Rationale
Physics takes place in three-dimensional space where many fundamental quantities are represented by vectors, which are mathematical quantities that have a magnitude (or length) and a direction. For the introductory Newtonian formulation of mechanics, vectors provide the mathematical framework. Any displacement, or change in position, can be thought of as a vector pointing from the initial position to the final position. Force is a vector quantity, as are acceleration, velocity and momentum. Further down the line, their rotational counterparts are all also vector quantities. Dealing with all of these quantities requires us to pay particular mind to their directions, but what is the mathematical context that surrounds a high school junior enrolled in a physics course? How can the treatment of vectors be best linked to concepts already present in a student’s math background?
My guess is that the best place to make the connection to their previous math education is the long abandoned number line. Students have not dealt with them in years, but the tidy row of numbers that was pasted high on the wall in every elementary classroom forms the very foundation of a one-dimensional coordinate system. The work of Galileo, Newton and Einstein was deeply focused on the idea of the reference frame. The coordinate system that is tied to an observation provides the information needed to translate measurements to other, equally viable, frames. Constructing coordinate systems from nothing and then making measurements within them are the processes that allow the quantification of observations, an essential tool for any student of physics. Without the understanding that physics is done relative to invisible number lines we construct for ourselves to help communicate observations, concepts of measurement and motion do not become intuitive to students.
The first time students are exposed to vectors is typically in a geometry class, which most students in my school take as sophomores. Geometry seems like the obvious place to take an extremely pictorial approach to vectors, but they are usually just only implemented to describe translations in the plane. This is natural and important, but physics requires more. The text we use at our school represents vectors in a component notation, <a,b>, that is meant to resemble an ordered pair. Students spend a week or so realizing that the vector moves the vertices of a shape a units in the x-direction and b units in the y-direction. Then, as far as students are concerned, the word vector is promptly forgotten and not used again.
I want to pick up my treatment of vectors from here. Viewing them as translations in the plane fits with what their implementation as states and changes. Getting them to this point in an efficient manner is the challenge. This will require a reimagining of the front matter of the course related to units and measurement. It seems that most of things are usually done in a very specific, and consequently not transferrable, context. Measurements are only made with a meter stick and a stopwatch in standard units. Coordinate systems are pre-prescribed instead of generated by students. Unit conversion is done in a way that feels like an arbitrary worksheet exercise as opposed to a necessary step in communicating information.
To do this differently, I want to look back toward elementary explorations of the number line, motion along it, and concepts of measurement. The purpose of dredging up these old sense-making tasks is to exhibit the unspoken connection between numbers in math class and quantities in physics. The learning my students do is often very isolated and they are frequently unable to connect concepts they have learned in math to things we do in physics unless the relationship is built up for them.
Once they have got this mathematical language to describe quantities, it is time to bring transformations into the picture. Composition of translations is vector addition. For students the connection is evident because they move quantities, but preserve scale. Dilation from the origin is multiplication by scalars. Students can see this because they stretch the vector and change the unit length, but preserve orientation (or at least restrict it to the original direction or the exact opposite direction). Seeing how quantities relate and change is at the heart of the study of physics and it is my hope that this dynamic representation of arithmetic will help students better visualize and understand physics.
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