Abstract
Learners’ difficulties with proof have been ascribed to their lack of understanding of functions that proof performs in mathematics, namely, verification, explanation, communication, discovery, and systematization. However, the extant mathematics education literature on validation of instruments designed to measure learners’ beliefs about the functions of proof is scant. The purpose of this mixed methods study was to use the Standards for Educational and Psychological Testing (as discussed by the American Educational Research Association/American Psychological Association/National Council on Measurement in Education (AERA/APA/NCME), 2014) as a theoretical and methodological platform to support arguments for the validity of the learners’ beliefs about the functions of proof (BAFP) instrument. Scale items were generated from de Villiers’ model and a review of the literature, panel of experts, and key informants. The resulting instrument, comprising 28 Likert-scale items and 5 open questions that assessed the five functions of proof, was administered to 87 grade 11 learners in one high school in the Pinetown Education District, KwaZulu-Natal, South Africa. The results put the validity of the scores on the BAFP instrument into question. The study was an exploration of validity arguments based on the evidence from various data sources. Confirmatory factor analysis with a larger sample in a different context needs to be conducted to warrant the use of the BAFP instrument. The key contribution of this study to the field is that it sheds light on the complexities of instrument validation; the scope of the effort may explain why comprehensive validation efforts have not been documented extensively in literature.
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Notes
I join Tall et al. (2012) in defining proof as an activity that “involves thinking about new situations, focusing on significant aspects, using previous knowledge to put new ideas together in new ways, consider relationships, make conjectures, formulate definitions as necessary and to build a valid argument” (p. 15).
The word “Dinaledi” means “stars” in the Sesotho language, which is one of the indigenous languages, along with IsiZulu, commonly spoken in South Africa.
The term “Colored” is used in the same sense as Isaacs-Martin and Petrus (2012) to identify a specific group in South Africa, most often attributed to learners popularly perceived as being of mixed racial and ethnic descent who, over time and due to specific historical, cultural, social, and other factors, have undergone various changes in their perceptions of their identity as Colored.
I use the phrase “group discussion” as being synonymous with focus group.
It is worth noting that after an EFA rerun, some of the items remained negatively loaded onto their factors. However, changing the rotation method from oblimin to promax changed these loadings to positive. An examination of item-rest correlations showed that these items correlated poorly with their factors. The source of the problem could be that, despite group discussion, the participants found the items to be highly ambiguous and confusing or that the data are inconsistent with the hypothesized factor structure. Cognitive interviews could shed more light on this problem.
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I wish to thank Vilma Mesa and the various anonymous reviewers for their helpful comments on earlier versions of this manuscript.
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Beliefs about the functions of proof instrument
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Shongwe, B. Learners’ beliefs about the functions of proof: building an argument for validity. Educ Stud Math 107, 503–523 (2021). https://doi.org/10.1007/s10649-021-10047-y
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DOI: https://doi.org/10.1007/s10649-021-10047-y