4.1.1 RQ1: Overall, do instruction treatments (picture/ AR) affect students’
understanding?
In order to ensure the effectiveness of the instruction, paired-samples t-tests were conducted to examine the difference of the mean scores between pretest and posttest scores in both picture and AR group.
In picture group, the paired t-test results showed a significant difference on scores of pretest (M= 4.66, SD= 1.24) and posttest (M= 5.24, SD= 1.51); t (40) = -3.23, p = .002< .005., as shown in Table 11 and Table 12.
Table 11
Descriptive Statistics of Picture Group
Score Mean N SD
Pretest 4.66 41 1.237
Posttest 5.24 41 1.513
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Table 12
Paired t-test on Pretest and Posttest Scores of Picture Group
t df p.
Pretest and posttest score
of picture groups -3.227 40 .002**
**p < .01.
Besides, the t-test results of AR group also exhibited a significant difference on the scores of pretest (M= 4.38, SD= 1.67) and posttest (M= 5.20, SD= 1.57); t (59) = -4.21, p= .000< .005., as shown in Table 13 and Table 14. The t-test results indicated that both pictures and AR Representations were effective.
Table 13
Descriptive Statistics of AR Representations Group
Score Mean N SD
pretest 4.38 60 1.678
posttest 5.20 60 1.571
Table 14
Paired t-test on Pretest and Posttest Scores of AR Representations Group
t df p.
-4.214 59 .000***
***p < .001.
4.1.2 RQ2: Is there a significant difference between learning with pictures and AR representations on students’ understanding?
Homogeneity of regressions were first examined and satisfied (representation, p=.679>.05; prior knowledge, p=.965>.05), following ANCOVA processes were executed.
Refer to Table 15 for the details. Afterward, Levene' s Tests of Homogeneity of variance were performed and satisfied (p= .353 > .05), as shown in Table 16.
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Table 15
Homogeneity Test of Regression Coefficients on prior knowledge and representations Students’ pretest scores
Source SS df MS F Sig.
Representations * pretest .243 1 .243 .173 .679 Prior knowledge * pretest .003 1 .003 .002 .965
Table 16
Levene' s Test of Equality of Error Variances
F df1 df2 Sig.
1.112 3 50 .353
The ANCOVA with the pretest score as the covariance was performed to examine statistically significant difference between representations (picture/ AR Representations) and prior knowledge (low/ high). Descriptive statistics of representations and prior knowledge are presented in Table 17. The ANCOVA results showed that no significant effect of representation was found on posttest scores, F(1, 49)= 1.206, p= .277 > .05, as shown in Table 18.
Table 17
Descriptive statistics of Representations and Prior knowledge
Prior knowledge
Low High
Adj. Mean SD Adj. Mean SD
Representations
Picture 5.31 1.69 5.22 1.04
AR
Representations 6.06 1.37 5.21 0.96
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Table 18
ANCOVA test on Representations and Prior knowledge Students’ understanding
Source SS df MS F Sig. exhibited no significant effect, F (1, 49)= .299, p= .587 > .05, indicating that no differences on understanding were found between students of high and low prior knowledge
4.1.4 RQ4: Whether the level of prior knowledge has an interaction effect with representations on students’ understanding?
Also refer to Table 18 , the ANCOVA results still exhibited no significant interaction effect of prior knowledge and representation on posttest, F(1,49) =1.209, p= .277 > .05, indicating that no inter-effects on understanding were found between levels of prior knowledge and two different representations
Results from the above ANCOVA indicated that none of them were significant. Therefore, two additional analyses were made to acquire more detailed information.
4.1.5 RQ A1: Is there a significant difference between high and low prior knowledge on understanding in picture representation group?
An independent-samples t-test was first performed to statistically analyze pretest of high and low prior knowledge in picture group. Results indicated a significant difference on the pretest between high and low prior knowledge of picture group, t (17.329) = -12.393, p= .000
< .005 (Table 19). The statistics indicated that students’ pretest scores of high and low prior
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knowledge were varied.
Table 19
Independent t-test on High and Low prior knowledge students’ pretest scores in picture group
t df p.
High and Low prior knowledge
students’ pretest scores -12.393 17.329 .000***
***p < .001.
A one-way ANCOVA with pretest scores as covariance was then performed to examine a statistically significant difference between high and low prior knowledge on posttest in picture group. Levene's Tests of Homogeneity of variance were also performed and satisfied (p= .555> .05), as shown in Table 20. Descriptive statistics of representations and prior knowledge are presented in Table 21. Results exhibited no significant effect of prior knowledge on understanding, F(1,19) = 1.608, p=.220 > .05., as shown in Table 22. The statistics indicated that, after the picture treatments, high and low prior knowledge student’s understanding tended to be similar.
Table 20
Levene's Test of Equality of Error Variances
F df1 df2 Sig.
.361 1 20 .555
Table 21
Descriptive Statistics of Achievements on High/Low Prior Knowledge in Picture Group
Prior Knowledge Adj. Mean SD N
low 6.38 1.69 11
high 4.44 1.04 11
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Table 22
ANCOVA test on Prior knowledge Students’ understanding in picture group
Source SS df MS F Sig.
Prior Knowledge 2.352 1 2.352 1.608 .220
Error 27.797 19 1.463
Total 707.000 22
4.1.6 RQ A2: Is there a significant difference between high and low prior knowledge on understanding in AR representation group?
In AR group, an independent-samples t-test was also first performed to analyze pretest of high and low prior knowledge. Results indicated a highly significant difference between low prior knowledge and high prior knowledge, t (28.832) = -13.900, p= .000 < .005., as shown in Table 23. The statistics indicated that students’ pretest scores of high and low prior knowledge were varied.
Table 23
Independent Samples t-test on High and Low prior knowledge students’ pretest scores in AR Representations group
A one-way ANCOVA with pretest scores as covariance was then performed to examine a statistically significant difference between high and low prior knowledge on posttest in picture group. Levene's Tests of Homogeneity of variance were also performed and satisfied (p= .308
> .05), as shown in Table 24. Descriptive statistics of representations and prior knowledge are presented in Table 25. The ANCOVA results showed that the effects of prior knowledge on understanding were nonsignificant, F (1,29) = .003, p= .954 > .05., as shown in Table 26. The
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statistics indicated that, after the AR treatments, high and low prior knowledge student’s understanding tended to be similar even much closer than the ANCOVA results of picture group (Table 22).
Table 24
Levene's Test of Equality of Error Variances
F df1 df2 Sig.
1.076 1 30 .308
Table 25
Descriptive Statistics of Achievement on High / Low prior knowledge in AR Representations Group
PriorKnowledge Adj. Mean SD N
High 5.50 .964 16
Low 5.56 1.37 16
Table 26
ANCOVA test on Prior knowledge Students’ understanding in AR Representations group
4.1.7 Interview data
In order to further investigate notable phenomena that the quantitative data did not tell, short interviews were carried out on random-chosen students in both picture and AR group. In the interview data, as few interesting findings were spotted, including manipulation and various analogical thinking related to life experiences.
Most of the participants in AR group reported that the manipulation process was beneficial for them to understand mathematics concepts more easily, while students in picture group merely copied what they saw in the textbooks. Furthermore, when asked how they would
Source SS df MS F Sig.
PriorKnowledge .004 1 .004 .003 .954
Error 36.324 29 1.253
Total 1055.000 32
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use these fraction concepts in lives, in picture group, both high and low prior knowledge students’ answers were constrained within the examples that appeared either in textbooks or the course of the present study, while students in AR groups showed more innovative examples.
The corresponding interview transcripts are listed below, as depicted in Table 27.
Table 27
Interview transcripts
Category Interview data
Manipulation
Picture group “I just copied all the numbers in the questions.”
(C17)
AR group
“I think it’s interesting to use tablet to scan objects.”
(B1)
“Also, the tablet shows the action step by step, so I realize how fraction is to equalize objects.” (A24)
“I think it’s really fascinating because I can play with teaching materials.” (B18)
Analogical thinking and life experiences
Picture group “Apple, onion, and cookies.” (C11)
AR group
“I’ll use fraction when I am sharing watermelon and egg tart.” (E26)
“I can use fraction to cut cakes, toast, and pizza.”
(B12)
“When I help my family fill the bowls with rice, I can use this fraction concept. I would divide rice into 5 portions, and then give one portion to each of them.” (A24)