Not by number alone: The effect of teachers’ knowledge and its value inevaluating “sins of omission”
Ilona Bass (ibass@wesleyan.edu)1 , Daniel Hawthorne-Madell (djthorne@stanford.edu)2 ,Noah D. Goodman (ngoodman@stanford.edu)2 , Hyowon Gweon (hyo@stanford.edu)2
1 Department of Psychology, Wesleyan University, Middletown, CT 06459
2 Department of Psychology, Stanford University, Stanford CA, USA

Abstract
What constitutes good teaching, and what factors do learners
consider when evaluating teachers? Prior developmental worksuggests that even young children accurately recognize andevaluate under-informativeness. Building on prior work, we
propose a Bayesian model of teacher evaluation that infers theteacher’s quality from how carefully he selected demonstrations given what he knew. We test the predictions of our modelacross 15 conditions in which participants saw a teacher who
demonstrated all or a subset of functions of a novel device andrated his helpfulness. Our results suggest that human adults
seamlessly integrate information about the number of functions taught, their values, as well as what the teacher knew,
to make nuanced judgments about the quality of teaching; thequantitative pattern is well predicted by our model.
Keywords: pedagogy, Bayesian models, social learning,causal learning, pragmatics

Introduction
Learning from others is beneficial, but the benefits comewith hazards. By learning from knowledgeable, helpful others, learners can draw powerful inferences that go beyond
the face value of information. However, because its power
hinges on the assumption that the teacher is knowledgeable
and helpful, learning can go awry when teachers are inaccurate or misleading. Thus it is critical for learners to besensitive to others’ quality as teachers.
Reasoning in pedagogical contexts can be formally described as a set of mutually constraining inferences (Shafto,
Goodman, & Griffiths, 2014). The teacher selects information that increases the learner’s belief in the target hypothesis (pedagogical sampling), and the learner rationally updates his beliefs with the assumption that the information has
been pedagogically sampled. This leads to a strong expectation that information provided by the teacher is not only
true but also sufficient for the learner to draw accurate inferences. For instance, when a teacher pedagogically demonstrates just one function of a multi-function device, a naı̈ve
learner would (inaccurately) infer that the toy has one, andonly one, function; if there were more, a knowledgeable andhelpful informant would have shown them.
Prior developmental work suggests that young children
draw strong inferences from pedagogically sampled information in ways that are consistent with the model predictions (Bonawitz et al., 2011; Xu & Tenenbaum, 2007), andappropriately respond to teachers who violate pedagogicalsampling (Gweon, Pelton, Konopka, & Schulz, 2014). For
instance, 6- and 7-year-olds rate a teacher as less helpful when he demonstrates one function of a multi-function
toy (thus being under-informative), compared to when hedemonstrates one function of a single-function (thus being
fully informative) (Gweon, Pelton, et al., 2014), and given abinary choice, even four-year-olds favor a fully-informative
teacher over an under-informative teacher (Gweon & Asaba,
2015; see also Barner, Brooks, & Bale, 2011 for similar inferences in linguistic communication). These results suggestthat even early in life, human learners recognize and evaluate “sins of omission”–under-informative pedagogy that misleads the learner to draw an inaccurate inference.
Intuitively, not all sins of omissions are equally blameworthy. What affects our evaluations of under-informative teaching? One simple possibility is that the degree to which we penalize sins of omission is inversely related to the learner’s degree of belief in the correct hypothesis. Thus the more information a teacher omitted, the more blame he would deserve.
However, there are reasons to believe that people make morenuanced judgments about under-informative teaching.
First, going beyond the amount of omitted information,
people might also consider the value of what was omitted.
Because not all knowledge is equally useful or valuable,omission can have varying consequences for the learner depending on the utility of the resulting knowledge. Consider
a gadget that has two buttons; one turns the gadget into an
auto-navigating robot vacuum and the other activates a blinking light on the side. Because learning about the former
would be more useful than the latter, showing just the vacuum function would be considered less blameworthy thanshowing just the blinking light.
Furthermore, people might be sensitive to the reason behind the omission (i.e., the teacher’s intentions). In evaluating harmful actions, both children and adults are sensitive to the intent of a perpetrator and appropriately exonerate
those who caused accidental harms due to their ignorance or
false beliefs (Hebble, 1971; Yuill & Perner, 1988; Young,
Cushman, Hauser, & Saxe, 2007). In linguistic communication, listeners flexibly adjust their interpretation of an utterance depending on the speaker’s knowledge; if the speaker
didn’t have the knowledge to justify the stronger alternative,
listeners don’t draw implicatures (Goodman & Stuhlmüller,2013). Thus even in the absence of explicit information
about others’ intentions, people spontaneously use information about the others’ epistemic states to infer their intent.
Similarly, in evaluating sins of omission, people might careabout whether or not the teacher knowingly omitted something. Consider someone who demonstrated just one function of a four-function gadget, simply because he was unaware of the other three functions. Although he might beindependently blamed for his ignorance, we might exonerate him from being guilty of a “sin of omission” because his
ignorance suggests that the omission was unintended.
In this study, we investigate whether people consider theamount and the value of information as well as the teacher’s
knowledge to evaluate under-informative pedagogy. Previous Bayesian models of pedagogical reasoning have been
extremely useful in formalizing our intuitions about what an
ideal learner might infer from pedagogically sampled data
(Shafto et al., 2014) and in how learners might choose between different teachers. For instance, Shafto, Gweon, Fargen, and Schulz (2012) looked at people’s evaluations of efficient vs. inefficient teaching, and modeled people’s relative preferences between two teachers as a ratio of two likelihoods. However, these models assume that teachers arealways fully knowledgeable and that all facts that could betaught are equally valuable. Furthermore, they do not provide explicit, systematic predictions about people’s evaluations of teachers based on their knowledge.
Building on prior work, here we provide a computationalmodel of teacher evaluation. The model captures the ideathat good teachers will work harder to teach well, where“teaching well” is specified by the pedagogical sampling assumption. We extend the underlying pedagogical model toaccount for incomplete knowledge of the teacher as wellas the value of information taught. To test the model predictions, we designed a behavioral task to get parametricevaluations of teachers’ ability in which we manipulated theknowledge of the teacher, the value of the knowledge, andthe amount of knowledge communicated to the learner.

U(d) = ∑ V ( fi ) ln pL ( fi |di )(2)iwhere di is 1 if the ith function was presented and 0 otherwise, pL ( fi |di ) is the teacher’s model of the learner’s beliefs
about function i after demonstration di , and V ( fi ) is the valueof function i. This pedagogical utility function is a generalization of previous accounts which examined the special casewhere all functions are equally valuable (effectively dropping V ( fi ) from the equation). Assuming that a demonstration of each function provides an equal amount of information to the learner, and that a priori belief in each function is
the same, Eq. (2) becomes (up to an additive constant which
does not impact decisions):
U(d) = ∑ kdiV( fi ),

Model

(3)

i

Reasoning about a pedagogical situation requires a notionof the causal relations between the world and the teacher’s
actions—an intuitive theory of pedagogy that captures theteacher’s goal of informing the learner, and specifies what
examples should be given to teach a particular fact or concept. This theory can be used in learning from teachers (asin Shafto et al., 2014), but also learning about the teachers themselves—the importance they place on informing thelearner and their ability to do so by choosing effective examples of things that will be useful to the learner. To formalizea theory of pedagogy, we first describe how a teacher mightchoose what demonstrations to give in different situations.
We do so with the standard softmax decision rule:p(d| f , α) ∝ eα U(d)−C(d) ,(α = 0) would choose demonstrations at random, preferring
less costly demonstrations; as α increases, a teacher would
be more likely to select demonstrations that have high pedagogical utility, as good teachers would.
For our model to predict how people evaluate the teacher,we must specify the utility of the teacher: what counts as
successful teaching? Shato et al. suggested that the teacher’sutility should be proportional to the log-probability that thelearner will guess the correct hypothesis (2014). To apply
this idea to cases where a device has different functions thatvary in value, we must generalize this utility to capture thefact that there are different aspects of the device that theteacher could convey, each of which has its own utility to
the learner. Thus we can define the pedagogical utility of aparticular set of demonstrations (d) as:(1)

where f is the set of functions known to the teacher, andd is the set of demonstrations he provides to the learner.
C(d), which we set equal to the number of demonstrations
provided, corresponds to the cost of a given set of demonstrations, and U(d) is the utility a learner is expected to accrue from the demonstrations (Eq. 2). The parameter α controls the degree to which the teacher chooses to maximize
the pedagogical utility of his demonstration. Thus α can beinterpreted as the teacher’s “quality.” An indifferent teacher

where k is a constant reflecting the change in the learner’s
belief that a function exists, resulting from a demonstration(this constant will be absorbed into the overall calibration ofutility below).
The demonstrations that a teacher selects using Eq. (1),which uses this pedagogical utility (Eq. 3), depends on the
precise balance between the cost of demonstrations, theteacher’s quality, and the value of each function. The demonstration cost pulls against the tendency of high-quality teachers to teach everything they know, such that a high-quality
teacher with high communication costs may only teach thehigh value functions.
To use this theory of pedagogy to evaluate (rather thanlearn from) a teacher, consider a person who knows thata teacher knows functions f and observes the teacher’s ddemonstration(s). This observer can use Bayes’ rule to invert their model of how teacher’s select demonstration (Eq.
(1)) to infer the teacher’s quality:p(α| f , d) ∝ p(d| f , α)p(α),(4)

where p(α) represents the person’s prior beliefs of theteacher’s quality (we assume p(α) ∼ Uniform(0, 1)). The
resulting estimates of teacher quality are sensitive to theteacher’s epistemic state and the value of the functions

demonstrated. For example, the quality estimate of a teacher
who demonstrates two low-value functions but also knew ofa high value function will be lower than someone who justknew the two low-value functions he taught. Similarly, ateacher who knows both a high- and a low- value function
would get the highest rating for showing both, a lower rating
for omitting the low-value function, and an even lower rating
for omitting the high-value function.

Experiment
Methods
Participants All participants were recruited from Amazon
Mechanical Turk, and were paid $0.50 for compensation.
We excluded a total of 462 participants from analyses basedon a priori criteria (responses to check questions; see Procedure) and 71 for participating more than once. The final sample consisted of 1024 participants (Age: µ(σ) = 36.2(12.6),
range: 18 - 72 yrs; 575 female).
Stimuli We generated cartoon scenarios in which one character (e.g., Paul) encounters a novel device and discovers all four, or a subset (one or two) of its functions, andthen shows another character (e.g., Laura) all or a subset of its functions. The device had four distinctive buttons that corresponded to each function. Pressing a red
circular button made the device verbally report the current
time; the purple lightning-shaped button reported the local
weather; the orange crescent-shaped button made the device
say “hello!”; and the green square button generated a beepsound. The relative values of these functions were validated
by a separate group of 52 mTurk participants who rankordered the four functions by their usefulness. Weather andTime (µ(σ) = 3.4(0.29) ranked higher than Beep and Hello(µ(σ) = 1.6(0.29);t(51) = 23.3p < .001), with no difference
within the high-value pair or the low-value pair (p’s> .298).
Design We varied the number of functions the teacher discovered (1, 2, or 4), the utility of the functions he discovered (H for high, L for low), and the number of functions hedemonstrated to the learner (1, 2, or 4). Crossing these variables while excluding impossible cases yielded 15 conditions(see Figure 2A for a full list of conditions). For instance, in
the KA TA condition the teacher Knew All and Taught All;in KHL TH, he knew two functions (one high-value and one
low-value) but taught only one high-value function. The exact function taught among the two equally valued functions(e.g. Weather vs. Time) was counterbalanced throughout.
Procedure Participants were randomly assigned to one of
the 15 conditions and were shown the cartoon scenario that
corresponded to that condition. The first phase of the cartoons introduced the device to ensure that the participants
knew about all the functions; the second phase showed theteacher discovering some or all of the device’s functions. In
the third phase, the learner entered the room1 and asked the
1 In conditions where the teacher did not know all the functions,the learner’s entrance interrupted the teacher’s discovery; this wasto minimize the possibility that the participants’ ratings were af-

Figure 1: Schematic of procedure in the KLL TL condition.
The teacher discovers two low-value functions and teaches
just one low-value function.

teacher to show her how the device works. In the fourthphase, the teacher demonstrated some, or all, of its functions(see Figure 1). Participants then answered the critical question: 1. Overall, how would you rate his teaching abilities?
We also asked additional questions in fixed order (2. Whichfunctions did he discover? 3. Which functions did he teach?
4. How well-intentioned do you think he was? 5. How nicedo you think he is? 6. Given what he knew, how good ajob did he do? 7. How willing would you be to learn fromhim?). Finally we asked the first question again (8. Overall, how would you rate his teaching abilities?). We used aseven-point Likert scale for questions 1 and 4-8. Questions
2 and 3 were used as check questions to exclude participants
who did not pay close attention.

Results and Discussion
Participants’ ratings of teacher ability in the first and the last
questions (Q1 and Q8: the teacher’s overall teaching abilities) were highly correlated (r(1021) = 0.833, p < .001). We
therefore used the average of the two ratings in our analyses. See Figure 2A for mean ratings of teacher ability infected by the teacher’s incompetence or failure to discover all functions.

all conditions. We conducted a linear regression to examine the effect of (1) the number of functions demonstrated
by the teacher, (2) the value of these functions, and (3) number of functions the teacher knew on participants’ estimateof his ability. We coded these three factors as continuous
variables (to code the effect of the value of the demonstratedfunctions as a continuous variable, we quantified it as the
proportion of the high valued functions known to the teacher
that were demonstrated). All three factors were highly significant predictors. The higher the number of demonstrations the higher the teacher was rated (β = 1.1,t(796) =17, p < .001); the higher the value of his demonstrations, thehigher he was rated (β = .66,t(796) = 4.8, p < .001); andthe more functions the teacher knew, the lower he was rated(β = −.3,t(796) = 17, p < .001); that is, when the teacher
did not know as many functions, people were more likely toexonerate him for his limited knowledge. We further investigated these main effects with targeted analyses of a subsetof the conditions that highlights the predicted effects of (a)naı̈ve omission, (b) the teacher’s prior knowledge, and (c) the
effect of prompting intention and knowledge.
a) Naı̈ve omission We first looked at conditions in which
the informant discovered only a subset of the device’s functions, but taught everything he knew (KL TL, KH TH,
KLL TLL, KHH THH). A 2 (number: 1 or 2) x 2 (value:
H or L) ANOVA found a significant main effect of number(F(1, 297) = 21.1, p < .001). We also observed a marginallysignificant main effect of value (F(1, 297) = 2.9, p = .088),
and the interaction was not significant (F(1, 297) = .27, p =.603). Thus even when the teacher communicated all heknew, participants’ ratings still reflected the number of functions taught.
b) Informant’s prior knowledge We then looked at conditions in which the informant taught two functions but either taught all he knew (KHH THH, KLL TLL) or a subsetof what he knew (KA THH, KA TLL). A 2 (prior knowledge: Knew Two (Taught All) or Knew All (Omitted)) x 2(value: HH, LL) ANOVA found a significant effect of Prior
Knowledge (F(1, 285) = 30.9, p < .001): Even though theteacher showed the same number of functions, participants
took into account the teacher’s knowledge when rating his
teaching abilities. See Figure 2B. We also saw a significant main effect of value (F(1, 285) = 12.0, p = .001) as
well as an interaction (F(1, 295) = 5.6, p = .019). Morespecifically, the value of functions taught affected ratings
only when the teacher taught just a subset of what he knew(t(132) = 3.8, p < .001); if he taught all he knew, the value
had no effect (t(153) = .084, p = .40).
c) The effect of prompting intention and knowledge Our
questionnaire had 8 questions, with the first and the last question serving as our main dependent measure. Between these
two identical questions, people were asked a few check questions as well as questions that prompted them to think about
the knowledge of the teacher and the intention of the teacher.
Thus it is possible that people are more likely to consider

others’ knowledge after having been explicitly prompted tothink about it; if so, answering these questions would leadto amplified differences in people’s ratings of omissions. To
test this idea we performed an exploratory analysis comparing conditions in which the informant always taught just one
function, with his knowledge base ranging from one to fourfunctions across conditions. We used the difference between
the two “Overall” questions as the DV (as opposed to themean) in a one-way ANOVA and found a significant main
effect of prior knowledge (F(2, 527) = 31.9, p < .001). Thatis, the less the teacher knew, the more likely participants
were to give a more generous rating for the teacher. Whenthe teacher taught just one function but knew just one or two
functions, people significantly increased their ratings relative
to their initial rating (K1: µdi f f (σ) = 0.77(1.1),t(145) =8.4, p < .001; K2: µdi f f (σ) = 0.25(.95),t(256) = 4.1, p <0.001). Conversely, when the teacher knew all functions but
just taught one function, participants gave a harsher rating in
the final question relative to their initial rating (µdi f f (σ) =−.12(.63),t(126) = 2.1, p = .035).
Model Fit The two free parameters in the model—the difference in the utility of knowing a high vs. a low valuefunction, and the cost the teacher incurs by communicating afunction—were fit to the mean of participants’ estimates of
the teacher’s overall ability. As seen in Figure 2C, the resulting model well predicts these ability estimates (R2 =0.96).

General Discussion
Here we presented a formal Bayesian account of how alearner might evaluate a teacher. Our model considers theamount of information a teacher provided, the value of that
information, and what he knew (but didn’t show) to infer his“quality” as a teacher. Going beyond using the amount ofcommunicated information as a proxy for evaluating teachers, our behavioral results showed that human learners spontaneously consider both the reason behind a teacher’s omission (i.e., he didn’t show because he didn’t know) as wellas the consequences for the learner (i.e., teaching X is morevaluable for the learner than teaching Y). Participants appropriately penalized or exonerated a teacher, generating gradedratings that reflect the teacher’s quality in ways that are consistent with our formal analysis.
Both knowledgeability and helpfulness are important traitsof good teachers, but not everyone around us is equallyknowledgeable and helpful. In fact, violation of pedagogicalsampling occurs frequently in real-world learning contexts.
For instance, a well-intended, helpful teacher might provide
insufficient information because he didn’t have all the relevant knowledge. A fully knowledgeable teacher might omit
information because she thought it is not worth teaching.
Furthermore, we have an intuitive sense that a “good” teacher
is not just someone who provides everything he knows; it issomeone who knows what’s best for the learner and prioritizes teaching “what matters”. Critically, what matters forthe learner might not be the same as what matters for theteacher. Although our model does not yet capture all theFigure 2: A. Mean teacher ability rating in all 15 conditions. B. The effect of value depends on whether or not the teacher
knew more than he taught. C. Comparison of behavioral data and the model predictions. Each colored point corresponds to abar in the same color in A. The fit between model and human data is R2 = 0.96.

complexities and sophistication in gauging someone’s quality as a teacher, it builds on, and importantly extends, priorcomputational work on pedagogical reasoning by assumingthat human learners integrate different sources of information to distinguish potentially misleading teachers from those
who are truly knowledgeable and helpful.
Our behavioral results were remarkably well predicted by
the model predictions. When the teacher knew all functionsof the device, people’s evaluations were rationally affectedby both the number and the value of the functions he taught;
the more he showed, and the more valuable the functions thathe showed, the higher people rated his teaching effectiveness. Furthermore, given two teachers who taught exactly thesame functions, people’s judgments reflected the knowledgeof the teacher; the less he knew (thus the less he omitted), themore generous the ratings were. While we used the averageof the first and the last questions for better reliability, theseeffects were robustly present even in people’s responses tothe first question (R2 = .93). These results suggest that human adults spontaneously consider both the potential intent
of the teacher’s omission and its consequences for the learnerto give nuanced, graded judgments about others’ qualities asteachers even without being explicitly prompted about theirintentions or knowledge.
One might wonder why we still observed an effect of number of demonstrated functions when the teacher taught all
he knew (see Naı̈ve Omission under Results). This is stillreasonable, because the teacher’s quality as judged by theobserver is still influenced by the pedagogical utility of thedemonstration. Consistent with what is predicted by ourmodel, in our behavioral task participants were asked to evaluate the teacher’s overall helpfulness, rather than the blameworthiness of the omission per se.
In the current model, the teacher’s quality was definedas the degree to which he considered pedagogical utility
when choosing demonstrations, with the assumption that thelearner would learn what was shown once. However, thisglobal weighting represents just one of many dimensions on
which teaching can be evaluated. Indeed, people’s evaluations of teaching may be even more nuanced than our model
presented here, and sensitive to factors that are left unexplored in our current work. For example, one of the hallmarks of a good teacher is sensitivity to the difficulty of thesubject material for the learner (i.e., “this is a difficult concept, so I should show this example multiple times”), as wellas the learner’s epistemic states, learning style, and abilities(i.e., “she is quick so I shouldn’t dwell on this”, or “I’ll skip
what he already knows”). Thus in some cases, omissions
might not only be forgivable but even desired, especiallywhen transmitting information can be costly (see Gweon,
Shafto, & Schulz, 2014 for children’s sensitivity to learner’s
epistemic states and cost of information). These abilities correspond to rich knowledge of the subject domain and a moresophisticated model of learners’ abilities and goals, both ofwhich were simple in the current model.
Furthermore, we have assumed that both the learner andthe teacher have identical utility functions. Indeed, this isoften not true in the real world, particularly in cases where
young children or students learn from adult teachers; children are often taught what adults think is valuable for children, rather than what children they themselves think is valuable (e.g., the value of math or history classes). Even within
learners and within teachers there are vast differences in how
they value and prioritize different kinds of knowledge. These
are subtle yet critically important considerations that go into
designing curriculums in real-world educational settings, andwhere future computational and empirical work can be useful in formalizing what constitutes “good teaching.”
In our current work, we have considered cases where ateacher omits functions with positive values. However, there
are cases where omission leads to negative consequences(e.g., a building manager who forgot to announce 3-day
water shutdown in your apartment). In fact, these are thekinds of omissions that can elicit the harshest evaluations,and perhaps the most indicative of informants who should
be avoided in the future. Understanding the asymmetries
in omitting positively-valued and negatively-valued information is an interesting topic for future work.
Recall that in our behavioral results, we observed a change
in people’s ratings between the first and the last questions(both of which were about his overall teaching abilities; see
The effect of prompting intention and knowledge under Results). Between these two questions, people answered questions that prompted them to think about the teacher’s intentions and his knowledge. Even though these prompts are notnecessary for accurate evaluation, the effect of these prompts
suggest that explicitly thinking about these factors can makethe differences even more pronounced. One interesting possibility is that people’s ability to think about others’ beliefs(Theory of Mind) is directly related to the extent to whichlearners consider the teachers’ knowledge and intentions.
Thus one might predict that people with impaired or low
Theory of Mind abilities might (a) show less considerationof the teacher’s knowledge and intent, but (b) benefit morefrom explicit prompting of these factors. Follow-up research
is under way to investigate these possibilities.
We note that although our behavioral experiments allowed
a precise manipulation of what was taught and what theteacher knew, it leaves an open question about whether students in real-world pedagogical settings are also sensitive tothese factors. Furthermore, our participants were third-party
observers of a teacher-student interaction rather than the students themselves. An interesting extension of this work is to
ask whether young learners, as students, also consider thesefactors to evaluate teachers in more realistic, live interactions
with a teacher.
As learners in the real world, we often face a challenge
of dealing with both uncertainty about the world as wellas the uncertainty about others’ qualities as teachers. How
real-world learners exploit diverse sources of information,
including their own exploration of the world, to simultaneously learn about both other people and about the world isan incredibly rich and exciting area for future computationaland empirical work. Our work provides an important step towards delineating the factors that we all consider in learningfrom whom to learn.

Acknowledgments
This work was supported by Varieties of Understanding grantfrom the John Templeton Foundation to HG and Jerome
Davis Research Fund at Oberlin College to IB.

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