That depends on the level of explanation the teacher requires and the level of the material. I’d say that at least until you get into calculus, you can work off of memorizing answers. I’d even go so far as to say that most students do, and succeed to greater or lesser degrees, based on my tutoring experiences. I am not sure to what degree you can “force” understanding: you can provide answers that require understanding, but it helps to guide that process.
I went to a lot of schools, so I can contrast here.
I had more than one teacher that taught me multiplication. One taught it as “memorize multiplication tables 1x1 through 9x9. Then you use these tables, ones place by ones place, ones place by tens place, etc.” One problem with this approach is that while it does act as an algorithm and does get you the right answer, you have no idea what you are trying to accomplish. If you screw up part of the process, there’s no way to check your answer: to a student in that state, multiplication just is “look up the table, apply the answer, add one zero to the end for every place higher than one that the number occupied.”
Whereas I had another teacher, who explained it in terms of groups: you are trying to figure out how many total objects you would have if you had this many groups of that, or that many groups of this. 25 is the right answer because if you have 5 groups of 5 things, you generally have 25 things in total. This is a relatively simple way of trying to explain the concept in terms of what you are trying to track, rather than just rote memorization. Fortunately, I had this teacher earlier.
The point being that you can usually teach things either way: actually, I think some combination of both is helpful. Teach the rote memorization but explain why it is true in terms of some understanding. Some memorization is useful: I don’t want to actually visualize groups of objects when I do 41x38. But knowing that is what I am trying to track (at least at the basic level of mathematical understanding I acquired in the 2nd grade) is useful.
Teach the rote memorization but explain why it is true in terms of some understanding.
Yes, but this only really works if, when the student is presented with an example they didn’t memorize, they can still solve it using their understanding. And to make sure they do understand, after they’ve practiced on the simple cases they can memorize, you routinely set problems that require understanding.
You can’t start with understanding because when solving a few simple cases (like 5x5), memorization really is effective, and students may choose to memorize even if you don’t explicitly tell them to.
That depends on the level of explanation the teacher requires and the level of the material. I’d say that at least until you get into calculus, you can work off of memorizing answers. I’d even go so far as to say that most students do, and succeed to greater or lesser degrees, based on my tutoring experiences. I am not sure to what degree you can “force” understanding: you can provide answers that require understanding, but it helps to guide that process.
I went to a lot of schools, so I can contrast here.
I had more than one teacher that taught me multiplication. One taught it as “memorize multiplication tables 1x1 through 9x9. Then you use these tables, ones place by ones place, ones place by tens place, etc.” One problem with this approach is that while it does act as an algorithm and does get you the right answer, you have no idea what you are trying to accomplish. If you screw up part of the process, there’s no way to check your answer: to a student in that state, multiplication just is “look up the table, apply the answer, add one zero to the end for every place higher than one that the number occupied.”
Whereas I had another teacher, who explained it in terms of groups: you are trying to figure out how many total objects you would have if you had this many groups of that, or that many groups of this. 25 is the right answer because if you have 5 groups of 5 things, you generally have 25 things in total. This is a relatively simple way of trying to explain the concept in terms of what you are trying to track, rather than just rote memorization. Fortunately, I had this teacher earlier.
The point being that you can usually teach things either way: actually, I think some combination of both is helpful. Teach the rote memorization but explain why it is true in terms of some understanding. Some memorization is useful: I don’t want to actually visualize groups of objects when I do 41x38. But knowing that is what I am trying to track (at least at the basic level of mathematical understanding I acquired in the 2nd grade) is useful.
Yes, but this only really works if, when the student is presented with an example they didn’t memorize, they can still solve it using their understanding. And to make sure they do understand, after they’ve practiced on the simple cases they can memorize, you routinely set problems that require understanding.
You can’t start with understanding because when solving a few simple cases (like 5x5), memorization really is effective, and students may choose to memorize even if you don’t explicitly tell them to.