Teaching Games:
Statistical Sampling Assumptions for Learning in Pedagogical Situations
Patrick Shafto
University of Louisvillep.shafto@louisville.edu

Noah Goodman
Massachusetts Institute of Technology
ndg@mit.edu

Abstract
Much of learning and reasoning occurs in pedagogical situations – situations in which teachers choose examples withthe goal of having a learner infer the concept the teacher has
in mind. In this paper, we present a model of teaching andlearning in pedagogical settings which predicts what examples
teachers should choose and what learners should infer given ateachers’ examples. We present two experiments using a novel
experimental paradigm called the rectangle game. The firstexperiment compares people’s inferences to qualitative model
predictions. The second experiment tests people in a situation
where pedagogical sampling is not appropriate, ruling out alternative explanations, and suggesting that people use contextappropriate sampling assumptions. We conclude by discussing
connections to broader work in inductive reasoning and cognitive development, and outline areas of future work.

Much of human learning and reasoning goes on in pedagogical settings. In schools, teachers impart their knowledge to students about mathematics, science, and literature
through examples and problems. From early in life, parents
teach children words for objects and actions, and cultural andpersonal preferences through subtle glances and outright admonitions. Pedagogical settings – settings where one agent ischoosing information to transmit to another agent for the purpose of communicating a concept – dominate human learningand reasoning.
If learners’ assumptions about how teachers sample information reflected this purposeful sampling, then learners
might be able to make much stronger inferences in pedagogical situations. Sampling assumptions are assumptions that alearner makes about the source of data, in order to better interpret the evidence for statistical learning. Recent researchsuggests that even infants are sensitive to the sampling processes that underlie observed data (Xu & Tenenbaum, 2007)
and young children make qualitatively different inferences
when data are sampled by a teacher (Gergely, Egyed, & Kiraly, 2007).
Computational models of human learning have not focusedon the sampling processes that generate observed data, andwhen they do the assumed processes are relatively simple.
For example, Fried and Holyoak (1984) modeled category
learning by assuming examples are generated uniformly at
random (weak sampling) and Tenenbaum (1999) modeled
learning from positive data by assuming examples are generated at random from the true concept (strong sampling).
Even simple assumptions gain a lot for a learner. For ex-

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Figure 1: Possible rectangle game scenarios. The top row shows apossible rectangle concept, and two possible pairs of examples that ateacher might choose to communicate to a learner. The bottom row
shows possible examples a learner may observe, and two possible
guesses about what rectangle the teacher had in mind. The middle
column shows better choices than the right column.

ample, Xu and Tenenbaum (2007) showed that the strong
sampling assumption allows learning the meanings of words
from positive examples alone. However, neither weak nor
strong sampling captures the purposeful sampling that underlies pedagogical situations. In this paper, we introduce apedagogical sampling model that formalizes which examples
teachers should give learners to most help them, and what
learners may infer from these helpful examples.
Consider a simple example which we call the rectangle
game: a game where the teacher thinks of a rectangle on aboard, and tries to communicate that concept to a learner bychoosing to label points inside and/or outside the rectangle(cf. Tenenbaum, 1999). In the rectangle game, the learner’s
job is to try to infer, given the labeled examples chosen by
the teacher, what rectangle the teacher is thinking of. Figure 1 presents potential teacher and learner scenarios. In eachcase, there seem to be choices which are obviously better thanothers. As a person trying to teach someone the rectangle inblue (top left), the examples in the middle panel seem better
than those on the right. Similarly, as a learner, given the examples in the bottom left, the rectangle in the middle panel
seems like a better guess than that on the right. Notice that inboth cases the examples on the top and the rectangles on thebottom are possible, however, our intuition tells us that the
middle panels are better guesses than the right panels. Peda-

gogical sampling results in samples that are representative of
the concept (Tenenbaum & Griffiths, 2001), in contrast withweak and strong sampling, which choose examples randomly.
In this paper, we formalize the problem of pedagogical reasoning from the perspectives of teacher and learner. For theteacher, the problem is to choose the examples that will mosthelp the learner infer the correct concept. For the learner, theproblem is, given that the teacher is choosing helpful examples, infer the correct concept. In two experiments, we compare model predictions to human behavior in a novel experimental paradigm (the rectangle game), showing strong correspondence between model predictions and human data. We
conclude by addressing implications of pedagogical sampling
and identifying areas of future work.

A computational model of pedagogicalsampling
We formalize pedagogical reasoning as an inference problem
based on the twin assumptions that learners and teachers act
as (approximately) rational agents. The rational learner assumption is that a learner will update their beliefs, given new
examples, according to Bayesian inference (a description of
optimal belief updating),p(h|d)learner ∝ p(d|h)teacher p(h).

(1)

The rational teacher assumption is that teachers choose examples that tend to increase the learner’s beliefs in the correct
hypothesis. We may formalize this via a Luce decision rule
(Luce, 1959),
p(d|h)teacher ∝ (p(h|d)learner )α ,(2)where the steepness parameter α governs the greediness ofthe teacher. (When α=0, pedagogical sampling recovers
weak sampling, as α becomes large the teacher chooses thebest examples.) Because Equations 1 and 2 are linked (withthe optimal teaching behavior depending on the learner, andvice versa), rational pedagogical reasoning is a solution tothis system of equations. 1
To understand this model it helps to consider one way ofsolving the system of equations: fixed-point iteration. Imagine that you are the learner, and wish to update your beliefs. To do so you will need an estimate of the likelihoodp(d|h)teacher of seeing the examples you are given. You canestimate this likelihood by assuming the teacher is rational—
Eq. 2—but to do this you need an estimate of the p(h|d)learnerused by the teacher. If you assume the teacher assumes thatyou are rational, you can use Equation 1 as such an estimate.... This recursive reasoning could carry on forever, but
1 Solutions

to this system of equations will depend on thelearner’s prior beliefs, p(h). For the current purposes, where we are
dealing with a completely novel game, we assume both the teacher
and the learner know that all hypotheses are equally likely. In general, it is reasonable to assume that the teacher makes a reasonable assumption about the learner’s prior. Interesting questions arise
when the teacher is uncertain or incorrect about the learner’s prior;these questions are beyond the scope of this paper.

eventually the estimated values from repeatedly using equations 1 and 2 will no longer change—we then say that theprocess has iterated to a fixed point, and this fixed point will
necessarily be a solution to the system of equations defining
rational pedagogical reasoning. Thus we can understand themodel as capturing the outcome of a recursive mental reasoning process, based on twin rational agent assumptions. However, it is worth emphasizing that rational pedagogical reasoning describes the outcome of this process (or rather the solution to the system of equations), and it is entirely possible that
this reasoning may be implemented by a psychological process that doesn’t require any explicit recursive reasoning. 2
There is one additional complication in modeling theteaching games used below. In addition to observing the location of positive and negative examples, learners also observe
the strategy the teacher used – how many positive and negative examples they use. The learner can observe the strategy
chosen by the teacher, and we simplify by treating this choiceof strategy as uninformative. Inference for both teacher andlearner are then simply conditioned on the observed strategy(number and kind of examples).
Applying this model to the examples in Fig. 1 results inpredictions that are consistent with intuition. In the case of
two positive examples, the prediction is that the teacher will
generally place examples in opposite corners of the rectangle,
and the learner will infer a rectangle such that the examples
are near opposite corners. To understand why this is a solution to equations 1 and 2, consider the recursive reasoning
described above (idealized to avoid complications of uncertainty): if the learner assumes that the teacher will choose examples at opposite corners, then Eq. 1 implies that the uniqueinference made by the learner is the tightest rectangle aroundtwo examples; if the teacher assumes that this is what thelearner is doing, then, according to Eq. 2, the teacher willusually choose examples in opposite corners of the true rectangle. In the case of one positive example and one negative
example, the reasoning is similar: the learner assumes that theteacher will choose a negative example close to the boundaryof the rectangle, enabling the learner to rule out larger rectangles; in turn the teacher will chose such examples under theassumption that this is how the learner will reason. We test
the model predictions in the following experiment.

Experiment 1: The rectangle gameIn this experiment, people played the rectangle game described in the introduction. People played the roles of teacher,choosing the examples given a rectangle, and learner, guessing a rectangle given examples. In our analyses, we will investigate how the models predict the qualitative and quantitative patterns observed in the human data. For the teaching task, we will ask whether the pattern of responses is ran2 To compute model predictions we use the fixed-point iteration
formulation of the model. That is, we initialize with the learner assuming weak sampling, then iterate Equations 1 and 2 until convergence. This is a straightforward, though computationally expensive,algorithm.

dom, as predicted by weak and strong sampling. We will alsocharacterize people’s preferred pattern of responses, and ask
whether the pedagogical sampling model correctly predicts
the pattern of examples generated by people. For the learningtask, we will ask comparable questions: are learner’s inferences consistent with the assumption that examples are randomly sampled? If not, do learner’s inferences reflect knowledge about the pedagogical nature of the data?

(a) Positive examples

MethodParticipants. 18 University of Louisville undergraduates
participated in exchange for course credit.
Design. The experiment consisted of two parts: teaching
and learning. Because pre-testing showed that participants
understood the learning task better when presented with theteaching task first, the two parts were presented in fixed order.
Procedure. Experiments were run using MATLAB. Participants were seated at a computer and told that they were going to learn about a game called the rectangle game. In thegame there is a teacher and a learner. It is the teacher’s job
to help the learner guess what rectangle he is thinking about
by choosing helpful examples, points that can be inside oroutside the true rectangle.
In the teaching task, participants were shown a rectangle
and asked to choose the best one, two, or three examples byclicking on the screen. A green circle automatically appeared
if the click was inside the rectangle, and a red X if the click
was outside the rectangle. Participants were shown 90 rectangles of various sizes and positions. Order of presentationof sizes and positions of the rectangles varied randomly.
In the learning task, participants were shown between one
and three examples and asked to draw the rectangle that they
thought the teacher was thinking of, by clicking on the screen
and dragging with the mouse. Examples were pre-chosen to
include all combinations of positive and negative examples.
Examples were generated based on a small set of patterns, andposition on the screen and distance between the examples was
randomized. The learning task consisted of 92 trials in total.
When participants completed the task, they were debriefed
and thanked.
Model implementation. To model the rectangle game, weused a hypothesis space of rectangles based on a 6x6 discretized approximation to the full board. Hypotheses included all rectangles from 2x2 up to 5x5, at all possible locations on the grid. 36 possible points, corresponding to all
possible blocks on the 6x6 board were considered. Examples were sets of positive, negative, or mixed pairs of thesepoints. To attempt to minimize strong boundary effects resulting from our discrete approximation, we modeled inferenceson an extended board upon which the rectangle or examples
were centered.

Results & Discussion
This experimental paradigm generates an extremely rich data
set. For teaching data, people could choose any point(s) on(b) Negative examples
Figure 2: Distributions of examples in the teaching task for (a) positive examples and (b) negative examples. Pictured from left to rightin each panel are the predictions of strong and weak sampling, theobserved human data, and the predictions of pedagogical sampling.
For the models, figures display the probability of an example in eachblock. For human data, the proportion of positive examples in eachlocation is plotted. People strongly preferred to give positive examples in the corners of the rectangle and negative examples near theboundaries, as predicted by pedagogical sampling.

the board. For the learning data, people could choose any
number of rectangles that vary in size and location such thatthey are consistent with the observed examples. Analyzing
the data required collapsing across examples to achieve adequate summaries of the patterns of behavior elicited. Also,
due to space constraints, we present only the subset of theteaching and learning data that include two examples.
We will consider performance on the teaching task first,
then the learning task. For the teaching task, we will consider
whether people’s data conform to the predictions of weak andstrong sampling or pedagogical sampling by separately analyzing the distributions of the positive and negative examples.
We will then investigate the specific pairs of examples chosen
by people, and compare these to the pedagogical sampling
model.
Pedagogical sampling predicts that for positive examples,examples in the corners are more informative than examples
in the middle, or along a single side. Both weak samplingand strong sampling predict that positive examples are distributed uniformly at random. To test these predictions, weanalyzed the examples that people generated, dividing eachrectangle into a 3x3 grid. Grids were normalized based on
the size of the rectangle, so the grid was finer for smaller
rectangles than for larger ones. This allowed us to ignore
the size of the rectangle and focus on the relative position of
the examples. Frequencies of examples in each area of thegrid were tallied and the proportion of examples in each location are shown in Figure 2a, middle panel. The left andright panels show the locations predicted by weak samplingand strong sampling (left) and pedagogical sampling (right).
Note that strong and weak sampling predict no differences
across locations – data should be distributed uniformly at
random. The examples people generated were highly nonrandom, χ2 (8) = 645.05, p < 0.0001. The pedagogical sampling predictions, right panel, are based on an average overall 3x3 hypotheses, and show that the model predicts a strong

Figure 3: Pairs of examples chosen in the teaching task for a randomly chosen rectangle, divided by type of pair. The top row contains pairs of negative examples, which are characterized by theirclose proximity to the boundaries, generally on opposite sides orcorners. The middle row shows mixed examples marking a single or
opposite corners. The bottom row shows pairs of positive examples,
which are predominantly examples in opposite corners.

preference for positive examples in the corners of the rectangle. A qualitatively similar pattern is observed in the humandata. Indeed, this qualitative correspondence is reflected ina strong correlation between model predictions and humandata, r = 0.98.
For negative examples, pedagogical sampling predicts
strong effects of distance – the most helpful negative examples are those that are near the boundaries. Weak sampling predicts that examples should be distributed uniformly
at random, while strong sampling makes no prediction. We
analyzed people’s choices by classifying examples based on
the relative distance from the boundary of the rectangle tothe outside of the board. For each of the four sides, we divided the area from the edge of the rectangle to the edgeof the board into three bins. For one quadrant, the result
is 15 bins, 3 to each side of the rectangle of the rectangle, and 9 for the intersection of these bins extending diagonally from the rectangle. The results are shown in Figure 2b.
The examples people chose were not randomly distributed,
χ2 (14) = 1268.82, p < 0.0001. Again, the apparent qualitative agreement is supported by a strong correlation between
model predictions and human data, 0.86.
Of primary importance from the perspective of our modelis which pairs of examples people choose. Figure 3 shows arandom subset of the data from a randomly chosen rectangle.
Examples have been divided into pairs of negative examples,
mixed example pairs, and pairs of positive examples. For thenegative pairs, people tend toward either choosing examples
on opposite corners or on opposite sides, and these are thetwo most likely strategies according to the model. For mixed
pairs, people tend to either mark a single corner with a positive example on the inside and a negative example on the outside, or opposite corners. These, along with pairs that mark aside on the inside and outside are most likely according to themodel. For positive pairs, people use the examples to markopposite corners. Pedagogical sampling very strongly predicts a preference for this strategy over other possible pairs of
positive examples.
The results of the teaching task suggest that people choose
particular examples to communicate different concepts. Of
equal importance is whether learners take advantage of thisknowledge to make stronger inferences. To answer this ques-

Figure 4: Results from the learning task. Plots show the positions of
the teacher’s examples, relative to the rectangles drawn by learners
for positive (left) and negative (right) examples. The results showthat learners clearly understand that teachers are sampling data pedagogically – positive examples indicate corners of the correct rectangle, and negative examples indicate the boundaries.

tion, we turn to the data from the learning task.
Pedagogical sampling predicts that learners’ should know
the strategies that the teacher will use to communicate different concepts. Therefore, if learners use pedagogical knowledge to guide inferences, we expect learners should draw rectangles that recover the patterns observed in the teaching data
(see Figure 2). However, weak and strong sampling predict that positive examples should be randomly distributedwith respect to the rectangles inferred by learners. Figure 4
shows the observed distributions for positive and negative examples, relative to the rectangles inferred by learners. To
test whether the examples where randomly distributed, we
computed the frequencies of different example locations relative to the inferred rectangle, as described for the teaching task. The results indicate that the inferred rectangles
where not randomly positioned with respect the the positive
examples, χ2 (8) = 104.37, p < 0.001, or negative examples,χ2 (14) = 465.10, p < 0.001.
To test whether people’s inferences were in accordancewith the predictions of pedagogical sampling, we correlated
the frequencies of examples in the grid with the probabilities
predicted by the pedagogical sampling model. For both thepositive and negative examples, we found a strong positive
correlation between the model predictions and human data,r = 0.90 and r = 0.87, respectively.
Finally, we present representative examples of people’s inferences in the learning task and maximum a posteriori model
predictions in Figure 5. The results show that inferred rectangles are consistent with the generating process that people
showed in the teaching task. In particular, given positive examples, learners infer rectangles for which the examples arein opposite corners. Similarly, when given a mixed pair, people tend to infer that the rectangle extends up to the negative
example and when given two negative examples, people inferrectangles whose corners are just inside the examples. The
model predicts these effects for the positive and mixed cases,
while in the negative case the model does not capture peoples
inferences with the strength that people show.

Experiment 2: Random rectangle game
One possible alternative explanation for the results of thelearning task (entertained by psychologists and other cynics)
is that people simply prefer scenarios where examples lie atthe corners of the rectangles for perceptual or other reasonsFigure 6: Density plots showing the relative position of the observed
examples with respect to the inferred rectangles in Experiment 2 forpositive (left) and negative (right) examples. Note that while there isvariation across the grids, in neither case is there a discernable pattern of responses (cf. Figure 4). The results show that when the data
are sampled by the learner, inferences about the true rectangle donot show the qualitative effects predicted by pedagogical sampling.
Figure 5: A sample of people’s inferences in the learning task (toprow) and the MAP solution according to the model (bottom row), divided by type of pair. The model clearly predicts the results for positive examples and mixed examples. For negative examples, people’s
guesses suggest that they assume negative examples mark corners,
while the pair of best results from the model does not capture thisprediction, though averaging over hypotheses closely corresponds topeoples resposes.

not related to how the data are sampled. This experiment introduces a variant of the game where the learner chooses theexamples to ask a teacher about. The teacher then labels theexamples as inside or outside the rectangle. The learner theninfers the rectangle the teacher is thinking of. Importantly, inthis version of the game, pedagogical sampling is no longer
appropriate. Because the learner does not know where therectangle is when they are guessing, the examples are effectively random with respect to the true rectangle. Thus, we do
not expect to find the pedagogical sampling effects observedin Experiment 1.

MethodParticipants 29 University of Louisville undergraduates
participated in exchange for course credit.
Procedure The design consisted of two parts, as in Experiment 1. In the first part participants were told that they weregoing to see all of the possible rectangles (the same rectangles that people saw in Experiment 1). They clicked (once)
anywhere on the screen to advance to the next rectangle. In
the learning task, participants played a version of the rectangle game in which they chose the examples that were labeled, then inferred the true rectangle concept. Participants
were told a number of examples and chose by clicking on thescreen. Each click was marked with a black dot. After they
made the pre-specified number of clicks, the black dots turned
into either green circles or red X’s depending on whether they
were inside or outside the unobserved rectangle. The unobserved rectangles used to label points were the same as thosein the teaching section, in random order. Participants werethen asked to guess where the true rectangle was by clicking
and dragging with the mouse. This manipulation insured thatparticipants were aware that locations of examples they observed were independent of the location of the true concept.
Participants were debriefed and thanked upon completion.

Results & Discussion
The focus of this experiment is to investigate learner’s inferences when data are not pedagogically sampled. In particular, we are interested in whether the pattern of inferences
observed in Experiment 1 are specific to pedagogical situations. Therefore, we will investigate the same qualitative andquantitative effects for the learning task as in the previous experiment.
First, we investigate the relationship between the positive
and negative pairs and the inferred rectangles. For positive
pairs, the question of interest is whether, when people see apair of positive examples, they draw a rectangle that places
those examples in the corners. That is, do people simply havea preference for drawing rectangles that have positive examples in the corners, or is this strategy observed in Experiment
1 due to the pedagogical nature of the examples? People’s
rectangles were divided into 3x3 grids, and we tallied the total number of examples in each bin. Figure 6, left side, shows
the distribution of positive examples. While they are not completely random, χ2 (8) = 16.03, p = 0.04, they do not show
any discernable pattern. Particularly, there is no evidence thatpeople chose rectangles such that the examples were placed in
the corners (compare with Figure 4). The lack of correspondence between the predictions of the pedagogical model andobserved data is reflected in their low correlation, r = 0.27.
For negative pairs, we are interested in whether the inferred rectangles tended to have negative examples near theedges. We categorized examples into bins based on the relative distance from the edge of the rectangle to the side of theboard, and tallied counts for each bin. Figure 6, right side,shows the distribution of examples was not completely random, χ2 (15) = 323.43, p < 0.0001, but it does not show themarked distance effects of the previous experiment (see Figure 4). The locations of negative examples do not correspondwith the predictions of the pedagogical model, r = 0.23. The
results show that people did not choose rectangles to be near
the negative examples – that the inferences drawn from nonpedagogically sampled data are qualitatively different thanthose based on pedagogically sampled data.
Together these results show that the predictions of the pedagogical model hold for data sampled by a teacher, but not fora related task where the learner chooses which point they get
to observe. The qualitative effects predicted by the model andobserved in Experiment 1 are not the result of simple perceptual preferences. Rather, people are sensitive to when pedagogical sampling applies. People generate examples differentially based on how helpful they are to a learner. Similarly,
when people receive data chosen by a teacher, people are able
to capitalize on this information to make stronger inferences.

General Discussion
Much of human learning occurs in pedagogical situations –social situations where one person chooses information forthe purpose of helping another learn. We have presented
a formal model of pedagogical reasoning, addressing which
examples teachers should choose to communicate ideas, andwhat inferences learners should make based on this purposefully sampled data. We also presented two experiments testing the predictions of this model in a simple experimental
setting. The first experiment showed that people’s choice ofdata as a teacher, and their inferences as a learner correspondto the qualitative and quantitative predictions of the pedagogical model. The second experiment provided an important
control condition, demonstrating that when data are chosen
by the learner and hence are not representative of the concept
to be learned, the predicted pedagogical effects are not observed. Together these results demonstrate a first step toward
capturing the important effects of pedagogical situations onlearning and reasoning.
Neither of the well-studied sampling assumptions (weakor strong sampling) captures the qualitative phenomena wehave observed because neither captures the intuitions aboutsampling in pedagogical situations. Both assume that data aresampled randomly, which does not account for the preference
for certain examples. As models of learning, both are unable
to capture the speed of learning in Experiment 1 – with justtwo examples, we have a strong intuition for which rectangle
the teacher has in mind – in contrast to the many examples
necessary for either of these models (Tenenbaum, 1999).
The pedagogical sampling assumption is not appropriate
in all circumstances. For many real-world learning situations, strong sampling is the right assumption; for example,
when learning about the laws that govern physical objects,
we never see situations that cannot happen and the situations
we do observe are random – certainly it is not reasonable toassume that nature is helpfully choosing examples to teachus something! Similarly, weak sampling applies to a variety of learning situations we might encounter in casinos.
Our evidence, with that of Tenenbaum (1999) and Xu andTenenbaum (2007) suggests that people are able to distinguish situations in which different sampling assumptions are
warranted.
Though modeling pedagogical reasoning in richer domains
is a significant challenge, it highlights a great strength of ourmodel. We have formalized pedagogical reasoning in the abstract language of probability and Bayesian reasoning, without reference to the specific details of the particular setting
we considered here. As a result we are able to derive predic-

tions in principle for any domain for which we can identify anappropriate set of hypotheses. Interesting domains to pursue
include word learning, where speakers choose words to communicate ideas, and causal learning, where a helpful teacher
may significantly reduce the number of interventions required
to learn latent causal structure. Different domains will havedifferent hypotheses and priors and as a result the model will
generate qualitatively different predictions.
One of the things that make people special is that we canteach others what we know (Csibra, 2007). However, evenwhen we are teaching others, we never communicate the complete idea in precise detail. As a result, people must resolve a
difficult inference problem in order to capitalize on the information teachers provide. Gergely et al. (2007) have shownthat children do capitalize on these situations, and they argue
that pedagogical reasoning is among the powerful tools children have for learning about the world. Much work remains
before the breadth of the implications of pedagogy for human learning are understood, but our work, providing a computational basis for understanding these kinds of inferences,
represents a step in this direction.

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