Analogies Are Like Boxes of Chocolates
I mentioned here that I would come back to analogies, which I talked about here and here.
The reason for my revisiting this idea is to clarify to some extent where I'm coming from when I write something like this (the analogy idea is highlighted):
It is obviously extremely important and useful for teachers and for curricula to try to "connect with where the students are" and to turn expertise "into building blocks for teaching." A sad sort of hilarity ensues, however, when "connection" and "building blocks" consume all thinking about mathematics education—as they have—to the extent that we completely disregard both the need for students to have a deep understanding of mathematics and the need for students to gain formal knowledge in the discipline of mathematics.
One can see this, I think, when one takes a look at just about any topic in elementary mathematics, but let's take a look again at division.
The need for "connection" to students' everyday thinking is so powerful in mathematics education in the United States that division as a topic is introduced and discussed almost exclusively as a process of "separating [dividend] objects into [divisor] equal groups and then counting the number in 1 group." We choose this perspective because students can almost instantly relate to dividing "things" into equal groups, and, more generally, they can relate to doing something to a group of objects in order to create a result.
This perspective is helpful, in my opinion, when explaining the long division algorithm, but relying on it solely may present serious problems.
First, it is much more difficult, under this perspective, to visualize the relationship between multiplication and division. We typically present multiplication as "starting with an amount and replicating that amount a certain number of times"—all activity, occurring from left to right. Then we talk about division in a separate perspective, still occurring from left to right. Each operation has a different "starting" point and a different "ending" point. Accessing one puts a student on a certain path that has nothing to do with the other.
Second, when students encounter division with fractions (say, 8 ÷ 1/2), they naturally become confused when they think, "Separate 8 objects into 1/2 group and then count the number in 1 group." What the hell does that mean? And when they see the solution, it is close to unbelievable—how can one divide a group of objects into equal groups, count the number in 1 group, and come up with an answer greater than the original number? We train students to think about division as a process whereby a group is shattered into so many equal groups. We shouldn't be surprised when they have difficulties with fraction division.
But what if we introduced and modeled the meaning of multiplication and division as follows, assuming an understanding that multiplication and division deal with equal groups:
6 ÷ 3: "If there are 6 in 3 groups, how many are there in 1 group?" This is the same as 6 × 1/3: "If there are 6 in 1 group, how many are there in 1/3 group?"
2 ÷ 1/3: "If there are 2 in 1/3 group, how many are there in 1 group?" This is the same as 2 × 3: "If there are 2 in 1 group, how many are there in 3 groups?"
8 ÷ 1/2: "If there are 8 in 1/2 group, how many are there in 1 group?" This is the same as 8 × 2: "If there are 8 in 1 group, how many are there in 2 groups?"
16 ÷ 2: "If there are 16 in 2 groups, how many are there in 1 group?" This is the same as 16 × 1/2: "If there are 16 in 1 group, how many are there in 1/2 group?"
I for one would be willing to sacrifice a little more time in the early stages presenting this understanding of multiplication and division if I knew that it would make more difficult topics easier for students down the road.
The reason for my revisiting this idea is to clarify to some extent where I'm coming from when I write something like this (the analogy idea is highlighted):
Hiding all the "mathy" stuff might very well be described as the central principle of modern elementary mathematics education, and those who promote mathless math education—consciously or unconsciously—have actually managed to convince people that they are the ones best concerned for the future of our children's brains. It's insane from any angle.For example, after reading the above and other examples like it, it may seem like I would disagree with the proudly un-pseudonymous Michael Paul Goldenberg when he writes, approvingly (emphasis is mine):
Operating inside this "frame" inevitably leads one to hold secretly fast to the assumption that students already know everything—that it is impossible to give them new concepts, new ways of thinking, and new knowledge. Instead we must "connect" everything we teach them with something we presume they already know, or with a way of thinking we presume they have.
In their book MAKING SENSE: Teaching and Learning Mathematics With Understanding, Hiebert, et al., propose three features for appropriate mathematical tasks: 1) the tasks make the subject problematic for students; 2) the tasks connect with where the students are; and 3) the tasks engage students in thinking about important mathematics.Or with Jenny D., who responds here to some criticism of a fifth grade math lesson presented by her dean (again, emphasis is mine):
As adults who are math fluent, we might not be able to know what it's like NOT to know math, and what it would take to get someone to learn math.But, in fact, I don't disagree with any of these thoughts.
This is why expert mathematicians might not be the best teachers of fifth graders. It's why great athletes are not always great coaches. Because taking that expertise and turning it into building blocks for teaching is sometimes difficult for some who [have] achieved mastery and beyond.
It is obviously extremely important and useful for teachers and for curricula to try to "connect with where the students are" and to turn expertise "into building blocks for teaching." A sad sort of hilarity ensues, however, when "connection" and "building blocks" consume all thinking about mathematics education—as they have—to the extent that we completely disregard both the need for students to have a deep understanding of mathematics and the need for students to gain formal knowledge in the discipline of mathematics.
One can see this, I think, when one takes a look at just about any topic in elementary mathematics, but let's take a look again at division.
The need for "connection" to students' everyday thinking is so powerful in mathematics education in the United States that division as a topic is introduced and discussed almost exclusively as a process of "separating [dividend] objects into [divisor] equal groups and then counting the number in 1 group." We choose this perspective because students can almost instantly relate to dividing "things" into equal groups, and, more generally, they can relate to doing something to a group of objects in order to create a result.
This perspective is helpful, in my opinion, when explaining the long division algorithm, but relying on it solely may present serious problems.
First, it is much more difficult, under this perspective, to visualize the relationship between multiplication and division. We typically present multiplication as "starting with an amount and replicating that amount a certain number of times"—all activity, occurring from left to right. Then we talk about division in a separate perspective, still occurring from left to right. Each operation has a different "starting" point and a different "ending" point. Accessing one puts a student on a certain path that has nothing to do with the other.
Second, when students encounter division with fractions (say, 8 ÷ 1/2), they naturally become confused when they think, "Separate 8 objects into 1/2 group and then count the number in 1 group." What the hell does that mean? And when they see the solution, it is close to unbelievable—how can one divide a group of objects into equal groups, count the number in 1 group, and come up with an answer greater than the original number? We train students to think about division as a process whereby a group is shattered into so many equal groups. We shouldn't be surprised when they have difficulties with fraction division.
But what if we introduced and modeled the meaning of multiplication and division as follows, assuming an understanding that multiplication and division deal with equal groups:
Multiplication: If there are [first factor] in 1 group, how many are there in [second factor] groups? Division: If there are [dividend] in [divisor] groups, how many are there in 1 group?
6 ÷ 3: "If there are 6 in 3 groups, how many are there in 1 group?" This is the same as 6 × 1/3: "If there are 6 in 1 group, how many are there in 1/3 group?"
2 ÷ 1/3: "If there are 2 in 1/3 group, how many are there in 1 group?" This is the same as 2 × 3: "If there are 2 in 1 group, how many are there in 3 groups?"
8 ÷ 1/2: "If there are 8 in 1/2 group, how many are there in 1 group?" This is the same as 8 × 2: "If there are 8 in 1 group, how many are there in 2 groups?"
16 ÷ 2: "If there are 16 in 2 groups, how many are there in 1 group?" This is the same as 16 × 1/2: "If there are 16 in 1 group, how many are there in 1/2 group?"
I for one would be willing to sacrifice a little more time in the early stages presenting this understanding of multiplication and division if I knew that it would make more difficult topics easier for students down the road.
Labels: education, mathematics, textbooks