Supporting primary students’ mathematical reasoning practice: the effects of formative feedback and the mediating role of self-efficacy

Figures & data

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1. Use the digits 4, 5, 6, 7, 8, and 9 to build two three-digit numbers and add them. You may use each digit only once. Example: 645 + 798 = 1443. With which numbers can you create the largest sum? Explain your answer.
Level 4	Correct largest number AND reason 864 + 975 = 1839; the largest numbers are the hundreds, the second largest are the tens, and the smallest are the ones. OR 865 + 974, 875 + 964 etc. AND justification, e.g. the largest numbers must be on the far left, the next largest in the middle, etc.
Level 3	Correct calculation, but not largest number AND reason. OR Correct reasoning, but wrong calculation. OR Correct, largest number and calculation, but missing reason.
Level 2	Correct calculation, not largest number, AND no/incorrect reasoning, e.g. 987 + 654 = 1641. OR Correct, largest number, but addition error, e.g. 1849 instead of 1839. No reasoning. OR Correct digits but the sum was not calculated. No reasoning.
Level 1	Calculation and reasoning are only rudimentary. OR A calculation, but wrong addition, no reasoning. OR Calculation and reasoning wrong.
2. Babies measure approximately 50 cm long when they are born. Until 12 years, a child grows on average □ cm each year. Pick the answer that is most appropriate and give a reason for your choice.A □ 2 cm B □ 4 cm C □ 9 cm D □ 14 cm
Level 4	Assumption of height at 12 years (my height): approx. 1.50 m 150 cm – 50 cm = 100 cm 100 cm: 12 = 8.3 cm (approx. 8 cm) Reason: 8 cm is closest to 9 cm -> C checked. OR Similar variant for other sensible size choices.
Level 3	Small calculation error, reasoning/calculation procedure correct. OR Correct calculation and reasoning/calculation procedure, but ticked selection unclear/wrong. OR Reasoning/calculation procedure small gaps, but correct. Correct solution ticked.
Level 2	Correct choice. Reasoning/calculation procedure incomplete/incorrect. OR Wrong choice. Reasoning/calculation procedure partially correct. e.g. I am 143 cm; 143:12 = 14 cm. 50 cm are forgotten and the calculation is wrong.
Level 1	Correct choice. No reasoning/calculation procedure. OR Wrong choice. No reasoning, only showing first steps. OR Wrong choice. Incomplete calculation.

Table 1. Descriptive statistics and correlations of students’ perceptions and mathematical reasoning competence.

	Mean	SD	ICC	1. Math reasoning t1	2. Self-efficacy t1	3. Formative feedback t1	4. Math reasoning t2	5. Self-efficacy t2	6. Formative feedback t2
1. Math reasoning T1	−.02	.66	.16	–	−.10	−.02	.87**	.15	−.05
2. Self-efficacy T1	3.99	.72	.06	.34**	–	.47*	−.11	−.24	.24
3. Formative feedback T1	3.85	.77	.13	.00	.16**	–	.10	.18	.67*
4. Math reasoning T2	0.60	.66	.15	.91**	.34**	.00	–	.52*	.11
5. Self-efficacy T2	4.08	.77	.04	.35**	.44**	.14**	.35**	–	.63*
6. Formative feedback T2	3.63	.88	.18	.01	.07**	.46**	.00	.21**	–

Note: N = 1261; **p < .01, *p < .05. Mathematical reasoning: Standardised ability estimates. Perceptions: 1–6 Likert scale; endpoints: fully not agree/fully agree or never/always. The correlations below the diagonal are within-values and those above the diagonal are between-values.

Figure 1. Two-wave longitudinal multilevel model of the standardised effects of perceived formative feedback on mathematical reasoning competence mediated by student self-efficacy perceptions. N_between = 71, N_within = 1261; **p < .01, *p < .05.

Table

Item	Formulation:
swim1	I think I will get a good mark for mathematics in my next report card.
swim2	I will understand everything well in mathematics this school year.
swim3	I will be able to solve all math problems well this school year.
swim4	I am good at solving word problems in mathematics.

Note. The response options were: 6 = absolutely agree, 5 = agree, 4 = somewhat agree, 3 = somewhat disagree, 2 = disagree, and 1 = absolutely disagree.

Table

Item	Formulation:
task1	My teacher gives feedback on whether I solved the task correctly or incorrectly.
process2	The teacher discusses with me whether my solutions lead to the goal.
process3	My teacher asks me to explain my solution.
selfreg1	My teacher tells me how I can check the solution myself.
selfreg4	My teacher asks me while working on the task whether I think I have already reached the goal.

Note. The root question and response options were: How often does the following occur during math lessons 6 = always, 5 = almost always, 4 = often, 3 = sometimes, 2 = seldom, and 1 = never.