Ph.D Student | Lachmy Royi |
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Subject | Developing and Using Productive Mathematical Ideas in Online Collaborative Problem Solving |

Department | Department of Education in Science and Technology |

Supervisor | Professor Boris Koichu |

Full Thesis text |

During the last decade, teachers and educators increasingly use online learning environments for different educational purposes, including collaborative mathematical problem solving. This study addresses the urgent need to deepen our understanding of the nature of the mathematical activities carried out in these environments. The goal of this study was to characterize the processes of developing and using productive mathematical ideas in online forums in which secondary school students collaboratively approached challenging geometry problems.

The study was conducted in two research environments. The first
environment was designed within three social networks (Facebook, Google,
WhatsApp) and the second one - within the Moodle learning management system;
174 students from six classes of 9^{th} and 10^{th} grades
worked on challenging geometry problems in these environments. Eighteen
problem-solving episodes were documented, and nine especially rich episodes
were analyzed quantitatively and qualitatively.

The main findings are:

• Students worked on each problem for 1-2 weeks; the lion's share of the work was performed during the first 2-3 days. The participation was distributed unequally among students: each group (ranged from 8 to 36 students) included a small sub-group of 3-5 active students who contributed on average 85% of student posts.

• In cases of
successful collaborative problem solving (four cases), three stages were
identified: (i) collaborative attempt to solve a problem by means of *accessible
mathematical resources* and facing a dead-end; (ii) collaborative attempt to
solve the problem by means of* less accessible mathematical resources* and
arriving, by a sole student, at a productive mathematical idea; (iii)
collaboratively exploring the productive mathematical idea.

• Five ways by
which students attempted to use less accessible mathematical resources on their
path to invention of a productive mathematical idea were identified: (i)
recalling a particular resource; (ii) attempting to find a geometry
configuration to employ the resource to; (iii) attempting to use the resource
in relation to a specific geometry configuration; (iv) actual use of the
resource in relation to a specific geometry configuration; and (v) deriving
conclusions based on the use of the resource and completing the solution.
Recalling and considering less accessible mathematical resources was sometimes instigated by a *bridging geometry
configuration*, i.e., by a sub-set of elements belonging to a geometry
configuration that had been previously used in relation to more accessible
resources.

• Five ways by which students used a productive mathematical idea formulated by a peer in relation to a particular claim were found: (i) inquiring the peer about her or his idea; (ii) acknowledging the idea’s productivity; (iii) using the idea for reconstructing the peer’s proof or for producing an additional proof to the same claim; (iv) using the idea for proving additional claims; and (v) using the idea for discovering new claims.

• Two types of interactions were identified - weakening interactions and strengthening interactions. These interactions may facilitate or impede the development of productive mathematical ideas during online collaborative problem solving.

This study has implications for further design of online learning environments and formulation of guidelines for moderating online discussions.