Education of a model student

Abstract

A dilemma faced by teachers, and increasingly by designers of educational software, is the trade-off between teaching new material and reviewing what has already been taught. Complicating matters, review is useful only if it is neither too soon nor too late. Moreover, different students need to review at different rates. We present a mathematical model that captures these issues in idealized form. The student's needs are modeled as constraints on the schedule according to which educational material and review are spaced over time. Our results include algorithms to construct schedules that adhere to various spacing constraints, and bounds on the rate at which new material can be introduced under these schedules.asymptotic analysis | scheduling I n his 2009 speech to the National Academy of Sciences, President Barack Obama exhorted the audience to imagine some of the things that could be made possible by a commitment to scientific research, including the invention of "learning software as effective as a private tutor" (1). This paper is a modest step in that direction.An important challenge for the design of educational software is to incorporate the results of empirical research on how people learn. Such research endeavors to provide principles for how to choose what is taught, how to present it, and how to sequence the material. Ultimately, educational software will require mechanisms for managing the constraints that arise when these principles are applied in different settings.Here we develop and analyze an idealized mathematical model for incorporating a fundamental class of such constraints into educational software-constraints arising from the importance of timing and review in the presentation of new material. For example, software for building vocabulary must determine when to introduce new words that the student has not yet learned, and when to review words whose definitions the student has successfully recalled in the past. The issue is similar to that faced by a high school math teacher deciding how often to schedule lessons that involve trigonometry, or by a piano teacher who needs to decide how much time a student should devote to practicing scales while also learning to play new pieces.The study of the importance of timing with respect to review dates to at least 1885 (2). The notion that it is better to spread studying over time instead of doing it all at once is called the spacing effect. In 1988, Dempster noted that "the spacing effect is one of the most studied phenomena in the 100-year history of learning research" (3). As Balota et al. point out in their review (4), "spacing effects occur across domains (e.g., learning perceptual motor tasks vs. learning lists of words), across species (e.g., rats, pigeons, and humans), across age groups and individuals with different memory impairments, and across retention intervals of seconds to months." See refs. 5 and 6 for reviews. For further results and background, see refs. 7-9. For work on exploiting the spacing effect to build vocabulary (in humans)...

Combinatorial Aspects of Flashcard Games

“…The first few observations are essentially trivial; we collect them in a single proposition. (Parts of this proposition were dubbed the "no passing property" and "slow marching property" in [NKS12], but we don't need these names here.) Proposition 3.1.…”

“…In later subsections, we prove some nontrivial results, including the resolution of a conjecture of [NKS12]. The first few observations are essentially trivial; we collect them in a single proposition.…”

“…In this section, we settle Conjecture 1 of [NKS12]; that is, we show that the functions T 1 (n) and T n (1) have growth rate Θ(n 2 ).…”

“…In their paper [NKS12], Novikoff, Kleinberg and Strogatz introduced a combinatorial process that we will call a flashcard game. These games are defined as follows: as initial data, we have a sequence (p k ) k∈Z>0 called the insertion sequence, and a deck of infinitely many cards 1, 2, 3, .…”

Does like seek like?: The formation of working groups in a programming project

“…It has even been theoretically modeled in recent data mining approaches applied to education (Bahargam, Erdös, Bestavros & Terzi, 2015). Unfortunately, its relevance is often forgotten in theoretical learning models (Novikoff, Kleinberg & Strogatz, 2012).…”