Students’ partitive reasoning

An analysis of elementary school children’s fractional knowledge depicted with circle, rectangle, and number line representations

“…While there could have been problems with students using partitioning to indicate unit fractions, reminding ourselves the fact that the 25-item test would already take about 40 minutes and the test would be too long with the additional three unit fraction problems, we decided not to include them. If students could represent proper fractions, instead of giving an extra 3 unit fraction problems for the Problem Type Two, we could infer their knowledge of working with unit fractions, which was also tested and confirmed by Norton's and Wilkins's (2010) study. As one would expect, the performances with this item dropped compared to the performances in the Problem Type One.…”

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

“…Doing so requires a disembedding operation, and it is equivalent to saying that the student can take a units of units as given-that the student is an MC2 student. However, students with only part-whole fraction schemes students do not view fractions as measurable extents-as lengths or quantities in their own right (Kerslake, 1986;Lamon, 2007;Norton & Wilkins, 2010;Steffe & Olive, 2010;Tzur, 1999). When students construct a partitive unit fraction scheme (PUFS), they go beyond solely part-whole ideas.…”

“…So, partitioning, disembedding, and iterating are all operations of this scheme. A PUFS is often investigated by assessing the reverse of it (Norton & Wilkins, 2010;Tzur, 2004). That is, students are posed a task like this: "Given this segment that represents 1/6 of a whole, can you draw the whole?"…”

Dylan’s units coordinating across contexts

“…Operations involved in coordinating whole number units include unitizing, disembedding, iterating, and partitioning. Individuals can coordinate such operations to equi-partition, which is requisite for constructing fractional units (Hackenberg, 2010;Lamon, 2007;Norton & Wilkins, 2010;Steffe & Olive, 2010). Equipartitioning involves partitioning a whole to form equivalent parts, removing a part without modifying the whole (disembedding), forming a unit size (unitizing), and then iterating that unit to re-form the size of the original whole (Steffe, 2001).…”

“…Some Stage 1 students' schemes for fractions situations instead involve partitioning without disembedding, resulting in a parts-within-wholes conception of a fraction as a ratio (Hackenberg, 2013). Stage 2 students sometimes construct parts-out-of-wholes ratio conceptions that incorporate disembedding but not equi-partitioning (Boyce & Norton, 2016;Hackenberg, 2013;Norton & Wilkins, 2010). Though these students mentally -set apart‖ a part from a whole without modifying the whole, they unitize the part as a discrete unit of ‗1' prior to counting activity instead of a unitizing a size to be iterated.…”