Learning how to Solve Problems from Video Lessons11 Mar 2016
If there is one thing modern students are blessed with, it is resources. The Internet has amazing free resources for learning almost anything, including math and sciences. Personally, as I grew up under a communist regime, the access to free, high quality information is a gift I cannot get tired of. For today’s students though, it is a bit different.
What the modern student expects the Internet to do, is to take the effort out of education. To a large extent, this is a positive and correct expectation, since the information can be found more rapidly, and can be accessed from anywhere, including home. So we can get someone to explain the solution to a problem to us whenever we have the time, wherever we are. But… can one learn how to solve problems by watching explanations of solutions? This is the question this article is trying to answer.
Mathematics is not like cooking: we watch a chef preparing a dish and we imitate the steps. This kind of learning does not apply to solving problems, since we would never have to prepare exactly the same dish as the chef. Each problem is different and requires its own special recipe. For the problem solver, the process of learning cannot be limited to sheer imitation, but must train the ability to handle different situations than the ones explained by the instructor in a video lesson. What is the video lesson good for? Definitely, it is not intended just for watching! The student who watches a video and says: ‘I got it. I now know how it’s done,’ is most often wrong. To see that the student did not ‘get it,’ just turn the video off, give them the same problem to solve, and a piece of paper. In most cases, the student will not be able to reproduce a complete solution, even if the requirement is to solve exactly the same problem as was shown in the video lesson.
Why does this happen? Why does the student believe that he “got it,” if it is not actually true? It is because, generally, the student has been able to follow the steps that the instructor has explained, given that the explanation has been very clear and very vivid. The steps, however, are not the most important part of the solution. Rather, the most important part of the solution is the idea: why does the solution consist of these steps and not others?
The instructor generally shows “how” it is done, not “why” it is done in that specific manner. The “why” is much more subtle and involves a high level of intimacy with the theoretical facts that underlie the creation of the problem. The instructor cannot possibly digress from the task of “showing the steps” into a dissertation about the theorems applied and about the execution decisions that were necessary when creating the solution. He merely shows the finished product, not the nitty-gritty of getting all the parts designed and assembled. The student looks at the finished product and decides that, yes, he understands all the setup, all the equation solving, all the number crunching, it is all correct and it makes sense. But understanding these steps is not the same as being able to solve the problem!
Does watching a video teach the student how to produce the idea behind the solution, how to make the decisions necessary to execute the idea and finalize the solution? Absolutely not. The student who tries to learn how to solve problems by watching videos of solutions is only getting to be a better referee without actually making progress as a player of the game. Most of the time, the student does not even realize how little they have actually learned from watching someone explain the solution to a problem.
Often times, students come to lesson and tell us that they do not need to review a certain problem anymore, since they have already reviewed it with an instructor and they “got it.” Upon this statement, our reply is “So how does it go? What would the general idea of the solution be?” In the vast majority of cases, students are not able to re-solve the problem, or even answer the most general questions about it. That is because the student, even though he has understood each step, does not have the “big picture” that calls for those steps and makes them necessary. The student does not have the needed degree of intimacy with the theory that underlies the problem. Therefore, they have learned very little from the video lesson.
It may seem from the above that it is a worthy goal to be able to memorize the solution that was presented. It sounds as if, by asking the student to retell and redo it, we are placing some value on the regurgitation of the solution as it was presented. We absolutely do not. The only reason we ask the student to show how the solution goes, is to verify that the video presentation has enlightened the student with respect to the theory to the extent that the student is now capable of re-solving the same problem from scratch. We never place value on memorizing solutions, but we do place a lot of value on deriving or re-deriving a solution from scratch. If the student can explain what the solution is based on, and re-design the execution steps as a result of decisions that he can explain, then there has been some benefit from having watched the presentation - the student had good theoretical background and the presentation has simply helped make one or two connections that the student was missing.
How, then, should we use these tremendous resources that the Internet offers? Of course, the answer is, in the same way in which we used the old fashioned resources. The Internet has only sped up things, made them more accessible, but has not fundamentally changed the spectrum of knowledge and skills that a human must develop in order to become a good problem solver. The student must understand the theory first, in order to be able to connect the idea of the solution to the facts that engendered it. Also, the student must have all the skills necessary to design and execute the solution: drawing, making a mathematical model based on given data, solving equations, systems, and inequalitites clearly, on paper, so that they can be read again and corrections, if needed, can be applied. To learn from a problem, the student must first identify which theoretical facts are to be learned before the solution idea emerges as a clear consequence. Then, he must study this theory - luckily, the Internet comes to the rescue again as, more often than not, simply typing a question in google will produce a couple of useful links to explanations. Lastly, he must go back to the problem and make sure he understands how the theory he just learned is the key to the puzzle. A video presentation might constitute 10% of this work, but not more. The rest must be done by the students themselves.
Most students do not have these skills. Even if someone gives them the idea for a solution, they have a difficult time executing it. Concerning the idea itself, students have difficulty generating it since they have a very poor and superficial knowledge of theory - this can be the case even for students who have an A in math and may be taking an accelerated math program. Both students and parents believe, mistakenly, that problem solving can be learned by solving problems alone and that learning more theory is a roundabout and wasteful way to progress. This minimalist approach is, even when somewhat successful, vastly less efficient than the more involved process of solving problems according to a theoretical basis.
So, when watching a video lesson:
- ask yourself what the idea of the solution was
- ask yourself which theoretical facts suggest this idea and are logically connected to it
- learn or re-learn this theory really well
- attempt to re-do the problem yourself, with pen on paper, from scratch
- analyze the execution steps (operations) and optimize them to the extent possible
- if unsuccessful, go back to step one and do everything again, only more in earnest
- if successful, ask yourself which other questions could be asked if certain data in the problem were different
- make up a problem similar to the one studied, and solve it
These are steps that must accompany the study of a problem. Of these, watching a video presentation is only a small part of the process of studying. Nobody can learn how to solve problems by watching videos just as nobody can learn how to play the violin by watching famous players. In mathematics, the part that goes unexplained is often the most important: how the idea for the solution occured. Watching people getting ideas is not a way of learning how to get ideas!