Statistical computing

Elements of statistical computing are discussed in this series. We begin with the notion of random numbers and their generation using computers. Using these computer generated random numbers for statistical computing and simulations is discussed in the later parts of the series.     

An indigenous and migrant critique of principles and innovation in education in Aotearoa/New Zealand

This paper questions notions of individualism underpinning technocratic approaches to education that marginalise indigenous and migrant peoples?? knowledges in tertiary education. Focusing on New Zealand (Aotearoa) with its colonial and immigrant history, its M??ori and Pacific Islander citizens, the authors ask whether education, as its process is being communicated there, leaves indigenous and migrant people vulnerable and marginalised in the dominant, English-speaking, New Zealand European (P??keh??) mainstream society. The question is whether education refers to capacity-building and strengthening the potential of marginalised students?? language and culture; or whether it is only geared towards sustaining English-language ascendancy and technical virtuosity. Taking on board the cultural heritage of Pacific Islanders (Pasifika) resident in New Zealand, a new teacher training diploma was introduced by the Auckland University of Technology in 2004. Both authors are involved in the panel meetings (Fono) where the papers presented during the diploma course are evaluated.     

A further study of productive failure in mathematical problem solving: unpacking the design components

This paper replicates and extends my earlier work on productive failure in mathematical problem solving (Kapur, doi:, 2009). One hundred and nine, seventh-grade mathematics students taught by the same teacher from a Singapore school experienced one of three learning designs: (a) traditional lecture and practice (LP), (b) productive failure (PF), where they solved complex problems in small groups without any instructional facilitation up until a teacher-led consolidation, or (c) facilitated complex problem solving (FCPS), which was the same as the PF condition except that students received instructional facilitation throughout their lessons. Despite seemingly failing in their collective and individual problem-solving efforts, PF students significantly outperformed their counterparts in the other two conditions on both the well-structured and higher-order application problems on the post-test, and demonstrated greater representation flexibility in working with graphical representations. The differences between the FCPS and LP conditions did not reach significance. Findings and implications of productive failure for theory, design of learning, and future research are discussed.     

Micro productive failure and the acquisition of algebraic procedural knowledge

Instructional Science - Productive failure has shown positive effects on conceptual and transfer measures, but no clear effects on procedural measures. It is therefore an open question whether, and...     

Is having more prerequisite knowledge better for learning from productive failure?

A critical assumption made in Kapur’s (Instr Sci 40:651–672, 2012) productive failure design is that students have the necessary prerequisite knowledge resources to generate and explore solutions to problems before learning the targeted concept. Through two quasi-experimental studies, we interrogated this assumption in the context of learning a multilevel biological concept of monohybrid inheritance. In the first study, students were either provided or not provided with prerequisite micro-level knowledge prior to the generation phase. Findings suggested that students do not necessarily have adequate prior knowledge resources, especially those at the micro-level, to generate representations and solution methods for a multilevel concept such as monohybrid inheritance. The second study examined how this prerequisite knowledge provision influenced how much students learned from the subsequent instruction. Although the prerequisite knowledge provision helped students generate and explore the biological phenomenon at the micro- and macro-levels, the provision seemingly did not confer further learning advantage to these students. Instead, they had learning gains similar to those without the provision, and further reported lower lesson engagement and greater mental effort during the subsequent instruction.     

Verbal self-instructional training: An examination of its efficacy, maintenance, and generalisation

The study examined the differential efficacy, maintenance and generalisation effects of verbal self-instructional (VSI) training on reading disabled children. Type of sub-culture (Tribal Vs. Non-tribal) is crossed with type of cognitive stage (concrete Vs. formal operation) to produce four training groups with 15 subjects per cell. The subjects were trained for 10 weeks and retested for post-test and follow-up data. While VSI training was beneficial to non-tribals and children of formal operational stage, it failed in case of tribals and children of concrete operational stage. VSI effects were maintained over four months time and evidenced clear generalisation effects.     

Productive failure in mathematical problem solving

This paper reports on a quasi-experimental study comparing a “productive failure” instructional design (Kapur in Cognition and Instruction 26(3):379–424, 2008) with a traditional “lecture and practice” instructional design for a 2-week curricular unit on rate and speed. Seventy-five, 7th-grade mathematics students from a mainstream secondary school in Singapore participated in the study. Students experienced either a traditional lecture and practice teaching cycle or a productive failure cycle, where they solved complex problems in small groups without the provision of any support or scaffolds up until a consolidation lecture by their teacher during the last lesson for the unit. Findings suggest that students from the productive failure condition produced a diversity of linked problem representations and methods for solving the problems but were ultimately unsuccessful in their efforts, be it in groups or individually. Expectedly, they reported low confidence in their solutions. Despite seemingly failing in their collective and individual problem-solving efforts, students from the productive failure condition significantly outperformed their counterparts from the lecture and practice condition on both well-structured and higher-order application problems on the post-tests. After the post-test, they also demonstrated significantly better performance in using structured-response scaffolds to solve problems on relative speed—a higher-level concept not even covered during instruction. Findings and implications of productive failure for instructional design and future research are discussed.     

Reverse osmosis

Osmosis is a phenomenon which regulates many biological functions in plants and animals. That the plants stand upright, or the water reaches the tip of every leaf of a plant is due to osmotic pressure. The fact that we cannot survive by drinking seawater is also linked to this same phenomenon. J H van’t Hoff showed in 1886 that osmotic pressure is related to concentration and temperature of the solution by a law that is similar to the gas law. An understanding of this phenomenon paved the way not only in explaining the biological functions which depend on osmosis, but also in creating conditions for reversing it known as ‘reverse osmosis’. Reverse osmosis has many applications, one of which is desalination of seawater. The inaugural Nobel Prize in Chemistry was awarded in 1901 to van’t Hoff for his seminal work in this area. The present article explains the principle of osmosis and reverse osmosis.     

Examining the preparatory effects of problem generation and solution generation on learning from instruction

The goal of this paper is to isolate the preparatory effects of problem-generation from solution generation in problem-posing contexts, and their underlying mechanisms on learning from instruction. Using a randomized-controlled design, students were assigned to one of two conditions: (a) problem-posing with solution generation, where they generated problems and solutions to a novel situation, or (b) problem-posing without solution generation, where they generated only problems. All students then received instruction on a novel math concept. Findings revealed that problem-posing with solution generation prior to instruction resulted in significantly better conceptual knowledge, without any significant difference in procedural knowledge and transfer. Although solution generation prior to instruction plays a critical role in the development of conceptual understanding, which is necessary for transfer, generating problems plays an equally critical role in transfer. Implications for learning and instruction are discussed.