Conceptualizing Multiple Coordinate Systems:
The Case of Hann
Inyoung Lee, Dr. Michelle Zandieh
Introduction
Research Question
Result
•Coordinate systems underlie many linear algebra
concepts such as span, linear (in)dependence, and
basis vectors.
• A coordinate system promotes students’ spatial
thinking of the key concepts because it is a
systemic way of coordinatizing geometric figures
such as vectors, lines, and planes in space.
•It also helps student understanding by imposing a
visual aspect to the nature of introductory linear
algebra courses which may be abstract and
computational.
•A matrix can be conceived as a string of numbers,
an ordered-list of coefficients in system of
equations (Larson and Zandieh, 2013), a
rectangular array of data points, a collection of row
vectors, or a collection of column vectors
depending on the context of application.
Conceptual Frame
•How does a student engage with tasks in which
space is described by multiple coordinate
systems?
Methods
•Any spatial objects are composed of points, and
any coordinate system describes those points’
locations.
•Data collection: Clinical interview with a student
who has taken linear algebra courses previously
•Data analysis: Grounded theory (Strauss & Corbin,
1990), Three Interpretations of a Matrix Equation
𝐴𝒙 = 𝒃 (Larson & Zandieh, 2013)
1) Naming the Treasure location in the Gulliver
system
• “The location of the treasure looks to be probably
about 1.3 in that direction(oasis), and then exactly
two in the waterfall direction”
•Two ways to represent it: vector and linear
combination (Fig. 1)
2) Locating [1.3, 0.5] in the Gulliver system
•“… going to place it on the map indicating where 1.3
and 0.5 so 1.3 that’s going to be in the oasis
direction. So that would be somewhere here and up
point five”
Visualizing the Native map: “to talk about the map, it’s
just going to be the same. But when we see these grids,
they’re just going to be a lot more dense. They’ll have a
lot more lines here and this way as well, I think only 3
times this way, 12 times that way.”
3) Relocating [1.3, 0.5] in the Native system
•Reciprocal reasoning : “then one 12th the length and
one third the vertical…So for 1.3 for them, .. Un it’s
going to be 1.3 twelfths of one of these. So that’s not
very far, that’s only maybe there and then point five
again, that’s half of a third.”
4) Renaming the Treasure location in the Gulliver,
Native, and Rock systems
•Proportional reasoning in the Native system: “So we
can just take those and multiply them, so it would be
12 times they would be horizontal direction, three
times vertical which would come up to… so I have
here 12 because one unit of Gulliver’s in the
horizontal direction is 12 units of the natives and then
it’s three for the vertical..” (Fig. 2)
•Reciprocal reasoning in the Gulliver system: “you
would have you know C and D then you go times one
over 12 and one over three.” (Fig. 3)
•Matrix as a system to correlate with the Rock system:
“So what I can do, I can make a system so I can plug
in, uh, for example…we'll go from Gulliver to the Rock
So I know that one and zero should correlate to six
and two sort of go over to the rectangular so I can
multiply this through and set up the system.” (Fig. 4 &
5)
<Fig. 1>
<Fig. 2>
<Fig. 3>
<Fig. 4>
<Fig. 5>
** The student was asked to compare/ coordinate three
coordinate systems: the Gulliver system, the Native
system, and the Rock system (=the Cartesian system)
References: Larson, C., & Zandieh, M. (2013). Three interpretations of the
matrix equation Ax = b. For the Learning of Mathematics, 33(2), 11–17.
Strauss, A., & Corbin, J. (1994). Grounded theory methodology: An
overview. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative
research (pp. 273 –285). Thousand Oaks: Sage Publications.
