I like Carol Burris and appreciate what she is doing to rally educators against what she sees as the errors being made by New York State officials and the implementers of Common Core Standards in general. As I have often said, it is up to every educator to be pro-active about change, whether it be in their room or nationally.

But a recent piece by her (and John Murphy), published in the Washington Post, does her cause no service at all. They argue that Common Core test questions are developmentally inappropriate. Alas, like too many other educators, she seems not to understand what Piaget meant about “concrete operational” thinking. She incorrectly complains that Piaget showed that pre-adolescents cannot engage in abstract and deductive thinking; therefore, the new Common Core tests have developmentally inappropriate questions.

First, here is the PARCC sample test item that she complains about:

Part B provides students with the following information:

The San Francisco Giant’s Stadium has 41,915 seats, the Washington Nationals’ stadium has 41,888 seats and the San Diego Padres’ stadium has 42,445 seats.

It then asks the following question:

Compare these statements from two students.

Jeff said, “I get the same number when I round all three numbers of seats in these stadiums.”

Sara said, “When I round them, I get the same number for two of the stadiums but a different number for the other stadium.”

Can Jeff and Sara both be correct? Explain how you know.

The authors report on an answer from a student they know:

This response by one of the 6th graders, an 11 year old, provides insight into how this age group thinks: “No, I know this because they all round to 42,000.”

We know her response is typical for her age group (7-11) because of the work of one of the greatest childhood psychologists of all time, Jean Piaget.  Piaget carefully and systematically studied the cognitive development of children.  Before him, it was assumed that when it came to thinking, kids were not as adept as adults, but their thought processes were essentially the same.

Piaget disagreed.  He discovered that the development of thinking is far more complex. He identified distinct stages of cognitive development that children go through as they mature, including, the ‘Concrete Operational’ stage (ages 7-11).

Students in this stage can engage in some inductive logic, but deductive logic, which is needed to solve problems such as the one described above, is beyond them. [emphasis added]

Alas, Burris and Murphy seem to have been taken in by the phrase “concrete operational” and thus seem to believe it to mean that young children cannot think abstractly and logically. On the contrary, “concrete operational” thinking is abstract and deductive, and refers to reversible mental operations like arithmetic and subtraction. (“Formal operational thinking” by contrast involves extended conditional logic e.g. algebraic thinking)

For example, knowing the answer to “How many apples were eaten if the group started with 8 and ended with 2?” is a concrete operational deductive multi-step exercise, typical of math work in grades 3 and 4. In fact, arithmetic, chess, Sudoku, and inferential reading would all be impossible for 11-year-olds if the authors were correct about students at this age not being able to think abstractly and deductively.

The authors then imply that because the test question and others like it are “developmentally inappropriate” children are being needlessly hurt by tests of the Common Core:

Many of the other [sample PARCC] tasks involve less abstraction, but are highly difficult. They are interesting questions that make adults stop and think. But as Piaget told us, children are not “mini-adults.” If a child is not developmentally ready, these problems will likely lead to frustration, discouragement and negative emotional reactions—which is exactly what parents are reporting.

But this is an odd argument. “If a student has to stop and think, then they will likely get discouraged and quit.” Really? Surely this under-estimates the ability of children to accept an intellectual challenge, while letting teachers off the hook for not routinely challenging kids to think about what they learn. By their argument, any ELA question that asks students to draw inferences about a character’s mood or motives is equally inappropriate.

More generally, I find the phrase “developmentally appropriate” a very squishy phrase,  used in too many cases to make learning less intellectually rich and challenging than it might be.

Let’s consider an alternative view, grounded in a class I just witnessed. Check out this picture:

2013-11-05 10.16.35

The teacher is soliciting from her students the characteristics of a good discussion, prior to asking them to engage in self-sustaining conversation. (A very abstract question). Students gave a bunch of good answers, as shown, and then were assigned the job of safeguarding these rules via a slip of paper given to them that had a symbol for one of the rules (e.g. lightbulb for ‘share ideas’, ear for ‘listen respectfully’, etc.), symbols that the students proposed. The students then proceeded to consider the question “What’s similar and what’s different in this author’s 4 books that we have just finished?” (also abstract) for 20 minutes, without any teacher facilitation except a few comments at “half-time.”

Some of the student answers included: they all had a moral or message. Most involved a change to the main character; finding help from a friend was a common theme. Notably, there were some disagreements that had to be ironed out by the group. Most impressively, students in charge of the ‘compass rose’ symbol – navigation – proposed a few times that they thought the conversation was off topic – a highly abstract inference.


2nd graders. Doing an author study of Lionni.

We do our kids no service to shield them from intellectual difficulty. We too often wrongly think that most kids cannot think really hard. (The teacher began the discussion by saying “This will be hard, but I think you can do it!”) Indeed, as one of the girls said at the end of this lively and completely student-self-sustained and monitored discussion, “Can we please do this again? This was fun!” leading to approving murmurs from most of her class-mates.

PS: Here are two released tests for 2nd grade that prove my point: kids are expected to do the kinds of problems I said above were appropriate for the grade tested, 4th grade:

California math test

Georgia math test

However, a propos released tests I am very disheartened to learn that Florida no longer releases its tests and item analyses, as it did for 10 years. (The links no longer work to find those tests, either). I will have more to say on test ‘security’ and fair accountability in my next post.

Meanwhile, here is an older test to show how releasing tests and item analysis can be so helpful – and should be done on moral as well as pedagogical grounds as I have long argued:



PPS: And here is a way cool video of high expectations of little kids:


PPPS: A few readers asked me for some references to Piaget to support my claim, above. A good source is one of Piaget’s early books, Judgment and Reasoning in the Child, which is far more readable than his later works which are laden with Boolean logic and fairly abstract analyses based on a lifetime of work. A second source is his paper on math education, one of the more important (but little-known) articles he wrote.

Here is what Piaget says about deductive ability in pre-formal children:

In formal thought the child reasons about pure possibility.  For to reason formally is to take one’s premises as simply given, without inquiring whether they are well-founded or not. Belief in the conclusion will be motivated solely by the deduction…. Between the years of 7-8 and 11-12 there is certainly awareness of implications when reasoning rests upon beliefs and not assumptions, in other words, when it is found on actual observation. But such deduction is still realistic, which means that the child cannot reason from premises without believing in them. [in The Essential Piaget, pp. 114-115.]

In fact, between the age of 7 and 11-12 years an important spontaneous development of deductive operations with their characteristics of conservation, reversibility, etc. can be observed. This allows the elaboration of elementary logic of classes and relations, the operational construction of the whole number series by the synthesis of the notions of inclusion and order, the construction of the notion of measurement, etc… Although there is considerable progress in the child’s logical thinking it is nonetheless fairly limited. At this level the child  cannot as yet reason on pure hypotheses, expressed verbally, and in order to arrive at a coherent deduction, he needs to apply his reasoning to manipulatable objects in the real world or in his imagination. [from “Comments on A Mathematical Education” in The Essential Piaget pp. 729-730.]

PPPPS: Here are released items from the MCAS in Massachusetts (with % correct) that show conclusively that, while challenging, such items are not inappropriate even though they demand deductive reasoning of a few steps of logic (and two are similar to the disputed PARCC item in the Post article):

Rita's shapes gr 4 MCAS MCAS gr 4 alg equation d Gr5 3 rounding MCAS MCAS rounding gr 4