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…

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…

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…

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

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)

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

Ten crucial mathematical concepts with which errors are made are listed, with methods used to teach them to high school students. The concepts concern order, place values, inverse operations, multiplication and division, remainders, identity elements, fractions, conversions, decimal points, and percentages. (MNS)

Presents an activity applicable at the beginning of the year to review necessary algebra concepts that are often misunderstood by students. Presents a true/false worksheet made up of examples illustrating common algebra and trigonometry errors made by students. (MDH)

Discusses the use of feedback from students and the analysis of students' error patterns to understand why students develop misconceptions in mathematics. Looks at learning vis-a-vis the abstract nature of mathematics and the students' cognitive development. (MDH)

Brief articles are included on exploring mathematical patterns, with two computer programs listed for finding happy numbers; a simple periodic function in trigonometry; and comments on humorous student errors. (MNS)