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How people see the world, even how they research it, is influenced by beliefs. Some beliefs are conscious and the result of research, or at least amenable to research. Others are largely invisible. They may feel like "common knowledge" (though myth, not knowledge), unrecognized premises that are part of the surrounding culture. As we will explain, people also hold ideas in both a detailed form and in a thumbnail image and may not notice when they are using the low-resolution image in place of the full picture. In either case, unrecognized myths about how young learners develop mathematical ideas naturally or with instruction are insidious in that they persist unconsciously and so sway research and practice without being examined rigorously. People are naturally oblivious to the ramifications of unrecognized premises (myths) until they encounter an anomaly that cannot be explained without reexamining those premises. Like all disciplines, mathematics education is shaped and constrained by its myths. This article is a conceptual piece. It uses informally gathered (but reproducible) classroom examples to elaborate on two myths about mathematics learning that can interfere with teaching and can escape the scrutiny of empirical research. Our goal is to give evidence to expand the questions researchers think to pose and to encourage thoughtful reappraisal of the implications of the myths. The myths we will discuss involve the order in which mathematical ideas are learnable and the "unity" of mathematical topics, with special attention to algebra. With examples, we will show that some ideas develop at a strikingly counterintuitive and early time. Taking advantage of such unexpectedly early developments can let educators devise pedagogies that build on the logic young children already have rather than predicating learning on statistically observed learning patterns or even the apparent structure of mathematics. Acknowledging such early developments might change the questions researchers ask and change how they study children's mathematical learning, with the possible result of changing how children are taught.

Natural language helps express mathematical thinking and contexts. Conventional mathematical notation (CMN) best suits expressions and equations. Each is essential; each also has limitations, especially for learners. Our research studies how programming can be a advantageous third language that can also help restore mathematical connections that are hidden by topic-centred curricula. Restoring opportunities for surprise and delight reclaims mathematics' creative nature. Studies of children's use of language in mathematics and their programming behaviours guide our iterative design/redesign of mathematical microworlds in which students, ages 7-11, use programming in their regular school lessons "as a language for learning mathematics." Though driven by mathematics, not coding, the microworlds develop the programming over time so that it continues to support children's developing mathematical ideas. This paper briefly describes microworlds EDC has tested with well over 400 7-to-8-year-olds in school, and others tested (or about to be tested) with over 200 8-to-11-year-olds. Our challenge was to satisfy schools' topical orientation and fit easily within regular classroom study but use and foreshadow other mathematical learning to remove the siloes. The design/redesign research and evaluation is exploratory, without formal methodology. We are also more formally studying effects on children's learning. That ongoing study is not reported here.

This article reports on an exploration of how second-graders can learn mathematics through programming. We started from the theory that a suitably designed programming language can serve children as a language for expressing and experimenting with mathematical ideas and processes in order to do mathematics and thereby, with appropriate tasks and teaching, learn and enjoy the subject. This is very different from using the computer as a teaching app or a digital medium for exploration. Children tackled genuine puzzles -- problems for which they did not already have a pre-learned solution. So far, we have built four microworlds for second-graders and tested them with a diverse population of well over three hundred children. The microworlds focus on the most critical second-grade mathematical content (as mandated in state standards), let children pick up all key programming ideas in contexts that make them 'obvious' (to maintain focus on the mathematics) and suppress all other distractions to minimize overhead for teachers or students using the microworlds. Because children see the results of the actions they articulate (in the computer language, Snap!), they can evaluate their methods and solutions themselves. The feedback is purely the outcome, not happy or sad sounds from the computer. Notably, nearly all children showed intense engagement, some choosing microworlds even outside of mathematics time. Teachers spontaneously reported this as well, with special mention of children whom they found hard to engage in regular lessons. We report our experiments and observations in the spirit of sharing the ideas and promoting more research.

A first-year algebra student's curiosity about factorials of negative numbers became a starting point for an extended discovery lesson into territory not usually explored in secondary school mathematics. In this article, the authors, math teachers in Massachusetts, examine how to solve for factorials of negative numbers and discuss how they taught this concept in their schools.

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 value in all endeavors that require analytic thinking. Unlike the content standards, which are tightly targeted to specific grades, the practice standards are intended to apply across all content areas and at all grades, acquiring greater depth and broader application over the years. For studying new problems or building new theories, mathematicians often use the practice of trying concrete examples, looking for regularity in the reasoning, and describing the pattern in the reasoning. The CCSS describes this practice as MP 8. To make sense of MP 8, this article illustrates what "repeated reasoning" means, why looking for and expressing regularity in it is such a valuable mathematical habit of mind, and how that differs from analyzing structure (MP 7) and from finding patterns in numerical results.

Formative assessment is a process used by teachers and students during instruction that provides feedback to adjust ongoing teaching and learning to improve students' achievements of intended instructional outcomes. Formative assessment means assessment embedded in instruction. That definition was adopted in 2006 by the Council of Chief State School Officers' (CCSSO) Formative Assessment for Students and Teachers State Collaborative on Assessment and Student Standards. Tasks are both instructional and informational; children learn mathematics and teachers have data and a framework that helps them continually notice and gather knowledge about their students' thinking. Formative assessment blends seamlessly into instruction, guiding teaching without stealing instructional time. In this article, the authors focus on only one component of formative assessment--the instructional task--and how teachers can design and choose tasks.

More important than infusing courses and curricula with modern content is to give students the tools to use and understand mathematics--including some that does not yet exist. A curriculum organized around habits of mind tries to close the gap between what users and makers of mathematics do and what they say. These cross-journal articles are written by Al Cuoco, E. Paul Goldenberg, June Mark, and Sarah Sword--a team of curriculum developers at the Educational Development Center who pioneered work using mathematical habits of mind central to the work of mathematicians for organizing school mathematics curricula. The TCM article considers the ideas, logic, techniques, and habits of mind that algebra entails, as well as when and to what extent they can be learned with intellectual integrity in the elementary school grades, before a formal course on algebra. (Contains 4 figures and 2 tables.)

Building coherence in the development of mathematical ideas across the grades is key to improving students' mathematical learning in the United States. Knowing the mathematical experiences, understanding, skills, and habits of mind that students bring to a grade level and what the expectations are for the following grades can help teachers bridge the transitions for students on each end. In this article, the authors discuss the implications of the habits-of-mind approach for high school curricula, using the Center for Mathematics Education (CME) Project as a source of examples. They examine general mathematical habits that are used across high school as well as specific geometric and algebraic habits. The authors found that using mathematical habits of mind as an organizer can bring genuine and often surprising coherence to a curriculum. (Contains 3 figures.)

"Mathematical habits of mind" include reasoning by continuity, looking at extreme cases, performing thought experiments, and using abstraction that mathematicians use in their work. Current recommendations emphasize the critical nature of developing these habits of mind: "Once this kind of thinking is established, students can apply it in the context of geometry, trigonometry, calculus, data and statistics, or other advanced courses" ("Achieve" 2008, p4). The authors argue that developing mathematical habits of mind in the middle grades is essential for students who are making the critical transition from arithmetic to algebra. These habits can also build on students' early algebraic thinking and help them move toward a more formal investigation of algebraic ideas typically found in an eighth-grade or ninth-grade algebra course. The authors present examples illustrating two particularly powerful mathematical habits of mind that focus on the critical transition from arithmetic to algebra. These habits take advantage of students' natural inclinations to find patterns and describe those patterns with language. The middle grades are an especially good time developmentally to build these skills. (Contains 1 figure.)

This article discusses divisibility tests for any prime number.

Poses the question, How can one find a polynomial function that agrees with a given input-output table? Describes one method that leads to some interesting mathematics. (Author/NB)

Presents one approach to teaching that involves taking particular ways of thinking seriously and giving them top priority among various principles one needs for organizing mathematics (or other) curricula, allowing teachers to directly help students acquire selected thinking skills and aid acquisition of concepts, skills, and facts in content area. Explores application of this approach from elementary school through early college and teacher education. (GR)

Explores the uses of technology, computers in particular, to teach mathematics. Examples are given of the ways technology can help students gather data and test, modify, and reject or accept conjectures as they think about mathematical concepts. Article asserts a restructuring of the mathematics curriculum is necessary to bring this about. (SLD)