-led class	Teacher B-led class
Number of strategies	24	8
Singular strategy	7	6
Multi-strategy	17	2

Figure 4. Number of multi-strategies featuring in each class

Table 3. Multi-strategies featuring in each class

	C2 + C4	C2 + C6	C4 + C6	Rc4+Rc6	Rc1 + Rc4+Rc6	C4 + Rc4+Rc5
Teacher A-led class	0	1	12	2	1	1
Teacher B-led class	1	0	1	0	0	0

Figure 5. Percentage of proof-production strategies featuring in the test responses

Table 4. Proof-production strategies featuring in the test responses

Class	Teacher A-led	Teacher B-led
Respondents	30	39
Number of strategies	103	106
Singular	46	90
Multi	57	16

Figure 6. Percentage of proof-production strategies observed in the test responses

Table 5. Proof-production strategies observed in test responses

	Rc1	Rc2	Rc3	Rc4	Rc5	Rc6	Rc1+Rc4	Rc4+Rc6	Rc1+Rc4+Rc6
Teacher A-led class	3	2	0	14	0	27	5	48	4
Teacher B-led class	0	14	0	21	0	55	0	16	0

Figure 7. Student KO’s test response

Figure 8. Scene from the first hour of the unit

Figure 9. Students’ work presented on the blackboard during the first-hour lesson (Teacher A-led class)

Figure 10. Student 5’s idea emerged in the interaction with the teacher

Miyazaki, M., Fujita, T., & Jones, K. (2017). Students’ understanding of the structure of deductive proof. Educational Studies in Mathematics, 94(2), 223–239. https://doi.org/10.1007/s10649-016-9720-9

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