walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf ·...

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Walking geometry: Using location-aware technologies to change the scale and modality of mathematical understanding Rogers Hall,Vanderbilt University Ricardo Nemirovsky, San Diego State University Support from Spencer Foundation (Hall, CASBS) and NSF Science of Learning Center catalyst grant (Hall & Nemirovsky) . Nr l . Nm svvk. iN & e, a N N r N m a i. a a l . .

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Page 1: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Walking geometry: Using location-aware technologies

to change the scale and modality of mathematical

understandingRogers Hall,Vanderbilt University

Ricardo Nemirovsky, San Diego State University

Support from Spencer Foundation (Hall, CASBS) and NSF Science of Learning Center catalyst grant (Hall & Nemirovsky)

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Page 2: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Scale, place and embodied mathematical experience

Conventional school geometry…

relies upon a generative relation between symbolic/graphical forms and physical models at graspable scale.

privileges extrinsic (POV from outside) over intrinsic (from inside) experience and imagination.

Hall,R.&Nemirovsky,R.(2008,September).Walkinggeometry:UsinglocaBon‐awaretechnologiestochangethescaleandmodalityofmathemaBcalunderstanding.InR.Hall&K.Leander(SymposiumLeaders),Place,mobility,embodiedac/onandlearning.InternaBonalSocietyforCulturalandAcBvityResearch,SanDiego,CA

Page 3: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Scale, place and embodied mathematical experience

If geometry were also taught and learned at a walking scale, could we…

highlight relations between extrinsic and intrinsic aspects of mathematical experience?

use changes in scale (and tools) to create different possibilities for activity, experience, and understanding?

Hall,R.&Nemirovsky,R.(2008,September).Walkinggeometry:UsinglocaBon‐awaretechnologiestochangethescaleandmodalityofmathemaBcalunderstanding.InR.Hall&K.Leander(SymposiumLeaders),Place,mobility,embodiedac/onandlearning.InternaBonalSocietyforCulturalandAcBvityResearch,SanDiego,CA

Page 4: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Conceptual understanding is “grounded” in modality-specific traces of body experience

Space is not isotropic in experience, but a composite of:

• personal space (proprioceptive),

• peripersonal space (graspable, extended by tools), and

• extrapersonal space (pointing, gaze following)

(Farne, Iriki &Ladavas, 2005)

(Kendon, 1976)

Hall,R.&Nemirovsky,R.(2008,September).Walkinggeometry:UsinglocaBon‐awaretechnologiestochangethescaleandmodalityofmathemaBcalunderstanding.InR.Hall&K.Leander(SymposiumLeaders),Place,mobility,embodiedac/onandlearning.InternaBonalSocietyforCulturalandAcBvityResearch,SanDiego,CA

Does changing scale matter for mathematical experience?

Page 5: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Drawing at walking scale “It is not down in any map; true places never are.” Herman Melville, Moby Dick

Jeremy Wood on drawing in London: The word “DOWN” was written in a cemetery. The quality of line is quite jagged since there were lots of sharp turns to negotiate to avoid the tombstones. The “D” started at the iron gates at the entrance, I turned left off a pebbled track, crossed the lawn over a dried up pond, continued round beyond the crematorium and along past memorial stones, gardens, and benches, and back to the gates.

http://www.gpsdrawing.com/gallery/land/meridians.html

Hall,R.&Nemirovsky,R.(2008,September).Walkinggeometry:UsinglocaBon‐awaretechnologiestochangethescaleandmodalityofmathemaBcalunderstanding.InR.Hall&K.Leander(SymposiumLeaders),Place,mobility,embodiedac/onandlearning.InternaBonalSocietyforCulturalandAcBvityResearch,SanDiego,CA

Page 6: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Walking scale geometry task • pre-service secondary math teachers

worked in small groups

• picked region of campus

• constructed geometric object

• performed a series of transformations (rotation, dilation, translation)

• displayed drawings to classmates using Google Earth

• guessed/discussed how objects were constructed, transformations performed

• data include drawings, written reflection, interview and video recording of re-constructed drawings

� �� � .� N� r � � � � � � l � � .� N� msvvk.� � � � � � � � � � iN� �� � � � � � � � � � � � � &� � � � � � � � � � � � � e� , �� � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � �� � � �� � � � � � �� � � � � � � � � � � � � � � �� � a � � � � � � �� � � � � N� � � � N� � �� � � r � � N� � � �� � � � � m� � � � � � a � � � � �� � � � i .� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � a � � a � �� � �� � � � � l � � � � � � �� � � .� � �� � � � � � � .� � � � �

Page 7: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Walking geometry group products

SHAPE group

• draws rectangle & circle

• disappointed with precision

NAME group • writes “PAUL” across

the “heart” of campus

• discovers scale inconsistencies and omissions in campus map

CLEF group • draws musical clef on

baseball field

• discovers extended analogy between geometry and marching band

Hall,R.&Nemirovsky,R.(2008,September).Walkinggeometry:UsinglocaBon‐awaretechnologiestochangethescaleandmodalityofmathemaBcalunderstanding.InR.Hall&K.Leander(SymposiumLeaders),Place,mobility,embodiedac/onandlearning.InternaBonalSocietyforCulturalandAcBvityResearch,SanDiego,CA

Page 8: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Constructing a musical clef

• paper (23 x 15 cm) vs. baseball out field (20 x 30 m)

• durable marks vs. marks hidden in device

• symbolic labels, fine-grasp, arm movement vs. bodies deployed as place marker, reeling in rope, moving stylus

• extrinsic POV on whole (above) vs. partial intrinsic POV (within)

• gestures fit figural scale vs. pointing and gaze following

• individual actions vs. required multi-party coordination (marking, handling rope, moving stylus)

• talk optional vs. talk required for coordination

Hall,R.&Nemirovsky,R.(2008,September).Walkinggeometry:UsinglocaBon‐awaretechnologiestochangethescaleandmodalityofmathemaBcalunderstanding.InR.Hall&K.Leander(SymposiumLeaders),Place,mobility,embodiedac/onandlearning.InternaBonalSocietyforCulturalandAcBvityResearch,SanDiego,CA

Page 9: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

How are line segments ‘held’ in the body?

Are line segments ‘held’ in the body?

You use a compass to ∆measure things?

It was the ∆same distance no matter where you were standing?

I realized that- it can ∆change length?

…while re-membering the clef

We can’t make it that ∆big. I’m gonna go until ∆HERE.

…while re-constructing the clef re-membering

appears to involve modality

-specific traces of personal body

activity

� �� � .� N� r � � � � � � l � � .� N� msvvk.� � � � � � � � � � iN� �� � � � � � � � � � � � � &� � � � � � � � � � � � � e� , �� � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � �� � � �� � � � � � �� � � � � � � � � � � � � � � �� � a � � � � � � �� � � � � N� � � � N� � �� � � r � � N� � � �� � � � � m� � � � � � a � � � � �� � � � i .� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � a � � a � �� � �� � � � � l � � � � � � �� � � .� � �� � � � � � � .� � � � �

Page 10: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Interplay between practices of geometric construction

Like in geometry class that- I'm in with Becka. We have to use compasses- I never thought about this before [this class. You use a compass to ∆measure things?

intended (by us)

discovered (by them)

� �� � .� N� r � � � � � � l � � .� N� msvvk.� � � � � � � � � � iN� �� � � � � � � � � � � � � &� � � � � � � � � � � � � e� , �� � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � �� � � �� � � � � � �� � � � � � � � � � � � � � � �� � a � � � � � � �� � � � � N� � � � N� � �� � � r � � N� � � �� � � � � m� � � � � � a � � � � �� � � � i .� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � a � � a � �� � �� � � � � l � � � � � � �� � � .� � �� � � � � � � .� � � � �

Page 11: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Does changing scale matter for mathematical experience? Shifting from paper to walking scale changed what could be seen (POV) by participants and how they used their bodies over time and space.

Changes in scale placed new demands on talk as a means to coordinate imagination (extrinsic and intrinsic POV), tool use, and body motion.

The participation structure of walking geometry must, itself, model (re-present) participants’ geometric imagination.

The CLEF group illustrates intended and discovered aspects of interplay or transfer at a collective level of analysis.

Hall,R.&Nemirovsky,R.(2008,September).Walkinggeometry:UsinglocaBon‐awaretechnologiestochangethescaleandmodalityofmathemaBcalunderstanding.InR.Hall&K.Leander(SymposiumLeaders),Place,mobility,embodiedac/onandlearning.InternaBonalSocietyforCulturalandAcBvityResearch,SanDiego,CA

Page 12: walking geo 080911 copysci.sdsu.edu/tlcm/all-articles/hall-nemirovsky-walking-geo-2008.pdf · Title: walking geo 080911 copy.ppt Author: Rogers Hall Created Date: 2/15/2010 10:11:43

Does changing scale matter for mathematical experience? Shifting from paper to walking scale changed what could be seen (POV) by participants and how they used their bodies over time and space.

Changes in scale placed new demands on talk as a means to coordinate imagination (extrinsic and intrinsic POV), tool use, and body motion.

The participation structure of walking geometry must, itself, model (re-present) participants’ geometric imagination.

The CLEF group illustrates intended and discovered aspects of interplay or transfer at a collective level of analysis.

Hall,R.&Nemirovsky,R.(2008,September).Walkinggeometry:UsinglocaBon‐awaretechnologiestochangethescaleandmodalityofmathemaBcalunderstanding.InR.Hall&K.Leander(SymposiumLeaders),Place,mobility,embodiedac/onandlearning.InternaBonalSocietyforCulturalandAcBvityResearch,SanDiego,CA