The COVID-19 pandemic has resulted in unprecedented amounts of information communicated to the public relating to STEM. The pandemic can be seen as a 'wicked problem' defined by high complexity, uncertainty and contested social values requiring a transdisciplinary approach formulating social policy. This article argues that a 'Critical STEM…

Two complementary processes involved in mathematical modelling are mathematising a realistic situation and applying a mathematical technique to a given realistic situation. We present and analyse work from two undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required…

In this study we utilize the notion of learner-generated examples, suggesting that examples generated by students mirror their understanding of particular mathematical concepts. In particular, we explore examples generated by a group of prospective secondary school teachers for a definition of a square. Our framework for analysis includes the…

Application of equality constraints on the categories of a variable is a simple and useful extension of multiple correspondence analysis. Equality is an easy way to incorporate prior knowledge. A procedure to deal with unequal category numbers and with subsets of variables is outlined and illustrated. (SLD)

A stochastic version of a knowledge space is developed in which knowledge states are considered as possible epochs in a subject's learning history. It specifies how a subject is channeled through and progresses along a "gradation." The model's application to artificial data is described, based on maximum likelihood methods. (TJH)

Diagnosing cognitive errors possessed by examinees can be considered as a pattern classification problem that is designed to classify a sequential input of stimuli into one of several predetermined groups. The sequential inputs in this paper's context are item responses, and the predetermined groups are various states of knowledge resulting from…

A metrics and mapping framework is proposed to account for how heuristics, domain-specific reasoning, and intuitive statistical induction processes are integrated to generate estimates. Results of 4 experiments involving 188 undergraduates illustrate framework usefulness and suggest when people use heuristics and when they emphasize…

A model is presented for the growth of knowledge reflected by 24 progress tests completed by approximately 600 students at the University of Limburg (Netherlands) Medical School. Based on the Rasch model, this model treats both the person's ability and the difficulty of the question as random variables. (SLD)

This paper focuses on the notion of human learners as being structure-determined. Alternate views of educational phenomena and educational research of G. Cziko (1989), D. Bohm and F. D. Peat (1987), H. Maturana and F. Varela (1980, 1987), and S. E. Pirie and T. Kieren (1989, 1990) are considered. A mathematical theory of chaos, random…

A methodology for calculating the return on training investment is described. The method covers attitudes, knowledge, and behavior of participants as well as effectiveness in organizational improvement. Output, time savings, quality improvements, and "soft data" on effectiveness are set against costs. (TJH)

The National Assessment of Educational Progress (NAEP) makes possible comparison of groups of students and provides information about what these groups know and can do. The scale anchoring techniques described in this chapter address the latter purpose. The direct method and the smoothing method of scale anchoring are discussed. (SLD)

Issues concerning the selection of judges for standard setting are discussed. Determining the consistency of judges' recommendations, or their congruity with other expert recommendations, would help in selection. Enough judges must be chosen to allow estimation of recommendations by an entire population of judges. (SLD)

Changes in information retention resulting from changes in reliability and number of intervals in scale construction were studied to provide quantitative information to help in decisions about choosing intervals. Information retention reached a maximum when the number of intervals was about 8 or more and reliability was near 1.0. (SLD)

Results of cognitive diagnoses were validated using the rule-space model, and the usefulness of cognitive diagnosis for instruction was demonstrated. Through preliminary studies, 33 stages of knowledge were defined as cognitive diagnoses to be applied to results from a test of knowledge about fractions of seventh and eighth graders (initial test…

The role of prior knowledge in skill acquisition is to enable the learner to detect and to correct errors. Computational mechanisms that carry out these two functions are implemented in a simulation model which represents prior knowledge in "constraints". The model learns symbolic skills in mathematics and science by noticing and…