I understand the difference. What I don’t understand is the odd conjunction.
so concerned with getting the right arithmetic answers that they weren’t thinking about arithmetic.
I parse this as saying: because they were so (strongly) concerned with getting the right answers, they weren’t thinking. If they had been less concerned with getting the right answers, they would have been thinking more.
What is the idea here? That if they didn’t spend so much time, effort, or worry on memorizing, then they would have had more time to think?
I think the worry is that they are only concerned about getting the answer that gets them the good grade, rather than understanding why the answer they get is the right answer.
So you end up learning the symbols “5x5=25,” but you don’t know what that means. You may not even have an idea that corresponds to multiplication. You just know that when you see “5x5=” you write “25.” If I ask you what multiplication is, you can’t tell me: you don’t actually know. You are disconnected from what the process you are learning is supposed to be tracking, because all you have learned is to put in symbols where you see other symbols.
But surely in math, of all subjects, it’s easily possible to construct problems that cannot be solved without thinking and understanding, that do not reduce to mere memory and recognition of a known question “5x5”. Then students who want the “right” answer will be forced to understand.
This isn’t absolute, of course. When learning elementary multiplication, pretty much all you can ask about is multiplying, and there are only a few dozen pairs in the 10x10 multiplication table, and students generally just remember them, they don’t calculate. But by the time you’re up to arbitrary size multiplication or long division, you need to apply an algorithm; that’s a step of understanding, because you can see that the algorithm also produces the results you memorized earlier. And so on.
When students are at the 5x5 level, I don’t think there is an answer to “why is 25 the right answer?” that they could understand—it just is the right answer, a brute fact about life, just like the sky is blue and sun comes up every day. But that doesn’t continue forever.
In my personal experience schools go way too far in the other direction, and keep asking for rote memorization when it’s already possible to ask for understanding.
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.
In elementary school, we had to memorize the multiplication tables (1-10 times 1-10).
We were then quizzed on this.
Being able to answer “5x5 = 25” meant we had memorized the tables
The teacher then asked what “5 x 12” was, which checked to make sure we actually understood multiplication.
Hopefully that clarifies the difference :)
I understand the difference. What I don’t understand is the odd conjunction.
I parse this as saying: because they were so (strongly) concerned with getting the right answers, they weren’t thinking. If they had been less concerned with getting the right answers, they would have been thinking more.
What is the idea here? That if they didn’t spend so much time, effort, or worry on memorizing, then they would have had more time to think?
I think the worry is that they are only concerned about getting the answer that gets them the good grade, rather than understanding why the answer they get is the right answer.
So you end up learning the symbols “5x5=25,” but you don’t know what that means. You may not even have an idea that corresponds to multiplication. You just know that when you see “5x5=” you write “25.” If I ask you what multiplication is, you can’t tell me: you don’t actually know. You are disconnected from what the process you are learning is supposed to be tracking, because all you have learned is to put in symbols where you see other symbols.
But surely in math, of all subjects, it’s easily possible to construct problems that cannot be solved without thinking and understanding, that do not reduce to mere memory and recognition of a known question “5x5”. Then students who want the “right” answer will be forced to understand.
This isn’t absolute, of course. When learning elementary multiplication, pretty much all you can ask about is multiplying, and there are only a few dozen pairs in the 10x10 multiplication table, and students generally just remember them, they don’t calculate. But by the time you’re up to arbitrary size multiplication or long division, you need to apply an algorithm; that’s a step of understanding, because you can see that the algorithm also produces the results you memorized earlier. And so on.
When students are at the 5x5 level, I don’t think there is an answer to “why is 25 the right answer?” that they could understand—it just is the right answer, a brute fact about life, just like the sky is blue and sun comes up every day. But that doesn’t continue forever.
In my personal experience schools go way too far in the other direction, and keep asking for rote memorization when it’s already possible to ask for understanding.
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.