Alan Maloney

Function probe

OUR goal in designing Function Probe was to encourage students to enter advanced mathematical thinking by using a tool built for them to investigate and model phenomena using mathematical functions. The software was designed to be used in courses on algebra, trigonometry, and precalculus starting from 14-to-18-year-old students and undergraduate college instruction. It has also proved effective in enhancing secondary teachers’ understanding of these topics.

Designing software for mathematical engagement through modeling

The framing theory guiding the work described here is that mathematics learning is facilitated through long-term student engagement in collaborative projects, integration of sustained emphasis on content knowledge, deep engagement of student interests, and support for student experience and progress, and commitment to learning through interactive microworlds that foster modeling and collaboration. We describe two case studies of software design/implementation, one an animation environment, and the other a game and game-design microworld. We describe each case in some detail, and compare the projects’ affordances, constraints, and design lessons, and persisting challenges.

Design Decisions in Developing Learning Trajectories–Based Assessments in Mathematics: A Case Study

This article analyzes the design decisions of a team developing diagnostic assessments for a learning trajectory focused on rational number reasoning. The analysis focuses on the design rationale for key decisions about how to develop the cognitive assessments and related validity arguments within a fluid state and national policy context. The study draws on ethnographic methods adapted from science, technology, and society studies to document key rationales for decisions. For this team, concerns about the validity of both the assessments and the hypothetical trajectory, anticipated uses of the assessments, and the available distribution of resources and expertise to different project activities were all considerations for significant design decisions throughout the project. The study findings suggest that success in the design of trajectories-based assessments depends on teams’ attention to balancing core design activities with engagement in the external policy environment, balancing precision with utility for diagnosis in defining the levels of a trajectory, and balancing the goals of supporting and assessing student learning.

Engineering [for] Effectiveness in Mathematics Education: Intervention at the Instructional Core in an Era of Common Core Standards

The initial purpose of the presentation that led to this essay was to identify “what are the potential best practices (“Best practices are generally accepted, informally standardized techniques, methods, or processes that have proven themselves over time to accomplish given tasks. Often based on common sense, these practices are commonly used where no specific formal methodology is in place or the existing methodology does not sufficiently address the issue.” (Wikipedia)) that define effective STEM schools?” The juxtaposition of “best practice” and “effective” here poses a particular challenge—“best practices” refers to the informal wisdom of the field delineating what “gets the job done.” But effectiveness demands an evidentiary base, with justification and warrant for a measured “effect” against some criteria for potential effects that are desirable as outcomes. As researchers, we claim that our charge can only be met definitively for “best practices” that have been researched relative to some common criteria. For this reason, our contribution to this volume is to review, discuss, critique, and refine what we have learned from a promising set of studies of curricular effectiveness, and to propose a means to refocus our approach to effectiveness from “curriculum” to the instructional core as a whole.

Breaching the Conditions for Success for a National Advisory Panel

The authors identify six conditions for success for the work of high-level national panels and identify breaches in these conditions in the recent Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008). They question the trustworthiness, validity, and intellectual integrity of its findings and advice to the nation because of (a) an inappropriate composition of Panel expertise and biased selection of literature (Condition 1); (b) failure to appropriately and consistently apply methodological standards (Condition 2); and (c) inconsistencies between the task group and subcommittee reports and the final report (Condition 6). In asking what difference these breaches make, the authors recount recent events suggesting that these breaches have already contributed to degrading the national discussion of curriculum standards for K–12 mathematics education.

Untangling the “Messy Middle” in Learning Trajectories

In “Tracing the Assessment TriangleWith Learning Progression-Aligned Assessments in Mathematics,” Lai, Kobrin, and DiCerbo present a learning trajectory on the measurement of area, a topic typically introduced in third and fourth grades. Proposing a five-level progression, they investigated student behavior and thinking in two studies: (1) cognitive think-aloud interviews using paper and pencil tasks and response-pattern coding and (2) computer-based tasks, digital data collection, and latent class analysis. Overall, the researchers find a reasonable degree of confirmation for the first and fifth levels of their LP, and support for the relevance, but nonrigid sequencing, of the middle levels. We commend the authors on the clear conceptualization of their ambitious scope of work. The well-organized and systematic report balances confirmatory and contrary evidence. Based on their conceptualization of the LP, they cleverly used digital-environment affordances to design data gathering on the underlying processes. We comment on two fundamental themes from LP research: (1) the structure of an LP relative to the “messy middle” and (2) the interaction of grain size and cognitive goals.