The draft of the NCTM *Standards* document states that problem solving should be the central focus of the mathematics curriculum (Commission on Standards for School Mathematics of the NCTM 1987). Now, more than ever, problem solving is being defined as a process. Akers (1984, 34) defined problem solving as “what you do when you don't know what to do,” and Schoenfeld (1988) wrote, “Indeed, ‘figuring it out’ is what mathematics is all about” (p. 8). Mathematics teachers, left with the task of determining how problemsolving skills should be taught, have the potential to play a key role in developing and sharing problems that interest students. I will share a certain type of problem that I think might be of some interest at the secondary and college levels.

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- Author or Editor: Randolph A. Philipp x

### Randolph A. Philipp

### Randolph A. Philipp

The concept of variable is one of the most fundamental ideas in mathematics from elementary school through college (Davis 1964; Hirsch and Lappan 1989). This concept is so important that its invention constituted a turning point in the history of mathematics (Rajaratnam 1957). However, research indicates that students experience difficulty with the concept of variable, a difficulty that might partially be explained by the fact that within mathematics, variables can be used in many different ways (Rosnick 1981; Schoenfeld and Arcavi 1988; Wagner 1983).

### Randolph A. Philipp

Up until recently I wasn't even aware that other people in the world did things [arithmetic algorithms] differently. I thought God sent these. That's the way of the world. The first day you [to another teacher] were talking about some way you did things differently in Ireland. It never occurred to me. I thought there was a world standard.

### Randolph A. Philipp and Casey Hawthorne

Using cups of sugar, this sequence of division tasks for K–grade 12 and adult learners highlights how seeing “wholes” results in fewer “holes” in reasoning.

### Victoria R. Jacobs and Randolph A. Philipp

Teachers demonstrated four categories of reasoning when deciding how to respond to students.

### Victoria R. Jacobs and Randolph A. Philipp

How did Misha solve the problems in figure 1, and what mathematical understandings do her strategies reflect? Misha's written work provides a rich context in which prospective and practicing teachers can discuss issues of mathematics, teaching, and learning. These discussions might include conversations about the similarities and differences between Misha's two strategies, what she must have understood about place value to generate these strategies, and what type of instruction likely preceded and should follow this problem-solving effort.

### Randolph A. Philipp and Bonnie P. Schappelle

Examples in which we relate the syntactic—including symbol manipulation—and semantic—including meaningful use of symbols—aspects of algebra and examine algebra as generalized arithmetic.

### Victoria R. Jacobs, Heather A. Martin, Rebecca C. Ambrose and Randolph A. Philipp

Avoid three common instructional moves that are generally followed by taking over children's thinking.

### Eva Thanheiser, Randolph A. Philipp, Jodi Fasteen, Krista Strand and Briana Mills

Helping prospective elementary school teachers (PSTs) recognize that they have something useful to learn from university mathematics courses remains a constant challenge. We found that an initial content interview with PSTs often led to the PSTs' changing their beliefs about mathematics and about their understanding of mathematics, leading to the recognition that (a) there is something to learn beyond procedures, (b) their own knowledge is limited and they need to know more to be able to teach, and (c) engaging in the mathematical activities in their content courses will lead them to learning important content. Thus, such an interview can set PSTs on a trajectory characterized by greater motivation to learn in their content courses.

### Ian Whitacre, Jessica Pierson Bishop, Randolph A. Philipp, Lisa L. Lamb and Bonnie P. Schappelle

A story problem about borrowing money may be represented with positive or negative numbers and thought about in different ways. Learn to identify and value these different perspectives.