Is counting hindering learning? An investigation into children’s proficiency with simple addition and their flexibility with mental computation strategies

Figures & data

Figure 1. An example showing the format of items on the MCF-A.

Table 1. Mean percentage (SD) for strategy frequency and accuracy by group

		Strategy
		Retrieval	Decomposition	Min count	Count from first	Count all
Proficient (n= 50)	Frequency	59.4 (16.2)***	39.8 (16.5)**	0***	0	0
	Accuracy	99.7 (1.2)**	97.9 (6.2)**	-	-	-
Almost-proficient (n= 48)	Frequency	54.5 (11.2) **	31.3 (14.8)	13.3 (12.4) **	0.1 (0.8)	0.3(1.6)
	Accuracy	98.5 (3.3)	95.2 (11.6)	98.4(4.2) **
Inaccurate-min-counting (n= 9)	Frequency	48.5 (13.1)***	30.2 (18.6)	15.1 (9.8) ***	1.9 (4.6)	1.5 (4.6)
	Accuracy	92.6 (6.2)	90.3 (10.6)	56.2 (14.2) ***
Accurate-min-counting (n= 53)	Frequency	32.0 (8.6)	28.4 (20.4)	37.7 (23.2)	0.6 (2.4)	0.2 (0.8)
	Accuracy	97.1 (6.4)	89.6 (17.6)	94.1 (7.6)
Inefficient-counting (n= 6)	Frequency	19.9 (15.0)	0.9 (2.3)	3.2 (4.1)	30.6 (34.8)	28.7 (34.4)
	Accuracy	96.6 (5.8)			43.8 (51.5)	40.6 (46.9)

Note. Groups are ordered according to frequency of retrieval (descending order) and this order is kept constant in all tables. Data are not shown for strategy accuracy if strategy frequency was ≤5%. Children in the inefficient-counting group responded with ‘don’t know’ on 16.7% of trials. *Results of planned comparisons between the accurate-min-counting group and the proficient, almost-proficient and inaccurate-min-counting group (***p < 0.001, **p < 0.01)

Table 2. Percent correct by condition and group, and differences in accuracy between conditions for each group

	Choice M (SD)	No choice M (SD)	Difference M (SD)
Proficient	98.7 (3.4)	89.1 (11.4)	9.6 (9.4)
Almost-proficient	97.7 (3.6)	76.6 (16.3)	21.1 (15.4)
Inaccurate-min-counting	84.9 (8.5)	54.3 (18.5)	30.5 (14.3)
Accurate-min-counting	94.0 (6.4)	58.2 (19.0)	35.7 (17.8)
Inefficient-counting	63.9 (36.5)	21.8 (16.0)	42.1 (29.2)

Figure 2. Association between percentage correct retrieval in choice condition and percentage correct in no-choice condition by groups. N = 36 problems for each scatterplot. ***p < .001. Problem Key: ●Subset 1 ▲Subset 2 * Subset 3.

Figure 3. Distribution of scores on the assessment of mental computation flexibility with addition (MCF-A) by group.

Table

Item	Strategy type	Correct response
4 + 7 = 10 + 1	Nonstandard partitioning	4 + 6 or 3 + 7
5 + 8 + 5 = 10 + 8	Noticing	5 + 5
8 + 3 + 2 = 10 + 3	Noticing	8 + 2
10 + 11 = 20 + 1	Standard partitioning	10 + 10
28 + 13 = 38 + 3	Standard partitioning (jump)	28 + 10
11 + 3 + 9 = 20 + 3	Noticing	11 + 9
6 + 18 + 34 = 40 + 18	Noticing	6 + 34
48 + 45 = 90 + 3	Nonstandard partitioning	45 + 45
36 + 37 = 72 + 1	Nonstandard partitioning	36 + 36
44 + 49 = 94–1	Compensating	44 + 50
56 + 33 = 80 + 9	Standard partitioning (split)	50 + 30
22 + 49 = 21 + 50	Compensating	22–1
23 + 44 = 64 + 3	Standard partitioning (jump)	20 + 44
45 + 67 + 82 = 180 + 14	Standard partitioning (split)	40 + 60 + 80
456 + 356 = 800 + 12	Standard partitioning (split)	450 + 350
955 + 445 = 1000 + 400	Standard partitioning (jump)	955 + 45

Table

Strategy	M	SD	No. of trials	% trials
Retrieval	2.4	1.5	2767	48.8
decomposition	5.1	4.4	1817	32.0
min-counting combined	6.4	5.0	933	16.5
min-addend 2	3.9	2.6	188
min-addend 3	4.9	3.0	215
min-addend 4	6.6	5.7	160
min-addend 5	7.7	5.8	130
min-addend 6	7.6	4.3	110
min-addend 7	8.8	4.9	77
min-addend 8	10.8	6.1	38
min-addend 9	13.2	10.1	15
count from first	9.4	4.8	58	1.0
count all	18.4	15.6	51	0.9
don’t know	8.3	6.2	45	0.7

Note. No. of correct trials = 5671 (94.9%). Min-addend 2 are trials that were reportedly solved using a min-counting strategy and problems contained an operand of 2 (2 + 2, 2 + 3, 2 + 4, …), min-added 3 signifies min-counting trials where problems contained an operand of 3 (3 + 3, 3 + 4 …) etc.