SCG-Teach: A Family of Games for Large Formal Science Courses

by Karl Lieberherr

February 2015 add on

Quasi-equilibria (see EC '15 submission) play an important role in teaching. We need tables with W,L,F,O and a timestamp for each game result. We look for the largest subset LP of players so that after time t all games between players in LP have the property that both players choose the same side, either O or P, and there is no fault.

Pyramid SCG: For large classes, the students are divided into teams of 3 and they determine the most meritorious players (full round-robin). We iterate the grouping and playing until no faults happen or a maximum level is reached. Note that all three players might win if there is no fault in a 3 person round-robin tournament.

Who gets a fault? If both have chosen freely: The one who makes the final mistake. The final mistake might be the side choice. If one is forced, only the non-forced player can make a fault. The non-forced player who makes the final mistake gets a fault. A mistake is either a wrong side-choice or a move which flips the truth value of the claim. This is a fair evaluation of the players.

by Ahmed Abdelmeged and Karl Lieberherr

March 2013

Abstract: We have used SCG-Teach over the last 5 years in various variations with increasing success. The goal of SCG-Teach is to engage the students in a systematic scientific discourse to defend their sides on claims and refute the sides of others on claims. The focus is on creating productive interactions, teaching from student-to-student and collaborative/competitive problem solving. But the instructor has the usual task of defining precisely the problem solving domains and the additional task of explaining SCG-Teach once. The learning of SCG-Teach is amortized over numerous problem solving domains that are covered in one course.

We believe that SCG-Teach, in a properly implemented form, is very useful for MOOCs. It encourages fruitful interactions between the students while relieving the teaching staff from uninteresting grading work.

An important benefit of SCG-Teach is that the instructor has to do less work. SCG-Teach monitors the students and they correct each other's misconceptions. The students drive each other into personalized contradictions. SCG-Teach uses a tournament to determine the "best" student in a group which decides whether the claim is true. The approach is meritocratic. At the end of a learning period, the instructor needs to check whether any claims are mislabeled. The instructor then plays the role of a very strong player who will clean out the knowledge base and refute false claims and defend true claims. This cleaning task is much less work than checking all the scientific discourse which is better done by SCG-Teach.

SCG-Teach is a self-organizing learning environment. (1) it adapts to the skill levels of the students. (2) the good students become teachers of the weaker students. (3) the winner of a game has more skill. (4) we can reliably measure the skill level of students. (5) when all students are perfect then there are only agreement games where the non-forced player is the winner.

To substantiate the time savings for the instructor, consider the following representative scenario: We have a lab with n students and a family of k claims in the lab where m games, m > n > k, are being played. During each game, each student is presented with a claim and the student decides on its side: verifier or falsifier. At the end of the game, one of the k claims is mislabeled as true but it is false. Let's assume that of the m games, 20% are verifier/falsifier and 40% verifier/verifier with half of them with contradiction for the non-forced player and 40% falsifier/falsifier with half of them with contradiction for non-forced player. 60% of m games lead to a personalized contradiction where students get effectively corrected by their peers. 40% (of the 60%) of the m games actively encourage innovation because the forced player cannot lose and can play agressively. If the instructor works manually, he has to inspect m game histories and check whether all rules have been followed and determine who wins and where the first mistakes (per game history) have been made. If the instructor uses SCG-Teach, then all m game histories are checked automatically or by the students. 60% of the m scientific discourses lead to a personalized contradiction where students get effectively corrected by their peers without any involvement by the instructor. The instructor has to teach one important lesson: how to refute the mislabeled claim.

Advantages of manual grading: In each game history, the instructor can point to the first mistake, e.g., that the wrong side was taken on a claim. Disadvantages of manual grading: Much more work.

Advantages of grading with SCG-Teach: Much less work: the instructor only needs to teach the correct side on mislabeled claims and show how to defend the correct side. For the correctly labeled claims, the instructor can engage the students to show their winning strategies. Disadvantages of grading with SCG-Teach: The students don't always get their first mistake marked. However, they get woken up by one of their peers who pushes them into a personalized contradiction which presents a good learning opportunity for the student who made a mistake.

Our approach is useful for large classes because the students tend to teach each other under the supervision of the teaching staff.


The instructor defines a problem solving domain by giving a notation for expressing claims using a predicate logic expression and a structure.

As the game proceeds, students get targeted feedback, from other students, about misconceptions they have about their ability to defend their sides.

SCG-Teach in a Nutshell

The students are given a claim out of an instructor-defined set of claims and take a side on the claim: verifier/falsifier. The Contradiction-Agreement Game (CAG) is elaborated determining the winners and losers.

The objective is to successfully defend your own sides on claims and to successfully refute the sides of others.

SCG Teach with Piazza

We are using SCG-Teach with Piazza in the Spring 2012 Algorithms course at CCIS, Northeastern University Algorithms and Data (CS 4800).

Piazza for Algorithm Students . This is a low-tech use of SCG-Teach because enforcing the game rules is done by the students collectively. (We also use a high-tech version of SCG-Teach where the students write avatars that play the game on their behalf. In that scenario the game rules are strictly enforced by software called SCG Court.) The low-tech use with Piazza is very successful. We had competitions about: "what is the worst-case input to Gale-Shapley" and "what is the worst CNF where any pair of clauses is satisfiable" and "what is the highest safe rung for a ladder with n rungs and k jars to break". In all three cases, the students collaboratively/competitively found the correct solutions.

The students learn about SCG-Teach from Piazza Interface for Algorithmic Problem_Solving. As communication language for the scientific dialog we use JSON so that the students can easily check the work of their peer using programs.

The features of Piazza that go well with SCG-Teach: All the scientific discourse about one claim is at one place. Because all students see the discourse they can check that the game rules are followed. We observed that students enjoy doing this policing because they help other students find flaws in their reasoning.

We have developed some patterns for writing interesting labs. (1) Use labs where claims can be strengthened. Seed the lab with a suboptimal claim and let the students take it from there. You will be surprised. (2) Occasionally define a lab with a small omission and give the impression that half of the claims are true. Then let the students find the omission which let's them refute all claims that were believed to be true. This encourages them to think critically. (3) Some labs are direct in that the topic to be covered appears directly in the claims. Other labs are indirect in that the topic to be covered is used in the solution approach.

It has happened on Piazza that students have released too much information through the scientific discourse. Therefore we introduced a new rule: If the scientific discourse reveals too much information about the homework solution the scientific discourse is only completed AFTER the due date for the homework. This way students get partial information which makes them think: "those girls have a solution of quality 0.8 while I only have a solution of quality 0.7. What is their clever technique?" They become motivated to learn what their peers already know. (see Ricochet Robots).


The following feedback is given from student to student:

[Misunderstanding: true but cannot defend] Alice proposes a true claim but she cannot defend it: signals a misunderstanding of Alice that is made concrete by the game. Bob clarifies the misunderstanding by refuting the true claim.

[Misunderstanding: false not true] Alice proposes a false claim which she thought to be true: signals a misunderstanding. Bob clarifies the misunderstanding by refuting the false claim.

[Misunderstanding: not optimal] Alice proposes a true claim but one which is not optimal. Bob defends a stronger version of Alice' claim which helps Alice to clarify her misunderstanding.

The workload for the teacher is greatly reduced because the students clarify several misunderstandings without teacher involvement.


What do the students learn from the above feedback? They get a more precise perception of their ability, both absolute and relative, to defend their sides.

They learn to generalize from their experience with specific objects to other objects. If they have to invent an algorithm the generalization is more difficult than if they have to invent just data for a given algorithm.

For more information, see the SCG Home Page: