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Butts, R.E. (2001). Galileo. In W.H. Newton-Smith (Author), a companion to the philosophy of science (pp. 149-152). Malden, MA: Blackwell Pub.

This excerpt from a reference work is a biographical sketch of Galileo, 17th

century Italian scientist. It outlines five crucial achievements that he made, a few of which include his divergence from Aristotelian theories of science, his advocacy of the real-world applications of mathematics, and his use of experimentation. The author outlines the origins of Galileo's scientific research, particularly in cosmology and his work with the telescope. His work also centered around making geometry less abstract. He pointed out how geometric laws worked in concert with both the natural and mechanical worlds. The most compelling point the author makes, which has application even today to teachers and researchers, is about Galileo's rejection of the dominant philosophies of the era. Though he created controversy, opposition drove his science to advance, leading ultimately to success.

Frodeman, R., & Parker, J. (2009). Intellectual merit and broader impact: The National

Science Foundation's broader impacts criterion and the question of peer review.

Social Epistemology, 23(3-4), 337-345.

The article examines how scientific discovery dictates social values, and how the philosophy of science has evolved. Science has historically been funded with an eye to how it will benefit society. The specific focus of the piece is on the NSF

peer review process and the change of criteria used to allocate funding that occurred in 1997. This change created two criteria: intellectual merit and broader impacts. The broader impacts criteria include education/outreach and an effort to broaden diversity. But the question remains: How is benefit to society determined and measured? The article also raises the question of whether these two criteria categories should be merged, and if intellectual merit is actually a subset of broader impact. This is brought up to point of the potential pitfalls of peer review and to call for a closer examination of its procedures. This is relevant to math education in that research into education practice should be viewed with a mind toward its application in the lives of students and its greater impact on society.

Lesser, L.M. (2000). Reunion of broken parts: Experiencing diversity in algebra.

Mathematics Teacher, 93(1), 62-67.

The author employs a central metaphor of the meaning of algebra, that being the reunion of broken parts. He compares this to the way students interact with algebra, given that they can feel disconnected from it and it is the job of teachers to provide real world applications that will make connections for the students. He takes aim particularly at the need to involve minority students in algebra, as it is foundational for future math study and thus, crucial to many career pursuits. He looks at three methods: history (melding information about the diverse geographical origins of algebra with the problems themselves), multiple representations (using notation, narrative, geometric, graphical, and other representations together to build understanding), and the object concept of function (teaching functions without generalizing about how traits of an individual relate to traits of a group). The article serves to offer some inventive solutions to a common problem in math education: How to make material relevant and compelling to a breadth of students.

Martinez, a.A. (2010). Triangle sacrifice to the gods. 1-11.

The article looks at Pythagoras, particularly the mythology surrounding his life and his most famous discovery, the Pythagorean theorem. It calls into question the historical evidence on which mathematics teachers base their teaching of this theory. The author points out how very little is known about Pythagoras and how he has been canonized by the math discipline because his theory is so useful. He outlines evidence that exists about Pythagoras, taking a critical eye to where and from whom the histories were derived. The tone of the article is playful but also makes an important point. It is critical that teachers take a close look at what information they are passing along as historical fact. It is not enough to simply believe that Pythagoras was a mathematician who gave the world one key principle. It is more advisable to do one's own research and avoid misleading students.

Westfall, R.S. (2001). Newton. In W.H. Newton-Smith (Author), a companion to the philosophy of science (pp. 320-323). Malden, MA: Blackwell Pub.