\input zb-basic \input zb-matheduc \iteman{ZMATH 2012a.00205} \itemau{Hackenberg, Amy J.} \itemti{Students' reasoning with reversible multiplicative relationships.} \itemso{Cogn. Instr. 28, No. 4, 383-432 (2010).} \itemab Summary: In an 8-month teaching experiment, I investigated how 4 sixth-grade students reasoned with reversible multiplicative relationships. One type of problem involved a known quantity that was a whole number multiple of an unknown quantity, and students were asked to determine the value of the unknown quantity. To solve these problems, students needed to produce a fraction of the known quantity that could be repeated some number of times to make the known'', rather than repeat the known quantity to make the unknown quantity. This aspect of the problems involved reversibility because students who do not make a fraction of the known quantity tend to repeat the known quantity. All four students constructed schemes to solve such problems and more complex versions where the relationship between known and unknown quantities was a fraction. Two students could not foresee the results of their schemes in thought -- they had to carry out some activity, review its results, and then carry out more activity in order to solve the problems. The other two could foresee results of their schemes prior to implementing them; their schemes were anticipatory. One of these two also constructed reciprocal relationships, an advanced form of reversibility. The study shows that constructing anticipatory schemes requires coordinating three levels of units prior to activity, a particular whole number multiplicative concept. The study also reveals that even students with this multiplicative concept will be challenged to construct reciprocal relationships. Suggestions for further inquiry on student learning in this area, as well as implications for classroom practice and teacher preparation, are considered. \itemrv{~} \itemcc{C33 F33} \itemut{mathematics instruction; grade 6; mathematics achievement; multiplication; numbers; concept formation; thinking skills; research; lower secondary} \itemli{doi:10.1080/07370008.2010.511565} \end