Euler's method for solving initial value problems is an excellent vehicle for observing the relationship between discretization error and rounding error in numerical computation. Reductions in stepsize, in order to decrease discretization error, necessarily increase the number of steps and so introduce additional rounding error. The problem is…

The development of patterning strategies during the year prior to formal schooling was studied in 53 children from 2 similar preschools. One preschool implemented a 6-month intervention focusing on repeating and spatial patterns. Children from the intervention group demonstrated greater understanding of unit of repeat and spatial structuring, and…

Distributions are the basis for an enormous amount of theoretical and applied work in statistics. While there are formal definitions of distributions and many formulas to characterize them, it is important that students at first get a clear introduction to this basic concept. For many of them, neither words nor formulas can match the power of a…

The objective of this paper is to study students' difficulties when they have to ascribe the same meaning to different representations of the same mathematical object. We address two theoretical tools that are at the core of Radford's cultural semiotic and Godino's onto-semiotic approaches: objectification and the semiotic function. The analysis…

This paper is an attempt to add to the foundation of our understanding of meaning making in mathematics education. This attempt seems to be necessary as a growing body of research, primarily in developmental psychology, begins to change our view of early human development. Empathy, reciprocity, and implicit understanding seem to be more suitable…

The least squares method of fitting a line is not one that naturally occurs to students. We present three tasks to understand student views on how lines may be fit.

This note shows how some density functions for continuous probability distributions can be constructed in a transparent manner to help students appreciate their development.

This article presents the views of 24 nationally recognized authorities in the field of mathematics, and in particular the calculus, on student understanding of the first-year calculus. A framework emerged that includes four overarching end goals for understanding of the first-year calculus: (a) mastery of the fundamental concepts and-or skills of…

We give an overview of why it is important to include sustainability in mathematics classes and provide specific examples of how to do this for a calculus class. We illustrate that when students use "Excel" to fit curves to real data, fundamentally important questions about sustainability become calculus questions about those curves. (Contains 5…

One of the often overlooked aspects of modern statistics is the analysis of time series data. Modern introductory statistics courses tend to rush to probabilistic applications involving risk and confidence. Rarely does the first level course linger on such useful and fascinating topics as time series decomposition, with its practical applications…

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…

This paper reports a teaching experiment in which two students engaged in tasks that challenged them to describe a final state for a variety of infinite iterative processes. The results from the study indicate that the students used multiple reasoning strategies for addressing these tasks. Refinements in the students' reasoning occurred as…

This article was inspired by a set of 12 cylindrical cups, which are volumetrically indexed; that is to say, the volume of cup "n" is equal to "n" times the volume of cup 1. Various sets of volumetrically indexed cylindrical cups are explored. I demonstrate how this children's toy is ripe for mathematical investigation, with connections to…

The method of propositional manipulation (MPM) aims to help students develop conceptual understanding of statistics by guiding them into self-explaining propositions. To explore task- and student-related factors influencing students' ability to learn from MPM, twenty undergraduate students performed six learning tasks while thinking aloud. The…

A storied math context helps students engage both emotionally and cognitively with mathematics and helps show that mathematics develops out of human experience. Children's literature also models mathematical storytelling for students, and creates opportunities for them to retell and extend stories. This article describes mathematics investigations…