Ariel's cycles of problem solving: An adolescent acquires the mathematics register

• Mathematics learning and teaching includes language and literacy challenges.
• Students develop mastery of the mathematics register over time.
• This case study presents Ariel's solving an algebraic equation over 18 months.
• Ariel uses an initial conjecture, reformulation and generalization of a rule.
• Ariel utilizes multiple modes of representation—linguistic, symbolic, and visual.
• Ariel demonstrates command of the mathematics register after solving the problem.

This analysis illustrates the interdependency between developing mathematical understanding and using the specialized language unique to the discipline to express those understandings employing the mathematics register. Our focus is on Ariel, a 7th grade bilingual student from an urban district, over an 18-month period. He develops knowledge of both mathematics and the language (both oral and written) required for his successful problem solving and communication of this process. Three questions are addressed: (1) How does Ariel display his mathematical understanding during problem solving? (2) How does Ariel use the mathematics register while solving the problem? (3) What systems of representation, including language (oral and written), does Ariel bring to solving the problem? The findings show that initially, Ariel displays his mathematical understanding during his attempt to solve a specific algebraic problem; in this exploration of understanding algebraic ideas, he seeks to identify patterns. We follow his mathematical and language learning from this initial conjecture, to his reformulation and generalization of a rule. Eighteen months later, after studying algebra, Ariel attends to the systems of representation, including both the oral and written expression of the mathematics register. The findings underscore the assertion that students may not be in full command—for production and comprehension—of the mathematics register until they understand the underlying mathematics. Teachers should be encouraged to create situations in which students deploy all of their diverse repertories of mathematics knowledge and skills to the instructional tasks at hand, and to refine their use of the mathematics register.