Intelligent Tutoring Systems are Missing the Tutor:
Building a More Strategic Dialog-Based Tutor
Neil T. Heffernan (email@example.com)
Kenneth R. Koedinger (firstname.lastname@example.org)
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
The mission of the Center for Interdisciplinary Research on Constructive Learning Environments (CIRCLE) is 1) to study human tutoring and 2) to build and test a new generation of tutoring systems that encourage students to construct the target knowledge instead of telling it to them (VanLehn et. al., 1998). Computer Aided Instruction systems were 1st generation tutors. They presented a page of text or graphics and depending upon the students answer, put up a different page. Model-tracing ITSs (Intelligent Tutoring Systems) are 2nd generation tutoring systems that allow the tutor to follow the line of reasoning of the student. ITS have had notable success (Koedinger et. al., 1997) despite the fact that human tutoring can look very different (Moore, 1996). One difference is that there is a better sense of a dialog in human tutoring. We, and others, think that is important; after analyzing over 100 hours of untrained tutors in naturalistic tutoring sessions, Graesser et. al. (1999) believe "there is something about interactive discourse that is responsible for learning gains."
The members of CIRCLE are working on 3rd generation tutoring systems that are meant to engage in a dialog with students to allow students to construct their own knowledge of the domain. We have built a new ITS, called Miss Lindquist, that not only can model-trace the students actions, but can be more human-like in carrying on a running conversation, complete with probing questions, worked examples, positive and negative feedback, follow-up questions in embedded sub-dialogs, and requests for explanation as to why something is correct. In order to build Miss Lindquist we have expanded the model-tracing paradigm; not only does she have a model of the student but also has a model of tutorial reasoning (e.g. Clancey, 1982) for our specific domain.
McArthur et. al. (1990) criticizes Andersons et. al. (1985) model-tracing ITS and model-tracing in general "because each incorrect rule is paired with a particular tutorial action (typically a stored message), every student who takes a given step gets the same message, regardless of how many times the same error has been made or how many other error have been made. Andersons tutor is tactical, driven by local student errors (p. 200)." They go on to argue for the need for a more strategic tutor. Miss Lindquist meets that criticism. Miss Lindquists model of tutorial reasoning is both strategic (i.e. has general multi-step plans that can be used to breakdown problems) and tactical (i.e. can recognize and respond to "teachable moments.") Merrill et. al. (1995) argued that human tutors are tactical in that "student-tutor dialogues were centered much more around student-initiated events, as they attempted to actively understand new instructional material and solve problems, [rather] than around tutorial presentation of material and subsequent querying of student understanding." Merrill et. al. went on to say "microanalysis of student-tutorial interactions in problem-solving situations suggests that tutors do more than simply reteach a correct procedure component when students encounter impasses or errors. Our tutors focused on guiding the error repair process rather then communicating their guesses about what students misconception." We argue that a tutor needs to be tactical in recognizing certain situations, and strategic enough to plan multi-step sequences of tutorial interaction. Therefore, it is important that Miss Lindquist can understand the student and come up with a strategy to respond with, even in the presence of multiple errors.
The task domain we are working on is symbolization, which is the task of writing an algebraic expression given a real-world problem context. (An example is shown below at line 240.) Symbolization is fundamental because if students cant translate problems into a mathematical algebraic model, they will not be able to apply algebra to solve real world problems. Fortunately, this domain makes it easy to avoid some difficult natural language issues because we can ask students to write algebraic expressions, and those expressions are easy for the computer student model to "understand". We take advantage of this property of the domain to avoid any serious natural language processing. Instead, we focus our energies on modeling tutorial reasoning which includes capturing the pedagogical content knowledge (Shulman, 1986) of an experienced human tutor. A good tutor not only knows the domain knowledge and general tutoring knowledge, but also has specific strategies that can help a student overcome common difficulties. We have set out to observe and model some of these strategies for our specific domain of symbolization. The following is an example of a content-specific pedagogical strategy for symbolization. It was collected and transcribed from a one hour long one-on-one tutoring session between an experienced human tutor and an eighth grade student.
240. Student: [reads the "bike-trip" problem] Cathy took a "m" mile bike ride. She rode at a speed of "s" miles per hour. She stopped for a "b" hour break. Write an expression for how long the trip took.
241. S: uhm [ writes "s/m+b" but should be "m/s+b"]
242. Tutor: How do you calculate the amount of time it takes you? If youre, if youre, if youre riding at, lets make it simple. If you are riding at 20 miles per hour, OK, and you go 100 miles, how many hours did that take you?
243. S: Um 5
244. T: 5. And how did you get that 5? How did you use the numbers 100 and
245. S: 100 miles divided by miles per hour
246. T: So you took the miles and divided it by the [garbled, but possibly "speed"]
247. S: Miles divided by s plus b equals time [writes m/s+b]
248. T: Right.[transcript available at http://www.pitt.edu/~circle]
We call the tutorial strategy displayed here the concrete articulation strategy, which Koedinger & Anderson (1998) refereed to as inductive support. McArthur et. al. also observed that human tutors often used what he called curriculum scripts and micro-plans, which often involved a series of questions designed to remediate particular difficulties. We call these scripts knowledge construction dialogs to emphasis the fact that we are trying to build a tutor that encourages students to build their own knowledge rather than being told it. Below, we will show how Miss Lindquist participates in an analogous dialog. We will also show three other tutorial strategies that Miss Lindquist can use.
We think that if you want to build a good ITS for a domain you need to:
We discuss each of these in turn. The fourth step is future work.
What Makes Symbolization Difficult?
Symbolization is a difficult task for students. For example, one month into an algebra class only 13% of students could answer the symbolization problem in the caption to Figure 1. To determine what makes symbolization difficult we conducted two difficulty factors assessments (Koedinger & MacLaren, 1997) which are paper and pencil tests that we gave to groups of 80+ students (Heffernan & Koedinger, 1997 and 1998). First, we identified three hypotheses about what makes symbolization difficult.
The first of these is the comprehension hypothesis. Much of the prior research (Cummins et. al., 1988; LeBlanc & Weber-Russell, 1996; Lewis & Mayer, 1987; Paige & Simon, 1979) on word problem solving has focused on students' comprehension abilities. Cummins et. al. "suggest that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language." The general conclusion from the above research is that comprehension rules are key knowledge components students must acquire to become competent problem solvers.
A second hypothesis is the generalization hypothesis. According to this hypothesis, symbolization is difficult because students must learn how to use variables to generalize arithmetic procedures..
More recent research by Koedinger and Anderson (1998), and which we confirmed (Heffernan & Koedinger, 1997 and 1998), showed that students could comprehend many problems well enough to find a numerical answer, but they nevertheless failed to correctly symbolize. Although this refutes the comprehension hypothesis it does not refute the generalization hypothesis because the symbolization problems had variables in them. Therefore, we compared students ability to symbolize a problem that contained a variable (with an answer like "800-40m") to their ability to symbolize a problem with just constants. In the "constants" case the students were asked to write an expression for their answer (i.e. "800-40*3") instead of finding a numerical solution (like "680"). Even if we counted as correct the very few students who did not follow the directions and evaluated the answer, we found that the presence of the variable in the problem did not make problems more difficult. Therefore, the generalization hypothesis was refuted.
So what can explain why symbolization is so difficult? We propose the articulation hypothesis which suggests that there is a "hidden" skill that is not obvious to most teachers and researchers. The hidden skill is the ability to produce symbolic sentences in the language of algebra. It appears that many students are able to figure out all the conceptual relations in a problem, but are not able to express those relationships in algebra. If we asked students to translate a story written in English into Greek we would not be surprised if many fail because they dont know Greek. But teachers and researchers often fail to realize that algebra too is a language. And a language that students have had relatively little practice in "speaking" By "speaking" we mean producing sentences of symbols, not verbalizing. This was demonstrated anecdotally by one of our students who when asked to symbolize a problem with the answer of "(72-m)/4" responded with "72-m=n/4=". Many commentators have noted that students will incorrectly use an equal sign in a way that makes sense if "=" means "results in." Sfard et. al. (1993) gives the following example "3*4=12-5=7." Another example is the student who when working on a problem with an answer of "550/(h-2)" answered with
This student means to suggest that first she would subtract 2 from "h." The arrow seems to indicate that this new decremented value of h should be assigned back to the symbol "h". Then 550 should be divided (indicated with the grade school way of expressing division) by this new value of "h." Both of these examples indicate students who probably understand the quantitative structure and the sequence of operations that should happen, but nevertheless, failed to express that structure in normative algebra. What does such a student need to learn? A computer scientist or linguist might say that the student needs to learn the correct grammar for algebraic expressions. The novice student knows how to write one-operator expression like "5+7" using the following simple grammar:
<expression> = <literal> <operator> <literal>
<literal> = 1|2|3|4 .
<operator> = "+" | "-" | "*" | "/"
But the competent student knows how to write multiple operator expression indicated by these grammar rules:
<expression> = <expression> <operator> <expression> | "(" <expression> ")" | <literal>
Phrased differently, what the student needs to be told is that "You can always wrap parentheses around an expression and substitute an expression anywhere you normally think a number can go. There are also rules for when you can leave out the parenthesis but you can always put them in to be sure that your expression wont be misinterpreted."
We found experimental evidence that supports the articulation hypothesis when we performed the following manipulation (Heffernan & Koedinger, 1997 and 1998). We started with a two-operator problem, like
Composed: Ann is in a rowboat in a lake. She is 800 yards from the dock. She then rows for "m" minutes back towards the dock. Ann rows at a speed of 40 yards per minute. Write an expression for Ann's distance from the dock. and decomposed the problem into two new separate questions like the following.
Decomposed: A) Ann is in a rowboat in a lake. She is 800 yards from the dock. She then rows "y" yards back towards the dock. Write an expression for Ann's distance from the dock.
B) Ann is in a rowboat in a lake. She then rows for "m" minutes back towards the dock. Ann rows at a speed of 40 yards per minute. Write an expression for the distance Ann has rowed.
Then we compared the ability of a student to answer the composed problem with their ability to get both decomposed parts correct. We found that the composed problems were much harder. Why? We speculated that many students could not compose the two decomposed expressions together; just because you know that you need to first add two quantities together and then multiply them by a number, doesnt mean you know how to express this correctly in the language of algebra. The following is an example of a student who appeared to be missing just this skill of composing expressions together. This example occurred while the first author was tutoring a student on the following "two-jobs" problem:
T: Debbie has two jobs over the summer. At one job she bags groceries at Giant Eagle and gets paid 5 dollars an hour. At the other job she delivers newspapers and gets paid 7 dollars an hour. She works a total of 30 hours a week. She works "g" hours bagging groceries. Write an expression for the total amount she earns a week.. [The correct answer is "5g+7(30-g)"]
S: A=5*g, B=30-g,C=7*B and D=A+C
This student clearly understands the 4 math operations that need to be performed, and the order in which to perform them. This student spontaneously introduced new variables (A, B, C, and D) to stand for the intermediate results. We were surprised to find that this student could not easily put this together and write "5g+7(30-g)". This student appears to be ready for a strategy that will help him on just one skill; combining expressions by substitution. (We also turn this idea into a tutoring strategy which is presented below in the section on the tutorial model)
To see if substitution really is a hidden component skill in symbolization, we designed the following transfer experiment. Thirty-nine students were given one hour of group instruction on algebraic substitution problems like the following:
Let X= 72-m. Let B= X/4. Write a new expression for B that combines these two steps.
The students were guided in practicing this skill. The students got better at this skill, but that is not the interesting part. By comparing pre-tests and post-tests, we found statistically significant increases in the students ability to do symbolization problems, even though they did not get instruction involving word problems! The students transferred knowledge of the skill of substitution to the skill of symbolization revealing a shared skill of being able to "speak" complicated (more than one-operator) sentences in the foreign language of algebra. This is strong supporting evidence for the articulation hypothesis.
This research has put a new focus on the production side of the translation process. This work also has ramifications for sequencing in the algebra curriculum. If learning how to do algebraic substitution involves a sub-skill of symbolization, perhaps algebraic substitution should be taught much earlier. In many curriculums (e.g. Larson, 1995) it is not taught until students get to systems of equations half-way through the year.
Cognitive Student Model
Our student model is a cognitive model of the problem solving knowledge that students are acquiring. The model reflects the ACT theory of skill knowledge (Anderson, 1993) in assuming that the problem solving skills can be modeled as a set of independent production rules. Our model has over 68 production rules. The cognitive model enables the tutor to trace the students solution path through a complex problem solving space. The cognitive model for symbolization is tied to the underlying hierarchical nature of the problem situation. We model the common errors that students make with a set of "buggy" productions. From our data of 218 individual student errors on eight different problems we found that the following list of errors accounts for over 75% of the errors that students made. We illustrate the errors in the context of a problem which has a correct answer of "5g+7(30-g)".
These "buggy" productions are used to allow us to make sense of a students input even if there are several incorrect steps. Traditional model-tracing tutors generate their feedback from templates associated with individual production rules which is why McArthur et. al. criticized then as being to tactical. Instead, our system passes the diagnosis to the tutorial model that reasons between tactical and strategic concerns.
The Tutorial Model
Our tutorial model is informed by observation of an experienced human tutor, but it doesnt pretend to model any one individual or make claims to being the most effective model. The tutorial model is implemented with 77 production rules that decide what to say and how to say it. Many of these productions manipulate the tutorial agenda which operates usually like a push-down stack. Most of the productions are concerned with implementing the various tutorial strategies but some are concerned with more mundane aspects including maintaining conversational coherence, giving positive feedback, and dealing with issues related to how to deal with multiple student errors or incomprehensible student input. This approach is similar to Freedmans (2000). We will now look at four different strategies that Miss Lindquist can use on any problem. The examples also show instances of more tactical behavior trigged by certain situations.
Concrete Articulation Strategy
The concrete articulation strategy usually uses three questions illustrated below in T2, T3, and T4. Here is an example of the dialog generated by Miss Lindquist which we believe closely parallels the experienced human tutor shown above.
T1: [Doing the "bike-trip" problem] Please write an expression for the total time the trip took. [The first question always asks the student to symbolize.]
T2: No. Let me ask you an easier question. Please compute the time actually on the bikes if the speed was 20 miles per hour, and the distance ridden was 100 miles? [ask to compute a concrete instance]
T3: Correct. What math did you do to get that five? [ask student to articulate in symbols the answer]
T4: Correct. Now write your answer of 100/20 using the variables given in the problem (i.e. put in m and s)? [ask student to generalize their concrete articulation]
S4: b+m/s [was expecting "m/s"]
We will walk through what Miss Lindquist did here. First, the student model diagnosed S1 ("s/m+b") as having the quantity "s/m" with reversed arguments (bug type #2.) The diagnosis also said that the student understood that the total trip time was the time of the break ("b") plus the time actually on the bikes. This diagnosis was passed to the tutorial model chosen from competing strategies. In this case the tutorial model decided to conduct a concrete articulation knowledge construction dialog. Therefore, the tutor model generated questions for the compute, explain/articulation, and generalization steps. The tutor put these three questions on the agenda.
At S2, the student answers the compute question correctly, and the tutor decides to continue with its plan. It has to be sensitive to what the student typed. If the student typed 100/20 instead of 5, then both the compute and explain questions would be removed from the agenda and the tutor would skip to the generalize step. An example of this sort of flexibility is demonstrated in T5. T4 was a question to articulate the time actually on bikes. The tutor was hoping the student would type "m/s" but instead the student answered "b+m/s." Only a pedantic tutor wouldnt accept that answer as it indicates an understanding that subsumes the understanding required for the sub-goal. This is where the tutorial agenda behaves differently than a stack because Miss Lindquist pops both of the questions off the agenda. To show you an example of how the previous situation could have come out differently, lets look at the following dialog also generated by Miss Lindquist.
T1: Hello. [Doing the "bike-trip" problem] Please write an expression for the total time the trip took. [symbolize]
S1: help! [The student can request help. The tutor treats such a request the same way it treats an totally incomprehensible student input.; it starts breaking down the problem into steps.]
T2: Let me ask you an easier question. Please compute the time actually on the bikes if instead the speed was 20 miles per hour, the time of the break was 2 hours, and the distance ridden was 100 miles? [compare to T2 for the above example]
T3: Correct. What math did you do to get that 7?
S3: 2+5 [student doesnt completely articulate]
T4: Okay, but how did you get the 5? [similar to T3]
T5: Correct Now lets try this again, what math did you do to get the 7? [same question as T3 but the phrase "lets try this again" thrown in for conversational coherence]
S5: 2+100/20 [correct articulation]
T6:Correct. Now write your answer of 2+100/20 using the variables given in the problem? [generalize like T4 from pervious example]
S6: b+m/s [of course accepts "m/s+b"]
This example is interesting in the way Miss Lindquist responds at T4. The system has bug rules to apply to questions other than symbolization questions. In this case, the student model reported that the students answer of "2+5" is a partially complete articulation but that the student didnt spell out where the "5" came from. The tutorial model is able to "dig-down" into the student's reasoning with an embedded sub-dialog. This involves adding the additional question of explaining how the student got the "5". We think that such tactical behavior, conditioned on particular student errors, is consistent with Merrills et. al. analysis of what human tutors do. Miss Lindquist can use embedded multi-step dialogs to respond to other specific situations such as parentheses errors, errors of omission, miscomprehension of the question, and argument ordering errors.
Substitution with Decomposition Strategy
A second example of a tutorial strategy is what we call the substitution with decomposition strategy. This strategy is motivated by previous cognitive analysis and experimentation that will not reviewed here (Heffernan & Koedinger 1998). Below we present an example of Miss Lindquist tutoring with this strategy. (As this article went to press we had just conducted our first pilot testing with human students. The following example is verbatim from this pilot test.)
T1: Hello. Jane is "j" years old. Peg is "p" years old. Mary is Peg's age minus Tom's age. Tom is 3 years younger than Jane. Peg is "x" years older that Mark. Write an expression for Mary's age.[correct answer is "p-(j-3)"]
T2: Hmm. ["Hmm" indicates students response was not diagnosed] No. Let me ask you and easier question. Use j and 3 to calculate Tom's age. [ask for the one subgoal]
S2: 3-j [This is a common error type that the tutor should be tactical enough to recognize. The tutor has since been modified to be able to respond with the strategy of asking the student "Should the answer be positive or negative?" followed by "Which is larger, the 3 or j?]
T3: Hmm. No. Try again please. What is Tom's age? Hint: pick one of these: j-3, 3-j, j+3, j*3, j/3, or 3/j. [This is a rephrasing as explained in the discussion section]
T4: Correct. Calculate Mary's age. Use A instead of j-3 for Tom's age.
T5: Correct. You are correct that p-A is Mary's age. Now, can you state this in terms of the numbers given in the problem. [Ask the student to algebraically substitute]
Two More Strategies that Foster Translating Between Multiple Representations
Mathematical communication (Schoenfeld et. al., 1992, Koedinger & Nathan, in press) is increasingly seen as a goal for students, and in particular, translating between algebraic (e.g. "m/s+h"), English (e.g. "the total trip time") and concrete (e.g. "7 hours") representations. So far we have discussed students translating from the English to algebraic or concrete representations. The next two strategies explore asking students to translate in different ways including: 1) translating from the algebraic to the English and 2) within the English representation.
Subgoal Reification Strategy
The subgoal reification strategy is inspired by Corbett & Andersons (1995) lisp tutor that showed learning gains when the students we asked to post (or reify) their subgoals before attempting to write lisp code. We have added to our tutor the ability to choose a similar strategy. At T2 below, the student is asked for the name of the subgoal. At T3 the student is asked to explain in English how to achieve that subgoal. At T4 the tutor asks for the subgoal in algebra. T5 and T6 repeat the pattern from T3 and T4.
T1: Hello. [Again, doing the "bike-trip" problem] Please write an expression for the total time the trip took.
T2: In order to find the total time, what do you first have to find out? [The student is presented with pull-down menu to select a quantity-noun phrase]
S2: [menu:] The time actually on the bikes
T3: Correct. Can you explain how you would find the time actually on the bikes)? [Presented with the following sentence to fill in the blanks: "The time actually on the bikes is equal to (select quantity) (select operator) (select quantity)." The student can select the 2 quantities and the operator from menus. ]
S4: The time actually on the bikes is equal to [1st menu:] the distance biked [2nd menu:] divided by [3rd menu:] the speed of the bikes.
T4: Good. Now say what that [note the simple pronoun substitution] in symbols.
T5: Correct, Can you explain how you would find the total time of the trip using the time on the bikes?
S6: The total time of the trip is equal to [1st menu:] the amount of time for the break [2nd menu:] added to [3rd menu:] the time actually on the bikes.
T6: Can you say that in symbols?
Self-Explanation of a Worked Example
The next strategy is inspired by the literature on worked examples and the self-explanation effect. Sweller (1988) proposed giving students worked out examples. Chi (1996) and Conati et. al. (1997) have both proposed asking students to self-explain. This strategy also gets at the goal of practicing translation from the algebraic representation to the verbal representation which Corbett et. al. (2000) have shown to be effective for learning symbolization. In this example, the student is told the answer and the asked to explain it.
T1: Hello. [Again, doing the "bike-trip" problem] Please write an expression for the total time the trip took.
T2: No. The correct answer is "b+m/s". Let me ask you some questions to make sure you understand that. Can you explain where the m/s came from? [This is similar to T3 from above. The student is presented with a sentence they must complete. The sentence begins "m/s is equal to" and is followed by three pull-down menus to compose the answer.]
S2: "m/s" is equal to [1st menu:] the distance ridden [2nd menu:] divided by [3rd menu:] the speed ridden.
T3: Correct. What does m/s represent?
S3: [menu:] The time actually on bikes
T4: Correct. Can you explain where the b+m/s came from?
S4: "b+m/s" is equal to [1st menu:] the time for the break [2nd menu:] plus [3rd menu:] the time actually on bikes.
T5: Correct. What does "b+m/s" represent?
S5: [menu:] The time actually on bikes
T6: Correct. It looks like you understand that problem.
How is Miss Lindquist different from traditional model-tracing tutors? Traditional model-tracing tutors offer feedback primarily through two mechanisms: 1) a "buggy" message that is triggered by the students input matching with a known buggy rule or 2) in response to the student requesting a hint. A student can usually continue to ask for a hint getting progressively more explicit messages. These "hint chains" sometime terminate in messages that tell the student exactly what to do, but the model-tracing paradigm does not require this.
Miss Lindquist is different in that she has a greater capacity to carry on a dialog by being able to handle embedded, multi-step dialogs (e.g. "Where did you get the 5?") Though many model-tracing systems phrase their hints as questions, they are usually just rhetorical questions. When Miss Lindquist ask you a question, she expects you to answer it. She does this because her strategies suggest that these questions are good ways to "break-down" complicated problems. If Miss Lindquist has run out of options on how to "break-down" a problem, then she adopts a strategy similar to model-tracing tutors and gives progressively more explicit rephrasings of a question. Most of Miss Lindquists rephrasings terminate with multiple choice questions rather then simply telling the student the answer. Through all of this, we hope to make the student more active.
Our experienced tutor was not always satisfied with a students answer, particularly when it looked like the student might have guessed. Sometimes she would ask reflective follow-up questions to test the students understanding. Miss Lindquist does a similar thing: when a student has likely guessed (indicated by having got the answer only after having reached the most explicit rephrasing) the student is asked one of a few types of follow-up questions. A long-term research goal is to learn which types of follow-up work best.
Another way Miss Lindquist is more similar to human tutors is in being more active. Others have also viewed tutor dialogs as primarily tutor initiated (Graesser et. al., 1999 and Lepper et. al., 1997.) Miss Lindquist does not wait for a student to ask for help. Our experience tutor made a comment on average once every 20 seconds! In summary, we view Miss Lindquist as capturing a balance between strategic and tactical responses. She is able to break down problems with different tutorial strategies while at the same time she can make more tactical decisions in response to particular situations (e.g. common errors or student guessing.)
We look forward to measuring the effectiveness of Miss Lindquist by comparing it to an effective benchmarked 2nd generation ITS (Koedinger et. al., 1997.) We also look forward to comparing the effectiveness of different strategies by allowing the tutor to pick randomly among the tutorial strategies. We can then measure the effectiveness of a strategy by seeing if the student correctly answers an isomorphic problem later in the curriculum. We also plan to compare these tutorial strategies to the strategies of a 1st generation tutor that in response to a student error simply tells the student the correct answer. This "cut-to-chase" strategy might me most effective because it allows the student to go on to the next problem quickly. These "fancy" dialogue strategies might waste kids time that would be better spent being told the correct answer and then doing more problems. If we control for time we can test this hypothesis that might suggest all of these other dialogue strategies are (or at least these strategies as they are implemented) not worth while.
This research was supported by NSF grant number 9720359 to CIRCLE and the Spencer Foundation. We would also like to thank for their helpful comments Steve Ritter, Adam Kalai, John Pane, Douglas Rhode, and Reva Freedman.
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