Is there a geometric imperative?

Summary
In this article, Tahta talks about the role and value of geometry. He states that unlike algebraic experience that lies in symbols, geometry involves mental images made through sight. He outlines three important aspects of studying geometry.
(1) imagining: seeing what is said
(2) construing: seeing what is drawn and saying what is seen
(3) figuring: drawing what is seen
Everyone has different ways of imagining geometric ideas when listening to verbal descriptions that invoke different types of images - kinesthetic, tactile, aural and visual. Next, Tahta provides two examples to illustrate "construing". The first example shows a 2-dimensional picture of a horse and cart in which some people may construe the wheels on the cart as being elliptical on paper and others may see them as being circular in 3-dimensional depth. To illustrate "saying what is seen", the second example shows a shape made of different triangles and bordered by hexagons. Tahta argues that people verbally describe this same picture differently based on what they see. Sharing your verbal descriptions of a shape or a solid makes other people realize that it is possible to consider the object from multiple acceptable viewpoints. In a geometrical sense, "figuring" is related to projective geometry which involves drawing 3-dimensional objects in 2-dimensions. Tahta states that until the 17th century, the Rennaissance architects and artists had influenced the development of projective geometry where they found new ways of representing projections of an object from different observation points before mathematicians considered cross-sections of solids from different projections. Lastly, the author thinks it is imperative that geometry be included in the math curriculum because children explore math ideas through action, intuition, and an awareness of imagery.

Reflections
A lot of my students are visual learners and enjoy learning algebra in a geometric context. Unfortunately, some important geometry topics have been removed from the math curricula. Some topics, such as conic sections should be brought back to the curriculum to deepen students' understanding of shapes and space essential for learning calculus. I feel that I constantly need to learn more geometry on my own to meet the needs of my students. In particular, I find it difficult to connect geometry to ideas, such as radical expressions and rational expressions.

In Math 8, whenever I teach a lesson on views and nets of prisms, students need to draw nets and do isometric drawings on paper. The students demonstrate their own ways of imagining transformations from 3D to 2D or vice versa. Those students who possess weak perceptual skills have difficulty in visualizing and drawing the sides of a solid from different vantage points. Nonetheless, their images drawn on paper, if fitted together properly, may produce some creative solids that are worthy of discussion. Another exercise that I do is to have one student describe verbally his/her mental image to others. Then, they are asked to sketch, compare, and contrast one another's drawings. This exercise gives the students an opportunity to talk about any common properties and relationships to algebra among their drawings. I find that students are very engaged in debating and construing each other's perspectives in a qualitative way. Whenever possible, I incorporate geometry into certain algebraic concepts through imagining, construing and figuring to improve students' learning.