schoolarithm.RdThe 23 fraction problems were presented to 191 first-level middle school students (about 11 to 12 years old). A subset of 13 problems is included in Stefanutti and de Chiusole (2017).
The eight subtraction problems were presented to 294 elementary school students and are described in de Chiusole and Stefanutti (2013).
data(schoolarithm)fraction17a person-by-problem indicator matrix representing the responses of 191 persons to 23 problems. The responses are classified as correct (0) or incorrect (1).
The 23 problems were:
p01 \(%
\big(\frac{1}{3} + \frac{1}{12}\big) : \frac{2}{9} = ?\)
p02 \(%
\big(\frac{3}{2} + \frac{3}{4}\big) \times \frac{5}{3} - 2 = ?\)
p03 \(%
\big(\frac{5}{6} + \frac{3}{14}\big) \times
\big(\frac{19}{8} - \frac{3}{2}\big) = ?\)
p04 \(%
\big(\frac{1}{6} + \frac{2}{9}\big) - \frac{7}{36} = ?\)
p05 \(%
\frac{7}{10} + \frac{9}{10} = ?\)
p06 \(%
\frac{8}{13} + \frac{5}{2} = ?\)
p07 \(%
\frac{8}{12} + \frac{4}{15} = ?\)
p08 \(%
\frac{2}{9} + \frac{5}{6} = ?\)
p09 \(%
\frac{7}{5} + \frac{1}{5} = ?\)
p10 \(%
\frac{2}{7} + \frac{3}{14} = ?\)
p11 \(%
\frac{5}{9} + \frac{1}{6} = ?\)
p12 \(%
\big(\frac{1}{12} + \frac{1}{3}\big) \times \frac{24}{15} = ?\)
p13 \(%
2 - \frac{3}{4} = ?\)
p14 \(%
\big(4 + \frac{3}{4} - \frac{1}{2}\big) \times \frac{8}{6} = ?\)
p15 \(%
\frac{4}{7} + \frac{3}{4} = \frac{?}{28}\)
p16 \(%
\frac{5}{8} - \frac{3}{16} = \frac{? - ?}{16}\)
p17 \(%
\frac{3}{8} + \frac{5}{12} = \frac{? \times 3 + ? \times 5}{24}\)
p18 \(%
\frac{2}{7} + \frac{3}{5} = \frac{5 \times ? + 7 \times ?}{35}\)
p19 \(%
\frac{2}{3} + \frac{6}{9} = \frac{?}{9} = \frac{?}{?}\)
p20 Least common multiple \(lcm(6, 8) = ?\)
p21 \(%
\frac{7}{11} \times \frac{2}{3} = ?\)
p22 \(%
\frac{2}{5} \times \frac{15}{4} = ?\)
p23 \(%
\frac{9}{7} : \frac{2}{3} = ?\)
subtraction13 is a data frame consisting of the following components:
Schoolfactor; school id.
Classroomfactor; class room id.
Genderfactor; participant gender.
Ageparticipant age.
Ra person-by-problem indicator matrix representing the responses of 294 persons to eight problems.
The eight problems were:
p1 \(73 - 58\)
p2 \(317 - 94\)
p3 \(784 - 693\)
p4 \(507 - 49\)
p5 \(253 - 178\)
p6 \(2245 - 418\)
p7 \(156 - 68\)
p8 \(3642 - 753\)
The data were made available by Debora de Chiusole, Andrea Brancaccio, and Luca Stefanutti.
de Chiusole, D., & Stefanutti, L. (2013). Modeling skill dependence in probabilistic competence structures. Electronic Notes in Discrete Mathematics, 42, 41--48. doi:10.1016/j.endm.2013.05.144
Stefanutti, L., & de Chiusole, D. (2017). On the assessment of learning in competence based knowledge space theory. Journal of Mathematical Psychology, 80, 22--32. doi:10.1016/j.jmp.2017.08.003
data(schoolarithm)
## Fraction problems used in Sefanutti and de Chiusole (2017)
R <- fraction17[, c(4:8, 10:11, 15:20)]
colnames(R) <- 1:13
N.R <- as.pattern(R, freq = TRUE)
## Conjunctive skill function in Table 1
sf <- read.table(header = TRUE, text = "
item a b c d e f g h
1 1 1 1 0 1 1 0 0
2 1 0 0 0 0 0 1 1
3 1 1 0 1 1 0 0 0
4 1 1 0 0 1 1 1 1
5 1 1 0 0 1 1 0 0
6 1 1 1 0 1 0 1 1
7 1 1 0 0 1 1 0 0
8 1 1 0 0 1 0 1 1
9 0 1 0 0 1 0 0 0
10 0 1 0 0 0 0 0 0
11 0 0 0 0 1 0 0 0
12 1 1 0 0 1 0 1 1
13 0 0 0 0 0 1 0 0
")
K <- delineate(sf)$K # delineated knowledge structure
blim(K, N.R)
#>
#> Basic local independence models (BLIMs)
#>
#> Number of knowledge states: 27
#> Number of response patterns: 115
#> Number of respondents: 191
#>
#> Method: Minimum discrepancy
#> Number of iterations: 1
#> Goodness of fit (2 log likelihood ratio):
#> G2(139) = 826.69, p = 0
#>
#> Minimum discrepancy distribution (mean = 1.69634)
#> 0 1 2 3 4 5 6
#> 60 35 36 34 17 7 2
#>
#> Mean number of errors (total = 1.69634)
#> careless error lucky guess
#> 1.0924962 0.6038394
#>
#> Error and guessing parameters
#> beta eta
#> 1 0.035470 0.247292
#> 2 0.199750 0.050746
#> 3 0.000000 0.047059
#> 4 0.320030 0.092397
#> 5 0.047000 0.187019
#> 6 0.056800 0.123800
#> 7 0.060740 0.168317
#> 8 0.086440 0.220812
#> 9 0.013050 0.244224
#> 10 0.299410 0.006645
#> 11 0.231360 0.008791
#> 12 0.057850 0.120558
#> 13 0.215260 0.080882
#>
## Subtraction problems used in de Chiusole and Stefanutti (2013)
N.R <- as.pattern(subtraction13$R, freq = TRUE)
# Skill function in Table 1
# (f) mastering tens and hundreds; (g) mastering thousands; (h1) one borrow;
# (h2) two borrows; (h3) three borrows; (i) mastering the proximity of
# borrows; (j) mastering the presence of the zero; (k) mental calculation
sf <- read.table(header = TRUE, text = "
item f g h1 h2 h3 i j k
1 0 0 1 0 0 0 0 0
2 1 0 1 0 0 0 0 0
3 1 0 1 0 0 1 0 0
4 1 0 1 1 1 0 1 0
4 0 0 0 0 0 0 0 1
5 1 0 1 1 1 1 0 0
6 1 1 1 1 0 0 0 0
7 1 0 1 1 1 1 0 0
8 1 1 1 1 1 0 0 0
")
K <- delineate(sf)$K
blim(K, N.R)
#>
#> Basic local independence models (BLIMs)
#>
#> Number of knowledge states: 18
#> Number of response patterns: 78
#> Number of respondents: 294
#>
#> Method: Minimum discrepancy
#> Number of iterations: 1
#> Goodness of fit (2 log likelihood ratio):
#> G2(222) = 200.57, p = 0.84604
#>
#> Minimum discrepancy distribution (mean = 0.64626)
#> 0 1 2 3
#> 149 103 39 3
#>
#> Mean number of errors (total = 0.64626)
#> careless error lucky guess
#> 0.4455788 0.2006808
#>
#> Error and guessing parameters
#> beta eta
#> 1 0.073427 0.000001
#> 2 0.124140 0.000001
#> 3 0.076601 0.083135
#> 4 0.000001 0.000001
#> 5 0.103035 0.193084
#> 6 0.045341 0.184319
#> 7 0.042343 0.273775
#> 8 0.048986 0.234973
#>