# EIRP Proceedings, Vol 10 (2015)

*Educating Integral Innovators in a European Academic Network*

**New
Tools for Spatial Intelligence Education:**

**the
X-Colony Knowledge Discovery Kit**

**Sorin
Alexe**^{1}**,
Gabriela Alexe**^{2}**,
Consuela Voica**^{3}**,
Cristian Voica**^{4}

**Abstract**.
This
study introduces a new framework for developing spatial education
programs based on a geometric language and manipulation of ensembles
of polyhedra, called *X-Colony
Knowledge Discovery Kit *(KDK).
The KDK main goals are to develop spatial intelligence, creativity,
strategic planning, forecasting skills, abstract reasoning,
self-confidence, and social skills. Landmark studies document that
spatial education plays a central role in driving performance in
science, technology, engineering and mathematics (STEM) occupations,
yet spatial education is under-studied and the infrastructure for
research on spatial learning is at the beginning. KDK introduces a
novel geometric language that allows visual communication and
develops spatial abilities by engaging students to perform creative
paper folding and various mental spatial transformations. KDK is
organized in program sessions consisting of cooperative open-end
paper construction activities that engage students to build modular
constructions of gradual complexity and to explore various strategies
for combining the constructs into novel configurations. KDK supports
the Core Math Standard and Science curricula and provides students
the opportunity to discover connections between mathematics, science
and various other disciplines. A pilot case-control study conducted
with fifth grade students indicates an average increase of 17% in
geometric reasoning after 8 hours of KDK activities.

**Keywords:**
Spatial Intelligence; Creativity; Geometric Language; Knowledge
Discovery Kit KDK; Modular Paper Constructions

**JEL
Classification:**
I250

**1. Introduction**

Landmark research studies document that spatial education at early age plays a central role in achieving performance in school and in Science, Technology, Engineering and Mathematics (STEM) occupations in the information age (Frick, et al., 2014; Jirout, et al., 2015; Newcombe, 2010; Kell, et al., 2013; Lubinski, 2010; Wai, et al., 2009). Despite the increased interest in training spatial abilities the infrastructure for research on spatial education is at the beginning and spatial education is under-studied. Current approaches for developing spatial learning tools and educational programs involve the use of puzzles, real or virtual manipulatives, and games like Lego, Rubik’ s Cube, Tetris, Chess, Origami.

Hereby we
present the *X-Colony
Knowledge Discovery Kit*
(KDK), a systematic, flexible and dynamic expandable platform for
developing spatial education tools based on a new “geometric
language” and manipulation of ensembles of polyhedra which was
invented by Sorin
Alexe (Alexe,
2012).

KDK was developed as a collection of open end paper construction activities which provides students the opportunity to unleash their imagination in assembling geometric modules of gradual complexity and to discover connections between mathematics and various other disciplines, from history of ancient Egypt, to aquatic life-forms, geology of planets in the Solar System, and possible life styles in the near future.

KDK main goals are to develop spatial intelligence, creativity, strategic planning and forecasting, abstract reasoning, and to enhance fine motor skills, focus, self-confidence and social skills. KDK programs support the Core Math Standard curriculum and and are designed to be used in after-school programs, math clubs and summer camps.

In a first
phase, the KDK programs were evaluated on culturally diverse 2-8
grade students, in several places worldwide: after-school programs
(Hulstrom K8, Northglenn, Colorado, USA; National Junior College,
Singapore; Middle School Herastrau, Bucharest, Romania, 2014), summer
camps (Romania, 2012-2013). The KDK effect in fostering spatial
abilities was evaluated in a pilot study with 5^{th}
grade students at Herastrau Middle School, Romania.

The evaluation results indicate that students enjoyed the KDK programs, became engaged and eager for new discoveries within ad-hoc brainstorming actions, gained more confidence, became more focused, recognized geometric forms with increased precision, and increased their performance in geometric reasoning. Overall the results suggest that the KDK educational activities are effective in educating spatial abilities for middle school students.

**2. The
KDK Educational Platform**

**2.1. KDK Basic Polyhedra**

The KDK
geometric language is defined over a universe of *states*
described as *geometric
configurations*
which are built with a set of *basic
polyhedral*
*modules*.
The geometric configurations evolve gradually based on *polyhedral
operations*
applied to states in a sequential manner.

KDK Basic
Polyhedra are obtained by truncating the three Platonic bodies:
regular *tetrahedron*
(*T*),
regular *octahedron*
(*O*)
and regular *icosahedron*
(*I*),
in such a way that all the faces of the truncated bodies are regular
*hexagons*.
The basic polyhedra *T*,
*O*
and *I
*are
shown in **Figure
1**.
The hexagonal faces created by cutting the corners of the Platonic
bodies are called *facets*.
The empty shapes associated to each corner are called *cofacets*.
Cofacets are module-specific: triangles for the *T*
module, squares for the *O*
module and pentagons for the *I*
module.

**Figure
1. The three KDK basic polyhedra: T, O, I.**

**2.2. KDK Polyhedral Operations**

KDK Polyhedral
Operations are polyhedral *connections*,
*rigid
rotations*
and *paper
folding*.
The KDK *connection*
operations join two geometric configurations by *juxtaposition*
of either facets or cofacets of equal shape, and are denoted *Delta*,
*Gamma*
and *Nabla*.
The *rigid
rotations*
are performed along the edges of the juxtaposed facets or cofacets,
by 60^{o}
for hexagonal facets and by 120^{o},
90^{o},
72^{o}
for *T*,
*O*,
*I*
cofacets, respectively. The *folding*
*operations
*are
used for compression, expansion, and for creating “dual”
configurations. The connection operations applied to pairs of basic
polyhedra are depicted in **Figure
2**.

**Figure
2. KDK connecting operations: ****Delta****,
****Nabla****,
****Gamma**

**2.3. KDK Geometric
Configurations**

KDK Geometric
Configurations are described by “geometric expressions”
created based on the*
*basic
modules *T*,
*O*,
*I
*and
sequential polyhedral operations. The operations are applied
sequentially, by first connecting the basic modules *T*,
*O*,
*I*
and then by creating configurations of increasing complexity. For
instance, **Figure
3**
illustrates the operation *Nabla*
applied 5 times to *T*
modules. Other complex geometrical expressions obtained by applying
join operations to 60 *T*
modules and 22 *O*
modules are shown in **Figure
4**.

**Figure
3. ****Nabla(T,T,T,T,T)**

**Figure
4. KDK geometrical expressions built with 60 T modules and 22 O
modules.**

**3. KDK
Euclidean Representation Rules **

KDK Euclidean
Representation Rules validate
the feasibility of the geometric representation in the 3D Euclidean
space. For example, while *Nabla
*(*T*,*T*,*T*,*T*,*T*)
in **Figure**
**3**
can be built by sequentially applying *Nabla*
to five *T
*modules,
an additional module *T*
cannot be connected to extend this expression.

**3.1. The KDK Educational
Program **

KDK implements
an effective interdisciplinary type of *learning
through playing*
for after-school programs, clubs and summer camps, which is focused
on fostering spatial intelligence, creativity, strategic thinking,
and stimulates the students to discover multiple connections between
math, science and arts. Instructors can use KDK programs as “research
labs” in which students can practice through hands-on paper
construction activities for various topics learnt in math, science
and other subjects.

A typical *KDK
session*
unfolds in groups of 4-6 students and starts with a short
presentation of the objective, materials and instructions. Students
understand the objectives, analyze the documentation and proceed with
developing optimal strategies for achieving the target. The
constructions are built incrementally by combining simpler modules
into objects of higher complexity. The modular objects are
investigated at each step for their geometric, kinetic and aesthetic
properties. The construction activity can be interrupted at any time
and continued from that point in the next session. At the end of the
program sessions the modular constructions can be displayed on stands
in classrooms and school exhibitions.

**Figures 5
and 6**
illustrate examples of KDK constructions realized by students at
Herastrau Middle School in Romania, in a series of 8 after-school KDK
sessions of 1hour/week. Students had access to documentation and
movies, and were challenged to propose new tasks and to answer
questions meant to evaluate the progress in enhancing comprehension,
reasoning and geometric creativity, e.g.: *How
else would you designate the construction you just made? How can you
use this construction? What other constructions do you think are
possible to be obtained with the same set of basic modular polyhedra?
*