ABC problem in elementary mathematics education: Arithmetic before comprehension

Abstract

Mathematical habits of prospective teachers affect problem comprehension and success and expose their beliefs about mathematics. Prospective elementary teachers (PSTs) (n = 121) engaged in a problem solving activity each week in class. Data were collected from PSTs enrolled in an undergraduate elementary mathematics methods course at a Southeastern State University over multiple semesters (six semesters, seven classes). PSTs' solution methods for one intentionally misleading mathematics problem were analyzed using a convergent parallel mixed methods content analysis. Two-thirds of PSTs misunderstood the problem scenario and directly translated numbers from the problem text. PSTs who answered correctly used a problem model strategy to comprehend the scenario and were more likely to use multiple models, draw a diagram, and draw a diagram before using another model. However, a large number of PSTs who answered incorrectly also used multiple models and drew diagrams. Self-correction was not common (8 of 121), because their equations did not provide feedback or support comprehension. Three kinds of imprecision also affected problem comprehension and were evident in both correct and incorrect solutions. Intentionally misleading problems helped PSTs see consequences of their mathematical habits and highlighted the importance of sense making and precision when creating problem models.

Prospective teachers anticipate challenges fostering the mathematical practice of making sense

“…Because I did this, I was often stressed and discouraged when I couldn't figure out the answer right away.” These responses reveal PTs' recognition that an unproductive belief that mathematics problems should be solved quickly (e.g., Boaler, ; Frank, ) interfered with their ability to engage in SMP1. This self‐reported tendency to jump into a strategy quickly also aligns with findings that a common problem‐solving “bad habit” exhibited by PTs is the tendency to directly translate surface features of the problem using a diagram or equation in a procedural way, missing the opportunity to understand the underlying meaning of the problem (Boote & Boote, ). This study builds upon those findings, as PTs' responses suggest that these unproductive habits and struggles translate to them anticipating difficulties in fostering SMP1 with their future students.…”

“…Third, this work advances the knowledge within the field of mathematics education by providing insight about specific areas of PTs' anticipated difficulties fostering engagement in SMP1. While prior studies have described PTs' difficulties fostering problem solving (e.g., Boote & Boote, ; Brown & Borko, ; Brown et al, ), research had not yet examined PTs' understandings in the context of decoding the meaning of SMP1. This knowledge connects to mathematics teacher educators' practice and informs the need for developing instructional interventions to equip PTs for fostering SMP1.…”

“…Only a third of PTs surveyed articulated that developing students' thinking processes for problem solving was an aspect of SMP1. Boote and Boote () also found that PTs brought “well‐practiced bad habits” (p. 118) for problem solving to methods courses, such as the direct translation of the surface features of a problem without first seeking to understanding the meaning within the problem situation. Students tended to view their diagrams and equations “as merely a procedural means of deriving the solution rather than a means of modeling a situation or generalizing the result” (Boote & Boote, , p. 116).…”

“…Boote and Boote () also found that PTs brought “well‐practiced bad habits” (p. 118) for problem solving to methods courses, such as the direct translation of the surface features of a problem without first seeking to understanding the meaning within the problem situation. Students tended to view their diagrams and equations “as merely a procedural means of deriving the solution rather than a means of modeling a situation or generalizing the result” (Boote & Boote, , p. 116). These procedural habits inhibited PTs' problem solving, and may also hinder PTs' efforts to foster this practice with their future students (Brown & Borko, ; Brown, Cooney, & Jones, ; Schoenfeld, ).…”

Evaluation of Primary School Mathematics Curricula of Northern and Southern Cyprus for NCTM Principles and Standards

“…In particular, the fact that knowledge is continually increasing makes it important that students who are at the age of basic education should be trained in the way they can access information rather than merely being equipped with the information. Particularly in a field shaped around the basic skills of reasoning and problem solving, such as mathematics, the demand to educate students in a better way increases the need for more efficient and effective education and training programs (Bråting et al, 2019;Ministry of National Education and Culture, 2018;Weinberg, 2019).…”