Literature Review
1.1 Discrete Mathematics used for Computer Science as a Fundamental basics to learn Algorithms
The Need for Computer Science
This is largely based on how exposed students are to computational thinking and computer science concepts. Additionally, educating students in computer science is beneficial for all students. With the digital age rising, there is a need to develop logical thinking and problem-solving which are all a part of learning computer science.
Computer Science Standards and Model Curriculum give students experiences that help them discover and take part in a world continually influenced by technology and to understand the role of computing
What is Discrete Mathematics?
Discrete Mathematics is an area of mathematics that most closely connects with the field of computer science. It is the study of mathematical structures that are countable or otherwise distinct and separable (as opposed to continuous quantities like in algebra or calculus).
What Is Discrete Mathematics?
Discrete Mathematics is a rapidly growing and increasingly used area of mathematics, with many practical and relevant applications.
• Because it is grounded in real-world problems, discrete mathematics lends itself easily to implementing the recommendations fo the National Council of Teachers of Mathematics (NCTM) standards. (The recently published Standards and Principles for School Mathematics notes that “As an active branch of contemporary mathematics that is widely used in business and industry, discrete mathematics should be an integral part of the school mathematics curriculum.”)
• Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be introduced at all grade levels, even with children who are not yet fluent readers.
Discrete mathematics will make math concepts come alive for your students. It’s an excellent tool for improving reasoning and problem-solving skills, and is appropriate for students at all levels and of all abilities. Teachers have found that discrete mathematics offers a way of motivating unmotivated students while challenging talented students at the same time.
Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be easily be introduced at the middle school grade level.
1.2 How Discrete Mathematics Can Help Students
The National Council of Teachers of Mathematics (NCTM) recommends that discrete mathematics be implemented into the curriculum as early as the seventh 4 grade, because it benefits students and helps maintain their interest in mathematics [5, p. 362]. But beyond what the NCTM recommends, is discrete mathematics really advantageous to students, and if so, why? Discussed below are some of the reasons. First, it keeps students interested in mathematics. It helps entice them to regularly attend and participate in class. When students are interested in the material, it is easier for them to learn and stay focused when presented with tough, complex problems. While solving problems in discrete mathematics can be complicated, the problems themselves can be easily understood. The students, therefore, are able to understand and work the problems, which gives them a much needed confidence [12, pp. 35-36]. Many students are lost from mathematics forever during high school. Discrete mathematics is a great way to help these students stay interested and involved in mathematics. Second, discrete mathematics benefits students by allowing them to see the connections between the mathematics they are studying and the real world. “For example, we might be able to convince our students that calculus can be used to help civil engineers build better bridges, but the students still might not see how it really works. But in graph theory [and other areas of discrete mathematics], we can explain the applications, the students can see how they work, and they can actually see real problems” [7, p. 94]. Teachers need to help students dismiss the idea that there is nothing new left to discover in mathematics and help them to look beyond basic arithmetic computation. Discrete mathematics is the mathematical foundation of computer science and is “used extensively in business, industry, and government. For example, difference equations are an essential mathematical tool for high-technology engineering firms, and matrices are indispensable for computer graphics” [4, p. 75]. 5 Another benefit of discrete mathematics is that it enriches the traditional curriculum. It places more emphasis on teaching students to think mathematically and less emphasis on certain computational skills [1, p. 83]. Discrete mathematics lends itself to group work more easily than does traditional mathematics. It is also helpful to teachers because it gives them a new way of teaching elements in the curriculum, which may make the traditional concepts easier to teach and learn. By using discrete mathematics to teach already existing elements in the curriculum, it can help to change the way students view mathematics altogether. Below are some of the ways that discrete mathematics complements topics already in the traditional secondary-school curriculum: · Algebraic skills are needed and reinforced throughout discrete mathematics. · In geometry, graph theory can be used to enrich the study of polygons and polyhedral. · Difference equations give use to the fascinating new geometry of fractals [4, p. 75]. The NCTM does not provide teachers with detailed guidelines. They only provide a set of goals and topics to cover in high school for discrete mathematics. Therefore, it is up to each individual teacher, or the mathematics department within each school, to decide how these topics should be implemented.
Unfortunately, many teachers are unfamiliar, and even uncomfortable, with discrete mathematics. Thus, it can be difficult for them to know how to incorporate these topics into the curriculum. Because discrete mathematics can be used to teach traditional elements in the curriculum, these topics can be covered in different ways throughout the school year, without having to set aside extra time to cover them. 6 While the implementation of discrete mathematics into the curriculum is not discussed here in detail, many references cited in this thesis give numerous ideas on how to do so. Some also give sample lessons and projects for different skill levels. Some excellent resources are: · DIMACS Series in Discrete Mathematics and Theoretical Computer Science 36 (eds. J.G. Rosenstein, D.S. Franzblau, and F.S. Roberts), American Mathematical Society; 1997. · Discrete Mathematics Across the Curriculum, K–12 (eds. M.J. Kenney, and C.R. Hirsch), NCTM, Inc, Reston, Virginia; 1991. · Mathematics Teacher, a monthly journal magazine.
These political battles over the mathematics curriculum resulted in discrete mathematics largely being ignored in these countries. Of the discrete mathematics topics specifed above, very few are part of the standard curriculum in most of the countries we are familiar with. Through personal communication, it appears that:
- Combinatorics is included in the secondary curriculum of several countries, including Spain, US, England, Germany, Hungary, Brazil, Israel.
- Connecting recursive patterns and sequences with algebraic formulas is taught, to some degree, in Spain, Germany, and the United States.
• Graph theory is included in Italy and some isolated state curriculum in the US. In England, students focusing on mathematics can take a special track in which they have extensive exposure to discrete mathematics. In many countries, it appears that the opportunities for dealing with discrete mathematics in schools, especially when it goes beyond combinatorics, are often only seen on the level of optional recreational mathematics (Colipan & Liendo, 2022; Gravier & OuvrierBufet, 2022; Greefrath et al., 2022). We wonder if one contributing issue for the lack of discrete topics taught in the schools may be that the termdiscrete mathematics’ is not well understood. Perhaps, each of the discrete topics mentioned above should be considered individually. For example, fair division algorithms and economic game theory are almost self-explanatory. While combinatorics sounds complicated, counting is clearly important. Instead of using the termgraph theory’ which may be misleading to some, we could talk about vertex-edge graphs or networks, which most people are familiar with. Iteration can be described in terms of simple recursive situations, such as repeatedly folding a piece of paper or compounding of interest, along with its accessibility through the use of spreadsheets. Therefore, it might be more productive to talk about the individual discrete topics than the discipline as a whole. This is also intended to describe the mentioned topic areas of discrete mathematics for school more clearly once again. We therefore shortly go into a little more detail on the most common discrete topics and how they may support mathematical competencies.
1.3 Potential benefits of discrete mathematics topics for mathematics education
We see the potential benefits of teaching discrete mathematical topics in three broad areas, and some of these benefits have been highlighted in existing literature. The first potential benefit is offered by the content, which is accessible and offers interesting and relevant topics for teaching and learning (Anderson et al., 2004; DeBellis & Rosenstein, 2004). 4 Potential benefits of discrete mathematics topics for mathematics education
The second potential benefit is the learning of mathematics and the acquisition of general mathematical competencies (Coenen et al., 2018; Vorhölter et al., 2019) including afect (Goldin, 2018) that influences the learning of mathematics and the third potential is the relevance of discrete mathematics for living in the modern world (Hart & Martin, 2018; Rosenstein, 2007). In this section, we will discuss general benefts and in Sect. 6, the benefts resulting from specifc discrete mathematics topics. 4.1 Accessible topics for teaching and learning As early as the end of the 1980s, there were calls to integrate discrete mathematics into teaching, not only in higher education but also in schools (Dolgos, 1990). Advocates of discrete mathematics have noted that problems in discrete mathematics are relatively accessible, in the sense that a student may be able to understand what a problem is asking or can explore a situation without needing a lot of prior mathematical experience (Anderson et al., 2004; Devaney, 2018; Ferrarello & Mammana, 2018; Rosenstein, 2007). This is often because the problems themselves do not require knowledge of technical definitions or specific mathematical knowledge, and students can exemplify and explore objects. The discrete nature of the objects would seem to lend itself to this accessibility. This accessibility of discrete mathematics is something that seems to be agreed upon by many mathematicians, mathematics educators and mathematics education researchers, and it is often used as an argument for the importance of the inclusion of discrete mathematics in curriculum (Anderson et al., 2004; Burghes, 1995; Dolgos, 1990; Hart & Martin, 2018). While we note that this is an aspect of discrete mathematics that would benefit from more systematic study and research, this accessibility has come through in some research studies, but not as an explicit focus of the study. Some of the research on the teaching and learning of discrete mathematics with younger students highlights not only the accessibility of topics but that this accessibility can help students make sense of the current curriculum. For example, iterative problems and difference equations can help students learn algebra (Amit & Neria, 2008; Blanton & Kaput, 2005; Carraher et al., 2008; Radford, 2008; Rivera & Becker, 2008; Sandefur et al., 2018; Steele, 2008; Yeap & Kaur, 2008). Even very young students can reason about combinatorial problems in meaningful ways (de Beer et al., 2015; Maher et al., 2011). English (1991, 1993) reports on young children’s strategies as they engage with combinatorial problems. As another example, students can naturally `invent’ graph theory to solve a problem (Ferrarello & Mammana, 2018; Greefrath et al., 2022; van den Heuvel & Krabbendam, 1991). We can only wonder how much better would the students’ work be if they already knew some graph theory or had previous experience with counting or recursive problems?
For many topics in discrete mathematics, students ranging from young children to undergraduate students can be posed similar questions and have a reasonable chance at investigating the problem at their different levels. This results in self differentiating tasks that allow individual approaches to the problem (Ostkirchen & Greefrath, 2022). For example, how many 2-color towers can I make of height 5, can be extended to more complicated problems for older students by increasing the number of colors and the height of the tower. Maher et al. (2011) describe the use of combinatorial problems in such contexts among students in a longitudinal study. Students as young as the third grade can investigate recursive structures they build with toothpicks and stickers while high school students can develop recursive models of bacteria growth and the spread of epidemics, which is a simplified version of models studied by epidemiologists (Radford, 2008; Sandefur & Manaster, 2022; Yeap & Kaur, 2008).
https://link.springer.com/article/10.1007/s11858-022-01399-7
Conclusion:
While we note that this is an aspect of discrete mathematics that would benefit from more systematic study and research, this accessibility has come through in some research studies, but not as an explicit focus of the study. Some of the research on the teaching and learning of discrete mathematics with younger students highlights not only the accessibility of topics but that this accessibility can help students make sense of the current curriculum. For example, iterative problems and difference equations can help students learn algebra (Amit & Neria, 2008; Blanton & Kaput, 2005; Carraher et al., 2008; Radford, 2008; Rivera & Becker, 2008; Sandefur et al., 2018; Steele, 2008; Yeap & Kaur, 2008). Even very young students can reason about combinatorial problems in meaningful ways (de Beer et al., 2015; Maher et al., 2011). English (1991, 1993) reports on young children’s strategies as they engage with combinatorial problems. As another example, students can naturally `invent’ graph theory to solve a problem (Ferrarello & Mammana, 2018; Greefrath et al., 2022; van den Heuvel & Krabbendam, 1991). We can only wonder how much better would the students’ work be if they already knew some graph theory or had previous experience with counting or recursive problems? For many topics in discrete mathematics, students ranging from young children to undergraduate students can be posed similar questions and have a reasonable chance at Investigating the problem at their different levels. This results in self differentiating tasks that allow individual approaches to the problem (Ostkirchen & Greefrath, 2022). For example, how many 2-color towers can I make of height 5, can be extended to more complicated problems for older students by increasing the number of colors and the height of the tower. Maher et al. (2011) describe the use of combinatorial problems in such contexts among students in a longitudinal study. Students as young as the third grade can investigate recursive structures they build with toothpicks and stickers while high school students can develop recursive models of bacteria growth and the spread of epidemics, which is a simplified version of models studied by epidemiologists (Radford, 2008; Sandefur & Manaster, 2022; Yeap & Kaur, 2008). This highlights what we mean by accessibility – students can have access to mathematical topics and ideas regardless of background and prerequisite knowledge (DeBellis & Rosenstein, 2004; Ferrarello & Mammana, 2018). Although we have several examples of empirical research that implicitly demonstrates the accessibility of discrete mathematics and the value of this accessibility, we note that there is also more work to be done. There is a need for the field to focus on the issue of accessibility more systematically and explicitly.
Reference:
https://repository.lsu.edu/
https://link.springer.com/article/10.1007/s11858-022-01399-7