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A Local Instructional Theory for the Guided Reinvention of the Quotient Group Concept Sean Larsen Portland State University

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Inspiration for the Project It has been observed that students struggle with the quotient group concept even though it is implicit in the elementary notion of parity. (Dubinsky, Dautermann, Leron, and Zazkis, 1994) Of course it has also been noted that “seeing the general in the particular’ is one of the most mysterious and difficult learning tasks students have to perform.” (Mason & Pimm, 1984) Our goal was develop instruction that would support students in reinventing the quotient group concept by building on their intuitive notions of parity.

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Theoretical Support for the Design Work The project was guided by the theory of Realistic Mathematics Education (RME) In particular it was motivated and guided by the design heuristic of guided reinvention. –The goal was “to allow learners to come to regard the knowledge that they acquire as their own private knowledge, knowledge for which they themselves are responsible” (Gravemeijer & Doorman, 1999)

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Skipping ahead… It is MUCH faster to write a definition on the board and give an explanation/examples than it is to have student reinvent a concept! So, is it worth the trouble to take the time to have students reinvent the quotient group concept? We did a bit of data collection to see if this reinvention approach was helpful…

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The Survey Surveys were administered to students in 9 sections of a junior level group theory course (8 PSU, 1 WOU). 4 sections used the TAAFU curriculum 5 sections used other approaches There were 5 questions. The last two referred to a group of six elements represented by a Cayley table. (Isomorphic to S 3 )

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The Questions

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Analysis Process Coding: 1/0 For correct answer on Question 4 1/0 For correct answer on Question 5 1/0 For partial normality/set op. check 1/0 For complete normality/set op. check Two coders coded independently and then resolved inconsistencies by reviewing responses together. Surveys were blinded for analysis – names were removed along with information about what section of the course the surveys came from.

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Example Surveys Question 4 Correct

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Example Surveys Question 5 Complete Normality

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Example Surveys Question 5 Complete Set Op

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Question Correctness Coding Results by Class Note: Section A & B scores are likely inflated because the students were permitted to trade their QG survey for an Isomorphism survey if they could not do the QG survey

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Question Correctness Coding Results by Curriculum

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Method Coding Results by Class

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Method Coding Results by Curriculum

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Conclusion TAAFU classes were a significant 35% more likely to get Question 4 correct. TAAFU classes were a significant 42% more likely to get Question 5 correct. TAAFU classes showed equal use of both methods (Normality and Set Op.), thus showed more versatility in their solutions. Does this prove that the reinvention approach is effective? No, but it does suggest efficacy. It does show that when implemented well, it can have a positive impact on students’ understanding of the quotient group concept.

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Now back to the design research… Recall that our goal was to develop a local instructional theory (LIT) for supporting the reinvention of the quotient group concept. To oversimplify, an LIT is an instructional sequence along with a theoretical and empirical rationale.

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Theoretical Support for the Design Work Two specific RME related theoretical notions guided our work: The emergent models heuristic: –A concept (model) emerges as a model-of students’ mathematical activity in specific task setting. –General activity emerges as students’ reasoning loses its dependency on situation-specific imagery. –In the process, a new mathematical reality is constituted in which the concept becomes a model-for more formal reasoning. Proofs and refutations is a process by which conjectures are revised (and concepts developed) as proofs are analyzed in light of counterexamples (Lakatos, 1976; Larsen & Zandieh, 2007).

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Quotient Groups as an Emergent Model: A Thought Experiment The reinvention process could begin by having students examine a specific (finite) group for properties like the Even/Odd parity in the integers. The quotient group concept could emerge as a model of the students’ mathematical activity in this task setting (i.e. the designers might see their activity as anticipating the formal ideas related to the quotient group concept.) The students could then further mathematize their mathematical activity by: –Generalize by finding a more complex partitioning of the finite group into subsets that form a group. –With an eye to a general situation, identify what properties are needed to make such a construction work. –Formalize the concept with definitions, theorems & proofs.

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Thought Experiment Questions: –How will the students come to see these Even/Odd partitions as groups of subsets? –How will they figure out how to partition a group (coset formation)? –How will they figure out the conditions needed for QG to work (e.g. normality)?

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Learning from the Students Gravemeijer (1999) described three key ingredients of a local instructional theory (modified): –Students’ informal knowledge and strategies that anticipate the more formal ideas. –Strategies for evoking these kinds of informal knowledge and strategies. –Strategies for leveraging these strategies and informal knowledge to support the development of the more formal mathematics. This framework guided our analysis (ongoing and retrospective) of the students’ mathematical activity.

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The Local Instructional Theory Key Steps/Signposts in the Reinvention Process: Seeing parity as a relationship between two sets relative to an operation. Seeing the Even/Odd partition as forming a group made up of subsets. Conjecturing/Proving that one of the subsets must be a subgroup (the identity subset). Generalizing to more complex partitions. Figuring out how to form the rest of the partitions after choosing a subgroup. Figuring out why some subgroups don’t work (that “normality” is a necessary condition.)

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Background The data I will share comes from two sources: A design experiment conducted with two undergraduate students (Sara and Rick) consisting of 10 (≈ 90 minute) sessions. Video from an implementation of an instructional sequence (based on the design experiment) in a junior level group theory course taught by a mathematician. I will use this data to illustrate the key signposts in the reinvention process, highlighting our findings related to the three ingredients of a LIT described by Gravemeijer.

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Background In both contexts the work with quotient groups was preceded by the reinvention of the group concept: The students explored the symmetries of an equilateral triangle and square. They developed their own symbols for these symmetries as well as a set of rules for computing combinations of symmetries. These rules anticipated the definition of group and also included the specific relations associated with these dihedral groups.

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Background Symbols for the symmetries of an equilateral triangle & a sample calculation:

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The Local Instructional Theory Key Steps/Signposts in the Reinvention Process: Seeing parity as a relationship between two sets relative to an operation. Seeing the Even/Odd partitions as forming a group made up of subsets. Conjecturing/Proving that one of the subsets must be a subgroup (the identity subset). Generalizing to more complex partitions. Figuring out how to form the rest of the partition after choosing a subgroup. Figuring out why some subgroups don’t work (that “normality” is a necessary condition.)

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Seeing parity as a relationship between two sets relative to an operation. Recall that students in both contexts had worked extensively with groups of geometric symmetries (D 6 & D 8 in particular). Task 1: Can you find anything like evens and odds in D 8 ? There were two different ways this task was interpreted by students - depending on whether students focused on the relationship between evens/odds or the forms of evens/odds.

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Parity in terms of forms of evens/odds: S1: Like evens equals... 2n. Parity as a relationship between subsets: S2: Add all the possible even numbers and all the possible even numbers and the result is all the possible even numbers. Ultimately it is the second way of thinking that will be important, but with a little creativity the first way of thinking can also be used to successful partition D8.

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Ways to partition D8 into evens and odds: EVENODD {I, R, R 2, R 3 }{F, FR, FR 2, FR 3 } {I, F, R 2, FR 2 }{R, R 3, FR, FR 3 } {I, FR, R 2, FR 3 }{R, R 3, F, FR 2 } It is easy to check that each such partitioning satisfies E + E = E, O + O = E, and E + O = O+ E = O. Students thinking about parity in terms of element form, sometimes come up with these by thinking in terms of even/oddness of exponents.

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The Local Instructional Theory Key Steps/Signposts in the Reinvention Process: Seeing parity as a relationship between two sets relative to an operation. Seeing the Even/Odd partitions as forming a group made up of subsets. Conjecturing/Proving that one of the subsets must be a subgroup (the identity subset. Generalizing to more complex partitions. Figuring out how to form the rest of the partition after choosing a subgroup. Figuring out why some subgroups don’t work (that “normality” is a necessary condition.)

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Students tended not to see their even/odd partition as forming a group whose elements were subsets. Instead they tended to see the 2x2 Even/Odd tables as shorthand for the whole operation table. It was helpful to discuss how many elements the Even/Odd groups had. Seeing the Even/Odd partitions as forming a group made up of subsets.

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How Many Elements?

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“Meta Groups” Video Removed for PDF Version

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The Local Instructional Theory Key Steps/Signposts in the Reinvention Process: Seeing parity as a relationship between two sets relative to an operation. Seeing the Even/Odd partitions as forming a group made up of subsets. Conjecturing/Proving that one of the subsets must be a subgroup (the identity subset). Generalizing to more complex partitions. Figuring out how to form the rest of the partition after choosing a subgroup. Figuring out why some subgroups don’t work (that “normality” is a necessary condition.)

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After constructing one or two of the even/odd groups from D8, the students conjecture that one of the subsets must be a subgroup - the one that acts as the identity element. This allows the students to be sure they have found all possible partitions of this type once they have exhausted all possible subgroups of order 4. It is also the first step in ascertaining what conditions are needed for a partitioning of a group to form a group. Conjecturing/Proving that one of the subsets must be a subgroup (the identity subset).

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S1: There’s not going to be any more because you have to have one be a subgroup… T: So the claim is that one of the two sets actually has to be a subgroup… Why do you think that one of them needs to be a subgroup? S1: If it’s a subgroup, then it with itself is just going make that subgroup back… T: The evens with itself is the evens. Ok, so you claim that based on that reason that some element with itself has to equal itself. You’re making the case that you need to have a subgroup to be one of your sets S1: Yeah, so that subgroup will act as the identity.

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If the identity element wasn’t in the identity subset, “something horrible would happen.” Video Removed for PDF Version

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Does the identity element of a quotient group need to contain the identity of the original group? Why? Think about this for the case of D 8 and then see if your reasoning makes sense for the general case (constructing quotient groups starting from any group). Hint: Suppose that the identity is in the yellow subset instead

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The Local Instructional Theory Key Steps/Signposts in the Reinvention Process: Seeing parity as a relationship between two sets relative to an operation. Seeing the Even/Odd partitions as forming a group made up of subsets. Conjecturing/Proving that one of the subsets must be a subgroup (the identity subset). Generalizing to more complex partitions. Figuring out how to form the rest of the partition after choosing a subgroup. Figuring out why some subgroups don’t work (that “normality” is a necessary condition.)

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The next step was to generalize the analysis of the D 8 to look for partition groups with more than 2 elements (subsets). Task 2: Can we make bigger groups by breaking D 8 into smaller pieces? Make as many four-elements groups as you can by breaking D 8 into two-element subsets: Using cards, build one that works. Using cards, build one that seems like it could work but does not (try to build it until you can explain why it can't work). Generalizing to more complex partitions.

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Interpretation of this task was non-trivial - students struggled a bit to move beyond looking for parity S1: What kind of property does this satisfy? Does this satisfy the even or odd? S2: No really it’s more complicated than even and odd. That’s the problem, they’re trying to get us away from even and odd. S1: But what’s the requirement to do this? You try to find subgroups?

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S2: You just try to find subsets that make kind of a… that map onto another one of the subsets. It’s kind of a group of subsets, sort of. It’s kinda weird. Once the students find a partitioning that works and explore other sensible partitions that do not work, they are ready to develop the idea of coset formation and necessary conditions for a partition of cosets to form a group.

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A 4-element Quotient Group {R, 3R} {F+R,F+ 3R} {F,F+ 2R} {I,2R} {R, 3R} {F+R,F+ 3R} {F,F+ 2R} {I,2R} {R, 3R} {F+R,F+ 3R} {F,F+ 2R} {I,2R}

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The Local Instructional Theory Key Steps/Signposts in the Reinvention Process: Seeing parity as a relationship between two sets relative to an operation. Seeing the Even/Odd partitions as forming a group made up of subsets. Conjecturing/Proving that one of the subsets must be a subgroup (the identity subset). Generalizing to more complex partitions. Figuring out how to form the rest of the partition after choosing a subgroup. Figuring out why some subgroups don’t work (that “normality” is a necessary condition.)

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In order to decide whether this is the only partition that works, it is important to figure out how many partitions need to be considered. So given a subgroup, it is important to figure how the rest of the group can be partitioned. It turns out this can only be done in one * way (via cosets) as Rick’s algorithm illustrates in the following clip. Figuring out how to form the rest of the partition after choosing a subgroup.

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Rick’s Partitioning Algorithm Video Removed for PDF Version

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The definition of coset can result from formalizing the kind of algorithm that Rick was using. This paves the way for two things: –Conclusively arguing that there is only one way to partition a group after selecting a subgroup to act as the identity element. –Formulating a necessary condition for such a partition to form a group

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The Local Instructional Theory Key Steps/Signposts in the Reinvention Process: Seeing parity as a relationship between two sets relative to an operation. Seeing the Even/Odd partitions as forming a group made up of subsets. Conjecturing/Proving that one of the subsets must be a subgroup (the identity subset). Generalizing to more complex partitions. Figuring out how to form the rest of the partition after choosing a subgroup. Figuring out why some subgroups don’t work (that “normality” is a necessary condition.)

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“It must be g+H = H+g” Video Removed for PDF Version Figuring out why some subgroups don’t work (that “normality” is a necessary condition.)

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Why does {I, R 2 } work and {I, FR} not work?

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What has to be true to make it work? C: I think that I got a pretty good explanation. The left coset and the right coset don’t match. L: Yeah C: Just use the one element and you get that. I used R and … I did RI and RFR and I got these, and then I did IR FRR and I got these two. So you can’t have F be a part of this subset and FR 2 be a part of … it doesn’t work…You started generating a ton of elements L: and it doesn’t work it on the right. It creates the yellow-purple group. Instead of just…

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What has to be true to make it work? Preview of whole class discussion - Video Coming Up!) A student claims that the {I, R 2 } subgroup works since it is the center of the group - because the identity element must commute. She also notes that the left cosets are the same as the right cosets. Another student explains how this “commutivity” is needed to make the identity of the quotient group work properly (gH H = H gH = gH). Yet another student argues that it isn’t necessary for the subgroup to be the center and gives a supporting counterexample. The instructor wraps up the debate before introducing the term “normal subgroup”.

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What has to be true to make it work? Video Removed for PDF Version

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Summary of the Reinvention Process The overall trajectory we observed seemed to be consistent with a combination of the two thought experiments we conducted. Emergent Models: The quotient group concept emerged first as a model of the students’ efforts to identify something like parity in D 8. Later, the students worked with this concept in a more general way - making & testing conjectures regarding conditions needed for this type of construction to work. Proofs & Refutations: The students were able to develop these conditions by considering examples/non-examples and analyzing attempts to show that specific partitions formed groups.

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In Case You Wondered… The reinvention process was much messier and more involved than what appears here. –Additional issues to work through include whether you must partition into equal sized subsets and whether you can use overlapping subsets. –Sometimes key issues emerged simultaneously (e.g. coset formation and normality.) The reinvention phase was followed by a more deductive sequence in which the details of the theory were worked out in a way that respected the students’ more informal activity: –Normality continued to be defined by gH = Hg. –Multiplication in terms of representatives arose naturally while proving that the set of cosets is closed under set multiplication (Students proved aHbH = abH). –Normality used to show that multiplication by representatives is well-defined.

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Ongoing Related Work Based on this design work, I developed an inquiry-oriented curriculum for a full ten- week group theory course. This curriculum is the focus of an NSF funded project in which we are: –Investigating the experiences of mathematicians teaching with the curriculum. –Investigating students’ learning as they interact with the curriculum. –Working to develop web-based instructor support materials: Click HereClick Here

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