I have to make this one quick because I have a cold and am trying to pack up my life and move uptown. But I wanted to finish my little “Nuggets” series with a thought inspired by Catherine Twomey-Fosnot and Maarten Dolk’s awesome Young Mathematicians at Work books. Twomey-Fosnot runs Math in the City, a math education think tank and professional development center run out of City College. She is writing a new book about algebra with the research mathematician Bill Jacob. I’m excited.
Anyway, people often talk like it’s a choice between developing students’ understanding of concepts and developing their technical ability. My experience is that everybody agrees that the two goals support each other, but there are major differences when push comes to shove in terms of where people believe the emphasis belongs. Maybe you all saw everything clearly from the jump, but I feel I spent a lot of years locked in this dichotomous framework. (Partly because of some experience with curricula like IMP that do a very good job with one of the goals and not the other.) What Young Mathematicians at Work did for me was to abolish the framework.
Nugget: You can develop conceptual understanding and technical ability (for example, computational ability) with the exact same lesson. The secret is to embed the technical instruction in the design of the problems you assign.
It’s necessary to take great care in designing the problems so that they support the development of skills over time. According to Fosnot (and I’ll take her word for it), very few American curricula have given adequate care to the sequence of problems and how it supports this development. My own consciousness was definitely raised by reading Young Mathematicians at Work about the extent to which (for example) the choice of the numbers matters.
For example, consider the following pair of questions:
At Sweet Virgo Desserts, a small chocolate cake costs $7.00. An apple dumpling costs $3.50.
1) How many chocolate cakes can you get for $49?
2) How many apple dumplings can you get for $49?
In 2007, a few months after reading Young Mathematicians at Work, I gave this pair of problems to a classroom of very-weak-skilled 6th graders, who would have balked at #2 (“you want me to divide by a decimal without a calculator?!”) if it had come first. They answered it easily and without any help after being asked and answering #1 first.
The two problems are formally identical. The only difference is the numbers. The important thing is not that #2 is harder; it’s that the way the numbers are chosen makes #1 a hint for #2. It’s also a hint with an applicability far beyond this problem: if n is hard to divide by, would 2n be easier? Pretty soon, the same class was using the technique to solve straight division problems accurately in their heads. (I’ve unfortunately lost the followup worksheet so I can’t tell you what problems; but they were things like 60 / 7.5 and 15 / 1.25.) This is a piece of computational technique; and teaching it this way supported the development of the students’ conceptual understanding of division at the same time that their proficiency with certain division computations was improving. The goals don’t have to be addressed separately.
Maybe you all think this is obvious. But I’m still constantly hearing folks (most recently, a college professor, a former high school principal, and the parent of a mathematically precocious 7-year-old) say things like “but at some point, they just have to memorize those times tables.” Meaning, “all this talk about understanding is really wonderful but you have to admit that there are some things you just have to bang into your head.” I used to be plagued by doubts of this form. Now I’m not. Yeah, you have to learn the times tables, but there’s never a reason to bang something into your head. Can’t remember 6×7? Great, do you know 6×6? How are they related? You go thru that a few times and not only will you remember 6×7 but you’ll be building the groundwork so that later it’ll seem intuitive that 6(x+1) = 6x+6.