What is “developmentally appropriate practice”? For many teachers, I think the definition is that school activities should be matched to children’s abilities—they should be neither too difficult nor too easy, given the child’s current state of development. The idea is that children’s thinking goes through stages, and each stage is characterized by a particular way of understanding the world. So if teachers know and understand that sequence, they can plan their lessons in accordance with how their students think.
In this column I will argue that this notion of developmentally appropriate practice is not a good guide for instruction. In order for it to be applicable in the classroom, two assumptions would have to be true. One is that a child’s cognitive development occurs in discrete stages; that is, children’s thinking is relatively stable, but then undergoes a seismic shift, whereupon it stabilizes again until the next large-scale change. The second assumption that would have to be true is that the effects of the child’s current state of cognitive development are pervasive—that is, that the develop mental state affects all tasks consistently.
Data from the last 20 years show that neither assumption is true. Development looks more continuous than stage-like, and the way children perform cognitive tasks is quite variable. A child will not only perform different tasks in different ways, he may do the same task in two different ways on successive days! [...]
The problem is not simply that Piaget didn’t get it quite right. The problem is that cognitive development does not seem amenable to a simple descriptive set of principles that teachers can use to guide their instruction. Far from proceeding in discrete stages with pervasive effects, cognitive development appears to be quite variable—depending on the child, the task, even the day (since children may solve a problem correctly one day and incorrectly the next). [...]
These experiments tell us that there is not a rapid shift whereby children acquire the ability to understand that other people have their own perspectives on the world. The age at which children show comprehension of this concept depends on the details of what they are asked to understand and how they are asked to show that they understand it. This pattern of task dependence holds for other hallmarks of Piagetian stages as well. The implication is that stages, if they exist, are not pervasive (i.e., they do not broadly affect children’s cognition). The particulars of the task matter. [...]
Until about 40 years ago, most thought of children’s minds as a set of machinery. As children developed, parts of the machine changed, or parts were discarded and replaced by new parts. The machinery didn’t work well during these transitions, but the changes happened quickly. Today, researchers more often think that there are several sets of machinery. Children have multiple cognitive processes and modes of thought that coexist, and any one might be recruited to solve a problem. Those sets of cognitive machinery undergo change as children develop, but in addition, the probability of using one set of machinery or the other also changes as children develop.
This conclusion doesn’t mean that there is no consistency across children in their thought, or in the way that it changes with development. But the consistency is only really evident at a broader scale of measurement. A geographic metaphor is helpful in understanding this distinction (Siegler, DeLoache, and Eisenberg 2003). If one begins a trip in Virginia and drives west, there are very real differences in terrain that can be usefully described. The East Coast is wet, green, and moderately hilly. The Midwest is less wet and flatter. The mountain states are mountainous and green, and the West is mostly flat and desert-like. There is no abrupt transition from one region to another and the characterization is only a rough one—if I tell you that I’m on the East Coast and you say, “Oh, it must be green, wet, and hilly where you are,” you may well be wrong. But the rough characterization is not meaningless. Similarly, all children take the same developmental “trip.” They may travel at different paces and take different paths. But at a broad level of description, there is similarity in the trip that each takes.
Obviously, the description of multiple sets of cognitive machinery rather than a single set complicates the job of the developmental psychologist who seeks to describe how children’s minds work and how they change as children grow. Worse, it negates the possibility that teachers can use developmental psychology in the way we first envisioned. There is a developmental sequence (if not stages) from birth through adolescence, but pinpointing where a particular child is in that sequence and tuning your instruction to that child’s cognitive capabilities is not realistic.
What I summarize from the above is that educators have decided that Piaget's theory is not helpful for deciding 'developmentally appropriate practice'. Perhaps because the transitions from one stage to another are fuzzy and overlapping, or because students of a particular age group are not necessarily in step. Furthermore, understanding of a concept is 'multi-dimensional' and there are many ways to approach it, and many ways for a child to think about it, rather than a unique pathway, so that a student might seem more or less advanced depending on how you ...
When I was a freshman in high school, I was a mediocre math student: I earned a D in second semester geometry and had to repeat the course. By the time I was a senior in high school, I was one of the strongest few math students in my class of ~600 students at an academic magnet high school. I went on to earn a PhD in math. Most people wouldn't have guessed that I could have improved so much, and the shift that occurred was very surreal to me. It’s all the more striking in that the bulk of the shift occurred in a single year. I thought I’d share what strategies facilitated the change.
I became motivated to learn more
I took a course in chemistry my sophomore year, and loved it so much that I thought that I would pursue a career in the physical sciences. I knew that understanding math is essential for a career in the physical sciences, and so I became determined to learn it well. I immersed myself in math: At the start of my junior year I started learning calculus on my own. I didn’t have the “official” prerequisites for calculus, for example, I didn’t know trigonometry. But I didn’t need to learn trigonometry to get started: I just skipped over the parts of calculus books involving trigonometric functions. Because I was behind a semester, I didn’t have the “official” prerequisite for analytic geometry during my junior year, but I gained permission to sit in on a course (not for official academic credit) while taking trigonometry at the same time. I also took a course in honors physics that used a lot of algebra, and gave some hints of the relationship between physics and calculus.
I learned these subjects better simultaneously than I would have had I learned them sequentially. A lot of times students don’t spend enough time learning math per day to imprint the material in their long-term memories. They end up forgetting the techniques that they learn in short order, and have to relearn them repeatedly as a result. Learning them thoroughly the first time around would save them a lot of time later on. Because there was substantial overlap in the algebraic techniques utilized in the different subjects I was studying, my exposure to them per day was higher, so that when I learned them, they stuck in my long-term memory.
I learned from multiple expositions
This is related to the above point, but is worth highlighting on its own: I read textbooks on the subjects that I was studying aside from the assigned textbooks. Often a given textbook won’t explain all of the topics as well as possible, and when one has difficulty understanding a given textbook’s exposition of a topic, one can find a better one if one consults other references.
I learned basic techniques in the context of interesting problems
I distinctly remember hearing about how it was possible to find the graph of a rotated conic section from its defining equation. I found it amazing that it was possible to do this. Similarly, I found some of the applications of calculus to be amazing. This amazement motivated me to learn how to implement the various techniques needed, and they became more memorable when placed in the context of larger problems.
I found a friend who was also learning math in a serious way
It was really helpful to have someone who was both deeply involved and responsive, who I could consult when I got stuck, and with whom I could work through problems. This was helpful both from a motivational point of view (learning with someone else can be more fun than learning in isolation) and also from the point of view of having easier access to knowledge.