Experimental simulation of mathematical learning process based on ‘chunk-objective’

Chunk is a concept in cognitive psychology, which means a combination of multiple components that are closely interconnected with each other [5]. The concept of chunk was initially proposed by George A. Mile, a well-known American psychologist, aiming to explain how information is encoded in short-term memory. In 1973, Chase and Simon raised the Chunk-learning Theory[6]: when a learner is studying, he/she encodes and extracts new knowledge with ‘chunk’ as unit. Chunk, as an effective strategy of information processing, integrates complex fragmentary knowledge as a whole, makes hard things simple, highlights the significance and effectively expands the space of memory.

The cognitive scientific research also indicates that knowledge is not dispersedly stored in human brain, but stored in form of ‘group’ or ‘block’ [7]. The surrounding information firstly enters the instantaneous memory through perception, and then, partial information enters the working memory via selective attention. The working memory is an information processing zone, which compares, infers and classifies the new information with the existing knowledge, so as to encode information and form meaningful chunk. After the chunk is stored in long-term memory, it can be transferred from long-term memory to the working memory at any time under the stimulus of related information, which is an extraction exercise.

According to the view of neuroscientists, learning is a process in which neurons transfer signals to each other through electric discharge in essence [8]. When two neurons transfer information, they are transferring neuronal discharge, where the signals are transferred from axon to the dendrite of receiving neuron, and these two neurons complete the transfer of signal by delivering neurotransmitter. The process is simulated in Fig. 1. When Neuron A receives a stimulus, it causes Neuron to action; if Neuron A discharges multiple times by repeat or exercise, a connection will be built between Neuron A and B; similarly, the discharge of Neuron B can also activate Neurons C, D and E. In case these activities repeatedly occur, these five neurons will form a network; through multiple repeated activities, the interconnected numerous neurons in different brain zones will form a chunk with specific relationship. Therefore, the formation of chunk depends on analysis, comparison and other thinking activities, and its nature is the process that multiple neurons transfer signals to establish a connection. The research shows that the connection between knowledge chunks is essentially a synthesis of protein[9]. At the same time, the formation of chunk is a process needing multiple and repeated consolidation. In this process, the memory gradually tends to stable from a unstable state, accompanied with the synthesis of new protein[10]. After the chunk is formed, when the neuronal activities caused by external stimulus is related or similar to the characteristics of neuronal activities in chunk, the chunk will be activated, causing the activities of neurons in related brain zones.

Fig. 1

First, chunks have the characteristics of reconstruction. Andre Giordan, a doctor of science in biology and education of university of Geneva, Switzerland, has proposed that learning is as complex as alchemy, which is an allotectonic process characterised by reconstruction. The reconstruction of chunk is initially manifested in its formation process. It not only requires that the sub-contents be digested and absorbed to clarify the internal principle and operation process of the chunk, but also requires learners to step out of the sub-contents and examine its whole functions, conditions and conclusions from the holistic perspective. It is this kind of transformation that combines its sub-content and connects them from different directions, forming the function chunk with specific relations, and producing the effect of ‘1+1>2’. In addition, its reconstruction also has the characteristic of construction. The formation of chunk, including the reflection on internal sub-contents and external reflection, will be affected by the knowledge, experience and cognitive structure of the learning subject. With the same unit knowledge, some learners may construct it into chunk, while others cannot. Some people may build the chunk tight, solid, with high quality, while some people may build it loose and unstable. Therefore, chunk is the embodiment of learners’ existing knowledge and unique mind.

Secondly, chunks have the property of encapsulation. Encapsulation refers to treating a chunk as a separate unit, hiding its internal details as much as possible, and using it only by calling its methods and properties. When the unit knowledge is formed into a chunk, they are encapsulated into a ‘box’ with specific functions which shows relative stability and the specific procedural knowledge in it. The box has some ‘interface’ methods and properties, such as what it can do, when to use it, and how it relates to other chunks. When analysing the problems by chunk, it is not necessary to pay too much attention to its internal knowledge, but only to its interface.

Blending is also an important feature of chunk. Primary and secondary school mathematics textbooks are arranged according to the order of content from simple to complex and from easy to difficult. Therefore, the chunks before and after show a strong hierarchical or inclusive relationship, such as the relationship between monomial multiplication and the power of the same base number, the power of product and the power of power, the relationship between polynomial multiplication and monomial multiplication. On the other hand, different chunks have transverse cross relations. Such as function, equation and inequality. The characteristic makes the mathematical chunks present a network relation of interrelation, and each one is connected and closely related, which provides several paths for solving mathematical problems. Studies have shown that the differences between experts and novices mainlyexist in their different ways of organising, storing and applying knowledge [11]. For example, experts can better extract and apply knowledge by means of chunk strategy [12], and can quickly identify some meaningful patterns and structures based on experience. They also tend to associate specific knowledge with deep principles or schemas, so as to realise knowledge transfer [13]. All of these have to do with the network of knowledge structures and cognitive structures of the experts. The research also shows that because their knowledge structure is reticular, even when a link in the structural chain is broken, it will be automatically repaired.

Combining mathematical knowledge with the above analysis of the formation process of chunks, we can see that the formation of each mathematical block includes two stages of internal understanding and external reflection. If the current knowledge is treated as a whole, internal understanding is an understanding of the internal sub-content, while external reflection is the reflection of the whole and other parts of the same level. When these two stages are completed, the current knowledge can be construed as a mathematical block.

Internal understanding stage

This stage refers to that basic knowledge should be deeply understood at first. In this process, current knowledge is analysed, compared and classified with existing concepts, formulas, theorems or methods in long-term memory, and it will be assimilated or adapted. This stage proceeds accompanied with the learning of basic knowledge.

External reflection stage

In this stage, the learners will look at knowledge from the perspective of a ‘third party’ rather than the knowledge itself: what problem can it solve? When do we use it? What's the condition to use it? When these questions are figured out, a chunk will initially form in the long-term memory, that's to say, distributed neuron links are formed in the brain. Of course, this chunk is not stable, and needs continuous practice and reflection to consolidate.

The calling of chunk is completed by extraction exercise. On one side, chunk is used to solve problem, on the other side, chunk should be consolidated and improved. Just as Dr. Karpicke from the Cognition and Learning Laboratory of Purdue University said that, ‘extraction is a key process to understand and promote learning, and extraction exercise can produce meaningful learning’ [14].

Some research [7] have demonstrated that the differences in students’ problem-solving ability mainly depend on the distinction of chunks and the application of adjustment strategy.

Mathematics textbooks shall follow strong logical relationship, and the contents of each section are displayed as module or a form similar to module. The major task of new lessons is to guide students to reconstruct modularised knowledge into ‘chunks’ in brain. The formation of chunk is both a deconstruction process and a construction process [16], which includes three sub-processes: component understanding, exercise application and reflective construction. Component understanding focuses on grasping the sub-component knowledge that constitutes the chunk from the angle of structure and procedural knowledge. Exercise application is mainly to carry out extraction exercises of the chunk through tasks, so as to consolidate sub-content knowledge and improve the application automation of chunk. Reflective construction refers to jumping out of chunk itself and thinking the problems like ‘what it is’ and ‘why it do in this way’ from outside of chunk, so it is a more essential to understand of chunk knowledge. For example, to give solution to practical problems by using equation set, first, it is required to understand the meaning of the question, find out equivalent relationships, and expresses them with unknown variables, to constitute equation set. Through repetition of exercises, the chunk of ‘equation thought’ will be formed. That's to say, as long as you set up an equation set according to the meaning of description, making the number of equations included in the equation set to equal to the number of unknown variables, you can complete the task.

This process of constructing chunk is consistent with Bloom's cognitive goals. The ‘component understanding’ and ‘exercise application’ correspond to ‘understanding’ and ‘application’ of Blooms’ respectively, while ‘reflective construction’ corresponds to higher objectives like ‘analysis’, ‘integration’, ‘evaluation’ and so on.

Exercise lesson refers to the class type of systematic practice of the learned knowledge after a stage of learning. Its purpose is to guide students to recognise the knowledge structure and promote digestion ability. From the perspective of chunk, it is also a process of applying and improving the knowledge chunk. On one hand, through the extraction practice of exercises, the students could consolidate the component knowledge inside the chunk, find out key points, further clarify the ‘muscles’ and ‘veins’ of the chunk, and reach to automatic level; on the other hand, it is essential to discover bias through errors in the learning process, sort out the difficulty and fallible points of chunk application, further repair chunk by reflection and induction. The knowledge chunk polished by exercises is an angular and clear-cut mathematical model, which is the basis of constructing new knowledge and solving model in the future.

The mathematical comprehensive abilities not only requires a deep grasping of learning content, but also needs to develop connections among various parts. In other words, this requires a horizontal and vertical association of chunk on the basis of deeply grasping the knowledge chunk. In the process of linking chunks by topic lessons, it is first to help students to sort out the knowledge structure of current part with chunk as unit, then guide students to understand the relationships between chunks via topic practices, and comprehensively use each chunk to solve problem. Similarly, confused and contrary knowledge chunk can be deeply analysed through connection, so as to make clear of the function and application of chunk from peripheral interface and thus to form a systematic knowledge network structure. This helps students to expand their divergent thinking ability. For example, after the unit ‘one-variable quadratic inequality’ is taught, the teachers could connect it with ‘one-variable quadratic equation’ and ‘one-variable quadratic function’ chunks, helping students to understand one-variable quadratic inequality with the help of ‘one-variable quadratic equation’ and ‘one-variable quadratic function’, form an one-variable quadratic function centred knowledge network, and generate a profound understanding of the thought ‘combination of numbers and figures’. Through extensive knowledge practices, the students will be able to deeply understand the parallel, progressive or inclusive relationships between knowledge chunks, form lucid logic knowledge structure and obtain basis to solve problem.

Before the experiment, the above three indexes of experimental class and control class were measured, and the last unit test score was taken as pre-test score. Then two (both) classes received independent sample T test. It can be seen from the results in Table 1 that there was no significant difference among the four variables (p>0.05), indicating that both classes are parallel class and suitable for experiments.

After experiment, correlated sample t-test was conducted on three indexes and unit test scores of both classes, and the results are shown in Table 2. It can be seen from the table that, each index of control class shows no significant difference (p>0.05), while each index of experimental class shows more significant difference (p<0.01). This indicates that the experiment exactly facilitate the promotion of all indexes and scores.

The sum of three indexes scores of experimental class is considered as the level of students’ knowledge chunk abilities, and a correlation analysis is made between the abilities and unit test scores. The results show that the correlation coefficient between them is 0.807 (p<0.01), so they are significantly correlated. This indicates that the experiment facilitates the raise of scores by promoting students’ knowledge chunk abilities.

As an ‘object’, mathematical chunk is a deep form of mathematical knowledge [16]. It is an understanding about ‘what it is’ formed after the knowledge is processed through ‘psychological activity’ and ‘operating procedures’. Chunks contribute to forming knowledge network more quickly and effectively, thereby providing foundation for deep thinking activity. There are three points to build a knowledge network by chunks. First, it is required to systematically grasp every chunk, from its internal structure to overall function. At the same time, appropriate exercises can help chunk application to achieve automation. Second, after an entire unit is taught, the knowledge should be sorted and summarised with the form of chunks, so as to grasp the structure of the part. Finally, the relationships between two chunks can be well understood with the help of comprehensive and topic training, and the important nodes of the chunks can be made clear by the reflection to wrong and typical questions. So that, the whole knowledge structure is clear, and the vein is distinct. To get a better effect of chunk learning, it is better to draw the knowledge structure by form of mind map, write it down by form of semantic knowledge, and make correction and improvement by continuous exercises.

Deep comprehension of mathematics mainly depends on students’ understanding, which can be achieved only by continuous reflection. The abstractness of mathematical objects, the exploration of mathematical activities, the preciseness of mathematical reasoning and the particularity of mathematical language determine that middle school students who are in the stage of thinking development cannot directly grasp the essence of mathematics, and must achieve it through reflection after learning. As American mathematician Polya said, ‘the solution of mathematics problem was only a half, and what more important was the retrospect after solution’ [17]. Reflection runs through the whole process of mathematics chunk learning. The process of chunk formation should reflect the relationships between current knowledge and previous knowledge and cultivate the connective thinking ability and network thinking ability in mathematics learning.

The process of chunk extraction exercise should reflect the generative condition knowledge and objective knowledge of chunk, and develop students’ reasoning ability and logical thinking ability. In the exercises for mistakes and difficulties, we should reflect on the relationships among the relevant mathematics chunks, train divergent thinking and comprehensive thinking ability, reflect on learning attitude and self-belief, and cultivate scientific concept of mathematics.

Problem solution is a process of exploration from ‘a known state’ to a target state’. When it comes to problem solving, it's important to frame the solution, not how to do it. Mathematical chunk is a kind of modular knowledge with specific functions and characteristics. When solving a problem, teachers should guide students to quickly reason the problem with chunks on basis of the analysis, build an idea and set up a solution to the problem quickly. Furthermore, the solution set up with knowledge chunks could clearly show the problem-solving process, which is conducive for students to study and analyse the problem-solving idea as well as the relationships among different parts, and to promote their understanding of mathematical thoughts and methods.

Based on the above factors, we realise that chunk theory has an important application in mathematics learning. The mathematical knowledge units can be seen as different chunks, and they are closely related to each other. When learning mathematics, students can understand the knowledge from different dimensions with the help of the chunk theory, such as internal and external, micro and macro, conditions and applications, so as to achieve a deep grasp of mathematical knowledge. Therefore, it is necessary for students to understand the chunk theory while learning mathematical knowledge. More importantly, teachers should fully comprehend the chunk theory. In teaching, teachers should not only teach students how to do exercises, but also teach students how to learn, which requires teachers to motivate students to think from a higher perspective rather than knowledge. The application of chunk theory provides an important method for it.