Diane Ravitch posted a comment from a reader on her blog yesterday, which drew this comparison:

"But both CCSS and standardized testing are trying to make teachers into KAPOs, a Nazi concentration camp prisoner who was given privileges if they would supervise work gangs."

Diane later noted that there was an "outpouring of rage on Twitter" against this post. She went on to say

"Several people said I should not have allowed it on the blog or words to that effect. I find this argument to be a form of political correctness that is used to censor opinion. If anyone wants to use an analogy to make a point, that is their choice."

Diane has continued to defend the posting on Twitter today, Dec. 27. I think this defense is off target in three ways.

First, I would not call this a case of political correctness. Diane has not said that the analogy is not offensive, but she has suggested that objecting to it is tantamount to political correctness. This will always be a judgment call, but I am on the other side of the fence on this one. There are certain events that are so important, so sensitive to the people in whose culture it is enmeshed that I find it unfeeling and insensitive to draw on that event for ones own purposes. That is especially true when the comparison minimizes the trauma and suffering associated with the event. Test-takers are not comparable to Holocaust victims, nor are students asked to perform public service comparable to slaves. This is a far cry from the hypersensitivity documented in The Language Police in which, for example, an elderly person could never be depicted as doing something stereotypically associated with the elderly.

Second, I disagree with Diane's characterization of people's objections. If you agree that some speech is ill-considered and offensive, telling people that is not censoring their opinion. It's just not the same thing. It's telling them you think their analogy is ill-considered and objectionable, and you are asking them to rethink. You're not forbidding them from saying it, obviously.

Third, more specific to Diane, if she had asked the author to change the analogy or had refused to post the piece because of the analogy, I would not call that censorship. The author does not have a guaranteed right to post what she likes in Diane's blog, a right that Diane would have been infringing. Diane was a offering a platform for this author's voice, and obviously she offers that platform to voices she thinks are worth amplifying. This situation is not comparable to that documented in The Language Police, in which enormous power was concentrated in the hands of few publishers. If an author wanted to publish a textbook they had to toe the line drawn by the publishers or give up on publishing the book. That power relationship does not exist in this case. This is the internet, for crying out loud.

Diane, I respectfully ask that you rethink your position on this matter. I don't think it was a good call and I don't think your defense of it holds up.

EDIT: 1:28 p.m EST. I referred The Language Police as Left Back.

Pop quiz: What’s the earliest age that children think abstractly?

• 2 years
• 4 years
• 7 years
• 9 years

In truth, it’s a bad question because the answer depends on the type of abstraction. If the subtext of the question is “what’s the earliest age at which children show understanding of an abstraction?” the best choice from those offered above is “2 years.” And very likely earlier. Here’s one example.

Caren Walker and Alison Gopnik (2013) examined toddlers ability to understand a higher order relation, namely, causality triggered by the concept “same.”

The experimental paradigm worked like this. The toddler was shown a white box and told “some things make my toy play music and some things do not make my toy play music.” The child then observed three pairs of blocks that made the box play music, as shown below. On the fourth trial, the experimenter put one block on the box and asked the child to select another that would make the toy play music. There were three choices: a block that looked the same as the one already on the toy, a block that had previously been part of a pair that made the toy play music, and a completely novel block.

Toddlers (21 to 24 months old) selected the identical block most often (61% of the time).

Further experiments showed that children as young as 18 months got the task right, and showed that children this age can use the concept “different” as well as they understand “same. “

What’s interesting about this finding? It would be easy to believe that children so young would fix attention on features of an individual block, rather than relations among blocks (i.e., red blocks make it work). Importantly, the child is not just appreciating sameness—he or she is using that property by understanding its causal role. And the child ignores other properties (e.g., shape or color) that are more salient. Furthermore, children learn this property readily, after exposure to just three trials. This finding may speak to the importance to our species of understanding causality.

I want to use this experiment to illustrate a broader point.

A dominant theme within cognitive developmental psychology over the last thirty years has been that children look more clever in proportion to the cleverness of experimenters. That is, as experimenters develop more subtle ways to evaluate children, it becomes clear that children understand more at a younger age than we appreciated. They were capable of learning it all the time. The problem lay in how we were looking.

The same applies to the concept explored by Walker & Gopnik. Dedre Gentner (Christie & Gentner, 2010) has reported data showing that even preschoolers seem to have trouble with reasoning tasks that call for higher-order relations—they seem to need scaffolding in which the important relation is labelled. Walker and Gopnik point out that Gentner used a task requiring verbal labeling, whereas their task merely called for an action. Seemingly small differences in what look like conceptually similar tasks can make a big difference in whether the child seems to understand.

This is one (but not the only) reason that I think it’s important to be cautious in making claims about what children are and are not ready for. The extent to which they appear ready to understand an abstraction depends partly on how we measure knowledge.

It is the normal state of developmental affairs that a child’s initial understanding of a concept looks fragile, fragmented, and uncertain. The child shows understanding on one task but is stumped by a conceptually similar task with (seemingly) trivial differences in format. He seems to understand one day, but not the next (e.g., Flynn & Sielger, 2007).

In fact, I’d suggest that complete mastery of concept across materials, types of query, and times is a good indication that the concept was introduced at a developmentally inappropriate time. We waited too long--the child probably already knew the concept.

Reference

Christie, S., & Gentner, D. (2010). Where hypotheses come from: Learning new relations by structural alignment. Journal of Cognition and Development, 11, 356–373.

Flynn, E., & Siegler, R. (2007). Measuring change: Current trends and future directions in microgenetic research. Infant and Child Development, 16, 135-149.

Walker, C. M. & Gopnik, A. (2013). Toddlers infer higher-order relational principles in causal learning. Psychological Science, published online doi: 10.1177/0956797613502983

A New York Times editorial on December 6 called for improved math instruction, calling the current system “broken.” Although I agree we could be doing a better job of teaching math, the suggestions in the editorial showed a striking naiveté about what it will take to improve.

The editors of the Times offered four suggestions:

1. A more flexible curriculum (especially better integration of engineering)
2. Very early exposure to numbers
3. Better teacher preparation
4. Experience in the real world (perhaps via collaborations with industry)

The editorial ignores the fact that 1 and 4 will be meaningless without 2 and 3. And it grossly underestimates the difficulty of implementing 2 and 3.

The editors’ idea regarding how to implement “early exposure to numbers” is to ensure better access to preschool. But that won’t do it because “exposure” won’t do it. Math is not learned like a language. Children can learn vocabulary and more complex syntax by mere exposure. They can’t learn math that way.

Still, the editorial is right to argue that very early learning is important. I’d argue it’s the key to math reform.

That National Math Panel concluded that three things need to be in place for students to be proficient in math: (1) you need to have memorized a small number of math facts to automaticity; (2) you need to know standard algorithms that apply to standard problems and; (3) you need a conceptual understanding of number, and of what algorithms do.

American kids are okay (not great) on math facts and okay (not great) on algorithms. On conceptual understanding, they are terrible. A student can get by for a time by memorizing algorithms and when they are to be applied, but eventually, not understanding what you’re doing catches up with you, meaning it affects your success in math.

But lacking conceptual understanding makes people hate math long before that. What could be more boring than executing algorithms you don’t understand? And if you don't understand what you're doing, why wouldn't you conclude "I'm not really good at math?"

This conceptual understanding ought to start in preschool with ideas like cardinality and equality. “Very early exposure to numbers” is not going to do it. That doesn’t mean taking what we had been doing in first grade and asking kids in pre-K to do it. That means putting activities into pre-K (e.g., games and puzzles that emphasize the use of space) that will provide a foundation for conceptual understanding so that first-graders will be in a better position to understand what they are doing. (Though first grade math will also have to change for that happen.)

The other recommendation that the editors of the Times get wrong is “Better teacher preparation.” They focus on high school, noting that many teachers of physical sciences did not major in these subjects. That may be a problem, but it ignores a much more serious teacher problem.

Most American teachers—like most American adults, including me--don’t have a deep conceptual understanding of math. They are a product of the system we are trying to change. You cannot teach what you don’t know.

So what’s my recommendation for American mathematics?

We need to pay much closer attention to preschool and to early elementary grades. That will entail developing methods of helping children understand the conceptual side of math—methods we now lack. It will also entail professional development to train teachers in the conceptual side of math.

The size of this undertaking is massive. But it directly addresses the issue encapsulated in the editorial’s title: “Who Says Math Has to Be Boring?” The editorial focuses on the idea that it’s boring to do things without knowing why you’re doing them. So the proffered solution is real-world application. But I think a worse problem is not understanding how math works, being asked to execute algorithms with no understanding of what is really happening. That’s a heavier lift but will ultimately be more rewarding.