Edited by: Katharina Loibl, University of Education Freiburg, Germany
Reviewed by: Chunxia Qi, Beijing Normal University, China; Laura Martignon, Ludwigsburg University of Education, Germany
This article was submitted to Educational Psychology, a section of the journal Frontiers in Psychology
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
This study aimed to examine the specific means and internal processes through which mathematical understanding is achieved by focusing on the process of understanding three new mathematical concepts. For this purpose interviews were conducted with 54 junior high school students. The results revealed that mathematical understanding can be achieved when new concepts are connected to at least two existing concepts within a student’s cognitive structure of. One of these two concepts should be the superordinate concept of the new concept or, more accurately, the superordinate concept that is closest to the new concept. The other concept should be convertible, so that a specific example can be derived by changing or transforming its examples. Moreover, the process of understanding a new concept was found to involve two processes, namely, “going” and “coming.” “Going” refers to the process by which a connection is established between a new concept and its closest superordinate concept. In contrast, “coming” is a process by which a connection is established between an existing convertible concept and a new concept. Therefore the connection leading to understanding should include two types of connections: belonging and transforming. These new findings enrich the literature on mathematical understanding and encourage further exploration. The findings suggest that, in order to help students fully understand new mathematical concepts, teachers should first explain the definition of a given concept to students and subsequently teach them how to create a specific example based on examples of an existing concept.
Mathematical understanding entails knowing, perceiving, comprehending, and making sense of the meaning and connotation of mathematical knowledge. Acquiring mathematical understanding plays an important and crucial role in mathematics learning. Bartlett contended that mathematical understanding can reduce the burden of memory, filter out invalid information in the brain, and maintain the longevity of memory (
Because mathematical understanding is very important and valuable, it has been widely researched since the middle of the last century (
Reviewing these few studies on the internal characteristics of mathematical understanding, especially on the characteristics of the internal psychological process of mathematical understanding, it can be seen that there are four different views at present. The first view, which is also the earliest one, holds that the internal process of mathematical understanding is one in which mathematical knowledge is comprehended and represented in the learners’ minds and links with each other are established. For example,
The second view is that the process of mathematical understanding refers to the transformation of mathematical knowledge representation For instance,
The third view believes that the internal process of mathematical understanding is a comprehensive, complex, and iterative process. The most famous scholars who hold this view are
The fourth point of view is the most widely held one, which proposes that in the internal process of mathematical understanding, knowledge enters the learner’s brain and interacts with the original knowledge to form a new cognitive structure, which means that it is a cognitive activity. For instance,
Apparently the above four views are different, although they all address the issue of the internal psychological process of mathematical understanding. The first view emphasizes representation and the connection between representations; the second emphasizes the transformation of representation; the third emphasizes comprehensive regression; and the fourth emphasizes the formation of new cognitive structures. However, they are undoubtedly of great help to our comprehension of mathematical understanding because they help us gain a more in-depth understanding of the internal process of mathematical understanding and shed light on methods for examining mathematical understanding.
Additionally, it is also obvious that these views fail to provide more specific and detailed information about the process of mathematical understanding to help us understand it completely. For instance, the second view believes that mathematical understanding is the process of transformation of mathematical knowledge or its representation, but how is such a representation transformed? What kind of transformation is most conducive to the generation of mathematical understanding? The fourth view insists that in the internal process of mathematical understanding, knowledge “enters” the learner’s brain and interacts with the original knowledge to form a new cognitive structure. But what kind of original knowledge is invoked to interact with the new knowledge? What are the characteristics of the new cognitive structure after the formation of a new mathematical understanding etc.? Due to the existence of such unmapped zones, many mathematics teachers find it difficult to apply these views to practical mathematics teaching (
When new mathematical knowledge is processed, under what specific internal situations does the understanding of it take place?
What kind of previously acquired knowledge present in the cognitive structure is essential for the formation of new mathematical knowledge?
It is therefore possible that the diverse perspectives on mathematical understanding reflect the various ways in which understanding has been conceptualized. For example,
However, most scholars and researchers generally agree that mathematical understanding falls within the purview of mathematics learning. Mathematical understanding is closely related to mathematical cognitive structures and processes. It is the process by which new mathematics knowledge becomes part of an individual’s internal cognitive structure by connecting with previously acquired mathematics knowledge and integrating it with the internal network (
The idea that understanding mathematics makes connections between ideas, facts, or procedures is not new. It is a theme that runs through classic works within mathematics education literature and emerges frequently in more recent discussions of representation and understanding in mathematics. Many of those who study mathematics learning agree that understanding involves recognizing relationships between pieces of information (
Many scholars and researchers have contended that as individuals continue to grow and develop, their level of mathematical understanding will transform accordingly. Indeed, “the degree of understanding is determined by the number and strength of the connections. A mathematical idea, procedure, or fact is understood thoroughly if it is linked to existing networks with stronger or more numerous connections” (
The easiest means to enlarge a network of mental representations is to connect a representation of a new fact or procedure to an existing network. Another method is reorganization, in which [R]epresentations are rearranged, new connections are formed, and old connections may be modified or abandoned. The construction of new relationships may force the reconfiguration of the affected networks. The reorganizations may be local, widespread, and dramatic, reverberating across numerous related networks. Reorganizations are manifested both as new insights, local or global, and as temporary confusion. Ultimately, understanding increases as the reorganization yields more richly connected cohesive networks
How are networks of mental representations configured? Current scholars and researchers believe that it consists of many nodes and connections and is very complex. The nodes in this network include elements such as concepts, signs, figures, formulae, axioms, and theorems (
Regarding the relationships that enhance mathematical understanding and those that are formed by the connections drawn between newly and previously acquired mathematics knowledge within an individual’s internal cognitive structure, scholars have contended that they can be classified as two types depending on whether they are based on (a) similarities and differences or (b) inclusion. The former type of relationship is established within a representation form by the noting of the correspondences between different external representational forms and within a given form. They are likely to be found in networks that resemble webs because the delineation of similarities and differences does not necessarily result in the emergence of higher-order relationships. The second type of relationship emerges when one mathematical fact or procedure is perceived to be a special case of another and is based on the notion of inclusion or general and specific cases. Accordingly, such relationships are likely to be found in hierarchical networks (
In accordance with this perspective, for a long time scholars and educators have always emphasized that, in order to help students understand mathematical concepts appropriately, definitions and specific examples should be presented and explained to students as a part of the teaching process. A definition of a mathematical concept is a statement or description of its connotation and characteristics, and it represents a generic construct that subsumes other lower-order constructs. It indicates the position of a concept within the entire conceptual system, its similarities to other domains, and the differences between them (
Mathematical understanding is an internal process. Therefore, how can we judge whether an individual has achieved mathematical understanding following mathematical learning, and how can we evaluate his/her degree and level of mathematical understanding? In general, scholars and researchers contend that this can be inferred based on external performance because internal psychological activities always manifest themselves externally (
However, scholars and researchers most commonly use the oral report method to ascertain an individual’s current level of mathematical understanding (
The general criteria for judging mathematical understanding using the oral report method are accuracy and clarity of an individual describing newly acquired knowledge in his/her own words and his/her awareness of the sources of this knowledge. Specifically, if an individual describes the meaning of newly acquired mathematics knowledge in their own words clearly and accurately and can also specify how they acquired this new mathematics knowledge, then they are considered to have understood the respective piece of knowledge. Otherwise, they are considered to have not adequately understood it (
In accordance with the above views and approaches toward mathematical understanding and the practical situation of teaching mathematical understanding, this study adopted the oral report method to assess students’ understanding of mathematical knowledge and hypothesized that: (1) mathematical understanding will be achieved when newly acquired mathematics knowledge is connected to multiple (rather than single) mathematics knowledge acquired previously. The newly acquired mathematics knowledge cannot be understood by merely drawing connections between itself and a single piece of previously acquired knowledge, even though it has entered an existing network and a new network is formed; and (2) one piece of previously acquired mathematics knowledge should be superordinate knowledge of the newly acquired mathematics knowledge. The connections between arbitrary previously acquired mathematics knowledge and newly acquired mathematics knowledge cannot help to realize a complete mathematical understanding.
We recruited 54 second-grade students from a junior high school in Jinan, China. Moreover, the academic performance of 14, 27, and 13 students was excellent, average, and poor, respectively. Participant characteristics are summarized in
Students’ information.
| Male | 5 | 13 | 8 |
| Female | 9 | 14 | 5 |
We recruited second-grade junior high school students because they are older than primary school students and possess foundational mathematics knowledge. Conversely, they are younger than high school students and are yet to acquire substantial amounts of mathematics knowledge. Therefore we speculated that it may be easier and more convenient to teach new mathematics knowledge and examine the extent of their understanding and the underlying processes.
The participants were divided into different groups based on their academic performance. Specifically, they had taken two-semester examinations in the past. The average of the two examination scores was computed and ranked. Using these ranks, they were divided into the following categories: excellent (top 25%), average (between 25 and 75%), and poor (bottom 25%). Because of the simplicity of such an operationalization, this is the most popular means of classifying school students (
This study was conducted in accordance with the recommendations of “The Guidelines of the International Committee of Medical Journal Editors” and “The Adolescent Mental Health Specialized Committee of Chinese Mental Health Association.” Prior to data collection, we obtained written informed consent from all the parents of non-adult participants and all adult participants (i.e., 37 teachers who participated in subsequent interviews). The parents and adult participants provided written informed consent in accordance with the Declaration of Helsinki. This study was approved by the ethics committee of the School of Psychology of Shandong Normal University.
Based on the discussion we had with the students’ mathematics teachers and analysis of previously acquired mathematics knowledge, we chose to focus on the following three mathematical concepts in this study: a twin prime (TP), a hetero-plane straight line (HPSL), and a semi-regular polyhedron (SRP). A TP is a pair of prime numbers with a numerical difference of 2. The HPSL consists of two straight lines in two different planes. The SRP is a convex geometrical figure enclosed by two or more types of polygons. In addition to the afore-mentioned explanations, another reason for focusing on mathematical concepts to study students’ mathematical understanding is that not only are they commonly found in mathematics textbooks for junior high school students, but they also constitute a major proportion of the mathematics knowledge contained in them.
As, according to the mathematics curriculum in China, junior high school students are yet to learn these concepts, we chose to focus on these three concepts. However, our discussions with five junior high school mathematics teachers revealed that it would not be difficult for junior high school students to learn these three concepts because they are closely related to the concepts that they have previously learned.
In order to enhance the brevity and effectiveness of the research instruments, we refined the questions that were used in the study based on the results of a preliminary investigation, in which we conducted interviews with 37 experienced junior high school mathematics teachers (the durations for which they had been teaching mathematics were > 10 years).
The main question that we used in the preliminary investigation was as follows: To help students understand mathematical concepts, what knowledge is it important to teach? Their responses included the names of concepts, their definitions, specific examples, the method of creating examples, graphics, relevant historical knowledge, relevant exercises, and practical applications. These results are consistent with those of
The frequency of teachers’ answers.
| Frequency | 37 | 36 | 23 | 7 | 14 | 15 |
| Percentage | 100 | 97.30 | 62.12 | 18.92 | 37.84 | 40.54 |
The method of creating examples is a concrete means of generating an example for a new concept based on examples of previous concepts. For example, the following means of deriving an ellipse is a method of creating an example: we can create an oblique section by using a plane to cut a cylinder obliquely. The edges of this oblique section are elliptical; therefore, an ellipse can be derived from a cylinder.
It can be inferred from
For TP, we provided the following method of creating examples: (a) find a prime number and (b) add 2 (larger number) and subtract 2 (smaller number) from this number. If the resultant larger (or smaller) number is also a prime number, then you have identified a pair of TPs. If this larger (or smaller) number is not a prime number, then try another prime number. With regard to the HPSL, we provided the following method of creating examples: (a) first finding a cuboid, (b) drawing two straight lines in their two adjacent planes, and (c) ensuring that they do not intersect on the intersection line of the two adjacent planes and have different angles when compared to the intersection line. The following method of creating examples of an SRP is provided by: (a) finding a cube, (b) connecting the midpoint of each edge, and (c) cutting off eight peripheral triangular pyramids with a plane. The resultant figure is an SRP because it is a convex geometrical figure that is enclosed by two types of polygons (i.e., a regular triangle and square).
We defined the three afore-mentioned mathematical concepts and generated three concrete examples and methods for creating concrete examples by interviewing five junior high school mathematics teachers. On four different cards, we wrote down the name of a concept (card A), its definition (card B), a specific example (card C), and a method of creating a specific example (card D). Twelve cards were used.
Individual interviews for collecting information on mathematical understanding were conducted in accordance with the following steps:
One of 54 students was chosen randomly.
One of the three afore-mentioned concepts was selected randomly. The student was shown the card with its name (card A) and asked if they have previously learned about this concept or can understand it. If they had learned about this concept or could understand it, then further exploration was terminated, another concept from the remaining cards was then randomly selected and the student was asked the above question again. If the student could understand all three concepts, the interview was terminated, and we reverted to step (1) choosing another student. This continued until all the students were interviewed. If they did not understand the mathematical concept, only then did we proceed to the next step.
One of the three study plans was randomly selected, namely, α, β, and γ (a detailed description of plans α, β, and γ are presented at the end of this section. Each plan was divided into two parts). The chosen study plan was executed in the following steps:
Implementation of the first part of the plan. When students finished the learning process, they were asked if they had understood the concept. If they said “yes,” they were asked to describe the meaning of this concept in their own words. Then we proceeded to step (b). If they said “no,” then we proceeded directly to step (b).
Implementation of the second part of the plan. When students finished the learning process, they were asked if they have understood the concept. If they said “yes,” they were asked to describe the meaning of this concept in their own words again. If they said “no” again, then the interview that focused on this concept was terminated and we proceeded to step (2) to choose another concept. This continued until all three concepts had been tested.
Students were asked to recount how they transitioned from a state of not understanding the mathematical concept to a state of understanding. They were asked to identify the cards that helped them understand the mathematical concepts, describe the role the contents on that card played in their learning, and how it helped them understand the concept.
Students were asked to identify any other content that helped them understand the concepts. If their responses were valid, they were asked to describe the role played by the specific content and how it helped them understand the concepts.
Students were asked to identify the contents that were unnecessary, and explain why they considered them to be unnecessary.
Students were asked to prioritize the presented content to help other students learn and understand this concept completely.
After the students finished it, the interview that focused on this concept was terminated.
Step (2) to choose another concept was proceeded to. This continued until all three concepts were tested. During this process, the concepts and plans to be used earlier were abandoned.
Once an interview with a student had been terminated, the interviewer reverted to step (1), chose another student, and repeated the afore-mentioned steps. This continued until all the students had been interviewed.
There are six types of complete permutations for cards B, C, and D. To enhance the efficiency of the research (under the condition of ensuring the effect), we chose three of these permutations according to the results of an advance investigation with 37 experienced junior high school mathematics teachers. They are permutations in BCD, BDC, and CDB. They are considered by most teachers to be helpful for students to understand mathematical concepts compared with the other three permutations. For these three selected permutations, we designed the following plans for the interview:
Part 1: The student was shown card B, which contained a definition of the mathematical concept, and card C, which presented a specific example of the concept. The student was allowed to read the contents and think about it aloud in order to understand the concept independently.
Part 2: The student was shown card D, which described a method of creating an example of a concept, and allowed the student to read the contents and think about it aloud in order to understand the concept independently.
Part 1: The student was shown card B, which contained a definition of the mathematical concept, and card D, which described the method of creating an example of the concept. The student was allowed to read the contents and think about it aloud in order to understand the concept independently.
Part 2: The student was shown card C, which presented conceptual examples, and allowed the student to read the contents and think about it aloud in order to understand the concept independently.
Part 1: The student was shown card C, which contained a specific example of the concept, and card D, which described a method of creating an example of the concept. The student was allowed to read the contents and think about it aloud in order to understand the concept independently.
Part 2: The student was shown card B, which contained a definition of the mathematical concept. The student was allowed to read the contents and think about it aloud in order to understand the concept independently.
The above procedures and plans are somewhat complicated; however, they can help researchers to identify the specific factors that affect students’ mathematical understanding and explore the internal processes that underlie their mathematical understanding. In view of this, this study firmly adopted them.
Three or four students were interviewed each day after class (i.e., in the afternoons) for a total duration of 3 weeks. Their mathematics teachers determined the order in which they would be interviewed. They were interviewed in a school campus activity room. Four mathematics teachers and four graduate students majoring in mathematics education served as research assistants with their consents. They were required to maintain a detailed and comprehensive record of the selected and used plans and the responses and behaviors of the students.
After the interviews were completed, we collected and collated the records of all the research assistants and validated each student’s answers and behaviors. Then we analyzed each student’s description of the concept they had learned in steps (a) and (b) and classified the extent of their understanding of the new concepts. To ensure the objectivity and reliability of the analytic process, we invited two experienced researchers (experts in the assessment of mathematical understanding) to independently analyze the data. Next, the other researchers checked and reviewed their results and discussed the combined results with the two expert researchers. The criteria used to assess students’ mathematical understanding were the accuracy, clarity, and comprehensiveness of their descriptions of the meanings of the newly learned concepts in their own words. If a student’s description of the meaning of a new concept was accurate, clear, and comprehensive, then he/she was considered to have understood the respective concept adequately. If a student’s description of the meaning of a new concept was inaccurate, unclear, or incomprehensive, then he/she was considered to have not understood it adequately. If a student could not describe the meaning of a new concept or his/her description was completely wrong or extremely unclear, then he/she was considered to have not understood it yet. Finally, we examined all the student responses that pertained to the contents they considered important for mathematical understanding and their priorities and analyzed the underlying meanings.
All the students completed their interviews within 30 min (
Following the implementation of part 1 of plan α, only 12.5 and 17.65% of students had fully understood the new concepts of TP and HPSL, respectively. Most of the students understood the new concepts only partially. Following the implementation of part 1 of plan β, about 70% of students (mainly students with excellent and average academic performance) fully understood the new concepts. Only some students (mainly students with poor academic performance) did not fully understand the concepts, and a few students had not yet understood the concepts. Following the implementation of part 1 of plan γ, only 10.53% of students had fully understood the concepts, and over 71% of students (mainly students with excellent academic performance) partially understood the concepts. In other words, almost none of the students fully understood the new concepts. The students’ understanding of the new concepts following the implementation of part 1 of the plans is summarized in
Students’ understanding after part 1 of the study plan.
| TP | Plan α | 2 (12.5) | 14 (87.5) | 0 (0) | 16 |
| Plan β | 17 (70.83) | 5 (20.83) | 2 (8.3) | 24 | |
| Plan γ | 0 (0) | 12 (85.71) | 2 (14.29) | 14 | |
| HPSL | Plan α | 3 (17.65) | 14 (82.35) | 0 (0) | 17 |
| Plan β | 11 (68.75) | 5 (31.25) | 0 (0) | 16 | |
| Plan γ | 0 (0) | 15 (71.43) | 6 (28.57) | 21 | |
| SRP | Plan α | 0 (0) | 12 (85.71) | 2 (14.29) | 14 |
| Plan β | 15 (71.43) | 6 (28.57) | 0 (0) | 21 | |
| Plan γ | 2 (10.53) | 17 (89.47) | 0 (0) | 19 |
Moreover, group comparisons of their level of understanding post-implementation of part 1 of the three plans for each new concept revealed no significant differences between male and female students and between students with excellent, average, and poor academic performance.
Following the implementation of part 2 of plans α, β, and γ, it could be seen that over 71.43% of the students had fully understood the new concepts. Only a few students (mainly students with poor academic performance) had not yet fully understood the new concepts. The students’ understanding of the new concepts post-implementation of part 2 of the plans is shown in
Students’ understanding after part 2 of the study plan.
| TP | Plan α | 13 (81.25) | 3 (18.75) | 0 (0) | 16 |
| Plan β | 20 (83.33) | 3 (12.5) | 1 (4.17) | 24 | |
| Plan γ | 10 (71.43) | 3 (21.43) | 1 (7.14) | 14 | |
| HPSL | Plan α | 14 (82.35) | 3 (17.65) | 0 (0) | 17 |
| Plan β | 14 (87.5) | 2 (12.5) | 0 (0) | 16 | |
| Plan γ | 15 (71.43) | 4 (19.05) | 2 (9.52) | 21 | |
| SRP | Plan α | 13 (92.86) | 1 (7.14) | 0 (0) | 14 |
| Plan β | 18 (85.71) | 3 (14.29) | 0 (0) | 21 | |
| Plan γ | 15 (78.95) | 4 (21.05) | 0 (0) | 19 |
Additionally, group comparisons of their level of understanding post-implementation of the second part of the three plans for each new concept revealed no significant differences between male and female students and between students with excellent, average, and poor academic performance.
The cards that students selected when they were required to identify the contents that played the most important role in helping them understand the new concepts were recorded. After collecting statistics and analysis, we found that all students (including students with excellent, average, and poor academic performance) considered the definitions of new concepts, examples, and the method of creating a specific example, to have played a very important role in helping them understand the new concepts. In particular, about 40% of students believed that definitions and over 31% of students believed that the method of creating a specific example, were important factors that facilitated this process. The statistical results are presented in
The most important contents for understanding.
| Card B | 24 (44.44) | 25 (46.3) | 21 (38.89) | 70 (43.21) |
| Card C | 13 (24.07) | 11 (20.37) | 12 (22.22) | 36 (22.22) |
| Card D | 17 (31.48) | 18 (33.33) | 21 (38.89) | 56 (34.57) |
Group comparisons of what the students considered to be the most important content that had helped them understand the new concepts revealed no significant differences between male and female students and between students with excellent, average, and poor academic performance.
The responses that students provided when they were required to prioritize the relevant contents that enhanced the understanding of learners, were recorded and analyzed. According to the results, over 50% of the students (including students with excellent, average, and poor academic performance) considered the order “A, B, C, and D” to be the best sequence of presentation of the contents pertaining to new concepts. In other words, they believed that successively presenting the names of concepts, their definitions, specific examples, and the method of creating a specific example would be the most helpful and effective means of helping learners understand a new concept. The detailed results are presented in
The best understanding priority.
| A, B, C, D | 38 (70.37) | 28 (51.85) | 31 (57.41) | 97 (59.88) |
| A, C, B, D | 4 (7.41) | 7 (12.96) | 5 (9.26) | 16 (9.88) |
| A, C, D, B | 4 (7.41) | 6 (11.11) | 3 (5.56) | 13 (8.02) |
| A, D, B, C | 2 (3.7) | 6 (11.11) | 5 (9.26) | 13 (8.02) |
| C, D, A, B | 6 (11.11) | 7 (12.96) | 10 (18.52) | 23 (14.2) |
Group differences in the prioritization of the contents that pertained to each new concept revealed no significant differences between male and female students and between students with excellent, average, and poor academic performance.
In the process of implementing part 1 of plan α, after reading the definition, most students looked at the following examples, and then returned to the description of definition, and started to repeat the keywords in a low voice several times “a pair,” “difference,” “straight line,” “two planes,” “two or more,” and “polyhedrons,” and then said “it should be like this,” “it should be correct,” “that is it.” In the process of implementing part 1 of plan β, after most students had read the definitions and methods continuously, most students turned to the definitions, read them silently again, and then whispered the keywords “a pair,” “different,” “straight line,” “two planes,” “more than two,” and “polygon,” then turned to the method, read it silently again, and then said “this way, this way., understand,” “this way.um., understood.” In the process of implementing part 1 of plan γ, most students went through the examples at first, then turned to the method, read it silently over and over, while whispering “this way. this way. um.,” “somewhat understanding” and “some understanding.”
In the process of implementing part 2 of plan α, after reading the description of the method, most students stared at the description, paused for a while, and then said “Oh, I understand,” “Oh, that’s it. I get it.” At this time, most students raised their heads and looked at the examiner with a smile. In the process of implementing part 2 of plan β, after reading the example, most students said “Well, yes, I understand,” “Yes, no problem. I understand.” In the process of implementing part 2 of plan γ, after reading the definition, most students stared at the definition narrative, and then confidently said, “I understand” At this point, students often looked up and leaned back.
Based on existing studies on mathematical understanding, this study aimed to explore the internal processes underlying mathematical understanding to enhance mathematics teaching and student learning. Using three mathematical concepts as instruments to investigate 54 students, we obtained many results and gained some insights.
The results showed that all students were able to understand the three new concepts following the implementation of parts 1 and 2 of the plans. This finding indicates that the definitions of concepts, use of examples, and the method of creating a specific example enabled students to transition from a state of not understanding to a state of understanding; consequently, these were the important contents that helped students understand the new concepts. This is consistent with the preliminary investigation results and confirms the perspective of
A careful analysis of the mathematical understanding of students post-implementation of part 1 of plans α, β, and γ revealed that the definitions of concepts and the method of creating a specific example were not only important but also necessary and indispensable. This is consistent with the findings of some researchers (
Similar conclusions can be drawn from the contents that the students considered important to their learning and the students’ oral reactions during the implementation of the plans. When they were asked to identify the contents that had played the most important role in helping them understand the new concepts, about 40% of the students named the definitions of the concepts, and over 31% of them named the method of creating a specific example. During the implementation of the plans, when the students learned the definitions of concepts and the method of creating a specific example, most of them said, “I understand,” especially when the definitions of concepts and the method of creating a specific example were arranged occurred after other contents.
As mentioned earlier, the definition of a mathematical concept is a statement or description of its connotations and characteristics. It indicates the position of a concept within the entire conceptual system, the concept that it is similar to, and the differences between them (
The same is true for the method of creating a specific example. In this method, an example of a new concept is created based on an example of a known concept. Consequently, the new example also reveals the relationship or connection between a related known concept and a new one, which promotes mathematical understanding. The only difference is that the relationship or connection here, was made through the process of creating a specific example that shows the link between the new concept and the known one (which shares a juxtaposed relationship with the new concept) (
Therefore it is the relationships, connections, and juxtapositions between new concepts and their superordinate concepts that promote mathematical understanding among students. This finding is consistent with previous studies (
If we further analyze the connection between the definitions of mathematical concepts and the method of creating a specific example, we find that they not only made the objects of connection of new concepts different but also altered the direction of its (connection. Definitions of mathematical concepts specify the superordinate concept category to which a new concept belongs; thus the resulting connection delineates the link between the new concepts and their superordinate concepts. When students learn the definition of a mathematical concept, they engage in superordinate learning activities (
Therefore the process of understanding new mathematical concepts should be based on the establishment of a connection between new and existing concepts in the minds of learners. This process can be accomplished in two different ways, namely, “going” and “coming.” “Going” refers to the process of connecting a new concept with an existing one whose inclusive level is higher than that of the new concept (which is a superordinate concept). Its connection direction is from a new concept to an existing concept. In contrast, “coming” refers to the process of connecting an existing concept (which is juxtaposed with the new concept) with the new concept. Its connection direction is from the old concept to the new concept.
The students indicated what they considered to be the best sequence of learning mathematical concepts after they had specified the contents that had played an important role in helping them understand the concepts. The results of statistical analyses of their responses revealed that most of the students considered the following sequence to be the most helpful to their process of understanding a new concept: learning the definition of a mathematical concept and then learning the method of creating a specific example. This means that to promote mathematical understanding among students, it may be best to allow them to connect new mathematical concepts to (a) the superordinate concepts and, subsequently, (b) existing juxtaposed concepts. Mathematical understanding is supposed to be a process of “going” proceeding “coming”.
Previous studies have shown that mathematical understanding is a process in which new concepts enter an individual’s cognitive structure, become a part of it, and form a network of relationships with previously acquired knowledge within the cognitive structure (
Additionally,
Mathematical understanding plays an important role in promoting student learning and the application of mathematical knowledge. Within the field of mathematics education, research on mathematical understanding has always been a popular topic (
Based on the contributions of (a) the meaning and role of the definitions of new concepts and (b) the method of creating a specific example of the processes that underlie mathematical understanding, several conclusions can be drawn. First, mathematical understanding is achieved when a connection between new concepts and at least two previously learned concepts is established within the cognitive structure of a learner. One of these two existing concepts should be the superordinate concept of the new concept, or more accurately, the superordinate concept that is most proximal to the new concept. The other concept should be convertible so that a specific example can be derived by changing or transforming its examples. Second, the process of understanding a new concept involves two processes, namely, “going” and “coming.” “Going” is the process in which a connection is established between a new concept and its closest superordinate concept. In contrast, “coming” is a process in which a connection is drawn between an existing convertible concept and a new one. Third, for a new concept to be understood, it should be situated in the middle of a connection between at least two concepts. Finally, in addition to the three dimensions that have been identified by
The present findings therefore suggest that in order to help students understand new mathematical concepts, teachers should first present students with the definition and subsequently teach them how to create a specific example based on an example of a previously learned concept. This will facilitate the formation of an interconnected network of new concepts in the minds of students. When teachers explain new concepts to their students, they should emphasize the superordinate concept to which the new concept belongs. Similarly, when teaching students the method of creating an example, they should emphasize that the example can be derived from the example of a previously learned concept.
The present findings (a) delineate the concrete processes that underlie mathematical understanding, (b) clarify the specific ways in which new concepts should connect to existing concepts so that mathematical understanding can be achieved, (c) illustrate the specific form of an internal network following the achievement of mathematical understanding, and (d) enrich the existing literature on mathematical understanding.
Although the mathematical concepts that were selected and examined in this study were all new to the participants, they were all closely linked to the mathematical knowledge that they had previously acquired. Will similar results be obtained if the mathematical concepts that are distant from previously acquired knowledge are presented to students? This is a noteworthy question for further study.
The datasets generated for this study are available on request to the corresponding author.
Prior to data collection, we obtained the written informed consent of all the parents of non-adult participants and all adult participants (i.e., 37 teachers who participated in subsequent interviews). The parents and adult participants provided written informed consent in accordance with the Declaration of Helsinki. This study was approved by the Ethics Committee of School of Psychology Shandong Normal University.
ZY designed the study and checked the data and the manuscript. ZY, KW, and YZ interviewed the students and analyzed the results. ZY and XY wrote the manuscript. GP and BX revised the manuscript and checked the results. All authors contributed to the article and approved the submitted version.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
This work was supported by the Shandong Provincial Education Department (Grant Number SDYJG21023) and Shandong Normal University (Grant Number 2016JG29).
We are extremely grateful to the reviewers for reading our manuscript and giving us many invaluable suggestions which have considerably improved the quality of our manuscript.