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
What is already known Active learning— Collaborative learning—doing Classroom discourse—focused, relevant Clear articulation of one's thinking, even just to oneself, helps develop that thinking. What this paper adds The common languages we use for classroom mathematics—natural language for conveying the meaning and context of mathematical situations and for explaining our reasoning; and the formal (written) language of conventional mathematical notation, the symbols we use in mathematical expressions and equations—are both essential but each presents hurdles that necessitate the other. Yet, even together, they are insufficient especially for young learners. Programming, appropriately designed and used, can be the third language that both reduces barriers and provides the missing expressive and creative capabilities children need. Appropriate design for use in regular mathematics classrooms requires making key mathematical content obvious, strong and the ‘driver’ of the activities, and requires reducing tech ‘overhead’ to near zero. Continued usefulness across the grades requires developing children's sophistication and knowledge with the language; the powerful ways that children rapidly acquire facility with (natural) language provides guidance for ways they can learn a formal language as well. Implications for policy and/or practice Mathematics teaching can take advantage of the ways children learn through experimentation and attention to the results, and of the ways children use their language brain even for mathematics. In particular, programming—in microworlds driven by the mathematical content, designed to minimise distraction and overhead, open to exploration and discovery As it aids the mathematics, it also builds computational thinking and satisfies schools' increasing concerns to broaden access to ideas of computer science.