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Abstract 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. 

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.