This article is a discovery-based instructional resource for faculty to use as a capstone course or exploratory project for undergraduates who are familiar with (but not necessarily fluent in) calculus, linear algebra, complex variables, and geometry. The explorations flow from a simple change in sign: in the complex numbers a + bi, where…

The authors describe their approach to teaching a course on finite fields and combinatorial applications, including block designs and error-correcting codes, using a hybrid of lectures and active learning. Under the discussed classroom model, there are two lecture days and two discovery-based discussion days each week. Discussions center around…

While it is commonly accepted that Learning Analytics (LA) tools can support teachers' awareness and classroom orchestration, not all forms of pedagogy are congruent to the types of data generated by digital technologies or the algorithms used to analyse them. One such pedagogy that has been so far underserved by LA is exploratory learning,…

Scaffolding is the asymmetrical social co-enactment of natural or cultural practice, wherein a more able agent implements or performs for a novice elements of a challenging activity. What the novice may not learn, however, is how the expert's co-enactments support the activity. Granted, in many cultural practices novices need not understand…

Motivation and math anxiety are crucial in performance and satisfaction, and augmented reality (AR) may be a useful tool in enhancing these factors because it provides users with interesting visual experiences. Since related empirical research is limited in investigating the effects of using free mobile AR apps integrating Keller's ARCS…

Whereas some educational designers believe that students should learn new concepts through explorative problem solving within dedicated environments that constrain key parameters of their search and then support their progressive appropriation of empowering disciplinary forms, others are critical of the ultimate efficacy of this discovery-based…

Forty 4th and 9th grade students participated individually in tutorial interviews centered on a problem-solving activity designed for learning basic algebra mechanics through diagrammatic modeling of an engaging narrative about a buccaneering giant burying and unearthing her treasure on a desert island. Participants were randomly assigned to…

Algebra is considered a linchpin for success in secondary mathematics, serving as a gatekeeper to higher-level courses. Access to algebra is also considered an important lever for educational equity. Yet despite its prominence, large-scale examinations of algebra instruction are rare. In my dissertation, I endeavor to better understand what…

After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…

This paper charts the design and application of a system to support 11-14 year old students' learning of algebraic generalisation, presenting students with the means to develop their understanding of the meaning of generality, see its power for mathematics and develop algebraic ways of thinking. We focus squarely on design, while taking account of…

Dynamic/interactive graphing applets can be used to supplement standard computer algebra systems such as Maple, Mathematica, Derive, or TI calculators, in courses such as Calculus, Differential Equations, and Dynamical Systems. The addition of this type of software can lead to discovery learning, with students developing their own conjectures, and…

Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…

Discovery learning is one of the instructional strategies sometimes used to teach Algebra I. However, little research is available that includes investigation of the effects of incorporating the graphing calculator technology with discovery learning. This study was initiated to investigate two instructional approaches for teaching slope of a line…

Two experiments were conducted, contrasting a minimally guided discovery condition with a variety of instructional conditions. College students interacted with a computer-based tutor that presented algebra-like problems in a novel graphical representation. Although the tutor provided no instruction in a discovery condition, it constrained the…

A key learning outcome in problem-solving domains is the development of flexible knowledge, where learners know multiple strategies and adaptively choose efficient strategies. Two interventions hypothesized to improve flexibility in problem solving were experimentally evaluated: prompts to discover multiple strategies and direct instruction on…