Improving a Mathematical Intelligent Tutoring System Experiments & Equation Solver Improvements

1. Improving a Mathematical Intelligent Tutoring SystemExperiments & Equation Solver Improvements July 27th, 2012 Jennifer Ferris-Glick & Hee Seung Lee

2. Mathematical Intelligent Tutoring Systems • How do we improve out dated textbook methods of learning and expand on intelligent tutoring systems?

3. Objective of Study • To investigate the role of verbal explanation- When does it help learning? • To Explore the degree to which we can teach without verbal instruction - Is verbal direction necessary? - How can we help students discover/learn underlying problem structure and induce correct rules for problem solving without explicit verbal directions?

4. Linearize task: solving algebra problems using data-flow diagrams: allows to study algebra learning anew in college population - 5 5 * 4 * + + 5 29 29

5. Findings from Linearize study • Showing intermediate cognitive steps • Preventing superficial analogy from an example to problem • Building appropriate problem representations • Avoiding excessive floundering • Giving a chance to reflect on solved problems

6. Showing Intermediate Cognitive Steps • Students still have to fill in gaps between steps when students study a worked example • Students learned better when intermediate cognitive steps were clearly revealed than when students had to determine hidden steps.  Force students to study step-by-step solutions

7. Preventing Superficial Analogy • Given a highly similar example, students tend to draw a superficial analogy from the example to problem. • When superficial solution is available, students fail to abstract deep problem structure. • When examples look too dissimilar from a target problem, students sometimes simply ignore or do not spend enough time to study provided examples.  Examples should be dissimilar, and structurally similar to a target problem.

8. Building Appropriate Problem Representations • Revealing hidden structure helped transfer- Making connections to Algebra activated prior knowledge and helped understand the problem structure • Highlighting relevant features (e.g., color coding)  Highlight features to direct student’s attention and reveal hidden problem structure

9. Restricting Floundering & Supporting Reflection Activity • When students show too much floundering, they may not remember solution paths. In order to reduce floundering: • Immediate feedback on wrong attempt • Providing hints when students show floundering • Auto-solving with a time limit • Giving students an opportunity to make sense of already solved problems  Allow only a certain number of off-track activities. When a certain step was solved automatically, show corresponding verbal explanations for a while.

10. Showing Intermediate Cognitive Steps Force step-by-step guided example before solving Maybe show previously solved equations Hint improvements Some changes but not full review Preventing superficial analogy Present side-by-side example Similarity of example Randomly selected from a pool of problems in each section (not too similar, not too dissimilar) Appropriate Problem Representation Aligning equations on the equals sign Coloring variable sides of equation Restrict Floundering Reduce off-path steps either allow 0 or 1 off-path step Reflection Time Provide side-by-side already worked out problems General Improvements to Carnegie Learning Cognitive Tutor (all conditions)

11. One-step linear equations Solving with addition and subtraction (no type in) o w + 6 = -1 Solving with addition and subtraction (type in) o -4 = 1 + z Solving with multiplication and division (no type in) o 2x = -1 Solving with multiplication and division (type in) o 8w = 5 Target Material

12. Two-step linear equations Solving One-step equations (Type in) o Mix of types from unit 5 Solving with multiplication (no type in) o -2 = 5x + 10 Solving with multiplication (type in) o -2 = 5x + 10 Solving with division (no type in) o x/6 + 7 = 8 Solving with division (type in) o x/6 + 7 = 8 Solving with a variable in the denominator (no type in) o 10 = -2/x Solving with a variable in the denominator (type in) o 2/z = -4 Target Material

13. Linear equations with similar terms Solving two-step equations (type in) o Mix of types from unit 6 Combining like variable terms and a constant with integers (no type in) o 2 – 9x + 6x = 2 Combining like variable terms and a constant with integers (type in) o -6x + 2 + 5x = -6 Combining like variable terms with decimals (no type in) o 9 = 9.7x – 2x Combining like variable terms with decimals (type in) o -2.1z – 8z = 2 Combining like variable terms and a constant with decimals (no type in) o 8.1 + 7.4x + 3.7x = 4.6 Combining like variable terms and a constant with decimals (type in) o -6.6z – 2.8 – 6.7z = 2.8 Target Material

14. Linear equations and the distributive property Solving Linear Equations (Type In) o 7 + 2x = -3 Using Multiplication and Integers (No Type In) o 4(z - 3) = 3 Using Multiplication and Integers (Type In) o -6(z – 6) = 10 Using Multiplication and Decimals (No Type In) o -4.5 = -2.6(z + 2.9) Using Multiplication and Decimals (Type In) o 8.3(z – 5.9) = 6.8 Using Multiplication and Large Decimals (No Type In) o -43.54 = -16.63(z + 24.90) Using Multiplication and Large Decimals (Type In) o -51.78(w + 97.21) = -11.63 Using Division (No Type In) Target Material

15. Experimental Conditions Kanawha County School District 3 Conditions divided by class (4 classes for each condition): Discovery, Instruction, Standard Presented in line with curriculum.

16. Experimental Conditions

17. Example Types • Step-by-Step Guided Problem One per section – selected based on sections “new”material Students simply follow directions in a step-by-step manner to solve the problem • Side-by-Side Worked Examples Already solved problems appear next to a target problem

18. Step-by-Step Guided ProblemInstruction To isolate w, subtract 6 from both sides. Select “subtract both sides” and enter 6. • Instruction condition • Provide “why” and “how” hints • Directly tell students what to do

19. Step-by-Step Guided ProblemDiscovery Select an item from Transform menu and enter a number. • Discovery condition • Provide “how” hints • Provide instruction on general interface

20. Color-Coding • Both conditions • Provide guided problem • Align equation on the equal sign • Coloring variable side of the equation • Highlighting changes between steps

21. Wrong Attempts This is not the best way to solve the problem. Click “undo” and ask for hint if you need help. • Both conditions • Reduce off-path steps using “undo” button.

22. Side by Side Worked ExampleInstruction & Discovery Example Problem • Both conditions • Provide already solved problems (screen-shot) • Selected from the pool of problems in the same section

23. Post-Test 2 types of test questions: (paper-based, within subject) -Plain equations -Equations with solver interface scaffolding