Recursive reasoning is a powerful tool used extensively in problem solving. For us, recursive reasoning includes iteration, sequences, difference equations, discrete dynamical systems, pattern identification, and mathematical induction; all of these can represent how things change, but in discrete jumps. Given the school mathematics curriculum's…

This is an article about abstraction, generalization, and the beauty of mathematics. We claim that abstraction and generalization in of itself may very well be a beauty of the human mind. The fact that we humans continue to explore and expand mathematics is truly beautiful and remarkable. Many years ago, our ancestors understood that seven stones,…

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector--WiiGraph--with 11-year-old children. In the selected episodes, the children used this motion detector to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) propose a new vision for the mathematics classroom with updated content standards and Standards for Mathematical Practice (SMP). These practices are founded on NCTM processes (Problem Solving, Reasoning and Proof, Communication, Representation, and Connections) and abilities…

The Common Core State Standards (CCSSI 2010) for Mathematical Practice have relevance even for those not in CCSS states because they describe the habits of mind that mathematicians--professionals as well as proficient school-age learners--use when doing mathematics. They provide a language to discuss aspects of mathematical practice that are of…

Students often struggle with concepts from abstract algebra. Typical classes incorporate few ways to make the concepts concrete. Using a set of woven paper artifacts, this paper proposes a way to visualize and explore concepts (symmetries, groups, permutations, subgroups, etc.). The set of artifacts used to illustrate these concepts is derived…

Making sense of mathematical concepts and solving mathematical problems may demand different forms of reasoning. These could be either domain-based, such as algebraic, geometric or statistical reasoning, while others are more general such as inductive/deductive reasoning. This article aims at giving visibility to a particular form of reasoning…

"Modeling Mathematical Ideas" combining current research and practical strategies to build teachers and students strategic competence in problem solving.This must-have book supports teachers in understanding learning progressions that addresses conceptual guiding posts as well as students' common misconceptions in investigating and…

For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…

This is the first of a two-part article that presents a theory of unit construction and coordination that underlies radical constructivist empirical studies of student learning ranging from young students' counting strategies to high school students' algebraic reasoning. My explanation starts with the formation of arithmetical units, which presage…

Tasks that develop spatial and algebraic reasoning are crucial for learning and applying advanced mathematical ideas. In this article, the authors describe how two early childhood teachers used stories as the basis for a unit that supports spatial reasoning in kindergartners and first graders. Having mathematical experiences that go beyond…

This article describes Realistic Mathematics Education (RME), a design theory for mathematics education proposed by Hans Freudenthal and developed over 40 years of developmental research at the Freudenthal Institute for Science and Mathematics Education in the Netherlands. Activities from a unit to develop student understanding of logarithms are…

In today's world, where the volume of knowledge everyone must deal with is increasing exponentially, many educators agree that schools must focus on developing skills for life-long learning. But what does that mean for an area such as algebra? Teachers' goal in school algebra should be to guide students to "work smarter" with algebraic symbols and…

High schools may still be anchored to 20th century expectations, but what are the critical guideposts for a 21st century high school education? There are many specific skills and competencies that young people will need to succeed, but more than particular skills, they will need the cognitive capacity to educate themselves throughout their entire…

This article describes the rich thinking that can go into solving a difficult equation. The authors also give some sources for challenging mathematical problems.