What's your problem?: Routine vs. Insight Problem Solving

Take a few minutes to solve the following problems:

1. Three hobbits and three orcs are running for their lives. They are being chased by Smaug the Dragon, who drives them towards a river. Fortunately, there is a boat that can ferry one or two creatures across the river. Unfortunately, if the orcs outnumber the hobbits, then they will be eaten. Strangely enough though, the orcs can be trusted to return the boat to the other side of the river. How can the hobbits safely cross the river?
2. You wake up in a cabin in the woods. On the table, you find a box of tacks, a candle, and a book of matches. Since it is getting dark, your goal is to attach the candle to the wall. 

While you were working through each of the problems, did you notice anything different about the way you approached the problem? Was there something different about the way you arrived at the solution? Once you had the solution, was it obvious? 

As you know, problems come in many different shapes and sizes. Some problems are easy, others are hard. Some need extensive domain-specific knowledge, while others require relatively little knowledge. Some problems are well-defined and have "right" or "wrong" answers, whereas others are ill-defined and have "better" or "worse" answers.

The distinction between the problems stated above is that the first is a routine problem and the second is an insight problem. What's the difference?

Just a Regular Day at the Office

A routine problem has the following characteristics. First, it starts with a well-defined initial state and goal state. In other words, we know what must be done, and we know when to stop once we have solved it. We can see that these two criteria hold up for the "hobbit and orcs" problem. The initial state is three hobbits and three orcs on one side of the river. The goal state is all three hobbits safely on the other side of the river. 

The second characteristic of a routine problem is that there exists an operator to move the problem from the initial state to the goal state. The operator in this case is extremely basic. One or two creatures get in the boat and move to the other side of the river. Once they have moved, then the problem is in a transitional state (i.e., somewhere between the initial and goal state).

Given the first two characteristics of routine problems, you can completely map out a problem space. The problem space for the "hobbits and orcs" problem is pretty small because there are only a few transitional states. But for more interesting problems, or games such as chess, the problem space can be enormous. 

If a problem can be characterized in this way, then solving the problem includes searching through the problem space for a path that connects the initial state to the goal state. With experience, people can become quite proficient at solving problems because they generate different methods that apply across many different types of problems. These methods are called problem-solving heuristics because they help solve the problem, but they do not guarantee a solution.

A Problem Less Ordinary

Insight problems may lack one (or more) of the defining characteristics of a routine problem. For example, it might not be entirely obvious what the operators are for an insight problem. Nor can you plausibly describe or articulate any of the transitional states. For example, if you're like me, you probably needed a sheet of paper to solve the "hobbit and orcs" problem. My sheet of paper looks like a filmstrip. Each frame of the film is a different state of the problem. With the candle problem, I don't really have any transitional states. I immediately leapt from the initial state to the solution via an aha! moment. 

Because insight problems lack the same characteristics of routine problems, does that mean there aren't any heuristics for solving them? That's a tough question, and I don't have a really solid answer. But I would say that you can train yourself to ask: What assumptions am I making? Which of those assumptions aren't true? This is the essence of "insight" because you have to question that which you cannot see [1].

Going back to the nine-dot problem from a previous post, notice that most of us initially make the assumption that you are required to stay within the bounds of the dots. Well, it turns out that was not a requirement of the problem. In other words, we have artificially constrained the problem. A problem constraint might be defined as the "rules of the game" or what counts as a valid solution. In the "hobbits and orcs" problem, there is an implied constraint that the hobbits can neither call in reinforcements nor get a bigger boat. I believe you can get better at solving insight problems by searching for problem constraints that are self-imposed.

The STEM Connection

Most math problems are pitched as routine problems. You have an initial and goal state, and the lesson for the day is to learn the "rules" or operators that help transition from the initial to goal state. Let's take the topic of summation as an example. You want your class to sum all of the integers between 1 and 100. The initial state is given, and the goal state is well-defined.

To motivate the lesson, you might talk about what the students already know. They already know the operator, which is adding up numbers, like this: 1+ 2 + 3 + 4 + ... + 98 + 99 + 100. Then, once they are done with that task, they learn about the summation symbol, its index, and the lower and upper bounds.

Solving this as a routine problem is generally necessary to go onto more advanced topics (e.g., computing the average of a dataset). But what if, after the solution is known, the problem is reframed as an insight problem? Wouldn't that make the lesson more interesting? Assuming your kids know about the commutative property of addition, the problem could be recast as: Can you find a more efficient way to calculate the sum of all the integers between 1 and 100? [2]

I realize that not all routine problems can be easily recast as insight problems; however, I do believe that knowing the distinction between routine and insight problems is useful because it highlights the fact that different mental operations are needed. When solving routine problems, the question one should be asking is: What operator do I apply? Alternatively, when solving insight problems, one needs to be asking: What constraints do I need to relax? Armed with these questions, there aren't any problems left that we can't solve!


Share and Enjoy! 

Dr. Bob

For More Information

[1] This is where Zen Buddhism meets Cognitive Psychology!

[2] Perhaps apocryphal, but one of my favorite stories of insight problem solving is little Karl Gauss, who, in primary school, summed the values of 1 through 100 in his head. How did he do it? Easy! By recognizing that 1+100=101, and that there are 50 pairs of these sums. I can almost guarantee he had an aha! moment when he arrived at his solution.

The Shape of an L: Mindset

Let's start by solving a super-easy puzzle. Given the figure below, can you make four equal triangles by re-arranging 3 of the matchsticks without overlapping any of them?

Did you solve it, yet? Yeah, I thought so. It's so easy that I almost didn't share it because I didn't want to insult your intelligence. [1] 

Whether you solved the puzzle or not:

• What thoughts (if any) were going through your head that weren't task related? Did you think about your ability to solve this problem? 
• Did you think about your innate intelligence and what the implication of solving (or not solving!) this problem had?

What makes you think you're so smart? 

This demonstration is meant to help you diagnose your own personal theory of intelligence. Do you believe that you were born with a fixed intelligence that does not change over the course of your life? Or do you instead believe that intelligence is something that changes due to the various experiences in your life? 

Psychologist Carol Dweck refers to these two theories of intelligence as opposing mindsets. The first theory is held by those with a fixed mindset. Those who have a fixed mindset believe their intelligence will not change. It's something that you are born with, like the color of your eyes. Engaging in a challenging task is uncomfortable because the difficulty of it could mean that you are not smart. Thus, people who have a fixed mindset gravitate toward easier tasks that they know they can solve. It also means that they tend to stay away from challenges because failing at something might reveal that they are not smart (or that they are a "loser"). 

The second theory of intelligence is held by those with a growth mindset. Those who have a growth mindset see intelligence as malleable. You can expand your intelligence through experience. A difficult problem isn't seen as a threat; instead, it is seen as a challenge. It might even be perceived as a chance to learn something new. According to a growth mindset, you can get smarter by trying hard, not giving up, and learning from your mistakes.

In a previous post, I mentioned the importance of being well-calibrated in terms of your meta-cognitive awareness. An individual who rightly knows what he does not know is at a supreme advantage because that person is better able to target gaps in his understanding. If you have a fixed mindset, it might be be uncomfortable to admit that you're not knowledgeable or highly skilled. But if you have a growth mindset, then admitting a lack of knowledge is a source of strength and not a weakness.

What if, at the beginning of this post, you held tightly to a fixed mindset? Is it possible to change your mindset? 

Mindset Characteristics

I think it's worth noting a couple of features of mindsets. First, they can change over time. If you grew up thinking that intelligence was a fixed trait, and later become convinced that they are not, then you can switch to a different mindset. In other words, a fixed or growth mindset is not itself a fixed trait. People can change their mindset at any time. 

Second, an individual might hold different mindsets about different domains. So far we have focused strictly on mindsets about intelligence, but you can have mindsets for any skill or characteristic. For example, you might hold a fixed mindset about artistic vision and creativity. Maybe you think that people are born artists. However, you might also have a growth mindset with respect to athletic ability. Maybe you were terrible at baseball the very first time you played. But after you joined a team and practiced with your friends, you became quite good. Not only are mindsets malleable, but they are also domain-specific. 

The STEM Connection

What are the educational implications of a fixed vs. growth mindset? Which mindset do you think educators want their students to adopt? If we surveyed educators, we would probably find that they want their students to gravitate toward challenging tasks, not to feel stupid if they can't solve a problem on the first try, and to learn from their mistakes. All those behaviors are aligned with having a growth mindset in the classroom. On the flip side, educators would probably also agree that the fixed mindset is prevalent in our society.  So what can be done to foster the growth mindset in the classroom? 

Probably the easiest change to make is to praise students for their effort instead of their intelligence. I realize it is fairly common for a teacher to say, "Well done, Bethany. You are extremely smart!" However, the recommendation from the mindset research is to praise the student's effort. So, instead, one should say, "Well done, Bethany. I noticed that you struggled at first. But you never gave up, and that was important because you eventually solved the problem and learned from the mistakes you made along the way." 

In addition, teachers should model problem-solving tenacity for their students. Even though it might be uncomfortable at first (and it goes against the idea that "teacher knows best"), teachers should let their students give them problems that cause them to struggle. I can see an advantage to admitting defeat and coming back another day with the solution. The goal is not to appear to have the right answers. The goal is to show that you don't back down from challenges and that you don't give up until you have tried every approach you can think of to solve a problem. 

It is tempting to think of kids as smart (or not). But we need to overcome that temptation and see them, instead, as starting at out at different baselines and trying hard to progress beyond what they already know. I would much rather see a student struggle and succeed than to fly through a task. Wouldn't you?

Share and Enjoy! 

Dr. Bob

For More Information

[1] If you want to see the solution, just send me a message. 

Rolling in the Deep: Explanatory Depth

Think about the following questions:

1. On average, how many pounds of coffee does Columbia export to the United States in a year? 
2. What is the formal procedure for filing a patent with the U.S. Patent Office?
3. How does a sewing machine work?

Instead of producing an answer for each question, rate each question on the following dimensions:

• How likely am I to know the answer?
• How confident am I that I know the answer? 
• How well could I explain the answer to a friend?

Getting Meta-cognitive About Question Types

In a previous post, we talked about two types of meta-cognition: the what (what we know) and the how (how we know what we know). The above questions all target the what, but can be further categorized into 3 different types of questions.

Declarative Questions: The first question is a simple, declarative fact that can be answered by going to a search engine and entering a query that will return a sensible answer. For declarative questions, figuring out if you know the answer or not is a relatively straightforward process. You look into long-term memory and search for an answer (or a close approximation). If you don't get any hits, then you know you don't know the answer.

Procedural Questions: The second question is procedural, in that you must know the steps of the process necessary to achieve an end goal. To learn how to file a patent, you could also interrogate your favorite search engine, ask an expert (e.g., a patent lawyer), or learn the process yourself by reading through the steps on the U.S. Patent website. Again, knowing if you know how to do this might be as simple as asking yourself, "Have I ever filed a patent?" If the answer is "no," then you know you don't know how to do it. 

Mental-model Questions: The third class of question has a definite declarative component. The facts, however, are not stated in isolation. Instead, they are linked together in some coherent way. For simplicity, I will call this type of question a "mental-model question" because it requires knowledge about causal connections between related facts. When asked if you know how a sewing machine works [1], what probably happens is that you think about a sewing machine and what it does. Then you think about the various components that are in a sewing machine and how they interconnect. Finally, you run a mental simulation and see how the various components interact. 

The interesting feature about mental-model questions is that you typically don't get a binary "yes/no" when you ask yourself if you know the answer. For example, when you ask yourself if you know how a sewing machine works, the answer is probably not "yes" or "no," but somewhere in between. The degree to which someone understands something is what I am calling explanatory depth. If a person has deep knowledge of a topic, then they are able to provide lots and lots of details. If they only have a cursory understanding, then they will give you a much shorter and more incomplete explanation.

The Deep End of the Spool

How might we cultivate the skill of assessing our own depth of understanding of a topic? One way to evaluate the depth of our knowledge is to try to provide an explanation and see how far you can get. For instance, if someone asks me, "Do you know how a sewing machine works?" I might be tempted to say "yes." To test whether my meta-cognitive awareness about my knowledge of the inner workings of a sewing machine is well-calibrated, however, I could attempt to provide an explanation. My explanation would go something like this: 

There are two primary parts of the upper portion of the sewing machine: the needle and a spool of thread. The thread is fed through a hole in the bottom of the needle, which moves up and down, piercing the fabric. Underneath the fabric is a hole that the needle descends into, where it meets another length of thread coming from something called a bobbin. Through some bit of magic, the thread from the needle combines with the thread from the bobbin to form a stitch. 

After actually trying to explain how a sewing machine works, I realize now that my meta-cognitive awareness of sewing machine mechanics was not well-calibrated. As you can plainly see, I do a fair bit of hand waving when it comes to the process by which the thread from the bobbin intertwines with the thread on the needle. Now that I am better calibrated, meta-cognitively speaking, I would have to say that I do not understand very well how a sewing machine works because the depth of my explanation is lacking precise details. [2]

The STEM Connection

Why is this important for learning? The concept of explanatory depth is important because it is often the case that we only have a partial understanding of many different topics. That partial understanding, however, makes us overconfident about what we know. As stated in a previous post, we want our students to be highly calibrated in judging how well they know something.

There are a couple of obvious ways to test and improve explanatory depth, and thereby increase meta-cognitive awareness. As is the case for any desired skill, practice always helps. To build meta-cognitive awareness about mental-model questions, we need to give our students as many opportunities to explain their reasoning as possible. This might be in the form of small groups so that students can explain to each other how things work. Working in small groups also exposes students to the ideas of their peers, which could be an added bonus in that students may better understand an explanation from a peer who has recently gone through the process of acquiring his or her own understanding. This is obviously in direct contrast to hearing an explanation from a teacher who has understood the topic for several years, and has thus forgotten why the topic is potentially confusing. It might also be beneficial to have whole-class discussions so students can model and test their reasoning in front of others.

Another, perhaps more controversial, recommendation would be to start using oral exams. It's likely that students will mutiny if they hear their next exam is going to be a face-to-face conversation with their teacher. But hearing a student talk about what they know is probably the best way to diagnose how deeply a student knows a topic. To prepare, the teacher would need to develop a robust scoring rubric so that it is obvious how deep a student's explanation is. 

The depth of an explanation is a pretty good proxy for how well you know something. I strongly recommend trying to explain things as much as possible because when you falter (as I did above), you uncover new opportunities to learn and refine your knowledge!


Share and Enjoy! 

Dr. Bob

For More Information

[1] Miyake, N. (1986). Constructive interaction and the iterative process of understanding. Cognitive science, 10(2), 151-177.

[2] For a great animated gif that demonstrates the sort of magic I couldn't explain, see the wikipedia entry on the sewing machine (hat tip: Mike Hasko). 

What was I thinking?!: Meta-cognition

Rate yourself on the following statements using the following five-point scale:

1 2 3 4 5 Strongly disagree Disagree Neutral Agree Strongly agree

1. I know a lot about the Civil War.
2. I know how to hit a baseball.
3. I know how to solve a system of linear equations.
4. I am an expert chef.
5. I am a right-brain thinker.
6. I am a visual learner.
7. I only need to see something once, and I can remember it forever.
8. I can multitask.

It's Time to Get Meta.

The concept that each of these statements intends to target is meta-cognition, or, your knowledge about what you know and how you think.

Statements 1-4: Let's start with the "what" first. Questions like, "Do I know this?" or "How well do I know this?" target your meta-cognition awareness of your warehouse of knowledge. Being aware of what you know is important because it can guide your behavior toward filling the gaps that you perceive in your knowledge. However, there is always a danger that we overestimate how much we know. Remember that demonstration with the visual blind spot? Well, that can also apply to meta-cognition. We sometimes don't perceive the gaps in our knowledge.

Statements 5-8: Meta-cognition also applies to the "how," or beliefs you have about your own thought processes and the way your mind works. For example, you may believe that your memory is like a video recording where all of the details are preserved in a pristine library somewhere in your brain. If you believe that, then you are going to have more trust in your memories than someone who believes that memories are actively reconstructed at the time of retrieval. Your understanding of how your mind works will also guide your behavior. For example, if you think your memory is like a video recording, then you may think you only need to read or hear something one time to remember it forever, rather than needing to write it down or review it a few times to make sure you can recall it later.

Putting the BAM! into Meta-cognition

If Emeril were a Cognitive Scientist, he'd say it's time to kick it up a notch. Hopefully you buy into the fact that meta-cognition operates on two levels. It applies to being able to declare what you know (the what). It also applies to the declarative knowledge that you have about how the mind works (the how). So here's how we can take it to the next level: Meta-cognition is a skill that can be developed with practice. BAM!

From my perspective, that is the most interesting feature about meta-cognition. We can become more accurate in our assessment of what we know or don't know. A simple way to start exercising this skill is to get into the habit of asking yourself: Does that make sense? If the answer is "sort of" or "no," then that becomes a trigger for asking deeper questions ("What is the root of my confusion?"). Because many people don't find it fun to admit they don't know something, it may not feel natural at first to go looking for things you don't know. But, if you embrace each gap that you identify in your knowledge as a new opportunity to learn and grow, then you'll soon reap the benefits of developing your meta-cognitive awareness. 

The STEM Connection

Why is the what and the how of meta-cognition important for learning? It is important for students to be accurate about what they know. For example, if you ask your class, "Does that make sense?" and they answer in the affirmative, then you might expect that they get it. However, when students who are not well-calibrated get home and start working on their homework, they will find that they did not understand the lesson as well as they thought they did. Now they have a new problem to solve. How do they get the support and guidance they need outside of the classroom? Do they ask their parents for help, do they go to the internet, or do they give up entirely? 

It is also important for students to have an accurate understanding of how the mind works. It's unrealistic to say that you will remember something if you've only seen it once (i.e., memory is not a video recording). Students should also understand that multitasking will fail under most conditions, so putting aside the cellphones while studying is probably a good idea. Finally, it would be helpful for students to understand the limitations of their working memory so they can tell their teacher to slow down or chop up the lesson into smaller pieces.

When do students get feedback on their meta-cognitive skill for assessing their knowledge? That generally happens during an assessment, which is often too late in the learning process to be helpful. Instead, it would be ideal if students could practice their skill of knowledge assessment before the big exam. Asking students to explain the content to each other in small groups is a great way for kids to determine whether or not they truly "get it." 

Meta-cognition is a skill that can be exercised and strengthened, and we need to give our students opportunities to assess their knowledge and receive feedback on their accuracy. As my friend Ido Roll once said, learning to gauge when you need help might be the secret sauce for learning. BAM!


Share and Enjoy! 

Dr. Bob

For More Information

[1] Roll, I., Aleven, V., McLaren, B. M., & Koedinger, K. R. (2007). Can help seeking be tutored? Searching for the secret sauce of metacognitive tutoring. In R. Luckin, K. R. Koedinger, & J. Greer (Eds.), Proceedings of the international conference on artificial intelligence in education (pp. 203-210). Amsterdam: IOS Press.