Chi Binh Le (chle0228@uni.sydney.edu.au)
John Spracklin (jspr0187@uni.sydney.edu.au)
Ruolan Li (ruli2623@uni.sydney.edu.au)
The lesson plan for MS-N1 Networks and Paths adopts the pedagogical approach of constructivist learning. The lesson content is carefully sequenced and planned to allow students to participate in the knowledge building process as active learners. This pedagogical choice is reflected in the following aspects:
The lessons are designed to engage students' prior knowledge
Prior knowledge is imperative for students' learning as it enables students to gain stronger motivation and deeper comprehension (Lin, Lin, & Huang, 2011). In lesson one, we deliberately select the game Treasure Hunt to elicit students' prior knowledge and their real-life experience. If students are familiar with the game and concept of mapping, they are more likely to bring their exisiting knowledge to network topic through this activity. Entry ticket is used in several lessons, with the purpose of checking students' understanding on the content of previous lessons, and the following activities are built upon that.
The lessons engage students with authentic task and real-life application
According to Tan & Nie (2015), real-life application generates students' positive attitude towards mathematics, as well as increases students' mathematical problem-solving competencies. The lesson plan not only incorporates several authentic tasks (e.g. Six Degrees of Kevin Bacon, building networks on inner Sydney railway map) in order to link mathematical knowledge to the current society, but also embeds reflection questions (e.g. the pros and cons of using each methodology in workforce) to enhance students' critical and metacognitive thinking. Investigative assessment serves the purpose of promoting students' autonomy. These approaches could best prepare students for the real-life problem solving processes after high school, and enable them to adopt a life-long learning atttitude.
Differentiation on content, process and product is implemented throughout the unit plan
According to Reys (2017), differentiation can be applied to classroom context via open-ended tasks that allows for multiple paths and multiple solutions. This engages students with different levels to start from where they are and motivates them to take responsibility of their own learning process. In this lesson plan, differentiation is achieved by offering students tasks with various difficulty levels, encouraging students to work in a group and engaging students with open-ended classroom discussion. By doing so, students are able to get equal access to learning opportunities, and achieve the same learning goal in the end of the class.
ICT use is highly integrated in the class
ICT use is incorporated in the lessons and is seen as one of the paramount factors to facilitate students' learning. The use of Six Maps in lesson Three and Geogebra in lesson Six is a good example. Students are exposed to different representations of content knowledge. They are also encouraged to use IT as one of the ways to express their own understanding.
Graph theory (of which Networks forms a part) has found applications in many fields including organic chemistry, biology, genealogy, geography and cartography, and linguistics to name just a few (Lessner, 2011). The extensive use of graph theory provides teachers the opportunity to motivate students to study Networks. When students are presented with real life applications that they can relate to, their prior knowledge and social background contribute to their learning.
Real life contexts are shown throughout this unit of work in the headers of each lesson and in the class exercises. Each lesson has a header graphic and description to showcase graph theory in practice. The examples are chosen from the contexts of:
the original Konigsberg bridge problem
computer networks
artificial intelligence
social networks
Rubik's cube
flight schedules
The class exercises further emphasise real world applications by posing problems in mapping, social networks, and infrastructure planning. The examples presented to students show the wide use of graph theory so that they can better appreciate the relevance of graph theory.
The social constructivist approach to teaching this topic is justified in that graph theory problems inherently lend themselves to reasoning and debate amongst peers (Ferrarello & Mammana, 2018). The class exercises promote discussion between students and with the teacher. Throughout the lessons, students construct their own networks, work in groups or pairs, and are given choice of subject material in the exercises.
There is minimal prior knowledge required for the topic of Networks so it is well suited for study in the Mathematics Standard course. Students will only require basic arithmetic skills of addition and subtraction, and comparing numbers to identify a smaller or larger number. Students should also be able to use tables to represent information. They would have been encountered this in earlier topics such as Data Collection and Representation (Stage 4) and Single and Bivariate Data Analysis (Stages 5.1, 5.2). The use of tables in the Networks topic will be different from that previously encountered in that the row headers and column headers will have the same items.
Students may fail to identify multiple minimum spanning trees that could be presented in a network diagram, they may also tend to think the minimum spanning tree is fixed, regardless if the weight of the edges changes.
Students may have the misconception that there is only one shortest path between two vertices.
Students may have the misconception that all vertices can be visited via shortest paths. They could also have the misunderstanding that the shortest path always refers to the shortest distance in length. Whereas the shortest path could also refer to the cheapest route/path or quickest route.
Students may have the misconception that the shortest path is the best path, or shortest path is contained in any minimum spanning tree (NSW Education Standards Authority, 2019).
Students may be unaware of the fact that networks can be used to represent many forms of conneciton, both simple and complex. The edges can be used to represent cost or time, not distances only.
The relative position of a vertex in a diagram is significant, therefore placing vertices in different positions results in different network (Rosenstein, 2018).
Students may have misconception that vertex cannot have an edge back to itself.