Keywords

1 Introduction

Prior knowledge is not only a foundation for new learning; it also supports learning mathematics as grandiose ideas. Insufficient or inaccurate prior knowledge cannot support learning connections. As learning is a type of construction, necessary prior knowledge should be available for new learning to build on. To make learning deeper, relevant prior knowledge is needed that allows rich connections to be drawn with the new knowledge. Rich connections can be supported by posing purposeful questions. In this way, teachers could make mathematics learning deeper and easier for students.

Little action research has been carried out seeking to improve mathematics teaching and learning in a Saudi context. The sample in this study is small, with only eight students in total. This is because the nature of the research requires observing student learning and stopping teaching at any time and talking with students who experience difficulties or have only a surface-level understanding of the knowledge. Therefore, a small sample was more appropriate. In addition, the aim of this action research is to improve a specific teacher’s teaching, not to produce generalizable results. The goal of this study is to improve my teaching of mathematics to allow deeper and easier learning for my students.

1.1 Sufficient Prior Knowledge Serves as a Foundation for New Learning

Sufficient prior knowledge is a prerequisite for new learning. In other words, when knowledge C1 is necessary to understand knowledge C2, then knowledge C1 is a prerequisite to knowledge C2 [17]. For instance, a machine learning course requires the completion of a linear algebra course first. This is because there are numerous concepts in a course on linear algebra that are prerequisites to several concepts in machine learning courses (e.g., principal component analysis requires knowledge of Eigen analysis; [25]). Hence, ensuring necessary prior knowledge is important for learning. In a previous study, I found that students with more prior content knowledge were able to control their learning better than their peers with less [1]. This could be because the students then interact with new learning with a lower working memory load. For example, in a single session, if a learner is attempting to learn the new concept ‘multiplication’ but does not know what ‘addition’ or ‘frequency’ is, then the learner may be unable to assimilate the new concept. The student must learn the meaning of the foundational concepts alongside the new concept, which could overload the student’s working memory.

In more detail, learning is a series of knowledge expansion processes carried out by a learner. When learners intend to learn new knowledge, they must activate and connect their existing knowledge to the latest information [19]. Activation of prior knowledge occurs when individuals bring relevant prior knowledge from their long-term memory to their working memory. Working memory capacity can be vastly increased when necessary relevant prior knowledge is stored and easily retrievable from long-term memory [22, 26]. However, if prior knowledge is insufficient, long-term memory cannot support new learning. This results in overloading working memory, giving up learning, or surface-level learning (e.g., rote memorization). Surface-level learning could occur if students cannot connect the new knowledge to their prior knowledge base [12].

Just as a lack of prerequisite knowledge or insufficient prior knowledge hinders the expansion of individuals’ knowledge, inaccurate prior knowledge could hinder new learning. Due to inaccurate prior knowledge, learners may even resist new learning. For example, when a student inaccurately counts numbers, it is impossible for the student to learn addition without correcting his mistakes. Indeed, several studies have indicated that the activation of inaccurate prior knowledge could impede new learning [3, 5]. Building on existing knowledge only works if the existing knowledge can be extended by new knowledge [20]. Thus, sufficient and accurate prior knowledge could contribute to making mathematics learning deeper and easier for students.

1.2 Relevant Prior Knowledge Supports Learning Mathematics as Grandiose Ideas

Students can learn when sufficient and accurate prior knowledge is available. However, the story does not end there. Prior knowledge must also relate to new knowledge to make learning deeper. In mathematics, this can be referred to as extensive ideas. Charles [9] defined a big idea as “a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole” (p. 10).

The concept of grandiose ideas makes mathematics coherent and allows teachers to practice effective teaching. It also allows students to develop a deep understanding of and the ability to apply mathematics. Research on the brain has shown that ideas that are connected are more readily implemented in new situations than unconnected ideas [14].

Therefore, mathematics curricula and teaching practices should be based on extensive ideas. Mathematics learning will be deeper, richer, and longer lasting when students make connections between different mathematical ideas [15]. Thus, a conscious effort is required to help students see mathematics as a highly connected structure of concepts across numerous topics rather than viewing them as unrelated concepts [27].

Students’ understanding of grandiose ideas results in more generalizable knowledge implementation and enables knowledge transfer [4, 10]. When learners provide explanations that apply ideas in several contexts, the ideas become less context-dependent and eventually more abstract [14]. Siemon [25] asserted that students’ mathematical progress may be limited without the development of extensive ideas. He argued that big ideas provide organizing themes that support additional learning and generalizations.

As students become more advanced in mathematics, they might be able to use different methods to address mathematical situations. For example, students in grade 3 can use counting in more diverse ways than students in lower grades [21]. For example, when students intend to solve 15 + 7, they can easily recall that 5 + 5 = 10. They use this prior knowledge to determine that 15 + 5 will result in 20, and then they add on the remaining 2 to make 22. These diverse ways of accounting are connected through the idea of equivalence. By regrouping the numbers into an equivalent representation, the students are using prior knowledge or known facts to figure out the unknown fact. For effective teaching, teachers should make such connections explicit. When students lack prior knowledge regarding such equivalent concepts, they will find it difficult to make these types of connections. Working memory would then not be supported by long-term memory, as discussed above. Therefore, sufficient prior knowledge to learn new knowledge and to support the development of big ideas should make learning deeper and easier for students.

1.3 Posing Questions to Assess and Address Relevant and Necessary Prior Knowledge

Posing questions is a powerful tool for engaging students in learning processes. When students seek the answers to questions, they are learning. Therefore, the teacher could use questions to make students seek answers to learn. Questions can be used for several purposes, including prompting thinking, checking students’ understanding, encouraging discussion, and facilitating critical thinking [18]. They can also be used for assessing prior knowledge [2]. Teachers can use questions not only to check for and address necessary prior knowledge, but also to make students connect relevant prior knowledge to learn and thereby develop extensive ideas.

Preparing a few key questions before class could be useful for assessing necessary prior knowledge, with other questions formulated on the spot in response to the discussion [18]. In this way, questions can be used for deeper learning. Posing questions stimulates the development of students’ higher-level thinking processes. However, it is important that the questions are precise and clearly phrased [16].

2 Methodology

2.1 Participants and Design

The study included two stages. The first stage involved data collection and analysis regarding my own teaching to develop a plan for improvement. The second stage involved implementing the planned improvement and reflecting on it to assess its effectiveness.

The participants consisted of eight students: four seventh-grade students and four sixth-grade students (age M = 12 years; SD = 1). Two were my own children, and six were my nieces. All were female. Although they differed in terms of academic level, their math levels were similar, and they had achieved similar grades in math (90–94%). The sample of each stage was four students (two seventh-grade students and two sixth-grade students in each stage). In this research, the small sample size is important to derive useful data because it would provide a researcher with an opportunity to deeply explore a research problem [11].

This type of research is action research with a single researcher. According to Ferrance [13] “Action research is a process in which participants examine their own educational practice systematically and carefully, using the techniques of research” (p. 9). In this research, there are five phases of inquiry carried out: 1. Identification of problem area, 2. Collection and organization of data, 3. Interpretation of data, 4. Action based on data, and 5. Reflection [13]. In the present research, qualitative design was applied using field observation.

2.2 Materials

2.2.1 Topics

‘Ratios, proportions, and percentages’ were selected as the topic of study. Some content related to the topic was not new to the students, particularly the basic knowledge; however, the advanced content was new. The content was developed by the researcher and given to two experts for validation. The instruction lasted for almost 10 days and was carried out during 10 h-long sessions totaling 10 h (five lessons for each stage).

2.2.2 Field Notes

Field observation notes were taken during the teaching. As I am both a teacher and a researcher, my intention was to monitor the students’ learning and document my observations, particularly difficulties the students encountered in learning as well as any surface-level learning. This was followed by reflecting on the notes and dialogs with the students about the reasons for their difficulties or surface-level learning. At the end of each lesson, students also received an assessment test. The teacher checked the assessment and discussed the answers with his students. The observation method used in this study consisted of two parts: descriptive followed by reflective information [23].

Data from the field observation notes were analyzed thematically. This involved the researcher becoming familiar with the data, generating the initial codes manually, searching for themes, reviewing the themes, defining, and naming the themes, and identifying emergent themes [6]. The data analysis examined learning problems, the reasons for those problems, and the plan for improving my own teaching based on the reasons for the learning problems. Then, planned teaching principles were extracted from the analysis.

2.3 Procedures

Although all participants were members of my family, formal consent to participate in the study was obtained. They were advised that they could withdraw from the study at any time with no explanation.

The content was developed by the researcher and validated by two experts. As noted above, the study included two stages. The first involved identifying surface learning and students’ difficulties in learning and developing a plan for improvement of my own teaching in the second stage. In the second stage, I implemented the plan and reflected on its effectiveness.

2.4 Collection, Organization, and Interpretation of Data and Action Based on Data

The findings showed that the students suffered from a lack of understanding, a lack of extensive ideas, a lack of necessary prior knowledge, and misconceptions. The data relating to these problems were analyzed, and a plan for improving my own teaching was developed. An important part of the planned improvement was posing questions across sufficient learning situations to deepen students’ learning, help them to develop extensive ideas, and assess prior knowledge to make their learning deeper and easier. The specific findings are presented and discussed in the following.

2.4.1 Lack of Understanding

As shown in Table 1, some students did not deeply understand the concept of percentage. For example, some did not understand what 10% means. It is possible that they did not understand the concept because they had not reflected on it. The teacher frequently found that some students would refocus on learning when they were asked to answer a question. When students are questioned, they must be active to respond to the question. When students seek the answers to questions, they are learning [18]. Thus, it is important to assess students’ learning by posing questions to stimulate their thinking. Indeed, posing questions is critical for effective teaching [18]. Even when some students knew how to work out 10% of 100, they did not know how to work out 10% of 10 or 1000, or they did not know how to work out 100% or 200% of 20. They may not have grasped the concept deeply because they had not been exposed to sufficient learning situations. Similarly, the students could use multiplication with X:Y, but they could not use division when necessary. They only had experience with multiplication, that is, the examples they were given only involved multiplication. To correct this, they need to be given examples of all situations. When learners are provided with explanations that apply ideas in several contexts, the ideas become less context-dependent to eventually become more abstract [14]. Importantly, the lack of conceptual knowledge of mathematics critically hinders students’ ability to transfer and generalize mathematics knowledge. Posing questions in more learning situations might be a reasonable solution to this issue. In fact, this solution worked well when applied to other students and the same topics. The students grasped the concepts and were able to utilize them in different situations.

Table 1 Lack of understanding
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2.4.2 Lack of Big Ideas

Table 2 shows that the students only knew one way to solve problems. For example, when they needed to work out 25% of 1200, they considered the 1200 and/or 25%. They did not try another way even when I asked. For example, they did not divide 1200 into 1000 and 200 or divide 25% into 10% and 10% and 5% to make it easier. They did not understand how mathematics concepts were related (lack of big ideas). As another example, the students understood X:Y, but they did not understand X:Y = X/Y. Supporting big ideas by encouraging students to connect concepts to related concepts by posing questions could help improve their mathematics connections and lead to deeper understanding. Students should be able to view mathematics as a coherent whole to construct the meaning of concepts and principles. Charles [9] defined a big idea as “a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole” (p. 10). Therefore, mathematics curriculum and teaching practices should be based on extensive ideas.

Table 2 Lack of big ideas
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Developing extensive ideas can help students learn mathematics deeply. Indeed, mathematics learning is deeper, richer, and longer lasting when students make connections between different mathematical ideas [15]. Thus, teachers can use the concept of big ideas to make their mathematics teaching more effective. This can also help students to develop a deep understanding of and ability to apply mathematics. Studies of brain activity have shown that ideas that are connected are more readily implemented in new situations than unconnected ideas [14]. Siemon [25] argued that students’ mathematical progress may be limited without the development of big ideas. Big ideas provide organizing themes that support additional learning and generalizations [25]. As students become more advanced in mathematics, they should be able to use different methods to address mathematical situations [21].

2.4.3 Lack of Necessary Prior Knowledge

As can be seen from Table 3, some students lacked the necessary prior knowledge. For example, they did not know how to read numbers, they did not know how to perform long division, they did not know that every number could be a rational number, and they had not memorized the multiplication table. This lack of necessary prior knowledge hindered their learning and kept the students dependent on the teacher. In a previous study, I found that students with more necessary prior knowledge were able to control their learning better than those with less [1]. This could be because the students are then able to engage in new learning with a lower working memory load. If prior knowledge is insufficient, long-term memory cannot support new learning. This leads to overloading working memory, giving up learning or surface-level learning (e.g., rote memorization). Surface learning may occur if students cannot connect the new knowledge to their prior knowledge base [12]. Accordingly, teachers must assess students’ prior knowledge related to the current learning by posing questions or asking them to solve certain problems. Continually assessing students’ learning has been found not only to help identify a lack of necessary prior knowledge, but also to keep students focused, as discussed above [2, 16, 18].

Table 3 Lack of necessary prior knowledge
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2.4.4 Misconceptions

Table 4 shows that some students had inaccurate prior knowledge. For example, the students thought that dividing a number by a fraction would give us the fraction of the number (e.g., 100 ÷ 1/2 = 50), whereas the correct method is to multiply 100 by the fraction. They also thought that dividing a number by a percentage would give us the percentage of the number (e.g., 120 ÷ 20% = 24), whereas the correct method is to multiply 100 by the percentage. Such inaccurate prior knowledge could hinder new learning. Moreover, due to inaccurate prior knowledge, learners might even resist new learning. Several studies have demonstrated that the activation of inaccurate prior knowledge could impede new learning [3, 5]. Building on existing knowledge only works if the existing knowledge can be extended by new knowledge [20].

Table 4 Misconception
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Making students aware of the discrepancy between their thinking and correct concepts is important for addressing this problem [2]. Teachers cannot predict this problem, but they can identify it and help to prevent it in the future. Continually assessing students helps to discover such issues [2].

3 Discussion

The planned improvement was posing questions across sufficient learning situations to deepen students’ learning, help them to develop big ideas, and assess prior knowledge to make their learning deeper and easier. Subsequent reflection on the planned improvement in the next teaching sessions indicated that the plan worked well, as no learning difficulties or surface-level learning were identified. When the teacher applied this with different students and then reflected on his teaching, he found that the students’ learning was smoother, the teacher became aware of the necessary prior knowledge and was able to address problems that emerged, and the students were able to deepen their learning. The aim of the research was achieved by making students’ learning process smoother by improving the teacher’s performance.

4 Conclusion and Limitations

In this research, I planned to improve my own teaching by using action research. The findings in this research are posing questions across sufficient learning situations to deepen students’ learning, help them to develop big ideas, and assess prior knowledge to make their learning deeper and easier. The sample was small for two reasons. First, a small sample was helpful for deeply assessing the students’ learning, as it enabled individual and immediate discussions when problems arose related to the students’ learning. Further, the students were all female and members of my family, and no high-stakes test was involved. This allowed me to discuss their understandings in a safe context. Second, the purpose of the research was to improve my own teaching, not to produce generalizable results.