Investigating the Advancement of Middle School Mathematics Teachers’ Meanings for Partitive Division by Fractional Values of Quantities

and when the divisors were not whole. For Task 11 (1/2÷3/4) in Phase 1, three of the teachers created models for multiplication instead of division, a phenomenon reported by other researchers (e.g., Jansen & Hohensee, 2016; Simon, 1993). RQ1.4: Partitive Scenarios I analyzed Phase 1 data to answer RQ1.4: What meanings do teachers reveal when they engage in tasks that I designed to elicit partitive conceptualizations of division, with varying degrees of abstraction, and with a focus on fractional divisors? The data from the tasks corresponding to RQ1.4 revealed more instances of partitive-to-quotitive 253 assimilation, which was the first type of partitive-quotitive interference. Additionally, I identified a second type, which I called quotitive-modeling interference, to refer to any difficulties in modeling partitive division that are caused by inappropriately applying meanings or strategies that are typically associated with quotitive division models. The data suggested that three teachers were impaired because of quotitive-modeling interference. This cognitive obstacle manifest itself in three different ways: (1) deliberately creating a quotitive model to cope with confusion during an attempt to model partitively, (2) confusion due to beginning a partitive model with a representation of the dividend (commonly done for quotitive modeling), and (3) modeling the dividend and divisor using a common unit (required for quotitive models, but errant for partitive models). During the water-container tasks, no teacher demonstrated partitive-quotitive interference, due to the specificity of the contexts. The primary cognitive obstacles for these tasks were related to mixed-number interference, re-unitization, and fractions as reciprocal measures of relative size. RQ1.5: Invert-and-Multiply Algorithm I analyzed Phase 1 data to answer RQ1.5: What justifications do the teachers provide for the invert-and-multiply algorithm after working through the (watercontainer) tasks? Leading up to Task 25 (explain the algorithm), all six teachers had the opportunity to generalize with two different fractional divisors in the water-container tasks; Task 21 (some amount of water fills 9/4 containers) and Task 24 (some amount of water fills 2/3 container). The data revealed that successful generalization in Tasks 21 and 24 did not guarantee a meaningful explanation for Task 25. Mindi, Mel, Uma, and Ursa were able to generalize both Task 21 and 24, but only Mindi and Mel were able to 254 use the same water-container context to generalize even further to explain Task 25. Uma was also technically successful although she believed she was not because she unfortunately said that a×c/b implied that multiplication by c should precede division by b, which was not the order she wanted. Ursa did not connect Task 25 to Tasks 21 and 24 and she tried unsuccessfully to resolve Task 25 using a quotitive conceptualization. Mark was not successful at generalizing in either water-container task, and consequently he could not generalize for Task 25. Linda generalized successfully only for Task 24, but she unsuccessfully attempted a quotitive explanation for Task 25. In summary, two teachers used the water-container tasks (partitive conceptualizations) to justify the algorithm, two teachers unsuccessfully attempted a quotitive explanation, and two teachers unsuccessfully attempted a partitive explanation. Additionally, three of the unsuccessful teachers exhibited partitive-quotitive interference during their attempts at Task 25. RQ1.6: Additional Cognitive Obstacles I analyzed Phase 2 data to answer RQ1.6: What cognitive obstacles do teachers further reveal as I actively attempt to promote the development of their meanings that are foundational to partitive division over the rational numbers? My findings reported in this dissertation only refer to Linda’s schemes. Phase 1 revealed that she had weak meanings regarding re-unitization, that she could not operate with fractional multipliers, and that did possess partitive-quotitive awareness. As I engaged her in activities that I designed to strengthen these schemes, I was able to discern additional cognitive obstacles for Linda. When she tried to model products with fractional multipliers and whole-valued multiplicands, she repeatedly experienced multiplier-switches. This happened even in 255 cases where she was deliberately trying to avoid it. Linda also revealed issues with not only re-unitizing, but with unitizing. By this I mean that she could not conjure appropriate units for the fractions she was trying to operate with, which suggested a disengagement from quantitative reasoning. Also, given one value and its unit, she could not conjure a new unit for a second value, which was detrimental to her success at a partitive model for 3÷1/4. Furthermore, she struggled to give a quantitative description of a general multiplicative statement, using language to describe the product that was effectively the same as the language she used to describe the multiplier. Consequently, she struggled with providing consistent quantitative meanings for division, including confusion about the correct ordering of the dividend and divisor. Additionally, Linda revealed weakness with language. She was vague when she described quantities and operations, and she nearly always omitted units when saying and writing values. She also struggled with interpreting the correct direction for a multiplicative comparison from written text. Lastly, she revealed an inability to retain information and ways of thinking, which raised questions about what is required to establish permanence for schemes. I now move on to summarize my findings for RQ2.1 and RQ2.2, which corresponded to Phase 3 of my study. These findings addressed my second primary research question: How do these teachers’ meanings change as a consequence of an instructional sequence that emphasized quantitative reasoning to aid in the advancement