What seven and five are...

Over on one of his blogs I didn't know he had, David Hogg spoke this morning about the problem of calculating without thinking.  David speculated that one of the missing pieces in our children's education, specifically math education, was that of common sense --  that we subvert a goal of thinking in favor of rote memorization; of testing oriented teaching; of decontextualized teaching of math. His inquiry was distinctly ungendered, but this piece of our children's education is unavoidable for me.

I have had a growing unease with my 8y old boy's math education. As I watch him work through third grade math I have begun to notice a distinct disinterest. My speculation has been focused on the extraordinarily slow pacing of his lessons, but I am unable to extract a clear reason from him. One piece of his math work that I have found interesting has been a focus on personal, written explanation. He is forced to take apart some of the harder problems, encode them into sentences, and, after doing so, reuse this written framework to create new problems (with new solutions). Perhaps this synchronization of writing and math is a good step towards thinking during calculation.

On the other hand, I still agree that our kids would benefit from developing a mental narrative about education that is more closely akin to David's main point:  that in their classrooms they would be rewarded by openly questioning illogical or simply unnatural contextualizations.  How to instill such an approach is hopefully easier than speculating on how to make the classroom welcoming to it.

So I will leave this idea of calculation with thinking with my own speculation about my children: I also have two girls, 4 and 2, and I wonder about their impending introduction to math. I often think that I will be beyond overjoyed by a parent teacher conference that someday suggests that one of my girls' most beloved (audio) books has inspired "calculation with thinking." That one of the seeds planted in their heads is a ringing desire to openly question their education's context in the voice of a little redheaded girl. That they will hear Pippi's open conflict with her brief math education to call bullshit on the learning vectors they are given.

I'll take my chances reproducing a version of the relevant, translated (copyrighted) text here:

"Pippi, can you tell me what seven and five are?"

Pippi, astonished and dismayed, looked at her and said, "Well, if you don't know that yourself, you needn't think I'm going to tell you."

All the children stared in horror at Pippi, and the teacher explained that one couldn't answer that way in school.

"I beg your pardon," said Pippi contritely. "I didn't know that. I won't do it again."

"No, let us hope not," said the teacher. "And now I will tell you that seven and five are twelve."

"See that!" said Pippi. You knew it yourself. Why are you asking then?"

The teacher decided to act as if nothing unusual were happening and went on with her examination.

"Well now, Pippi, how much do you think eight and four are?"

"Oh, about sixty-seven," hazarded Pippi.

"Of course not," said the teacher. "Eight and four are twelve."

"Well now, really, my dear little woman," said Pippi, "that is carrying things too far. You just said that seven and five are twelve. There should be some rhyme and reason to things even in school. Furthermore, if you are so childishly interested in that foolishness, why don't you sit down in a corner by yourself and do arithmetic and leave us alone so we can play tag?"

The teacher decided there was no point in trying to teach Pippi any more arithmetic. She began to ask the other children the arithmetic questions.

"Can Tommy answer this one?" she asked. "If Lisa has seven apples and Axel has nine apples, how many apples do they have together?"

"Yes, you tell, Tommy," Pippi interrupted, "and tell me too, if Lisa gets a stomach-ache and Axel gets more stomach-ache, whose fault is it and where did they get hold of the apples in the first place?"

The teacher tried to pretend that she hadn't heard and turned to Annika. "Now, Annika, here's an example for you: Gustav was with his schoolmates on a picnic. He had a quarter when he started out and seven cents when he got home. How much did he spend?"

"Yes, indeed," said Pippi, "and I also want to know why he was so extravagant, and if it was pop he bought, and if he washed his ears properly before he left home."

The teacher decided to give up arithmetic altogether.