'Introduction to Vectors' is a Year 12 topic located in the NSW Mathematics Extension 1 Senior Syllabus. It has the following intended outcome: “ME1-12-02 operates with 2D and 3D vectors and uses 2D vectors to solve problems involving motion in two dimensions” (NESA, 2025). It is important that students learn this topic as vectors are a fundamental concept in mathematics with many real-world applications. According to the NSW Education Standards Authority (2025), the topic 'Introduction to Vectors' teaches students about the behaviour of objects in two dimensions and how it can be expressed. This includes the consideration of position, displacement and motion. Additionally, students are exposed to a variety of notations to represent objects in two dimensions. In learning this topic, students can learn how to use these notations effectively to explore the geometry of a situation and solve complex problems. Furthermore, 'Introduction to Vectors' is an important topic to teach as many of its concepts are introduced through a geometrical perspective. This can help develop students' spatial reasoning and geometric thinking skills, which are crucial for various disciplines.
Given the declining enrolments within tertiary mathematics courses (Brown, 2009; OCS, 2012), it is important that students are taught the topic 'Introduction to Vectors' as it serves as a bridge to upper level mathematics. Vectors are foundational to advanced mathematical topics such as linear algebra, quantum mechanics, analytical geometry and more. In addition, vectors are implemented in many fields of study and play a critical role in various real-world contexts. For example, in physics and engineering, vectors are commonly used to represent position, displacement, velocity, acceleration, forces and other physical phenomena. In video game development, vectors are often used to represent properties of objects (e.g., rotation, position) within the game world and determine how two objects interact. In 3D modelling, vectors are essential for defining light directions and simulating realistic shadows. In data science, vectors are used to represent data samples which make it more easier to understand, process and analyse. In navigation, vectors are used to calculate the distance, direction and movement of aircrafts and thus help guide them to their desired destinations. Overall, vectors are especially useful in real-life and offer a powerful language for expressing and understanding the localisation of people, places, and objects.
This unit will focus on the first three groups of syllabus content indicators in the 'Introduction to Vectors' topic: (1) 'Vector representation and notation', (2) 'Introduction to 2D and 3D vectors', and (3) 'Operating with vectors'. The aim of this unit is to consolidate students' foundational knowledge of vectors and foster a deeper understanding of the topic's critical concepts through the pedagogical approaches listed below.
Concrete, Representational, Abstract (CRA) model
This unit of work applies the Concrete, Representational, Abstract model which is based on Jerome Bruner’s (1964) concept of enactive (action-based), figurative (image-based) and symbolic (language-based) expression methods. Bruner (1964) suggests that the manipulation of concrete materials can help anchor students' understanding before moving to abstract, symbolic representations. A recent Swiss study by Zhao (2024) implemented the CRA approach to teach eighth-grade students vector addition. Zhao (2024) found that the fading of 'concreteness' in the CRA model facilitates the building of procedural knowledge and promotes mathematical sense-making. Thus, the CRA sequence can be used as a powerful tool to enhance students' learning experience on complex mathematical topics, such as vectors (Zhao, 2024). The CRA approach is used in the '3D Coordinate System Model' activity in Lesson 3, as students have to construct their own paper model and use this to represent points and vectors in 3-dimensional space. Students are guided to progress from concrete sensory properties of the 3-dimensional axis system to abstract properties, such as finding the equation of planes. This activity engages students in a meaningful learning experience as they are guided to make connections between the physical model, drawings and abstract notation of 3D vectors, rather than simply following a memorised procedure or algorithm.
Using ICT
As mentioned previously, 'Introduction to Vectors' introduces many fundamental concepts through a geometrical perspective. Literature has shown that vectors is a difficult topic to learn due to its highly abstract nature and a lack of conceptual understanding (Knight, 1995; Nguyen & Meltzer, 2003; Zavala & Barniol, 2010). Recently, research has been conducted on the implementation of information and communication technology (ICT) in the teaching and learning of vectors. For example, an Australian study by Angateeah et al. (2017) investigated the use of interactive tools developed through Geogebra to teach advanced vector concepts, including vectors in three-dimensional space. The results of this study indicate that the use of ICT tools can improve students' conceptual understanding through providing visual and concrete representations of abstract vector ideas (Angateeah et al., 2017). Similar findings were shown in a German study by Donevska-Todorova (2015) which explored the use of dynamic geometry software to teach the dot product of vectors. The study suggests that technology can be used to support students' conceptual understanding of vectors by providing a "dynamic visualisation, including geometric and arithmetic-algebraic modes of descriptions" (Donevska-Todorova, 2015, p. 195). Thus, this unit plan incorporates Desmos to help students conceptualise various vector procedures, such as addition and subtraction of vectors in Lesson 4. The aim of these activities is to support students' conceptual understanding through interactive means, such as changing lengths and directions of vectors and seeing its effects.
The 'Introduction to Vectors' topic assumes students know the following:
Basic algebra (e.g., algebraic notation, arithmetic operations, expansion)
Bearings – can be used to describe the direction of a vector
Distance-speed-time relationships – used in time-dependent motion problems in vector form
Coordinate geometry in 2 dimensions (e.g., Cartesian coordinates, midpoint formula, distance formula, parallel and perpendicular lines) – many of these concepts are extended into 3 dimensional space
Trigonometry (e.g., trigonometric ratios, degrees and radians, properties of right-angled triangles, Pythagoras' formula) – fundamental knowledge to calculate the direction of vectors in 2 dimensions
Cosine rule – used to derive the geometric and algebraic formulas for the scalar (dot) product
Functions and graphs (e.g., equations and graphs of parabolas, exponentials, hyperbolas, circles, other polynomial curves and their transformations) – used to represent the path of an object as well as projectile motion
Differential calculus – used to find the velocity vector and acceleration vector of an object given its position vector
Integral calculus – used to find the position vector and the velocity vector of an object given its acceleration vector
Literature on the mathematical topic of vectors suggests that students have difficulty understanding the concepts and mathematical ideas underlying computational vector methods (Stewart & Thomas, 2009, p. 951). In the words of Sierpinska et al. (2002), the abstract nature of vectors makes it a "highly theoretical knowledge, and its learning cannot be reduced to practicing and mastering a set of computational procedures" (p. 1). Research has revealed the following common misconceptions and student errors related to this topic:
Treat vectors as scalars during vector operations (Flores et al., 2004; Gagatsis & Demetriadou, 2001)
Confuse vector addition and subtraction algorithms (Flores et al., 2004; Knight, 1995; Nguyen & Meltzer, 2003). Examples from Nguyen and Meltzer's (2003) study include:
Incorrectly adding vectors by going from tail-to-tail or tip-to-tip
Reorienting vectors in order to use the triangle law of vector addition
Imprecisely executing the tip-to-tail rule or parallelogram addition rule
Failing to apply vector addition procedures in the absence of a grid
Treat vectors as fixed objects in space during vector operations (Nguyen & Meltzer, 2003)
Equating two vectors if they point to the same general region (Nguyen & Meltzer, 2003)
Misunderstand the dot product of two vectors as a vector (Zavala & Barniol, 2010)
Misunderstand the magnitude of x and y component vectors of a vector (Zavala & Barniol, 2010)
References
Angateeah, K. S., Thapermall, S., & Jawahir, R. (2017). Developing interactive ICT tools for the teaching and learning of vectors at A-level [Paper presentation]. The Annual Meeting of the Mathematics Education Research Group of Australasia, Melbourne, Victoria, Australia. https://ejmt.mathandtech.org/Contents/eJMT_v9n3a1.pdf
Bruner, J. S. (1964). The course of cognitive growth. American Psychologist, 19(1), 1-15. https://doi.org/10.1037/h0044160
Brown, G. (2009). Review of education in mathematics, data science and quantitative disciplines. Group of Eight. http://amsi.org.au/wp-content/uploads/2014/07/33_go8mathsreview_Dec09.pdf
Flores, S., Kanim, S. E., & Kautz, C. H. (2004). Student use of vectors in introductory mechanics. American Journal of Physics, 72(4), 460-468. https://doi.org/10.1119/1.1648686
Gagatsis, A., & Demetriadou, H. (2001). Classical versus vector geometry in problem solving. An empirical research among Greek secondary pupils. International Journal of Mathematical Education In Science & Technology, 32(1), 105-125. https://doi.org/10.1080/00207390150207103
Knight, R. D. (1995). The vector knowledge of beginning physics students. The Physics Teacher, 33, 74-77. https://doi.org/10.1119/1.2344143
Nguyen, N., & Meltzer, D. E. (2003). Initial understanding of vector concepts among students in introductory physics courses. American Journal of Physics, 71(6), 630-638. https://doi.org/10.1119/1.1571831
NSW Education Standards Authority. (2025). Mathematics Extension 1 11–12 Syllabus: Introduction to vectors. https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-1-11-12-2024/content/year-12/fa682e9b47?show=advice%2Cexample&ta_scroll=no
Office of the Chief Scientist (2012). Mathematics, engineering and science in the national interest. Canberra: Australian Government. http://www.chiefscientist.gov.au/wpcontent/uploads/Office-of-the-Chief-Scientist-MES-Report-8-May-2012.pdf
Sierpinska, S., Nnadozie, A., & Okta, A. (2002). A study of relationships between theoretical thinking and high achievement in linear algebra. Concordia University.
Stewart, S. & Thomas, M.O.J. (2009). A framework for mathematical thinking: The case of linear algebra. International Journal of Mathematical Education in Science and Technology, 40 (17), 951-961. https://doi.org/10.1080/00207390903200984
Zavala, G., & Barniol, P. (2010). Students’ understanding of the concepts of vector components and vector products. AIP Conference Proceedings, 1289(1), 341-344. https://doi.org/10.1063/1.3515240