This month’s focus is on the curricular competency: visualize to explore mathematical concepts.
In the 2007 WNCP mathematics curriculum, visualization is defined as involving “thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the visual-spatial world”. Concepts such as number, spatial relationships, linear relationships, measurement, and functions and relations can be explored and developed through visualization.
In the new BC grades 10-12 courses, the elaborations for this curricular competency are:
- create and use mental images to support understanding
- visualization can be supported using dynamic materials (e.g., graphical relationships and simulations), concrete materials, drawings, and diagrams
Visualization and spatial reasoning involve the relationship between 2D and 3D shapes as well as dynamic imagery such as different perspectives, movement, rotations and reflections. Visualizing involves an interplay between internal imagery and external representations (Crapo cited in NRICH article below). Students need experience with concrete and visual representations/pictures/models as well as being able to visualize something in their minds, often referred to as the “mind’s eye”.
Canadian and International research has shown that there are links between strong abilities to visualize and success in mathematics. One widely used psychological assessment for visualization involves “The Paper Folding Test” in which a paper is folded and a hole is placed through a specific location and the participant is asked to visualize what the paper will look like when it is unfolded, utilizing the ability to generate, maintain and manipulate a mental image, (Lohman, 1996 cited in Moss et al 2016). A recent study also found a link between the ability to visualize and success with solving mathematical word problems, citing the ability to mentally visualize and make sense of the problem contributed to success in diagramming and solving problems (Boonen et al 2013 cited in Moss et al 2016). The Canadian work of (Moss et al 2016 ) and their Math for Young Children research project focuses on spatial reasoning and the importance of developing students’ flexible use of visualization skills and strategies.
The book Taking Shape (referenced below) provides several visualization tasks on pages 30-35 but visualization is an important component of most of the spatial reasoning tasks in the book.
Quick Images is an instructional routine that supports the visualization of quantities and shapes. Dot patterns and composition of shapes are often used as quick images. More information and videos can be found on the TEDD website HERE.
A short article from the NCTM explaining the connection between visualization and subitizing can be found here:
Fawn Nguyen has compiled a collection of visual patterns HERE. Visual patterns provide the first three steps of the pattern and then students are asked to visualize the next steps, which involves both arithmetic, algebraic and geometric thinking.
Desmos in an online graphing calculator that allows for students to predict,
visualize and graph linear relationships and functions and relations.
So what does it mean to be proficient with visualizing?
As we begin to work with the new proficiency scale across BC, we need to consider what it means to be proficient with visualizing to explore mathematical concepts in relation to the grade level curricular content. As more teachers across the provinces the the scale, we will have examples of student proficiency that demonstrates initial, partial, complete and sophisticated understanding of the concepts and competencies involved.
For example, a grade six student at the end of the year would be considered proficient with visualizing geometric transformations if they were able to follow directions to mentally translate, rotate and reflect a 2D shape and show or describe the resulting orientation/position.
Some questions to consider as you plan for learning opportunities to develop the competency of visualizing:
How is the core competency of communication developed through the process of visualization? What different ways can students show and explain what they are visualizing – using materials, pictures or words?
How do the competencies of estimating and visualizing complement each other to support reasoning and analyzing in mathematics? How can using visual referents support estimating?
How can we help students understand the purpose and usefulness of developing visualization skills and strategies? What examples can we share of scientists and inventors that used visualization to develop theories and ideas?
What opportunities are we creating for students to practice and use visualization skills and strategies across different mathematical content areas such as geometry, measurement, number, algebra and functions?
Thinking Through and By Visualizing (NRICH)
The Power of Visualization in Math by Jeremiah Ruesch
Spatial Reasoning in the Early Years: Principles, Assertions, and Speculations by Brent Davis and the Spatial Reasoning Study Group, 2015
Taking Shape: Activities to Develop Geometric and Spatial Thinking by Joan Moss, Catherine D. Bruce, Tara Flynn and Zachary Hawes, 2016