This page will discuss approaches to teaching of mathematics (as it is usual in the subject, with emphasis on the pre-university level, where the care in didactical aspects is the most relevant) and the outstanding sources of the relevant materials about teaching.
Scope
Pedagogically well written introductory books in mathematics, rather than about pedagogical matter, are also of our concern, but they will preferably be posted under elementary mathematics, introductions to mathematics?, elementary geometry? and related pages.
Overview
Most traditionally, teaching methods were improvized adaptations of communication of subject matter from the knowledgeable teacher to a learner.
Modern educational theory is greatly influenced by the works on child psychology. In particular, it has been investigated which cognitive aspects can be achieved at certain age, or within certain educational or other cognitive experiences.
It is now commonly accepted that the advanced and long term knowledge is better achieved if the learner is also a discoverer. This means in mathematics that the emphasis on procedural knowledge should be replaced by wider experience in which a student discovers her own ways to approach the problems which make up the subject. The teacher and the learning environment hence have to anticipate and foster also specific processes in learning the subject rather then only the goals and supposed content matter. Many authors however acknowledge the importance of balance with more traditional coaching and somewhat standardized procedural techniques (micromanagement being counterproductive). While in most educational taxonomies application of knowledge comes only at very hi stages in taxonomy, applying and experiencing concepts in practice, applications and in interaction with technology is considered necessary even at initial steps, and lower degrees of learning the subject. This should be therefore taken into account when creating the goals of mathematics curricula.
Douglas A. Grouws ed., Handbook of research on mathematics teaching and learning: a project of the National Council of Teachers of Mathematics, Macmillan Library Reference 1992
Liping Ma, Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series)
Liping Ma, A critique of the structure of U.S. elementary school mathematics, Notices AMS, Oct 2013 pdf
Harold W. Stevenson (educational psychologist), Mathematics learning in early childhood, NCTM, 1985
John A. Van de Walle et al. Elementary and middle school mathematics. Teaching developmentally, Pearson 2004, 2007, 2010, 2013
J. Mayberry, The Van Hiele levels of geometric thought in undergraduate preservice teachers, Journal for Research in Mathematics Education 14 (1): 58–69 (1983) doijstor
Richard Askey?, Good intentions are not enough, pdf
Mariya Boyko, The “New Math” Movement in the U.S. vs Kolmogorov’s Math Curriculum Reform in the U.S.S.R., html
R. Balian, A. Connes, Bismut, Lafforgue, Serre, Les savoirs fondamentaux au service de l’avenir scientifique et technique, pdf, a text lamenting the current state of the scientific part of education in France
G. Ziegler, Teaching and learning “What is mathematics?”, in Proc.ICM 2014, Seoul, vol. 4
Alexander Karp, Bruce R Vogeli (eds.), Russian mathematics education, 2 vols, World Sci. Publ.
Lingguo Bu, Robert Schoen (eds.), Model centered learning, Pathways to mathematical understanding using geogebra, vol. 6 of Modeling and simulations for learning and instruction
Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckle, The advantage of abstract examples in learning math, Science Magayibe 2012 pdf
Discussing elementary mathematical notions avoiding “pleasure principle” in education:
Carl E. Linderholm, Mathematics made difficult, Wolfe Publishing 1972
The following epistemiology of math article offers also discussion on educational issues and has related bibliography
Zbigniew Semadeni, The triple nature of mathematics: deep ideas, surface representations, formal models, article
Cognitive, linguistic and cultural aspects of mathematics (which are of relevance for learning) are emphasized in