Card tricks based on mathematical principles can be a great way to get students interested in exploring some important mathematical ideas. Bonomo (2008) describes several variations of a card trick that rely on nested floor functions, but these generally go beyond the reach of beginning algebra students. However, a simple spreadsheet implementation shows students why the card trick works and allows them to explore several variations. As an added bonus, students are introduced to composite functions, the floor function, and iteration, and they learn how to use formulas and the INT function in Microsoft Excel. The depth of the mathematical explanation can be varied according to students' background.

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### Scott Steketee and Daniel Scher

Experience with multiple representations fosters students' robust understanding of what functions are, how they behave, and how they can be composed.

### Gregory D. Foley

Ellipses vary in shape from circular to nearly parabolic. An ellipse's eccentricity indicates the location of its foci, but its aspect ratio is a direct measure of its shape.

Students analyze a photograph to solve mathematical questions related to the images captured in the photograph. This month the editors consider photographs of African bowls made of recycled telephone wire. The mathematics involves trigonometry, parametric equations and their graphs, and linear regression.

### Low Chee Soon

Use freedom of choice to promote students' mathematical flexibility.

### Kristen Lew and Juan Pablo Mejía-Ramos

This study examined the genre of undergraduate mathematical proof writing by asking mathematicians and undergraduate students to read 7 partial proofs and identify and discuss uses of mathematical language that were out of the ordinary with respect to what they considered conventional mathematical proof writing. Three main themes emerged: First, mathematicians believed that mathematical language should obey the conventions of academic language, whereas students were either unaware of these conventions or unaware that these conventions applied to proof writing. Second, students did not fully understand the nuances involved in how mathematicians introduce objects in proofs. Third, mathematicians focused on the context of the proof to decide how formal a proof should be, whereas students did not seem to be aware of the importance of this factor.

### Scott Steketee and Daniel Scher

Students see that geometric transformations—dilation and translation—correspond to algebraic parameters—m and b—in the familiar equation for a linear function.

A set of problems of many types.

A set of problems of many types.