Making practice more perfect

The ground was rocky where my two-year-old granddaughter and I walked in the woods, looking for blueberries, and there were so many treacherous roots and branches in the way as we searched. Yet my granddaughter persevered, falling, getting up again, practicing what was already easy for me, navigating the forest floor. As I watched her I thought about engineering professor Barbara Oakley’s article on girls and math practice in the August 7 issue of the New York Times.

I have been a math educator in NYC public schools for 25 years, both as a classroom teacher of elementary school children and as a math coach for elementary school teachers. Girls and mathematics, children and mathematics, is something I think about a great deal.

A huge majority of elementary school teachers are women. All of them had mathematics in elementary school, middle school and high school, and many of them find mathematics teaching, if not terrifying, then the least rewarding part of their teaching day.

Ms. Oakley posits that one reason for this is that these former elementary school students found easy success in reading and writing, and—compared with themselves—a harder time with mathematics. Or, rather, they soared above their male counterparts in reading and writing but were only even with them in mathematics. She says that the basis for success in mathematics is practice, which boys get more of than girls, because girls choose things at which they are superior, rather than equal, on which to expend their energy. Therefore, Ms. Oakley says, parents and educators need to make sure their girls and girl students practice mathematics.

To this point I am intrigued and wondering about what significance this has for action in the classroom. I have seen myself that in Kindergarten classes there are often more girls than boys in, for example, the play-acting corner, while more boys than girls are building with blocks. I have considered that we would be better off as a society if boys had more experiences building relationships and girls more experience building structures. But what does it mean to practice mathematics?

The author’s answer is to liken practice to chords and scales, and to disparage the emphasis on conceptual understanding. As someone who learned to play the piano without a conceptual understanding of the relationship of notes to each other, I don’t think separating conceptual understanding from developing neural pathways, both physical and mental, is helpful.

Take the counting sequence. There is no doubt that it is important to memorize the sequence, so that the mental pathway is so smooth and well lit that no effort is needed to find or navigate it. But consider the different task confronting a speaker of English (or Spanish or German) and a speaker of Chinese (or Japanese, or Quechua). Children speaking English routinely consider that they must learn the order of 100 numbers names before they start over with “one hundred one, one hundred two,” etc. But Chinese children learn the names of 10 numbers, and because their language mirrors the base 10 system that the whole world uses to organize large and small quantities, they start repeating those names with 11. In English the equivalent Chinese name for 11 would be ten one (or one ten one). 26 is named two ten six, and so on. You want to add 24 and 35? No need to rename twenty and thirty as fifty – instead two ten and three ten is called five ten, and 4 + 5 is another 9, for five ten 9.

The quantity denoted by the word twelve is always and forever one more than the quantity denoted by eleven and internalizing the eternal nature of this relationship is key to navigating mathematics, but demystifying the word used to denote the quantity 1 ten 2 (twelve? What’s up with that?) and connecting it to its place in the base 10 structure we all use, goes a long way toward making it possible to connect the sequence memorization with the organizational structure that names all numbers in terms of groups of ten and not yet tens.

Mathematics is all about relationships. Experience (practice) with those relationships is necessary. But it isn’t separate from the urge to gain insight into the structures that make mathematics so beautiful.

The Common Core standards in mathematics have gone a long way toward helping teachers see the relationships between (formerly discrete) topics for themselves so they can help students make the connections that will ensure the most robust development of neural pathways.

Mathematics asks the questions, What is the pattern? Will this always be true? Can you convince us of your thinking? If we ask these questions about observed patterns in elementary school girls and boys can dig in and strive to prove or disprove, and finally generalize, with over and over again experiences. For example:

In first grade asking “Is it always true that we can find ten inside of every addition problem? (8 + 7 is the same as 10 + 5, which we already know is 15. What about 9+4? 17 + 6)?”

In 4th grade, Can every fractional amount be expressed as the sum of unit fractions? (1/7 + 1/7 + 1/7 is three 1/7s (3/7) just like 1 + 1 + 1 = 3 ones. What about 13/15?)

In 3rd grade, Is it ever possible to have an odd product when one of the multipliers (factors) is even? Why or why not?

Imagine conjectures like these debated in mathematics class, arguments grounded in the experience of many examples. Imagine the resultant internalization of relationships, each relationship opening doors to others, like cresting a rise in the woods, which then reveals further vistas to explore, practice and embrace.

It’s the discovery of your own power to understand these connections and then make more of them that makes mathematics like wanting to learn to walk unaided in the beauty of the woods, even though you aren’t very good at it yet.

 

 

 

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