Transparencies were commonplace in the 1980s when the first NCTM Standards were released. This article reflects on how the use of an overhead projector and transparencies helped to enact the Process Standards - and make the real purpose of learning mathematics more transparent to students.

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## On My Mind: 5 Language Substitutions When Teaching Fractions

### readers speak out

### Jennifer M. Bay-Williams

Being mindful of language used during discussions of fractions can promote greater understanding and retention.

### Jennifer M. Bay-Williams

Through Brilliant, Clever Poems and prose, Shel Silverstein has worked his way into many classrooms and into the hearts of students and teachers. He wrote poems with an obvious knowledge that a child's mind is active and curious, and middle school students love reading and listening to his words. Imbedded in many of his poems and prose are opportunities to do mathematics in ways that will get students' minds “flickerin'.” Using a poem, picture book, or portions of a novel can raise the curiosity of middle school students and can increase their desire to solve mathematics problems. As students engage in solving literature-based mathematics lessons, they are applying mathematics in different contexts and making connections among mathematical ideas, which are expectations outlined in the Connections Standard in Principles and *Standards for School Mathematics* (NCTM 2000). In the following sections, four delightful Silverstein works prompted engaging mathematics explorations. Each selection focused on a different mathematical strand, although there is much integration of other concepts in the problems.

### Jennifer M. Bay-Williams

In his song “Still Crazy after All These Years,” Paul Simon (1974) writes, “I seem to lean on old familiar ways.” Shipley's article “Algebra in the Elementary Schools,” written in 1912, offers the opportunity for us to reflect on what we have accomplished, what we have learned, and what remains *hazy* even 100 years later with respect to teaching and learning mathematics. Do we still lean on old familiar ways of teaching algebra? This article has many messages that are relevant to us today in thinking about algebra and about teaching and learning mathematics in general. In addition, it raises the following questions: How does Shipley's conception of algebra compare with our conception of it today? What broader issues does he address that are still prevalent today? How far have we come in our implementation of his ideas? What aspects of teaching and learning mathematics in elementary school remain hazy after all these years?

### Jennifer M. Bay-Williams

Patterns have long been part of early mathematics experiences. The K–4 Patterns and Relationships Standard in Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) was replaced in Principles and Standards for School Mathematics (NCTM 2000) with a K–12 Algebra Standard. This Standard encompasses patterns, functions, and some topics that are beyond what traditionally was considered to be algebra. However, the word algebra, often associated with content covered in a traditional middle school or high school course, can evoke feelings of anxiety and raise questions of appropriateness when discussed in relation to elementary school children. What is algebra in elementary school if it is more than identifying and extending patterns in the early grades yet is not the abstract content of an algebra course?

### Jennifer M. Bay-Williams

While talking with parents and other noneducators, I try to be careful when using education jargon; however, that does not always mean that we are communicating about the same thing.

### Gina Kling and Jennifer M. Bay-Williams

Basic fact fluency has always been of interest to elementary school teachers and is particularly relevant because a wide variety of supplementary materials of varying quality exist for this topic. This article unpacks eight common unproductive practices with basic facts instruction and assessment.

### Jennifer M. Bay-Williams and Stefanie Livers

Learn when and how teachers can use rich mathematical vocabulary to develop and maximize students' learning, particularly English Language Learners and struggling readers.

### Jennifer M. Bay-Williams and Graham Fletcher

Putting a twist on a popular mathematical tool, this collection of activities shows how placing a number 1 in the bottom-left cell and a 100 in the top-right cell can better support student reasoning.