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00:04:44 The Law of the Instrument

00:09:05 How to Develop Better Mental Models

00:09:50 To develop your ability to use and create better models, try the following: Expose Yourself to a Variety of Ideas The key is variety.

00:10:40 Always Connect

00:11:24 Cultivate Metacognition

00:16:43 In the Lankavatara (a third- or fourth-century Mahayana Buddhist sutra), there is a passage

00:23:55 How to Make Your Own Analogies

00:29:39 How to Teach Like Feynman

00:30:24 The Protégé Effect

00:37:29 How to Explain Anything to Anyone

00:39:07 I is for Introduce

00:40:00 R is for Relate

00:40:31 A is for Apply

00:41:29 D is for Demonstrate

00:42:05 E is for Examine

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• A model is a map of reality and a scaled down, simplified representation of something in the world. Feynman’s genius rested on the power of his mental models and maps of reality and his ability to switch between them. Kaplan’s law of the instrument says that “when all you have is a hammer, every problem looks like a nail.” To better solve problems, have more tools in your inventory and become better at choosing which one to use when.

• Developing good mental models is not the same as refining your personal worldview or philosophy. It’s about learning to use your own brain as your primary instrument, and use worldviews as secondary instruments. To develop better models, expose yourself to a range of ideas, constantly seek to connect different ideas, and cultivate metacognition.

• Science is a form of imagination that uses language, imagery, metaphor, symbolism, and analogy to engage with reality. We can understand more by connecting the unknown to the known—but without getting confused about what is “finger” and what is “moon”!

• The protégé effect is simply when teaching (or even pretending to teach or preparing to teach) helps us understand our material in a deeper and more insightful way. Use simplified, ordinary language, get your ego out the way, and teach someone else/yourself to get to the heart of a concept.

• The IRADE method is an approach used in physics education, and it stands for introduce, relate, apply, demonstrate, and examine. It can help you not only teach and learn, but communicate your ideas more clearly.

#Analogy #Bessel #BetterMentalModels #CultivateMetacognition #ExposeYourself #Feynman #AuthorAbrahamKaplan #IRADE #Lankavatara #Mahāmati #MahayanaBuddhist #Mental #NeilsBohrA #Nestojko #ProtégéEffect #RichardFeynmansMentalModels #RussellNewton #NewtonMG #PeterHollins #TheScienceofSelf #RichardFeynman’sMentalModels

Bader knew I had studiedCalculus for the Practical Mana little bit, so he gave me the real works–it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn’t know anything about. That book also showed how to differentiate parameters under the integral sign—it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when the guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school.

If it was a contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me." —Richard Feynman,Surely You’re Joking, Mr. Feynman!(pages 86–87) Scientists love models. A model is a map of reality—it’s a scaled down representation of something out there in the world. A map, for example, is itself a model of the terrain it is describing. It is not the same thing as the landscape, and has been greatly simplified, but it really helps in navigating that real landscape when you’re in it.

The picture you have in your head right now of what an atom looks like is not really what an atom looks like—it’s a model invented by Neils Bohr. A computer simulation of evolution or the economy are both models, as are mathematical representations of certain physical phenomena. Established scientific theories come to depend on well-known conventional models that become a part of culture, but Feynman shows that each of us can also have our own mental models (i.e., perspectives and points of view). Feynman’s strategy of differentiating under the integral sign was a mental model that he had in his toolbox that was missing in the toolboxes of his peers. The result was that he could sometimes solve problems they couldn’t. It’s not that Feynman was a supersmart genius who could use the same tool better—rather, he had another, different tool. Even better, he knew when and how to switch tools to best tackle the task in front of him. This is the power of mental models and why it’s so important to consciously choose the ones you’re using.

The Law of the Instrument Author Abraham Kaplan describes what he calls the law of the instrument: “Give a small boy a hammer, and he will find that everything he encounters needs pounding." This is not unlike the old proverb that says that when all you have is a hammer, every problem looks like a nail. The implication is obvious: If you only have one framework through which to view the world, you will find that every problem you encounter seams to fit into the framework somehow. The world will narrow to fit your limited conception of it. And as Feynman described, if you simply never occupy a certain mental framework, you may live forever without knowing what a certain problem might have looked like through that lens. Every model is a simplification—it crunches reality down so we can more readily manage it. But this is not a problem so long as we are aware of the limits of the models we’re using.

We need to always be cognizant of the fact that we are using mental models in the first place, that these models are by their nature limited, and that we can always abandon our model for another one if it serves our needs better. Remember the paradox of expertise? It may be that the deeper into one field our expertise grows, the more at risk you are of “confusing the map for the territory,” or assuming that your particular skill set is the answer to every problem you encounter. You apply it to everything. Confirmation bias allows you to remember all the times it works, so your assumptions get further entrenched, and when it doesn’t work, you conclude the problem is unsolvable. In other words, you have a hammer, and you go around bashing everything. One day you are challenged to do open heart surgery, but you kill the patient with your hammer and then say sadly, “Well, you can’t save everyone.

There are limits to what medical science can do!" Your mistake is to think that your model equals medical science. Your mistake was to not have a range of different tools at your disposal. Here, “tools” does not mean a flashy new app or gadget, an updated theory, or a machine that can do more than you can. This is just more of the same, a little like becoming really, really good at using a hammer. What you want to do is look beyond the hammer entirely—not just inventing literal new tools, but theoretical and philosophical frameworks that spawn the very idea of those new tools in the first place. In the movie inspired by the life of the mathematician John Nash, A Beautiful Mind, Nash is attempting to hold a lecture while construction workers are making a noise outside.

Nash closes the windows, but the students complain of getting hot. One of the students then gets up to open the window and politely asks the workers to take a break until the class is over. Nash then says, “Well, as you will find with multivariate analysis, there are a number of solutions to any given problem." Regardless of how accurately the movie depicts mathematical genius (probably not very well), the point is neatly made. One way of looking at the problem is to imagine it mathematically: The window can be open or closed, and there are two distinct problems with each possibility. But, this “either/or” mental model is not the only one we can use. The student shows that there is another one, where the construction workers outside are not taken as a fixed variable.

What’s more, the solution of the mathematician (close the window, get hot) is not necessarily the solution of someone with better social skills (negotiate and cooperate). The question of whether the window should be open or closed is most satisfactorily resolved when one relinquishes the mental model that says that open and noisy versus closed and hot are the only options possible. How to Develop Better Mental Models Developing good mental models is not the same as refining your personal worldview or arriving at a workable philosophy for life that you are happy with. Rather, it’s about learning to use your own brain as the primary instrument and reminding yourself constantly that the best way to think is the way that allows you to best understand what you are seeking to understand. Many people are surprised to learn that the idea of an ecosystem is just a model and not truly the way that the natural world works. Likewise most things they are taught in high school physics is a model—not actually what is happening. To develop your ability to use and create better models, try the following: Expose Yourself to a Variety of Ideas The key is variety.

Read things that are not on the ordinary prescribed reading list. If you process only that information that everyone else processes, you will end up with a mental model much like theirs. Read little known material within your field, but also read material from other fields. Try to expand the depth and breadth of what you consider your field in the first place. If you are studying science, study also the history and philosophy of science. Look at how your topics of interest overlap with other disciplines, with politics, with culture, and over time. Always Connect Your brain is a pattern-finding machine.

Every time you learn something new, ask how it fits into the web of meaning you already have. Look for links and connections (though, don’t force them!). Switch your perspective and look at the same problem through another expert’s eyes—what do they see that you didn’t notice before? Intelligent people are always cross-referencing ideas this way, but it’s something you can teach yourself to do. The real world is not like school, where each “subject” is cut into neat, separate chunks. Everything is happening all at once! Cultivate Metacognition Get into the habit of thinking about your thinking.

Ask questions about the questions you are asking. Think about Feynman’s response to the question about magnets; he didn’t just talk about magnets on the level the question was asked. He pointed to the different ways one might look at magnets, and the different ways one might evaluate those ways. He pointed to the act of questioning itself as an object for deeper understanding. To the degree that you can think reflexively in this way, you will be able to think in, under, around, about, without, and outside the box, and spare yourself from becoming one of those people who don’t know they’re in a box at all. Consider this (admittedly simplified) example. Johnny has come to make the following observation: Every time he drinks a single alcoholic drink, he feels completely and utterly ruined—he gets a fearsome headache, he feels on the brink of fainting, and it takes him two days to recover.

He starts with a natural question: Why? You might have a few suggestions after making the same observation, but from within Johnny’s map of the world, he comes to this conclusion: You are becoming very healthy and strong, and that means your body is more intolerant to toxins and poisons than ever before. This is a good thing. Johnny goes a bit further and starts to frame his alcohol intolerance as a subtle sign of his moral and spiritual development, too. It’s as though the very cells of his body are rejecting what is bad! But Johnny could have thought about the problem in many different ways. He could have asked why can’t I process alcohol? and then investigated the properties of alcohol itself, trying to deduce its effect on him.

Or, he could have concluded that there was a psychological reason—perhaps he has some deep, unconscious guilt and shame around alcohol and what it represents, and his intolerance is really a psychosomatic manifestation of his unresolved anger at his father, an alcoholic. Perhaps he becomes curious about his belief that there is a problem at all. Is the issue that he is only perceiving that he is unwell after alcohol, but isn’t in reality? Or perhaps he takes a completely different approach again and says that his inability to tolerate alcohol is a sign that his body is weak, and the only way forward is to toughen up a bit and drink more regularly. And on and on ... One day, purely by accident, a blood test reveals that Johnny has a very common and harmless genetic mutation that means his liver doesn’t process alcohol quite as efficiently as normal, and this is why alcohol is especially hard on his system. He realizes all at once that he never considered this pretty obvious mental model—the medical one.

Now, the question is, which mental model was the “right” one? In Johnny’s case, the problem is an arbitrary one, and he can solve it without necessarily understanding why it happens (i.e., just stop drinking alcohol). In this case, it doesn’t really matter which mental model he uses. But if the substance was not alcohol but instead some common food item that he was deathly allergic to, then it definitely would matter which model he used, since it would heavily inform how he chose to “solve” the problem! Get into the habit of stepping back, looking at your mental models as tools, and asking yourself, “Is this way of thinking actually helping me understand the problem? Is this really useful?" Once you’ve done that, you can begin to ask, “If this doesn’t help, then what kind of mental model would?

How do I have to think if I wish to understand what’s in front of me?" It’s only then that things start to get interesting ... Science is a Form of Imagination Perhaps you have had the opportunity to watch two very intelligent people talk. If one is trying to explain something to the other, you might see that they say things like, “Imagine this saltshaker is a three-dimensional object, right? But this napkin we’ll call a field, and we’ll put the saltshaker here. Now, if this fork represents gravity ... ” When describing things that are unknown, it is inevitable that you need to rely on the language, images, and ideas of the known. In other words, you use metaphor, symbolism, and analogy.

And you don’t know the rules of the game, but you’re allowed to look at the board from time to time, in a little corner, perhaps. And from these observations, you try to figure out what the rules are of the game, what [are] the rules of the pieces moving. You might discover after a bit, for example, that when there’s only one bishop around on the board, that the bishop maintains its color. Later on you might discover the law for the bishop is that it moves on a diagonal, which would explain the law that you understood before, that it maintains its color. And that would be analogous to us discovering one law and later finding a deeper understanding of it. Ah, then things can happen—all of a sudden some strange phenomenon occurs in some corner, so you begin to investigate that, to look for it. It’s castling—something you didn’t expect.

We’re always, by the way, in fundamental physics, always trying to investigate those things in which we don’t understand the conclusions. The thing that doesn’t fit is the thing that’s most interesting—the part that doesn't go according to what you'd expect. Also we can have revolutions in physics. After you've noticed that the bishops maintain their color and that they go along on the diagonals and so on, for such a long time, and everybody knows that that's true; then you suddenly discover one day in some chess game that the bishop doesn't maintain its color, it changes its color. Only later do you discover the new possibility that the bishop is captured and that a pawn went all the way down to the queen's end to produce a new bishop. That could happen, but you didn't know it." With that simple analogy, he was able to elucidate the entire history of scientific evolution, of the way that science progresses over time, and the deeper underlying worldview that informs scientists’ approach to studying the world.

He took something that people knew (chess) to describe something they didn’t (science’s overarching orientation toward nature) and connected them so that people could transfer their understanding from one to the other. In other words, so they could learn. Feynman was noted as a brilliant communicator (when many physicists are known to be precisely the opposite) for a few reasons. 1. He was enthusiastic and genuinely had a passion for his work. This inspired and excited others. His wonder and awe at the world was infectious and made others see the magic in even the most ordinary phenomena. Watch any Feynman interview or lecture and you never get the sense that he is bored of anything.

Compared to the sense of tedium that many science educators bring to their material, it’s no wonder this gave him such an edge! 2. He was able to use analogy, and use it wisely. Feynman was not an intellectual snob, and he met people where they were—if someone knew nothing of what he was talking about, he didn’t assume they never could understand; rather, he figured out what “language” they spoke and addressed them accordingly. This ability to use whatever tool will work is precisely what makes him a good scientist. Theoretical physicists work in strange realms; their world is mathematical, abstract, and probabilistic. They understand the value of converting these sometimes bizarre findings into more intuitive and concrete metaphors to better convey them to, well, the people who live in the real world! 3. He was able to make science human—he used stories, narratives, and emotional appeals to make people care about what he was saying.

Science is not some dead, neutral thing. Rather, curiosity and the thirst to understand is a deeply intimate human experience that digs right into the most poignant questions of life. To acknowledge this human element—rather than drily dismiss it—makes science something that belongs to everyone. 4. Finally, as we’ve already seen, Feynman was good at not letting ego get in the way. The technical term may be “vibration,” but if he can make the point more quickly by saying “jiggling,” then there isn’t a reason not to. Let go of stiff, inaccessible language—it may be “right” and make you feel like an expert, but it’s useless if it communicates little other than your superiority. Language is there for the using!

It’s allowed to be colorful and eccentric—especially if that eccentricity breaks apart stale old conceptions and invites real, genuine insight. How to Make Your Own Analogies Analogies can help any time you want to communicate what you know, teach someone something they don’t understand, or (perhaps best of all) teach something to yourself so that you understand it better. The good news is that analogy-making comes naturally to human beings. That said, there are some steps to follow to be more deliberate in your creation of high-quality analogies. Step 1: Figure out the property or characteristic you are most trying to convey about a particular domain Let’s say you are Feynman and want to explain to a group of laypeople how the human eye experiences and processes light waves. It’s an extremely complex concept. But he starts by homing in on the property he most wants to communicate—the fact that waves are experienced in a particular way when you are immersed in them, and that you can infer characteristics about the entire waves’ motion by experiencing it in just one point.

Step 2: Find something concrete and ordinary to map this characteristic onto You are probably already so used to thinking of sound or light waves as water waves—because scientists have been pointing to this analogy for centuries. So you draw on this association and say that light waves behave similarly to waves in a pool of water. Now, how do you communicate the above characteristic? Feynman once said that you could imagine a little fly floating on the water. The fly would bob up and down as the waves passed over it. This bobbing up and down would be its experience of the wave, but if it was a smart fly, it could gather all this data and begin to make a theory about what the waves were and where they were coming from, even if, being so small, it couldn’t gain a bigger picture. Feynman went on to explain that this is precisely what the eye does when light waves enter It: the eyes detect the stimulus (analogous to bobbing up and down) and the brain pieces the data together to create a mental map of where they are coming from.

In this way the brain constructs an internal model of the visible world around it. It’s the same as the fly coming to understand the waves that are moving it up and down. Step 3: Adjust or discard your analogy if it isn’t working! Your analogy’s success will depend on the quality of the relationship between the two things you are comparing. If you are trying to describe light waves hitting the eye using an analogy of an octopus in a supermarket buying cantaloupes, your listeners will exhaust all their brainpower just trying to guess how the two things fit together. Be aware, as well, that an analogy is limited—it is highlighting just a single aspect of relationship or similarity. In this example, the wavelike characteristic of both water waves and light waves is relevant—other properties are not.

In this example, a fly has very little in common with a human eye, other than the fact of it intelligently processing data. If your listener or student is hung up on the idea of a fly, then forget the fly and choose something else! Other interesting science metaphors and analogies you might find helpful—look closely and see if you can identify the single characteristic that is being highlighted, and the exact relationship between the things being compared. •The atom is like a tiny solar system (naturally, this analogy is useless if the person hearing it doesn’t know what a solar system is ... ). •The spread of an infectious disease is like the spread of a wildfire. Only a small spark is needed, but if enough “fuel” is encountered, the spread could be exponential. •Receiving a fecal transplant is a little like sowing wildflower seeds into bare soil.

•Taking a probiotic supplement to boost the gut microbiome is like pouring a bucket of water into the ocean to boost sea levels. •A proper science lesson is like an enzyme—it instigates changes in the student’s mind akin to a chemical reaction by using a device (such as an analogy) to speed up and trigger that reaction. But the understanding was always something the student was capable of, and the raw material was always there! The teacher/device, just like a catalyst, remains unchanged. •Metabolism is like an engine. •Determining redshift by spectroscopy is a bit like identifying a fingerprint on a balloon that has been stretched (note, this analogy is so good that you probably understand the concept being explained even though you don’t know what redshift or spectroscopy is!). How to Teach Like Feynman Perhaps you read the title of this section and thought, “Well, this bit probably doesn’t apply to me since I’m not a teacher."

But even if you aren’t, hopefully by the end of this section you will see that teaching is a life skill that all of us need to master, whether we are formally called teachers or not. Knowing how to teach means you know how to communicate. And we can only communicate to the limits of our own understanding. Therefore, teaching can help other people shine light on their own blind spots and remove them, but it can also be the one thing that lets us see our own blind spots and overcome them in the same way. The Protégé Effect Theprotégé effectis simply when teaching (or even pretending to teach or preparing to teach) helps us understand our material in a deeper and more insightful way. Teachers and lecturers have for years recommended that their students study and prepare for exams by pretending to teach a fellow student. Not only does this drill and reinforce the material they need to know, it also quickly highlights gaps in understanding or areas where the illusion of understanding is replacing genuine understanding.

How do you take advantage of the protégé effect in your own life and learning? Tip 1: You don’t have to literally teach someone. It’s enough to simply imagine that you are teaching them, or need to prepare some materials in order to teach them. If you have a willing participant, by all means try to explain what you’re learning to them—it will certainly be helpful. But you can achieve the same effect by delivering a “lecture” to nobody in particular (or, if you like, teach a row of stuffed toys or draw a face on your notebook and talk to that—it’s silly, but it will help). The main thing is that you approach and digest the material as though you would be required to share it with someone else later. It may even help to imagine that this person is less knowledgeable than yourself, so you are forced to come up with your own simplified paraphrasing, novel analogies, and answers to anticipated questions.

Tip 2: Imagine a universal student. What can you imagine would be most tricky for people to understand? How might they phrase their questions? Imagine yourself answering this question, and then imagine that the student isn’t actually satisfied with this. Try to imagine the nature of their concerns and confusions and how you might address them. Remember that we can often fool ourselves into thinking that we have knowledge about something, or that our model actually has some explanatory power. But by imagining a challenging student, we poke at our own potentially circular arguments or lazy definitions.

The simplest way to do this is to conjure in your mind’s eye a student who won’t stop asking “But why?" In time, you might find that you internalize this “universal student” and have mini-dialogues with yourself as you’re working methodically through a problem. “But why do I keep coming up with the wrong answer? Well, let’s take a closer look ... When did the calculation start to go wrong? I’m not sure—maybe this line here ... What am I really saying in this line? Let me see if I can find any assumptions I’ve made that might be wrong ... ” In this way, you become your own teacher, because you have completely internalized the powerful and generative teacher/student dynamic as your own. Tip 3: Invite and engage with feedback.

This teacher/student dialogue is so useful because it is a living, breathing conversation. It’s a dialectic, and in the to-and-fro, the student (and often the teacher, too) moves from ignorance to understanding. In a way, it’s not different from Socratic dialogue, which frames the discovery of knowledge as a joint conversation that arrives at sound conclusions step by step. One underrated way of building this sense of dialogue is through feedback. Sadly, conventional educational approaches focus heavily on achievement and tend to use feedback as a way to score and rank a student, rather than to merely elucidate a path forward on their learning. This means that most of us are fairly resistant to receiving feedback because we see it as criticism, an attack, or a humiliation. This is a pity because feedback is extremely useful.

A good teacher will regularly respond to a student by telling them the effect their efforts are having, things they may be overlooking, or hints for how to master what they are struggling with. If a scientist hopes to truly understand the mechanism of the scientific method (having their hypothesis rejected or not), they must first be comfortable with the trial-and-error nature of learning (i.e., sometimes you will make an error). Do not be afraid of feedback. Identify the smartest, most competent people you know and be courageous enough to ask for their input. Don’t try to always be the smartest person in the room—you will learn best if everyone around you knows more than you do! Remember the point is not to be ranking and rating yourself or making comparisons—instead you are seeking information about the quality of your approach, the usefulness of your question, the soundness of your method. How to Explain Anything to Anyone Some smart people leave others completely baffled.

When they talk, it’s as though they’re speaking in another language. You have no doubt they’re very clever, but what on earth do they mean? Feynman wasn’t one of those people. For him, being able to teach others wasn’t some nice side-effect of his own understanding—he saw his ability to clearly articulate what he knew to be instrumental in him knowing it. Many people assume that if something is very smart and important, only a few people will be able to understand it. Feynman turned this on its head—anyone can understand anything if it’s pitched at the right level, and if you can’t explain it so they understand, it’s not because the material is too complex—it’s because you don’t have a thorough enough grasp on it. Let’s take a look at the IRADE method, which is a multi-pronged approach used in physics education.

It was inspired in part by Feynman’s words: "The best way to teach is to be very chaotic, in the sense that you use every possible way of doing it." Imagine the concept you’re trying to teach is revealed to the student in separate snapshots—the greater the number of snapshots, and from the greatest number of different angles, the more fully they can see the concept. I is for Introduce Use narrative and story to begin. Give some context and explain why you’re learning the concept and how your particular learning fits into the bigger scheme. How might it connect to what you already know, and how does it help you achieve what you’re trying to achieve? If you’re a good teacher, you can use story, humor, and emotion to pique interest and make the concept seem relevant. Feynman himself was famous for getting even science-phobes to admit, “That does sound pretty fascinating, now that you mention it."

You can also begin with the dictionary definition of the terms you’ll be using to get acquainted with the topic and find your footing. R is for Relate Next, show what the concept is by giving real-world examples—give at least three, if you can. If you start abstract and end abstract, it’s difficult for anyone to see how it connects to the “real world” in any way. Especially when it comes to physics, but also with other more abstract topics, you need to stay grounded in why the concept matters at all. A is for Apply There is (hopefully!) some practical application to what you’re learning about. The method/idea/theory is a tool—what things does it help you achieve out there in the world? If it’s a mathematical formula, show how this tool can be used to literally work your way through some problems.

You might like to connect the examples you gave to classical problems and then show how the new knowledge solves those problems. Again, you are embedding this new knowledge in a pragmatic way. Accept questions and get the students to try their hand, too. Work through a range of practical applications, from easy to difficult, so that the students see the principle in action, rather than just hear you talk about it. D is for Demonstrate This is connected to the earlier point. Visual demonstrations can be really powerful. Students always remember the cool hands-on experiments in class over the dry and dull lessons from textbooks.

As with writing, show, don’t tell. You could re-create a mini experiment or set up a demonstration that shows the principle at work. A picture paints a thousand words—a real-life demonstration paints a thousand pictures! E is for Examine You want to test that what you’ve transmitted has been received. It doesn’t have to be a formal exam or even a pop quiz. Just make sure that you are giving the student the opportunity to reflect on what they’ve learned. Keeping the protégé effect in mind, ask them to teach back to you what you’ve just taught them.

Then listen carefully—the places they struggle with or get confused are areas you need to revisit. The questions students ask you can also reveal a lot about how much they’ve absorbed. Get them to paraphrase, make summaries, or debate the concept to show they understand the underlying themes. Or, more classically, give them a problem of the type you’ve covered in the examples and demonstrations, and see if they now know what to do. To make sure they really understand, however, don’t just rely on them regurgitating the right words. One great way to test comprehension is to show them a problem solved incorrectly or a wrong answer. Get them to tell you why it’s wrong.