**Author:** Kathy Eastwood

**Affiliation:** Northern Arizona University

**Country:** United States

**Co-Author(s):** Wanda Diaz

**Affiliation:** Office of Astronomy for Outreach

**Country:** Japan

**Title:** Targeting Conceptual Mathematical Barriers in Astronomy

Many astronomical concepts use advanced mathematical concepts. Cognitive neuroscientists believe that humans possess basic capabilities for numbers and geometry from birth, an ability they share with other animal species. As astronomers, we tend to think of advanced three-dimensional mathematical concepts as visual, because that is how most of us learned them, i.e. in a mono-sensorial manner. However, in a recent study comparing the brain activity of both sighted and blind mathematicians, Amalric et al. (2018) concluded that Òvisual experience is not necessary for the development of a normal cerebral representation of advanced mathematical concepts.Ó Blind and visually impaired (VI) students can be trained in all such concepts, and sighted students who struggle with the concepts may also benefit from a multi-sensorial approach. In order for students to study astrophysics successfully at university, they must have a good understanding of advanced mathematical concepts.

Any concept involving direction or changes in position is more difficult to teach to blind and VI students. Most lessons for blind students focus on changing the presentation of the lesson such that the content of the lesson is accessible, but may not address any underlying difficulty of understanding the concept. We have identified several such concepts and have started writing lessons to address those concepts.

The first lesson, written for the three-dimensional tactile grid called the CARDIS, addresses surface area and volume. We intend for the next lesson to address the projection of a three-dimensional object onto a two-dimensional plane. The latter concept is crucial for understanding what is recorded in the field of view of a telescope. This concept can be addressed by tracing the outline of different solid objects onto a sheet of paper, and noting how the two-dimensional shape varies with the presentation angle of the solid.

Another series of lessons starts with the concepts of scaling and ratio. The second lesson in the series develops a scale model of the Earth-Moon system. The next lesson planned for this series will be a scale model of the Sun-Earth-Moon system, with particular attention paid to the possible relative orientations of those three objects. This lesson will eventually lead to the concept and possibility of eclipses, but only after we have addressed the issue of light (ÒenergyÓ) traveling in straight lines, and the concept of how projected size diminishes with distance.

As is often the case with curriculum developed for special needs students, these lessons may well prove useful for sighted students as well. As any instructor can tell you, many sighted students have a difficult time with concepts involving three-dimensional visualization.

**References:**

[1]: Amalric, M., Denghien, I., & Dehaene, S., 2018, ÒOn the role of visual experience in mathematical development: Evidence from blind mathematiciansÓ, Developmental Cognitive Neuroscience 30 (2018) 314-323.