Introduction

      Looking at the historical development of mathematics, it is clear that mathematics is inherently visual. Visualization, as an act of creation and a process of reflection upon pictures, images, or diagrams modeling mathematical objects, is a productive way to support mathematics teaching and learning. Consequently, it is important to emphasize this powerful cognitive instrument in the preparation of secondary mathematics preservice teachers. Increased use of computer-generated graphics in educational practice in recent years may lead some to believe that hand-drawing mathematical objects, or notions, is a relic of the past.  However, as evidenced by researchers in cognitive psychology and neurobiology, a learner’s exposure to predetermined and ready-to-use images of mathematical objects is not a substitute for an active creation of pictures of these objects (Castro-Alonso et al.,2019). The process of illustrating mathematical content by hand fosters cognitive and holistic development, promoting essential eye-hand-mind coordination that helps to concretize abstract mathematical objects. In other words, visualization aids in making students’ thinking more explicit. In this article, we discuss how the visualization of mathematical objects through hand drawing can be addressed in practice by drawing three types of images.  These images can help support students’ sense-making of mathematics and their ability to make connections among various mathematical notions.

What is Visualization?

      The word “visualization” has different meanings depending on the context in which it has been used. It can be used as a noun or as a verb. Therefore, it is important to make a distinction between a) Visualization objects refer to material objects created to understand something about things or ideas not explicitly presented. These include pictures, diagrams, or animations. b) Introspective visualization refers to an introspective activity in one’s brain that a person makes about visualized objects, and c) Visualization as a verb, that depicts an act of creating a model and provides an interpretation of a mathematical object or idea (Philips et al,2010). 

We see hand-drawing as an interpretive visualization activity that creates models of mathematical objects, and we adopt Quillin and Thomas’ (2015) definition of drawing as “a learner-generated external visual representation depicting any type of content, whether structure, relationship, or process, created in two dimensions on any medium” (p.2).

      When preservice teachers engage in hand drawing, they elucidate and clarify their understanding of mathematical concepts or ideas. For example, we observed that secondary mathematics preservice teachers always construct an altitude of a triangle in the interior of the triangle. Thus, their concept of the altitude of a triangle is often limited to the context of acute triangles. The altitude from the vertex A in the triangle presented in Figure 1 will be sketched inside the triangle, making an obtuse angle with the side opposite the vertex A.

                                Figure 1. Altitude construction in an obtuse triangle ABC

      Therefore, instances in which a given triangle is obtuse create opportunities for the preservice teachers to visualize the altitude of a triangle in various contexts. The diverse drawings utilized deepen their understanding of this mathematical concept.

Our observation of preservice teachers’ drawings in undergraduate courses

      We observed the drawings of secondary preservice mathematics teachers in the Geometry course, and we recognized the importance of enhancing their visualization skills through drawing. Based on years of observation of preservice teachers, many of them lack experience in using drawing as a modeling or cognitive tool.  They often justify this as being “bad at drawing” (Author, 2019; Philips et al, 2010), confusing mathematical notions with their material representations. To illustrate the power of an image, for many preservice teachers, the notion of a circle coincides with the material object, a drawing on a whiteboard, a screen, or a piece of paper. So, in Figure 2, only the drawing on the left is a circle, the one that will be used in a given problem.

                                                                             Figure 2.

Visualization of mathematical concepts in mathematics lessons can help students solve problems and make sense of the content addressed.  In this process, the role of the instructors is of utmost importance. To help students develop their modeling skills, drawing instruction should be simple and always accompanied by explicit demonstration from the instructor.

Three types of images

To support preservice teachers in developing a conceptual understanding of mathematical notions, following Quillin and Thomas (2015) and our observations across the years, we suggest three stages when creating images during enacted lessons, namely, drawing a primary image, drawing a secondary image, and if necessary, drawing an auxiliary image. 

Primary image

      The primary image is a drawing/visualization of the given data of a problem or activity, which aligns with visualization as a verb.  We illustrate this with the image in Figure 3, created for the following problem: What is the angle between the line passing through the points A(1,4,4) and B(2,2,1) and the xy-coordinate plane? To illustrate the given data of this problem instructor should sketch the first octant of the 3D coordinate system, the two points A and B, and the portion of the line between these points. Additionally, as students work on the task, it will be helpful to guide them by providing useful strategies to address challenges associated with drawing models in a 3D coordinate system. For example, it helps when visualizing the position of a point in 3D space, to see it as a vertex of a box whose edges are parallel to the coordinate axes.

                                              Figure 3. The primary image of a calculus problem

      While the primary image is displayed, probing questions could be posed to help preservice teachers develop an inner visual image of a line in space. Sample questions may include: What are the distances of these points from the coordinate planes? How can we determine their distances from the origin?  

Secondary image

      The secondary image should illustrate the ideas leading toward the solution of the given problem. More detail is provided in this image (right part of Figure 4). Particularly, we can see the extension of the line segment AB, the projection of this line segment on the xy-plane, an additional line in the yz-plane parallel with the z-axis, and points of intersection of line AB with xy- and zy-coordinate planes respectively. The additional information helps preservice teachers visualize a path that leads towards the solution: the desired point of intersection of the line AB with its projection in the xy-plane and with the line in the yz-plane parallel to the z-axis and passing through the point on the y-axis marked with the question mark on the second image. 

                                  Figure 4. Primary and secondary images of the calculus problem

 Auxiliary image

The auxiliary image is designed to clarify a particular argument related to the secondary image. We illustrate this with the left image in Figure 5, which will help students recall the definition of the trigonometric functions.

                                                Figure 5. Auxiliary and build-on image

      The right image in Figure 5, which is a “build-on image”, can help preservice teachers make sense of a sequence of steps that lead to the solution of an additional question: What is the equation of the plane containing the lines AB and the line passing through the point A and being parallel to the y-axis? This problem builds on the previous problem, and preservice teachers may use the secondary image for scaffolding that will help them gain insight into the best strategy leading to the solution of this problem.

Conclusion

 Mathematics and mathematics education courses should intentionally aim to develop preservice teachers’ visualization skills by encouraging them to create images and interpret the information gained. Effective questioning strategies can assist preservice teachers in building on their primary images, whether with their own secondary or auxiliary images. Assigning tasks that allow preservice teachers to draw can help them understand mathematical concepts and conceptualize meaningful connections, thereby fostering a deeper understanding of the content addressed.

References

Castro-Alonso, J. C., Ayres, P., & Sweller, J. (2019a). Instructional visualizations, cognitive load theory, and visuospatial processing. In J. C. Castro-Alonso (Ed.), Visuospatial processing for education in health and natural sciences (pp. 111–143). Springer. https://doi.org/10.1007/978-3-030-20969-8_5

Philips, L., Norris, S. & McNab, J. (2010). Visualization in mathematics, reading, and science education. Models and modeling in science education, Vol 5. Dordrecht, The Netherlands: Springer.

Quillin, K., & Thomas, S. (2015). Drawing-to-learn: A framework for using drawings to promote model-based reasoning in biology. CBE—Life Sciences Education, 14(1), 1-16.