Asymmetry in student achievement on multiple-choice and constructed-response items in reversible mathematics processes

Abstract

In this paper we report the results of an experiment designed to test the hypothesis that when faced with a question involving the inverse direction of a reversible mathematical process, students solve a multiple-choice version by verifying the answers presented to them by the direct method, not by undertaking the actual inverse calculation. Participants responded to an online test containing equivalent multiple-choice and constructed-response items in two reversible algebraic techniques: factor/expand and solve/verify. The findings supported this hypothesis: Overall scores were higher in the multiple-choice condition compared to the constructed-response condition, but this advantage was significantly greater for items concerning the inverse direction of reversible processes compared to those involving direct processes.

Mathematical Reasoning Required when Students Seek the Original Graph from a Derivative Graph

“…Combined direct-reversiblereasoning pathway Specifically, the two types of thinking are needed in dealing with problems in mathematics, however, the reversible-reasoning type is more conceptual than directreasoning, because the subjects understand the causal relationship between functions and derivatives based on inherited traits and integration processes. This is in line with the opinion (Sangwin & Jones, 2017) that a problem that is reversible can encourage a problem-solver to conduct an in-depth analysis, forming the critical and creative thinker characters. The inability of students to change the direction of their thinking from direct to reversible results from their previous experience that was focused on symbolic processes rather than visual processes (Thomas & Stewart, 2011) and most of the obstacles faced by the students are due to the dominance of this type of thinking.…”

Exploring the Potential Role of Reversible Reasoning: Cognitive Research on Inverse Function Problems in Mathematics

“…The focus of previous researchers are more on the operational aspects, for example by identifying error reversals that students make for the problem of "students and professors" (González-Calero, Arnau, & Laserna-Belenguer, 2015;Soneira, González-Calero, & Arnau, 2018;Tunç-Pekkan, 2015), reversible multiplication relationships (Hackenberg, 2010), cognitive conflict and insufficient mental processes to reverse problem situations (Ramful, 2014), the type of task that causes reversible reasoning (B. Dougherty, Bryant, Bryant, & Shin, 2017;B. J. Dougherty, Bryant, Bryant, Darrough, & Pfannenstiel, 2015;Sangwin & Jones, 2017;Simon, Kara, et al, 2016;Vilkomir & O'Donoghue, 2009). Likewise, with investigations about inverse functions, most focus more on errors made by students in solving problems (Carlson et al, 2015;Kontorovich, 2017;Paoletti et al, 2018;Zazkis & Kontorovich, 2016;Zazkis & Zazkis, 2011).…”

Two-Stage Collaborative Exams have Little Impact on Subsequent Exam Performance in Undergraduate Mathematics

“…Different patterns of results across stages 1-3 for different questions suggest that there may be a 'question effect' on the effectiveness of the two-stage format, with certain questions more amenable to group discussion and to prompting individual student learning. For mathematics in particular, it could be the case that more procedural questions offer less opportunity for fruitful discussion than conceptual questionsfor instance, there may be a different effect for questions in two-stage exams based on their categorisation in the MATH taxonomy (Smith et al 1996), or depending on whether the questions are multiple-choice or constructed-response (Sangwin and Jones 2017). This was not something I set out to explore in the studies reported here, and a post-hoc analysis of the questions based on whether they saw gains following group discussion does not reveal any obvious pattern.…”