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An example of how middle-school engineering projects can be used as contexts for students’ exploration of previous and new mathematics topics is shared here. In this example, students algebraically determined where to place frets on the fretboard of a stringed instrument they were designing and along the way made use of a fractional exponent that was beyond what they experienced in prior courses. Students engaged in reasoning and problem solving, had extended meaningful discourse with each other, stayed motivated through productive struggle, and actively built new mathematical understandings from new experiences and prior mathematical knowledge. 

The original functions first described in Paul Goldenberg's "Exploring Language with Logo" are compared with modern-day implementations of the same concepts in Snap!, a block computing language developed at the University of California at Berkeley. 

Logo was the first computing language explicitly designed for children. The name, Logo, was chosen to emphasize its intended use for symbolic processing and exploration of language, among other subjects. (“Logos” is the Greek word for word.) This focus reflected a belief that this might offer a more inviting entry point into programming for novices than traditional school mathematics, and thereby making it more broadly appealing to diverse audiences. This article reviews Paul Goldenberg's landmark work, Exploring Language with Logo, placing it in context. 

Incorporating modeling activities into classroom instruction requires flexibility with pedagogical content knowledge and the ability to understand and interpret students’ thinking, skills that teachers often develop through experience. One way to support preservice mathematics teachers’ (PSMTs) proficiency with mathematical modeling is by incorporating modeling tasks into mathematics pedagogy courses, allowing PSMTs to engage with mathematical modeling as students and as future teachers. Eight PSMTs participated in a model-eliciting activity (MEA) in which they were asked to develop a model that describes the strength of the magnetic field generated by a solenoid. By engaging in mathematical modeling as students, these PSMTs became aware of their own proficiency with and understanding of mathematical modeling. By engaging inmathematical modeling as future teachers, these PSMTs were able to articulate the importance of incorporating MEAs into their own instruction.