### Viewing the Third Dimension

```Viewing the Third Dimension
methods) can help students view
and understand the third dimension.
Overview
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Spatial Thinking
Student Feedback (very informative)
Computer Software
The Challenge of the 3rd Dimension
• How does one mentally “see” spatial
dimensions?
• Can it be taught?
• Students in STEM tracks vs. non-technical
• Use of technology … pros and cons.
Visual and Analytical Approach
Difficulty of “Teaching” the Third
Dimension
• It requires strong spatial thinking skills.
• It is not a “procedure” that can be learned by
following an algorithm.
• Some are stronger than others at seeing the
third dimension.
• It is an innate skill that can be enhanced, but
can it be taught from scratch?
Common Modes of Teaching the 3rd
Dimension
• Non-Technology Based:
• Hand-drawn sketches on the board.
• Static examples of pictures/graphs of common 3-D
surfaces.
• Hand-held manipulates.
• Technology Based:
• Computer Graphics
• What do students think of these
various methods?
Questionnaire Background
• I have taught multi-variable calculus for many
years, to both math/science majors and prebusiness majors. I have tried many modes of
delivery and have been curious how they are
perceived by the students. Anecdotally, I
receive feedback that they sometimes “see”
something entirely different than what I am
trying to convey.
Questionnaire
• I recently polled a number of students who recently took our
MAT-211 (Math for Business Analysis) at ASU.
• I polled my own students as well as those of other instructors.
Total polled: 187 students.
• Participation was optional and anonymity was guaranteed so that
students would feel comfortable in expressing their true opinion.
• I deliberately timed the poll to come after they had been tested
on this material, so that is was still fresh, but also so that they
were relaxed.
• The feedback was desired to establish some baseline ideas of
what was working, and what was not.
• The questionnaire ran 14 questions and took about 5 minutes.
First Question
• When discussing a 3-dimensional surface, various methods of delivery by
the instructor are used. Please rate these delivery modes on their
effectiveness to you in helping you perceive the third dimension (1 = no
help, 3 = neutral, 5 = very helpful. Use 0 if you never saw this mode)
• Images generated by Maple (Mean: 2.673)
• Images generated by the free 3-d grapher on the Mac computer (Mean:
3.65)
• Hand-drawn images by the instructor (Mean: 3.525)
• Manipulates & physical examples of various shapes (Mean: 3.964)
• Contour Maps (Mean: 3.207)
• Conclusions?
Second Question
• When viewing any contour map, how do you
rate your ability to mentally “see” the three
dimensional features? Use 1 for the contour
map is no help at all, up to a 5 for very
(Examples include hand-drawn maps, weather
maps, topography maps, etc)
Third Question
• Look at the contour map (right)
How readily do you “see” the
following features? (1 = I don’t
see it at all, 3 = I see it only after
staring at it for awhile, 5 = I see it
immediately)
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Highpoint(s) (Max points) (Mean:
3.76)
Areas that are steeper than other
areas (Mean: 3.612)
Questions 4-7
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(4) In class the instructor sometimes uses the method of “traces” where he/she
sketches the cross-section parts of the surface on the xz and yz planes, then
explains how to infer the rest of the shape based on its two cross-sections. How
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(5) In class the instructor may refer to a set of core surfaces: the paraboloid, the
plane, the saddle, the hemisphere, and a few other common surfaces. Did
having these shapes serve well in helping you understand more complicated
surfaces? (Mean: 3.467)
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(6) In class, when discussing partial derivatives on a contour map, the instructor
may use the “little arrow” visual method to convey how to infer the sign of the
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(7) In class, when discussing a path constraint, the instructor usually draws a
dashed line on the contour map, then sketches a 2-d graph showing the profile of
the path. Did you find this useful? (Mean: 3.237)
Questions 8 & 9
• How would you describe yourself when it
comes to rating your visual skills (i.e.
“seeing” the picture) in mathematics? (1 =
poor, 3 = average, 5 = good) (Mean: 3.237)
• How would you describe your analytical skills
(i.e. “doing” the actual math)? (1 = poor, 3 =
average, 5 = good) (Mean: 3.375)
Question 10
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Free response: if you have any comments or insights that you
think may be interesting or useful, please feel free to write them
below.
The way the teacher explained it was fine as long as you try to
imagine it you’ll get the basic feel … The one thing I found very
helpful was when my professor turned our room into a 3d surface
(the z-plane was one wall, the x-plane a table and the y-plane the
floor) … It was helpful when our professor physically made a grid
out of tables and jumped around as the point … When my
instructor brought in a pringle chip it really explained the saddle
point … Translating contours into cross sections really helped me
in my geology class … I think computer technology would be very
useful to help average to poor math student to understand the
concepts better. … From seeing the 3-D images in the textbook, I
would have loved the opportunity to have manipulated them on a
computer program … This is the first time I have seen 3d graphs
being plotted & even after trying to study the graphs I had a hard
time understanding 3D in a 2D graph … Physical models are
helpful because the 3-D graph is displayed non-2-dimensionally ...
I really understood 3-D figures when our instructor showed us an
actual 3-D figure.
What Works
• Physical hand-held models.
• Computer graphics and animations.
• Training and persistence, both from the
teacher and the student.
• Multiple synchronous approaches.
• Spatial thinking has been shown to be
“teachable”.
Spatial Thinking Teaching Skills
• Use descriptive terms such as “inside”,
“outside”, “north”, “south”, “parallel”, etc, as
often as possible.
• Gesture! It has been shown to work. “Talk
• Stress visualization. Even a few moments of
directed thought works.
There’s Hope for the “Low Ability”
Spatial Thinkers!
• “For low-ability participants, there is an
initial hump to get over. They improve slowly,
if at all, for the first half dozen or so sessions.
But if they persevere, faster improvement
comes, so it’s important that students (and
teachers) not give up.”
– Nora Newcombe, Temple University, “Picture This”,
American Educator, Summer 2010.
What Apparently Does Not Work
• Hand-drawn images by the instructor on the
board.
• Purely verbal descriptions.
• Attempts to explain the 3rd dimension by
using analogs from the 2nd dimension
(although it seems to be useful as an
• A single approach only.
Common Problems and
Misperceptions
• Inability to infer “remaining” shape of
surface based on its xz and yz traces.
• Contour Maps are meaningless to some
students (they simply do not “see” the
undulations of the surface).
• Paths drawn on a contour map (e.g. for
constrained optimization) are not viewed as
being “on the surface” but instead exist
somewhere else.
A student’s model
The Challenge of Technology
in the Classroom
• Not all rooms equipped the same.
• Not all in-room computers have the same
software, especially for software-dependent
demonstrations.
• Some software (e.g. Maple, Mathematica)
very effective, but requires training for
students to use.
Jing (and other
video-capture programs)
• A simple way to capture real-time animations
• Also makers of popular video-capture
programs such as camtasia and snagit.
• Videos stored at off-site server, you just need
a web connection in the classroom.
Other Free 3-D Graphing Programs
• www.runiter.com by Saied Nourian
• http://www.intmath.com/vectors/3dgrapher.php (contour mapper) by B. Kaskosz
and Doug Ensley
• http://www.wolframalpha.com/
Graphing Apps for the iPhone
• SpaceTime by Pomegranate Software (\$9.99).
• MyCalculatorPro by Pomegranate Software
• Graphicus by Seraphim Chekalkin (\$1.99).
Not a 3D grapher, but a high-level 2D grapher
that can do implicit plots. May be useful as
• Grafly … RIP. 
Something I Created
• http://math.la.asu.edu/~surgent/mat211/co
ntourmaps.pdf
• Static displays, shows various topo maps of
places around Arizona. Hope that students
can relate their experiences to the maps
• Other uses, such as in weather maps and
medical mapping.
• Received well by the students. Feedback
positive.
General Conclusions
• Do not rely on one mode.
• Combine the traditional with the
technological.
• Be cognizant of your language: use “spatial”
terms often, be descriptive and be persistent.
• Stress visualization and practice.
Credits
• Scott Surgent
– Senior Lecturer, Mathematics, ASU
– math.asu.edu/~surgent
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