Mental Computation and Estimation

Lemonidis, Charalampos

doi:10.4324/9781315675664

Cited by 13 publications

(10 citation statements)

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“…3. Students were given A4 white paper which they could use, if they wished, for "short notes" that support memory (Lemonidis, 2015;B. Reys et al, 1993;R.…”

Section: Methodsmentioning

confidence: 99%

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The 8<sup>th</sup> Grade Students’ Competencies in Alternating Different Symbolic Representations of Rational Numbers

Lemonidis

Pilianidis

2020

INT ELECT J MATH ED

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One of the attributes of rational numbers that make them different from integers are the different symbolic modes (fraction, decimal and percentage) to which an identical number can be attributed (e.g. 1 4 , 0.25 and 25%). Some research has identified students' difficulty in mental calculations with rational numbers as has also the switching to different symbolic representations between fractions and decimals. However, pupils' performance, and repertoire of strategies have not been systematically studied in mental calculations with rational numbers expressed in different symbolic representations. The principal question of this research: how is the ability of students to perform mental calculations with rational numbers affected when the same number changes in fraction, decimal and percentage? For the purpose of the study 62 8 th grade students were interviewed to examine how this symbolic shift in the number of operations affects the success and type of strategies they use, and the ability to alternate the rotation of these symbolisms. The results of the research show that the symbolic change of the rational numbers affects the success and the type of strategies that students use in mental calculations. Another result of the study demonstrated that students are not flexible when switching between the different symbolic representations of rational numbers as benchmark while performing mental calculations.

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“…3. Students were given A4 white paper which they could use, if they wished, for "short notes" that support memory (Lemonidis, 2015;B. Reys et al, 1993;R.…”

Section: Methodsmentioning

confidence: 99%

“…Mental calculation is an essential component of number sense and leads to a better understanding of rational numbers (Lemonidis, 2015). The acquisition of number sense has been recognized as a fundamental component of learning mathematics.…”

Section: Introductionmentioning

confidence: 99%

The 8<sup>th</sup> Grade Students’ Competencies in Alternating Different Symbolic Representations of Rational Numbers

Lemonidis

Pilianidis

2020

INT ELECT J MATH ED

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“…Summarizing the studies mentioned in the section above, children barely use subtraction strategies flexibly and adaptively to solve multi-digit subtraction problems. This may be explained by the usual blocked practice , which is the common approach for teaching subtraction strategies in elementary school classrooms (e.g., Selter, 2001; Department for Education and Skills [DfES], 2006; Common Core State Standard Initiative [CCSSI], 2010; Lemonidis, 2016). In the blocked practice, the strategies are introduced and practiced successively (firstly the number-based, afterward the standard written algorithm).…”

Section: Introductionmentioning

confidence: 99%

Interleaved Learning in Elementary School Mathematics: Effects on the Flexible and Adaptive Use of Subtraction Strategies

et al. 2019

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Empirical findings show that students are often not capable of using number-based strategies and the standard written algorithm flexibly and adaptively to solve multi-digit subtraction problems. Previous studies have pointed out that students predominantly use the standard written algorithm after its introduction, regardless of task characteristics. Interleaved practice seems to be a promising approach to foster the flexible and adaptive use of strategies. In comparison to the usual blocked approach, in which strategies are introduced and practiced successively, they are presented intermixed in interleaved learning. Thus, the students have to choose an appropriate strategy on the basis of every task itself, and this leads to drawing comparisons between the different strategies. Previous research has shown inconsistent results regarding the effectivity of interleaving mathematical tasks. However, according to the attentional bias framework, interleaved practice seems to be a promising approach for teaching subtraction strategies to enhance the students’ flexibility and adaptivity. In this study, 236 German third graders were randomly assigned to either an interleaved or blocked condition. In the interleaved condition the comparison processes were supported by prompting the students to compare the strategies (between-comparison), while the students of the blocked approach were encouraged to reflect the adaptivity of a specific strategy for specific subtraction tasks (within-comparison). Both groups were taught to use different number-based strategies (i.e., shortcut strategies and decomposition strategies) and the standard written algorithm for solving three-digit subtraction problems spanning a teaching unit of 14 lessons. The results show that the students of the interleaved condition used the shortcut strategies more frequently than those of the blocked condition, while the students of the interleaved condition applied the decomposition strategies as well as the standard written algorithm less frequently. Furthermore, the students of the interleaved condition had a higher competence in the adaptive use of the shortcut strategies and the standard written algorithm. A subsequent cluster analysis revealed four groups differing in their degree of adaptivity. Being part of clusters with a comparatively high level of adaptivity was positively related to the prior arithmetical achievement and, even more so, to the interleaved teaching approach.

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“…Thus, we are primarily concerned with active, applied knowledge, as opposed to more passive “interpretable” or “explainable”; and we emphasize “fast and easy to compute” as opposed to “simulatable” or “humanly-reproducible” AI. Human ability to memorize and to compute has been a subject of extensive research in psychology and cognitive sciences [ 30 – 33 ], leading us to the following principal HKM constraints:…”

Section: Methodsmentioning

confidence: 99%

Human knowledge models: Learning applied knowledge from the data

Dudyrev

Semenkov²,

Kuznetsov³

et al. 2022

PLoS ONE

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Artificial intelligence and machine learning have demonstrated remarkable results in science and applied work. However, present AI models, developed to be run on computers but used in human-driven applications, create a visible disconnect between AI forms of processing and human ways of discovering and using knowledge. In this work, we introduce a new concept of “Human Knowledge Models” (HKMs), designed to reproduce human computational abilities. Departing from a vast body of cognitive research, we formalized the definition of HKMs into a new form of machine learning. Then, by training the models with human processing capabilities, we learned human-like knowledge, that humans can not only understand, but also compute, modify, and apply. We used several datasets from different applied fields to demonstrate the advantages of HKMs, including their high predictive power and resistance to noise and overfitting. Our results proved that HKMs can efficiently mine knowledge directly from the data and can compete with complex AI models in explaining the main data patterns. As a result, our study reveals the great potential of HKMs, particularly in the decision-making applications where “black box” models cannot be accepted. Moreover, this improves our understanding of how well human decision-making, modeled by HKMs, can approach the ideal solutions in real-life problems.

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Mental Computation and Estimation

Cited by 13 publications

References 0 publications

The 8<sup>th</sup> Grade Students’ Competencies in Alternating Different Symbolic Representations of Rational Numbers

The 8<sup>th</sup> Grade Students’ Competencies in Alternating Different Symbolic Representations of Rational Numbers

Interleaved Learning in Elementary School Mathematics: Effects on the Flexible and Adaptive Use of Subtraction Strategies

Human knowledge models: Learning applied knowledge from the data

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