To appear in IEEE Transactions on Visualization and Computer Graphics. 1616
Research Representations & Interaction S. Sandra Bae, Rishi Vanukuru, Ruhan
Yang, Peter Gyory, Ran Zhou, and Ellen Yi-Luen Do are with CU Boulder. E-mail:
{suba2574, riva3436,ruya6408, pegy8859, razh6791<EMAIL_ADDRESS>Danielle Albers Szafir is with UNC-Chapel Hill. E-mail:
<EMAIL_ADDRESS>Bae et al.: Cultivating Visualization Literacy for
Children Through Curiosity and Play

# Cultivating Visualization Literacy for Children
Through Curiosity and Play

S. Sandra Bae  Rishi Vanukuru  Ruhan Yang  Peter Gyory  Ran Zhou  Ellen Yi-
Luen Do  and Danielle Albers Szafir

###### Abstract

Fostering data visualization literacy (DVL) as part of childhood education
could lead to a more data literate society. However, most work in DVL for
children relies on a more formal educational context (i.e., a teacher-led
approach) that limits children’s engagement with data to classroom-based
environments and, consequently, children’s ability to ask questions about and
explore data on topics they find personally meaningful. We explore how a
curiosity-driven, child-led approach can provide more agency to children when
they are authoring data visualizations. This paper explores how informal
learning with crafting physicalizations through play and curiosity may foster
increased literacy and engagement with data. Employing a constructionist
approach, we designed a do-it-yourself toolkit made out of everyday materials
(e.g., paper, cardboard, mirrors) that enables children to create, customize,
and personalize three different interactive visualizations (bar, line, pie).
We used the toolkit as a design probe in a series of in-person workshops with
5 children (6 to 11-year-olds) and interviews with 5 educators. Our
observations reveal that the toolkit helped children creatively engage and
interact with visualizations. Children with prior knowledge of data
visualization reported the toolkit serving as more of an authoring tool that
they envision using in their daily lives, while children with little to no
experience found the toolkit as an engaging introduction to data
visualization. Our study demonstrates the potential of using the
constructionist approach to cultivate children’s DVL through curiosity and
play.

###### keywords:

Data visualization literacy, children, constructionism, informal learning

Data is Yours. (a) The assembled toolkit with the physicalization bar chart
panel currently in use and a smartphone displaying the rendered visualization.
The toolkit supports interchangeable chart panels (i.e., line, pie) that are
not in use. (b) Physical components that make up the chart panels and each
chart’s corresponding paper template. Paper templates for the line and bar
charts consist of tear-off rectangles. (c) Physical components that make up
the support structure of the toolkit.  Introduction

.. Visualizations serve a critical role in public discourse [16, 43],
information dissemination [40, 22, 77], civic engagement [1, 8], and decision
making [44, 52]. The COVID-19 pandemic has emphasized the importance of data
visualization literacy (DVL)—the ability to constructively reason with data—as
experts and news outlets frequently use visualizations to increase public
awareness, communicate public health decision-making, and drive action within
different communities [66]. The proliferation of visualizations from the
pandemic has highlighted how the general public’s limited DVL in tandem with
poor visualization design (e.g. misleading visualizations [69, 21]) is causing
a rapid spread of both accurate and inaccurate information [20, 46].

We investigate how physically crafting visualizations (i.e., making and
assembling with craft materials) through play and curiosity may serve to make
visualization more comfortable and familiar to young children. Despite
significant attention to statistical literacy and, increasingly, general data
literacy in elementary curricula [48], we still have limited knowledge of how
to best foster DVL at early ages. Most work in DVL for children relies on a
more constrained, formal educational context (i.e., a teacher-led approach).
In addition, we lack formalized methods (i.e., visualization literacy tests
for children) to evaluate the efficacy of these approaches. Broadly
constraining DVL education to formal environments, children are restricted in
where they learn and the datasets they work with (e.g., predefined datasets
that children may not generally be interested in). Educational research
highlights that allowing children to make artifacts driven by play and their
own curiosity (known as a constructionist approach) serves as a more
approachable entry point for children to better grasp complex topics (e.g.,
mathematics [64, 63, 71], artificial intelligence [41], science [74], and
programming [2]). Many aspects of DVL develop over prolonged study and
experiences, such as the ability to critically reason with and design
visualizations. To ground future pedagogical developments, our work explores
the constructionist approach to provide preliminary steps towards
familiarizing children with visualization.

We investigate Data is Yours as a design probe to explore how children can
accumulate playful learning experiences and personalize visualizations made
from everyday materials (e.g., paper, cardboard, mirrors) (Cultivating
Visualization Literacy for Children Through Curiosity and Play). Data is Yours
as a toolkit stems from the constructionist [56] and embodied cognition
theories [75, 4] and lets children lead the learning/creation process. By
crafting and playing with physical materials, children can give physical form
to data. These physical actions can enable children to consciously reason
about a visualization’s visual encoding (e.g., colors and marks) freely and
without the constraint of finding a “right” answer to any guiding task.

The toolkit’s workflow is illustrated in Figure 5: children assemble the kit,
collect data, create a paper template, and then construct a chart panel to
represent their data. Though past methods for fostering children’s DVL show
promise [3, 17, 7], these tools typically rely on exclusively digital
solutions. Digital tools may come off as “black boxes” to young children and
limit their embodied experiences with visualizations [60]. In contrast,
physical representations enable children to engage in embodied learning and
can serve as a more approachable introduction to visualization. Thus, we focus
on enabling children to craft physicalizations, where the act of making
scaffolds their understanding of different visualization components. Our
approach maintains the benefits of past digital approaches by using computer
vision to dynamically render, update, and save digital versions of the
physical charts.

Using Data is Yours, our objectives are to investigate (1) how children might
respond to physical crafting experiences in the context of working with
visualizations and (2) how to expand DVL learning to informal learning spaces
(i.e., learning that takes place outside of a structured classroom [54]). The
toolkit is a product of a three-phase design process. We first conducted a
formative evaluation using a preliminary prototype with educators and children
to better understand current teaching practices. We then refined the prototype
based on observations and feedback. Lastly, to investigate our two objectives,
we deployed the toolkit in a series of in-person workshops with children
($n=5$; aged 6–11) and conducted interviews with educators ($n=5$).

Our observations reveal that the toolkit helped children creatively engage and
interact with visualizations. Children with prior knowledge of data
visualization reported our toolkit serving as an authoring tool that they
envision using in their daily lives, while children with little to no
experience found the toolkit as an engaging introduction to data
visualization. Educators spoke of the benefits of Data is Yours, emphasizing
how it can be easily integrated into existing activities and spaces (e.g.,
museums, classrooms, homes). From these insights, we provide a future vision
outlining how to expand the ways children can creatively engage with the data
and visualizations found in their everyday lives.

##  1 Background and Related Work

Our work draws on practices both in education and in visualization. We explore
prior approaches and investigate the potential of using constructionist
practices to cultivate children’s DVL.

###  1.1 Tools to Improve Children’s Data Visualization Literacy

Focusing on core DVL components[47, 10, 13, 15], prior tools for children’s
DVL help children improve their abilities to interpret and extract information
from visually represented data. These tools primarily differ based on their
types of activities (i.e., guided vs. child-driven) and intended context
(i.e., formal vs. informal instruction).

The majority of existing DVL tools are actively guided by an educator and used
within a formal learning context (i.e., classrooms). For example, C’est La Vis
[3] teaches young children about pictographs and bar charts through
concreteness fading—a pedagogical method where new concepts are first
presented with concrete examples before progressively abstracting them—via
animations. Similarly, Construct-A-Vis [7] helps children create free-form
visualizations within a classroom setting. Both tools consider how teachers
can configure datasets and visual encodings for children to work with. While
this approach enables teachers to provide guidance and feedback, it largely
limits children’s engagement with data. Visualizations and data exist in
various contexts beyond the classroom (e.g., news, games, books, infomercials,
and advertisements). Yet this approach constrains children with which datasets
they work with and where they learn.

Other approaches are child-driven and used within informal contexts (i.e.,
learning that takes place outside of a structured classroom [54]).
Gamification approaches, such as Huynh et al.’s role-playing game [38] and
Diagram Safari [26], foster curiosity and experimentation by allowing children
to learn through experiential play [53]. While both games let children pace
themselves, they are limited to predefined datasets. In contrast, Scratch
Community Blocks [17] and data sculptures [6] help children work with datasets
that the children think are important. This educational approach stems from
the constructionist theory, which advocates for student-centered learning by
having children build artifacts that are personally relevant and meaningful
[56] (see subsection 1.2).

Like data sculpture [6] and Scratch Community Blocks [17], we follow the
constructionist approach. However, we investigate this approach from a do-it-
yourself (DIY) perspective, drawing on constructive visualizations [35].
Manually constructing visualizations has helped adults explore [35, 37], learn
[58, 70], reflect [29, 68], and play [32] with data. Children may similarly
benefit from this approach. By constructing an artifact, children have an
“object-to-think-with” to externalize their mental model, observe when their
designs succeed and fail, and discuss their insights with others [55].
Research also shows that while modern technology provides rich
functionalities, it may come off as “black boxes” to young children as its
inner mechanisms are often hidden and thus poorly understood [60]. To
circumvent this challenge, like data sculpture [6], we rely on low-cost
physical materials that children can manipulate to create interactive
physicalizations and learn from their embodied experiences.

Working with physical materials leverages the embodied cognition theory, which
states that motor action and cognition are highly interrelated, and hence,
mutually dependent upon each other [75, 4]. However, working with physical
materials can be challenging and tedious when needing to update datasets and
perform other data interactions (e.g., filter, select, encode). As a result,
physicalizations traditionally limit children to represent mainly static
datasets [65, 72]. Our approach overcomes this limitation and differs from
data sculptures in that we still leverage computational capabilities where
children can dynamically render, update, and save their visualizations (see
subsection 3.2).

###  1.2 Using Toolkits to Construct Visualizations

Papert postulated that making and designing are critical to learning: they
position the learner as an active agent in the creation process, rather than
as a passive recipient of materials designed for the learner [56]. D’Ignazio &
Bhargava [18, 19] recognized the benefits of the constructionist model when
teaching visualizations to novices. They focused on hands-on learning with
creative activities (e.g., analyzing music lyrics) to introduce new data
concepts. However, their work explicitly targets adults. Research with
children requires a different set of considerations [59, 23], making D’Ignazio
& Bhargava’s activities difficult to apply to children. Alternative
considerations are needed to create meaningful artifacts for children’s
diverse interests and experiences.

Toolkits can help achieve the twin goals of meeting children’s idiosyncratic
needs and constructing visualizations in a creative, playful manner. Learning
technologies illustrate that creativity support toolkits can support a wide
range of explorations powered by children’s interests [61]. The success of
these toolkits is driven by design principles, such as being approachable and
supporting exploration, where children can construct diverse artifacts even
with low-cost, easily available materials (e.g., Cricket [60], Makey Makey
[67]). Visualization also highlights the benefits of toolkits, such as the
InfoVis Toolkit [24], prefuse [30], and D3 [9]. They can simplify creating
complex visualizations using methods that facilitate critical thinking and
ownership [51]. Creative use of visual toolkits, such as danceON [57], shows
how creative activities can engage novices (e.g., young women of color) in
technical computational and data concepts while still providing personal and
culturally relevant learning experiences. Data is Yours has similar
objectives. We explore how to best support children to playfully and
creatively construct visualizations while broadening their engagement with
data through physical construction.

We investigate the potential of physically constructing visualizations for DVL
as it can help children become more comfortable and familiar with
visualizations. By learning from embodied experiences (see subsection 1.1),
children may understand the different components of a variety of
visualizations. In turn, this approach may help children read and interpret
existing visualizations. However, toolkits to help children author
visualizations driven by their interests are limited and sparse (subsection
1.1). We take inspiration from both education and visualization to create a
toolkit using everyday objects (e.g., cardboard, paper, mirrors) and paper
templates to empower children to create, customize, and personalize their own
visualizations. Grounded by constructionist and embodied cognition theories,
we use Data is Yours to explore how to leverage child-driven activities to
author interactive visualizations and expand DVL learning to informal learning
spaces. Our approach offers a new perspective on how to best cultivate
children’s DVL through creativity and play.

Figure 1: (a) Diagram of inner workings of Data is Yours. (b) Fiducial markers
(FM) behind chart panels face the phone’s rear-facing camera. Detected FM
(outlined in pink) require a high-contrast background. FM are detected based
on their (c) orientation and (d) position.

##  2 Formative Study

We developed an initial prototype and used it to conduct a formative study
with children and educators. The prototype was designed to reach a broad
audience using prevalent technology (e.g., smartphones). The objectives of the
formative study were to (1) understand the methods and challenges associated
with teaching data visualizations in elementary school and (2) collect
feedback on the prototype to inform the development of a design probe for an
exploratory study.

###  2.1 Formative prototype

Inspired by the various tangible interactions printed fiducial markers enable
[76, 49], our preliminary prototype explored these markers as control points
for children to map, read, and update paper-based visualizations [5]. Each
fiducial marker provides information based on its identity, position, and
orientation (Figure 1). Our initial setup included a single USB camera mounted
above the paper graph. The camera was connected to a laptop which presented
the digital visualizations.

However, to enable participation from a broader set of backgrounds, we
designed a new prototype that used a smartphone. The smartphone acted both as
a camera to track changes in markers as well as an interactive screen to show
a digital representation of the physical visualization. Relative to personal
computers, smartphones are considered to be more widely accessible hardware—in
2019, 95% of U.S children had internet access at home, but 6% (primarily low-
income minorities) accessed the internet only via smartphones [25]. The
prototype consisted of five vertical sliders that were mapped to five bars of
a bar chart (Figure 2a). Fiducial markers were placed behind each slider, and
the smartphone was mounted at an angle that can view the fiducial markers via
two mirrors enclosed in the cardboard casing (Figure 1a). To further enhance
authoring capabilities, we introduced paper templates that children could draw
or write custom chart titles and labels. The templates would first be scanned
by the phone camera and then applied to the digital chart (Figure 2b).

###  2.2 Method

After designing a phone-based prototype, we conducted a formative study with
children and educators, which was approved by CU Boulder IRB. All study
materials are available in the supplemental materials111https://osf.io/7u93z/.

Children. We interviewed two children (ages 10 & 11; 1 female, 1 male). We
first conducted a pre-interview using images of visualizations from popular
media to determine their familiarity with visualizations and asked how the
children currently create visualizations. We then used a think-aloud protocol
asking participants to work through two activities for approximately 30
minutes. For each activity, we asked what data observations children noticed
from the charts they created. In the end, we conducted a semi-structured
interview to understand the children’s experience in creating their own
visualizations and learn how the prototype could be improved. We took notes
and captured images and video recordings for analysis.

The first activity taught the children how to use the prototype. Children
first sorted different cards with fruit images. Then the children entered the
number of each fruit type to create a bar chart using the prototype’s sliders.
In the second activity, children generated their own data from a game.
Children were given toy disks to slide toward a target (see Figure 5b). The
closer the disk was to the center, the more points children were rewarded. The
game was played under five different conditions. Four of the conditions were
predefined by the research team (left hand, right hand, closed eyes, five
steps back), and one was child-defined (i.e., the child created their own game
condition). Children created a bar chart of the points they scored per
condition using the prototype and the paper templates.

Educators. To complement our formative study with the children, we interviewed
5 educators (3 female, 2 male) who work in a variety of settings: educational
research organizations, formal classrooms, after-school programs, and online
teaching. The objectives with the educators were to understand current
practices, the challenges they faced when teaching data visualization, and how
the prototype might help address those challenges. We conducted a semi-
structured interview and an artifact walk-through where we shared the same
prototype the children used and demonstrated the two aforementioned
activities. We took notes and captured images and video recordings for
analysis.

Figure 2: (a) Formative prototype. (b) A child from formative study is moving
cardboard sliders to author a bar chart. His paper template consists a mix of
words and icons and is reflected on the smartphone display (indicated by
dotted outlines). See subsubsection 3.2.2 for more details.

###  2.3 Key Findings and Design Implications

We identified three themes from the formative study: a need for
approachability, engagement, and scaffolding and embodied learning.

Approachability. Prior studies and our conversations with educators note that
children even of the same general age range are likely to have a variety of
experiences with data, drawn from both their personal experiences and
educational backgrounds. This variety means that there is no standard method
to teach children about data visualization [48].

Though it is difficult to determine how to make visualization approachable for
all children, we noted the promise of everyday materials. Educators commented
that because the prototype was made from materials that children are familiar
with (e.g., paper), it can readily foster experimentation (i.e., if a child
tried something and did not like the results or made a mistake, they could
simply start over). We observed this during the study where children freely
engaged with the prototype without fear of breaking something.

Engagement. Educators and children alike expressed a need to center the
child’s learning around their interests. This is particularly pertinent as
young children have short attention spans, yet for them to learn, they need to
be driven by frequent, long-term engagement. Educators address this challenge
by empowering students with autonomy and increasing their sense of ownership
(e.g., letting children choose their own project topics). We saw this
reflected in the formative investigation where both children found the first
activity with the predefined fruit dataset as “cheesy” and one they would not
be interested in doing again. Educators also suggested letting children
construct parts of the prototype themselves to increase their investment in
the learning activities. From these comments, the final design probe expanded
children’s ability to author their own visualizations in two ways. First, we
designed the toolkit such that children can freely author visualizations using
data of their choice [31]. Second, we designed the probe for easy assembly and
provided more options to personalize each chart (e.g., choosing colors and the
number of data points) to increase their sense of ownership in the activities.

Scaffolding & Embodied Learning. Educators highlighted how students struggle
when initially designing visualizations (e.g., determining what information
should go on which axis, setting up appropriate scales), remembering
visualizations components (e.g., titles, axes labels), and interpreting visual
representations (e.g., reading values between y-axis tick marks). These
remarks generally aligned with our observations that children recognized
common visualizations (bar, line, pie) but sometimes struggled to articulate
what they were conveying. Educators also noted that though the prototype’s
input method of connecting physical movements to abstract changes in the bar’s
height supports embodied learning (see subsection 1.1), more explicit
associations between the physical form and visualization would be useful.

Based on these comments, we designed the probe to tightly integrate contextual
scaffolding into the physical artifact. We changed the design from a pre-
assembled tangible interface to a DIY toolkit where children would physically
construct the box and charts to create physicalizations. The chart panels and
paper templates of the design probe were designed to be a complete visual
representation on their own by containing all necessary information (e.g.,
visual encoding, title, axes) while still corresponding to the digital
representation of the visualization. Scaffolding is built into the design of
the physical charts that children would reflect on while constructing (e.g.,
choosing the number of sliders as a proxy for defining the number of data
points; subsection 3.1).

###  2.4 Design Goals

We derived three core principles from our formative investigation and prior
literature: the design probe should (1) be approachable to children; (2)
provide children an engaging experience when authoring visualizations and (3)
support scaffolding and embodied learning. We further divide these principles
into six design goals to understand the usability, inclusiveness, and
widespread applicability of the design probe in our exploratory study.

Approachability

G1.

Foster experimentation: Use low-cost materials to encourage children to
experiment and iterate on designs without having to worry about losing or
damaging materials.

G2.

Replicable: Create a design with materials that can be replicated at home or
informal learning spaces (e.g., after-school programs, libraries) to expand
data engagement.

Engagement

G3.

Ease of use: Create a low barrier of entry for children to build the toolkit
and explore a wide range of datasets.

G4.

Support creative expression: Enable children to create their own designs when
authoring visualizations.

Scaffolding & Embodied Learning

G5.

Craft visualizations: Provide a contextualized, embodied experience to help
children understand different visualization components.

G6.

Physical interactions: Focus on making kit mechanisms transparent, connect
physical manipulations to data operations, and help children understand why
different features are necessary.

##  3 Design Probe: Constructing Visualizations with Everyday Materials

To address these goals, we created Data is Yours: a design probe to study how
constructionist practices can engage children to create visualizations in a
playful, creative manner. It is a toolkit made out of everyday materials
(e.g., paper, cardboard) and uses fiducial markers to help children author
visualizations. Figure 5 illustrates the workflow of using the toolkit. After
constructing the kit (G6), children mount a smartphone on top of the box and
place a chart panel with a paper template onto the acrylic surface. Then
children manipulate the physicalization to reflect their collected data and
are able to see a corresponding visualization on the smartphone display. The
toolkit is open source222https://www.instructables.com/Data-Is-Yours-Toolkit-
Assembly-Instructions/ to enable children, parents, and educators to adopt and
fabricate the toolkit on their own (G2).

###  3.1 Physical Components

The physical components of the toolkit include interchangeable chart panels,
paper templates, and a cardboard-based box. These components are made from
readily accessible materials that can be easily replicated (G2, G3). Children
first decide the number of data points required, assemble the chart panel, and
then create a paper template to label the chart. The combination of the chart
panel and the paper template results in a complete physicalization (Figure 3).

####  3.1.1 Chart Panels

The design probe supports three chart types (bar, line, pie) (Figure 3) (G5).
These charts were chosen based on their common use in media and our insights
from the formative study. Each chart panel consists of a cardboard base,
paper, and 3D printed sliders. To support scaffolding and embodied learning,
children use these physical pieces to assemble a visualization. Children can
draw connections of the existing idioms behind common visualizations through
their physical interactions as opposed to being presented with a predefined
chart. Note that though Figure 3 shows five data points for each chart, charts
can be customized so that children can remove and add data points accordingly.
A paper-based alternative to the 3D printed components is provided in our
Instructable documentation.

Bar chart. The bar chart panel uses two cardboard panels, which are held
together by 3D printed snap presses. Strips of colored paper are inserted
between the cardboard and the handle base, acting as the physical “bars” of
the bar chart. To support creative expression and help children think about
visual attributes, children can choose different color strips of paper while
authoring a bar chart (G4). For example, children can reason about visual
semantics when five bars each have a unique color as opposed to the same
color. Fiducial markers are attached to the back of the 3D printed snaps to
track the height of the bars while children are physically manipulating them.

Line chart. The line chart panel uses the same cardboard body and 3D printed
handles as the bar chart panel. This approach allows children to think about
the relationship between a bar and line chart (i.e., how a bar chart can be
transformed into a line chart with a temporal emphasis) through hands-on
assembly (G5). Instead of wide paper strips, the line chart uses narrow paper
strips that are placed between the cardboard and the 3D handle. These paper
strips have pre-punched folding lines that shrink and stretch as the 3D
handles move either closer or further away from each other. The paper strips
are intentionally designed to help children observe and experiment with the
strips as they move the 3D printed handles (G1).

Pie chart. The pie chart panel requires a cardboard base and a transparent
plastic sheet. Children assemble the “slices” of a pie chart using cardstock
circles. Each cardstock circle corresponds to a section of the pie chart and
has a unique fiducial marker at the back. These cardstock circles are spirally
inserted into the slit of the transparent plastic sheet and pinned by a 3D
printed clip. Children can then rotate and manipulate the pie chart around the
3D printed clip. Children can choose pie sections of any color and arrange
them in order of their preference.

Figure 3: (a) Frontside of each chart panel (bar, line, pie). Blank paper
templates layered on top. (b) Backside of each chart panel, illustrating where
fiducial markers are placed when facing the camera (Figure 1).

####  3.1.2 Paper Templates

Each chart panel requires a paper template, which is a layer of paper that
covers the top of the chart panel (Figure 3a). The paper templates are made
from regular printing paper (G2), and children can be readily provided with
multiple copies of the same paper templates to foster experimentation (G1).
The paper templates’ design subtly scaffolds children about visualization
design by having them fill in the slots for different chart components (e.g.,
titles, axes labels, y-axis scale). The blank templates also enable children
to creatively customize the charts (e.g., by drawing, writing) (G4).

The paper template design for the bar chart and line chart is the same. It
consists of a title block, five x-axis label blocks, and tear-off rectangles.
The left-most rectangle folds back so children can label the y-axis scale. The
tear-off rectangles are to encourage children to think about the data they are
representing (G5). Before creating the physicalization and rendering the
visualization, children must decide how many rectangles to tear off depending
on the number of their data points. Children can tear off up to four
rectangles, and those that are not torn will cover the sliders of the
respective chart panel. The low cost of paper allows children to experiment
and try again if they make the mistake of ripping off too many rectangles
(G1).

The paper template for the pie chart similarly includes a title block but
features a color legend and a cut-out circle. The color legend helps children
consider what the different colors represent within the pie chart. Children
can also take notes on their pie chart in the white space of the template. For
unused colors, we provided white stickers for children to cover the
corresponding legend blocks.

####  3.1.3 Box

The main support structure of the toolkit is a folded cardboard box
(Cultivating Visualization Literacy for Children Through Curiosity and Playc).
Four other cardboard components are required to create the final structure.
These five components can be quickly assembled and disassembled by simply
inserting and unfastening tabs without needing glue or other adhesive
materials. The box was designed so that the assembly does not require fine
motor skills, making it more generally approachable and appropriate for
children (G3). In addition, the cardboard material allows children to decorate
the box, promoting creative expression while providing a sense of ownership
(G4).

Inside the assembled box, children place two mirrors to capture the fiducial
markers with the phone camera (Figure 1a). Afterward, children insert an
acrylic plate from either side of the box. The acrylic panel acts as a support
for the interchangeable chart panels. Then the chart panels and paper
templates are directly placed on top of the acrylic plate.

###  3.2 Digital Components

Data is Yours supports a web app where children can dynamically render,
update, and save their visualizations. The app consists of three user flows
(Figure 4) that children enter after constructing the toolkit and creating
their physicalizations. The first flow teaches children how to interact with
the chart panels to author a visualization, the second flow enables children
to scan and save their own chart, and the third flow allows children to
revisit previously saved charts. All three flows start via their respective
buttons on the home screen.

####  3.2.1 Flow 1: Interacting with the Chart Panels.

Flow 1 provides a tutorial of the system’s digital functions and represents
the first experience children have when connecting the physicalization to the
digital visualization. After constructing and placing the physical chart panel
on top of the acrylic panel, children can explore the basic features of their
respective digital charts. Children can move the physical components and see
how the digital representations change as a result. They can also change the
color of the digital chart by tapping on the visualization components, such as
an individual bar, to match the chosen colors represented in the physical
panel.

Children can also “pause” the chart in its current position and lift the phone
to share their rendered visualizations. The pause functionality prevents the
digital chart values from incorrectly changing as the phone camera’s field of
view is moved around. Given Flow 1’s purpose as a tutorial, the data types and
scales in Flow 1 are the same as those in the formative study: visualizing
different types and quantities of fruit with a predefined dataset.

Figure 4: Screenshots of web app. (a) Home page (b) Bar chart tutorial in Flow
1 (c) Menu in Flow 2 (d) Scanning instructions for paper templates (e) Example
of a scanned template (f) Saved visualizations in Flow 3.

####  3.2.2 Flow 2: Authoring Visualizations

Once children have created the physical panels, collected relevant data of
their choice, and prepared the corresponding paper templates, they can enter
Flow 2 to author interactive visualizations. Flow 2 requires two steps:
scanning and authoring.

Scanning and defining the chart. Children are guided on how to scan their
paper template through a series of GIFs and instructional text displayed on
the screen. To scan their paper templates, children remove the physical panel
and place the paper template face-down on the acrylic panel. The app displays
a “Scan” button to initiate the process. Once the scanning is complete (i.e.,
taking a picture), children are instructed to flip the paper template upright,
reposition the physical panel, and place the template on top of the chart
panel to complete the physical chart. For only the line and bar chart,
children are prompted to enter information corresponding to the x- and y-axes
of the charts, such as the number of data points and the maximum value of
their collected dataset.

Authoring and interacting with the chart. To author visualizations, children
follow the same general process as Flow 1. The predefined dataset is replaced
with information from the paper template that the children scanned into the
chart. Information from the templates, such as the hand-written chart title
and x-axis labels, are displayed at the corresponding locations on the screen
by taking cropped images. The scale of the y-axis is calculated based on the
input from the previous step. Children can then move the physical sliders to
represent the data values they have collected. This information is translated
onto the digital visualization. In addition to the “Pause” button, Flow 2 also
allows children to save a chart. Clicking the “Save” button results in a small
visual indicator that the chart in its current state has been saved.

####  3.2.3 Flow 3: Revisiting Saved Charts

Once children have saved their charts, they can view them within the web app
by selecting the “Saved Charts” button from the home screen. They can browse
through a list of chart images in order of the day and time at which they were
saved. These images are non-interactive and meant to serve as “snapshots” of
data collected in time.

###  3.3 System Architecture & Implementation

Once the toolkit is fully constructed, we have two mirrors inside which direct
the phone’s rear-facing camera at the chart panels (Figure 1a). This
configuration enables a web-based application to detect the markers on the
interface and render the charts. This approach allows us to make a variety of
interfaces that function with just paper. The front-end is implemented with a
combination of HTML5, CSS, JavaScript, and a Javascript port [39] of the ArUco
Computer Vision Marker library [27, 62]. The library offers up to 1000 unique
markers for tracking a variety of values (e.g., position, rotation).

Figure 5: Workflow for children in the formative and exploratory study. (a)
Construct a kit (b) Collect data by playing the game of disk darts (c) Author
a paper template (d) Scan the paper template (d) Manipulate the
physicalization chart panel (bar, line, pie) to match collected data.

##  4 Exploratory Study

To investigate how constructionist practices can foster children’s comfort and
familiarity with data visualization, we conducted an exploratory study using
the design probe in a series of workshops with children (subsection 4.1) and
interviews with educators (subsection 4.2). Workshops can elicit rich
qualitative insights for early stages of applied visualization research [42].
Using guidelines by Kerzner et al. [42], Huron et al. [34, 36], and Terre des
hommes [50], we designed our workshop around the constraints of working with
children in a limited timeframe while still evaluating the research objectives
of the toolkit. Note: the workshop is a means of investigating the
constructionist approach where we primarily collected qualitative data rather
than quantitatively testing children’s learning outcomes. Current DVL tests
are meant for adults and their efficacy for children has not been fully
researched333Existing standardized tools[10, 47, 13] target adults. E.g., two
components of VLAT focus on determining range and extremum for various
visualizations. Such concepts are generally not taught till children are older
(11–14 y/o)..

All activities with the children and educators were approved by the CU Boulder
IRB. Study materials are available in supplemental materials. A comprehensive
overview of the participants can be found in Table 1.

Participant |  Role |  Age / Occupation
---|---|---
P1 |  Child |  6 years old
P2 |  Child |  6 years old
P3 |  Child |  7 years old
P4 |  Child |  8 years old
P5 |  Child |  11 years old
E1 |  Educator |  After-school program
E2 |  Educator |  After-school program
E3 |  Educator |  After-school program
E4 |  Educator |  Educational Researcher
E5 |  Educator |  Elementary School Librarian
Table 1: Summary of participants’ backgrounds in the exploratory study.

###  4.1 Workshops with Children

Participants. We conducted two two-hour workshops with children ($n=5$)
between 6–11 years old to understand how the constructionist approach may
differently impact different age groups. This age range aligns with the
development stage where children are able to formulate representational
thought through symbolic thinking (i.e., words and pictures) [28]. The first
workshop was a group workshop at an after-school program with four children (2
male, 2 female; aged 6–8 years old), and the second workshop involved one
child (1 female; aged 11 years) at a public museum. We recruited the
participants by sharing virtual flyers and emails about the workshop at
informal learning spaces (e.g., public museums, local makerspaces, and after-
school program venues).

Procedure. The workshop was divided into four stages, where the first stage
consisted of icebreaker activities, the second and third stages focused on
constructing and using the toolkit, and the last stage was for concluding and
reflection. Figure 5 illustrates the general workflow of the workshop. Before
engaging in the activities, the research team first provided an overview of
the workshop to the parents and guardians of the children, gained consent from
parents, and collected assent from the children. We took photographs and
collected video-audio recordings as the children were constructing and
interacting with the toolkit.

Stage 1. Introduction and Icebreaker Activities. The workshop facilitators
introduced themselves to the children. Children were asked about their
perception of data and visualization in general. Afterward, they were shown a
series of common visualizations (bar, line, pie) and asked if they recognized
the charts. Then children engaged in an open-ended discussion about what they
noticed in each of the images to gauge their familiarity with these
visualizations more concretely.

Stage 2. Toolkit Construction. In the second stage, children were introduced
to Data is Yours. Workshop facilitators first provided a finished toolkit for
children to play with and encouraged children to inspect the sliders and
mirrors. Children were instructed on how to use the Data is Yours by moving
the sliders. At this time, children played with the toolkit using the
predefined fruit dataset used in the formative study, observing changes
displayed on the smartphone. Afterward, children moved on to building their
own Data is Yours. Each child was given a cardboard fold-out template and a
packet containing the various components required for assembly. Facilitators
guided the children through the construction process by demonstrating the
physical operations required for assembly. After constructing the kit,
children were provided craft materials (e.g., pens, pipe cleaners, stickers)
to decorate the outer body of the toolkit.

Stage 3. Using the Toolkit. In the third stage, children used their toolkit to
author visualizations. Working under the constraints of a guided and timed
workshop, children were asked to generate and collect data using the same game
from the formative study for three rounds (c.f., subsection 2.2). In Round 1,
children played the game under five different conditions to author a bar
chart. Two of the conditions were predefined by the research team (throwing
disks using the left hand and right hand) and the remaining three conditions
were defined by the children. In Round 2, from the five conditions, children
were asked to repeat one game condition of their choice four more times. This
activity was intended to help children generate data to create a line chart.
In Round 3, children calculated the total score for all five conditions and
visualized each condition’s contribution (i.e., proportion) to the total score
for the pie chart. After finishing the game of disk darts, children were shown
how to use the paper templates to translate their collected data into chart
axes and other chart elements (e.g., title, legend). They then scanned the
templates and authored the different visualizations (i.e., bar, line, pie)
using the toolkit. Afterward, children analyzed the resulting graphs and
compared their results to others.

Stage 4. Conclusion and Reflection. In the fourth stage, we quickly recapped
the activities the children engaged with. Children were asked semi-structured
questions to gain their overall impression of the toolkit and data
visualization in general.

###  4.2 One-on-One Interview with Educators

Participants. We recruited five educators (3 female, 2 male) from a local
school district and via our network of education researchers. All educators
had experience with teaching visualization to young children in the context of
either mathematics or science curricula. Our aim with the educators was to
gain feedback on how Data is Yours might be integrated into informal learning
spaces. Following the guidelines by Kerzner et al. [42], we recruited
educators with diverse experiences.

Procedure. Each participant provided a verbal introduction of their background
at the start of the interview. The facilitators then performed a demo of the
toolkit, illustrating how to build the toolkit and showcasing the different
features (e.g., scanning with paper templates, switching panels to create
different visualizations). After the demo, educators partook in a concurrent
think-aloud protocol where they were encouraged to build and interact with the
toolkit in an open-ended session for approximately 20 minutes. Educators then
engaged in a semi-structured interview where they answered additional survey
questions about their impressions, ideas, and concerns about the toolkit.
Educators were audio-recorded and photographed working with our toolkit.

##  5 Results

We qualitatively analyzed materials (i.e., notes, images, videos) from our
exploratory study. We conducted a content analysis on these materials using
the design goals that emerged from the formative study as a priori categories
[73]. Two researchers performed this categorization independently, after which
a final set of observations corresponding to each design goal was agreed upon
(subsection 2.4). Findings were extracted from these grouped observations. For
the interviews with the educators, we summarized their impressions, feedback,
and concerns also through the lens of our design goals and research
objectives.

###  5.1 Workshops with Children

Children provided positive impressions of the workshop and enjoyed engaging
with Data is Yours, despite having greatly varying baseline knowledge of data
visualizations (as assessed by the ice-breaker discussions). In the beginning,
the younger children (P1-P4) were not able to assign names to the
visualizations, explaining they did not frequently interact with or create
visualizations. Yet, they found the workshop particularly fun, with P4
expressing, “I really loved it!” P3 strongly expressed wanting a version of
the toolkit that would work with his iPad, so that he could use it anytime
without relying on his parents’ smartphones. In contrast, P5, the oldest child
of the two workshops, was able to identify, interpret, and explain the
function of all charts and found the workshop with the toolkit interesting,
but only a “little fun.”

####  5.1.1 Approachability

G1: Foster Experimentation. Initially, P1-P4 were hesitant to perform certain
physical operations, such as folding the flaps of the cardboard base, and even
caused minor tears in the cardboard. We then explained to the children that
these issues are normal and were, in fact, the reason why we used cardboard
and everyday materials: mistakes were a welcome part of the process. After our
explanation, P1-P4 began to more confidently interact with the cardboard base
and intentionally experiment with different assembly configurations of the
components, such as having the phone camera cutout be on the right instead of
the left (Figure 6a). Similarly, with the paper templates, three of the four
younger children (P1, P2, P4) initially ripped the template in half when
trying to remove the tear-off rectangles. After providing more copies of the
template, they were able to successfully author a template. These examples
speak to the children’s willingness to try again despite the challenges they
faced and to experiment with the toolkit in unexpected ways to learn more
about the content and function it provided.

G2: Replicable. P1-P4 mentioned they envisioned themselves attempting to
recreate the toolkit at home with support from their family. Though children
were given pre-packaged kits to construct their own toolkit during the
workshop, they provided suggestions on how to further improve the
replicability of Data is Yours. P2 and P3 suggested that the overall toolkit
should be bigger so it can support larger screen interfaces like a tablet.
P3’s motivation for this request stemmed from wanting increased agency by
using a device he had access to at home. P5, in contrast, felt more
comfortable with the idea of constructing a toolkit by herself and
specifically envisioned how the toolkit could be used by children of her age
at her school’s library. She explained how the school librarian would often
make children do activities related to data visualization (e.g., create a pie
chart to track genres of books read), and that the toolkit might make for an
interesting way to author multiple charts related to that information. Data is
Yours’s design showcases the breadth of how different children envision the
toolkit being reused in different contexts.

####  5.1.2 Engagement

G3: Ease of Use. In both workshops, we provided verbal instructions on how to
construct the box and also demonstrated relevant actions in real-time. The
components of the toolkit were designed so that it would be easy for children
to assemble the box by themselves. This was true for P5, but the younger
participants all required more time and some hands-on assistance to complete
the process. The younger children found it difficult to securely insert some
cardboard flaps into the pre-cut slots especially when the pieces were not
already connected. They also needed instructions while arranging fiducial
markers on the back of the panels in the right order. P5 took 15 minutes to
complete the box assembly, while P1-P4 needed around 35 minutes.

Despite this variance, Data is Yours successfully enabled all children to
transform their data into digital charts. All participants were able to
successfully create visualizations even from the workshop’s first activity:
translate sorted stacks of fruit images into a digital bar chart using the
toolkit slider. The interface also encouraged the participants to construct
the kit by themselves and promoted collaboration. In the group workshop,
despite no instructions to work together, P1-P3 started to naturally
collaborate to help each other create and provide comments on each other’s bar
chart (Figure 6b).

G4: Support creative expression. After constructing the toolkit, children
personalized their box using a range of craft materials including sketch pens,
colored paper, and stickers. Children particularly enjoyed this step, spending
significant time (20 minutes on average) deciding which materials to use and
how to design their box. The younger children began by writing their names on
the box, adding stickers, coloring the sides, and drawing faces on the box to
personify it. P5, who had the greatest familiarity with visualizations,
focused on many functional additions, such as textual instructions for where
to place the mobile phone, a handle to carry the box made using pipe cleaners
that were inserted into the flutes of the cardboard, and two small knobs to
keep the paper template in place (Figure 6d).

Children’s creative expression was not limited to solely the box but also
included the paper templates and paper strips for the bar charts. With the
paper templates, children came up with a range of titles for their
visualizations, such as “Frisbee,” “Throw-it Game,” “[P3’s] Game,” and
“Targeting.” Children also exhibited creative variations in how they labeled
their x-axes: three children expressed the x-axis labels with full words
(Figure 6c), one used a combination of letters and numbers, and one drew
icons. With the paper strips, children used three types of color schemes: all
of the same colors (P3–to use their favorite color), all different colors (P1,
P2, P5–to make use of all available options), or two alternating colors (P4–to
imitate the facilitator’s toolkit). Even the simple decisions of choosing
paper strips, deciding titles, and naming x-axis labels highlight how children
were excited to showcase their creative expressions.

####  5.1.3 Scaffolding & Embodied Learning

G5: Craft visualizations. Although children faced some issues during the
construction process, most children were able to create, analyze and draw
meaning from their visualizations. Across both workshops, children spent about
30 minutes building and interacting with the chart panels. Towards the end of
the workshop, P2 and P5 inferred from their charts that their performance with
one hand was better than the other due to better control of their dominant
hand. P5 noticed a slight upward trend in her line chart as she performed
better over time as she played the game. She elaborated that the resemblance
of the physical line chart to its digital form drew her attention to the
line’s shape. Similarly, P1-P4 were also able to draw parallels between the
physical bar chart and the digital representation. While playing with the bar
chart control panels (moving sliders up and down), they noted how the height
of the paper strips proportionally matched the heights of the digital bars.
The ability to physically craft the visualization and the manner in which the
digital representation mirrored the movement of the 3D printed sliders helped
reinforce the various visualization components that they interacted with.

Figure 6: (a) P3 assembling the cardboard base (b) P1–P3 collaborating (c) P1
interacting with her authored bar chart (d) P5’s decorated toolkit.

G6: Physical Interactions. After interacting with the facilitators’ toolkit,
children immediately raised questions about and tried to explain the box’s
mechanism. P3 hypothesized the mirrors were necessary for the camera to see
fiducial markers “…to provide some kind of coding that’s connect[ing] the
markers and phone.” P3 continued to question the toolkit’s mechanism, asking
“So basically you build the data [visualization] with [the toolkit], and then
you print [the visualization] out?” This curiosity and fascination motivated
the younger children to take the lead in the learning and creation process.
For example, P1-P4 expressed great excitement in building their own Data is
Yours and chose to construct the toolkit before playing the game of disk darts
(Figure 5b) when offered a choice. Though the construction process took longer
for the younger children, they were engaged throughout the entire process.
They continuously questioned how certain components and features were related
to data visualizations. For example, P4 asked, “Why are there these slots when
we aren’t using them?” when he noticed the two slots at the base of the box
that were meant to let in light to the bottom mirror. Similarly, P2 questioned
why there were multiple colors for the paper strips and why it mattered for
visualization.

While the toolkit fostered their curiosity, children also noted improvements
in the toolkit’s construction process. P5 wished that the toolkit box would be
more stable while she was decorating her box. Additionally, while constructing
visualizations, she noticed that the sliders did not smoothly move for all
panels (bar, line, pie). Consequently, for the bar and line charts, she spent
approximately three minutes rapidly moving the sliders up and down the
channels in an attempt to manually smooth the cardboard structure. P1-P4 faced
a bottleneck during the panel assembly process when they tried to affix the
two parts of the sliders to the chart panel. The grooves within some of the
sliders were worn down after a few cycles of opening and closing, and the
younger children requested assistance during this step.

Figure 7: Sketches of incorporating Data is Yours into various scenarios to
expand ways children engage with data.

###  5.2 Interview with Educators

Educators were overall appreciative of Data is Yours. E1, E2, and E5
specifically appreciated the DIY aspect of the toolkit. E1’s and E2’s after-
school programs emphasize making and physical activities, and they noted the
toolkit was a clever solution to introduce children to data visualizations in
line with those pedagogical goals. All three educators also felt that the
ability to personalize both the overall appearance of the box and the charts
can help children better identify with the activities and take more ownership
of the results. As a researcher for early childhood education, E4 emphasized
the importance of physical manipulation for young children’s education,
pointing out that even adults use physical operations to help them think more
clearly (e.g., using their hands to count). Based on these comments, E4 noted
Data is Yours could help students have another mental metaphor to work with
when reasoning about abstract concepts in data.

However, both E3 and E5 commented on the toolkit’s limitations concerning the
mobile phone. While E5 commented that the toolkit is a powerful introduction
for younger children (i.e., 6-8 years old), “the [current] toolkit is going to
[reach] a ceiling where [children] have already reached to highest capability
they can go [with the phone].” Similarly, E3 questioned the choice of a mobile
phone. She felt smartphones provide a useful introduction to the common
visualizations but will be quickly limited in their ability to create complex
charts later on. She suggested a similar toolkit scheme could incorporate a
tablet or laptop in the future. We assume that these concerns arise from the
perceived limitation of the compact physical form in addition to the mobile
phone’s capability relative to laptops when visualizing data. Both educators
found it difficult to imagine how the toolkit can incorporate more than five
data points in its current form while retaining sufficiently large marks for
children to work with.

For future work, E3–E5 stressed how Data is Yours could be integrated into
existing curriculum (e.g., connecting with schools or after-school programs).
They commented the strength of Data is Yours stems from how easily it can be
integrated into existing activities. E1 and E4 suggested integrating Data is
Yours into informal learning spaces that are likely to already have their own
instructional content relating to visualizations (e.g., science museums). E1
explained that in her past experience as a middle school teacher, she
frequently asked her students to record their daily homework progress using
stickers in a notebook, similar to the process of creating and saving charts
in Flow 3. E1 also expressed the challenges she experienced when students
misunderstood or were unwilling to create visualizations for their science lab
reports. She stated a toolkit like Data is Yours might have helped her
students with those two activities more easily, while also resulting in
engaging, informative visualizations throughout the school year.

##  6 Discussion

We used Data is Yours as a design probe to understand two core research
questions: (1) how might children respond to physical crafting experiences in
the context of working with visualizations and (2) how can we expand DVL
learning to informal learning spaces? We found that Data is Yours helped
younger children learn more about visualization components, while older
children who were more familiar with data visualizations found Data is Yours
to be more of an engaging authoring tool (subsection 6.1). Educators
immediately saw the benefits of our approach and how it could be expanded and
integrated with other activities in informal learning spaces (subsection 6.3).

###  6.1 Data is Yours

Integrating physical crafting into a visualization authoring context was
useful for children but that utility differed based on the children’s age and
prior knowledge of data visualization.

Older children, who are likely to be more familiar with visualizations, viewed
the toolkit primarily as a platform for creating their own visualizations with
more agency. At the end of the workshop, P5, the oldest child, clearly stated
she did not learn anything new about data visualization from the workshop.
However, she noted how Data is Yours was a more engaging method of authoring
charts that she could see herself using in her daily life. Although she had
used spreadsheet programs like Google Sheets to visualize her reading habits
as pie charts, she described the limitations of the “black box” nature of
purely digital tools: “[With Google sheets] you just type in a bunch of
numbers and [the visualization] appears, but with Data is Yours I can actually
create charts with my friends and talk about it.” P5 also spoke positively of
how she could use Data is Yours and the paper templates to author charts using
any dataset (e.g., exercise time, how long she spends coding). P5’s statements
support the idea that the physical and interactive nature of the Data is Yours
results in a more engaging and empowering process compared to exclusively
digital visualization software.

###  6.2 Limitations & Future Work

Our exploratory study does not aim to provide any direct pedagogical evidence
of our approach improving children’s DVL. Rather the qualitative observations
we collected provide preliminary evidence that the children enjoyed
interacting with data physically, can familiarize themselves with foundational
concepts of visualization (e.g., reason about visual semantics, incorporate
chart elements, set up appropriate scales) through curiosity and play, and
were empowered to creatively engage with data. We currently lack more formal
assessment methods that are appropriate for children, such as revised
visualization literacy assessment tests, that would allow for a more formal
DVL measurement.

Though Data is Yours is designed to help children author visualizations using
data of their choice, the tool’s expressiveness was not fully explored within
the constraints of a two-hour workshop. For example, the importance of working
with personally meaningful datasets varied based on age. P1-P4 did not mind
interacting with the predefined fruit datasets. P5 expressed a similar
sentiment as the two formative children, who were similarly aged, indicating
that the fruit datasets felt childish and the self-generated game datasets
were more enjoyable.

Additionally, a majority of the time in the workshop was dedicated to
assembling Data is Yours. While further observations of how children engaged
with their visualizations would have provided more in-depth insights, we found
that assembling Data is Yours helped children accumulate playful learning
experiences that sparked an interest in and new comfort with visualization.
Extensive DVL encompasses complex knowledge and reasoning processes that go
beyond the two-hour activities described here. Thus, further studies will be
required to see how children will use Data is Yours to author data
visualizations in an unconstrained and longitudinal setting.

Assessing children’s DVL frequently focuses on reading, creating, and
interpreting visualizations (subsection 1.1). However, DVL also encompasses
critical engagement with visualizations, such as understanding when charts are
misleading [14, 45]. Our toolkit currently requires facilitator feedback to
let children know about misleading aspects of their visualizations. While we
did not observe such errors, they are possible given the open-ended authoring
functionality (e.g., using misleading axis values). To develop critical design
literacy, future solutions could incorporate visual linting and similar
verification strategies [33].

###  6.3 A Research Agenda for Children’s DVL

With any learning technology for children, research stresses the importance of
three principles: “low floor” (how easy is the technology for children to
learn and use?); “high ceiling” (does the technology advance children in
making more complex projects?); and “wide walls” (how inclusive is the
technology in supporting all children’s interests?) [61]. Our observations
from the workshops indicate that Data is Yours successfully accomplished a low
floor. Young, inexperienced children were able to craft and interpret basic
visualizations using the system. But our approach requires more consideration
as to how to create a high ceiling. E3 and E5 expressed concerns about the
toolkit’s current dependency on a smartphone. However, as children become
older (e.g., teenagers), we anticipate they will not need the scaffolding from
the physical panels to learn and interact with visualizations. Rather,
research shows various ways to interact with data that are particularly suited
for mobile devices [11, 12]. E3’s and E5’s comments indicate that future
iterations of the toolkit could benefit from further considerations on how to
better leverage state-of-the-art visualization techniques to engage older
children.

Expanding Data is Yours to additional contexts would also increase the variety
and complexity of charts children can create. Children and educators suggested
several ways the toolkit can support wide walls, including integrating the
toolkit into activities that children naturally enjoy (Figure 7). For
instance, children can have a more sustained engagement with data at home by
working with their family to recreate visualizations from news media or
progressively tracking changes around them over time (e.g., tracking
children’s height). These activities, in turn, can seamlessly introduce
children to different types of data (e.g., qualitative, geographic,
categorical, temporal). Data is Yours can also be incorporated into group
activities that already take place in informal learning spaces (e.g., science
museums). The constructionist approach shows how produced artifacts can be
“shown, discussed, probed, and admired” [55], serving as a locus for
collaborative discussions. These actions align with our observations from the
group workshop where children collaborated with each other and provided
feedback on their peers’ visualizations without being prompted to. We believe
expanding Data is Yours into daily contexts and collaborative settings can
help children be more critical of the data found around them.

##  7 Conclusion

We explored how a constructionist approach can foster children’s data
visualization literacy (DVL) using Data is Yours. Our observations from a
series of workshops with children ($n=5$; 6–11 years old) and interviews with
educators ($n=5$) reveal that physical crafting with visualizations benefits
children in different ways depending on their past experiences with data. This
exploratory study confirms the need to further investigate how to expand the
ways children can interact with data and visualizations. We reflect on
findings and current research gaps to identify new challenges in children’s
DVL research. We hope that our approach can inspire new ways for children to
creatively and playfully interact with the visualizations and data that are
woven into the fabrics of everyday life.

###### Acknowledgements.

The authors would like to thank Dr. Shaz Zamore for their assistance on the
project. This material is based upon work supported by the National Science
Foundation under Grant No. IIS-2040489, IIS-2046725, & STEM+C 1933915.

## References

  * [1] D. Airinei and D. Homocianu.  Data visualization in business intelligence.  2010 Proceeding of WSEAS MCBEC 2010 - Recent Advances in Mathematics and Computers in Business, Economics, Biology & Chemistry, 2010\.
  * [2] D. Alimisis and C. Kynigos.  Constructionism and robotics in education.  Teacher education on robotic-enhanced constructivist pedagogical methods, pp. 11–26, 2009.
  * [3] B. Alper, N. H. Riche, F. Chevalier, J. Boy, and M. Sezgin.  Visualization literacy at elementary school.  In Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems, CHI ’17, p. 5485–5497. Association for Computing Machinery, New York, NY, USA, 2017. doi: 10 . 1145/3025453 . 3025877
  * [4] M. L. Anderson, M. J. Richardson, and A. Chemero.  Eroding the boundaries of cognition: Implications of embodiment 1.  Topics in cognitive science, 4(4):717–730, 2012.
  * [5] S. Bae, R. Yang, P. Gyory, J. Uhr, D. A. Szafir, and E. Y.-L. Do.  Touching information with diy paper charts & ar markers.  In Interaction Design and Children, IDC ’21, p. 433–438. Association for Computing Machinery, New York, NY, USA, 2021. doi: 10 . 1145/3459990 . 3465191
  * [6] R. Bhargava and C. D’Ignazio.  Data sculptures as a playful and low-tech introduction to working with data.  In Proceedings of the 2017 Designing Interactive Systems. Association for Computing Machinery, 2017.
  * [7] F. Bishop, J. Zagermann, U. Pfeil, G. Sanderson, H. Reiterer, and U. Hinrichs.  Construct-a-vis: Exploring the free-form visualization processes of children.  IEEE Transactions on Visualization and Computer Graphics, 26(1):451–460, 2020. doi: 10 . 1109/TVCG . 2019 . 2934804
  * [8] S. Bohman.  Data visualization: an untapped potential for political participation and civic engagement.  In International Conference on Electronic Government and the Information Systems Perspective, pp. 302–315. Springer, 2015.
  * [9] M. Bostock, V. Ogievetsky, and J. Heer.  D³ data-driven documents.  IEEE Transactions on Visualization and Computer Graphics, 17(12):2301–2309, 2011. doi: 10 . 1109/TVCG . 2011 . 185
  * [10] J. Boy, R. A. Rensink, E. Bertini, and J.-D. Fekete.  A principled way of assessing visualization literacy.  IEEE Transactions on Visualization and Computer Graphics, 20(12):1963–1972, 2014. doi: 10 . 1109/TVCG . 2014 . 2346984
  * [11] M. Brehmer, B. Lee, P. Isenberg, and E. K. Choe.  A comparative evaluation of animation and small multiples for trend visualization on mobile phones.  IEEE Transactions on Visualization and Computer Graphics, 26(1):364–374, 2020. doi: 10 . 1109/TVCG . 2019 . 2934397
  * [12] M. Brehmer, B. Lee, J. Stasko, and C. Tominski.  Interacting with visualization on mobile devices.  In Mobile Data Visualization, pp. 67–110. Chapman and Hall/CRC, 2021.
  * [13] K. Börner, A. Bueckle, and M. Ginda.  Data visualization literacy: Definitions, conceptual frameworks, exercises, and assessments.  Proceedings of the National Academy of Sciences, 116(6):1857–1864, 2019. doi: 10 . 1073/pnas . 1807180116
  * [14] J. D. Camba, P. Company, and V. L. Byrd.  Identifying deception as a critical component of visualization literacy.  IEEE Computer Graphics and Applications, 42(1):116–122, 2022. doi: 10 . 1109/MCG . 2021 . 3132004
  * [15] F. Chevalier, N. Henry Riche, B. Alper, C. Plaisant, J. Boy, and N. Elmqvist.  Observations and reflections on visualization literacy in elementary school.  IEEE Computer Graphics and Applications, 38(3):21–29, 2018. doi: 10 . 1109/MCG . 2018 . 032421650
  * [16] S. Claes and A. V. Moere.  The role of tangible interaction in exploring information on public visualization displays.  In Proceedings of the 4th International Symposium on Pervasive Displays, PerDis ’15, p. 201–207. Association for Computing Machinery, New York, NY, USA, 2015. doi: 10 . 1145/2757710 . 2757733
  * [17] S. Dasgupta and B. M. Hill.  Scratch community blocks: Supporting children as data scientists.  In Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems, CHI ’17, p. 3620–3631. Association for Computing Machinery, New York, NY, USA, 2017. doi: 10 . 1145/3025453 . 3025847
  * [18] C. D’Ignazio and R. Bhargava.  Databasic: Design principles, tools and activities for data literacy learners.  The Journal of Community Informatics, 12(3), 2016.
  * [19] C. D’Ignazio and R. Bhargava.  Creative data literacy: A constructionist approach to teaching information visualization.  Digital Humanities Quarterly, 2018.
  * [20] T. L. I. Diseases.  The covid-19 infodemic.  The Lancet. Infectious Diseases, 20(8):875, 2020.
  * [21] S. Doan.  Misrepresenting covid-19: lying with charts during the second golden age of data design.  Journal of Business and Technical Communication, 35(1):73–79, 2021\.
  * [22] M. Dörk, P. Feng, C. Collins, and S. Carpendale.  Critical infovis: exploring the politics of visualization.  In CHI’13 Extended Abstracts on Human Factors in Computing Systems, pp. 2189–2198. Association for Computing Machinery, New York, NY, USA, 2013.
  * [23] A. Druin.  The role of children in the design of new technology.  Behaviour and information technology, 21(1):1–25, 2002.
  * [24] J.-D. Fekete.  The infovis toolkit.  In IEEE Symposium on Information Visualization, pp. 167–174, 2004\. doi: 10 . 1109/INFVIS . 2004 . 64
  * [25] N. C. for Educational Statistics.  Children’s internet access at home, 2021.  https://nces.ed.gov/programs/coe/indicator/cch.
  * [26] J. Gäbler, C. Winkler, N. Lengyel, W. Aigner, C. Stoiber, G. Wallner, and S. Kriglstein.  Diagram safari: A visualization literacy game for young children.  In Extended Abstracts of the Annual Symposium on Computer-Human Interaction in Play Companion Extended Abstracts, CHI PLAY ’19 Extended Abstracts, p. 389–396. Association for Computing Machinery, New York, NY, USA, 2019. doi: 10 . 1145/3341215 . 3356283
  * [27] S. Garrido-Jurado, R. Muñoz-Salinas, F. Madrid-Cuevas, and R. Medina-Carnicer.  Generation of fiducial marker dictionaries using mixed integer linear programming.  Pattern Recognition, 51:481–491, 2016. doi: 10 . 1016/j . patcog . 2015 . 09 . 023
  * [28] U. E. Goswami.  Blackwell handbook of childhood cognitive development. Blackwell publishing, 2002.
  * [29] P. Gourlet and T. Dassé.  Cairn: A tangible apparatus for situated data collection, visualization and analysis.  In Proceedings of the 2017 Conference on Designing Interactive Systems, DIS ’17, p. 247–258. Association for Computing Machinery, New York, NY, USA, 2017. doi: 10 . 1145/3064663 . 3064794
  * [30] J. Heer, S. K. Card, and J. A. Landay.  Prefuse: A toolkit for interactive information visualization.  In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI ’05, p. 421–430. Association for Computing Machinery, New York, NY, USA, 2005. doi: 10 . 1145/1054972 . 1055031
  * [31] K. Hirsh-Pasek, J. M. Zosh, R. M. Golinkoff, J. H. Gray, M. B. Robb, and J. Kaufman.  Putting education in “educational” apps: Lessons from the science of learning.  Psychological Science in the Public Interest, 16(1):3–34, 2015\. doi: 10 . 1177/1529100615569721
  * [32] A. Hopkins, N. A. Vladis, and A. Satyanarayan.  Data crafting: Exploring data through craft and play.  In VisActivities: IEEE VIS Workshop on Data Vis Activities at IEEE VIS 2020, 2020.
  * [33] A. K. Hopkins, M. Correll, and A. Satyanarayan.  Visualint: Sketchy in situ annotations of chart construction errors.  In Computer Graphics Forum, vol. 39, pp. 219–228. Wiley Online Library, 2020.
  * [34] S. Huron, S. Carpendale, J. Boy, and J.-D. Fekete.  Using viskit: A manual for running a constructive visualization workshop.  In Pedagogy of Data Visualization Workshop at IEEE VIS 2016, 2016\.
  * [35] S. Huron, S. Carpendale, A. Thudt, A. Tang, and M. Mauerer.  Constructive visualization.  In Proceedings of the 2014 Conference on Designing Interactive Systems, DIS ’14, p. 433–442. Association for Computing Machinery, New York, NY, USA, 2014. doi: 10 . 1145/2598510 . 2598566
  * [36] S. Huron, P. Gourlet, U. Hinrichs, T. Hogan, and Y. Jansen.  Let’s get physical: Promoting data physicalization in workshop formats.  In Proceedings of the 2017 Conference on Designing Interactive Systems, DIS ’17, p. 1409–1422. Association for Computing Machinery, New York, NY, USA, 2017. doi: 10 . 1145/3064663 . 3064798
  * [37] S. Huron, Y. Jansen, and S. Carpendale.  Constructing visual representations: Investigating the use of tangible tokens.  IEEE Transactions on Visualization and Computer Graphics, 20(12):2102–2111, 2014. doi: 10 . 1109/TVCG . 2014 . 2346292
  * [38] E. Huynh, A. Nyhout, P. Ganea, and F. Chevalier.  Designing narrative-focused role-playing games for visualization literacy in young children.  IEEE Transactions on Visualization and Computer Graphics, 27(2):924–934, 2021. doi: 10 . 1109/TVCG . 2020 . 3030464
  * [39] jcmellado.  Javascript port of the aruco libray.  https://github.com/jcmellado/js-aruco, 2018.
  * [40] A. M. Jones.  Data visualization and health econometrics.  Foundations and Trends in Econometrics, 2017.
  * [41] K. Kahn and N. Winters.  Constructionism and ai: A history and possible futures.  British Journal of Educational Technology, 52(3):1130–1142, 2021\.
  * [42] E. Kerzner, S. Goodwin, J. Dykes, S. Jones, and M. Meyer.  A framework for creative visualization-opportunities workshops.  IEEE Transactions on Visualization and Computer Graphics, 25(1):748–758, 2019. doi: 10 . 1109/TVCG . 2018 . 2865241
  * [43] S. Knudsen, J. Vermeulen, D. Kosminsky, J. Walny, M. West, C. Frisson, B. Adriel Aseniero, L. MacDonald Vermeulen, C. Perin, L. Quach, P. Buk, K. Tabuli, S. Chopra, W. Willett, and S. Carpendale.  Democratizing open energy data for public discourse using visualization.  In Extended Abstracts of the 2018 CHI Conference on Human Factors in Computing Systems, CHI EA ’18, p. 1–4. Association for Computing Machinery, New York, NY, USA, 2018. doi: 10 . 1145/3170427 . 3186539
  * [44] J. Kokina, D. Pachamanova, and A. Corbett.  The role of data visualization and analytics in performance management: Guiding entrepreneurial growth decisions.  Journal of Accounting Education, 38:50–62, 2017.
  * [45] C. Lauer and S. O’Brien.  How people are influenced by deceptive tactics in everyday charts and graphs.  IEEE Transactions on Professional Communication, 63(4):327–340, 2020. doi: 10 . 1109/TPC . 2020 . 3032053
  * [46] C. Lee, T. Yang, G. Inchoco, G. M. Jones, and A. Satyanarayan.  Viral Visualizations: How Coronavirus Skeptics Use Orthodox Data Practices to Promote Unorthodox Science Online.  In ACM Human Factors in Computing Systems (CHI), 2021. doi: 10 . 1145/3411764 . 3445211
  * [47] S. Lee, S.-H. Kim, and B. C. Kwon.  Vlat: Development of a visualization literacy assessment test.  IEEE Transactions on Visualization and Computer Graphics, 23(1):551–560, 2017. doi: 10 . 1109/TVCG . 2016 . 2598920
  * [48] V. R. Lee and V. Delaney.  Identifying the content, lesson structure, and data use within pre-collegiate data science curricula.  Journal of Science Education and Technology, 31(1):81–98, Feb 2022\. doi: 10 . 1007/s10956-021-09932-1
  * [49] L. Liu, J. Guo, C. Zhang, Z. Wang, P. Zhu, T. Fang, J. Wang, C. Yao, and F. Ying.  Electropaper: Design and fabrication of paper-based electronic interfaces for the water environment.  Electronics, 10(5), 2021. doi: 10 . 3390/electronics10050604
  * [50] S. Mareschal and S. Delaney.  Using focus group discussions with children and adolescents.  Terre des Hommes, 2020.
  * [51] G. G. Méndez, U. Hinrichs, and M. A. Nacenta.  Bottom-up vs. top-down: Trade-offs in efficiency, understanding, freedom and creativity with infovis tools.  In Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems, CHI ’17, p. 841–852. Association for Computing Machinery, New York, NY, USA, 2017. doi: 10 . 1145/3025453 . 3025942
  * [52] S. A. Murphy.  Data visualization and rapid analytics: Applying tableau desktop to support library decision-making.  Journal of Web Librarianship, 7(4):465–476, 2013.
  * [53] D. Oblinger.  The next generation of educational engagement.  Journal of interactive media in education, 2004(1), 2004.
  * [54] D. Oblinger.  Leading the transition from classrooms to learning spaces.  Educause quarterly, 1(7-12), 2005.
  * [55] S. Papert.  The children’s machine: Rethinking school in the age of the computer.  Basic Books, Inc., 1993.
  * [56] S. A. Papert.  Mindstorms: Children, computers, and powerful ideas.  Basic books, 2020.
  * [57] W. C. Payne, Y. Bergner, M. E. West, C. Charp, R. B. B. Shapiro, D. A. Szafir, E. V. Taylor, and K. DesPortes.  Danceon: Culturally responsive creative computing.  In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems, CHI ’21. Association for Computing Machinery, New York, NY, USA, 2021. doi: 10 . 1145/3411764 . 3445149
  * [58] C. Perin.  What students learn with personal data physicalization.  IEEE Computer Graphics and Applications, 41(6):48–58, 2021. doi: 10 . 1109/MCG . 2021 . 3115417
  * [59] S. PUNCH.  Research with children: The same or different from research with adults?  Childhood, 9(3):321–341, 2002. doi: 10 . 1177/0907568202009003005
  * [60] M. Resnick, R. Berg, and M. Eisenberg.  Beyond black boxes: Bringing transparency and aesthetics back to scientific investigation.  The Journal of the Learning Sciences, 9(1):7–30, 2000.
  * [61] M. Resnick, B. Myers, K. Nakakoji, B. Shneiderman, R. Pausch, T. Selker, and M. Eisenberg.  Design Principles for Tools to Support Creative Thinking.  1 2005. doi: 10 . 1184/R1/6621917 . v1
  * [62] F. J. Romero-Ramirez, R. Muñoz-Salinas, and R. Medina-Carnicer.  Speeded up detection of squared fiducial markers.  Image and Vision Computing, 76:38–47, 2018. doi: 10 . 1016/j . imavis . 2018 . 05 . 004
  * [63] J. Sarama.  Technology in early childhood mathematics: Building blocks as an innovative technology-based curriculum.  Engaging young children in mathematics: Standards for early childhood mathematics education, pp. 361–375, 2004.
  * [64] J. Sarama and D. H. Clements.  Building blocks for early childhood mathematics.  Early Childhood Research Quarterly, 19(1):181–189, 2004.
  * [65] J. Schwabish.  Teaching data visualization to kids.  In VisActivities: IEEE VIS Workshop on Data Vis Activities at IEEE VIS 2020, 2020.
  * [66] B. Shneiderman.  Data visualization’s breakthrough moment in the covid-19 crisis.  Medium, 2020.
  * [67] J. Silver, E. Rosenbaum, and D. Shaw.  Makey makey: Improvising tangible and nature-based user interfaces.  In Proceedings of the Sixth International Conference on Tangible, Embedded and Embodied Interaction, TEI ’12, p. 367–370. Association for Computing Machinery, New York, NY, USA, 2012. doi: 10 . 1145/2148131 . 2148219
  * [68] A. Thudt, U. Hinrichs, S. Huron, and S. Carpendale.  Self-reflection and personal physicalization construction.  In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems, CHI ’18, p. 1–13. Association for Computing Machinery, New York, NY, USA, 2018. doi: 10 . 1145/3173574 . 3173728
  * [69] M. Trajkova, A. Alhakamy, F. Cafaro, S. Vedak, R. Mallappa, and S. R. Kankara.  Exploring casual covid-19 data visualizations on twitter: Topics and challenges.  Informatics, 7(3), 2020. doi: 10 . 3390/informatics7030035
  * [70] A. Vande Moere and S. Patel.  The physical visualization of information: Designing data sculptures in an educational context.  In M. L. Huang, Q. V. Nguyen, and K. Zhang, eds., Visual Information Communication, pp. 1–23. Springer US, Boston, MA, 2010.
  * [71] B. N. Verdine, R. M. Golinkoff, K. Hirsh-Pasek, N. S. Newcombe, A. T. Filipowicz, and A. Chang.  Deconstructing building blocks: Preschoolers’ spatial assembly performance relates to early mathematical skills.  Child development, 85(3):1062–1076, 2014.
  * [72] P. Verhaert, G. A. Panagiotidou, and A. Vande Moere.  ‘datablokken’: Stimulating critical data literacy of children through the use of a life-size data physicalisation game.  In VisActivities: IEEE VIS Workshop on Data Vis Activities at IEEE VIS 2021, 2021.
  * [73] R. P. Weber.  Basic content analysis, vol. 49.  Sage, 1990.
  * [74] C. Wickham, C. Girvan, B. Tangney, A. Sipitakiat, and N. Tutiyaphuengprasert.  Constructionism and microworlds as part of a 21st century learning activity to impact student engagement and confidence in physics.  Constructionism, pp. 34–43, 2016.
  * [75] M. Wilson.  Six views of embodied cognition.  Psychonomic Bulletin & Review, 9(4):625–636, Dec 2002. doi: 10 . 3758/BF03196322
  * [76] C. Zheng, P. Gyory, and E. Y.-L. Do.  Tangible Interfaces with Printed Paper Markers, p. 909–923.  Association for Computing Machinery, New York, NY, USA, 2020.
  * [77] A. Zinovyev.  Data visualization in political and social sciences.  arXiv preprint arXiv:1008.1188, 2010.


# Orbital Peierls mechanism of the formation of molecular clusters in
Na2Ti3Cl8 and in similar systems

Daniel I. Khomskii  II. Physikalisches Institut, Universit$\ddot{a}$t zu
K$\ddot{o}$ln, Z$\ddot{u}$lpicher Stra$\ss$e 77, D-50937 K$\ddot{o}$ln,
Germany  Takashi Mizokawa  Department of Applied Physics, Waseda University,
Tokyo 169-8555, Japan  Sergey V. Streltsov∗ Institute of Metal Physics, S.
Kovalevskoy St. 18, 620990 Ekaterinburg, Russia  Ural Federal University, Mira
St. 19, 620002 Ekaterinburg, Russia<EMAIL_ADDRESS>
(August 27, 2024)

Materials containing well-defined clusters of several atoms - “molecules in
solids”, are very interesting systems. Sometimes these clusters - dimers,
trimers or larger clusters - exist just by the very structure of a material.
In other cases they are formed spontaneously, as a result of structural phase
transitions at decreasing temperature. They display very diverse properties,
and their origin is an important problem.

Recently an interesting class of such systems attracted big attention. These
are layered systems with basically kagomé lattice, but with inequivalent,
small and large, triangles – “breathing kagomé”. This is, specifically,
Na2Ti3Cl8 Kelly et al. (2019), but also materials of the class of
(Zn,Fe,Ni)2Mo3O8, which contain (breathing) kagomé layers of Mo ions. Recently
the structural transition in Na2Ti3Cl8 at 200 K with the formation of
triangular Ti3 clustersKelly et al. (2019) was studied by ab-initio
calculations in Paul et al. (2020). It was concluded on the basis of the DFT+U
results that the usual band effects are not sufficient to explain this
transition. The authors then invoked magnetic mechanisms, including
complicated factors such as biquadratic and ring exchanges. In this short
paper (the still shorter version was recently published in PRL as a
CommentKhomskii et al. (2021)) we want to point out that this overcomplicated
explanation is unnecessary, and that the formation of trimers, or breathing
kagomé lattice in Na2Ti3Cl8, and in many other systems, has very simple,
almost trivial explanation, which exactly follows the line of reasoning first
presented in Khomskii and Mizokawa (2005).

The simple picture is that of the orbital-driven Peierls distortion. Ti2+ ions
have an electronic configuration $d^{2}$. In the kagomé structure of Na2Ti3Cl8
the direct $d-d$ hopping between $t_{2g}$-orbitals would lead to the formation
of three one-dimensional ($1D$) bands in three directions, formed by $xy$,
$xz$, and $yz$-orbitals, Fig. 1. Each Ti atom lies on the intersection of two
such chains and would participate in two bands. With two electrons per Ti,
each $1D$ band is half-filled similar to VO2, which would immediately lead to
Peierls dimerization. And the breathing kagomé structure of Na2Ti3Cl8
corresponds exactly to such dimerization in all three directions! (Fig. 1).

Exactly the same mechanism can explain the formation of similar trimers in
other family of materials - in Zn2Mo3O8, Fe2Mo3O8 etc, which contain basic
kagomé layer of Mo ions with the same electronic configuration, Mo4+ ($d^{2}$)
Khomskii and Streltsov (2021). In these systems we also have the same
dimerization in all three directions, leading to the formation of Mo3
triangular clusters. It is interesting that the same basic superstructure
survives even at some doping, e.g. for LiZn2Mo3O8Sheckelton et al. (2012);
Mourigal et al. (2014).

Figure 1:  Top: Formation of Ti3 trimers in kagomé lattice of Na2Ti3Cl8 due to
dimerization of chains. Bottom: formation of trimers in triangular lattice of
LiVO2 due to the same mechanism: trimerization in 1D chains. Inset shows 3
$t_{2g}$ orbitals (active lobes are colored) direct overlap of which results
in formation of $1D$ bands.

This “orbital-driven” Peierls mechanism can also explain the origin of
clusters observed in many other transition metal compounds including not only
the formation of dimers (with the resulting beautiful total superstructures)
in spinels CuIr2S4 and MgTi2O4Radaelli et al. (2002); Schmidt et al. (2004);
Khomskii and Mizokawa (2005), but also, e.g., trimerization in another $d^{2}$
material, LiVO2 Pen et al. (1997). This system has not kagome, like Na2Ti3Cl8,
but triangular lattice. Consequently in this case each V lying on the
intersection of three $1D$ chains, participates not in two but in three $1D$
bands, which, with $d^{2}$ occupation of V3+, would give these bands 1/3
filled each. Thus one might expect in this case not Peierls dimerization, but
trimerization. And indeed the structure of the low-temperature phase of LiVO2
shown in Fig. 1 corresponds exactly to that - it is a trimerization in all
three directions. In Pen et al. (1997) this superstructure was explained
proceeding from more localised point of view, as a result of a particular
orbital ordering. But it is interesting that exactly the same conclusion can
be reached also proceeding from the band picture. Also experimentally it was
shown Katayama et al. (2009) that the same trimerization occurs also in LiVS2,
in which the corresponding transition is really a metal-insulator transition.

There are also other examples of nontrivial superstructures in solids which
can be explained in a similar way. One of such is the ‘diamond necklace”
lattice in ReS2 and ReSe2Wilson and Yoffe (1969), see Fig. 2. We have a
triangular lattice of Re ions and again common edge sharing of ReS6 octahedra.
There are three $d$ electrons at each Re and three half-filled “one-
dimensional” bands, each of them half-filled, which may provide dimerization
in all three directions. This is exactly what is observed experimentally:
going along each direction (Re-Re chains) we see dimerization.

Figure 2:  “Diamond necklace” lattice of ReS2. Re (S) ions are shown by red
(blue) balls. Short Re-Re bonds are thick solid blue lines, while long bonds
are dashed. Note dimerization in all three straight Re-Re chains.

Of course, not all superstructures observed in transition metal compounds can
be explained by this picture, especially those with transition metals that are
in prismatic coordination rather than in octahedra. In some cases one should
rather use more “chemical” description, operating with valence bonds formed by
available electrons at different orbitals. This is e.g. the situation in
AlV2O4, in which such picture gives a natural explanation of the formation of
trimers made of V3+($d^{2}$) and of V tetrahedra made by V2+($d^{3}$)Browne et
al. (2017); Komleva et al. (2021). But it is gratifying that the simple
Peierls picture described above gives a good description of the formation of
molecular clusters in so many cases.

Interestingly the Peierls-like instability in Na2Ti3Cl8 clearly manifests
itself already in the calculations of Paul et al. (2020). The authors of Paul
et al. (2020) focus on the DFT+U results, which do not give lattice
instability for the undistorted phase. However, one may notice that the phonon
frequencies get imaginary, i.e. the lattice becomes unstable, in simple
nonmagnetic DFT calculations of Paul et al. (2020), see Tab.I in SIPaul et al.
(2020). Thus, already on the band level the systems tends to distort. This is
nothing else but a manifestation of the Peierls instability. This is a typical
situation that including correlations in a simplified way via DFT+U
approachLiechtenstein et al. (1995) we often worsen the description of a
system tending to form clusters. The famous example is VO2, for which DFT
reproduces dimerized phase observed experimentallyWentzcovitch et al. (1994),
while DFT+U stabilizes equally spaced V ionsWickramaratne et al. (2019). The
tendency to form molecular orbitals, which is well described by a kinetic
energy in DFT and which is of order of hopping parameter $t$, is substituted
in DFT+U by a strong exchange coupling, which is typically much weaker $\sim
t^{2}/U$. DFT+U (in contrast to cluster-DMFTBiroli and Kotliar (2002)) was
constructed in a way to localize electrons not on molecular bonds, but on
atomic orbitals and therefore it is not surprising that for Na2Ti3Cl8 this
method suppressed tendency to formation of trimers due to dimerization in Ti
chains. Kelly et al. (2019).

Summarising, we see that in quite a few cases the Peierls physics, with
orbital structure taken into account, can successfully explain the formation
of “molecular clusters” in correlated solids, and this simple physics can be
used to rationalise the results of many experiments on many systems.

## Acknowledgements

We acknowledge German Research Foundation (277146847-CRC-1238), JSPS
(19H01853), Russian Foundation for Basic Researches (RFBR 20-32-70019), the
Russian Ministry of Education and Science (program “Quantum”
AAAA-A18-118020190095-4) for financial support.

## References

  * Kelly et al. (2019) Z. A. Kelly, T. T. Tran, and T. M. McQueen, Inorg. Chem. 58, 11941 (2019), ISSN 1520510X.
  * Paul et al. (2020) A. Paul, C. M. Chung, T. Birol, and H. J. Changlani, Phys. Rev. Lett. 124, 167203 (2020), ISSN 10797114, URL https://doi.org/10.1103/PhysRevLett.124.167203.
  * Khomskii et al. (2021) D. I. Khomskii, T. Mizokawa, and S. V. Streltsov, Phys. Rev. Lett. 127, 49701 (2021), ISSN 1079-7114, URL https://doi.org/10.1103/PhysRevLett.127.049701.
  * Khomskii and Mizokawa (2005) D. I. Khomskii and T. Mizokawa, Phys. Rev. Lett. 94, 156402 (2005), ISSN 0031-9007, URL http://link.aps.org/doi/10.1103/PhysRevLett.94.156402.
  * Khomskii and Streltsov (2021) D. I. Khomskii and S. V. Streltsov, Chem. Rev. 121, 2992 (2021).
  * Sheckelton et al. (2012) J. P. Sheckelton, J. R. Neilson, D. G. Soltan, and T. M. Mcqueen, Nat. Mater. 11, 493 (2012), ISSN 1476-1122, URL http://dx.doi.org/10.1038/nmat3329.
  * Mourigal et al. (2014) M. Mourigal, W. T. Fuhrman, J. P. Sheckelton, A. Wartelle, D. L. Abernathy, T. M. Mcqueen, and C. L. Broholm, Phys. Rev. Lett. 112, 027202 (2014).
  * Radaelli et al. (2002) P. G. Radaelli, Y. Horibe, M. J. Gutmann, H. Ishibashi, C. H. Chen, R. M. Ibberson, Y. Koyama, Y.-S. Hor, V. Kiryukhin, and S.-W. Cheong, Nature 416, 155 (2002), ISSN 0028-0836, URL http://dx.doi.org/10.1038/416155ahttp://www.nature.com/nature/journal/v416/n6877/suppinfo/416155a{_}S1.html.
  * Schmidt et al. (2004) M. Schmidt, W. Ratcliff, P. G. Radaelli, K. Refson, N. M. Harrison, and S. W. Cheong, Phys. Rev. Lett. 92, 056402 (2004), URL https://link.aps.org/doi/10.1103/PhysRevLett.92.056402.
  * Pen et al. (1997) H. F. Pen, J. van den Brink, D. I. Khomskii, and G. A. Sawatzky, Phys. Rev. Lett. 78, 1323 (1997), ISSN 0031-9007, URL http://link.aps.org/doi/10.1103/PhysRevLett.78.1323.
  * Katayama et al. (2009) N. Katayama, M. Uchida, D. Hashizume, S. Niitaka, J. Matsuno, D. Matsumura, Y. Nishihata, J. Mizuki, N. Takeshita, A. Gauzzi, et al., Phys. Rev. Lett. 103, 146405 (2009), ISSN 00319007, eprint 0908.1244.
  * Wilson and Yoffe (1969) J. A. Wilson and A. D. Yoffe, Adv. Phys. 18, 193 (1969).
  * Browne et al. (2017) A. J. Browne, S. A. J. Kimber, and J. P. Attfield, Phys. Rev. Mat. 1, 052003 (2017).
  * Komleva et al. (2021) E. Komleva, D. I. Khomskii, and S. V. Streltsov, unpublished (2021).
  * Liechtenstein et al. (1995) A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, 5467 (1995), URL http://prola.aps.org/abstract/PRB/v52/i8/pR5467{_}1.
  * Wentzcovitch et al. (1994) R. M. Wentzcovitch, W. W. Schulz, and P. B. Allen, Phys. Rev. Lett. 72, 3389 (1994).
  * Wickramaratne et al. (2019) D. Wickramaratne, N. Bernstein, and I. I. Mazin, Phys. Rev. B 99, 214103 (2019).
  * Biroli and Kotliar (2002) G. Biroli and G. Kotliar, Phys. Rev. B 65, 155112 (2002), ISSN 0163-1829, URL http://link.aps.org/doi/10.1103/PhysRevB.65.155112.