Abstract: Modules for Math is a program for improving the foundations of learning math. This is a cognitive enhancement program that was developed by Canadian Psychologist J. P. Das. The theory framework of Modules for Math is Planning- Attention- Simultaneous Processes- Successive Processes (PASS), which is based on the Luria’s working of the brain and Vygotsky’s principles of educating children, mainly internalization of speech that guides self-learning, development of strategies with assistance or prompting, often called ‘scaffolding’ and the idea that learning occurs in collaboration with others. There are five modules in the program; and they present some 50 activities for enhancement of each of the corresponding five basic skills. Each module involves training a specific basic skill, widely used in doing math. Some examples are Number Line, Differentiating Size of numbers and Value, Visual search, Selective attention, Numerosity (counting), Simultaneous verbal and nonverbal reasoning, and Working Memory. Modules in the manual have two parts: a focus on the broad cognitive foundations of these skills (global part) followed by their application to basic curricula in math — a ‘Bridging’ part that closes the gap between cognitive principles and math curriculum by transferring global concepts learned to applications. The program selects for training ‘the elements that are active in the beginning of children’s math learning’ as Geary (2013) mentioned, such as the Approximate number system, Numeral magnitude mapping, and Early explicit knowledge of accurate number system. Geary describes the overall process as ‘attentional control’. The concept can be understood as essentially Planning and Executive processes. A series of related models provide the basic theoretical and applied structure of the cognitive training tasks in the modules. Broadly, Math Proficiency is divided into two major components that depend on each other. These are computing and solving word problems. Planning and Executive Functioning (EF) are the predominant cognitive process for computations or step by step calculations that need to be followed. Similarly, Simultaneous processing is required for comprehension of word problems, but with some help from planning strategies. Attentional control is subsumed in Planning and EF. Simultaneous processing comprises logical-grammatical relations. The last level highlights the two components of EF, Inhibition and Shifting. For word problems, logical and grammatical divisions are to be trained by both non-verbal (matrices-type tests), and verbal–simultaneous tasks and activities in The Modules. Planning/EF and Simultaneous processing are the major processes for learning math, with Working Memory as a sub-set of attentional control. Working Memory may be regarded as an integral part of EF. However, more empirical studies should be designed to understand the different roles played by these components and how they interact with each other. Modules for Math are designed for children beginning to learn math. That is for kindergarten and early years of primary school. However some older children might have missed out in comprehending the basic principles of math and the skills that are required to learn math. The tasks in the modules therefore need to be adjusted upwards for increasing their difficulty levels for older children. Math Modules have a flexible structure to allow additional tasks and activities to expand their use. In order to further understand the basic cognitive functions in math, cognitive neuroscience research should be designed, and evidence for their neural correlates must be examined. Additionally, the effect of cognitive training may be demonstrated in corresponding changes in measures of brain imaging techniques. Such research is on the way together with behavioral evidence and such neural correlates are critical for assuming that changes in both behavior and brain are recorded after a period of cognitive training as in Modules for Math. Thus Modules for Math is a program that opens the door to a better understanding of the basic foundations of Math learning, and consequently for cognitive training.

Key words: Modules for Math, PASS Theory, Cognitive Processing

摘要： 数学训练方案（Modules for Math）是加拿大心理学家J.P.Das提出的促进数学学习的训练方案。该方案的理论基础是基于Luria神经三级网络的PASS（计划-注意-同时性加工-继时性加工）模型以及Vygotsky主张以言语内化方式进行自我学习的教育原则。数学训练方案包括模式转换；学习数轴；数一数；模仿、画路径和估算；数字记忆广度等五个训练模块，通过大约五十多个活动任务实现一般认知加工的促进，同时迁移到数学学习的课程中。数学训练方案今后将结合行为实验与认知神经影像学数据，证实训练的效果及变化。

关键词: 数学训练方案, PASS理论, 认知过程