Understanding Of Mathematical Equivalence Research Articles

The benefits of a metacognitive lesson on children's understanding of mathematical equivalence, arithmetic, and place value.

Metacognition is theorized to play a central role in children's mathematics learning. The primary goal of the current study was to provide experimental evidence in support of this role with elementary school students learning about mathematical equivalence. The final sample included 135 children (59 first-graders and 76 second-graders) who participated in the study across three sessions in their classrooms. They completed a pretest during session one, a lesson and posttest during session two, and a two-week delayed retention test during session three. For session two, children were randomly assigned to receive a lesson on mathematical equivalence with or without integrated metacognitive questions. Relative to the control lesson, children who received the metacognitive lesson demonstrated higher accuracy and higher metacognitive monitoring scores on the posttest and retention test. Further, these benefits sometimes extended to uninstructed items targeting arithmetic and place value. No condition effects were observed for children's metacognitive control skills within any of the topics. These findings suggest a brief metacognitive lesson can improve children's mathematics understanding.

Factors associated with children’s understanding of mathematical equivalence: An investigation across six countries.

Factors associated with children’s understanding of mathematical equivalence: An investigation across six countries.

Mathematical equivalence assessment: Measurement invariance across six countries

Within mathematics education research, there has been a strong focus on students’ understanding of mathematical equivalence because of its key role in the development of mathematics skills. One of the most frequently used tools to assess students’ understanding of equivalence has been the Mathematical Equivalence Assessment (MEA) (Rittle-Johnson et al., 2011). In this study, we investigate for the first time the cross-cultural measurement invariance of an adaptation of the MEA. This included a sample (N = 2760) of students aged 8–12 years old from China, England, New Zealand, South Korea, Turkey, and US to examine whether the same construct is being measured across all countries. Configural and partial scalar invariance was established for a two-factor, 11-item version of the adapted MEA. There were significant mean differences across countries, with students from China performing better and students from New Zealand performing worse than the rest of the sample.

Young students’ understanding of mathematical equivalence across different schools in South Africa

Background: Mathematical equivalence is a critical element of arithmetic understanding and a key component of algebraic thinking which is necessary for success in all levels of mathematics. Research studies continue to highlight misconceptions related to equivalence and reveal that many primary school students have a narrow and limiting view of the equals sign as an operation.Aim: This study aims to investigate young students’ understanding of mathematical equivalence in South Africa with a particular focus on their interpretations of the equals sign.Setting: Research data was obtained from students across six schools from different contexts within the Western Cape.Methods: We gave students an adapted standardised assessment containing 15 items related to equivalence.Results: Our analyses indicated that students focus more on the equals sign as an operation which involves calculating an answer. While some referred to equivalence as meaning the same as, most of them were inclined to accept the operational definition of the equals sign (i.e. the answer to the problem) as a better and preferred definition. In addition, student performance was poor on equation-solving problems and they rarely used comparative relational strategies in their solutions.Conclusion: The findings of this research confirmed that difficulties with equivalence reported by earlier research is widespread across this group of grade 4 students. This has implications for both curriculum, textbook and materials design and teacher professional development.

Features of a pan balance that may support students’ developing understanding of mathematical equivalence

ABSTRACT Elementary school students struggle in interpreting the equal sign as a symbol denoting equivalence. Although many have advocated using a pan-balance scale to help students develop this understanding, less is known about what features associated with this model support learning. To attempt to control and examine these features, the investigators developed a digital, pan-balance computer applet. This allowed for experimentally manipulating three relatively grounded instructional conditions (involving the core principle of making two sides the same; a balancing analogy; or both, along with a dynamic demonstration), compared to a relatively idealized control condition. Results indicated that the relatively more grounded conditions promoted a relational understanding of mathematical equivalence among 148 second- and third-grade students and further suggest that providing dynamic, grounded support may not be as optimal as less-enriched supports to promote students’ learning.

Improving children’s understanding of mathematical equivalence via an intervention that goes beyond nontraditional arithmetic practice.

Elementary school children (ages 7–11) struggle to understand mathematical equivalence, a foundational prealgebraic concept. Some manipulations to the learning environment, including well-structured nontraditional arithmetic practice alone, have been shown to improve children’s understanding; however, improvements have been modest. The goal of this study was to test an iteratively developed supplemental intervention for second grade that was designed to yield widespread mastery of mathematical equivalence. The intervention included three components beyond nontraditional arithmetic practice: (a) lessons that introduce the equal sign outside of arithmetic contexts, (b) “concreteness fading” exercises, and (c) activities that require children to compare and explain different problem formats and problem-solving strategies. After the development process, a small, randomized experiment was conducted with 142 students across eight second grade classrooms to evaluate the effects of the intervention on children’s solving of mathematical equivalence problems, encoding of mathematical equivalence problems, and defining of the equal sign. Classrooms were randomly assigned to the intervention or to nontraditional arithmetic practice alone. Analyses at the classroom level demonstrated that the intervention classrooms performed better than the active control classrooms both in terms of pre-to-post change in understanding of mathematical equivalence (g = 1.87) and accuracy on transfer problems at posttest (g = 2.07). Nonparametric analyses led to the same conclusions. Results suggest that the comprehensive intervention improves children’s understanding of mathematical equivalence to levels that surpass equivalent levels of well-structured arithmetic practice alone, as well as business-as-usual benchmarks from previous studies. (PsycINFO Database Record (c) 2019 APA, all rights reserved)

An integrative data analysis of gender differences in children’s understanding of mathematical equivalence

This study examined gender as a potential source of variation in children’s formal understanding of mathematical equivalence. The hypothesis was that girls would perform more poorly than boys. An integrative data analysis was conducted with 960 second and third graders across 14 previously conducted studies of children’s understanding of mathematical equivalence. Measures included problem solving, problem encoding, and equal sign definition. Overall, children performed poorly on all measures. As predicted, girls were less likely than boys to solve mathematical equivalence problems correctly, even though there were no gender differences in calculation accuracy. In addition, girls were more likely than boys to use the “add-all” strategy, an incorrect strategy that has been shown to be more resistant to change than other incorrect strategies. There were not statistically significant differences for encoding or defining the equal sign, suggesting that deficits may reflect girls’ tendency to follow taught algorithms.

Moving to Learn: How Guiding the Hands Can Set the Stage for Learning.

Previous work has found that guiding problem-solvers' movements can have an immediate effect on their ability to solve a problem. Here we explore these processes in a learning paradigm. We ask whether guiding a learner's movements can have a delayed effect on learning, setting the stage for change that comes about only after instruction. Children were taught movements that were either relevant or irrelevant to solving mathematical equivalence problems and were told to produce the movements on a series of problems before they received instruction in mathematical equivalence. Children in the relevant movement condition improved after instruction significantly more than children in the irrelevant movement condition, despite the fact that the children showed no improvement in their understanding of mathematical equivalence on a ratings task or on a paper-and-pencil test taken immediately after the movements but before instruction. Movements of the body can thus be used to sow the seeds of conceptual change. But those seeds do not necessarily come to fruition until after the learner has received explicit instruction in the concept, suggesting a "sleeper effect" of gesture on learning.

Benefits of Practicing 4 = 2 + 2: Nontraditional Problem Formats Facilitate Children’s Understanding of Mathematical Equivalence

This study examined whether practice with arithmetic problems presented in a nontraditional problem format improves understanding of mathematical equivalence. Children (M age = 8;0; N = 90) were randomly assigned to practice addition in one of three conditions: (a) traditional, in which problems were presented in the traditional "operations on left side" format (e.g., 9 + 8 = 17); (b) nontraditional, in which problems were presented in a nontraditional format (e.g., 17 = 9 + 8); or (c) no extra practice. Children developed a better understanding of mathematical equivalence after receiving nontraditional practice than after receiving traditional practice or no extra practice. Results suggest that minor differences in early input can yield substantial differences in children's understanding of fundamental concepts.