A Dynamic Evaluation of the Process of Solving Mathematical Problems, according to N.F. Talyzina's Method

Background

The process of teaching mathematics represents a challenge for primary education, due to the different perspectives and disciplines involved. In addition, as an active and flexible process, it requires feedback on what the students actually achieved. An analysis of the different learning and development outcomes allows the teacher to understand the mathematical content and the method of teaching it in the classroom, with the objective of promoting the students' conceptual development.

Objective

The objective of our study was to analyze the general skills for problem solving which students developed, by applying dynamic evaluation.

Design

A verification method was used to identify the students' abilities and difficulties. A protocol for evaluating the process of solving mathematical problems was organized. The assessment included four simple problems and four complex ones. The participants were 15 students in the third grade of primary school attending a private school located in Mexico City.

Results

The results showed that the students identified the types of mathematical operations (addition, subtraction, multiplication, and division) required to solve the problems as their objective. Therefore, their preparation of a solution plan, its execution, and its verification were based only on empirical thinking and quantitative information.

Conclusions

We concluded that problem-solving is an intellectual activity that requires conceptual development to carry out a solution plan, execute it, and verify it, in addition to the main objective of answering the question posed by the problem. We propose that these characteristics be included in the organization of mathematics teaching in order to develop mathematical thinking.

Proportional Reasoning Deficit in Dyslexia

Dyslexia has been linked to an altered perception of metrical structures in language, but no study to date has explored the link between reading impairments and other forms of metrical thinking (e.g., proportional reasoning). In the present study, we assessed proportional reasoning in 16 dyslexic children and 16 age-matched controls from 7 to 10 years of age in order to investigate whether dyslexia might be also linked to an altered form of metrical thinking. We found that dyslexic children are less accurate in performing judgments about proportionality compared to typical peers and that reading accuracy correlates with proportional reasoning abilities for 7-8-year-old children. Overall, these findings suggest that a link exists between reading and proportional reasoning abilities. We might speculate that fostering reasoning based on the meter can facilitate reading because it permits the segmentation of words in syllables and that dyslexia can be identified early with alternative non-reading tasks such as the proportional reasoning task used in this work.

Associations between Sleep, Depression, and Cognitive Performance in Adolescence

The relevance of cognitive performance during adolescence requires further studies that analyze potential associated factors. This study aimed to analyze inductive reasoning, reading comprehension, and mathematical thinking (problem-solving and number and calculation) in relation to sleep and depression in 244 students aged 12-17 years (47.6% boys and 52.4% girls). Daytime sleepiness, sleep quality, dysthymia, and euthymia (state and trait) were assessed by self-reported questionnaires. Moreover, correlations between these variables and cognitive performance, and differences depending on sociodemographic variables (sex, age, or academic year) were analyzed using non-parametric tests. Robust regression models were also conducted to evaluate the predictive role of significant variables on cognitive performance. The results showed significant bidirectional relationships between sleep- and depression-related variables, and between the latter ones and cognitive performance. Depression-trait was more related to cognitive performance than depression-state, and euthymia more than dysthymia, but neither daytime sleepiness nor sleep quality significantly correlated with it. As for sociodemographic variables, girls reported worse sleep and more depressive symptoms than boys did, and younger students reported better sleep but performed worse than the older ones. Although these findings should be further explored in forthcoming studies adding other promising variables, they highlight the importance of promoting euthymia to improve cognitive performance in adolescents.

The neural correlates of retrieval and procedural strategies in mental arithmetic: A functional neuroimaging meta-analysis

Mental arithmetic is a complex skill of great importance for later academic and life success. Many neuroimaging studies and several meta-analyses have aimed to identify the neural correlates of mental arithmetic. Previous meta-analyses of arithmetic grouped all problem types into a single meta-analytic map, despite evidence suggesting that different types of arithmetic problems are solved using different strategies. We used activation likelihood estimation (ALE) to conduct quantitative meta-analyses of mental arithmetic neuroimaging (n = 31) studies, and subsequently grouped contrasts from the 31 studies into problems that are typically solved using retrieval strategies (retrieval problems) (n = 18) and problems that are typically solved using procedural strategies (procedural problems) (n = 19). Foci were compiled to generate probabilistic maps of activation for mental arithmetic (i.e., all problem types), retrieval problems, and procedural problems. Conjunction and contrast analyses were conducted to examine overlapping and distinct activation for retrieval and procedural problems. The conjunction analysis revealed overlapping activation for retrieval and procedural problems in the bilateral inferior parietal lobules, regions typically associated with magnitude processing. Contrast analyses revealed specific activation in the left angular gyrus for retrieval problems and specific activation in the inferior frontal gyrus and cingulate gyrus for procedural problems. These findings indicate that the neural bases of arithmetic systematically differs according to problem type, providing new insights into the dynamic and task-dependent neural underpinnings of the calculating brain.

The influence of ageing on complex brain networks: a graph theoretical analysis

OBJECTIVE

To determine the functional connectivity of different EEG bands at the "baseline" situation (rest) and during mathematical thinking in children and young adults to study the maturation effect on brain networks at rest and during a cognitive task.

METHODS

Twenty children (8-12 years) and twenty students (21-26 years) were studied. The synchronization likelihood was used to evaluate the interregional synchronization of different EEG frequency bands in children and adults, at rest and during math. Then, graphs were constructed and characterized in terms of local structure (clustering coefficient) and overall integration (path length) and the "optimal" organization of the connectivity i.e., the small world network (SWN).

RESULTS

The main findings were: (i) Enhanced synchronization for theta band during math more prominent in adults. (ii) Decrease of the optimal SWN organization of the alpha2 band during math. (iii) The beta and especially gamma bands showed lower synchronization and signs of lower SWN organization in both situations in adults.

CONCLUSION

There are interesting findings related to the two age groups and the two situations. The theta band showed higher synchronization during math in adults as a result of higher capacity of the working memory in this age group. The alpha2 band showed some SWN disorganization during math, a process analog to the known desynchronization. In adults, a dramatic reduction of the connections in gray matter occurs. Although this maturation process is probably related to higher efficiency, reduced connectivity is expressed by lower synchronization and lower mean values of the graph parameters in adults.