This is a practice-led, conceptual paper describing selected means for action learning and concept motivation at all levels of mathematics education. It details the approach used by the authors to devise insights for practitioners of mathematics teaching. The paper shows that this approach in mathematics education based on action learning in conjunction with the natural motivation stemming from common sense is effective. Also, stimulating questions, computer analysis (internet search included), and classical famous problems are important motivating tools in mathematics, which are particularly beneficial in the framework of action learning. The authors argue that the entire K-20 mathematics curriculum under a single umbrella is practicable when techniques of concept motivation and action learning are in place throughout that broad spectrum. This argument is supported by various examples that could be helpful in practice of school teachers and university instructors. The authors found pragmatic cause for action learning within mathematics education at virtually any point in student academic lives.

The availability of sophisticated computer programs such as Wolfram Alpha has made many problems found in the secondary mathematics curriculum somewhat obsolete for they can be easily solved by the software. Against this background, an interplay between the power of a modern tool of technology and educational constraints it presents is discussed. Using topics from algebra (equations) and elementary number theory (summation of powers of integers), the paper suggests ways of developing problems that are both technology-immune and technology-enabled in the sense that whereas software can facilitate problem solving, its direct application is not sufficient for finding an answer. Stemming from the author's work with secondary mathematics teacher candidates, this paper highlights the appropriate use of technology as support system for multiple ways of knowing and knowledge construction in the modern classroom.

The paper is a reflection on the author’s work with students enrolled in undergraduate and graduate elementary mathematics content and methods courses. Two specific pedagogical issues make up the focus of the paper. The first issue deals with demonstrating to future teachers the diversity of mathematical ideas behind a contextual question asked by a second-grade pupil allowing for multiple solution strategies to be used in addressing the question. The second issue deals with the use of Wolfram Alpha in aiding different mathematical features of this demonstration. The paper is congruous with mathematics teaching standards used across six continents and illustrated by reflective comments of the author’s students regarding their mathematics teacher education experiences. Teaching ideas shared in the paper may be of interest to instructors who want to explore elementary mathematics in depth with teacher candidates.

The paper is written to demonstrate the applicability of the notion of triangulation typically used in social sciences research to computationally enhance the mathematics education of future K-12 teachers. The paper starts with the so-called Brain Teaser used as background for (what is called in the paper) computational triangulation in the context of four digital tools. Computational problem solving and problem formulating are presented as two sides of the same coin. By revealing the hidden mathematics of Fibonacci numbers included in the Brain Teaser, the paper discusses the role of computational thinking in the use of the well-ordering principle, the generating function method, digital fabrication, difference equations, and continued fractions in the development of computational algorithms. These algorithms eventually lead to a generalized Golden Ratio in the form of a string of numbers independently generated by digital tools used in the paper.

Motivated by work done with pre-teachers of mathematics in a problem-solving course, this paper shows how computing technologies, including a spreadsheet and Maple, facilitate an informal journey into a hidden aspect of the formal content of the pre-college curriculum dealing with the arithmetic of partitions. By using three problems from different grade levels within a state curriculum as an example, the paper suggests that a deeper perspective on seemingly disconnected problem-solving contexts may serve as a powerful didactical tool in helping teachers to appreciate mathematics and its pedagogy as an integrated whole. The connection of the hidden aspect of the curriculum to the concept of mathematical play is also explored.