# Representational Alignment
Supports Effective Machine Teaching

Ilia Sucholutsky
Princeton University Katherine M. Collins∗
University of Cambridge Maya Malaviya
Stevens Institute of Technology Nori Jacoby
MPI Weiyang Liu
University of Cambridge
MPI Theodore R. Sumers
Anthropic Michalis Korakakis
University of Cambridge
The Alan Turing Institute Umang Bhatt
NYU Mark Ho
Stevens Institute of Technology Joshua B. Tenenbaum
MIT Brad Love
University College London Zachary A. Pardos
UC Berkeley Adrian Weller
University of Cambridge
The Alan Turing Institute Thomas L. Griffiths
Princeton University Contributed equally

###### Abstract

A good teacher should not only be knowledgeable; but should be able to
communicate in a way that the student understands – to share the student’s
representation of the world. In this work, we integrate insights from machine
teaching and pragmatic communication with the burgeoning literature on
representational alignment to characterize a utility curve defining a
relationship between representational alignment and teacher capability for
promoting student learning. To explore the characteristics of this utility
curve, we design a supervised learning environment that disentangles
representational alignment from teacher accuracy. We conduct extensive
computational experiments with machines teaching machines, complemented by a
series of experiments in which machines teach humans. Drawing on our findings
that improved representational alignment with a student improves student
learning outcomes (i.e., task accuracy), we design a classroom matching
procedure that assigns students to teachers based on the utility curve. If we
are to design effective machine teachers, it is not enough to build teachers
that are accurate – we want teachers that can align, representationally, to
their students too.

##  1 Introduction

What makes an effective teacher? Massive Open Online Courses (MOOCs) promised
to revolutionize education by having educators from top research institutions
record lectures in their domain of expertise and make course materials
available online for any learner to interact with. However, after many trials,
this expert-first approach to online learning proved not as generally
effective and accessible as hoped for [1], with courses delivered more
traditionally by local teachers often showing better outcomes, in-person and
online [2].

Similarly, many can relate to having teachers who are brilliant experts in
their domains, but who struggle to instill knowledge in their students [3, 4].
These two pedagogical phenomena share a common thread – for their teaching to
be effective, teachers should not only be knowledgeable in their domain but
should also be able to appropriately map what they know into a space
accessible to their audience of learners. As powerful AI systems become
increasingly prevalent in society, it becomes increasingly pressing for us to
understand how these systems can be turned into effective teachers for their
human users – thought partners that enhance us rather than supplant us.
However, as these systems begin to outperform humans on some tasks [5, 6, 7],
their internal representations, at times, seem to become less human-like [8,
9], setting up a tension between domain expertise and the ability to map
knowledge into human-understandable spaces.

In this work, we bridge ideas from the burgeoning subfield of representational
alignment [10] with work on machine teaching, and the cognitive science of
pedagogy, beginning to address the question: is a teacher who is an expert but
representationally misaligned from a student better than a teacher who is less
of an expert but more representationally aligned? We hypothesize that there is
a tradeoff between expertise and representational alignment. Clarifying this
tradeoff is important in practice, as there may be design pressures in
building machine teachers wherein practitioners need to choose between
increasing accuracy versus increasing alignment. Through simulations, we trace
out a utility curve expressing this tradeoff. We validate our curve through
experiments in which machines teach humans, demonstrating that expert
(machine) teachers may not be as valuable as peer (machine) teachers who are
representationally aligned to their students.

Figure 1: Schematic of various teaching set-ups. (Left:) Dyadic interaction
between a single teacher and a single student. The teacher selects an example,
or set of examples, to communicate to the student. Teachers be “student-
centric” and represent their student when deciding which examples to select
(left bottom). Middle: “‘Classroom” setting wherein a single teacher selects a
set of examples to provide to all students in their “class” at once; student-
centric teachers reason about all students in their classroom to make their
selections (middle bottom). Right: “School” setting wherein many different
teachers are matched with different students; each student is matched with a
single teacher.

Alignment is not necessarily a fixed quantity of a teacher. Skilled human
teachers can often infer their students’ representations and adapt their
teaching accordingly. To compare such “student-centric” teachers to the “self-
centered” ones discussed above, we extended our analysis to “classroom”
settings where a single teacher has to teach multiple students using the same
set of examples (see Figure 1). While we find that, as expected, student-
centered teachers have better teaching outcomes than self-centered teachers,
we show that they are nonetheless impacted by the representational alignment
of their students with each other, and that the average student performance
drops as class size increases. This means that simply assigning all students
in a school to the most expert student-centered teacher is not always an
optimal strategy for maximizing student group outcomes.

Combining these insights, we leverage our utility curve to design a classroom
matching procedure that, when given a pool of students and teachers (i.e., a
“school”), strives to maximize student outcomes by assigning students to
teachers based on their accuracy and representational alignment. We test our
proposed algorithm with various learning outcome metrics used in real-world
settings including average performance, worst-case performance , and passing
rate. We show that our matching algorithm significantly improves learning
outcomes for students compared to both random matching and standardized
curricular materials (e.g., a recorded MOOC lecture distributed to all
students). Furthermore, we find that having student-centered teachers
available at a school, even if these teachers have very little expertise in
the domain being taught, can greatly improve student outcomes, especially for
students who are representationally unique relative to the other students and
teachers. Our proposed framework and experimental results provide early
insights that could inform the general study of pedagogy and human learning,
as well as the specific problem of designing effective machine teachers in an
age of models trained from human feedback (i.e., reinforcement learning with
human feedback [11]).

##  2 Problem Setting

Figure 2: Schematic of teaching and representational alignment. Teachers and
students have distinct representational spaces ($X,Y_{s}$) with some mapping
between them ($T_{s}$). There is a true label function ($f$) that can be
projected onto both the teacher and student spaces, but a teacher may not
perfectly know this true label function and have their own, diverging label
function ($f^{\prime}$). The teacher designs curricular materials ($L_{0}$; a
set of examples paired with labels) that are projected to each student’s space
($L_{s}$), where each student uses them to learn a label function ($g_{s}$).
Each student’s performance ($V_{s}$) is then measured as the divergence
between the learned label function and the hidden true label function
($T^{\prime}_{s}(f)$).

Figure 2 shows a schematic of our teaching and representation alignment
framework. Consider a space $X$ of stimuli. We consider the case where the
teacher tries to teach the students some function $f:X\rightarrow C$. We
illustrate a simple case in Figure 2 wherein $C$ is a binary classification
$C=\\{0,1\\}$ dividing $X$ into two regions ($C=0$ and $C=1$ are represented
in Fig. 2 in light and dark gray, respectively). The teacher observes label
function $f^{\prime}:X\rightarrow C$, which may be different from $f$.

The teacher chooses $n$ points from the space ${x_{1},x_{2},\ldots,x_{n}}\in
X$ and assigns labels to the points $l_{i}$. The teacher materials can be
represented by the labeled points : $L_{0}=(x_{i},l_{i})_{i=1,\ldots,n}$. To
represent the fact that students’ representations may differ from that of the
teacher, we assume that the student $s$ has a space $Y_{s}$ that corresponds
to the student’s representations. Note that each student is part of some
classroom or population of students $s\in\mathcal{S}$.

Next, we assume there is some transformation $T_{s}:X\rightarrow Y_{s}$. We
assume that the function $T_{s}$ is also selected from some parametric
function $T_{s}\sim\mathcal{T}$. The student $s$ observes stimuli presented by
the teacher $y_{i}=T_{s}(x_{i})$ and labels $l_{i}$. The student learning
input (i.e., the teaching materials mapped into the student’s space) is thus
$L_{s}=(T_{s}(x_{i}),l_{i})_{i=1,\ldots,n}$. From that, the student infers the
labeling for the rest of the space, which can be represented as the learning
function $g_{s}(y|L_{s})$. The classification performance of the student is
tested over additional test points where the expected performance of the
student is $V_{s}=D(g_{s}(y|L_{s})||T_{s}(f(x)))$. Here, $D$ represents some
distance measure.

##  3 Simulations

###  3.1 Setup

Task. We construct a grid environment of size $N\times N$ where there are $K$
categories. Each grid cell is defined by its location within the grid with a
label $k\in\\{1...,K\\}$. The student’s goal is to correctly label each point.
To support the student, the teacher can reveal labels on the grid (i.e.,
creating teaching materials $L_{0}$ which consist of grid points and their
associated labels). We consider two different label structures ($f$): one
wherein there are $N$ classes (one class per column) and one with 4 classes
(partitioning the grid into quadrants). Our experiments are reproducible and
all the implementations for all computational experiments will be made
available open-source upon publication.

Representation. The coordinates of grid cells form the student ($Y_{s}$) and
teacher’s ($X$) representations. We compute representational alignment as
correlation between pairwise Euclidean distances. We corrupt the teacher’s
representation (in order to create misalignment between teacher space $X$ and
student space $Y_{s}$) by sampling a number of grid cells to swap such that
cell $(i,j)$ for the student appears as cell $(i^{\prime},j^{\prime})$ for the
teacher (i.e., the mapping $T_{s}$ from the teacher to the student would
reverse these swaps).

Student and teacher models Following our theoretical framework in Section 2,
we instantiate our student ($g_{s}$) as a 1-nearest neighbor (1-NN)
classifier, who takes as input the teacher’s revealed examples ($L_{s}$) and
classifies each of the unlabeled points. Student performance ($V_{s}$) is
computed as the accuracy of their classifications over the unlabeled points.
The teacher chooses $K^{\prime}$ points intended to maximally help the student
(whom the teacher “knows” is a 1-NN classifier) to achieve high accuracy on
the remaining points. We assume the teacher has access to labels for all
cells; however, the “erroneous” teacher with some probability assumes the
wrong label on a cell (i.e., $f^{\prime}$ is different from $f$, which can
ripple into their selections accordingly). The teacher computes the centroid
of each class (using its own believed labels $f^{\prime}$, which may have
errors) and selects one example per class to reveal to the student. The
teacher reveals its believed labels to the student for the selected points.

Figure 3: Representational (mis) alignment between students and teachers can
impact student learning outcomes. (Left:) Sample student representation.
(Middle:): A misaligned teacher. Color indicates the identity of the stimuli
as a visual aid to illustrate different representations of the same set of
stimuli. The same stimuli are indicated with the same color between student
and teacher. (Right:) Tracing out a utility curve between teacher error rate
and representational alignment between teacher-student in simulations (where
representations are grid locations, as in the left and middle plots). Points
are colored by representational alignment (higher/darker is more aligned).
Student learning outcomes are operationalized as accuracy at classifying grid
cells (higher is better).

###  3.2 Relationship between representational alignment and teacher
performance

In Figure 3, we trace out a relationship between student-teacher
representational alignment, teacher error rate, and student accuracy. We
uncover instances wherein students can achieve higher performance by learning
from teachers who are more erroneous (“less expert”) provided the teachers are
representationally aligned with the students than comparatively more expert
but representationally misaligned teachers, underscoring that it is not just
the accuracy of a teacher that matters for student learning outcomes. However,
we note that there is a tradeoff: more representationally aligned teachers are
not always better if the teacher is too noisy or misguided in their labels.
Yet, for a fixed teacher error rate, higher representational alignment is
always better for a student (provided the error rate is not too high). We
uncover similar curves across grid sizes and the number of categories (see
Appendix B.2).

##  4 Experiments with Machines Teaching Humans

###  4.1 Setup

Participants. We recruit 480 participants from Prolific [12]. We filtered the
participant pool by country of residence (United States), number of completed
studies ($>100$), and approval rate ($>95\%$). Participants gave informed
consent under an approved IRB protocol.

Task. We design a task for our machines to teach humans about categories, in
which participants see a grid of stimuli; for each cell in the grid, there is
an underlying true category. Our simulated teacher model selects labels based
on these underlying categories, and participants see these labels with the
grid. Participants must then categorize all the unlabeled stimuli on the grid
using the teacher’s labels. We do not inform participants of the number of
examples per category. We investigate two structures of categories: one class
per column of the grid (“cols”) and one class per quadrant of the grid
(“quad”). Note that these categories induce labeling functions that the
students should be able to learn; they are tractable (column structure and
block structure). There were two different stimuli sets. The first (simple-
features) is the closest analog to our simulated experiments, in which
participants saw a $6\times 6$ grid with blank cells, so the features are
completely expressed via the coordinates of the grid. The second (salient-
dinos) is a more rich set of stimuli, wherein participants see a $7\times 7$
grid of dinosaur (“dino”) images from [13]. Dino stimuli were defined by nine
different features (e.g., body length, neck length, neck angle) and organized
on the grid by two principle components of those underlying features. For a
visualization of the participant’s view, see Figure 9. For each condition,
different teachers were generated from our model, sampling across varying
levels of alignment. This structure leads to 24 different conditions (2
stimuli sets $\times$ 2 category structures $\times$ 6 teacher alignment
levels) for which we collect 20 participants each.

Models and Example Selection. We employ the same model types as in our
simulations. Teachers are self-centered and assume that students are 1-NN
classifiers111We acknowledge such an assumption is highly simplistic for
students and encourage future work to explore alternate models of students..
In both settings, we assume the representations of teachers and students can
be expressed through their two-dimensional grid locations. For the simple-grid
setting, there are no features for the human to use for their categorization
beyond grid cell location; and in the salient-dinos setting, features were
defined by two principal components (which we can use as grid coordinates). We
again induce representation misalignment between teacher and student by
shuffling the stimuli on the grid. We sample a set of teachers spanning a
range of representational alignments. We select a single set of points for
each teacher, assuming the teacher has perfect accuracy. We explore alternate
labeling functions to simulate alternate teacher error rates post-hoc (see
Appendix C).

###  4.2 Results

Figure 4: Tracing out a utility curve between teacher error rate and
representational alignment between machine teachers and human students. Points
are colored by representational alignment (higher/darker is more aligned).
Student learning outcomes are operationalized as classification accuracy
(higher is better). (Left:) Results when classifying grid points with no
images (the simple-features task). (Right:) Results when classifying dino
images (the salient-dinos task). (Top:) Results with one class per quadrant.
(Bottom:) Results with one class per column (6 for simple-features, 7 for
salient-dinos).

Representational alignment between machine teachers and human students can
affect learning outcomes. We construct a utility curve paralleling our
simulations by post-hoc varying teacher error rate (see Appendix C.3). We find
in Figure 4 that across both tasks, generally, higher representational
alignment induces higher average student accuracy, and report correlations in
Table 1. By computing a simple linear regression on the points with no teacher
error, we find that a 0.1 increase in representational alignment corresponds
to a 2.1% increase in student accuracy for the simple-features conditions
($r=0.59,p=0.054$) and a 1.4% increase in student accuracy for the salient-
dinos conditions ($r=0.63,p=0.037$).

Table 1: Pearson correlations (with associated $p$-values) of average human student accuracy and representational alignment of the machine teacher across the various conditions. | Quadrants | Columns | Both
---|---|---|---
simple-features | 0.91 ($p$=0.013) | 0.59 ($p$=0.221) | 0.59 ($p$=0.054)
salient-dinos | 0.52 ($p$=0.286) | 0.86 ($p$=0.027) | 0.63 ($p$=0.037)

Thus, we find that even large increases in teacher error rate can be offset by
increasing representational alignment (e.g., a teacher with error rate $0$ and
representational alignment of $0.3$, has similar student outcomes as a teacher
with error rate $0.4$ and representational alignment $0.8$). However, we note
that the ordering of high representational alignment is less clear,
particularly for the settings where each class corresponds to a column. We
posit that people have a strong prior against classes being distributed as
columns, and find that especially for the columns conditions, participants
would often label using strategies that did not correspond to nearest neighbor
classification (e.g., several participants labeled in a way that corresponded
to different types of tilings). This motivates future work extending our
framework to account not only for representational alignment between teachers
and students but also for computational or algorithmic alignment.

##  5 Classroom Matching

We have demonstrated that a teacher’s representational alignment with a
student matters alongside teacher accuracy for student outcomes. Yet, in real
teaching settings, rarely is a teacher matched with a single student; rather,
they may need to select examples that suit a cohort, or group, of students at
once. Selecting such examples grows all the more challenging when students
have different representations. When provided with a “school” of teachers and
students, how can we simultaneously construct student groupings and which
teacher to allot for which group of students? We begin to explore this
question through a series of “classroom matching” experiments. In particular,
we develop a classroom matching procedure which, given a pool of students and
teachers, pairs groups of students to teachers based on our representational
alignment-teacher utility curve. But, some students may still be left behind.
To begin to address lower-performing students in our pool who may not have a
representationally close teacher, we depart from our self-centered teacher
set-up to consider a student-centric teacher: one who is able to select
examples in recognition of the students’ representations of the problem. We
find that we can get a boost of up to an extra 10% of accuracy points for a
subset of students who are representationally distinct from the rest of the
pool of students and teachers – crucially, provided there are not too many
students grouped into the classroom. We emphasize though, that our analogy to
“classrooms” is explored in simulation with machines teaching machines;
substantial future work is needed – and motivated by this work – to cache out
possible links between representational alignment, teacher error rate, and
classroom properties like classroom size in practice.

###  5.1 Setup

Student and teacher pools. We focus on our simple-features setting and extend
our dyad (single teacher, single student) setting to simulated pools of
teachers and students over our same $6\times 6$ grid. Here, we consider row-
based labels (a new $f$). We include generalization experiments to the
salient-dinos setting in Appendix B.5. We explore two different pools of
students and teachers: (i) unstructured pools spanning a range of
representational alignments and error rates, and (ii) clustered sets of
students and teachers. For the latter, we construct a generative model over
student-teacher populations wherein we have a set of clusters, with a fixed
number of students per cluster share similar representations. We sample one
similar teacher from each cluster with some error rate (sampled from a uniform
distribution over $0$ to $0.5$). We then deliberately downsample from the
available teachers to simulate the case where some students are
representationally distinct from the other students and available pool of
teachers. Additional details are included in Appendix B.4. For each
experiment, we sample $10$ different teacher pools.

Matching procedures. We propose to match students using our utility curve to
estimate their accuracy. That is, we compute the representational alignment
between a student and teacher and index into a bucketed version of the utility
curve222We recompute the curve by also averaging over samples of corrupted
students as our first utility curve was constructed for dyad setting wherein
the student’s grid was never corrupted (see Appendix B.3). that we construct
in Section 3 using both the representational alignment and teacher’s expected
error rate (which we assume we have access to). The resulting performance
provides an expected performance that we anticipate the student may attain if
paired in a classroom with that teacher. We iterate over all teachers for each
student and select the teacher who helps the student achieve the highest
expected performance. We refer to this method as Ours.

Additionally, we consider three baselines: (i) Random matching of students to
teachers, (ii) MOOC which matches all students to the lowest error rate
teacher to simulate the case non-local expert teacher arrangements, and (iii)
Optimal wherein we match students to the highest attainable accuracy, giving
an indication of the upper limit of performance that a matching algorithm
could achieve. Importantly, gaps between (ii) and (iii) further drives home
the importance of more than just teacher accuracy for informing student
outcomes.

###  5.2 Results

Table 2: Student learning outcomes (accuracy) from different classroom matching approaches in the unstructured pool setting. We compute the average student performance across all $N=1000$ pooled students (paired with potentially $M=30$ teachers), as well as accuracy over the bottom and top 10% of students in each matching, respectively. We additionally compute the proportion of students who achieve “passing” marks (set to a moderately high threshold of 45% accurate, given chance guessing is 16.6% on our 6x6 grid). Higher is better for all metrics. $\pm$ indicates standard errors computed over 10 sampled pools and associated assignments. Method | Avg Acc | Bottom 10% | Top 10% | Pass Rate
---|---|---|---|---
Random | 0.32 $\pm$ 0.01 | 0.17 $\pm$ 0.01 | 0.51 $\pm$ 0.01 | 0.08 $\pm$ 0.01
MOOC | 0.38 $\pm$ 0.01 | 0.25 $\pm$ 0.00 | 0.54 $\pm$ 0.04 | 0.18 $\pm$ 0.05
Ours | 0.39 $\pm$ 0.01 | 0.24 $\pm$ 0.01 | 0.59 $\pm$ 0.04 | 0.23 $\pm$ 0.03
Optimal | 0.49 $\pm$ 0.01 | 0.36 $\pm$ 0.00 | 0.70 $\pm$ 0.03 | 0.53 $\pm$ 0.01
Table 3: Student learning outcomes (accuracy) from different classroom matching approaches in the structured (clustered) pool setting. We have 10 representationally distinct clusters, each with 50 students, and sample 5 available teachers across the clusters (i.e., 50% of the clusters will not have a representationally aligned teacher available). Method | Avg Acc | Bottom 10% | Top 10% | Pass Rate
---|---|---|---|---
Random | 0.33 $\pm$ 0.01 | 0.2 $\pm$ 0.02 | 0.49 $\pm$ 0.02 | 0.12 $\pm$ 0.03
MOOC | 0.37 $\pm$ 0.02 | 0.26 $\pm$ 0.02 | 0.54 $\pm$ 0.06 | 0.17 $\pm$ 0.07
Ours | 0.39 $\pm$ 0.02 | 0.27 $\pm$ 0.02 | 0.57 $\pm$ 0.06 | 0.23 $\pm$ 0.06
Optimal | 0.43 $\pm$ 0.02 | 0.32 $\pm$ 0.02 | 0.6 $\pm$ 0.05 | 0.30 $\pm$ 0.06

Putting our utility curve to work to design “classrooms”. We find that our
matching algorithm, which groups students to teachers based on their
representational alignment and teacher error rate, suggests that prudent
student-teacher pairings are better than random matching and, particularly for
top-performing students, are better than having assigned the student to an
expert (minimal error rate teacher; MOOC) who may be representationally
distinct (see Tables 2 and 3). This observation is intriguing – students may
not achieve their full potential when paired with a representationally
misaligned teacher, even if that teacher is an expert. We note that in the
structured case (Table 3), wherein students and teachers have distinct
clusters of representational alignment, assigning all students to the most
accurate teacher (MOOC) does not do substantially better than random matching
across some of the metrics. We observe performance gains for our utility
curve-based matching across both pool types. However, we note that we do not
yet attain optimal matching performance; such discrepancy can be attributed to
a mismatch in our utility curve.

Student-centric teachers can cover students who are representationally “left
behind”. Even with our matching approach, students can be left behind if there
is no suitable representationally aligned teacher for them. Can we better
train teachers by drawing on our findings that representational alignment
appears to be a key factor in student learning outcomes? Thus far, our
teachers have all been self-centered; that is, they have a single
representation that they operate over to select points. This setting may be
realistic in practice for machine teachers that are not equipped with the
ability to adapt to the students at hand. However, real human teachers – and
the machine teachers we may want to design – ought to be able to adapt to the
student at hand. We next consider student-centric teachers that select points
to maximize the average performance of the students in their pool (see
Appendix B.6).

We begin to explore student-centric teachers by appending a second stage to
our matching procedure. After matching using our utility curve (as noted
above), we greedily attempt to pair the lowest performing students with a
student-centric teacher who chooses points by optimizing for the students in
their pool (i.e., taking the students’ representations into account). We
continue incorporating the next lowest-performing students into the student-
centric teacher’s classroom until a student’s attained accuracy with the
original pairing is not improved by the student-centric teacher. We apply our
procedure to the clustered pool structure noted above and find that it is
beneficial to continue adding students up to a point: if the teacher is an
expert (zero error rate), we can add all students from one cluster before we
see detrimental performance across the pool of students assigned to said
teacher. As the student-centric teacher’s error rate increases, fewer students
can be pooled before performance dropoff (see Figure 5).

Figure 5: (Left:) Group sizes from greedily incorporating the lowest
performing students into the classroom of a single student-centric teacher.
(Right:) Average accuracy gains (out of 1.0) in performance for students
grouped with the student-centered teacher, on top of what they would have
achieved from a self-centered teacher. Error bars are standard errors over
$20$ seeds of student-centric teacher groupings for a sampled structured pool
of students and teachers.

These results indicate the student-centric teachers can cover students who are
representationally distinct and help boost their learning outcomes. However,
classroom size matters,corroborating prior works in machine and human teaching
[frank2014, 14, 15, 16]. We conduct a deeper dive into the relationship
between classroom size and student outcomes in our setting when student-
centric teachers are available in Appendix B.7. Herein, we see that teachers
who may try to overalign to all students at once in a large classroom induce
poorer outcomes for the classroom writ large.

##  6 Related Work

Learning Sciences. The extended learning sciences community has studied
aspects of what makes for a good teacher or computer-based teaching system.
The expertise or quality of the teacher with respect to excellence of
schooling, certification, and a teacher’s own test scores have been observed
to positively affect student learning [17]. More classroom-adaptive qualities,
like a teacher’s amount of experience in classrooms and teaching strategies
employed (i.e., pedagogy) are also top contributing attributes [17]. Closeness
of representation to students with respect to demographic features has been
shown to lead to more effective student performance [18], in part due to the
role model effect, but also because teachers closer in these dimensions can
serve as sociocultural interlocutors, helping translate the relevance of
material to students [19, 20]. Intelligent Tutoring Systems [21], growing out
of the cognitive and learning sciences, have been a consistently effective
paradigm of computer-based teaching [22], primarily utilizing the pedagogy of
mastery learning [23]. They adapt the amount of prescribed practice based on a
representation of the student’s level of mastery of the skill being worked on
and provide procedural remediation in the problem-solving context. In a two-
year, large-scale evaluation, a commercial ITS was found to be effective
overall, but only demonstrated superior learning gains to standard classroom
instruction in the second year. It was hypothesized that this may have been
due to teachers needing to learn how best to align their classroom to the
technology [24].

Machine teaching. Machine teaching aims to study the problem of teaching
efficiency by characterizing such efficiency as the minimal number of
effective data examples that is needed for a learner to learn some target
concept. It has an intrinsic connection to optimal education [25], curriculum
learning [26, 27] and optimal control [28]. Depending on the type of learner,
machine teaching can be performed in a batch setting [25, 16, 29] or an
iterative setting [26, 30, 31, 32, 33]. The batch teaching aims to find a
training dataset of minimal size such that a learner can learn a target
concept based on this minimal dataset. The iterative teaching seeks a sequence
of data such that the learner can sequentially learn the target concept within
minimal iterations. Complementary to these works, our findings indicate that,
alongside the quality of examples that the teacher selects, it is also
critical for both the teacher and the student to share similar
representations.

Pragmatic communication. Successful communication rests on our ability to
understand others’ beliefs and intentions [34, 35]. Indeed, even young
children are sensitive to others’ knowledge and competence when teaching [36,
37] and learning [38, 39, 40] from others. Inspired by Gricean pragmatics
[41], recent computational models have formalized this process as recursive
reasoning about others’ latent mental states [42, 43, 44]. Such pragmatic
models have been used to facilitate human-AI interaction [45, 46, 47, 48, 49,
50, 51, 52, 53, 54]. Crucially, however, when either party _fails_ to
accurately model the other’s beliefs or perspective, human-human [55, 56] and
human-AI [57, 46] communication can be significantly degraded. Our work adds
to this literature by formalizing and analyzing the effect of
_representational_ misalignment on communication.

Representational alignment. Representational alignment [10] offers a
conceptual and grounded mathematical framework for characterizing teaching
settings wherein two or more agents engage on some task. Already, ideas from
representational alignment are providing new ways of thinking about machine
learning efficiency [58], value alignment [59, 60], disagreement [61], and
applications like human & machine translation [62]. In this study, we show
that representational alignment is a key dimension in predicting and
optimizing student outcomes, with similar importance as the teacher’s subject
expertise.

##  7 Discussion and Limitations

We have begun to characterize qualities of effective machine teaching through
the lens of representational alignment. Our computational and human
experiments elucidate that teacher accuracy is not all that matters for
student learning outcomes: we also ought to care about a teacher’s
representational alignment with a student, a message with resonance to
learning sciences literature.

Our results herald practical implications. For machines to be effective
thought partners who help us grow and learn, we may want them to be able to
successfully model our representations; but, humans vary in our
representations of the world [63, 64]. Our classroom matching studies motivate
designing for diverse pools of teachers if you want the best outcomes for
diverse pools of students. “Representations should be representative” – i.e.,
distribution of teacher representations should be representative of the
students with care taken to classroom size. Further, our results link back to
some of the challenges of deploying uniform curricular materials like MOOCs;
even if machine teachers are superhuman experts in some respects, some
students may benefit from less expert teachers who are more representationally
aligned. We emphasize that our results in no way suggest that the MOOC
approach of having top educators design teaching materials is somehow bad, but
rather that some students will be better served by instead being matched with
representationally aligned teachers, even if those teachers have less domain
expertise. Appropriately matching students to teachers may become all the more
important as the space of available AI systems expands rapidly (we already see
a suite of large language models on offer for widespread use [65, 66, 67, 68,
69]).

Yet, we emphasize that our work is a first step in the study of the
relationship between representational alignment, teacher efficacy, and
student-teacher matching. Our simulations always assume that students are 1-NN
classifiers, which grossly undercuts the richness of human behavior. Further,
we always assume that the student’s representations are fixed; in practice,
students may adapt their representations of the world over time, and it may be
advisable for them to do so. Additionally, we consider only single-lesson
settings and settings wherein students have no indication of the reliability
of the teacher. We plan to extend our simulations and human experiments to
multi-lesson sequences wherein students get feedback on teaching accuracy in
the future.

##  8 Conclusion

Expertise on a task is not sufficient to be a good teacher. Representational
alignment between a teacher and student matters too. We trace out a utility
curve between teacher accuracy, teacher-student representational alignment,
and student accuracy to characterize the crucial relationship between
representational (mis)alignment and student learning outcomes. We find that
the most expert teacher is not always the most optimal and that the utility of
representation is approximately 2% (absolute) improvement in student accuracy
for every 10% (absolute) increase in representational alignment. This
empirical result from a machine teaching lens, underscores the importance of
having teachers, human or machine, capable of representing a diversity of
students, or facilitating pools of teachers such that there is at least one
teacher that any student can effectively learn from. Our work motivates
further investigation into the relationship between human-human
representational alignment and pedagogical effectiveness, not just for
teacher-student interactions but peer-to-peer learning.

## Acknowledgements

KMC gratefully acknowledges support from the Marshall Commission and the
Cambridge Trust. AW acknowledges support from a Turing AI Fellowship under
grant EP/V025279/1 and the Leverhulme Trust via CFI. This work was supported
by a grant from NOMIS Foundation to TLG.

## References

  * Reich and Ruipérez-Valiente [2019] Justin Reich and José A Ruipérez-Valiente.  The MOOC pivot.  _Science_ , 363(6423):130–131, 2019.
  * Kelly [2014] Andrew P Kelly.  Disruptor, distracter, or what? a policymaker’s guide to massive open online courses (MOOCs).  _Bellwether Education Partners_ , 2014.
  * Carter et al. [1987] Kathy Carter, Donna Sabers, Katherine Cushing, Stefinee Pinnegar, and David C. Berliner.  Processing and using information about students: A study of expert, novice, and postulant teachers.  _Teaching and Teacher Education_ , 3(2):147–157, 1987.  ISSN 0742-051X.  doi: https://doi.org/10.1016/0742-051X(87)90015-1.  URL https://www.sciencedirect.com/science/article/pii/0742051X87900151.
  * Hinds et al. [2001] Pamela J Hinds, Michael Patterson, and Jeffrey Pfeffer.  Bothered by abstraction: the effect of expertise on knowledge transfer and subsequent novice performance.  _Journal of Applied Psychology_ , 86(6):1232, 2001.
  * Strachan et al. [2023] James Strachan, Dalila Albergo, Giulia Borghini, Oriana Pansardi, Eugenio Scaliti, Alessandro Rufo, Guido Manzi, Michael Graziano, and Cristina Becchio.  Testing theory of mind in GPT models and humans.  2023\.
  * [6] Dave Van Veen, Cara Van Uden, Louis Blankemeier, Jean-Benoit Delbrouck, Asad Aali, Christian Bluethgen, Anuj Pareek, Malgorzata Polacin, Eduardo Pontes Reis, Anna Seehofnerova, et al.  Clinical text summarization: Adapting large language models can outperform human experts.  _Research Square_.
  * Thirunavukarasu et al. [2024] Arun James Thirunavukarasu, Shathar Mahmood, Andrew Malem, William Paul Foster, Rohan Sanghera, Refaat Hassan, Sean Zhou, Shiao Wei Wong, Yee Ling Wong, Yu Jeat Chong, Abdullah Shakeel, Yin-Hsi Chang, Benjamin Kye Jyn Tan, Nikhil Jain, Ting Fang Tan, Saaeha Rauz, Daniel Shu Wei Ting, and Darren Shu Jeng Ting.  Large language models approach expert-level clinical knowledge and reasoning in ophthalmology: A head-to-head cross-sectional study.  _PLOS Digital Health_ , 3(4):1–16, 04 2024.  doi: 10.1371/journal.pdig.0000341.  URL https://doi.org/10.1371/journal.pdig.0000341.
  * Fel et al. [2022] Thomas Fel, Ivan F Rodriguez Rodriguez, Drew Linsley, and Thomas Serre.  Harmonizing the object recognition strategies of deep neural networks with humans.  _Advances in Neural Information Processing Systems_ , 35:9432–9446, 2022.
  * Muttenthaler et al. [2022] Lukas Muttenthaler, Jonas Dippel, Lorenz Linhardt, Robert A Vandermeulen, and Simon Kornblith.  Human alignment of neural network representations.  _arXiv preprint arXiv:2211.01201_ , 2022.
  * Sucholutsky et al. [2023] Ilia Sucholutsky, Lukas Muttenthaler, Adrian Weller, Andi Peng, Andreea Bobu, Been Kim, Bradley C. Love, Erin Grant, Jascha Achterberg, Joshua B. Tenenbaum, Katherine M. Collins, Katherine L. Hermann, Kerem Oktar, Klaus Greff, Martin N. Hebart, Nori Jacoby, Qiuyi, Zhang, Raja Marjieh, Robert Geirhos, Sherol Chen, Simon Kornblith, Sunayana Rane, Talia Konkle, Thomas P. O’Connell, Thomas Unterthiner, Andrew K. Lampinen, Klaus-Robert Müller, Mariya Toneva, and Thomas L. Griffiths.  Getting aligned on representational alignment, 2023.
  * Christiano et al. [2017] Paul F Christiano, Jan Leike, Tom Brown, Miljan Martic, Shane Legg, and Dario Amodei.  Deep reinforcement learning from human preferences.  _Advances in Neural Information Processing Systems_ , 30, 2017.
  * Palan and Schitter [2018] Stefan Palan and Christian Schitter.  Prolific.ac—A subject pool for online experiments.  _Journal of Behavioral and Experimental Finance_ , 17:22–27, 2018.
  * Malaviya et al. [2022] Maya Malaviya, Ilia Sucholutsky, Kerem Oktar, and Thomas L Griffiths.  Can humans do less-than-one-shot learning?  In _44th Annual Meeting of the Cognitive Science Society: Cognitive Diversity, CogSci 2022_ , 2022.
  * Yeo et al. [2019] Teresa Yeo, Parameswaran Kamalaruban, Adish Singla, Arpit Merchant, Thibault Asselborn, Louis Faucon, Pierre Dillenbourg, and Volkan Cevher.  Iterative classroom teaching.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 33, pages 5684–5692, 2019.
  * Ma et al. [2018] Yuzhe Ma, Robert Nowak, Philippe Rigollet, Xuezhou Zhang, and Xiaojin Zhu.  Teacher Improves Learning by Selecting a Training Subset.  In Amos Storkey and Fernando Perez-Cruz, editors, _Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics_ , volume 84 of _Proceedings of Machine Learning Research_ , pages 1366–1375, 09–11 Apr 2018.  URL https://proceedings.mlr.press/v84/ma18a.html.
  * Zhu et al. [2018] Xiaojin Zhu, Adish Singla, Sandra Zilles, and Anna N Rafferty.  An overview of machine teaching.  _arXiv preprint arXiv:1801.05927_ , 2018.
  * Rice [2003] Jennifer King Rice.  _Teacher quality: Understanding the effectiveness of teacher attributes._ ERIC, 2003.
  * Dee [2004] Thomas S Dee.  Teachers, race, and student achievement in a randomized experiment.  _Review of Economics and Statistics_ , 86(1):195–210, 2004.
  * Egalite et al. [2015] Anna J. Egalite, Brian Kisida, and Marcus A. Winters.  Representation in the classroom: The effect of own-race teachers on student achievement.  _Economics of Education Review_ , 45:44–52, 2015.  ISSN 0272-7757.  doi: https://doi.org/10.1016/j.econedurev.2015.01.007.  URL https://www.sciencedirect.com/science/article/pii/S0272775715000084.
  * Harfitt [2018] G Harfitt.  The role of the community in teacher preparation: Exploring a different pathway to becoming a teacher.  _Front. Educ. 3: 64. doi: 10.3389/feduc_ , 2018.
  * Anderson et al. [1985] John R Anderson, C Franklin Boyle, and Brian J Reiser.  Intelligent tutoring systems.  _Science_ , 228(4698):456–462, 1985.
  * Wang et al. [2023] Huanhuan Wang, Ahmed Tlili, Ronghuai Huang, Zhenyu Cai, Min Li, Zui Cheng, Dong Yang, Mengti Li, Xixian Zhu, and Cheng Fei.  Examining the applications of intelligent tutoring systems in real educational contexts: A systematic literature review from the social experiment perspective.  _Education and Information Technologies_ , 28(7):9113–9148, 2023.
  * Bloom [1984] Benjamin S Bloom.  The 2 sigma problem: The search for methods of group instruction as effective as one-to-one tutoring.  _Educational Researcher_ , 13(6):4–16, 1984.
  * Pane et al. [2014] John F Pane, Beth Ann Griffin, Daniel F McCaffrey, and Rita Karam.  Effectiveness of cognitive tutor algebra I at scale.  _Educational Evaluation and Policy Analysis_ , 36(2):127–144, 2014.
  * Zhu [2015] Xiaojin Zhu.  Machine teaching: An inverse problem to machine learning and an approach toward optimal education.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 29, 2015.
  * Liu et al. [2017] Weiyang Liu, Bo Dai, Ahmad Humayun, Charlene Tay, Chen Yu, Linda B Smith, James M Rehg, and Le Song.  Iterative machine teaching.  In _International Conference on Machine Learning_ , pages 2149–2158. PMLR, 2017.
  * Korakakis and Vlachos [2023] Michalis Korakakis and Andreas Vlachos.  Improving the robustness of NLI models with minimax training.  In Anna Rogers, Jordan L. Boyd-Graber, and Naoaki Okazaki, editors, _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), ACL 2023_ , pages 14322–14339, 2023.  doi: 10.18653/V1/2023.ACL-LONG.801.  URL https://doi.org/10.18653/v1/2023.acl-long.801.
  * Lessard et al. [2019] Laurent Lessard, Xuezhou Zhang, and Xiaojin Zhu.  An optimal control approach to sequential machine teaching.  In _The 22nd International Conference on Artificial Intelligence and Statistics_ , pages 2495–2503, 2019.
  * Liu and Zhu [2016] Ji Liu and Xiaojin Zhu.  The teaching dimension of linear learners.  _Journal of Machine Learning Research_ , 17(162):1–25, 2016.
  * Liu et al. [2018] Weiyang Liu, Bo Dai, Xingguo Li, Zhen Liu, James Rehg, and Le Song.  Towards black-box iterative machine teaching.  In _International Conference on Machine Learning_ , pages 3141–3149, 2018.
  * Liu et al. [2021] Weiyang Liu, Zhen Liu, Hanchen Wang, Liam Paull, Bernhard Schölkopf, and Adrian Weller.  Iterative teaching by label synthesis.  _Advances in Neural Information Processing Systems_ , 34:21681–21695, 2021.
  * Qiu et al. [2023] Zeju Qiu, Weiyang Liu, Tim Z Xiao, Zhen Liu, Umang Bhatt, Yucen Luo, Adrian Weller, and Bernhard Schölkopf.  Iterative teaching by data hallucination.  In _International Conference on Artificial Intelligence and Statistics_ , pages 9892–9913, 2023.
  * Zhang et al. [2023] Chen Zhang, Xiaofeng Cao, Weiyang Liu, Ivor Tsang, and James Kwok.  Nonparametric iterative machine teaching.  In _International Conference on Machine Learning_ , pages 40851–40870, 2023.
  * Gweon [2021] Hyowon Gweon.  Inferential social learning: Cognitive foundations of human social learning and teaching.  _Trends in Cognitive Sciences_ , 25(10):896–910, 2021.
  * Vélez et al. [2023] Natalia Vélez, Alicia M Chen, Taylor Burke, Fiery A Cushman, and Samuel J Gershman.  Teachers recruit mentalizing regions to represent learners’ beliefs.  _Proceedings of the National Academy of Sciences_ , 120(22):e2215015120, 2023.
  * Liszkowski et al. [2008] Ulf Liszkowski, Malinda Carpenter, and Michael Tomasello.  Twelve-month-olds communicate helpfully and appropriately for knowledgeable and ignorant partners.  _Cognition_ , 108(3):732–739, 2008.
  * Bridgers et al. [2020] Sophie Bridgers, Julian Jara-Ettinger, and Hyowon Gweon.  Young children consider the expected utility of others’ learning to decide what to teach.  _Nature Human Behaviour_ , 4(2):144–152, 2020.
  * Bass et al. [2022] Ilona Bass, Elizabeth Bonawitz, Daniel Hawthorne-Madell, Wai Keen Vong, Noah D Goodman, and Hyowon Gweon.  The effects of information utility and teachers’ knowledge on evaluations of under-informative pedagogy across development.  _Cognition_ , 222:104999, 2022.
  * Csibra and Gergely [2009] Gergely Csibra and György Gergely.  Natural pedagogy.  _Trends in Cognitive Sciences_ , 13(4):148–153, 2009.
  * Bonawitz et al. [2011] Elizabeth Bonawitz, Patrick Shafto, Hyowon Gweon, Noah D Goodman, Elizabeth Spelke, and Laura Schulz.  The double-edged sword of pedagogy: Instruction limits spontaneous exploration and discovery.  _Cognition_ , 120(3):322–330, 2011.
  * Grice [1975] Herbert P Grice.  Logic and conversation.  In _Speech Acts_ , pages 41–58. Brill, 1975.
  * Chen et al. [2022] Alicia M Chen, Andrew Palacci, Natalia Vélez, Robert Hawkins, and Samuel J Gershman.  A hierarchical Bayesian approach to adaptive teaching, Dec 2022.
  * Goodman and Frank [2016] Noah D Goodman and Michael C Frank.  Pragmatic language interpretation as probabilistic inference.  _Trends in Cognitive Sciences_ , 20(11):818–829, 2016.
  * Shafto et al. [2014] Patrick Shafto, Noah D Goodman, and Thomas L Griffiths.  A rational account of pedagogical reasoning: Teaching by, and learning from, examples.  _Cognitive Psychology_ , 71:55–89, 2014.
  * Sumers et al. [2021] Theodore R Sumers, Mark K Ho, Robert D Hawkins, Karthik Narasimhan, and Thomas L Griffiths.  Learning rewards from linguistic feedback.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 35, pages 6002–6010, 2021.
  * Sumers et al. [2022] Theodore Sumers, Robert Hawkins, Mark K Ho, Tom Griffiths, and Dylan Hadfield-Menell.  How to talk so AI will learn: Instructions, descriptions, and autonomy.  _Advances in Neural Information Processing Systems_ , 35:34762–34775, 2022.
  * Lin et al. [2022] Jessy Lin, Daniel Fried, Dan Klein, and Anca Dragan.  Inferring rewards from language in context.  _arXiv preprint arXiv:2204.02515_ , 2022.
  * Andreas and Klein [2016] Jacob Andreas and Dan Klein.  Reasoning about pragmatics with neural listeners and speakers.  _arXiv preprint arXiv:1604.00562_ , 2016.
  * Dale and Reiter [1995] Robert Dale and Ehud Reiter.  Computational interpretations of the Gricean maxims in the generation of referring expressions.  _Cognitive Science_ , 19(2):233–263, 1995.
  * Fried et al. [2018] Daniel Fried, Ronghang Hu, Volkan Cirik, Anna Rohrbach, Jacob Andreas, Louis-Philippe Morency, Taylor Berg-Kirkpatrick, Kate Saenko, Dan Klein, and Trevor Darrell.  Speaker-follower models for vision-and-language navigation.  _Advances in Neural Information Processing Systems_ , 31, 2018.
  * Wang et al. [2016] Sida I Wang, Percy Liang, and Christopher D Manning.  Learning language games through interaction.  _arXiv preprint arXiv:1606.02447_ , 2016.
  * Wang et al. [2020] Pei Wang, Junqi Wang, Pushpi Paranamana, and Patrick Shafto.  A mathematical theory of cooperative communication.  _Advances in Neural Information Processing Systems_ , 33:17582–17593, 2020.
  * Ho et al. [2016] Mark K Ho, Michael Littman, James MacGlashan, Fiery Cushman, and Joseph L Austerweil.  Showing versus doing: Teaching by demonstration.  _Advances in Neural Information Processing Systems_ , 29, 2016.
  * Zhi-Xuan et al. [2024] Tan Zhi-Xuan, Lance Ying, Vikash Mansinghka, and Joshua B. Tenenbaum.  Pragmatic instruction following and goal assistance via cooperative language-guided inverse planning, 2024.
  * Aboody et al. [2023] Rosie Aboody, Joey Velez-Ginorio, Laurie R Santos, and Julian Jara-Ettinger.  When naive pedagogy breaks down: Adults rationally decide how to teach, but misrepresent learners’ beliefs.  _Cognitive Science_ , 47(3):e13257, 2023.
  * Sumers et al. [2023] Theodore R Sumers, Mark K Ho, Robert D Hawkins, and Thomas L Griffiths.  Show or Tell? Exploring when (and why) teaching with language outperforms demonstration.  _Cognition_ , 232:105326, 2023.
  * Milli and Dragan [2020] Smitha Milli and Anca D Dragan.  Literal or pedagogic human? analyzing human model misspecification in objective learning.  In _Uncertainty in Artificial Intelligence_ , pages 925–934. PMLR, 2020.
  * Sucholutsky and Griffiths [2023] Ilia Sucholutsky and Thomas L Griffiths.  Alignment with human representations supports robust few-shot learning.  _NeurIPS_ , 2023.
  * Rane et al. [2023] Sunayana Rane, Mark Ho, Ilia Sucholutsky, and Thomas L Griffiths.  Concept alignment as a prerequisite for value alignment.  _arXiv preprint arXiv:2310.20059_ , 2023.
  * Wynn et al. [2023] Andrea Wynn, Ilia Sucholutsky, and Thomas L Griffiths.  Learning human-like representations to enable learning human values.  _arXiv preprint arXiv:2312.14106_ , 2023.
  * Oktar et al. [2023] Kerem Oktar, Ilia Sucholutsky, Tania Lombrozo, and Thomas L. Griffiths.  Dimensions of disagreement: Unpacking divergence and misalignment in cognitive science and artificial intelligence, 2023.
  * Niedermann et al. [2024] Jakob Niedermann, Ilia Sucholutsky, Raja Marjieh, Elif Celen, Thomas L Griffiths, Nori Jacoby, and Pol van Rijn.  Studying the Effect of Globalization on Color Perception using Multilingual Online Recruitment and Large Language Models, Feb 2024.  URL osf.io/preprints/psyarxiv/3jvxw.
  * Peterson et al. [2018] Joshua C Peterson, Joshua T Abbott, and Thomas L Griffiths.  Evaluating (and improving) the correspondence between deep neural networks and human representations.  _Cognitive Science_ , 42(8):2648–2669, 2018.
  * Marti et al. [2023] Louis Marti, Shengyi Wu, Steven T Piantadosi, and Celeste Kidd.  Latent diversity in human concepts.  _Open Mind_ , 7:79–92, 2023.
  * OpenAI [2023] OpenAI.  GPT-4 Technical Report, 2023.
  * Touvron et al. [2023] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_ , 2023.
  * Jiang et al. [2023] Albert Q Jiang, Alexandre Sablayrolles, Arthur Mensch, C Bamford, DS Chaplot, Ddl Casas, F Bressand, G Lengyel, G Lample, and L Saulnier.  others, Mistral 7B.  _arXiv preprint arXiv:2310.06825_ , 2023.
  * Chowdhery et al. [2022] Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles Sutton, Sebastian Gehrmann, Parker Schuh, Kensen Shi, Sasha Tsvyashchenko, Joshua Maynez, Abhishek Rao, Parker Barnes, Yi Tay, Noam Shazeer, Vinodkumar Prabhakaran, Emily Reif, Nan Du, Ben Hutchinson, Reiner Pope, James Bradbury, Jacob Austin, Michael Isard, Guy Gur-Ari, Pengcheng Yin, Toju Duke, Anselm Levskaya, Sanjay Ghemawat, Sunipa Dev, Henryk Michalewski, Xavier Garcia, Vedant Misra, Kevin Robinson, Liam Fedus, Denny Zhou, Daphne Ippolito, David Luan, Hyeontaek Lim, Barret Zoph, Alexander Spiridonov, Ryan Sepassi, David Dohan, Shivani Agrawal, Mark Omernick, Andrew M. Dai, Thanumalayan Sankaranarayana Pillai, Marie Pellat, Aitor Lewkowycz, Erica Moreira, Rewon Child, Oleksandr Polozov, Katherine Lee, Zongwei Zhou, Xuezhi Wang, Brennan Saeta, Mark Diaz, Orhan Firat, Michele Catasta, Jason Wei, Kathy Meier-Hellstern, Douglas Eck, Jeff Dean, Slav Petrov, and Noah Fiedel.  PaLM: Scaling Language Modeling with Pathways.  _arxiv:2204.02311_ , 2022.  URL https://arxiv.org/abs/2204.02311.
  * Pardos and Bhandari [2024] Zachary A Pardos and Shreya Bhandari.  ChatGPT-generated help produces learning gains equivalent to human tutor-authored help on mathematics skills.  _PLOS ONE_ , 19(5):e0304013, 2024.
  * Reich [2020] Justin Reich.  _Failure to disrupt: Why technology alone can’t transform education_.  Harvard University Press, 2020.
  * Fischer et al. [2020] Christian Fischer, Zachary A. Pardos, Ryan Shaun Baker, Joseph Jay Williams, Padhraic Smyth, Renzhe Yu, Stefan Slater, Rachel Baker, and Mark Warschauer.  Mining big data in education: Affordances and challenges.  _Review of Research in Education_ , 44(1):130–160, 2020.  doi: 10.3102/0091732X20903304.  URL https://doi.org/10.3102/0091732X20903304.

##  Appendix A Broader Impact and Societal Risks

As we discuss in Section 7, our work portends broader implications for the
design of machine-human teaching setups where machines are intentionally built
with representation alignment in mind, as well as representation diversity to
safeguard against threats to inclusivity. It is possible that our simulations
could support interventions in real classrooms, e.g., informing classroom size
decisions drawing on measures of the representational diversity of a classroom
pool and the expertise of the teacher. However, we heed caution in over-
generalizing our results to settings where real student experiences and
learning potential is at stake. Broad-brush application of AI systems in
education has not been met with universal success [70] and inappropriately
incorporated can have unintended impacts on student success [71].

##  Appendix B Additional Details on Computational Simulations

###  B.1 Compute Details

All experiments were run on an 8-core, 16 GB memory laptop. Experiment were
run exclusively on CPUs and were all runnable within at most three hours.

Figure 6: Utility curves on a $6\times 6$ grid for different label structures.
(Left:) 4 class underlying label-per-quadrant; (Right:) 6 class underlying
label-per-column.

###  B.2 Constructing the Dyadic Utility Curve

To construct our utility curve in Section 3, we sweep over a range of possible
teacher error rate parameterizations (from $0$ to $0.9$ in increments of
$0.1$) and representation corruption levels (from $0$ to $1.0$ in increments
of $0.01$). We always use the same “student” and corrupt teachers over the
respective student grid. We compute the representation alignment between the
student and corrupted teacher; as the pairwise swaps (“corruptions”) are
randomly made over a fraction of the grid parameterized by the corruption
level, we bucketize the resulting observed representation alignment between
student and teacher. We then sample $10$ different seeds of selections for
each teacher and average student performance. We repeat the same sweeps over
teacher parameterizations for our two labeling schemes: grids wherein each
column corresponds to one label ($N$ labels for an $N\times N$ grid), and one
where each quadrant corresponds to one label (four labels). We average the
resulting utility curves across label types.

#### Impact of Underlying Label Structure

We depict the separate utility curves in Figure 6. Notably, we observe
different utility curves for different label structures. While there are some
minor rank swaps between teachers across the structures, we see high Spearman
rank correlation ($\rho=0.994,p<<10e-48$) between the two settings
underscoring general consistency in teacher orderings.

###  B.3 Utility Curve over Classrooms

The utility curves that we construct in Section 3 and B.2 were always
constructed with respect to a single student (in a respective,
“dyadic”333Pairing two agents – one student and one teacher.). However, in our
classroom settings, we also corrupt the students’ representations to simulate
representational diversity. We sample a new utility curve, wherein, for each
teacher parameterization (same error rate parameterization as above, with
representation corruptions now in increments of $0.1$), we sample $10$
different student corruptions ranging over $0$ to $0.9$ in increments of
$0.1$. We build this curve only for the column label type. We then bucketize
the teacher error rate as well as the representation alignment such that we
can index into the curve to extract an “expected average performance” for any
student-teacher pair.

###  B.4 Constructing Classroom Pools

We construct two different classroom pools in Section 5: unstructured and
structured. Here, we provide additional details on how we sampled students and
teachers for each pool type.

#### Unstructured Pool

All students and teachers are sampled independently. We sample $1000$ students
and $30$ teachers with corruption levels (pairwise swaps) sampled from a beta
distribution ($\alpha=1.5,\beta=2.5$) to ensure that we have some students
that are reasonably aligned. We sample teacher error rate uniformly over the
range $0-0.5$.

#### Structured Pool

In the structured setting, we construct clusters of similar students and
teachers. We prespecify a number of clusters $M$ and number of students per
cluster. Clusters are designed to span a range of levels of representation
alignment over the “original” grid. We loop over possible representation
alignments corruptions ranging from $0$ to $1$ in increments of $1/M$. For
each cluster, we sample a “seed” student using that corruption level. We then
sample [notation?] students ontop of this cluster with a representation
corruption of $0.01$ ontop of the base student to ensure students share
similar (but some variation) in their representation. For each cluster, we
sample a teacher with error rate uniformly from $0-0.5$ and representation
with a similar slight possible corruption (sampled uniformly from $0-0.01$)
ontop of the seed student, thereby ensuring that there would be a
representationally similar teacher for each student in each cluster if
provided. However, to simulate gaps in coverage of particular representation
characterezations, we randomly drop some teachers from the pool.

###  B.5 Generalization to the Dino Stimuli

We explore generalization of our utility curve constructed in the simple-
features setting to our salient-dino stimuli. We repeat our two different pool
types, which we depict in Tables 4 and 5, respectively. We find that our
utility curves generalize nicely to different grid sizes and stimuli type,
yielding student outcomes that on average appear to boost student accuracy
particularly for the students in the top-performing group than baselines which
do not account for representation misalignment (MOOC).

Table 4: Student learning outcomes (accuracy) from different classroom matching approaches in the unstructured pool setting for the dino stimuli. We again compute the student performance across all $N=1000$ pooled students (paired with potentially 30 teachers). Error bars are again computed over 10 different sampled pools. Method | Avg Acc | Bottom 10% | Top 10% | Pass Rate
---|---|---|---|---
Random | 0.29 $\pm$ 0.00 | 0.16 $\pm$ 0.01 | 0.46 $\pm$ 0.01 | 0.04 $\pm$ 0.01
MOOC | 0.34 $\pm$ 0.02 | 0.22 $\pm$ 0.00 | 0.51 $\pm$ 0.07 | 0.09 $\pm$ 0.05
Ours | 0.36 $\pm$ 0.01 | 0.22 $\pm$ 0.00 | 0.64 $\pm$ 0.08 | 0.15 $\pm$ 0.03
Optimal | 0.44 $\pm$ 0.01 | 0.31 $\pm$ 0.00 | 0.69 $\pm$ 0.08 | 0.32 $\pm$ 0.02
Table 5: Student learning outcomes (accuracy) from different classroom matching approaches in the structured (clustered) pool setting for the dino stimuli. We again have 10 representationally distinct clusters, each with 50 students, and sample 5 available teachers across the clusters. Method | Avg Acc | Bottom 10% | Top 10% | Pass Rate
---|---|---|---|---
Random | 0.29 $\pm$ 0.02 | 0.18 $\pm$ 0.02 | 0.44 $\pm$ 0.03 | 0.06 $\pm$ 0.03
MOOC | 0.34 $\pm$ 0.03 | 0.23 $\pm$ 0.02 | 0.50 $\pm$ 0.08 | 0.10 $\pm$ 0.06
Ours | 0.35 $\pm$ 0.03 | 0.23 $\pm$ 0.02 | 0.54 $\pm$ 0.08 | 0.13 $\pm$ 0.06
Optimal | 0.38 $\pm$ 0.02 | 0.28 $\pm$ 0.01 | 0.56 $\pm$ 0.07 | 0.15 $\pm$ 0.06

###  B.6 Additional Details on Student-Centric Teacher

In contrast to our self-centered teacher, our student-centric teacher does not
use its own representation to select examples to provide to the student.
Instead, the student-centric teacher is endowed with an inner optimization
loop over the students assigned to it, whereby the teacher loops $T$ times
over the students and randomly selects one point per category (using the
teacher’s believed class – the teacher may not know the true categories) and
measures the expected performance of each student if that set of examples were
revealed. Note, the teacher computes the expected accuracy of each student
using against its belief of the true categorization (which may be incorrect).
The teacher then chooses the set of examples that attains the highest average
accuracy over students. Here, we set $T$ to 100; exploring the impact of
varied $T$ is a sensible next step. Exploration of alternate optimization
functions, e.g., optimizing over the minimum attained performance over the
students in the teacher’s classroom rather than average classroom performance
are also ripe ground for future work.

###  B.7 Relationship Between Classroom Size and Student-Centric Teacher
Utility

We conduct a deeper dive into the relationship between classroom size and
student performance by sampling sweeping over a range of teacher error rates
(between $0$ and $0.5$ in increments of $0.1$) and classroom sizes. We sweep
over 10 samples for each setting, where each self-centered teacher optimizes
for their pool through $T=100$ inner loop iterations. Students are drawn from
our structured, clustered pool. We then marginalize over our sampled error
rates to compute an expected average classroom accuracy per classroom size. We
find that the performance of students in a classroom with a student-centric
teacher falls off rapidly as a function of classroom size and then plateaus in
Figure 7.

Figure 7: Average performance of students paired with a student-centric
teacher as a function of classroom size. Error bars depict standard error over
runs.

##  Appendix C Additional Human Experiment Details

###  C.1 Participant Recruitment and Compensation

Participants were recruited from Prolific and were paid $12/hr plus a 10%
bonus if they responded reasonably (i.e., did not select labels randomly or
choose the same label for all stimuli). The research did not contain risks to
participants, and they were able to opt out at any time. The institution of
the principal investigator obtained IRB approval for this experiment, and
participants gave informed consent under this protocol.

###  C.2 Task Instructions

We include the full set of instructions provided to participants in Figure 8
and sample interfaces in Figure 9.

Figure 8: Experiment instructions displayed to all participants, introduced
paragraph by paragraph. The only changes to instructions were to modify the
type of stimuli (“empty cells”, “images that represent stick figure
dinosaurs”), size of the grid ($6\times 6$, $7\times 7$), the number and names
of categories (4; A-D or 6/7; A to F/G).

Figure 9: Above are two example views of the experiment. All participants,
after viewing the instructions in Figure 8 were taken to a page that contained
a grid and the labeled stimuli. They were asked to categorize stimuli via a
dropdown menu selection. Finally, they rated their confidence using a scale
below the stimulus grid. Left: salient-dinos, 7 (“col”) categories, medium-
alignment teacher. Right: simple-features, 4 (“quad”) categories, high-
alignment teacher.

###  C.3 Further Analyses

#### Simulating teacher error in human experiments

All human experiments were run with machine teachers set to zero error, as
collecting all combinations of teacher error and representational alignment
would be prohibitively expensive. Instead, we simulate the effect of teacher
error in a post-hoc analysis by corrupting the true underlying labels in the
same way we corrupted the teacher labels for the simulation experiments (i.e.,
error rate corresponds to the probability with which we flip each true label
to be a different label). Human student accuracy was then recomputed against
these corrupted true labels. The original human student results with no
simulated teacher error are reported in Figure 10.

Figure 10: Average human student classification accuracy at various levels of
representational alignment. Error bars correspond to one standard error.
(Left:) Results from simple-features setting. (Right:) Results from salient-
dinos setting. (Top:) One class per quadrant. (Middle:) One class per column
(6 for simple-features, 7 for salient-dinos). (Bottom:) Combined results.