Common situations, like planning air travel, can become grist for mathematical modeling and can promote the mathematical ideas of variables, formulas, algebraic expressions, functions, and statistics.

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- Author or Editor: Zalman Usiskin x

### Zalman Usiskin

A strong curriculum is not the sole reason for Singaporean students' success on international assessments.

### Zalman Usiskin

This article briefly describes the timing of the first concentrated study of algebra over the 100 years of NCTM, from a 9th-grade course taken by only about 1/5 of students to a course taken by virtually all students, with almost half taking it in 8th grade.

### Zalman Usiskin

Elementary or first-year algebra is the keystone subject in all of secondary mathematics. It is formally studied by students from grade levels as early as seventh grade and as late as college, but begun and completed more often in ninth grade than at any other time. The main purpose of this article is to question that timing. The conclusion to be argued here is that most students should begin the study of algebra one year *earlier* than they now do. This conclusion is contrary to a recommendation currently subscribed to by the National Council of Teachers of Mathematics and to the views of a number of leaders in mathematics education. I attempt to show here that these leaders have been misguided.

### Zalman Usiskin

Little reason exists for engaging in enrichment for enrichment's sake or in problem solving merely for mental practice unless one wishes to waste precious time. The curriculum is too crowded to allow such luxuries even for gifted students, because we want these students to be aware of a much greater range of mathematical concepts than their less knowledgeable peers.

### Zalman Usiskin

Mathematics is commonly believed to be difficult. Tell a person that you teach mathematics, and the response will reflect a view that you are smarter than most people (unless the person you're talking to is a mathematician!). We remember the hardest mathematics course we ever took or the one we didn't understand, and we read journals and feel that much is beyond us.

### Zalman Usiskin

The subfield of pure mathematics that has grown most significantly in the past few decades is that of algebra, by which is meant “higher” or “abstract” algebra and linear algebra. Twenty years ago courses in algebra were at the advanced undergraduate and graduate level, and it was easy to become a certified mathematics teacher without having any knowledge of groups, rings, fields, or vector spaces. Today virtually all prospective teachers take a course in which some of these structures are studied.

### Zalman Usiskin

It is possible to form a set in an infinite additive group by beginning with any nonzero real *a* and adding it to itself over and over again, then including zero and the opposites of all numbers in the set. We call such a set the set of integral multiples of *a.* If *a =* 3, then here is such a set.

### Zalman Usiskin

In this article, two problems are called *equivalent* if they can be solved using the *same* mathematics.