In the 1970s, the movement to the metric system (which has still not completely occurred in the United States) and the advent of hand-held calculators led some to speculate that decimal representation of numbers would render fractions obsolete. This provocative proposition stimulated Zalman Usiskin to write “The Future of Fractions” in 1979. He challenged this opinion by illustrating how it was based on flawed assumptions and erroneous reasoning. Some twenty-eight years later, Usiskin follows up on his fraction remarks in “Some Thoughts about Fractions.” In this work, he bolsters his previous arguments concerning the welfare of fractions and asserts that their study will remain an important part of school mathematics.

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- Author or Editor: Zalman Usiskin x

### Zalman P. Usiskin

The writing of “The future of fractions” in 1978 was motivated by a single incident, the quote that begins the article. That quote—indicating that fractions would become obsolete—is from an editorial essay in the *Virginia Mathematics Teacher* by Lucien Hall. Hall was an active, responsible, and knowledgeable member of the mathematics education community, and when I saw what he had written, I knew that his statement represented the beliefs of many other mathematics teachers and educators. I felt that his statement and this broader view of the impending demise of fractions had to be addressed.

### 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

Arguing against topics already in the curriculum—what this article tries to do—is certain to arouse emotions. Every topic in the curriculum is beloved by some mathematicians or mathematics teachers. If there were not such a constituency, how would the topic have become standard in the first place? We see today how difficult it is it to change practice. Every topic entered the curriculum at some time with what were, for that time, compelifng reasons for its inclusion.

### 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

A collection of problems for classroom use.

### Zalman Usiskin

The “crutch premise” is essentially that if you allow students to use a calculator for arithmetic problems that can be done by hand, then the students will be unable to do arithmetic when the calculator is absent. A corollary to this premise is that calculators should not be used with young students who are still learning arithmetic. Another corollary is that calculators should not be used with older students who have not yet learned arithmetic— that is, calculators should not be used in remedial classes.