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Training the Approximate Number System Improves Math Proficiency
Humans and nonhuman animals share an approximate number system (ANS) that permits estimation and rough calculation of quantities without symbols. Recent studies show a correlation between the acuity of the ANS and performance in symbolic math throughout development and into adulthood, which suggests that the ANS may serve as a cognitive foundation for the uniquely human capacity for symbolic math. Such a proposition leads to the untested prediction that training aimed at improving ANS performance will transfer to improvement in symbolic-math ability. In the two experiments reported here, we showed that ANS training on approximate addition and subtraction of arrays of dots selectively improved symbolic addition and subtraction. This finding strongly supports the hypothesis that complex math skills are fundamentally linked to rudimentary preverbal quantitative abilities and provides the first direct evidence that the ANS and symbolic math may be causally related. It also raises the possibility that interventions aimed at the ANS could benefit children and adults who struggle with math.
Journal Article
Making graphs
\"A class is learning a lesson on making graphs and interpreting data, and the class clown, Logan, has some off-the-wall answers to his classmates' surveys.\"-- Provided by publisher.
Book
Design of experiments for engineers and scientists
The tools and technique used in the Design of Experiments (DOE) have been proved successful in meeting the challenge of continuous improvement over the last 15 years. However, research has shown that applications of these techniques in small and medium-sized manufacturing companies are limited due to a lack of statistical knowledge required for their effective implementation. Although many books have been written in this subject, they are mainly by statisticians, for statisticians and not appropriate for engineers.Design of Experiments for Engineers and Scientists overcomes the problem of statistics by taking a unique approach using graphical tools. The same outcomes and conclusions are reached as by those using statistical methods and readers will find the concepts in this book both familiar and easy to understand. The book treats Planning, Communication, Engineering, Teamwork and Statistical Skills in separate chapters and then combines these skills through the use of many industrial case studies. Design of Experiments forms part of the suite of tools used in Six Sigma.Key features:* Provides essential DOE techniques for process improvement initiatives* Introduces simple graphical techniques as an alternative to advanced statistical methods - reducing time taken to design and develop prototypes, reducing time to reach the market* Case studies place DOE techniques in the context of different industry sectors* An excellent resource for the Six Sigma training programThis book will be useful to engineers and scientists from all disciplines tackling all kinds of manufacturing, product and process quality problems and will be an ideal resource for students of this topic.Dr Jiju Anthony is Senior Teaching Fellow at the International Manufacturing Unit at Warwick University. He is also a trainer and consultant in DOE and has worked as such for a number of companies including Motorola, Vickers, Procter and Gamble, Nokia, Bosch and a large number of SMEs.
eBook
Zombies read graphs!
\"Zombies are truly terrifying monsters--but they've never been described as helpful before! This high-interest book shows readers and zombie hunters alike how the walking dead can help interpret data in different kinds of graphs, including picture graphs and bar graphs. Young mathematicians will love the creepy illustrations as well as the fun fact boxes detailing more information about the origins and legends concerning these scary creatures.\"-- Provided by publisher.
Book
Wave scattering by small bodies of arbitrary shapes
This book presents analytical formulas which allow one to calculate the S-matrix for the acoustic and electromagnetic wave scattering by small bodies or arbitrary shapes with arbitrary accuracy. Equations for the self-consistent field in media consisting of many small bodies are derived. Applications of these results to ultrasound mammography and electrical engineering are considered. The above formulas are not available in the works of other authors. Their derivation is based on a mathematical theory for solving integral equations of electrostatics, magnetostatics, and other static fields. These equations are at a simple characteristic value. Convergent iterative processes are constructed for stable solution of these equations. The theory completes the classical work of Rayleigh on scattering by small bodies by providing analytical formulas for polarizability tensors for bodies of arbitrary shapes.
eBook
Making bar graphs
Guides readers through the different parts of a bar graph, including its title, labels, and scale.
Book
Solving olympiad geometry without human demonstrations
Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning
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, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges
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, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.
A new neuro-symbolic theorem prover for Euclidean plane geometry trained from scratch on millions of synthesized theorems and proofs outperforms the previous best method and reaches the performance of an olympiad gold medallist.
Journal Article
Making tally charts
Tally charts are an excellent way to organize, represent, and interpret data, a key skill in today's math classrooms. Readers will learn the components of a tally chart and how each is essential in understanding the information the chart holds. Entertaining topics, questions and answers, and a colorful design make this book a comprehensible companion to the mathematics curriculum. Step-by-Step Instructions, Quiz, Glossary, For Further Information Section, Index.
Book
Relationships between magnitude representation, counting and memory in 4- to 7-year-old children: A developmental study
Background
The development of an evolutionarily grounded analogue magnitude representation linked to the parietal lobes is frequently thought to be a major factor in the arithmetic development of humans. We investigated the relationship between counting and the development of magnitude representation in children, assessing also children's knowledge of number symbols, their arithmetic fact retrieval, their verbal skills, and their numerical and verbal short-term memory.
Methods
The magnitude representation was tested by a non-symbolic magnitude comparison task. We have perfected previous experimental designs measuring magnitude discrimination skills in 65 children kindergarten (4-7-year-olds) by controlling for several variables which were not controlled for in previous similar research. We also used a large number of trials which allowed for running a full factorial ANOVA including all relevant factors. Tests of verbal counting, of short term memory, of number knowledge, of problem solving abilities and of verbal fluency were administered and correlated with performance in the magnitude comparison task.
Results and discussion
Verbal counting knowledge and performance on simple arithmetic tests did not correlate with non-symbolic magnitude comparison at any age. Older children performed successfully on the number comparison task, showing behavioural patterns consistent with an analogue magnitude representation. In contrast, 4-year-olds were unable to discriminate number independently of task-irrelevant perceptual variables. Sensitivity to irrelevant perceptual features of the magnitude discrimination task was also affected by age, and correlated with memory, suggesting that more general cognitive abilities may play a role in performance in magnitude comparison tasks.
Conclusion
We conclude that young children are not able to discriminate numerical magnitudes when co-varying physical magnitudes are methodically pitted against number. We propose, along with others, that a rather domain general magnitude representation provides the later basis for a specialized representation of numerical magnitudes. For this representational specialization, the acquisition of the concept of abstract numbers, together with the development of other cognitive abilities, is indispensable.
Journal Article