Even so, his intertemporal model became a vital part of macroeconomics, particularly the consumption function, which many well-known economists advanced with increasingly acceptable

mathematical rigor.

The essential element of

mathematical rigor lies in the fact that individuation and concretization of the objects are developed out of the definitions and principles from an external source.

He brought new

mathematical rigor to his field, helped transform the MIT economics department into one of the strongest in the world, wrote a weekly column for Newsweek for 15 years, and in 1970 became just the second person to win the Nobel Prize for economics.

The class was filled with analytical and

mathematical rigor (one did not dare miss a class) and aside from the exams there was a term paper requirement that everyone (including Clarence) took very seriously.

These concepts are (hopefully) now well known to high school students, but in 1944 they represented a new perspective: that living systems could be analyzed using similar

mathematical rigor as had led to enormous advances in high-energy physics, which provided the foundations to search for previously undiscovered elemental particles.

An ideal algorithm uses

mathematical rigor to reduce code size, memory requirements, and ultimately, cost, while still beating current industry standards for accurate and precise diagnoses.

Whereas the

mathematical rigor of Descartes and later philosophers proved its worth in natural philosophy, it was absolutely unsuitable for grasping the many ambiguities, or even paradoxes, of the human condition or of religious belief.

If you crave more

mathematical rigor, however, you can explore most of the topics further in boxed sections woven throughout the chapters.

Yet, Gutstein has argued effectively and insightfully that tying mathematics knowledge to students' cultural and experiential backgrounds, while helping them develop the tools of critical thinking and

mathematical rigor, empowers them in all areas of their lives, inside and outside the classroom (Gutstein, 2003; Gutstein, Lipman, Hernandez, & de los Reyes, 1997).

Consequently, the customary sequences of definitions, lemmas and theorems will be omitted; nevertheless a

mathematical rigor in all the arguments has been kept.

An important supplemental conclusion for preservice teachers to draw from this lesson is that constructivism yields mathematically correct solutions that hold students responsible for the same level of

mathematical rigor that they have always been expected to attain.

Despite the fact that it covers a wide range of material, the book presents the concept gradually in accessible and self-contained stages and consistently steers away from the deeper theoretical side of learning machines without sacrificing too much

mathematical rigor.