Research literature is reviewed, identifying a need to design instruction to support multi-digit mental computation and critiquing conventional place value instruction. A distinctive instructional domain is proposed, called conceptual place value (CPV)--incrementing and decrementing numbers by units of ones, tens and hundreds. A design research study is reported, focusing on refining the instructional design for CPV with low-attaining 3rd and 4th graders. Two cases of extensive instruction in CPV are described, drawn from 10-week, 1-on-1 teaching cycles of 29 and 32 25-minute lessons. Three key features of instruction that appear to support the two students' learning of CPV are identified: advancing the level of tasks over the lessons, micro-adjusting tasks within lessons and the linking of number words, numerals and quantity. Three dimensions for advancing the level of tasks are identified: extending the range of numbers, making increments and decrements more complex and progressively distancing materials. (Contains 1 figure and 2 tables.)