21.+Problem+Solving

May 5, 2015
 * 20. Problem Solving**

Deep structure Math concepts Brainstorming Means-ends analysis
 * Outline**

Chi et al. (1981): Asked physics novices and experts to classify problems into categories
 * Deep structure**

Surface features Deep structure
 * Novices used surface features (inclined planes) as their basis for classification
 * Experts classified them based on principles like the conservation of energy (deep structure)
 * Details that are not central to the problem
 * e.g., inclined plane
 * The central underlying concepts
 * e.g., conservation of energy

The fortress problem
 * A dictator has planted mines on each road leading to the fortress, so a full-scale direct attack would be impossible.
 * Solution: The general divided his army into small groups and dispatched each group to the head of a different road. [[image:lol.png width="284" height="249" align="right"]]When they attacked, the small groups passed safely over the mines.
 * Shallow structure
 * Attacking a fortress
 * Deep structure
 * Small forces from many directions can mount a large attack on a central feature

Gick and Holyoak (1980)
 * Asked subjects to solve:
 * 1. The fortress problem
 * 2. The radiation problem (focus a bunch of low radiation rays on one spot to kill the tumor without damaging healthy cells)
 * Conditions:
 * Analogy group
 * Attempt fortress problem
 * Told solution
 * Attempt radiation problem
 * Analogy + hint group
 * Fortress problem
 * Told solution
 * Radiation problem
 * Hint: "think back to the fortress problem"
 * No analogy group
 * Radiation problem
 * [[image:analogy.png width="294" height="178"]]
 * Why did they do so poorly?
 * Because the surface features were different.
 * It's hard to internalize deep structure.

Two ways to do math Richland et al. (2012):
 * Math problem solving**
 * Memorize procedures
 * Doesn't require understanding
 * Understand concepts
 * Deep knowledge
 * Participants: community college students taking remedial math.
 * High school grads who continued their schooling
 * The majority of CC students take remedial math
 * Examined conceptual knowledge (do they understand deep structure?)
 * Quotes from CC students:
 * "Math is just all these steps"
 * "Sometimes in math you have to just accept that that's the way it is and there's no reason behind it"
 * "I don't think [being good at math] has anything to do with reasoning. It's all memorization"
 * That's the opposite of math...

Brainstorming's 4 rules: Osborn (1957): "the average person can think up twice as many ideas when working with a group than when working alone"
 * Brainstorming**
 * The more ideas the better
 * The wilder the ideas the better
 * Improve or combine ideas already suggested
 * Don't be critical

Diehle & Stroebe (1987): Brainstorming doesn't work The problem is:
 * Participants were told the four rules of brainstorming. Asked to produce ideas for an issue for 15 minutes (into microphones
 * Conditions:[[image:brainstorming.png width="295" height="222" align="right"]]
 * Real group condition
 * Worked in groups of four
 * Nominal group condition
 * Worked individually
 * Told they were part of a group of 4 whose ideas would be pooled
 * Results: Nominal group produced more ideas than the real group.
 * Conclusion: Brainstorming is a terrible way to solve problems
 * 3 possible explanations:
 * Explanation 1: Freeloading? (people slack off and expect others to carry the load?)
 * In the first study, participants were told their assessment was based on group performance (they can freeload and still do well)
 * In a follow up: participants were told assessment was based on //your// performance (freeloading should go away)
 * [[image:freeloading.png width="261" height="164"]]
 * Nominal group still did better
 * Explanation 2: evaluation apprehension? (scared to say dumb stuff in a group?)
 * Low apprehension
 * Think of some ideas
 * High apprehension (the real and nominal groups should be equally anxious, so the effect should go away)
 * One way mirror
 * You'll be judged by experts
 * Video taped
 * Your response will be used in class demos
 * [[image:apprehension.png width="296" height="187"]]
 * Nominal group still did better
 * Explanation 3: blocking? (maybe they just don't get as many chances to talk?)
 * Real group
 * Nominal group
 * Nominal with blocking group (a red light comes on when "someone else is talking"
 * [[image:blocking.png width="335" height="192"]]
 * Nominal with blocking performed similarly to the real group--blocking is the problem
 * Individuals combined are better than groups
 * Freeloading
 * Evaluation apprehension
 * Blocking

Problem solving
 * Means-ends analysis**
 * The problem space
 * The set of all states that can be reached in solving a problem
 * Problem space is too huge to manage for most problems (even well-defined problems)
 * Evaluation the first four moves in chess, at 1 move per second, would take 10,000,000 years
 * We don't actually do it psychologically
 * Means-ends analysis
 * Break problem into sub-problems
 * Solve the sub-problems
 * Means: the tools at your disposal
 * End: What you want (i.e., the goal)

Sometimes problem solving is mainly a matter of asking yourself the right __question__.

The Monte Hall Problem: There are 3 doors, one with a prize behind it. Pick one. They'll show you that there's a goat is behind one of the doors that you didn't pick. Should you change your choice, or should you stay?
 * Ask the right questions:
 * What happens if your original choice is right?
 * You always win if you stay.
 * Chance of winning: 33%
 * What happens if your original choice is wrong?
 * You always win if you switch.
 * Chance of winning: 67%

Most real-life problems are ill-defined:
 * No clearly defined goal
 * No clearly defined means
 * No clearly defined problem space