Low levels of student success in introductory college math courses are rampant. There are several reasons for low pass rates in these courses, with the most serious issue being lack of concept mastery. To aid students in concept mastery, adaptive learning systems were created. These systems adapt instruction for each student and create an…

We detail a learning progressions approach to early algebra research and how existing work around learning progressions and trajectories in mathematics and science education has informed our development of a four-component theoretical framework consisting of: a curricular progression of learning goals across big algebraic ideas; an instructional…

In this dissertation, the following "network computing problem" is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function f of the messages. The objective is to maximize the average number of times f can be computed per network usage, i.e., the "computing…

The method of "Double False Position" is an arithmetic approach to solving linear equations that pre-dates current algebraic methods by more than 3,000 years. The method applies to problems that, in algebraic notation, would be expressed as y = L(x), where L(x) is a linear function of x. Double False Position works by evaluating the described…

This paper reports on a study investigating students' ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace. In interviews conducted with eight undergraduates, we found students' initial descriptions of subspace often varied substantially from…

We investigated how students use the representation of data in a given example appearing in an interactive diagram (ID) and how they create additional examples with the ID. Students who worked with the ID that offered limited representations and tools ("illustrating ID") looked for ways to bypass the designed constraints: they changed the…

When learning the order of operations, students are instructed to adhere to a directive when determining the numerical value of an arithmetic expression. A more typical approach is the use of a popular mnemonic called PEDMAS (parentheses, exponents, division, multiplication, addition, subtraction). The literature is scant on conceptual approaches…

This two-part investigation (a) assessed the impact of the Jaime Escalante Math Program (JEMP), a structured summer mathematics intervention program, on the math achievement of urban middle school students, (b) identified the characteristics of the program that the administrators and teachers perceived to contribute to student achievement, and (c)…

In 2003, Chicago launched "Double-Dose Algebra," requiring students with pretest scores below the national median to take two periods of math--algebra and supplemental coursework. In many schools, assignment to Double Dose changed the peer composition of the algebra classroom. Using school-specific instrumental variables within a…

Our 1997 article in "IJAIED" reported on a study that showed that a new algebra curriculum with an embedded intelligent tutoring system (the Algebra Cognitive Tutor) dramatically enhanced high-school students' learning. The main motivation for the study was to demonstrate that intelligent tutors that have cognitive science research…

The purpose of this study was to examine child-level pathways in development of prealgebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early…

Research supports special educators' use of progress-monitoring data for instructional decision-making purposes as an evidence-based practice for improving student achievement. This article describes the Professional Development for Algebra Progress Monitoring (PD-APM) system. PD-APM, is an online system that includes two "hubs" that…

The purpose of this study was to examine child-level pathways in development of prealgebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early…

We extend the well-known identity, log det A = tr log A, for any square non-singular matrix A.

The purpose of this study was to explore and compare the proficiency of Grade 11 Mathematics and Mathematical Literacy (ML) learners in solving two problems based on the temperature conversion formula. The first was a result-unknown problem that asks for the result of a formula for a given input. The second was a start-unknown problem in which the…