Supplementary (Online) Tables
In this file, we report results in which we used an ordinary least squares path analytic framework to evaluate the moderator effects of
pretest working memory, language comprehension, and reasoning on the effect between each intervention condition versus the control
group. We conducted three analyses on the comparison accuracy score. In each analysis, one of the three possible moderators was tested as
the moderator, while controlling for the other two cognitive/linguistic variables. We ran three analogous models for the word-problem
outcome, while also controlling for pretest word-problem skill and the interaction between pretest word-problem score and condition.
Sample-based standard scores across the two conditions were employed for all measures. In Table S1, we report results of the analyses for
the contrast between EXP versus CON, and below the table, we identify significant moderator effects and classify them as compensatory
or prerequisite ability moderators. In Table S2, we do the same for the contrast between WP versus CON.
Table S1
Moderation Analyses on Magnitude Comparisons and Word Problems for the Explanations (EXP) versus Control (CON) Conditions
Outcome/Moderator
Magnitude Comparisons
Word Problems
Working
Memory
t (p)
Language
Comprehension
Reasoning
Working
Memory
Language
Comprehension
Reasoning
t (p)
t (p)
t (p)
t (p)
t (p)
Constant
-0.02 (.984)
0.07 (.943)
-0.02 (.983)
0.16 (.875)
0.74 (.460)
0.15 (.884)
Moderator
0.52 (.603)
1.25 (.213)
2.55 (.011)
1.75 (.082)
1.73 (.087)
0.36 (.717)
Condition
8.49 (<.001)
8.47 (<.001)
8.84 (<.001)
-0.14 (.885)
0.38 (.703)
-.27 (.789)
Moderator x Condition
-1.06 (.291)
-0.81 (.420)
-3.75(<.001)
-2.54 (.012)
1.44 (.154)
-2.80 (.006)
Pretest WPs
NA
NA
NA
4.00 (<.001)
4.28 (<.001)
3.92 (<.001)
Pre-WPs x Condition
NA
NA
NA
-0.40 (.687)
-1.07 (.287)
-0.27 (.785)
Working Memory
NA
0.50 (.620)
0.54 (.593)
NA
2.20 (.030)
1.92 (.056)
Language Comp
1.28 (.202)
NA
1.97 (.051)
1.17 (.243)
NA
1.89 (.061)
Reasoning
2.75 (.007)
2.84 (.005)
NA
0.61 (.543)
0.43 (.667)
NA
Effects
Control Variables
On Magnitude Comparisons, EXP received relevant intervention via the core multicomponent fraction program and the EXP
component; CON only received the schools’ fraction program. Reasoning was a significant a compensatory moderator of the effect of
EXP over CON. Reasoning was correlated with outcomes in CON but not in EXP.
On Word Problems, both conditions received word-problem instruction through the schools’ fraction program; however, the EXP
condition received intervention on fraction understand and procedures (and the focus of the word problems was fraction). Although
there was no significant effect between conditions on word problems, reasoning and working memory were significant a
compensatory moderators of EXP versus CON. Reasoning and working memory were correlated with learning in CON but not in
EXP.
Table S2
Moderation Analyses on Magnitude Comparisons and Word Problems for the Word Problem (WP) versus Control (CON) Conditions
Outcome/Moderator
Magnitude Comparisons
Word Problems
Working
Memory
t (p)
Language
Comprehension
Reasoning
Working
Memory
Language
Comprehension
Reasoning
t (p)
t (p)
t (p)
t (p)
t (p)
Constant
-0.01 (.993)
0.12 (.907)
-0.14 (.885)
0.62 (.539)
0.83 (.407)
0.71 (.482)
Moderator
1.63 (.106)
0.93 (.354)
3.67 (<.001)
2.95 (.004)
1.14 (.257)
2.95 (.004)
Condition
5.59 (<.001)
5.62 (<.001)
5.73 (<.001)
6.13 (<.001)
6.30 (<.001)
6.18 (<.001)
Moderator x Condition
0.07 (.947)
-0.81 (.418)
-2.53 (.012)
-0.50 (.615)
1.06 (.291)
0.31 (.756)
Pretest WPs
NA
NA
NA
3.04 (.003)
3.24 (.002)
3.11 (.002)
Pre-WPs x Condition
NA
NA
NA
-0.64 (.523)
-0.98 (.331)
-0.74 (.458)
Working Memory
NA
1.61 (.110)
1.72 (.088)
NA
2.96 (.004)
2.91 (.004)
Language Comp
1.07 (.286)
NA
1.20 (.231)
0.94 (.351)
NA
0.96 (.338)
Reasoning
3.26 (.001)
3.33 (.001)
NA
3.02 (.003)
2.97 (.004)
NA
Effects
Control Variables
On Magnitude Comparisons, WP received relevant intervention relevant explanation via the core multicomponent fraction program;
CON only received the schools’ fraction program. Reasoning was a significant a compensatory moderator of the effect of EXP over
CON. Reasoning was correlated with outcomes in CON but not in WP.
On Word Problems, WP received relevant intervention via the WP intervention component; CON only received the schools’ fraction
program. We identified no moderators of the effect between WP and CON.