In mathematics, as in baseball, the conventional wisdom is to avoid errors at all costs. That advice might be on target in baseball, but in mathematics, avoiding errors is not always a good idea. Sometimes an analysis of errors provides much deeper insights into mathematical ideas. Certain types of errors, rather than something to be eschewed, can…

Asked to complete a decimal-ordering task, several preservice teachers were unable to arrange the values from smallest to largest. Even more surprising to the authors were the number who could solve this task correctly but could not justify their solution by representing each decimal in an area model using a decimal grid. Their preservice teachers…

Ask any teacher what he or she needs more of, and it is a good bet that time will top the list. Anything that promises to recoup a little bit of their workday time is sure to be a best seller. One overlooked time-saver is in how they use feedback. Teachers know that feedback is important for teaching and learning. Unfortunately, most secondary…

Instructional design can be more effective if it is as fixedly dedicated to the accommodation of individual differences as it currently is to the accommodation of subject matters. That is the hypothesis. A menu of accommodation options is provided that is applicable at each of three stages of instructional development or administration: before,…

An inquiry-based laboratory exercise is described in which introductory chemistry students measure the density of water five times using either a beaker, a graduated cylinder, or a volumetric pipet. Students are also assigned to use one of two analytical balances, one of which is purposefully miscalibrated by 5%. Each group collects data using…

Early in their study of one-dimensional kinematics, my students build an algebraic model that describes the effects of a rolling ball's (perpendicular) collision with a wall. The goal is for the model to predict the ball's velocity when it returns to a fixed point approximately 50-100 cm from the wall as a function of its velocity as it passes…

The purpose of this article was to extend the use of standard errors for equated score differences (SEEDs) to traditional equating functions. The SEEDs are described in terms of their original proposal for kernel equating functions and extended so that SEEDs for traditional linear and traditional equipercentile equating functions can be computed.…

As partners in a professional development project, the authors jumped at the opportunity to use a real-life problem to engage elementary and middle school teachers in a one-day exploration of the concept of area. "Length times width"--a common response to the question, "What is area?"--is a rote formulaic expression that applies only to certain…

This article describes how a free, web-based intelligent tutoring system, (ASSISTment), was used to create online error analysis items for preservice elementary and secondary mathematics teachers. The online error analysis items challenged preservice teachers to analyze, diagnose, and provide targeted instructional remediation intended to help…

In a provocative and influential paper, Jesse Rothstein (2010) finds that standard value-added models (VAMs) suggest implausible future teacher effects on past student achievement, a finding that obviously cannot be viewed as causal. This is the basis of a falsification test (the Rothstein falsification test) that appears to indicate bias in VAM…

Increased English learner accuracy can be achieved by leading students through six stages of awareness. The first three awareness stages build up students' motivation to improve, and the second three provide learners with crucial input for change. The final result is "sustained language awareness," resulting in ongoing…

Teachers can incorporate topics in forensic psychology into lower level courses to increase student interest and to show how psychological processes influence outcomes in high-stakes applied contexts. One such topic is eyewitness identification, which teachers can use to show how stress affects memory and how memories can become distorted during…

This article examines the question of finding the best quadratic function to approximate a given function on an interval. The prototypical function considered is f(x) = e[superscript x]. Two approaches are considered, one based on Taylor polynomial approximations at various points in the interval under consideration, the other based on the fact…

The methods and models for categorical data analysis cover considerable ground, ranging from regression-type models for binary and binomial data, count data, to ordered and unordered polytomous variables, as well as regression models that mix qualitative and continuous data. This article focuses on methods for binary or binomial data, which are…

We present a new feedback strategy implemented in a natural language generation-based e-learning system for German as a second language (L2). Although the system recognizes a large proportion of the grammar errors in learner-produced written sentences, its automatically generated feedback only addresses errors against rules that are relevant at…