Ask a Scientist: When Are Children Ready to Learn Abstract Math?
Zane Wubbena is a doctoral candidate in education at Texas State University. He studies cognition as it relates to early mathematics. As a former special education teacher, Wubbena wanted to know how brain development affected students' ability to comprehend the math curriculum for their grade level. The conversation below has been edited for length and clarity.
What led you to be interested in studying early-childhood math?
I was troubled by this problem that I found in almost every grade.
For example, in basic addition and subtraction problems, [teachers would ] maybe hand out a sheet and have children work through these addition and subtraction problems without really having a background into each child and whether or not they have developed the concrete skills to be able to do more abstract reasoning.
Do they have one-to-one correspondence where they understand that every time I say if I touch the number 1, this means 1? [Do they know] when I hold two marbles in my hand that means there are two marbles? From 1 to 9, are they able to understand that 1 comes before 2, and 3 comes after 2?
That led me to my research question for the study I conducted: How can we ensure that the expectations we place on children are appropriate for each child at that grade level?
Please explain how your experiment worked.
I wanted to look at 1st grade children. That's a very pivotal year when kids are really expected to become fluent in mathematics, specifically addition and subtraction. Fluency is really indicative of skill mastery, being able to master something or to suggest that I'm ready to move on to more complex mathematical operations.
So when you talk about fluency in addition, would that mean they could say, "2 plus 2 is four" and "2 plus 3 is 5" and "2 plus 5 is 7"? And they can do it quickly?
Right. And abstractly, too. So for example, if I was to write "1 plus 1" on a piece of paper then they could answer that with 2. They wouldn't need to have manipulatives, blocks, or something like that.
Or counting on their fingers?
I went to three elementary schools [in central Texas] and I tested every single 1st grader. So the first thing I did was [assess their] cognitive development in math.
Basically you have two cups of water and each of those cups of water has the same amount of water in them, equal amounts. Then you'll ask a child, "Do these have the same or a different amount of water?" And they'll usually respond that they're the same. Almost always they'll respond that they're the same. Then you take one of those glasses of water and you pour it into another glass that is wider or thinner, so it looks like it's a different amount of water but it's actually the same. And I did this right in front of the kid's eyes, each kid individually, and I compared those two glasses of water again.
And so one [for which] the diameter is not as large, a kid who was "nonconserving" would say the glass that's skinnier has more water, whereas a kid who would be "conserving" would say that both of the glasses still had the same amount of water.
And conserving means they understand that the amount of water didn't change?
Right, right. That even though the glass changed they were still able to reject perceptual inconsistencies, meaning differences in the amounts of water that they perceived, to think more logically. [They realized] that actually no water has been added or taken away; that it's just a different glass.
Those two responses suggest children are in two different levels of development. [Jean] Piaget, [an early childhood researcher], outlines four different levels of development: preoperational, which is from being born up to maybe 4 or 5 years old, and then preconservation, and then concrete operation, and then formal operation.
So you're saying a kid in 1st grade could be in the second or third stage and still be fine? Like you wouldn't have to be worried about the development if they were still in the second stage because it switches over at some point during those years? Is that right?
That's correct. And I think that's a caveat of having grade levels, how they're structured. And it's in addition to trying to prepare children not necessarily for just learning in and of itself, but trying to prepare them for say passing a test in another grade. Something in the future, always pushing them to know something else that they don't need to know. It can push people to not account for those differences in cognitive development.
Given that 1st grade teachers today have the standards to work toward, would you suggest that that they ignore those standards? And if so how would a teacher readjust her classroom in order to teach to the different developmental levels?
I wouldn't necessarily say "ignore the standards," because they serve as kind of a norm, especially at the earliest levels of education, that really serve as a foundation for all the years of education that come after that. But children can be in these same grade levels and in different levels of cognitive development.
In the three schools [he tested in May 2015], it was basically split almost 50/50 between conserving and nonconserving children, meaning that they're going to the next grade expected to be fluent in these mathematical skills when developmentally [half] are just not ready yet.
And does that break—that 50/50 break—line up with socioeconomic standing or race or anything else?
No, it did not. And that's in line with previous research.
After I measured each of their levels of cognitive development, I tested their mathematical fluency.
What I found was that cognitive development had an effect on their mathematical fluency in both addition and subtraction. And then age had an additional effect above and beyond the level of conservation or level of development. So if we're just looking at their cognitive development, kids who were 6 years old who were conserving scored higher than kids who were 6-year-olds who were nonconserving. And kids who were 7 years old who were conserving scored higher than kids who were 7 years old and nonconserving. And kids who were 7 years old who were nonconserving scored more fluently than kids who were 6 years old and nonconserving.
Is there any reason to think that a 7-year-old who is nonconserving at the end of 1st grade is in need of special education services?
Just because a kid is not yet fluent in these skills doesn't mean that there's necessarily something wrong. Perhaps even in the classroom their developmental needs are not being attended to appropriately.
With the nonconserving kids, my suggestion is [to ensure] that they've mastered all of the necessary skills to be able to become fluent when they're ready.
So a 1st grader who's nonconserving should still be using manipulatives rather than being asked to show fluency on a written test?
Right. And also [the teacher should] still push. Have the expectations, but know what's appropriate at what stage.
It sounds like the basic idea is that there's no point in doing something that you're not actually cognitively able to do, and the practice should be aligned with the developmental stage, right?
What implications would your research have for a parent who wants to help their children be prepared for the math standards that do exist?
Ensure they understand the basics in terms of what we take for granted. You don't have to have any type of materials at all. Say you're walking in a park; you can count rocks. One, two, and then you could open your hand and say "Two. How many do we have? We have two." And then you'd add another rock. Or putting rocks in order, 1, 2, 3, 4, and then having them go back and count. Ensure that they can master that and do it by themselves forward and backward, meaning they can count to ten and then they can count from ten back to one.