Mistakes can be a source of frustration for teachers and students in mathematics classrooms because they reveal potential misunderstandings or a lack of learning. However, increasing evidence shows that making mistakes creates productive pathways for learning new ideas and building new concepts (Boaler 2016; Borasi 1996). Learning through…

Despite repeated discussions and practice, algebra students continue to make variable errors, in many ways, year after year. This same problem appeared thirty years ago in a list of common errors that math teachers today would immediately recognize, many involving exponents and distribution (Marquis 1988). Similar complaints even appeared in the…

This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…

In the criminal justice system, defendants accused of a crime are presumed innocent until proven guilty. Statistical inference in any context is built on an analogous principle: The null hypothesis--often a hypothesis of "no difference" or "no effect"--is presumed true unless there is sufficient evidence against it. In this…

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…

Mathematics teachers are often asked, "What is the most difficult topic for you to teach?" Their answer is teaching students to count. The concepts can be challenging and slippery to apply in problems. Many times, no rigid procedures or formulas can be used to solve the problems directly, and students simply do not know where or how to approach…

The mathematics that students see in their textbooks is highly polished. The steps required to solve a problem are all clearly laid out. Thus, students are denied what could be a valuable learning experience. Often when students meet a problem that differs only slightly from the ones in the book, they are unable to proceed and afraid to "play…

The purpose of this article is to point out some of the problems that arise in the use of calculators to illustrate derivatives, due to rounding or truncation, by the calculator. Several illustrations of numerical differentiation techniques are also given. (Author/MP)

Several examples of mathematical diagnosis using microcomputer software are presented, with possible future considerations suggested. Three limitations of the software are discussed. (MNS)

This article, a reprint from 1930, offers an interesting perspective on the importance of diagnostic and prescriptive teaching as well as the role of drill and the use of objectives in mathematics. (Author/MK)

An approach to teaching factors and terms is offered that builds directly on students' knowledge of arithmetic. Errors arising from factor and term confusions are noted, followed by the pedagogical strategy of exploration, invention, and discovery. (MNS)

A lesson in editing was learned from a study of errors made on a national mathematics test in Israel. Analysis indicated that the errors were often caused by careless printing of the test, misleading figures, poor designations, or ambiguous phrasing. (MNS)

Several common errors reflecting difficulties in probabilistic reasoning are identified, relating to ambiguity, previous outcomes, sampling, unusual events, and estimating. Knowledge of these mistakes and interpretations may help mathematics teachers understand the thought processes of their students. (MNS)

The author mentions several broad types of student elementary algebra errors, including incorrect cancellation and cross-multiplication, notes confusing terms and factors, and gives several suggestions to help students avoid these. (MN)

Three theorems for Pythagorean triples are presented, with discussion of how students can amend their ideas about such numbers. (MNS)