This study compared how students engage in inductive and deductive geometric reasoning using augmented reality (AR) by analyzing their geometric thinking elements, gestures, and cognitive functions of actions while collaboratively producing geometric proofs. High school students (N=14; seven dyads) completed inductive (pyramid) and deductive (cylinder) tasks using AR. Students’ speech and actions were coded for geometric thinking elements (operational, logical, and generalizable), action types (pointing, depictive gestures), and cognitive functions (pragmatic, epistemic, complementary), then analyzed using Epistemic Network Analysis (ENA). Results revealed distinct reasoning patterns: inductive reasoning relied more on external resources through pragmatic actions and pointing gestures to generate insights and form logical and generalizable thought, while deductive reasoning involved complementary actions and dynamic depictive gestures that externalized mental processes to construct logical and operational thought. These findings illustrate how AR interactions differentially support inductive and deductive geometric reasoning, informing the design of AR-based learning experiences.

Computer based instructional systems either direct students so modelling their actions is tractable, or provide them with total autonomy, but give little support to learning and problem solving processes. Instructional principles for empowering the student are emerging whereby more of the responsibility of diagnosis and goal-setting is placed on the student. Critical to this view is providing an environment which makes the ramifications of students' actions clear so students can meaningfully assess their own performance. In the domain of word algebra, the meaning of formal expressions can be reflected in computer animation which depicts the corresponding situation. An unintelligent tutor — knowing nothing of the problem being solved and possessing no student model — helps students to understand problems and debug formal expressions.

A method for hypothesizing the locations of features that may have been undetected by low level three-dimensional image processing algorithms is presented. The procedure relies on a predicate logic based approach to object recognition in which viewpoint dependent (observed) data is transformed into a set of viewpoint independent assertions. The detected features in these viewpoint independent assertions are variabilized and the assertions are combined to form a clause which is then tested against known models for consistency. If the features in a known model are able to instantiate the free variables in the variabilized clause, then a subgraph isomorphism is established between the features in the observed data and those of the model. Missing or undetected features in the observed data are then sought by examining the semantic relationships between model features that are instantiated and those that are not. Results of applying the method to synthetic range data are presented.

A robotic vision system is being developed which uses three-dimensional laser range data to sense its environment. The recognition subsystem incorporates topological as well as geometric information to identify viewed objects. Theorem-proving techniques are used to produce symbolic pattern matches. The major contributions of the recognition subsystem are 1) the use of viewpoint independent descriptors as the basis for representing known object models and 2) the use of theorem proving techniques to hypothesize object identities and recognize the viewed object as an instance of the appropriate viewpoint independent model descriptor. The representation scheme permits describing objects at a variety of topological and geometric levels. Furthermore, the use of viewpoint independent descriptors facilitates object recognition from a single arbitrary view despite missing information or the inclusion of viewpoint dependent artifacts. The theorem-proving approach establishes a symbolic correspondence between viewpoint independent features in the (recognized) model and features in the observed data. The recognition process uses a three-phase approach. First, hypotheses are generated which correspond to model descriptors that are likely to match the data. Evidence is applied to viable hypotheses to produce a partial match. The partial match is then used to constrain the full recognition process which leads to object identification. This strategy has been found to constrain strongly the search space of possible matches and leads to large reductions in recognition times. Results of the recognition process on synthetic and actual laser range data are presented for several objects. The system is shown to operate with robustness and alacrity.

An object recognition system has been developed which incorporates topological as well as geometric information to match viewpoint dependent object descriptors. Theorem proving techniques are used to produce symbolic pattern matches. The recognition process uses a three phase approach. First, hypotheses are generated which correspond to model descriptors that are likely to match the data. Evidence is applied to viable hypotheses to produce a partial match. The partial match is then used to constrain the full recognition process which leads to object identification. This strategy has been found to strongly constrain the search space of possible matches and leads to large reductions in recognition times. The major contributions of the system are the representation scheme and the use of theorem proving techniques to verify object identities. This approach permits describing objects at a variety of levels and facilitates recognition despite missing information or the inclusion of artifactual data. Results of the recognition process on synthetic and actual laser range data are presented for curved and planar objects. The system is shown to operate with robustness and alacrity.