There’s been a whole lot of fuss about the word “abstract” appearing in the Common Core K-11 Standards for Mathematical Practice (SMP). For some, that word means “age inappropriate” for the Common Core. After all who knows better than Piaget and he said kids can’t think abstractly until age 11, right?
Well, not exactly. Piaget describes that stage, Formal Operational, as a period of increased ability to think in more logical and abstract ways. The key to understanding Piaget is to notice the word “more.” Throughout the stage descriptions there is a progression, no mile markers. Ylvisaker (2006) says it well when he reflects upon “Green Eggs and Ham” where, after a long fight, a person changes his mind and decides he likes green eggs and ham.
At a more abstract level of understanding, it is about people in general being capable of modifying their thoughts and desires … This more abstract level of understanding can be appreciated by two and three year old children only if the higher level of meaning comes out of a discussion of the book with a more mature adult. At older ages and higher levels of thinking, this same process of more mature thinkers facilitating higher levels of abstraction in less mature thinkers characterizes the process of teaching abstract thinking. For example, this is how great philosophers, like Socrates and Plato, taught their pupils how to think abstractly.
The SMP are consistent in every grade through out the Common Core. They are goals toward which students should strive their entire lives. The SMP require students to “reason abstractly” in the same way religion tells people to “be good.” No one ever completely achieves that, but it is something we can all work toward. This does not mean kindergarteners will be expected to explain Pythagorean Theorem. What the word “abstract” means also varies with age (Ylvisaker, 2006):
Abstraction is a relative concept, related to the age of the child. For a two year old, “the day after tomorrow” is a highly abstract concept. For a college student, the day after tomorrow is relatively concrete, as opposed to highly abstract ideas like Heisenberg’s Indeterminancy Principle. And of course there are many degrees of abstraction between these two extremes. A major component of intellectual development is this process of gradually moving from extremely concrete thinking to increasingly abstract thinking in an ever increasing array of content areas.
My service this morning in a large church nursery brought to mind several common examples of abstract reasoning that I have witnessed at a toddler level over many years of caring for nearly a hundred children. These examples can be related to topics freshmen often struggle to master in Algebra if they have not explored such activities in real life:
- One toddler tries to take another baby’s toy but that is disallowed so he walks over to the toy box and explores until he finds one to match (substitution)
- One toddler sees another baby wants a third baby’s toy. The toddler goes to toy shelves, stares at them for several seconds, and very thoughtfully chooses a similar toy and offers it as a substitute (second order substitution)
- Toddler humor includes making the wrong animal noise while the others laugh (substitution with a non-solution)
- One toddler picks up a large hollow object and wears it as a hat. Moments later, six toddlers have found large hollow objects and are wearing them as hats (generalization, estimation)
- On toddler tries to fit several different shapes into a shape sorter until one finally makes it through (trial and error)
- One toddler notices a baby freaks out when she loses her binky. The toddler later corners the child and takes the binky. (This borders on scientific method.)
Many toddler toys are designed to scaffold abstract thinking. One book in the nursery where I currently serve has holes through the entire book such that the eyes of the animals are missing. Toddlers quickly embrace the fun of substituting their or my eyes by holding the book to our faces. Function mapping is another commonality: Put the ball in the blue hole and the toy sends it out an exit. Put the ball in the red hole, and the toy sends it to a different exit…consistently.
One does not need to read volumes of research to debunk the claim that the Common Core Standards are inappropriate because they require abstract thinking. It only requires a clear understanding of the definition of “analyze” and a morning spent serving in the nursery.
Ylvisaker, Mark, Hibbart, Mary, Feeney, Timothy (2006). What are Concrete and Abstract thinking? Retrieved from http://www.projectlearnet.org/tutorials/concrete_vs_abstract_thinking.html