"Algebraic Thinking and Geometric Thinking", Thomas Banchoff, Brown University, Providence, RI, In Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook, National Council of Teachers of Mathematics 2008. Banchoff shows how geometric shapes and visualization can be used to make algebraic procedures understandable by students.
"Developing Algebraic Thinking in Earlier Grades", Jinfa Cai & John Moyer, University of Delaware and Marquette University, In Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook, National Council of Teachers of Mathematics 2008. Cai and Moyer report that the Singapore curriculum (NG 2004, Ng and Lee) feature "pictorial equation solving. "It is common to use pictures to model problem situations in the first and second grades. Later, these pictures are replaced by more abstract rectangles. To help third, fourth, and fifth graders engage with algebra word problems while avoiding the necessity of using algebraic equations, students are taught to use rectangles to solve algebra word problems…"
Learning Model
"Farewell, Lecture?, Eric Mazur, Harvard University, Science 2, Jan 2009, P 50. "Data obtained in my class and in classes of colleagues worldwide, in a wide range of academic settings and a wide range of disciplines, show that learning gains nearly triple with an approach that focuses on the student and on interactive learning (7,8)"... "So evidence is mounting that readjusting the focus of education from information transfer to helping students assimilate material is paying off. My only regret is that I love to lecture."
Competition:
"A Personal Tutor for Algebra, Yudhijit Bhattacharjee, Science 2, January 2009,Vol 323 P64.
"Yet there's no guarantee that CT (Cognitive Tutor from Carnegie Learning) will improve learning. In at least two other studies the product led to no significant rise in student test scores"
Productive Practice
Anders Ericsson, a Professor at Florida State University, and his colleagues, have spent years researching expert performance in many different fields. What he found is that most great athletes, or musicians or academics he studied were not born great. They attained greatness through dedication, hard work and what he terms "deliberate practice". This type of practice must push you to higher and higher levels of performance, requiring more and more difficult mental or physical excursion. And, you must receive timely and accurate feedback on your performance. It is no surprise that elite athletes have coaches to ensure that they are receiving this feedback and constantly striving to reach their potential. The elements of "deliberate practice" are much the same as those necessary for what we call "productive practice", under the guidance of a teacher or Personal Learning Manager.
Ericsson, K.A., Charness, N., Feltovich, P., and Hoffman, R.R. (Eds.). Cambridge Handbook of Expertise and Expert Performance. Cambridge, UK: Cambridge University Press.