Curriculum spaces and mathematical models for curriculum design

Highlights

• Knowledge space theory is complemented by the notion of a curriculum space.

• Bipartite graphs model learners and tasks in a curriculum.

• Greedy algorithm for compatible curriculum maps.

• Prerequisite structure given by a partially ordered set of tasks in a curriculum.

• Students who remain underprepared after completing prerequisite courses.

Abstract

The area of mathematical psychology concerning knowledge spaces is complemented by the theoretical notion of a “curriculum space,” which is developed as a mathematical model for curriculum design in order to illuminate the interplay between learners and the curricula in which they enroll. The analysis is carried out in terms of concepts from graph theory and order theory, yielding a precise language in which to communicate aspects of curricular influences on student-achievement. Emphasis is placed on the structure of tasks that are addressed in a curriculum, the structure of prerequisite-course assignments within the curriculum, and the compatibility between these structures. Among the main results, conditions are given under which the tasks in a curriculum induce a partially ordered set, curricula that admit “compatible” prerequisite-structures are completely characterized, and a greedy algorithm is presented for categorizing courses into “curriculum maps” that respect the compatibility condition.

Introduction

The concept of a knowledge structure was first defined in Doignon and Falmagne (1985) to be a set $Q$ of “problems” together with a collection $\mathcal{K}$ of subsets of $Q$ such that if $K\in \mathcal{K}$ then $K$ is realizable as the precise set of problems in $Q$ that a student could be capable of solving. Knowledge spaces are then defined as knowledge structures such that $\mathcal{K}$ is closed under union. Since the seminal work in Doignon and Falmagne (1985), knowledge space theory has received a substantial amount of interest from many researchers, and is the focus of attention across a significant body of literature (e.g., Cosyn and Uzun, 2009, Dowling, 1993, Düntsch and Gediga, 1995, Schrepp, 1997, Stefanutti and Koppen, 2003, Suck, 2003, Suck, 2004, Suck, 2011, and references to many other related papers can be found in the extensive bibliography provided in Falmagne, Albert, Doble, Eppstein, & Hu, 2013).

The present study shares a common feature with knowledge space theory in that it is concerned with students’ preparation to learn. The inception of knowledge space theory in Doignon and Falmagne (1985) was motivated by the desire to provide a more comprehensive description of a learner’s “knowledge state” than what is uncovered by assessment-metrics. In short, assessment-metrics are often used to categorize individuals according to “how much” they know, whereas knowledge space theory aims to evaluate individuals according to “what” they know.

While the origins of knowledge space theory can be traced to student assessment, the concepts introduced in the current investigation are based on curriculum assessment. Instead of gauging the effectiveness of a curriculum according to “how much” success is achieved, curricula are evaluated based on their structural properties that affect “what” success may be achieved. Sets of tasks serve as “courses” of a curriculum,¹ learners are identified with sets of “cognitive experience”, and mild assumptions are imposed that lead to the existence of a structure that encodes the interactions between learners and curricula. The theory accepts cognitive experience as an attribute of learning-preparedness (e.g., it admits the possibility that two learners who are able to complete the same tasks in a course need not be equivalently prepared to complete subsequent tasks), and it acknowledges the existence of “non-prerequisite” tasks that may facilitate preparedness. Furthermore, arguments are given to suggest that the purely mathematical theory inspired by the applications is interesting in its own right.

During the last century, academia has experienced an increase in demands from accrediting bodies to implement designs of curriculum assessment that address student-achievement (a brief history on accreditation in the United States is given in Ewell, 2015). The concept of curriculum assessment has been defined broadly, among other suitable interpretations, as a “process of gathering and analyzing information from multiple sources in order to improve student learning in sustainable ways” (Wolfe, Hill, & Evers, 2006). Curricular goals and the research devoted thereto vary widely from the appraisal of program-level learning outcomes, to success in post-graduate occupational placement (e.g., Chance and Peck, 2015, Maples et al., 2008, respectively). A wealth of resources and methods for facilitating the evaluation of curricula are readily available, and are typically geared toward strategies for collecting and interpreting data (e.g., a complete curriculum assessment plan is accompanied by several surveys in the appendices of Wolfe et al., 2006).

The evaluators of a curriculum are faced with the fundamental challenge of defining and measuring success. Usually, a collection of objectives that are deemed relevant to a given construct are introduced, and then assessment-metrics are utilized to gauge “success” by the extent to which these objectives are met. Revisions to the curriculum are then implemented with the purpose of improving levels of achievement toward selected goals.

Assessment-metrics provide a means by which statistical tools become available to evaluate a curriculum, and the exploration and analysis of these metrics is an active area of research (e.g., Ahmed and Bhatti, 2015, LaForge and Hodge, 2011, Mendez et al., 2014). Unsurprisingly, statistical analyses make up the bulk of the quantitative literature on curriculum assessment. While these data-driven approaches expand on observations that reflect the nature of a curricular system, the present investigation is concerned with the general underlying structure that governs the outcomes of such analyses.

The contrast between data-driven approaches to curriculum assessment and the perspective taken in this study mimics that of the commonly accepted depictions of experimental physics, where the nature of a physical system is documented through empirical evidence, and theoretical physics, which seeks to explain the experimental findings through abstract mathematical models of the system. These mathematical models are often based on idealized prototypes of real-world objects. The physical existence of such a prototype need not be experimentally provable, but its theoretical existence is valued in accordance with its capacity to approximate that which is known to exist.

In this paper, the purely-mathematical notion of a “curriculum space” is developed. Concepts from graph theory and order theory are applied to illuminate the interplay between learners and the tasks that they are required to complete in a curriculum. The focus is placed on curricular properties rather than the characteristics of learners; in fact, learners are regarded as being boundlessly intelligent (although, not omniscient), which is conducive to isolating curricular effects on student-success. At least in part, the investigation is offered in response to an apparent demand for a precise language in which to communicate aspects of curricular structure that impact student-achievement. The manner in which prerequisite courses are structured within a curriculum is of particular interest.

The theory is developed from the ground up, beginning with primitive notions (including undefined terms and assumptions), and gradually moving toward rigorous definitions and theorems. Most of the main results are found in Sections 3 Curriculum space, 4 Curricular structure of courses, with the majority belonging to the latter. From a purely order-theoretic perspective, most of Section 4 is independent of the investigations in Sections 2 Learners, tasks, and curricula, 3 Curriculum space (see the second paragraph of Section 4), and readers who are primarily interested in the structure of partially ordered sets are invited to forgo these two sections (although, see Theorem 3.6).

The work in Sections 2 Learners, tasks, and curricula, 3 Curriculum space is driven by the idea that there are sometimes many different ways to perform a task, so that one’s ability to complete a given task may not be necessary for, but merely contributes toward the realization of one’s ability to complete a subsequent task (see the discussion at the beginning of Section 2.2). A mathematical portrayal of this notion is offered in Section 2 and, under the ensuing “contributing relation”, it is observed that arbitrary collections of tasks cannot, at the outset, be assumed to be partially ordered (cf. Remark 2.4 and Example 3.3). Section 2 supplies contextual understanding of the abstract mathematical model that is presented in Section 3, and these sections serve to enable order theory for the applicability of Sections 4 Curricular structure of courses, 5 Further considerations: permissive curricula (where tasks of a curriculum are always assumed to be partially ordered).

In Section 2, the premises of the theory, and the mathematical objects that serve as prototypes of learners, curricula, courses, and the tasks that learners encounter in a curriculum are introduced. The concept of a task “contributing” toward a learner’s ability to complete another task is given formal meaning (Definition 2.3). Also, circumstances that influence whether a learner is adequately supported by a curriculum are considered (Condition 1).

In Section 3, the concepts from Section 2 are translated into graph-theoretic terms. A “curriculum space” is defined to be a bipartite graph whose neighborhoods satisfy special conditions that reflect the premises introduced in Section 2 (Definition 3.1). Curricula can then be regarded as particular collections of subsets from within a curriculum space, and it is shown that the tasks associated with “adequate curricula” (which, essentially, are self-contained curricula) are partially ordered according to a rule that respects the “contributing relation” (Theorem 3.4). Moreover, the mathematical qualities of curriculum spaces are considered in their own right, and it is proved that every partially ordered set can be realized from the partial order induced by the tasks of an adequate curriculum (Theorem 3.6).

Section 4 addresses the structure of prerequisite courses of a curriculum. A condition is presented for a prerequisite-structure to be “compatible” with the structure of the tasks covered by the curriculum (Condition 2), and prerequisite-structures that satisfy this condition are completely characterized (Theorem 4.7, Theorem 4.8). Also, a greedy algorithm (Algorithm 1) is given for categorizing courses according to “levels”, providing a rule for the construction of “curriculum maps” that respect compatible prerequisite-structures. The study concludes in Section 5, where obstacles related to curriculum standards are considered, and are left open to further investigation.

References on graph theory and order theory can be found in Bollobás (1998) and Davey and Priestley (2002), respectively.