Prior expectations in pedagogical situations
Patrick Shafto1 , Noah D. Goodman2 , Ben Gerstle1 , & Francy Ladusaw1
1

2

Department of Psychological and Brain Sciences, University of Louisville
Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology

Abstract
Much of human learning occurs in social situations, andamong these, pedagogical situations may afford the mostpowerful learning. In pedagogical situations, a teacher
chooses the concept that they are going to teach and theexamples that they use to teach the concept. If learners
know that a teacher is helpful and understands the implications, this could support strong inferences. In previous work, Shafto and Goodman (2008) proposed andtested a model of pedagogical data selection. We integrate special-purpose pedagogical expectations in thisframework, and derive a task that allows independent
assessment of pedagogical expectations. Two experiments contrast people’s expectations about pedagogical and communicative situations. The results showthat people’s expectations differ in these situations, andthat in pedagogical situations people expect teachers topresent generalizable and semantically coherent knowledge. We discuss the implications for modeling learningin pedagogical settings, as well as for understanding human learning more broadly.
Keywords: Pedagogy; Learning; Bayesian Model

Much of human learning occurs in social contexts. We
learn from siblings, parents, friends, and teachers by observing, imitating, and teaching. Among these social
learning settings, pedagogical situations stand out aspotentially the most important. Pedagogical situations
are situations in which one person, a teacher, chooses
information for the purpose of helping another person,a learner, arrive at some belief. Pedagogical situations
might provide uniquely powerful learning situations, especially if learners are privy to, and understand the implications of, teachers’ intentions to help.
Indeed, recent theories argue that an intuitive understanding of pedagogical situations may be what sets us
apart from other animals (Csibra, 2007). Under this
proposal, learners’ intuitive understanding of pedagogical situations consists of two components: inferences
how teachers choose examples to teach a concept, andexpectations about what kinds of concepts teachers are
more likely to teach.
The issue of how teachers choose information andlearners’ understanding of these situations have been investigated in detail (for a review, see Csibra and Gergely,2009). Recently, Shafto and Goodman (2008) have proposed a computational model of reasoning in pedagogical1
Please address correspondence to Patrick Shafto,
p.shafto@louisville.edu

situations. This account provides a formal explanation of
why and how teachers decide which examples to choose,and how learners can capitalize on the teacher’s intent
to make stronger inferences.
Researchers have also argued that young children
come prepared with expectations about what kinds ofknowledge to expect in pedagogical situations. Specifically, Csibra and Gergely (2009) argue that very young
children expect that knowledge provided in pedagogical contexts is semantically generalizable. For instance,
Topal et al. (2008) show that children make A-not-B
errors in pedagogical contexts, but not in neutral contexts. They argue that the perseverative errors are a consequence of children misinterpreting initial pedagogicaldemonstrations as indicating that the A box is where theball belongs. While these results are quite compelling,
they contain influences of both the learner’s inference
about the teachers’ choice of data, and the learners’ expectations about what kinds of properties are likely to
be taught.
In the current paper, we investigate the hypothesis that people expect semantically generalizable knowledge in teaching situations. We begin by discussing therole of prior knowledge in pedagogical reasoning, andhow this can be integrated with Shafto and Goodman’s(2008) model of pedagogical reasoning. We then use
this framework to develop a method for separating therole of pedagogical priors from pedagogical data selection. Two experiments use this method to investigate
whether adults expect generalizable knowledge (Experiment 1) and whether adults expect semantically coherent knowledge (Experiment 2). In each case, we contrast
pedagogical situations with communicative situations toaddress whether these prior expectations are specific topedagogical contexts. We conclude by discussing implications for modeling human learning and understanding
reasoning in social situations.

The role of priors in pedagogicalreasoning
The proposal that learners expect generalizable information can be integrated naturally into a Bayesian reasoning framework. From this perspective the problem oflearning is one of inferring the probability of different
hypotheses, h, given observed data, d. Bayes’ theorem
provides a way of updating our posterior beliefs about
hypotheses, P (h|d), given prior beliefs, P (h), and as-2182

sumptions about how data are sampled, P (d|h),
P (h|d) ∝ P (d|h)P (h).

In standard Bayesian learning, it is typically assumed
that the prior, P (h), is determined by some stochastic
generative process, and data are sampled from a hypothesis, P (d|h), randomly; however, these standard assumptions are not appropriate for pedagogical situations. In
pedagogical situations, a teacher chooses both the hypothesis that they are going to teach, and the data thatthey use to teach the hypothesis. It is, therefore, reasonable for the leaner to expect that teachers’ choices are
not random, but are instead purposeful.
Recent work has formalized pedagogical sampling –how teachers may choose data for the purpose of teachinga hypothesis (Shafto and Goodman, 2008). This model
suggests that learners may use the knowledge that theteacher is choosing data for the purpose of teaching thema hypothesis, to replace the random sampling with anassumption that the teacher will choose data that tend
to make the learner believe the correct hypothesis,
P (d|h) ∝ P (h|d).

Grasshopper

(1)

(2)

Their results show that this pedagogical sampling assumption allows for stronger inferences than random
sampling. They also provided evidence that people’s inferences differ in pedagogical and ostensibly random contexts, and the pedagogical model accurately predictedpeople’s intuitive pedagogical inferences.
Here, the question is whether people have prior expectations about which kinds of hypotheses should betaught. That is, are there a specific set of prior probabilities, P (h), that apply to teaching situations. Intuitively, the question is whether learners expect teachers
to choose particular hypotheses that are important or
worthy of teaching. For example, more general hypotheses might be expected because the knowledge is more
likely to be applicable in future situations. Formally, we
capture these expectations as a utility function which
defines how hypotheses are weighted in pedagogical situations, U (h; pedagogy). Integrating this prior into Equation 1 would allow us to capture how prior expectations,specific to pedagogical situations, affect inferences.

Investigating pedagogical priors
In this paper, we attempt to answer two questions: (1)what are the characteristics of people’s pedagogical priorexpectations, and (2) are these expectations specific topedagogical situations? To address the second question,
we need to choose appropriate (non-pedagogical) control
conditions. To address the first question, we can ask
people to make judgments about which of two teachers
they would rather have teaching them, while varying theparticular hypothesis that each is teaching. However,given our above analysis, this requires that we separate
the contribution of pedagogical data sampling from thecontributions the pedagogical prior expectations.
Judgments about the teachers can be formalized as aninference about whether the teacher chooses hypotheses

Ant

IguanaOwl

Frog
OstrichLeopardSeal

Figure 1: A tree representing the intuitive taxonomic
relationships among 8 animals.

that the learner expects to be taught about. More formally, we assume a parameter, θ, which specifies how
systematically a teacher chooses her examples. The
probability of choosing hypotheses depends on how systematic the teacher is,
P (h|θ) ∝ e−θU (h) ,(3)where systematic teachers tend to choose hypotheses
that have higher utility (Luce, 1959).
The learner can then infer how systematic a teacher
is, given some data,X
P (θ|d) ∝P (d|h)P (h|θ)P (θ)(4)
hwhere P (θ) is a prior distribution specifying whether
people tend to be systematic or not. This equation statesthat teachers are considered systematic to the degree
that they choose hypotheses that agree with the learner’s
prior expectations, as specified by Equation 3. However,in this inference, the influence of P (h|θ) is not isolated.
To see how we could isolate the effects of P (h|θ), consider hypotheses about properties of the the animals in
Figure 1. The set of possible hypotheses can be defined
extensionally, by enumerating all possible combinationsof animals that have or do not have the property. For example, one hypothesis about a property is {ostrich=true,
owl=false, grasshopper=false, ant=false, iguana=false,
frog=false, leopard=false, seal=false}. Our goal is toeliminate the contribution of the sampling assumption,
P (d|h). Assuming that we want to teach the learner
the hypothesis that all of the animals have a particularproperty, how would we choose which animal or animals
to provide as examples? By presenting all of the data –each animal labeled as having or not having the property– we essentially choose one hypothesis. Thus, the contribution of the sampling of data is to simply indicate2183

a particular hypothesis. Formally, the P (d|h) = 1 forthe true hypothesis, and zero for all others. Equation 4
reduces to,
P (θ|d) ∝ P (h|θ)P (θ).
(5)
Given the fully labeled data, the learner’s judgments
about the teacher’s systematicity depend on whether thelearner expects that hypothesis to be chosen, and theirprior expectations about systematicity.
To isolate the influence of learners’ prior expectations about hypotheses P (h|θ), we can ask learners tochoose between two teachers. Because each teacher isequally likely to be systematic a priori, judgments about
which of two teachers is preferred isolate the effects ofa learner’s prior expectations. Formally, the judgment
becomes a ratio of two inferences, each individually specified by Equation 5,
P (θ1 |d1 )P (h1 |θ)P (θ)
P (h1 |θ)∝=.
P (θ2 |d2 )P (h2 |θ)P (θ)
P (h2 |θ)(6)

In the following, we present two experiments in whichpeople make judgments about which of two teachers theywant to have teaching them in the future (presumably
the one that chooses a hypothesis that is more consistent
with their expectations). In our investigations, we have
two goals: (1) identifying the prior expectations thatpeople bring to pedagogical situations, and (2) establishing whether these expectations are unique to pedagogical situations. The experiments test two claims related
to prior expectations about pedagogical situations: thatlearners expect more generalizable information, and thatlearners expect semantically coherent information.

Experiment 1: Testing the bias toward
generalizability
Experiment 1 investigated whether people have an expectation that teachers would teach generalizable information. To investigate this question, we choose a domain for which we have a good understanding of thepossible hypotheses, the domain of animals. Figure 1
shows the animals, and the intuitive taxonomic relations
among these animals.2 We operationalize generalizable
concepts here as a concept that is true of a broader classof animals.
To investigate whether people expected generalizableknowledge, we presented participants with scenarios inwhich pairs of teachers taught concepts of different levelsof generality. The generalizable teacher taught a property that was consistent with the tree structure and wastrue of a greater number of exemplars. For instance, thegeneralizable teacher might teach a property that wastrue of all 8 animals, while the less generalizable teacher
might teach a property that was true of only ostriches
and none of the other animals. If people expect teachers to teach generalizable information, we expect to findthat people choose the teacher who teaches properties
that were true of broader sets of examples.
2
The tree was derived using the tree learning algorithm
and a subset of the animals used in Kemp and Tenenbaum(2008).

Methods:
Participants: Twenty-four university undergraduates
participated in this experiment in exchange for course
credit. Participants were randomly assigned to the pedagogical or the communication scenarios.
Procedure: In the pedagogical situation, people were
presented with a series of questions asking them to decide which of two teachers they would like to learn fromin the future. Each teacher was presented as teaching
about a novel enzyme, e.g. “Teacher 1 is teaching about
enzyme P23T.” The names of the enyzmes were random
combinations of letters and numbers. This was followedby lists indicating which of the eight animals had the
enzyme and which did not. Each question contrasted
two teachers, where teachers differed in the generality ofthe properties taught. For instance, one teacher mightteach a property that was true of owls, ostriches, leopards, and seals, but not of grasshoppers, ants, iguanas,
and frogs, while the other was teaching a property thatwas true of all eight animals. Paired teachers always
taught concepts where one was a subset of the other, so
the more generalizable concept included all of the positive examples of the property in the less generalizable
concept, with additional positive examples (e.g. ostrich
versus ostrich and owl). Participants indicated whichteacher they would rather have teaching them about new
enzymes using a Likert scale ranging from −10 to 10,
where the extremes indicated the teacher on the left or
the right and zero indicated indifference. Participants
rated all possible pairings of teachers, resulting in a total of 34 questions. Order of the questions, as well as
the side (right or left) of the more general concept, wererandomized.
The communication condition was identical to thepedagogical condition, with the exception of some of thewording. Instead of teaching about enzymes, the situations described people who were talking about enzymes.
For example, “Person 1 is talking about enzyme P23T.”
Additionally, participants were asked to provide ratings
about which one they would rather talk to in the future. Otherwise, the questions and response sheets wereidentical.

Results & Discussion
We coded people’s judgments as positive if they were in
the direction of the more generalizable teacher and negative if they were in the direction of the teacher with theless general property. To test whether people expected
more general properties, we compared the average ratings to chance (zero). In the pedagogical condition, people chose the teacher with the more general information,mean = 0.66, t(407) = 2.06, p < 0.05. In contrast, in thecommunication condition, people choose the less general
information, mean = −0.56, t(407) = −2.09, p < 0.05.
The difference between the two conditions was significant, t(814) = 2.84, p < 0.01. These results suggest thatpeople expect that more general properties will be taught
in pedagogical situations, in contrast with communicative settings, where people expect less general properties.
To follow up on these results, we investigated the pat-

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1.5

-1.5

Rating6

RatingRating6-4Pedagogy

Communication-40

Difference in Generality

(a)

(b)

300

0

Difference in Generality

300

(c)

Figure 2: Experiment 1 results: (a) Average human ratings in the pedagogy and communication conditions. Positive
ratings indicate the more generalizable teacher. (b) Scatterplot showing the relationship between the difference ingeneralizability for pairs of teachers (x axis) and people’s ratings (y axis) for the pedagogy condition. The strengthof people’s ratings increases with an increasing difference in generalizability, r = 0.51, indicating that they expect
more generalizable concepts in pedagogical settings. (c) Scatterplot showing the relationship between difference ingeneralizability and people’s ratings for the communication condition. The strength of ratings decreases with anincreasing generalizability, r = −0.66, indicating that people expect less generalizable concepts in communicative
settings.
tern of ratings for individual items. If people choose
more generalizable concepts, then pairs for which there
was the greatest gap between the more and less generalizable teacher should have the strongest ratings. To investigate this question, we needed to quantify how generaleach hypothesis was. We consider two possible measuresof generality: the number of positive examples, and thesum of the distances among items in the tree. To test
whether ratings indicated an expectation that properties would be generalizable, we collapsed individual judgments into a single average rating for each question, resulting in 34 ratings. To investigate which measure ofgenerality best predicted people’s judgments, we conducted a stepwise regression with item averages as thedependent variable. The independent variables included
the number of positive examples in more general concept,
the number of positive examples in the less general concept, the difference in number of positive examples, aswell as the summed distances for the more and less general concepts, and the difference in the summed distance.
The two difference scores allowed us to test whether people’s judgments take into account both teachers, or justa single teacher when making their judgments. Stepwise regression greedily selects the variable that accounts
for the greatest variance, and iterates until no variables
account for significant variance. Analysis of the pedagogy condition showed that the difference in summed
distances accounted for the greatest variance, r = 0.51,
F (1, 32) = 11.49, p < 0.01, and that no other variables
accounted for significant residual variance. The correlation indicates that the bigger the difference in generalizability was, the stronger people’s ratings were toward
the more generalizable teacher. In contrast, regression
analyses on the communication condition showed thatwhile the difference in summed distance was a significant predictor of ratings, the relationship was negative,r = −0.66, F (1, 32) = 24.52, p < 0.001. This suggeststhat in communicative settings, people’s expectations
about generalizability are the opposite of their expectations in pedagogical settings.

The number of positive examples, while a straightforward measure of generality for this task, is undesirable
for two reasons. First, if this leads to an accurate characterization of people’s inferences, then one might wonder to what degree the results are a consequence of task
demands (given that people were answering questions
about lists of animals). Second, the number of positive
examples is not a very good measure of generality because it bears no necessary relationship with actual semantic generalizability. As can be seen in Figure 1, many
possible sets with the same number of positive examples
differ markedly in their coverage of the tree. Instead,
we prefer to measure the generalizability of a concept interms of the sum of distances between all pairs of positive
examples. This provides a measure that is not subject totask demands, and is related to the semantic generalityof the concept. Our analyses show that distance in thetree provides a better description of people’s behavior,providing evidence that people’s judgments do not simply reduce to task demands, and that their judgments
are based on semantic generalizability.

It appears that people have strong prior expectations
that they bring specifically to pedagogical situations. In
pedagogical situations, learners expect that teachers will
choose to teach generalizable information. In contrast,
when in communicative situations, people expect thatspeakers are likely to talk about specific information.
Our analyses showed that people’s judgments are better
predicted by distance in a semantic tree, consistent witha bias toward semantically generalizable information.

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Experiment 2: Testing the bias toward
semantic coherence
Experiment 2 investigated whether people have an expectation that teachers will choose semantically coherent concepts. To investigate this, we presented participants with scenarios in which two teachers each taught
concepts with two positive exemplars. The semantically coherent teacher taught a property that was trueof two tree-consistent exemplars, such as owl and ostrich. They were contrasted with a semantically incoherent teacher who taught a property that was true of twotree-inconsistent exemplars, such as ostrich and leopard.
If people expect teachers to teach semantically coherent concepts, we expect to find that people choose theteacher who teaches tree-consistent properties.

Methods:
Participants: Twenty university undergraduates participated in this experiment in exchange for course
credit.
Procedure: The procedure was identical to that usedin Experiment 1 with the exception of the questions used.
Each scenario provided information taught by two teachers. All properties were true of two animals, but wereabsent in the other six. In each scenario, one teacher
taught a property that was semantically coherent – itwas consistent with the structure of the tree – and theother taught a property that was semantically incoherent – it was inconsistent with the structure of the tree.
For instance, a semantically coherent property might betrue of owls and ostriches, but no other animals. Contrarily, a semantically incoherent property might be trueof owls and leopards but no other animals. Questions
were designed such that semantically coherent pairs were
contrasted with all minimally different semantically incoherent pairs that overlapped one animal. For example,
owls and ostriches were contrasted with owls and leopards, owls and seals, ostrich and leopards, and ostrichand seals. This resulted in a total of 16 questions. Orderof the questions, as well as the side of the semanticallycoherent pair (left or right), were randomized.

Results & Discussion
Do people expect teachers to teach semantically coherent
concepts? To address this question, we coded people’s
ratings as positive numbers if they were in the direction of the semantically coherent teacher, and negative
numbers if they were not. We then ran separate t-tests
comparing the means in the pedagogical and communicative conditions to zero. In the pedagogical condition, people tended to choose teachers of semanticallycoherent concepts, mean = 0.97, t(159) = 2.04, p < 0.05,
one-tailed. In the communication condition, people
also chose teachers of semantically coherent concepts,
mean = 2.21, t(159) = 6.76, p < 0.001. The difference between the two conditions was also significant,t(308) = 2.16, p < 0.05.
To further investigate the role of semantic coherence,
we computed the distance between all of the positive

examples in each scenario (see Figure 1). If people expect semantically coherent concepts, then more semantically coherent pairs – those with shorter distances –should have the strongest ratings. We ran a stepwise regression with people’s ratings as the dependent variable,
and independent variables including distance between
the positive examples in the more and less coherent sets,
and the difference in the distances. For the pedagogical condition, the distance between positive examples
in the coherent concept was the only predictor selected,
r = −0.70, F (1, 14) = 13.24, p < 0.01. Of the coherent hypotheses, the teachers teaching the more coherent
concepts were rated more strongly. For the communication condition, regression analyses showed that distance
between positive examples of coherent pairs did notstrongly predict people’s ratings, r = 0.23, F (1, 14) =0.80, p > 0.3. 3Interestingly, unlike in Experiment 1, people’s judgments in Experiment 2 were best predicted by the coherence of the more coherent hypothesis alone (as opposedto the difference in coherence). This suggests that thesemantically incoherent hypotheses did not play a largerole in people’s judgments. This may reflect an explicit
judgment that these cases are so unexpected that they,in effect, have zero weight.
The evidence suggests that people expect teachers toteach semantically coherent concepts: overall, people
chose teachers of more semantically coherent concepts,
and the strength of people’s ratings decreased as thestrength of coherence decreased. The evidence also suggests that people’s expectation of coherence may apply
across more than just pedagogical situations. Resultsfrom the communication condition showed that people
tended to choose the more coherent speaker, but thestrength of their ratings was not related to the degreeof coherence. These results suggest that people’s expectation of semantic coherence may not be limited topedagogical situations.

Discussion
Pedagogical situations play a central role in humanlearning. In pedagogical situations, teachers choose
which concepts to teach and which examples to use toteach the concept. We have presented an extension of
Shafto and Goodman’s (2008) model of pedagogical data
selection that incorporates specific expectations aboutpedagogical situations. Using this framework, we havederived a method for isolating the effects of prior expectations about pedagogical situations. The results ofExperiment 1 showed that people expect teachers to provide generalizable knowledge, and that this expectation
does not apply in more general communicative settings.
The results of Experiment 2 showed that people expect
teachers to provide semantically coherent information,although this appears not to be specific to pedagogicalsituations. Taken together, these results provide evidence that people have specific expectations—intuitive3
A separate stepwise regression showed that none of the
independent variables accounted for significant variance inpeople’s judgments.

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7

RatingRatingRating

7

2

8

2

8

-3

Pedagogy

Communication-5

Distance(a)-5Distance(b)(c)

Figure 3: Experiment 2 results: (a) Average human ratings in the Pedagogy and Communication conditions. Positive
ratings indicate the more semantically coherent teacher. (b) Scatterplot showing the relationship between distanceamong positive examples of coherent hypotheses (x axis) and people’s ratings (y axis) in the pedagogy condition.
People’s ratings increase with decreasing distance, r = −0.70, suggesting that people expect coherent hypotheses inpedagogical settings. (c) Scatterplot showing the relationship between distance among positive examples of coherent
hypotheses (x axis) and people’s ratings (y axis) in the communication condition. People’s ratings are only weakly
related to distance, r = 0.23.
theories of pedagogical situations.
Our results provide additional evidence in supportof Csibra and Gergely’s (2009) claim that people expect generalizable information in pedagogical contexts.
Where previous results focused on young children, ourresults suggest that this expectation continues into
adulthood. Our results also provide evidence that semantic coherence, while expected in pedagogical situations, is not specific to these contexts. Rather, the expectation of semantically coherent concepts extends tocommunicative, as well as pedagogical situations.
Here we have focused on learners’ expectations, but
for these pedagogical expectations to be reasonable, it isimportant that teachers meet their expectations. Specifically, do people choose to teach concepts that are moregeneralizable and more coherent? If so, what are theimplications of these matching (or mismatching expectations) in terms of the kinds of concepts that can belearned, the speed at which they are acquired, and therobustness of knowledge transmission? Future research
will aim to answer these questions.
Our experiments have provided information about
people’s prior expectations in pedagogical situations, but
it is also important to explain why people have these biases. There is work to be done in formalizing computational models that explain why certain hypotheses would
be more or less likely to be taught. This may not turnout to be entirely straightforward because while there isa reasonable motivation for teaching generalizable concepts, there are also motivations for teaching other kindsof concepts. For instance, one might also want to teachsparse concepts because they may be difficult to discoveron one’s own. Further empirical research may help narrow down the possibilities and provide guidance for more
explanatory models.
More generally, previous approaches to modeling human learning have focused on a single unitary set ofprior expectations that apply generically across situa-

tions (but see Shafto et al., 2006). However, this approach seems obviously too simple. We all intuitively
understand that we have different expectations that apply when, for example, we talk to children as opposed
to adults. Pedagogical situations are but one case of amore general problem. Understanding how social situations affect learning will require understanding how different contexts affect both learners’ prior expectations
and learners’ assumptions about how information is selected.

Acknowledgments
Thanks to Russell Warner and Carissa Shafto for helpful comments and suggestions during the writing process.

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