Practicing the Water Jar Problem...With a Vengeance!
Your goal is to measure out a precise amount of water, given three empty jars. The first problem is a warm-up with only two jars. Jar A holds 29 units of water, and Jar B holds 3 units. Your goal is to measure out 20 units of water exactly. How do you do that? First, you fill Jar A completely. Then you use Jar A to fill Jar B three times (i.e., 3 units x 3 refills = 9 total units). After you fill Jar B three times, you are left with 20 units of water (29 units - 9 units = 20 units).
Now it's your turn . Use the interactive widget to solve problems 2 through 11. (Hint: be sure to empty a Jar before attempting to refill it.)
Einstellung: The Mechanization of ThoughtWhile solving the problems, did you notice a general pattern that started to emerge? They all followed a standard solution template, namely: Fill B, then use B to fill A, and then use B to fill C twice. Stated symbolically, it might look like this: B-A-2C. That seems to work really well until you hit Problem 9, but it works for all the other problems (e.g., Problems 2-8, 10-11).
If you have been reading this blog for a while, then you probably guessed there must be a twist. Indeed there is. The twist is that, even though the B-A-2C pattern works for all but Problem 9, there is a more efficient solution for Problems 7-11. Go back and see if you can figure out it .
If you used the B-A-2C algorithm for any of the problems past 6, then you fell prey to Einstellung, or the "mechanization of thought." How does this happen? When you approach a new problem for the first time, and you don't know what to do, you start applying general problem-solving heuristics. Once you have some success with a solution strategy, you try it again. Lo and behold, it works! You are rewarded by applying the same algorithm over and over. This sets up a mental bias against trying new solution strategies. It is almost like a set of mental blinders.
Mental SetEinstellung occurs when you create a solution strategy where none existed previously, and you keep using it as you try to solve new problems. In this case, the knowledge that is hindering you from trying new problem-solving tactics is currently held in working memory. Can our prior knowledge, stored in long-term memory, have similar effects? Consider the following brain teasers :
- The 22nd and 24th presidents of the United States had the same mother and the same father, but were not brothers. How could this be?
- Picture two plastic jugs filled with water. How could you put all of this water into a barrel, without adding the jugs or any divider to the barrel, and still tell which water came from which jug?
- As I was going to St. Ives, / I met a man with seven wives. / Every wife had seven sacks, / Every sack had seven cats, / Every cat had seven kittens. / Kittens, cats, sacks, wives, / How many were going to St. Ives?
If these questions tripped you up, it's likely that your prior knowledge artificially created a constraint that did not exist in the problem itself. For example, the first question is tricky because the wording of the question caused you to activate your knowledge of your concept of brother. That concept requires two or more people. Once you recognize that you have made a faulty assumption, that the 22nd and 24th president were two different people, then you can easily solve the problem.
When prior knowledge causes us to be blind to our own assumptions, we call that a mental set. We might blame a mental set for a problem that we encountered in a previous post where we were trying to arrange matchsticks to form four equilateral triangles.
The STEM ConnectionHow do you avoid getting stuck in a mental rut? How can you help your students avoid set effects?
Fortunately, the recommendation for avoiding Einstellung is somewhat simple. In the original study, the experimenters noticed that the participants were surprised when they were shown the more efficient solutions. Their reactions included: "How stupid of me" and "How blind I was." The experimenters decided to re-run the experiment, but this time they reminded the second batch of participants: Don't be blind! This advice was effective because problem solvers adopted the more efficient solution at a much higher rate than the first batch of volunteers. Our advice should be the same to our students. Remind them that there may be a more efficient solution lurking out there, just waiting to be discovered.
Another way to avoid Einstellung is to walk away from the problem and return after some time has passed. The goal is to let all of those items in working memory lose their activation so you come back to the problem with a fresh pair of eyes (and a clean working memory!).
Avoiding mental set is a little more tricky because prior knowledge is almost always useful. We don't want to encourage our students to forget what they know because that would run completely counter to our educational mission! This is probably easier said than done, but I think the advice here is to be open to your prior knowledge, but just don't be constrained by it. In other words, we should be neither blind nor self-constrained!
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For More Information Luchins, A. S. (1942). Mechanization in problem solving: the effect of Einstellung. Psychological Monographs, 54(No. 248).
 In the movie, the main characters had to measure out four gallons of water using a 5- and a 3-gallon jug. Here is the problem statement and its solution.
 Another special thanks to Josh Fisher for creating the interactive version of the Luchins water jar problem.
 It turns out that they can be solved by completely ignoring B and adding or subtracting A and C. Problems 7, 9, 11 all use A-C; and Problems 8 & 10 both use A+C.
 I stole the first two brain teasers from a card game that we own called MindTrap. If you like these kind of puzzles, then you might enjoy this game. The third was stolen from Simon Gruber from Die Hard.