{"id":18796,"date":"2026-10-28T12:42:05","date_gmt":"2026-10-28T20:42:05","guid":{"rendered":"https:\/\/www.scotthyoung.com\/blog\/?p=18796"},"modified":"2026-10-28T12:42:05","modified_gmt":"2026-10-28T20:42:05","slug":"solving-hard-problems","status":"publish","type":"post","link":"https:\/\/www.scotthyoung.com\/blog\/2026\/10\/28\/solving-hard-problems\/","title":{"rendered":"Should You Study The Solutions Or Solve Problems?"},"content":{"rendered":"
\n

Watch the video<\/a> of this essay.<\/p>\n<\/blockquote>\n

Learning requires effort. But not all effort is useful for learning.<\/p>\n

Consider the following situation. You have some problem sets for a class you\u2019re taking. You have two ways you can study them:<\/p>\n

    \n
  1. You can solve the problems on your own.<\/li>\n
  2. You can look at the solutions given by your teacher.<\/li>\n<\/ol>\n

    Which method would result in more general learning, letting you perform better on new types of problems?<\/p>\n

    \n
    \"\"<\/figure>\n<\/div>\n

    Interestingly, in many cases the answer is actually option two: looking at the solution. And this can be true even if you\u2019re capable of solving the problems on your own.<\/p>\n

    This seems to go against what a lot of us were taught about how to study effectively: the only way to build skill is by solving hard problems. No pain, no gain, right? I sympathize with this perspective because, at least until I came to understand the science, that\u2019s largely how I felt too.<\/p>\n

    In this essay, I want to share with you the research that changed my mind. I want to show you what goes on inside your head as you solve difficult problems, as well as give you the key to know when problem solving works (and when it underperforms).<\/p>\n

    The Puzzle<\/strong><\/h2>\n

    Is it possible to solve a problem without learning how you did it?<\/p>\n

    In 1982 the psychologist John Sweller was conducting an experiment on problem solving with undergraduate students.<\/p>\n

    The problems the students were given were fairly simple: given a starting number and a goal number, the aim is to go from the start to the goal. But there are only two possible moves: either multiply the number you have right now by three, or subtract twenty-nine from it.<\/p>\n

    For instance, suppose you have the number 15 and you\u2019re told to get to 16. You could multiply it by 3, to start, which gets you to 45. And then you subtract 29, which gets you to 16. Done.\u2028\u2028<\/p>\n

    In all the problems there was a trick. Every problem Sweller gave could be solved by alternating between multiplying by three and subtracting twenty-nine some number of times. In other words, you just had to repeatedly apply those two rules and you\u2019d always get the answer eventually.<\/p>\n

    But as Sweller later recounted<\/a>:<\/p>\n

    \n

    \u201cMy undergraduates found these problems relatively easy to solve with very few failures, but there was something strange about their solutions. While all problems had to be solved by this alternation sequence \u2026 very few of the students discovered the rule \u2026 Whatever the problem solvers were doing to solve the problems, learning the alternating solution sequence rule did not play a part.\u201d<\/p>\n<\/blockquote>\n

    How could students solve these problems, but not learn the method<\/em> they were using to solve them?<\/p>\n

    How We Solve Problems<\/strong><\/h2>\n

    Just ten years before Sweller\u2019s experiment, future Nobel laureate Herbert Simon<\/a> and Allen Newell<\/a> published a landmark book, Human Problem Solving<\/a>. In it, the duo studied the mental processes people use to solve hard problems.<\/p>\n

    They found that when people tackle a problem they don\u2019t know how to answer, they engage in a search process through a problem space. This is a little bit like trying to navigate a maze, except instead of going through a physical space with walls and dead-ends, you\u2019re navigating through the possible states of the problem itself.\u2028\u2028<\/p>\n

    In Sweller\u2019s experiment, there were only two moves\u2014either multiply by three or subtract twenty-nine. Of course, in most problems, there are more possible moves. A Rubik\u2019s cube has eighteen different moves, leading to a possible total of forty-three quintillion<\/em> configurations. That\u2019s part of why just twisting the cube around randomly doesn\u2019t solve it\u2014there are too many possibilities. Instead, we need to do something smarter.<\/p>\n

    Simon and Newell found that subjects reduced this complexity by using heuristics, or general-purpose procedures for solving problems. These heuristics are innate. We don\u2019t need to be taught them. We apply them spontaneously, even if we aren\u2019t aware of it.<\/p>\n

    Simon and Newell identified several of these heuristics. But there was a particular one that Sweller thought might be playing a role in his students surprising behavior: means-ends analysis.<\/p>\n

    This heuristic has a fancy name, but it\u2019s really easy to understand. Means-ends analysis starts by looking at a goal, then looking for actions that could let you reach it. If you can\u2019t perform that action directly, you then create a sub-goal to get to the point where you can take that action. If you repeat the process enough times, you\u2019ll get to the solution.<\/p>\n

    Let\u2019s make that clearer with an example. Assume you need to get from New York to Paris. What\u2019s a way you could get to Paris? A plane. How do take a plane? You buy a ticket. How do you buy a ticket? You can buy one from the airline at the airport. How do you get to the airport? Take a taxi. You get the idea.<\/p>\n

    \n
    \"\"<\/figure>\n<\/div>\n

    Means-ends analysis works. It\u2019s a good strategy for solving unfamiliar problems, but it has a drawback. Thinking up all those goals and subgoals takes up a lot of mental bandwidth, what psychologists call your working memory<\/a>. And, if your working memory is overloaded, you may not have much capacity left for noticing useful patterns in what you\u2019re trying to learn.<\/p>\n

    To Sweller, this suggested an experiment. If you could suppress means-ends analysis, would you learn to solve the problems better?<\/p>\n

    Problems without Goals?<\/strong><\/h2>\n

    Sweller tested this in an experiment with students learning to solve trigonometry puzzles. Essentially, he asked students to solve for the missing angle or side length of a shape, much like you probably did in high school.<\/p>\n

    The classic formulation for these puzzles is to designate one angle or side-length and ask the student to find it. To do that, students typically rely on means-ends analysis. Starting from the goal, they work backward to figure out what other values they need in order to calculate the answer. This approach uses a lot of working memory, which might distract from learning the actual principles of geometry.<\/p>\n

    To suppress this, Sweller came up with \u201cgoal-free\u201d problems. These were the same trigonometry puzzles, but, instead of finding a particular angle, students were asked to find as many angles and side lengths as they could.<\/p>\n

    \n
    \"\"<\/figure>\n<\/div>\n

    Sweller and his colleague, Elizabeth Owen, tested the students’ understanding<\/a> of the puzzles by giving them a new type of trigonometry puzzle they had never seen before. The goal-free group got eight out of ten right, while the group who practiced traditional problem solving only managed three.<\/p>\n

    The Worked Example Effect<\/strong><\/h2>\n

    Goal-free problems helped avoid the mentally demanding means-ends analysis, but they were of limited use. The problem spaces of most real-world problems are just way too large for this approach to scale.<\/p>\n

    To deal with this, Sweller opted for a different technique: worked examples. These were written-out solutions to a problem, typically followed by a practice attempt on a very similar problem to ensure you actually studied it.<\/p>\n

    In a follow-up experiment, Sweller and colleagues tried this approach with algebra problems. Half the students were given worked examples. The other half were given the problems to solve, without solutions. \u2028\u2028To ensure that the success rate of the two groups was matched, any student from the problem-solving group who failed to solve the problem after five minutes was just given the solution. So we know that the problem solvers wouldn\u2019t underperform simply because they never got to the answer.<\/p>\n

    When tested on similar problems, both groups performed about equally. But here\u2019s the kicker\u2014when a new type of problem was introduced, three-quarters of the worked-example group solved the problem, while none of the problem-solving group could.<\/p>\n

    Far from resulting in brittle memorization, studying the worked examples seemed to result in better understanding and generalization of the technique than problem solving did.<\/p>\n

    Cognitive Load Theory<\/strong><\/h2>\n

    These experiments, along other experimental effects, led Sweller and his colleagues to formulate cognitive load theory<\/a>. This theory basically states that learning involves at least two different kinds of demands on our limited cognitive capacity.<\/p>\n

    The first, intrinsic load, is the working memory that is required for learning. Without this effort, learning doesn\u2019t happen.<\/p>\n

    In contrast, extrinsic load is the working memory that isn\u2019t strictly required for learning, but may be imposed by the task. We\u2019ve already seen one type of this created by problem solving\u2014means-ends analysis requires you to keep all those goals and subgoals in mind as you work toward the solution.<\/p>\n

    But it\u2019s not the only kind. Extrinsic load also occurs in other cases, like when you have to visually move around materials to make sense of a diagram, or if you have irrelevant information you need to ignore.<\/p>\n

    If the total load exceeds your working memory capacity, some of it will be dropped. And if the part that was dropped was intrinsic to learning, you\u2019ll end up learning less. Optimizing cognitive load\u2014ensuring you\u2019re maximizing the mental effort spent on genuine learning\u2014is the key to learning well.<\/p>\n

    Part Two: The Reversal<\/strong><\/h2>\n

    Okay, I can imagine you saying, that\u2019s a neat theory, but it\u2019s not how you get good at difficult skills. You don\u2019t get good at math, programming, physics or anything just by looking at the teacher\u2019s answer\u2014you have to think hard and do it for yourself.<\/p>\n

    Recently I published a video about the power of retrieval practice<\/a>. Recall strengthens memory more than review, and students learn better when they\u2019re forced to remember with a closed book rather than rereading a text.<\/p>\n

    Am I contradicting myself?<\/p>\n

    Well, no. Because, as it turns out, the power of goal-free problems, worked examples, and even teaching itself starts to reverse as you continue to learn.<\/p>\n

    I wrote above that people automatically use heuristics, like means-ends analysis, when they\u2019re facing hard problems. As it turns out, this is only partly correct.<\/p>\n

    Novices Use Heuristics, Experts Retrieve Knowledge<\/strong><\/h2>\n

    \u2028\u2028Novices who are encountering problems for the first time solve problems this way. As we\u2019ve seen, this is often the only way to solve them.<\/p>\n

    But experts, who have learned how to solve the problem, or one like it before, have a different process. Instead of solving a problem by working backward, they can work forward from their starting point, using the knowledge they already possess to get to the destination.<\/p>\n

    \n
    \"\"<\/figure>\n<\/div>\n

    Take the example from before, of trying to go from New York to Paris. When I asked how you\u2019d solve this problem, a lot of you probably didn\u2019t go through the kind of backwards reasoning I described of first deciding on air travel, then buying a ticket, getting to the airport, hailing a taxi, and so on.<\/p>\n

    Instead, you\u2019ve been on trips before, so you didn\u2019t need to do this. Maybe you opened up your phone and went to your favorite travel website and started looking up flights.<\/p>\n

    This kind of forward reasoning doesn\u2019t have the same heavy, extrinsic load that means-ends analysis creates. By using stored knowledge in your head, what psychologists call schemas, you can automatically recognize what sort of problem you\u2019re dealing with and apply solutions from your starting point to get to your destination.<\/p>\n

    For experts, solving a familiar problem doesn\u2019t require heuristic search. It simply requires retrieving the appropriate schema from memory and using that knowledge to get you to the next step.<\/p>\n

    Retrieval practice strengthens memory more than review does. As a result, under these conditions, we should expect the worked-example effect to flip. As you practice more and more, the benefits of studying the solution will get weaker, and the benefits of retrieving the schema to solve the problem yourself will go up.<\/p>\n

    But there\u2019s more. Even the intrinsic load required to learn changes over time. Through a process known as \u201cchunking<\/a>\u201d our effective working memory goes up as we learn within a subject. Thus, even for totally new problems, experts benefit more from solving problems than studying examples.<\/p>\n

    \n
    \"\"<\/figure>\n<\/div>\n

    The expertise-reversal effect is a well-documented finding that shows that as we continue to learn within a field, the value of worked examples decreases until problem solving becomes the more effective strategy.<\/p>\n

    Practice and problem-solving matter for mastery. But when<\/em> they matter is just as important as how much. Good instruction should focus on clear explanations and worked examples at the start, slowly fading that support to focus on independent problem solving as students progress.<\/p>\n

    Part Three: The Application<\/strong><\/h2>\n

    How can you apply this theory to learn better?<\/p>\n

    The simplest way is simply to realize that we need both good examples and ample practice to learn. Early on, examples are more potent. Later on, practice problems should take up more of our time. By cycling between the two, we can prevent some of the extreme failure cases that come from omitting one type of learning experience entirely.<\/p>\n

    But, some researchers have tried to do better. Cognitive load is a sliding scale, so we should find ways to smoothly transition between studying examples and fully independent problem-solving. Here are three:<\/p>\n

    1. The first strategy is to use self-explanations.<\/strong> In this approach, you start with fully worked examples, but you have a prompt to give an explanation for each of the steps taken. Researchers Alexander Renkl and colleagues<\/a> found that this extra step of eliciting self-explanations resulted in deeper learning than worked examples alone in an experiment.<\/p>\n

    2. The second strategy is fading.<\/strong> With this approach, we smoothly transition from worked examples to problem solving by omitting more and more of the example and requiring the student to fill it in for themselves. Researchers<\/a> found that backwards fading, where the final steps of the problem-solving process are omitted first, works best.<\/p>\n

    3. The third strategy is stepwise retrieval problems.<\/strong> In this case, the solution to the problem is revealed step-by-step. This gives you an opportunity to retrieve knowledge, but doesn\u2019t require you to produce the complete solution. Researchers Gesa van den Broek and colleagues<\/a> found this improved learning, although in their experiment students did take more time.<\/p>\n

    If you\u2019re interested in using this in your own studies, I\u2019ve made a tool that uses AI to apply these techniques automatically. Check it out here.<\/p>\n

    The Debate<\/strong><\/h2>\n

    The basic findings of cognitive load theory\u2014that working memory is a major constraint in learning, that good instruction and worked examples are important, particularly to those just starting out, and that the value of independent problem-solving and practice grows as we gain more expertise\u2014are, by now, beyond question.<\/p>\n

    However, I\u2019d be lying if I said that these findings have always been well-received. The idea that clear guidance, instruction and practice are essential for learning has often run against deeply ingrained beliefs. Often, beliefs that run afoul of what the data actually support.<\/p>\n

    And as I mentioned at the start, this is a topic in which digging in to the original research largely changed my mind.<\/p>\n

    Effective learning is effortful. But not all efforts are equally effective. Learning well requires not only finding your own path, but also following the lead set by good examples.<\/p>\n","protected":false},"excerpt":{"rendered":"

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Learning requires effort. But not all effort is useful for learning. Consider the following situation. You have some problem sets for a class you\u2019re taking. 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