Techniques for teaching mathematics terminology allow adolescents to expand their abstract reasoning ability and move beyond operations into problem solving.

## Informing Practice: A Hybrid Perspective on Functions

### research matters for teachers

### Eric Weber

Formal notions of function, which appear in middle school, are discussed in light of how teachers might complement the input-output notion with a covariation perspective.

### Ann C. McCoy, Rita H. Barger, Joann Barnett and Emily Combs

While filling vases with water and observing volume and height relationships, students learn the fundamentals of functions.

### Scott Steketee and Daniel Scher

Transformations using dynamic software can provide a unique perspective on a common topic.

### Low Chee Soon

Use freedom of choice to promote students' mathematical flexibility.

### Courtney Starling and Ian Whitacre

Introduce your students to a fun and innovative game to encourage precise communication

MT's letters to the editor department. Readers comment on published articles and share their mathematical interests.

### Arnulfo Pérez, Bailey Braaten and Robert MacConnell

A hands-on, project-based modeling unit illustrates how real-world inquiry deepens student engagement with function concepts.

### 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.

### David Rock and Mary K. Porter

A monthly set of problems is aimed at a variety of ability levels.