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The relationship between domain- and task-specific self-efficacy and mathematical problem posing: a large-scale study of eighth-grade students in China

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Abstract

This study explored 1634 Chinese eighth-grade students’ domain- and task-specific self-efficacy and its relationship to their problem-posing performance. In particular, the linear regression model, generalized additive model (GAM), and piecewise regression model (PRM) were used to detail the linear and non-linear relationships between these variables. The findings indicate that most (92.5%) of the students could pose mathematical problems in all tasks, but the effect of their domain-specific self-efficacy on their problem-posing performance was lower than the effect of their task-specific self-efficacy. Students’ problem-posing performance and their task-specific self-efficacy were not always matched when the requirements of the problem they posed varied in difficulty. As the level of difficulty increased, the correlation coefficient between task-specific self-efficacy and problem posing declined from 0.22 to 0.06. Furthermore, PRM confirmed that there were significant changes of the slope around the cut-point of the relationship between task-specific self-efficacy and students’ problem-posing performance. Moreover, the relationship between task-specific self-efficacy and posing performance was different for easy and difficult problems, as the cut-point and slopes before and after the point varied. The findings of this study contribute both to understanding self-efficacy as well as advancing understanding about the characteristics of problem posing from a non-cognitive perspective.

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Authors and Affiliations

  1. Collaborative Innovation Center of Assessment toward Basic Education Quality, Beijing Normal University, Beijing, China

    Qimeng Liu, Jian Liu & Zhikun Zhang

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Jinfa Cai

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  1. Qimeng Liu
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  2. Jian Liu
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  3. Jinfa Cai
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Correspondence to Jian Liu or Jinfa Cai.

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Appendices

Appendix 1

1.1 Problem-posing self-efficacy items in instrument 1

Domain-specific items

1. I can easily construct mathematical problems.

A. Strongly disagree B. Disagree C. Not certain D. Agree E. Strongly agree.

2. My teacher believes that I am able to construct mathematical problems.

A. Strongly disagree B. Disagree C. Not certain D. Agree E. Strongly agree.

3. I believe that I can pose problems without any assistance.

A. Strongly disagree B. Disagree C. Not certain D. Agree E. Strongly agree.

4. Posing problems is more difficult than solving problems.

A. Strongly disagree B. Disagree C. Not certain D. Agree E. Strongly agree.

5. In case I fail to construct a mathematical problem in 5 min, then I give up.

A. Strongly disagree B. Disagree C. Not certain D. Agree E. Strongly agree.

6. I think that the construction of mathematical problems is a difficult task.

A. Strongly disagree B. Disagree C. Not certain D. Agree E. Strongly agree.

7. I found it very difficult when asked to construct mathematical problems.

A. Strongly disagree B. Disagree C. Not certain D. Agree E. Strongly agree.

8. I need much assistance in order to construct mathematical problems.

A. Strongly disagree B. Disagree C. Not certain D. Agree E. Strongly agree.

Task-specific items

Task 1: Doorbell task.

In a party, guests enter when the doorbell rings.

The first time the doorbell rings, 1 guest enters.

The second time the doorbell rings, 3 guests enter.

The third time the doorbell rings, 5 guests enter.

The fourth time the doorbell rings, 7 guests enter.

The pattern keeps on going in the same way. On the next ring, a group enters that has 2 more persons than the group that entered on the previous ring.

1. Please rate how certain you are that you can pose an easy problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain.

2. Please rate how certain you are that you can pose a moderately difficult problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain.

3. Please rate how certain you are that you can pose a difficult problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain.

Task 2: Cake task.

Here are some children and cakes. Seven girls share 2 cakes equally and 3 boys share 1 cake equally.

figure a

4. Please rate how certain you are that you can pose an easy problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain.

5. Please rate how certain you are that you can pose a moderately difficult problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain.

6. Please rate how certain you are that you can pose a difficult problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain.

Task 3: Pattern task.

figure b

The pattern continues as it was shown above.

7. Please rate how certain you are that you can pose an easy problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain.

8. Please rate how certain you are that you can pose a moderately difficult problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain.

9. Please rate how certain you are that you can pose a difficult problem in the situation described above.

A. Very uncertain B. Uncertain C. Not sure D. Certain E. Very certain

Appendix 2

1.1 Problem-posing items in instrument 2

Task 1: Doorbell task.

(The situation is the same as in Appendix 1.)

Please make up three problems based on the above situation: an easy problem, a moderately difficult problem, a difficult problem.

Task 2: Cake task.

(The situation is the same as in Appendix 1.)

Please make up three problems based on the above situation: an easy problem, a moderately difficult problem, a difficult problem.

Task 3: Pattern task.

(The situation is the same as in Appendix 1.)

Please make up three problems based on the above situation: an easy problem, a moderately difficult problem, a difficult problem.

Appendix 3

1.1 Statistical parameters of the instruments

For the problem-posing self-efficacy instrument (instrument 1), the item-total correlation ranged from 0.346 to 0.719, whereas the inter-item correlations were distributed from 0.132 to 0.627. Cronbach’s alpha coefficients were 0.88 for task-specific self-efficacy and 0.80 for domain-specific self-efficacy. The measurement model showed acceptable fit with CFI and TLI beyond 0.9 and RMSEA and SRMR lower than 0.08 (Marsh & Hau, 1996). All standardized factor loadings in the model were significant at the α = .05 level and ranged in magnitude from 0.43 to 0.81.

For the problem-posing instrument (instrument 2), Cronbach’s alpha for the entire assessment was 0.83. Mean square information weighted fit statistic (INFIT MNSQ) and logits are also reported. According to Wolfe and Chiu (1999), the INFIT MNSQ should fall within 0.7 to 1.3. All of the items on the problem-posing tasks (instrument 2) fell into this range, which means that the items all contributed to the underlying construct and were not muted.

Table 7 Mean, standard deviation, factor loading, item-total correlation, and inter-item correlations of task-specific self-efficacy of problem posing in instrument 1
Full size table
Table 8 Mean, standard deviation, factor loading, item-total correlation, and inter-item correlations of domain-specific self-efficacy of problem posing in instrument 1
Full size table
Table 9 Summary of goodness-of-fit statistics for task-specific and domain-specific self-efficacy of problem posing in instrument 1
Full size table
Table 10 Infit MNSQ and logits of each item on the problem-posing in instrument 2
Full size table

Appendix 4

1.1 Fitting graph generated by GAM

Fig. 5
figure 5

The fitting graph of self-efficacy and problem posing

Full size image
Fig. 6
figure 6

The fitting graph of self-efficacy and problem posing under different level of difficulties

Full size image

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Liu, Q., Liu, J., Cai, J. et al. The relationship between domain- and task-specific self-efficacy and mathematical problem posing: a large-scale study of eighth-grade students in China. Educ Stud Math 105, 407–431 (2020). https://doi.org/10.1007/s10649-020-09977-w

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