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Viewing the Third Dimension How technology (and traditional methods) can help students view and understand the third dimension.

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Overview Spatial Thinking Student Feedback (very informative) Modes: Traditional and Technology Computer Software iPhone/iPad Software Conclusions, Discussion, Comments

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The Challenge of the 3 rd Dimension How does one mentally “see” spatial dimensions? Can it be taught? Students in STEM tracks vs. non-technical tracks, e.g. business/economics majors. Use of technology … pros and cons.

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Visual and Analytical Approach

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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?

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Common Modes of Teaching the 3 rd 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?

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Questionnaire Background I have taught multi-variable calculus for many years, to both math/science majors and pre- business 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.

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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.

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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?

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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 helpful (Mean: 3.097) (Examples include hand-drawn maps, weather maps, topography maps, etc)

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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) Highpoint(s) (Max points) (Mean: 3.76) Saddles (Mean: 2.572) Areas that are steeper than other areas (Mean: 3.612)

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Questions 4-7 (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 effective was this to you? (Mean: 2.761) (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) (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 derivative in a particular direction. Was this helpful to you? (Mean: 2.613) (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)

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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)

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Question 10 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.

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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”.

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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 with your hands.” Stress visualization. Even a few moments of directed thought works.

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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.

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What Apparently Does Not Work Hand-drawn images by the instructor on the board. Purely verbal descriptions. Attempts to explain the 3 rd dimension by using analogs from the 2 nd dimension (although it seems to be useful as an adjunct). A single approach only.

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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.

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A student’s model

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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.

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Jing (and other video-capture programs) A simple way to capture real-time animations from your desktop. Free download at techsmith.com/jing 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.

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Other Free 3-D Graphing Programs www.runiter.com by Saied Nourian www.runiter.com http://www.math.uri.edu/~bkaskosz/flashm o/graph3d/ by Barbara Kaskosz http://www.math.uri.edu/~bkaskosz/flashm o/graph3d/ http://www.intmath.com/vectors/3d- grapher.php (contour mapper) by B. Kaskosz and Doug Ensley http://www.intmath.com/vectors/3d- grapher.php http://www.wolframalpha.com/

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Graphing Apps for the iPhone SpaceTime by Pomegranate Software ($9.99). Received very high reviews. MyCalculatorPro by Pomegranate Software ($1.99). Highly received also. 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 an adjunct. Received very high reviews. Grafly … RIP.

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Something I Created http://math.la.asu.edu/~surgent/mat211/co ntourmaps.pdf 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.

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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.

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Credits Scott Surgent – Senior Lecturer, Mathematics, ASU – math.asu.edu/~surgent

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