Motivation- An Example
What is the core difference between these two math geometry problems?
Find the midpoint of the line segment joining A
(2,2) and B(4,6) ?
Qualitative versus Quantitative
Thanks to Descartes!
How could we build an intelligent tutoring system to help students understand
analytical geometry math problems, solve them, and provide scaffolding
An iterative Method
Challenges & Solutions
Challenge 1 How could we represent quantitative information from analytical geometry problems?
Solution 1 Computational Semantic
Challenge 2 How could we perform quantitative reasoning using the semantic representation?
Solution 2 Planning and Constraint Solving
Challenge 3 How could we generate context-sensitive tutoring scaffolds for the problem?
Solution 3 Program Synthesis
Challenge 4 How could we analyze user input and give adaptive scaffold feedback?
Solution 4 Model Tracing Tutor
Challenge 5 How could we let students interact with the tutoring engine as naturally as possible?
Solution 5 Evidence based User Interface and Interactions
Solution 1: Problem Representation
Syntax Level: knowledge entity (such as a point, line or circle),
knowledge attribute (y-coordinate of a point, slope of a line),
knowledge relation (such as perpendicular lines, parallel lines,
distance between two points). These three knowledge
components can be related with a label.
Semantic Level: A math problem can be described using the
above syntax as a directed acyclic graph.
Human Annotation: (4,3), (2,v), m=?, m*m1=-1, m1=½, v=?.
Solution 2: Quantitative Reasoning Scenarios
Scenario 1: Forward Reasoning
● Find the distance between two points A(2,3) and B(4,5).
Scenario 2: Backward Reasoning
● There are two points A(2,4) and B(5,v), the distance between A and B is 5. What is the value of v?
Scenario 3:Embedded Reasoning
● Line A passes through two points (4,3) and (2,v). The line is perpendicular to line B in which the
slope of line B is 1/2. what is the value of v?
Solution 3: Tutoring Scaffold Auto-Generation
Executing the solving procedure will fire logic rules. Each fired logic rule maps to a
scaffold step, which records two statuses before and after the rule execution, the
applied rule, and meta-rule.
Current Rule Types: Arithmetic Rules, Algebraic Rules, Algebraic Equation Rules, Geometry
Pattern Match Rules, Reification Rule, Geometrical Unary/Binary Relation Rules.
Meta-Rule: A meta rule is the context-free version of the applied rule. For example, the meta rule for
the algebraic distributive law is “Consider using the distributive law.”
Solution 4: Model Tracing Tutor
The scaffolding trace is a linear sequence of geometric
scaffoldings, where each scaffold could contain a linear
sequence of algebraic scaffoldings. The scaffolding trace can
be seen as a two-dimensional linear structure.
Solution 4: Stepwise Verification
The match-and-verify algorithm to check the user’s answer using the model-
Match State on the Scaffolding Trace Graph
Match State on the Semantic DAG
Solution 4: Scaffold Adaptation
The current deterministic method:
Status flag to track the current status in the scaffolding trace graph.
Solution 5: Keyword Highlighting and Dragging Interaction
There exists two points A(2,4) and B(5,v), the distance between A and B is
5. What is the value of v?
There exists two points A(2,4) and B(5,v), the distance between A and B is 5.
What is the value of v?
What we got from dragging upon those visual highlight words?
Solution 5: Input and Reasoning
Reasoning: Inferential reasoning upon the sketch modality (Oviatt et.al.):
● Gesture-based question-mark.
Input (sketching): Algebraic expressions in the algebraic side. Geometric
shapes in the geometric side.
Tutoring & User Interaction
1: Tutoring Performance (verification feedback and scaffold adaptation feedback).
2: UI keyword highlighting and dragging.
3: Overall usability.
For 2, we evaluated its effect using a repeated measure design. We evaluated tutoring performance and
overall usability in both conditions. Ten participants solved 10 problems with the keyboard highlighting
and dragging, and other 10 problems without it.
Tutoring (marginal of keyword highlighting and dragging independent variable):
● Participants used the system to check their answer step.
● Participants used the system to ask the next step.
● Participants made the error-rectification once during the study session.
UI keyword highlighting and dragging:
● Participants made less query for the next step using this technique.
● Participants made less error-rectification using this technique.
● Participants preferred to use with the keyword highlighting and dragging UI.
Overall usability (marginal of keyword independent variable):
● Dual Canvases Effect (log file analysis and post questionnaires)
● Participants felt it was effective to use pen-and-gesture input-and-query flow
(5.6 out of 7).
● Participants were satisfied with the current tutoring system (5.4 out of 7).
Discussion and Future Work
● Participants felt that the system has constraints to let them freely input and query the knowledge
due to the sketch recognition errors in both canvases.
● Though the system showed both meta-scaffold and scaffold explanations, participants felt not that
useful and effective toward their problem-solving due to:
Prediction accuracy, repeated scaffold explanations without considering the individualization (such
as skipping certain scaffolds using the same KCs) and explanation verbal narratives.
AnalyticalInk is an interactive tutoring system toward analytical geometry to
facilitate the quantitative reasoning.
Semantic graph representation makes the connection between the algebraic
reasoning and the geometric concept reasoning.
The system is useful in terms of interaction techniques and its user
Thank you! Questions?
Back-end source code and data: https://github.com/buptkang/Behavior.Model
Front-end source code: https://bitbucket.org/buptkang/math-tutor-ui
Special thanks to Fei Liu, Pamela Wisniewski and all anonymous reviewers. This work is supported in part
by NSF CAREER award IIS-0845921.