The word “hypotheses” is used synonymously with “given conditions” in many high school geometry texts. This procedure gives the students an acquaintance with only one of the meanings of the word. There is another sense in which the word is used which is significant enough to warrant its consideration. This sense is the one used by the scientist when he speaks of a possible answer to a problem. As such, it is one of the steps in the thinking process.

### Florian Cajori

A problem as fascinating as the puzzle of the origin of language relates to the evolution of the forms of our numerals. Proceeding on the tacit assumption that each of our numerals contains within itself, as a skeleton so to speak, as many dots, strokes, or angles as it represents units, imaginative writers of different countries and ages have advanced hypotheses as to their origin. Nor did these writers feel that they were indulging simply in pleasing pastime or merely contributing to mathematical recreations. With perhaps one exception, they were as convinced of the correctness of their explanations, as are circle squarers of the soundness of their quadratures.

### Scott Smith

Two problems in December 2014's Calendar (MT, vol. 108, no. 5) set the stage for further investigation. Problem 21 states: There exist many points inside a triangle such that segments can be drawn from the point to the sides of a triangle (not at the vertices) so that the three resulting quadrilaterals have equal areas.

### Christine Phelps-Gregory and Sandy M. Spitzer

One goal in teacher education is to prepare prospective teachers (PTs) for a career of systematic re_ ection and learning from their own teaching. One important skill involved in systematic re_ ection, which has received little research attention, is linking teaching actions with their outcomes on student learning; such links have been termed hypotheses. We developed an assessment task to investigate PTs' ability to create such hypotheses, prior to instruction. PTs (N = 16) each read a mathematics lesson transcript and then responded to four question prompts. The four prompts were designed to vary along research-based criteria to examine whether different contexts in_ uenced PTs' enactment of their hypothesizing skills. Results suggest that the assessment did capture PTs' hypothesizing ability and that there is room for teacher educators to help PTs develop better hypothesis skills. Additional analysis of the assessment task showed that the type of question prompt used had only minimal effect on PTs' responses.

### Don R. Warkentin

Inspired by Jason Slowbe's article “Pi Filling, Archimedes Style” (“Activities for Students,” *Mathematics Teacher* 100, no. 7 [March 2007]: 485–89), I decided to “transform an ordinary [trigonometry] lesson into a rich mathematical exploration” that would be interesting and accessible to my students (Slowbe 2007, p. 487).

### Marshall Lassak, Brad Heller and Steven Siegel

This month's “Delving Deeper” brings you two investigations.

### Lloyd I. Richardson Jr.

An invitation to speak before the fourth-grade class of one of my former students had been accepted. As an enrichment lesson, I had decided to use the Möbius strip and to relate William H. Upson's story (Fadiman 1962), “Paul Bunyan versus the Conveyor Belt,” to the class. Beyond telling the story, any attempts toward hypothesis testing or looking for patterns would depend on class reactions.

### David A. Coffland

Questions about closure on sets of numbers provide a context for hypotheses and proofs.

### James E. Johnson

This article focuses on a particular cognitive process involved in problem solving, namely, the ability to create hypotheses and the ability to relinquish hypotheses (Wason 1960). This process is illustrated with activities suitable for children and adults—activities that illustrate how to stres the importance of producing and evaluating new ideas while solving a problem.

### Thomas J. Cooney

The purpose of this article is to examine the relevance of the International Study of Achievement in Mathematics to those mathematics educators in volved in the preparation of teachers. While the explicit purpose of the Study is not to provide directions for teacher-education programs, the Study nevertheless examines questions which indirectly involve the training of mathematics teachers. While all of the hypotheses in the Study deal with some aspect of education as they relate to mathematics, some hypotheses, namely Hypotheses 12–26, arc involved with probl ems more directly related to the mathematics education community. In particular, Hypotheses 12, 13, 14, 15, 18, 19, and 20. in the opinion of the writer, take on added significance when considering the preparation of mathematics teachers.