## Practicing the Water Jar Problem...With a Vengeance!

If you are into puzzles, this one might sound familiar because it's been around for about 70 years [1]. It is so famous that it was actually used in the movie Die Hard: With a Vengeance [2]. In the movie, the two main characters are attempting to save a school full of children from being blown up while the bad guy has them solving some seriously challenging problems all over the city. To make sure you are prepared if you ever find yourself in this situation, you can get some practice by working through the problems below.

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 [3]. Use the interactive widget to solve problems 2 through 11. (Hint: be sure to empty a Jar before attempting to refill it.)

A: 21
B: 127
C: 3
Target: 100
0
0
0

## Einstellung: The Mechanization of Thought

While 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 [4].

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 Set

Einstellung 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 [5]:
1. 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?
2. 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?
3. 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 Connection

How 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!

Share and Enjoy!

Dr. Bob

[1] Luchins, A. S. (1942). Mechanization in problem solving: the effect of Einstellung. Psychological Monographs, 54(No. 248).

[2] 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.

[3] Another special thanks to Josh Fisher for creating the interactive version of the Luchins water jar problem.

[4] 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

[5] 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.

## Thursday, September 17, 2015

### I Work Out!: Brain Training

Editorial Note: I'm really excited about this week's topic because we are going to hear from a very good friend of mine, Dr. Jason Chein. In today's post, our guest writer is going to discuss a highly controversial and extremely interesting topic: brain training. Dr. Chein has conducted research in this area [1-3], which is why I'm so excited that he agreed to write this week's post. Take it away, Jason!

## Is it time to hit the gym…for your brain?

With so many advertisements and pop-culture books claiming that you can achieve a “smarter you” in just a few minutes a day of “brain training,” you might be thinking about hitting the cognitive gym. But don’t start strapping on your brain workout gear just yet. In today’s post I’ll take you through a brief history of some brain training research, and tell you about where the field stands today. (Hint: it’s not ready for primetime.)

If you were going by what had been the conventional wisdom in experimental psychology for the last several decades, then brain training — engaging in regular mental exercises that are intended to enhance your general cognitive functioning — would seem like a pretty silly idea. Just about all of the research from the mid-1960’s up through the turn of the millennium indicated that, while you could get incredibly good at just about any task (even really demanding ones) with enough practice, the benefits would be observed only for that specific task, and wouldn't transfer to other mentally challenging activities. Take for example the seminal work of Chase and Simon (1973) exploring the amazing memory of chess experts [4]. With just a few seconds to glance at the arrangement of pieces on the chessboard, advanced chess players can reconstruct the position of nearly every piece. Pretty impressive stuff! But, that’s true only if the pieces are in positions that “make sense” in the course of actual game play. If you change things up so that the pieces are placed randomly on the board (not in positions that would occur in a real game), then the experts’ memory drops to near novice levels (see for yourself in this video posted by psychologist Daniel Simons). And, it turns out that playing all that chess doesn't make someone generally smarter than others, or any better at problem solving in other situations. All those thousands of hours of practice and all it’s good for is beating someone at chess? Yep.

## Core Strength

In the ensuing years, many psychologists have tried to find a mentally engaging activity that would leave a bigger footprint on the landscape of cognitive functioning, but time and time again the results suggested that practice with a given skill just doesn't transfer to other skills. So, you'd think everyone would have given up on the idea of brain training long ago (and many had). But in the early 2000's a new(ish) idea started to gain some traction. What if, just as performance with many physical activities can be enhanced by focusing exercises on "core" musculature, intellectual functioning could be generally improved by focusing mental exercises on "core" cognitive systems. Makes sense, right? And, based on a large body of prior behavioral experiments, and corroborating neuroimaging studies, researchers had a pretty good idea what some of those “core” cognitive abilities might be. One that seemed especially promising was working memory; the topic of an earlier post from Dr. Bob. Working memory is supposed to serve as a general workspace for the mind, and a slew of studies show that individual differences in working memory capacity can explain why some people excel while others lag behind on a very wide range of cognitively demanding tasks. If the capacity of this general mental workspace could somehow be expanded, perhaps through repeated exercises that target working memory, this could have a profound impact on overall intellectual functioning!

With this basic idea in mind, a few pioneering researchers decided to throw caution to the wind and to try their hand once again at the brain training enterprise. And, to many scientists great surprise (especially those who were pretty settled on the conclusion that practice just doesn’t transfer), the early results looked really promising. First came a pair of studies showing that training focused on working memory and other executive processes was effective in improving cognitive performance among kids diagnosed with ADHD, and it turned out, even among the healthy kids and college students who had been included as the comparison groups in those studies [5, 6]. Those exciting early results inspired another study [7] that really captured the imagination of the field, showing that scores on a test of general fluid intelligence (the closest thing we have to an index of someone’s general intellectual ability) were improved by working memory training, and in a dose dependent fashion (more training = more improvement). At the time that paper was published, I was myself already engaged in another working memory training study [1], which ultimately showed that a month of training could enhance both attention control and reading comprehension in college students (we looked, but didn’t find any evidence of improved fluid intelligence in this group).

## Drinking From the Firehose

What started as a trickle of papers on working memory training soon turned into a deluge. Study after study seemed to be finding the same basic thing: that mental exercises targeting core functions of the mind (not just working memory, but also other “executive” and attentional functions) could produce meaningful transfer to important intellectual abilities. Yay, brain training works!…right?

Well, that depends on who you ask and what you mean by “works.” This is where the story gets interesting (and complicated, but don’t worry, I’ll keep it simple). After some of the initial excitement wore off, reports of failed replication attempts and null results (studies showing no benefits of training) started to come in. Others trying to reproduce the most impressive findings, like the gains in fluid intelligence and improvements in ADHD symptoms, weren’t always meeting with as much success. It seemed like the field was dividing into camps: let’s call them the ‘believers’ and the ‘doubters’. The doubters were understandably worried about failed replications, and raised some really important concerns about the methods used in earlier studies (like whether the groups that completed training and those that didn’t just had different expectations about how they should perform, similar to the placebo effect that can arise in drug studies). The believers kept at it, improved their studies to address the doubters’ concerns, and, even with these more careful measures in place (e.g., better control groups), many of their studies continued to produce exciting results.

So, which camp is right, the believers or the doubters? In situations like this we need to take a step back and look at the overall pattern and weight of the evidence. One way to do that is through meta-analysis – pull all of the relevant studies together, account for the size of the study sample (how many people participated) by giving more weight to larger studies, and then look to see where the “truth” lies. But here too the doubters and believers come to different conclusions. That’s because the answer you get depends on which specific studies you think should count, which methods you use to gauge the size of the training effect produced by each study, and most importantly, which behavioral outcomes you decide to focus on. There isn’t much debate about the benefits of training on tasks that are really similar to those that made up the training regime (we call these “near transfer” measures). In general, training does seem to improve performance on closely related tasks. So, if by “works” you mean “makes you better able to remember lists of things” (and indeed, that might be an important skill in some scenarios), then yes, it looks like training works. But does it boost your IQ, sharpen your attention, and improve your overall cognitive acumen (does it lead to “far transfer”)? I’d say we just don’t know yet. While there is some evidence that it can do these things, the overall body of evidence isn’t unequivocally favorable. But on the flip side, there also isn’t enough evidence that it doesn’t work (getting a little technical here, Bayesian factor analysis suggests that there is neither enough evidence to accept the claim nor to reject it). So pick your favorite metaphor – the jury is still out, the dust hasn’t settled, the waters are still too muddy – and maybe wait until the next New Year before you make a brain training resolution.

Dr. Jason M. Chein is currently a faculty member at Temple University, where he is the principle investigator of the Neurocognition Lab. I met Jason in 1998 when we were both graduate students at the Learning Research and Development Center. While in grad school, Jason became an expert in cognitive neuroscience, which included learning cool methodologies like conducting studies using fMRI. While in grad school, Jason also became quite proficient at frisbee golf.

[1] Chein, J., & Morrison, A. (2010). Expanding the mind’s workspace: Training and trans- fer effects with a complex working memory span taskPsychonomic Bulletin & Review, 17(2), 193–199.

[2] Morrison, A. B., & Chein, J. M. (2011). Does working memory training work? The promise and challenges of enhancing cognition by training working memoryPsychonomic Bulletin & Review, 18(1), 46-60.

[3] Morrison, A. B., & Chein, J. M. (2012). The controversy over CogmedJournal of Applied Research in memory and Cognition, 1(3), 208-210.

[4] Chase, W. G., & Simon, H. A. (1973). Perception in chessCognitive psychology, 4(1), 55-81.

[5] Klingberg, T., Forssberg, H., & Westerberg, H. (2002). Training of working memory in children with ADHD. Journal of clinical and experimental neuropsychology, 24(6), 781-791.

[6] Klingberg, T., Fernell, E., Olesen, P. J., Johnson, M., Gustafsson, P., DahlstrÃ¶m, K., Gillberg, C.G., Forssberg, H., & Westerberg, H. (2005). Computerized training of working memory in children with ADHD-a randomized, controlled trial. Journal of the American Academy of Child & Adolescent Psychiatry, 44(2), 177-186.

[7] Jaeggi, S. M., Buschkuehl, M., Jonides, J., & Perrig, W. J. (2008). Improving fluid intelligence with training on working memory. Proceedings of the National Academy of Sciences, 105(19), 6829-6833.

## Thursday, September 10, 2015

### There and Back Again: Near and Far Transfer

Imagine for a moment that you are landing in a city you have never visited before, and you have to find your way from the airport to your hotel. When you land and get off the plane, what do you do? What steps might you have to take to navigate your way from the airport terminal all the way to the comfort of your 4-star hotel room? If you have travelled before, what aspects from your prior experiences, if any, might you draw upon to get you to your hotel in this new, unfamiliar city?

## Near vs. Far Transfer

Hopefully you will be able to take advantage of some of your previous travel experiences in the above scenario thanks to what is known as transfer. Transfer is when knowledge learned in one domain is applied to a different domain. A domain is the topic or the subject matter of the to-be-learned knowledge. An everyday example of transfer comes from navigating a mass transit system. Suppose you grew up in a place where the only mass transit option was the city bus. When learning how to ride the bus, you figured out that a key to getting on the right bus was the sign found at the top of each bus, which displayed the line number and the terminal destination for that bus. This information was critical because it indicated which route the bus was going to follow, and whether the bus was heading towards the stop you want or away from the stop you want. This information is useful because it tells you if the bus is heading in the direction you want to go.

Now suppose you then find yourself in a new city, such as Washington, DC, and you want to ride their Metro, which is their subway system. You would demonstrate transfer by applying what you know about riding the bus (i.e., the source domain) to riding the subway (i.e., the target domain). Subway trains also display the line number and the terminal destination at the front of each train.

We might say that transferring knowledge from riding a bus on one line to riding a bus on a completely different line isn't much of a stretch because they are both busses, and they both use a destination sign to communicate with the rider. That is an example of near transfer because it involves learning within the same domain (i.e., riding the bus). What would be an example of far transfer? Learning how to ride the subway would be an example of far transfer because the surface features vary slightly and so does the setting or context (e.g., bus stops vs. subway stations).

## Why Might Transfer Fail?

Although transfer can be very useful, it can also be very hard to accomplish. Why is transfer hard, and what are some of the ways that it might fail?

Transfer might fail when there is a mismatch between the learning setting and the application setting. A lot of learning occurs in the classroom, and it is the hope of every teacher that students transfer their lessons to real life. A semi-famous counterexample is a study of Brazilian street children making change [1]. When working on the streets, these children were able to perform fairly complicated computations in their heads. But when they were asked problems that had the same deep structure, and only the surface features changed (i.e., isomorphic problems), then they failed to solve the problems. In other words, the Brazilian street merchants were not able to transfer what they knew from selling candy to the classroom environment.

Another way in which transfer might fail is when the surface features of the problem change. One of my favorite examples of transfer failure comes from geometry [2]. In this example, children learn how to calculate the area of a parallelogram. When learning this particular skill, the problem is accompanied by the following diagram (see Fig. 1).

 Figure 1. To calculate the area, drop two perpendicular bisectors.

The student is shown that when you drop two perpendicular bisectors (lines 1 & 2), which form two triangles of equal size that essentially equate the parallelogram with a rectangle. Since students already know how to compute the area of a rectangle, they can easily solve this problem. However, when the students are asked to compute the area of the following parallelogram (see Fig. 2), the children claim that they never learned how to solve this type of problem!

 Figure 2. How do you calculate the area of this parallelogram?

This is a rather tragic example because it means that the students cannot see the applicability of their knowledge in the two different situations. This is an example of the failure of near transfer.

## The STEM Connection

To reiterate, transfer is hard and can fail for multiple reasons. It can fail when the learning and application settings do not match. It can also fail when the learner does not recognize the connection between the surface features of what they learned (e.g., a parallelogram resting on its base) and a slightly different case (e.g., a parallelogram that is rotated and resting on its side).

Why does the setting and surface features matter? In a previous post, we learned about procedural knowledge, which can be modeled using what is called a production rule. Production rules have two parts: a condition and an action. When I see this (condition), then I should do that (action).

Production Rule: [ Condition ] => [ Action ]

We can model problem solving in geometry as a series of production rules. We might, for example, say:

Condition: If I see a parallelogram,
AND: My goal is to compute the area;
Action: Then, calculate the product of the base and the height.

One potential explanation why transfer fails is because the condition (i.e., the left side of the production rule) is not general enough for the learner to see when his or her knowledge applies. The goal of education is to help students generalize their knowledge to the point where they can see how it applies across settings and across seemingly disparate situations.

Share and Enjoy!

Dr. Bob

[1] Carraher, T. N., Carraber, D., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3, 21-29.

[2] Wertheimer, M. (1945). Productive thinking. New York, NY: Harper.

## Thursday, September 3, 2015

### They Call Me the Working Man: Working Memory (Part 2)

Editorial Note: This is a continuation of a previous post where we discussed a  model of a short-term memory buffer we called "working memory." In this post, we explore the underlying mechanisms for how working memory-capacity can change over the course of a lifetime.
Here is a fun game called the dual n-back task. For the first task, you have to remember the spatial location of a series of squares. For the second task, you need to hold a few numbers in working memory. It becomes a dual task when you combine the tasks and do them at the same time. Sounds hard, right? It is. Try it for yourself.

## Working Memory Expansion Pack: Adding Capacity

How did you do? Did you feel like your working memory was being taxed as you added the second task? What if you tried the n-back task when you were a kid? Would your adult self beat your younger self? In other words, does working-memory capacity change as we grow older?

There are roughly three different influences that can change working-memory capacity: neural development, knowledge, and recall strategies.

Neural Development. At age five, children can remember about four random digits or letters. By the time they are 20, they can remember upwards of seven or eight arbitrary digits or letters. It seems, therefore, that our brain adds capacity as it naturally develops.

Knowledge. Age, however, is hopelessly confounded with knowledge and experience. As we grow older, our brain undergoes massive changes and we file away volumes of new experiences and skills. So what would happen if we could somehow dissociate age and knowledge/experience? To do so, we would need to find areas in which kids, despite their young age, know more or have more experience than adults.

As it turns out, there are indeed areas in which kids know more than adults. Chess is just such a domain, as it is easy to find kids who have vastly more chess-related knowledge than adults. What if we pitted kids who are young, but highly knowledgeable about chess, against adults who are older, but less experienced when it comes to chess? Who would be able to remember more positions of chess pieces on a chess board? Who would be able to remember more numbers from a list of random digits? It turns out that researchers have investigated this and the results are plotted below [1].

As you can see, children were able to recall about nine chess positions, but only about six numbers. The results for adults were completely flipped. Namely, they remembered fewer chess positions than the children, but recalled more numbers. This suggests that children don't necessarily have a lower working-memory capacity. Instead, it indicates that they have less experience to help structure their recall.

Recall Strategies. You might be skeptical of that last statement. Why would experience increase your capacity to recall a list of digits? A perfect example of how experience can enhance the recall of random digits was demonstrated in a previous post in which we used the years of significant historical events in American history to form four-digit chunks. Also, we encountered a person who underwent deliberate training to increase his working memory capacity to a startling 79 items. Finally, we learned strategies for memorizing arbitrary lists of words, numbers, and phrases. Because adults have more experience temporarily storing information, we have come to develop our own strategies. Kids, on the other hand, have had fewer opportunities to figure out ways to hack their own memories.

## The STEM Connection

Working memory capacity is obviously important for education. Research on this topic suggest a couple of conclusions. First, individuals have different working-memory capacities. We refer to this as an individual difference (like height or eye color). We also learned that working memory can change as a function of normal development. Some estimate that kids between the ages of 5-7 can remember four words and about the same number of digits. College-age students, on the other hand, can store upward of six words and approximately eight digits.

We can't do anything to expedite normal development, but we can help students gain familiarity with the symbols and structures of information from a particular domain. That is where the real growth can happen. As we saw in the chess example, if students put in the time, they can expand their working-memory capacity to a point where they can outperform adults.

In conclusion, working-memory capacity seems to change over the course of a lifetime, and there are three potential explanations. First, we add capacity naturally as a consequence of natural brain maturation. Second, we learn new techniques and strategies for remembering, like repeating the same items over and over. Finally, we acquire new knowledge, which we can then use to help structure the to-be-remembered information. Kids who are chess experts can form larger chunks than adults who are novice chess players. Hopefully, this will supply you with the tools you need to go out and get a mental upgrade!

Share and Enjoy!

Dr. Bob