.

# Representation of convex geometries of convex dimension 3 by spheres

Kira Adaricheva  ,  Arav Agarwal  and  Na’ama Nevo

###### Abstract.

A convex geometry is a closure system satisfying the anti-exchange property.
This paper, following the work of Adaricheva and Bolat (2019) and the Polymath
REU (2020), continues to investigate representations of convex geometries with
small convex dimension by convex shapes on the plane and in spaces of higher
dimension. In particular, we answer in the negative the question raised by
Polymath REU (2020): whether every convex geometry of $cdim=3$ is
representable by the circles on the plane. We show there are geometries of
$cdim=3$ that cannot be represented by spheres in any $\mathbb{R}^{k}$, and
this connects to posets not representable by spheres from the paper of
Felsner, Fishburn and Trotter (1999). On the positive side, we use the result
of Kincses (2015) to show that every finite poset is an ellipsoid order.

###### Key words and phrases:

Convex geometry, anti-exchange closure operator, convex hull operator, convex
dimension, poset dimension, sphere order, ellipsoid order

###### 2020 Mathematics Subject Classification:

06A07, 06A15, 52A37, 52C05, 52C07

The work on this paper was initiated while the second and third authors
attended the New York Discrete Mathematics REU in the summer of 2022 and were
mentored by the first author. The REU was supported by NSF grant # 2051026 and
led by PI Adam Sheffer and Co-PI Pablo Soberon Bravo (both CUNY Baruch
College). We appreciate the welcoming atmosphere of the Mathematics Department
at Baruch College that generously hosted the REU, as well as the support of
other mentors and student participants.

##  1\. Introduction

This paper addresses a question raised in the first paper of the convex
geometries team [9] developed during the Polymath REU-2020 project, see more
details about the project in [6].

######  Question 1.1.

Is it possible to represent all finite convex geometries of convex dimension 3
by circles in the plane using the convex hull operator acting on circles?

Convex geometries are closure systems whose closure operator satisfies the
anti-exchange property. _Convex dimension_ ($cdim$) refers to a certain
parameter of such closure systems defined by Edelman and Jamison in [12].

The representation of convex geometries by the circles on the plane was first
introduced by Czédli [10], and the first convex geometry which cannot be
represented that way was found in Adaricheva and Bolat [1]. The obstruction to
such representation came in the form of the Weak Carousel Property that failed
for some convex geometry on a 5-element set and whose convex dimension = 6.

In [9] it was discovered that among 672 non-isomorphic geometries on a
5-element set there are seven that cannot be represented due to the Triangle
Property. In particular, some among these seven had $cdim=4,5$. At the same
time, all geometries of $cdim=3$ on a 5-element set had circular
representations on the plane.

This left open Question 1.1 regarding the representation of arbitrary convex
geometries of $cdim=3$: either by circles on the plane, or by the spheres in
spaces $\mathbb{R}^{k}$ of dimension $k>2$. In this paper, we answer the
question in the negative by reducing this problem to another problem that
brought considerable attention of many combinatorialists and order specialists
in the 1990s.

In 1989, Brightwell and Winkler [8] asked whether every finite poset is a
sphere order and suggested that the answer was negative. As usual, by a
_poset_ we assume a pair $(X,\leq)$, where $X$ is a set and $\leq$ is a
partial order: a reflexive, anti-symmetric and transitive binary relation on
$X$. A poset is a _sphere order_ , when every $x\in X$ can be represented by a
sphere $F(x)$ in $\mathbb{R}^{k}$ so that $x\leq y$ iff $F(x)\subseteq F(y)$.

Indeed, the question was solved in the negative by Felsner, Fishburn and
Trotter [14] by presenting a 3-dimensional poset that cannot be a sphere order
in any space $\mathbb{R}^{k}$. Note that _poset dimension_ is defined as the
smallest number $t$ such that the poset embeds into a product (with component-
wise ordering) of $t$ chains. Equivalently, the partial order is recovered as
an intersection of at most $t$ of its linear extensions. It is well-known that
2-dimensional posets are representable by closed intervals in $\mathbb{R}$,
i.e., by spheres in one-dimensional space.

We observe that the closure space of any convex geometry can be thought of as
a poset of closed sets of its associated closure operator. However, poset
dimension and convex dimension of a convex geometry are fairly different
parameters. Recently, the relation was studied in Knauer and Trotter [19], who
presented a series of convex geometries which have poset dimension = 3, while
their convex dimension grows unboundedly.

In the Propositions of Section 3, we show that the poset from [14] which is
not a sphere order can be made isomorphic to the poset of join-irreducible
elements of some convex geometry of $cdim=3$. In Proposition 2.18 we connect
the elements of the base set of the convex geometry with join-irreducible
elements of its closure space, which allows to conclude that the convex
geometry will not be represented by the spheres in any $\mathbb{R}^{k}$,
proving our main result in Theorem 3.1.

Note that the result of [14] does not identify the smallest size of a poset
which is not a sphere order. This suggests a question about _the smallest size
$\rho$ of (the base set of) a convex geometry of $cdim=3$ that is not
representable by spheres in any_ $\mathbb{R}^{n}$.

We propose that at least $\rho>6$:

######  Hypothesis 1.2.

Every convex geometry of $cdim=3$ on a 6-element set is representable by
circles on the plane.

Note that according to the Online Encyclopedia of Integer Sequences
(OEIS.org), there are almost 200,000 non-isomorphic convex geometries
(equivalently, _antimatroids_) on a 6-element set, with the exact number being
given in sequence A224913.

Much less was known about _ellipsoid orders_ , i.e., finite posets that are
representable by ellipsoids in some $\mathbb{R}^{k}$. We use a result of
Kincses [18] regarding the representation of convex geometries by ellipsoids
using the convex hull operator for ellipsoids to conclude that every finite
poset is an ellipsoid order.

##  2\. Terminology and Known Results

A convex geometry is a special case of a closure system. It can be defined
through a closure operator, or by means of a closure space.

######  Definition 2.1.

Let $X$ be a set. A mapping $\varphi\colon 2^{X}\to 2^{X}$ is called a
_closure operator_ , if for all $Y,Z\in 2^{X}$:

  1. (1)

$Y\subseteq\varphi(Y)$,

  2. (2)

if $Y\subseteq Z$ then $\varphi(Y)\subseteq\varphi(Z)$,

  3. (3)

$\varphi(\varphi(Y))=\varphi(Y)$.

A subset $Y\subseteq X$ is _closed_ if $\varphi(Y)=Y$. The pair $(X,\varphi)$,
where $\varphi$ is a closure operator, is called a _closure system_.

######  Definition 2.2.

Given any (finite) set $X$, a closure space on $X$ is a family $\mathcal{F}$
of subsets of $X$ which satisfies two properties:

  1. (1)

$X\in\mathcal{F}$,

  2. (2)

if $Y,Z\in\mathcal{F}$ then $Y\cap Z\in\mathcal{F}$.

Closure systems are dual to closure spaces in the following sense.

If $(X,\varphi)$ is a closure system, one can define a family of closed sets
$\mathcal{F}_{\varphi}:=\\{Y\subseteq X:\varphi(Y)=Y\\}$. Then
$\mathcal{F}_{\varphi}$ is a closure space.
If $\mathcal{F}$ is a closure space, then define
$\varphi_{\mathcal{F}}:2^{X}\rightarrow 2^{X}$ in the following manner: for
all $Y\subseteq X$, let
$\varphi_{\mathcal{F}}(Y):=\bigcap\\{Z\in\mathcal{F}:Y\subseteq Z\\}$. Then
$(X,\varphi)$ is a closure system.

######  Definition 2.3.

A closure system $(X,\varphi)$ is called a _convex geometry_ if

  1. (1)

$\varphi(\emptyset)=\emptyset$,

  2. (2)

for any closed set $Y\subseteq X$ and any distinct points $x,y\in X\setminus
Y,$ if $x\in\varphi(Y\cup\\{y\\})$ then $y\not\in\varphi(Y\cup\\{x\\})$.

Property (2) above is called the _Anti-Exchange Property_.

We can use duality between closure operators and closure spaces to provide
another definition of a convex geometry.

######  Definition 2.4.

A closure system $(X,\varphi)$ is a convex geometry iff the corresponding
closure space $\mathcal{F}_{\varphi}$ satisfies the following two properties:

  1. (1)

$\emptyset\in\mathcal{F}_{\varphi}$,

  2. (2)

if $Y\in\mathcal{F}_{\varphi}$ and $Y\neq X$, then there exists $a\in
X\setminus Y$ such that $Y\cup\\{a\\}\in\mathcal{F}_{\varphi}$.

We now need to discuss an important parameter of convex geometries known as
_convex dimension_ , first introduced in [12].

######  Definition 2.5.

A closure space $\mathcal{F}$ is called monotone if sets of $\mathcal{F}$ form
a chain under inclusion.
(See Definition 2.12 for a chain or linear order.)

######  Remark 2.6.

Note that, due to Definition 2.4, a monotone closure space $\mathcal{F}$ is a
convex geometry iff $\mathcal{F}$ has exactly $|X|+1$ subsets of $X$:
$\emptyset\subset\\{x_{1}\\}\subset\\{x_{1},x_{2}\\}\subset\\{x_{1},x_{2},x_{3}\\}\subset\dots\\{x_{1},x_{2},\dots,x_{n-1}\\}\subset
X=\\{x_{1},\dots x_{n-1},x_{n}\\}$. Therefore, every monotone convex geometry,
or _linear geometry_ , on $X$ is uniquely associated with some linear order on
$X$: $x_{1}<x_{2}<\dots<x_{n-1}<x_{n}$.

######  Definition 2.7.

Given two closure spaces $\mathcal{F},\mathcal{K}$ on the same base set $X$,
their join $\mathcal{F}\vee\mathcal{K}$ is defined to be the smallest closure
space $\mathcal{T}$ such that $\mathcal{F},\mathcal{K}\subseteq\mathcal{T}$.

It can be easily verified that $\mathcal{F}\vee\mathcal{K}=\\{F\cap
K:F\in\mathcal{F}\text{ and }K\in\mathcal{K}\\}$.

A known result in [12] about joins of convex geometries follows.

######  Theorem 2.8.

Let $X$ be a finite set.

  * (1)

If closure spaces $\mathcal{F},\mathcal{K}$ on $X$ are convex geometries, then
$\mathcal{F}\vee\mathcal{K}$ is a convex geometry as well.

  * (2)

Closure space $\mathcal{F}$ of any convex geometry on set $X$ can be expressed
as the join of some collection of monotone convex geometries on the same base
set.

This motivates the following definition.

######  Definition 2.9.

[12] The _convex dimension_ of a convex geometry $\mathbf{G}=(X,\varphi)$ is
the minimal number $k$ such that closure space $\mathcal{F}_{\varphi}$ can be
expressed as the join of $k$ monotone convex geometries on set $X$.

To compute the convex dimension of a convex geometry, we can examine maximal
cardinality antichains of meet-irreducibles in its closure space
$\mathcal{F}_{\varphi}$, as discussed in Edelman and Saks [13]. Thus,
informally, the $cdim$ parameter of a convex geometry represents the diversity
of closed sets with respect to the closure operator $\varphi$.

A particular example of a closure operator on a set is the _convex hull
operator_ , where the base set $X$ is a set of points in Euclidean space
$\mathbb{R}^{k}$.

######  Definition 2.10.

  * (1)

A set $S$ in $\mathbb{R}^{k}$ is called _convex_ if for any two points $p,q\in
S$, the line segment connecting $p$ and $q$ is also contained in $S$.

  * (2)

Given a set $S$ of points in $\mathbb{R}^{k}$, the _convex hull_ of $S$,
denoted $\operatorname{CH}(S)$, is the intersection of all convex sets in
$\mathbb{R}^{k}$ which contain $S$. That is, it is the smallest convex set
containing $S$.

Comparing with Definition 2.1, we see that $\operatorname{CH}$ is a closure
operator acting on $\mathbb{R}^{k}$.

Finally, we recall the definition of the convex hull operator for spheres
introduced in [10]. If $x$ is a sphere in $\mathbb{R}^{k}$, then by
$\tilde{x}$ we denote the set of points belonging to $x$. It is allowed that a
sphere has a radius $0$, in which case it is a point.

######  Definition 2.11.

Let $X$ be a finite set of spheres in $\mathbb{R}^{k}$. Define the convex hull
operator for spheres, $\operatorname{ch}_{s}:2^{X}\rightarrow 2^{X}$, as
follows:

$\operatorname{ch}_{s}(Y)=\\{x\in
X:\tilde{x}\subseteq\operatorname{CH}\bigg{(}\bigcup_{y\in
Y}\tilde{y}\bigg{)}\\},$

for any $Y\in 2^{X}$.

See the figure below for an illustration of the $\operatorname{ch}_{s}$
operator in $\mathbb{R}^{2}$. Observe that
$\operatorname{ch}_{s}(\\{a,b,c\\})=\\{a,b,c,d,e\\}$ and
$\operatorname{ch}_{s}(\\{a,c\\})=\\{a,c,e\\}$.

(a) Circles $d,e$ are in the convex hull of $a,b,c$

(b) Circle $e$ is in convex hull of $a,c$ but $b,d$ are not

Figure 1. Convex hull operator for spheres

It was established in [10] that the closure operator $\operatorname{ch}_{s}$
satisfies the Anti-exchange Property. Therefore, the closure system
$(X,\operatorname{ch}_{s})$ on the set of spheres $X$ in $\mathbb{R}^{k}$ is a
convex geometry.

We say that a finite convex geometry $(X,\varphi)$ is _represented by spheres
in $\mathbb{R}^{k}$_, when there are $|X|$ spheres in $\mathbb{R}^{k}$ such
that the action of $\varphi$ is identical to $\operatorname{ch}_{s}$. We can
similarly define convex hull operators in $\mathbb{R}^{k}$ on convex shapes
different from spheres, such as ellipsoids.

The survey paper [12] was instrumental to start off the study of finite convex
geometries, which are also the dual systems to _antimatroids_. The study of
infinite convex geometries was initiated in Adaricheva, Gorbunov and Tumanov
[4]. To see the development of the topic, including infinite convex
geometries, one needs to consult the more recent survey by Adaricheva and
Nation [2].

In [10] the representation of finite convex geometries was proposed by
interpreting elements of base set $X$ as circles in the plane, and closure
operator $\varphi$ as a convex hull operator acting on circles. The main
result of the paper is that all finite convex geometries with _convex
dimension_ at most 2 can be represented by circles on the plane.

In [1] it was found that there is an obstruction for representation of convex
geometries by circles on the plane, which allowed the authors to build an
example of a convex geometry on a 5-element set of $cdim=6$. More obstructions
for representation of geometries by circles were found in two papers written
by subgroups of PolyMath-2020 team [9, 5].

Passing to the terminology on posets, we recall that a set $X$ with a binary
relation $\leq$ is called a partially ordered set (poset) if $\leq$ is
reflexive, anti-symmetric and transitive. An important example of a poset is
the family $\mathcal{F}$ of closed sets of a closure operator with relation
$\subseteq$.

######  Definition 2.12.

A _linear order_ , or _chain_ , is a partial order $(X,\leq)$ where any two
elements of $X$ are comparable, that is either $u\leq v$ or $v\leq u$, for any
$u,v\in X$. We freely use the terms linear order and chain interchangeably.

A relevant parameter of a poset is the _the order dimension_ , also known as
the Dushnik-Miller dimension. Recall that a linear extension $(X,\leq^{*})$ of
poset $(X,\leq)$ is a poset with $\leq\subseteq\leq^{*}$, where $\leq^{*}$ is
a linear order.

######  Definition 2.13.

The order dimension of a poset $(X,\leq)$ is the least integer $t$ for which
we have a family of $t$ linear extensions $\leq_{1},\ldots,\leq_{t}$ of $\leq$
such that $\leq=\bigcap_{i=1}^{t}\leq_{i}$.

Equivalently, order dimension is the minimal number of chains such that the
poset embeds into their direct product. For a comprehensive monograph on the
topic, see Trotter [17].

######  Definition 2.14.

Given a poset $\mathbf{P}=(X,\leq)$, a function $F$ which assigns to each
$x\in X$ a set $F(x)$ is called an inclusion representation of $\mathbf{P}$
when $x\leq y$ if and only if $F(x)\subseteq F(y)$.

In other words, an inclusion representation is the mapping which realizes
$\mathbf{P}$ as an inclusion order of some objects.

######  Example 2.15.

Any poset has an inclusion representation. Given $\mathbf{P}=(X,\leq)$, we can
associate to each element $x\in X$ the set

$F(x)=\downarrow x=\\{y\in X\,|\,y\leq x\\}.$

This is the _down-set_ generated by the element $x$. A down-set in general is
a set $S\subseteq X$ such that if $s\in S$ and $x\leq s$, then $x\in S$.

The transitivity of $\leq$ tells us that

$\downarrow a\subseteq\downarrow b\text{ iff }a\leq b.$

Indeed, the idea of the above example is fundamental, and key to Birkhoff’s
representation theorem, often referred to as the fundamental theorem of finite
distributive lattices.

######  Theorem 2.16.

[7] The lattice of down-sets of a poset is distributive. Any finite
distributive lattice $L$ is isomorphic to the lattice of down sets of the
partial order of the join-irreducible elements of $L$.

In the theorem, the join-irreducible elements of the lattice of all down-sets
of the poset are precisely the down-sets generated by singletons as
illustrated in Example 2.15. Note that mapping $\varphi:Y\mapsto\downarrow Y$,
that maps any subset $Y$ of partially ordered set $(X,\leq)$ into smallest
down-set $\downarrow Y$ containing $Y$ is a closure operator on $X$ satisfying
the Anti-Exchange Property. Thus, the lattice in Theorem 2.16 is also the
lattice of closed sets of this closure operator, and finite distributive
lattices are convex geometries.

Similar to the representation of convex geometries by convex shapes, we have
the concept of representation of posets by spheres.

######  Definition 2.17.

A poset $\mathbf{P}=(X,\leq)$ is a sphere order if there exists $k\geq 1$ such
that $\mathbf{P}$ has an inclusion representation using spheres in
$\mathbb{R}^{k}$. That is, $F(x)$ for any $x\in X$ is required to be a sphere
in $\mathbb{R}^{k}$.

Before discussing results concerning representation of posets as sphere
orders, we establish the key connection between representation of convex
geometries by spheres and posets as sphere orders.

######  Proposition 2.18.

Suppose a convex geometry $(X,\varphi)$ is represented by spheres in some
$\mathbb{R}^{k}$. Then, the poset of join-irreducible elements of the
associated closure space $\mathcal{F}_{\varphi}$ is a sphere order in
$\mathbb{R}^{k}$.

###### Proof.

By the definition of representation of convex geometries by spheres, each
element $x\in X$ is given by some sphere $F(x)$ in $\mathbb{R}^{k}$ so that
$\varphi$ acts on spheres as the $\operatorname{ch}_{s}$ operator. It is well-
known that in _standard_ closure systems $(X,\varphi)$ there is a one-to-one
correspondence between elements of $X$ and
$\mathrm{Ji}(\mathcal{F}_{\varphi})$ (see, for example, [3, Lemma 4-2.8]),
where $\mathrm{Ji}(\mathcal{F}_{\varphi})$ denotes the set of join-irreducible
elements of the closure space $\mathcal{F}_{\varphi}$ associated to $X$.
Convex geometries are standard closure systems, so this correspondence works
in convex geometries. Specifically, the one-to-one correspondence between $X$
and $\mathrm{Ji}(\mathcal{F}_{\varphi})$ is given by
$x\mapsto\varphi(\\{x\\})$. Ordering by set inclusion, we obtain the poset
$(\mathrm{Ji}(\mathcal{F}_{\varphi}),\leq)=(\\{\varphi(\\{u\\}):u\in
X\\},\leq)$, and we now show it is a sphere order.

Consider $\varphi(\\{x\\})\in\mathcal{F}_{\varphi}$. Since we represented the
geometry by spheres, in terms of the sphere representation $\varphi(\\{x\\})$
is exactly $\operatorname{ch}_{s}(F(x))$. By definition of
$\operatorname{ch}_{s}$, we obtain

$\operatorname{ch}_{s}(F(x))=\\{F(y)\,|\,F(y)\subseteq F(x),y\in X\\}.$

Indeed, we have arrived at the case of Example 2.15, for if we consider the
inclusion order of the spheres, we observe that

$\operatorname{ch}_{s}(F(x))=\,\downarrow F(x),$

and we have $\downarrow F(u)\subseteq\downarrow F(v)$ iff $F(u)\subseteq F(v)$
for any $u,v\in X$. We can now write for any $u,v\in X$:

$\varphi(\\{u\\})\leq\varphi(\\{v\\})\iff\operatorname{ch}_{s}(F(u))\subseteq\operatorname{ch}_{s}(F(v))\iff\downarrow
F(u)\subseteq\downarrow F(v)\iff F(u)\subseteq F(v).$

Hence, we have shown that $\varphi(\\{u\\})\leq\varphi(\\{v\\})$ iff
$F(u)\subseteq F(v)$. Therefore, the sphere representation of a convex
geometry $(X,\varphi)$ simultaneously provides us the representation of the
poset of join-irreducibles $(\\{\varphi(\\{u\\}):u\in X\\},\leq)$ of the
closure space $\mathcal{F}_{\varphi}$, proving the poset is a sphere order. ∎

As per the record in [14], the question of whether every finite 3-dimensional
poset has an inclusion representation using circles in $\mathbb{R}^{2}$ was
raised by Fishburn and Trotter at the Banff conference of 1984. In Sidney et
al. [21] a finite 4-dimensional poset was found that was not a _circle order_
(i.e., not represented by spheres on the plane), and in Scheinerman and
Wierman [20] it was shown by a Ramsey theoretic argument that the countably
infinite 3-dimensional poset $\mathbb{Z}^{3}$ is not a circle order.

These results prompted the following more general question:

######  Question 2.19.

[8] Is every finite 3-dimensional poset representable as an inclusion order of
spheres in some $k$-dimensional space?

In [14] we find the culmination of the search to answer this question, in the
negative.

With the next simple definition in hand, we can formally state the main
theorem from [14].

######  Definition 2.20.

For positive integers $n,t$, let $\mathbf{n}$ denote the chain

$0<1<\cdots<n-1$

and $\mathbf{n}^{t}$ the cartesian chain product of $t$ copies of
$\mathbf{n}$, so that we obtain the following canonical partial ordering on
$\mathbf{n}^{t}$:

$(a_{1},a_{2},\ldots,a_{t})\leq(b_{1},b_{2},\ldots,b_{t})\text{ iff }a_{i}\leq
b_{i}\,\,\text{for all }i.$

######  Theorem 2.21 (2.1 of [14]).

There exists an integer $n_{0}$ such that if $n\geq n_{0}$, the finite
3-dimensional poset $\mathbf{n}^{3}$ is not a sphere order.

In Section 3, we will use this theorem and Proposition 2.18 to show that not
every convex geometry of $cdim=3$ is representable by spheres.

We note that a convex geometry $(X,\varphi)$ can be thought of as a poset
$(\mathcal{F}_{\varphi},\subseteq)$, which could be measured using order
dimension. The relationship between order dimension and the convex dimension
pertinent to convex geometries was studied in [19]. While in 2-dimensional
geometries the convex dimension is the same and equals 2, the picture is quite
different for $3$-dimensional geometries. In particular, there are convex
geometries $\mathbf{P}_{n}$ with the $dim(\mathbf{P}_{n})=3$ and
$cdim(\mathbf{P}_{n})=n+1$.

##  3\. Existence of Convex Geometry with cdim=3 Not Representable by Spheres

Our goal in this section is to prove the following result.

######  Theorem 3.1.

There exists a convex geometry with $cdim=3$ that is not representable by
spheres in any $\mathbb{R}^{t}$.

To achieve this, we provide here an explicit construction of a convex geometry
with $cdim=3$, such that its poset of join-irreducibles is isomorphic to
$\mathbf{n}^{3}$. Representing this convex geometry by spheres in
$\mathbb{R}^{t}$ would also result in an inclusion representation of
$\mathbf{n}^{3}$ by those spheres, as shown in Proposition 2.18, which we know
for large enough $n$ is not possible by Theorem 2.21. Hence, this would
suffice in proving the existence of a convex geometry not representable by
spheres in the space of any dimension.

For the remainder of this section, we set the following notation. Fix
$X=\\{0,1,2,\ldots,n-1\\}^{3}\subseteq\mathbb{Z}^{3}$, and set $\mathbf{P}$ to
be the poset $\mathbf{P}=(X,\leq)$, where $\leq$ is the natural ordering
induced from $\mathbf{n}$ in the manner of Definition 2.20.

We now consider linear extensions of this natural ordering in $\mathbf{P}.$ In
particular, we examine lexicographic orderings on the set $X$.

######  Definition 3.2 ($123$-lex-ordering).

We define the poset $(X,\preceq_{1})$. Given any two points
$x=(x_{1},x_{2},x_{3})$ and $y=(y_{1},y_{2},y_{3})$ in $X$, we say

$\displaystyle(x_{1},x_{2},x_{3})\preceq_{1}(y_{1},y_{2},y_{3})$
$\displaystyle\text{ iff }x_{1}\leq y_{1}$ $\displaystyle\text{OR
}x_{1}=y_{1},x_{2}\leq y_{2}$ $\displaystyle\text{OR
}x_{1}=y_{1},x_{2}=y_{2},x_{3}\leq y_{3}$

This is precisely lexicographic ordering where we prioritize the values of the
first coordinates, and if those are equal then the second coordinate, and
finally the third.

We can change the order of axes comparisons to get analogous but different
lexicographic orderings as follows.

######  Definition 3.3 ($231$-lex-ordering).

We define the poset $(X,\preceq_{2})$. Given any two points
$x=(x_{1},x_{2},x_{3})$ and $y=(y_{1},y_{2},y_{3})$ in $X$, we say

$\displaystyle(x_{1},x_{2},x_{3})\preceq_{2}(y_{1},y_{2},y_{3})$
$\displaystyle\text{ iff }x_{2}\leq y_{2}$ $\displaystyle\text{OR
}x_{2}=y_{2},x_{3}\leq y_{3}$ $\displaystyle\text{OR
}x_{2}=y_{2},x_{3}=y_{3},x_{1}\leq y_{1}$

This time we look first at the second coordinates, then the third, and finally
the first.

######  Definition 3.4 ($312$-lex-ordering).

We define the poset $(X,\preceq_{3})$. Given any two points
$x=(x_{1},x_{2},x_{3})$ and $y=(y_{1},y_{2},y_{3})$ in $X$, we say

$\displaystyle(x_{1},x_{2},x_{3})\preceq_{3}(y_{1},y_{2},y_{3})$
$\displaystyle\text{ iff }x_{3}\leq y_{3}$ $\displaystyle\text{OR
}x_{3}=y_{3},x_{1}\leq y_{1}$ $\displaystyle\text{OR
}x_{3}=y_{3},x_{1}=y_{1},x_{2}\leq y_{2}$

This time we look first at the third coordinates, then the first, and finally
the second.

It is not hard to see that the lexicographic orderings are not just partial
orderings, but rather organize the elements of $X$ into a chain. For example,
if we consider $\mathbf{2}^{3}$, using $123$-lex-ordering instead gives us the
following chain:

(0,0,0)(0,0,1)(0,1,0)(0,1,1)(1,0,0)(1,0,1)(1,1,0)(1,1,1)

Now, we can use precisely the orderings $\preceq_{1},\preceq_{2}$ and
$\preceq_{3}$ (from Definitions $2,3$ and $4$) – respectively being the
123-lex-ordering, 231-lex-ordering, and 312-lex-ordering – to generate a
convex geometry following Remark 2.6 and Theorem 2.8. More specifically, each
of these three linear orders can be thought of as a monotone convex geometry
on $X$ in the manner of Remark 2.6: so we obtain three linear geometries which
we denote $(X,\mathcal{F}_{1}),(X,\mathcal{F}_{2}),(X,\mathcal{F}_{3})$.
Taking the join of these three linear geometries in the manner of Definition
2.7 results in another convex geometry by Theorem 2.8. Finally, by Definition
2.9, this resultant convex geometry has $cdim\leq 3$. Call this geometry
$(X,\mathcal{F})$, with $X$ the base set and $\mathcal{F}$ as the closure
space. Let us now prove that the poset of join-irreducibles of $\mathcal{F}$,
denoted by $(\mathrm{Ji}(\mathcal{F}),\subseteq)$, is isomorphic to
$\mathbf{P}=(X,\leq)$.

It is well known that any join-irreducible in a standard closure system is
precisely the minimal closed set generated by some singleton $x$ from base set
$X$. Indeed, in standard closure system there is a one-to-one correspondence
between elements of the base set and join-irreducibles. Denote by $J(x)$ the
join-irreducible convex set in $\mathcal{F}$ corresponding to $x$. Recall the
procedure of generating a convex geometry of $cdim\leq m$ as the intersection
of closed sets of $m$ linear convex geometries on $X$: this is how we
constructed $(X,\mathcal{F})$ from our three linear lexicographic geometries.
It follows that the smallest closed set generated by $x\in X$, equivalently
$J(x)$, is the intersection of minimal closed sets generated by $x$ in each
linear geometry. In summary, our observations in this paragraph state the
following:

######  Proposition 3.5.

$J(x)$ will be obtained by intersecting the smallest convex set containing $x$
in each of the three generating linear geometries.

Note importantly that the smallest convex set containing $x$ in the linear
geometry $(X,\mathcal{F}_{i})$ is precisely the down-set with respect to
$\preceq_{i}$. That is, the minimal convex set containing $x$ in the $i$th of
the three generating geometries is exactly $\\{y\in X:y\preceq_{i}x\\}$. The
intersection over $i$ gives us $J(x)$.

So, we know there is a one-to-one correspondence between elements of $X$ and
$\mathrm{Ji}(\mathcal{F})$, and we also understand the nature of this
correspondence. The final two propositions show that the correspondence
preserves ordering, and hence conclude proving that
$(\mathrm{Ji}(\mathcal{F}),\subseteq)$ is isomorphic to $\mathbf{P}=(X,\leq)$.

######  Proposition 3.6.

If $x\leq y$ in $\mathbf{P}$, then $J(x)\subseteq J(y)$.

###### Proof.

It suffices to show that $x\preceq_{i}y$ for each $i$, because of how we
obtain $J(x)$ and $J(y)$ this would clearly imply $J(x)\subseteq J(y)$.

Write $x=(x_{1},x_{2},x_{3})$ and $y=(y_{1},y_{2},y_{3})$. We are told $x\leq
y$, so we must have $x_{i}\leq y_{i}$, but this directly implies
$x\preceq_{i}y$ by definition of the lex-ordering. This holds for any
$i=1,2,3$, and so finishes our proof. ∎

######  Proposition 3.7.

If $x,y$ are incomparable in $\mathbf{P}$, then $J(x),J(y)$ are incomparable
(in terms of set-inclusion).

###### Proof.

Write $x=(x_{1},x_{2},x_{3})$ and $y=(y_{1},y_{2},y_{3})$. For $x,y$ to be
incomparable in $P$, we must precisely have the state that $x_{i}<y_{i}$, and
$y_{j}<x_{j}$, for some $i\neq j$. Without loss of generality, assume that
$x_{1}<y_{1}$ and $y_{2}<x_{2}$.

Now, because $x_{1}<y_{1}$, we have $x\prec_{1}y$ by definition of $123$-lex-
ordering. This means $y\notin J(x)$. Similarly, because $y_{2}<x_{2}$, we have
$y\prec_{2}x$. This further implies that $x\notin J(y)$.

So, we have $y\notin J(x)$ and $x\notin J(y)$, but of course $x\in J(x)$ and
$y\in J(y)$, and we have hence proved that $J(x),J(y)$ are incomparable, when
ordered by set-inclusion. ∎

Finally, we are ready to provide a quick proof of Theorem 3.1.

###### Proof of Theorem 3.1.

For a given $n$, we constructed in this section a convex geometry
$(X,\mathcal{F})$. We showed that the join-irreducibles of $\mathcal{F}$,
denoted $\mathrm{Ji}(F)$, are in bijection with elements of $P$, and
Propositions 3.6 and 3.7 further show that they are in fact isomorphic as
posets.

Now, by Proposition 2.18 we know that any representation by spheres of
$(X,\mathcal{F})$ will result in $\mathrm{Ji}(\mathcal{F})$, and hence,
$\mathbf{P}$, being a sphere order. But Theorem 2.21 tells us that for some
$n_{0}$, if $n>n_{0}$ then $\mathbf{P}$ is not a sphere order. So, for large
enough $n$, $(X,\mathcal{F})$ cannot be representable by spheres.

Finally, $(X,\mathcal{F})$ by construction has $cdim\leq 3$. However, in [10]
it is shown that all convex geometries of $cdim\leq 2$ are representable by
circles on the plane. So, in fact for $n>n_{0}$, $(X,\mathcal{F})$ must have
$cdim=3$, which proves Theorem 3.1. ∎

##  4\. Ellipsoid Orders

In the survey on geometric containment orders, Fishburn and Trotter [16]
discuss various results related to poset representations via containment
orders of different geometric objects such as angles, polygons and spheres.
They also mention that representations by ellipsoids were not intensely
studied. In [15] they have shown that any 2-dimensional poset can be
represented as a containment order by a family of _similar_ ellipsoids that
share the same center.

As for to convex geometries, the following result was shown by J. Kincses.

######  Theorem 4.1.

[18] Any finite convex geometry with convex dimension $t$ can be represented
in $\mathbb{R}^{t}$ with ellipsoids which are arbitrary close to a sphere.

The construction of ellipsoids in Theorem 4.1 does not assume similarity of
ellipsoids, but all of them contain a unit sphere in $\mathbb{R}^{t}$ and are
themselves contained in the sphere of radius $s$. Taking $s$ close to $1$
allows to make them close to the unit sphere.

Another result in the same paper [18] formulates an Erdös-Szekeres type of
obstruction from Dobbins et al. [11] that shows that not all convex
geometries are represented by ellipses on the plane.

We also mention the result in [5] about representation of all geometries on
5-element set by ellipses on the plane. Thus, it would be interesting to learn
what is the smallest size of base set of a geometry so that representation by
ellipsoids exists only in $\mathbb{R}^{k}$ with $k>2$.

We can now connect representation of geometries by ellipsoids and the notion
of ellipsoid order. We mention that Proposition 2.18 could be formulated for
ellipsoids in place of spheres, which connects representation of convex
geometries and posets.

######  Theorem 4.2.

Every finite poset $\mathbf{P}=(X,\leq)$ is an ellipsoid order.

###### Proof.

First, we show that every poset is realized as the poset of join-irreducible
elements of some convex geometry. Indeed, starting from $\mathbf{P}=(X,\leq)$
we can build the lattice $\mathbf{D}=\operatorname{Down}(X,\leq)$ of down-sets
of $\mathbf{P}$, which is a distributive lattice by Birkhoff’s Theorem 2.16.
Every finite distributive lattice is a convex geometry, since related closure
operator satisfies the the Anti-Exchange Property. (See more general
description of lattice properties of finite convex geometries in [2, Theorem
5-2.1]). Since the join-irreducibles of this convex geometry are precisely the
down-sets of singletons, i.e. down-sets of the form $\downarrow x$ for $x\in
X$, we can conclude in the manner of Example 2.15 that $\mathbf{P}$ is
realized by join-irreducible elements of this convex geometry.

By Theorem 4.1 this geometry can be represented by ellipsoids in some space
$\mathbb{R}^{t}$, where $t=cdim(\mathbf{D})$. In particular, the set of join-
irreducible elements of $\mathbf{D}$, which is isomorphic to $\mathbf{P}$,
will provide the ellipsoid containment representation of $\mathbf{P}$. ∎

## References

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  * [2] K. Adaricheva and J.. Nation  “Convex Geometries”  In _Lattice Theory: Special Topics and Applications_ Springer International Publishing, 2016, pp. 153–179  DOI: 10.1007/978-3-319-44236-5˙5
  * [3] K. Adaricheva and J.. Nation  “Lattices of Algebraic Subsets and Implicational Classes”  In _Lattice Theory: Special Topics and Applications: Volume 2_ Springer International Publishing, 2016, pp. 103–151  DOI: 10.1007/978-3-319-44236-5˙4
  * [4] K.. Adaricheva, V.. Gorbunov and V.. Tumanov  “Join-semidistributive lattices and convex geometries”  In _Advances in Mathematics_ 173.1, 2003, pp. 1–49  DOI: https://doi.org/10.1016/S0001-8708(02)00011-7
  * [5] Kira Adaricheva et al.  “Convex geometries representable with colors, by ellipses on the plane, and impossible by circles”, 2022  DOI: 10.48550/arXiv.2206.05636
  * [6] Kira Adaricheva et al.  “When Life Gives You Lemons, Make Mathematicians”  In _Notices of the American Mathematical Society_ 68.3, 2021
  * [7] Garrett Birkhoff  “Rings of sets” Publisher: Duke University Press  In _Duke Mathematical Journal_ 3.3, 1937, pp. 443–454  DOI: 10.1215/S0012-7094-37-00334-X
  * [8] Graham Brightwell and Peter Winkler  “Sphere Orders”  In _Order_ 6, 1989, pp. 235–240  DOI: 10.1007/BF00563524
  * [9] PolyMath REU Convex Geometries Collaboration et al.  “Convex geometries representable by at most 5 circles on the plane”, 2021  arXiv:2008.13077 [math.CO]
  * [10] Gábor Czédli  “Finite convex geometries of circles”  In _Discrete Mathematics_ 330, 2014, pp. 61–75  DOI: https://doi.org/10.1016/j.disc.2014.04.017
  * [11] Michael Gene Dobbins, Andreas F. Holmsen and Alfredo Hubard  “Regular systems of paths and families of convex sets in convex position”  In _Transactions of the American Mathematical Society_ 368.5 American Mathematical Society, 2016, pp. 3271–3303  URL: https://www.jstor.org/stable/tranamermathsoci.368.5.3271
  * [12] Paul H. Edelman and Robert E. Jamison  “The theory of convex geometries”  In _Geometriae Dedicata_ 19.3, 1985, pp. 247–270  DOI: 10.1007/BF00149365
  * [13] Paul H. Edelman and Michael E. Saks  “Combinatorial representation and convex dimension of convex geometries”  In _Order_ 5.1, 1988, pp. 23–32  DOI: 10.1007/BF00143895
  * [14] Stefan Felsner, Peter C. Fishburn and William T. Trotter  “Finite three dimensional partial orders which are not sphere orders”  In _Discrete Mathematics_ 201.1, 1999, pp. 101–132  DOI: https://doi.org/10.1016/S0012-365X(98)00314-8
  * [15] P.. Fishburn and W.. Trotter  “Containment orders for similar ellipses with a common center”  In _Discrete Mathematics_ 256.1, 2002, pp. 129–136  DOI: 10.1016/S0012-365X(00)00366-6
  * [16] P.. Fishburn and W.. Trotter  “Geometric Containment Orders: A Survey”  In _Order_ 15.2, 1998, pp. 167–182  DOI: 10.1023/A:1006110326269
  * [17] Peter C. Fishburn  “Combinatorics and Partially Ordered Sets: Dimension Theory (William T. Trotter)” Publisher: Society for Industrial and Applied Mathematics  In _SIAM Review_ 35.3, 1993, pp. 519–520  DOI: 10.1137/1035116
  * [18] János Kincses  “On the representation of finite convex geometries with convex sets”  In _Acta Scientiarum Mathematicarum_ 83.1, 2017, pp. 301–312  DOI: 10.14232/actasm-017-502-z
  * [19] Kolja Knauer and William T. Trotter  “Concepts of Dimension for Convex Geometries”, 2023  arXiv:2303.08945 [math.CO]
  * [20] Edward R. Scheinerman and John C. Wierman  “On circle containment orders”  In _Order_ 4.4, 1988, pp. 315–318  DOI: 10.1007/BF00714474
  * [21] J.. Sidney, S.. Sidney and Jorge Urrutia  “Circle orders, N-gon orders and the crossing number”  In _Order_ 5.1, 1988, pp. 1–10  DOI: 10.1007/BF00143891

KIRA ADARICHEVA
Hofstra University
Hempstead, New York, USA
E-mail address<EMAIL_ADDRESS>
ARAV AGARWAL
Bowdoin College
Brunswick, Maine, USA
E-mail address<EMAIL_ADDRESS>
NA’AMA NEVO
Northeastern University
Boston, Massachusetts, USA
E-mail address<EMAIL_ADDRESS>


# “It Can Relate to Real Lives”: Attitudes and Expectations in Justice-
Centered Data Structures & Algorithms for Non-Majors

Anna Batra  0009-0007-5771-3270 University of WashingtonSeattleWAUSA
<EMAIL_ADDRESS>,  Iris Zhou  0009-0003-4219-2768 University of
WashingtonSeattleWAUSA<EMAIL_ADDRESS>,  Suh Young Choi  0009-0006-2247-2254
University of WashingtonSeattleWAUSA<EMAIL_ADDRESS>,  Chongjiu Gao
0009-0000-9547-4740 University of WashingtonSeattleWAUSA<EMAIL_ADDRESS>,
Yanbing Xiao  0009-0003-6335-5730 University of WashingtonSeattleWAUSA
<EMAIL_ADDRESS>,  Sonia Fereidooni  0009-0008-4522-8953 University of
WashingtonSeattleWAUSA<EMAIL_ADDRESS>and  Kevin Lin  0000-0001-9946-3635
University of WashingtonSeattleWAUSA<EMAIL_ADDRESS>
(2024)

###### Abstract.

Prior work has argued for a more justice-centered approach to postsecondary
computing education by emphasizing ethics, identity, and political vision. In
this experience report, we examine how postsecondary students of diverse
gender and racial identities experience a justice-centered Data Structures and
Algorithms designed for undergraduate non-computer science majors. Through a
quantitative and qualitative analysis of two quarters of student survey data
collected at the start and end of each quarter, we report on student attitudes
and expectations.

Across the class, we found a significant increase in the following attitudes:
computing confidence and sense of belonging. While women, non-binary, and
other students not identifying as men (WNB+) also increased in these areas,
they still reported significantly lower confidence and sense of belonging than
men at the end of the quarter. Black, Latinx, Middle Eastern and North
African, Native American, and Pacific Islander (BLMNPI) students had no
significant differences compared to white and Asian students.

We also analyzed end-of-quarter student self-reflections on their fulfillment
of expectations prior to taking the course. While the majority of students
reported a positive overall sentiment about the course and many students
specifically appreciated the justice-centered approach, some desired more
practice with program implementation and interview preparation. We discuss
implications for practice and articulate a political vision for holding both
appreciation for computing ethics and a desire for professional preparation
together through iterative design.

attitudes; confidence; sense of belonging; non-majors; diversity; social
justice; data structures; computing education; iterative design

††journalyear: 2024††copyright: rightsretained††conference: Proceedings of the
55th ACM Technical Symposium on Computer Science Education V. 1; March 20–23,
2024; Portland, OR, USA††booktitle: Proceedings of the 55th ACM Technical
Symposium on Computer Science Education V. 1 (SIGCSE 2024), March 20–23, 2024,
Portland, OR, USA††doi: 10.1145/3626252.3630754††isbn:
979-8-4007-0423-9/24/03††ccs: Social and professional topics Computing
education

##  1\. Introduction

This experience report presents a study of the attitudes and expectations of
postsecondary students enrolled in a justice-centered Data Structures and
Algorithms (DSA) course designed for non-
majors111https://courses.cs.washington.edu/courses/cse373/23wi/ at the
University of Washington, Seattle campus, a large, research-intensive
university in the United States. Our university is located in a growing high-
tech area with many large tech companies and emerging startups that employ a
majority white and Asian workforce. The University of Washington plays a key
role in enabling this tech workforce by teaching computing skills to students
from all majors across campus. At our university, there are two DSA courses:
one only open to computer science majors, and one only open to non-computer
science majors. Our paper focuses on the latter course.

Given that justice-centered approaches have not been significantly studied in
the postsecondary context, and because sociological phenomena like career
funneling operate on larger scales beyond the scope of an individual course or
even individual university, we wanted to focus on a more immediate unit of
analysis: our students enrolling and taking computing courses today. The end
goal of this study is to inspire future work to create, improve, and evaluate
pedagogies for justice-centered approaches to higher computing education. We
focus on these 2 research questions:

  1. RQ1

How do students’ computing confidence, perceptions of computing, and sense of
belonging change (if at all) before and after taking a justice-centered DSA
course?

  2. RQ2

What are students’ sentiments and experiences around their goals and
expectations in a justice-centered DSA course?

Prior work has explored the factors affecting confidence and sense of
belonging for students, some contextualizing these factors in relation to
students’ gender and racial identities (Winter et al., 2021; Irani, 2004). A
study by Hogan et al. explored differences in fears and confidence between
students (primarily non-majors) taking two introductory programming courses:
one more traditional course and one intentionally designed for non-majors.
They found that students taking the course intentionally designed for non-
majors, which included more creative programming assignments, experienced
greater increases in confidence than students taking the more traditional
course (Hogan et al., 2023). Moya et al. surveyed and interviewed high school
students taking introductory CS courses revealing that, although women
experienced greater increases in self-efficacy than men, they still had lower
efficacy than men overall at both the start and end of the term. Women
initially had lower sense of belonging than men, but significantly increased
in sense of belonging in the CS student community, with no significant
differences at the end of the quarter (Moya et al., 2023). Another study by
Krause-Levy et al. found that women consistently had lower sense of belonging
across three quarters of lower-level computing courses, but that Black,
Latinx, Native American, and Pacific Islander (BLNPI) students did not have a
lower or higher sense of belonging (Krause-Levy et al., 2021).

Although prior work has examined student expectations with respect to several
dimensions of computing education with the relationship between students’
computing experiences and their decisions to pursue a computer science major
(Lewis et al., 2011), to our knowledge there has not been prior work examining
student expectations for DSA specifically. DSA might be one of the more
challenging cases for implementing a justice-centered approach in the
postsecondary computing curriculum because (1) it often follows multiple terms
of introductory computing that might reinforce dominant narratives, and (2) it
is often preceded by its reputation of being the course for preparing students
for the “technical interview.” For these reasons, evaluating student goals and
expectations might be able to surface a particularly nuanced view of the
broader sociological forces shaping students’ experiences in the course.

##  2\. Justice-Centered Approach

At the heart of a justice-centered approach is an emphasis on students and the
potential role that education can play in helping students understand the
world around them, find their place within it, and empower them to effect the
change that they wish to see. There is a growing body of work in culturally-
relevant, culturally-responsive, and culturally-sustaining computing
pedagogies in primary and secondary computing education that can help realize
the goals of justice-centered computing education (Vakil, 2018, 2020; Ryoo,
2019; Ryoo et al., 2020; Davis et al., 2021; Madkins et al., 2019).

However, postsecondary computing education poses a particularly challenging
context for this transformative approach to computing education. The dominant
narrative for pursuing computing education emphasizes maximizing efficiency
and business profit over “do[ing] something good” (Vakil, 2018, 2020; Ko et
al., 2020; Malazita and Resetar, 2019). Recognizing the disciplinary values
interpretation (Vakil, 2020) at work, more recent work has called on computing
educators and researchers to address inequalities by drawing on diverse forms
of computer science capital (Kallia and Cutts, 2021) and by addressing
critical questions about the goals of computing education (Washington, 2020;
Lin, 2022; Grosland, 2019; Philip et al., 2018; Philip and Sengupta, 2021;
Vakil and Ayers, 2019; Tissenbaum et al., 2021; Ko et al., 2020).

Our particular course largely follows _Critical Comparative Data Structures
and Algorithms_ (CCDSA), a justice-centered pedagogy for DSA that emphasizes a
motif of “critical comparison” as a means of realizing ethics in the
curriculum, identity in the learning environment, and political vision (Lin,
2022).

Ethics via epistemological comparison:

by teaching not only asymptotic analysis for data structure and algorithm
implementations, but also affordance analysis (Lin, 2021) for ethical
evaluation of data structure and algorithm interfaces.

Identity via cultural comparison:

by drawing on core components for _Culturally Responsive-Sustaining Computer
Science Education_ (Davis et al., 2021), such as providing equitable
opportunities for creative and meaningful ways to learn and express learning.

Political vision via narrative comparison:

by foregrounding diverse career paths that students can pursue after taking
the course, refuting the narrative that DSA is (only) for fueling the elite
tech sector.

Our course makes a novel contribution in how it bridges two key concepts that
were left separate in CCDSA: analyzing implementations versus analyzing
interfaces. In the dominant approach to DSA, students only analyze
implementations for their runtime and use their analysis to create more
efficient implementations. In CCDSA, students do all of the above but also
learn to analyze interfaces for their affordances and evaluate the
relationship between those affordances and real-world applications that
utilize those interfaces. Our approach goes one step further to connect the
two skills: to respond to affordance analysis, the software designer (the
student) will need to design a new interface, which will then require new
implementations that respond to asymptotic analysis.

This process then forms a nested loop: by iterating through both the outside
loop (affordance analysis) and the inside loop (asymptotic analysis), our goal
was for students to develop not only a greater appreciation for the role of
ethics in the curriculum, but also a better understanding of their
responsibility as software creators going beyond asymptotic analysis. When
future interviewers ask one of our students to solve a coding challenge, we
hoped they would be able to integrate both ‘inside the box’ thinking
(developing more efficient solutions) and ‘outside the box’ thinking
(surfacing unstated assumptions and designing new interfaces). In doing so,
they would be able to not only preach ethics, but also put their ideas into
practice too.

To structure this process of iterative design, we drew on a technique for
teaching critical software design skills called _CIDER_ (Oleson et al., 2023).
In CIDER, students learn to address design biases and think beyond ‘average’
users by  (1) Critiquing a real-world technology to identify a list of
assumptions about users,  (2) Imagining how one of the identified assumptions
could lead to exclusion,  (3) (re)Designing the technology to no longer rely
on the assumption,  (4) Expanding their knowledge by sharing their list of
assumptions with others, and  (5) Repeating steps (2) and (3) to practice the
work of addressing assumptions.  Iterative design has the potential to not
only teach students a way to address design biases, but also to empower
students to make decisions with an appreciation and more nuanced understanding
of their capacity to change human conditions.

Most coursework required students to submit self-recorded video explanations
of their work. More specifically, we have:

  * •

conceptual assessments that evaluated student understanding of data structures
and algorithms as they were introduced throughout the quarter;

  * •

software projects where students implemented and analyzed solutions to real-
world problems using the CCDSA approach;

  * •

midterm interview questions where students were asked to come up with data
structure and algorithm designs to address software problems;

  * •

and a final portfolio where students explained how to implement a feature
using graphs and apply iterative design to bring together all the
aforementioned skills they learned in the course.

##  3\. Study Population

Our study population was comprised of students across two quarters: Autumn and
Winter. Pre-quarter and post-quarter surveys were administered comprising
questions about the following areas of interest: confidence in computing
ability, perceived usefulness of computing, sense of belonging in CS,
expectations and experiences, and students’ identities. Out of the 426 total
students still enrolled by the end of Autumn and Winter quarters, 334 students
filled out both the pre-quarter and post-quarter surveys, giving us a response
rate of 78.4%. Only data from the 334 students who filled out both surveys
were included in our analyses for our research questions. Both surveys were
required for completion in the class, but participation in the study was
optional.

Students reported their gender and racial identities using multiple option
selection on the survey with an additional option to fill-in a response.
Racial identities were grouped according to the 2020 census state
redistricting issued by the US Department of Commerce in June 2021. Notable
changes from the old census is that Middle Eastern and North African (MENA)
are now in their own category (of Commerce, 2021) due to them not perceiving
themselves as white, and others not perceiving them as white either (Neda
Maghbouleh, 2022). Note that our course has both majority white and Asian
student populations.

For quantitative analysis, we chose to group students with minority identities
together when there were fewer than 30 observations. In terms of racial
minorities, we grouped Black, Latinx, Middle Eastern and North African, Native
American, and Pacific Islander (BLMNPI) students together. This practice was
informed by Krause-Levy et al., which grouped Black, Latinx, Native American,
and Pacific Islander (BLNPI) students together (Krause-Levy et al., 2021).
Those of whom are mixed races are considered as part of every group they
identified as we are using binary one-hot data encoding for our analysis in a
one versus all approach. This explains why the racial identities do not all
add up to 100%. Similarly, we grouped women, non-binary, and other students
not identifying as men (WNB+) because NB+ has less than 30 students in this
group and may have more similar experiences to women rather than the majority
gender, men.

Table 1. Study participants’ gender and racial identities

Men | WNB+
---|---
69% | 31%

White | Asian | BLMNPI
---|---|---
14% | 70% | 10%

##  4\. RQ1 — Student Attitudes

The same 11 Likert items from the Attitudes Towards Computing Scale (Wanzer et
al., 2019) were posed at both the start and end of each academic quarter. The
11 statements fall under three categories, which we will refer to as the three
computing attitudes:

  1. (1)

Confidence in computing and problem-solving abilities

  2. (2)

Computing importance and perceived usefulness

  3. (3)

Sense of belonging in the computing community

Each student indicated their level of agreement with each statement on a
Likert scale from 1 to 5, where 1 was “Strongly Disagree” and 5 was “Strongly
Agree.” Some items in the scale are reverse-coded. During analysis, all items
were standardized so that 1 represents a negative attitude and 5 represents a
positive attitude.

Responses across each category of statements were averaged to create one mean
composite score per category. This method of using mean composites within
categories of questions was tested for reliability in the Attitudes Survey
Validation Study (Wanzer et al., 2019). Within each category, we were
interested in answering two questions:

  1. (1)

Did the class experience any significant changes in attitudes between the pre-
quarter and post-quarter surveys?

  2. (2)

Did students of different gender or racial identities experience any
significant differences in attitudes between the pre-quarter and post-quarter
surveys?

Separate statistical tests were used to answer each of the preceding
questions. To determine whether any of the three computing attitudes
increased, we used a 2-sample dependent t-test for each category of
statements, pairing scores for each student from their pre-quarter response to
their post-quarter response. To determine whether there were significant
differences between responses across the three attitudes for different gender
and racial groups, we used a 2-sample independent t-test. For each grouping,
we tested to see if significant differences existed at the start and end of
the quarter. We applied the Bonferroni correction for each grouping of
students to account for multiple comparisons across the three categories of
statements ($\alpha_{\textit{new}}=0.05/3\approx 0.017$).

###  4.1. Results

Table 2 shows the data. We summarize the results below with more information
about the source of statistically significant differences.

Table 2. Pre-quarter and post-quarter mean composite scores for question
categories where * indicates the “yes” score is significantly less than “no”
score with $\alpha\approx 0.0167$. The pre and post averages for all students
are displayed on the right.

|  | WNB+ | White | Asian | BLMNPI
---|---|---|---|---|---
|  | Yes | No | Yes | No | Yes | No | Yes | No
Confidence | pre | 3.15* | 3.54 | 3.53 | 3.4 | 3.32* | 3.64 | 3.41 | 3.42
post | 3.41* | 3.77 | 3.76 | 3.64 | 3.61 | 3.75 | 3.92 | 3.63
Importance | pre | 4.44 | 4.42 | 4.52 | 4.41 | 4.41 | 4.47 | 4.47 | 4.42
post | 4.35 | 4.37 | 4.52 | 4.34 | 4.37 | 4.35 | 4.52 | 4.35
Belonging | pre | 3.39* | 3.67 | 3.66 | 3.58 | 3.53 | 3.7 | 3.86 | 3.55
post | 3.55* | 3.8 | 3.88 | 3.7 | 3.72 | 3.73 | 3.94 | 3.7
All
---
3.42
3.65
4.43
4.36
3.58
3.72

####  4.1.1. Confidence in computing ability

Across the class, students’ confidence in their computing abilities
significantly increased (p ¡ .001) from the start of the quarter to the end of
the quarter. Across the groupings for gender and racial identities,
significant differences in confidence were found in WNB+ and Asian students.
WNB+ students reported significantly lower levels of confidence (p ¡ .001)
than men in both the pre and post-quarter surveys. Asian students also
reported significantly lower levels of confidence at the start (p = .001), but
not at the end of the quarter (p = .142) (Table 2). Specifically, Asian
students initially agreed less with the statement “I think I could handle
difficult computing problems,” and agreed more with the statement “I’m not
good at computing.”

No significant differences in confidence were found when grouping white
students or BLMNPI students.

####  4.1.2. Computing importance and perceived relevance

No significant changes or differences were found in perception of computing
importance across the class (p = .072) or with respect to gender and racial
identities (p ¿ .0167).

####  4.1.3. Sense of belonging in CS and in STEM

Across the class, sense of belonging in CS and in STEM significantly increased
(p ¡ .001) from the start to the end of the quarter. Although median scores
across the class did not change, students’ responses to the statement, “I feel
like I ’belong’ in computer science,” were generally more neutral (median = 3)
than for other statements about pride in computing abilities and self-
consideration as STEM professional (median = 4). WNB+ students, additionally,
reported significantly lower senses of belonging at both the start (p = .002)
and end (p = .005) of the quarter. This was largely due to a more neutral
response to the statement “I feel like I belong in computer science” in WNB+
students (mean = 3.05) than in men (mean = 3.54), at the start of the quarter.

No significant differences in sense of belonging were found when grouping
white, Asian, or BLMNPI students.

###  4.2. Discussion

As this course is likely many students’ first exposure to a collaborative,
discussion-based computing course, the overall increase in confidence and
sense of belonging is a welcome though unsurprising result. Encouraging
collaboration and discussion may have led to a greater sense of community
between students. Our results replicate prior studies that indicate a
consistently lower sense of belonging in women and lack of difference between
BLMNPI and non-BLMNPI students (Krause-Levy et al., 2021; Sax et al., 2018).

Findings from open-ended interviews conducted by Moya et al. revealed that
sense of belonging for female students was more related to collaboration with
peers and female CS teachers (Moya et al., 2023). Our course’s high percentage
of 67% WNB+ identifying teaching assistants might have contributed toward
breaking down stereotypes about men being more suited toward technical roles,
a perspective explored by Irani through qualitative interviews (Irani, 2004).
Implicit in the designation “non-major” is an out-group status label, which
likely contributed to the baseline levels of belonging.

Although prior work has explored differences in confidence and belonging with
respect to gender, less work has examined differences in these attitudes for
Asian or BLMNPI students. We found that Asian students had significantly lower
computing confidence than non-Asian students at the start but not the end of
the quarter (Table 2). These results raise new questions considering the
relative over-representation of Asians in our course compared to both the
computing industry and broader demographics in the United States.

##  5\. RQ2 — Student Expectations

In the post-quarter survey, we asked students to respond to a free-response
question: _Thinking back to the beginning of the quarter and your goals for
taking this course, have you gained what you hoped to gain from this course?
Was your experience in this class what you wanted or expected?_ The open-ended
nature of this question allowed students to discuss anything about the course,
and students were also encouraged to write longer thoughts or reflections.

We inductively coded student responses to this question to understand their
satisfaction with the course and how the course compared to their
expectations. The first two passes were carried out by the first author to
generate a preliminary codebook. Afterwards, three other authors, under
supervision of the first author, each individually performed an additional
pass adding onto the preliminary codebook. Preparation of the final results
involved another review by the first author and merging of repeated codes.

###  5.1. Results

Our inductive coding produced an analysis of student sentiments (5.1.1) and an
analysis of student experiences (5.1.2).

####  5.1.1. Overall Sentiments

Each student response received exactly 1 of the following overall sentiment
codes.

Positive:

(86% of class) for students who described an overall positive experience in
the class, and indicates that they had a typical positive experience of
satisfying their goal, such as “Yes, the different algorithms.”

Negative:

(13% of class) for students who described an overall negative experience in
the class, such as “Harder than expected, intellij was kind of confusing for
me.”

Inconclusive:

(1.5% of class) for students who described something that had an inconclusive
sentiment, such as “I slacked off a lot in this class so I didn’t gain what I
hoped to get out of this course.”

Table 3. Overall sentiments by identity Code | Men | WNB+ | White | Asian | BLMNPI
---|---|---|---|---|---
Positive | 84% | 90% | 87% | 89% | 84%
Negative | 14% | 10% | 11% | 9% | 13%

Table 3 shows the breakdown of positive and negative overall sentiments by
identity. Due to inconclusive sentiments (not shown), not all columns sum to
100%.

####  5.1.2. Specific Experiences

Many students also received a code for their specific experiences in the
class. A student response could receive 0, 1, or more than 1 of the following
codes. Due to the open and optional nature of the prompts, not all students
contributed specific experiences that corresponded to our research questions.

Appreciated Justice Approach (A-Justice):

12% — for students who appreciated the justice-centered approach, such as
“This allow me to see how the code or technology we use effects the world we
live in.”

Appreciated Change in Expectations (A-Change):

3% — for students who initially expected more programming, but still
appreciated what they learned in the course despite not matching their initial
expectations, such as “Discourse involves less coding assessment, than I
thought, but I learned a much more useful thing.”

Appreciated Programming and Interview Prep (A-Prep):

9% — for students who appreciated improving their programming skills overall
and preparation for software job interview questions, such as: “LeetCode
problems feel way easier now which is something I was hoping for.”

Desired Programming and Interview Prep (D-Prep):

16% — for students who desired more implementation of data structures and
algorithms concepts, usually toward the end goal of software jobs, such as “it
would be beneficial to have the professor share code with the class, and show
us how to implement things.”

Desired Different Approach (D-Approach):

2% — for students who desired a different course structure.

###  5.2. Discussion

####  5.2.1. Programming and Interview Preparation

We identified 3 types of desires under D-Prep: implementation practice, Java
syntax, and technical interview preparation.

Students wanting more implementation felt that implementing data structures
and algorithms, rather than learning through pseudo-code or conceptual
problems, would best help them comprehend concepts. This desire for
implementation may stem from the programming-heavy focus of our university’s
CS1 and CS2 courses. This is a focus that students may need to dis-acclimate
from DSA in general, justice-centered or not, as portrayed by a student who
“expected a [computer science] course to be code focused,” but realized that
computing is “more about the methods and analysis.”

Some students expected a continued focus on gaining familiarity with Java as a
programming language. This expectation somewhat conflicted with our
assumptions and intents; we assumed comfort with object-oriented programming
in Java, and used Java as a context for learning rather than the focus of
learning.

Additionally, some students also expected “preparation and facilitated
practice for coding interviews,” since DSA concepts commonly appear in coding
challenges and technical interviews. While this was not a focus of our course
design, we found that students expressed valid concerns that they did not
“feel ready to pursue a professional career at many levels.” Anecdotally, we
note that our prior approach to DSA also failed to fully address these
concerns.

In contrast to the unmet expectations in D-Prep, students in A-Prep expressed
satisfaction with the amount of implementation and Java practice. Some
students reported that they could now “solve LeetCode problems by [themselves]
and know how to design an algorithm.” Additionally, the self-recorded video-
based explanation assessments helped them “convey [their] thought process
which… help[ed] in preparing for technical interviews.” It is noteworthy that
some students felt more prepared for technical job interviews after taking our
justice-centered computing course, as we don’t spend time talking about how to
prepare students for technical interviews that can be learned through geared
practice of similar questions. Instead we focus on the on-the-job skills such
as asymptotic and affordance analysis.

While incorporating more programming and interview-inspired assignments could
be one way to meet student expectations in future quarters for D-Prep, there
may be other ways to address student expectations. With the immense focus on
the dominant software job narrative in the preceding CS1 and CS2 courses, our
findings suggest that more effort could be directed to show how existing
course practices prepare students for jobs of all kinds—in the tech industry
and beyond.

####  5.2.2. Change in Expectations and Different Approach

A-Change represents students that expected this course to be purely technical
and found the justice-centered concepts to be “unique” or “unexpected,” but
ultimately appreciated that the inclusion of ethics gave them “a new
perspective” on how “it can relate to real lives.” For example, one student
stated “my goal was to use this class to be prepared for technical interviews,
which I did not accomplish because of the lack of coding, but I think I have
since found an importance in understanding computing and the implications
around it as I was taking the class.” Importantly, the student does not seem
to be expressing that the justice-centered approach caused their
dissatisfaction or detracted from their other goals. Rather, their discovery
of computing ethics seems to have changed their expectations. This suggests
that there is room in the course design to hold both goals together: improve
preparation for technical interviews and realize our political vision for
students to view themselves as people whose actions can change the world.
Another student recounts that they now “view programming as an art that has
many different effects on society.”

D-Approach included a student who had the inverse experience, where the
inclusion of ethics was not a new perspective and thus “did not challenge
[their] thinking,” since their major already focused on societal impacts.
While only 0.6% of students mentioned this, other students of similar
backgrounds may also find the justice-centered content redundant, as our
approach aims to introduce these ideas assuming no prior knowledge. Future
offerings can consider presenting additional opportunities for deeper
engagement with computing ethics by, for example, partnering with or
highlighting current research efforts for socially-responsible computing at
the institution.

####  5.2.3. Appreciated Justice-Centered Approach

Additionally, other students in A-Justice expressed that aspects of assignment
and course design presented avenues for personal growth or expressing their
interests. One student noted a positive view of video-based explanation
assessments. They described how, in previous programming courses, they “had so
many questions about those specs and the details/special cases” and “teachers
were always like: ‘ignore it… we don’t require that for this assignment.”’ In
contrast, they felt that video submissions “encouraged [them] to express
[their] thoughts and assumptions” and that it felt “nice to see how those
concerns for our programs are valued”—suggesting that their professional
identity as a programmer could be better supported by assessment modalities
that allowed for greater self-expression. They added that in-class discussion
of affordance analysis, such as through lecture or iterative design
activities, were “very helpful and important for our study in CS.”

Another student noted that the open-ended, design-focused exams allowed them
to “connect what I learned in this course to my own personal interest in
health informatics… on the topic of hospital bed management.” Some students
noted that they learned a lot from this exam because they were “able to review
[their] ‘toolkit’ of data structures and algorithms to answer different
questions and find different approaches to them.” In fact, one student
expressed a desire for even more design questions as “most of the projects
already told us what structures to use” and that more feedback on the design
assignment may also help “to better understand why [they] would use a specific
data structure in a specific environment.”

Collaboration and teamwork were not only encouraged, but required for many of
small-group discussions in larger programming assignments and lecture problem-
solving activities. In particular, one student constructively criticized,
“sometimes the emphasis on teamwork out-shadowed the focus on the CS
concepts,” possibly referring to either the programming assignments or to
lecture. They instead suggested that “really spending time having a class
discussion on those [key concepts] and welcoming people to participate in that
whole class discussion,” would be a “wonder experience to share with everyone”
and have the “‘Aha’ moment together about how fundamental some concepts are.”
This suggests that the entire-class community can be a space for more
generative and spontaneous interactions beyond those provided in the small-
group discussions.

One way this idea could be realized is by the structure of CCDSA. Another
student commented, “I loved that the concepts that we learned were taught in a
way that always tied back to each other.” The comparative data structures
approach in the CCDSA pedagogy provides many opportunities for content to be
connected to each other and revisited in different ways. These opportunities
could be leveraged in class to spark discussion among students, rather than
explained away in the individual pre-class preparation readings.

##  6\. Threats to Validity

This experience report captures a very local snapshot of our offerings of a
justice-centered DSA course for non-majors as we offered them over two
quarters at our university.

For the quantitative analysis of RQ1 in particular, note that our findings
replicate effects found in dominant-approach courses. It is not impressive to
us that taking another computing course would correlate with an increase in
student confidence and sense of belonging given the circumstance and potential
selection bias at work.

To provide a more fine-grained evaluation of the effects of instruction, it
may be desirable to survey students more frequently than just at the start or
end of the quarter. Another constraint of the survey format is the use of the
Likert scale, which forced students to describe their attitudes and feelings
within a 5-option discrete format without further justification. Additionally,
even though students described their identities using multiple-option check-
boxes with an optional fill-in box, their identities were analyzed within a
categorical system of racial and gender classification—one that encodes
mutually-exclusive binaries during its analysis. Nonetheless, the quantitative
analysis offers a (limited) basis for future researchers to replicate.

For the qualitative analysis of RQ2 in particular, one-on-one interviews or
focus groups with students would provide a process to refine recommendations
through dialogue rather than inference. These two offerings were among the
first justice-centered offerings of any course offered by the computer science
unit at our university, so most students likely had limited prior experience
with critical computing concepts, which could increase discomfort and impact
results.

##  7\. Conclusion

This experience report makes three contributions to the field:

  1. (1)

an evaluation of postsecondary students in a justice-centered computing
course, and recommendations for the future (Lin, 2022),

  2. (2)

a method for bridging the two conceptual halves of implementations and
interfaces in CCDSA with CIDER (Oleson et al., 2023),

  3. (3)

a justification, method, and set of results for evaluating postsecondary
student expectations with inductive coding.

Regarding student attitudes (RQ1), we replicated results from non-justice-
centered courses in the literature about computing confidence and sense of
belonging. Regarding student expectations (RQ2), we presented codes for
overall sentiments and specific experiences of students after taking the
course, as well as an in-depth discussion and reflection specific student
experiences with potential implications for practice in both justice-centered
DSA and dominant approaches to teaching DSA.

Future work should investigate justice-centered computing for in-majors, study
the role and position of instructional staff in creating the conditions for
justice-centered DSA, and lower barriers to adoption of critical computing
pedagogies. Justice-centered computing has the potential to not only “relate
to real lives,” but also empower students to provide better conditions for
them too.

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