I happened to be in a second-grade classroom last week in which the math specialist was teaching the children to play a game that had them composing and decomposing numbers to make a total of 20. It’s a very common thing to play games in a progressive math program, but I think of all progressive practices, the way we teach math is always least understood.
I have often wondered when we will get to a tipping point at which the way parents were taught math reflects the way most schools now teach math. We are not there yet! Most parents with whom I speak were taught math in the same way that I was at elementary school, which relied heavily on numbers and algorithms.
In math terms, “algorithm” is just another word for a procedure we use to solve equations. As a first grader, I would have a page of problems to complete to practice, say, the subtraction algorithm. If the problem was 25 minus 19, I would have to put the larger number on top of the other one; starting in the right hand column, I would know that I couldn’t subtract 9 from 5, so I had to “borrow” 1 from the left-hand column and then “pay it back.”
Algorithms work: if I asked you to do a similar equation, many of you would imagine the number stacked in your head and do exactly what you would have done on paper. There’s nothing wrong with this, but it’s not necessarily efficient. In fact, it’s much easier when solving 25 minus 19 to count up from 19, adding 1 to get to 20 and then adding another 5 to get to 25, for a final answer of 6. When problems start going into the hundreds and thousands, it gets really hard to keep those numbers visualized in your head while you patiently work on each column.
The other issue with algorithms is that they allow you to solve problems without having any sense of the magnitude of a number. The algorithm “works” for numbers into the billions just as well as it does for numbers in the tens. But it doesn’t teach children to be numerate — in other words, to develop “math sense.” To compare this with reading, it’s a little like teaching children how to decode the words they read without teaching them the comprehension skills to make sense of what they’re reading.
We often talk about 21st-century skills, and the ability to be numerate is one of these. I was part of the last generation that needed to be able to manipulate numbers on paper. I vividly remember my father bringing home a calculator sometime in the mid-1970's and how Space Age it felt. We turned out the lights and marveled at the way the numbers lit up — this was before LED displays — and how we could make words if we entered certain numbers. Once calculators became mainstream, the need to be able to solve problems on paper diminished. (For many years, however, the accusation of “dumbing down” was leveled at the use of calculators in schools. If children used the calculator, so the charge went, they would no longer be using the part of the brain that was reserved for numerical understanding.)
The world changed, too. School curricula were developed during the Industrial Age, when gaining specific skills created a functional workforce. In our current time, these skills no longer help a person be successful — there are too many jobs that are automated and outsourced. Our job as educators is to prepare students for this new world by helping them become persevering problem-solvers who can communicate their ideas and apply them to novel situations.
What happened in math instruction then was that the emphasis shifted from being able to compute on paper to becoming a creative thinker and flexible problem-solver. Manipulating numbers relies on the ability to understand the relationships between numbers. For example, when I was taught the algorithm for long multiplication, I was taught that when you multiplied by the tens number, you added a zero. I had no idea why, but that was the “trick” that was needed for the algorithm to work. It was only when I began teaching that I started to understand why this was so (and that the terms “borrowing” and “paying back” had no mathematical meaning).
In addition — excuse the pun — math instruction now emphasizes that there are many ways to solve problems and that what works best for me might not work best for someone else. For those who are terrified of math, it’s often because of a fear of not getting the right answer or not doing it “the right way.” We now emphasize that wrong answers can be useful information and that there is no one way to get the correct answer. Research shows that mistakes literally help the brain grow. If children aren’t making mistakes, they aren’t learning. This has liberated many children from math-phobia.
In What's Math Got to do With It, Jo Boaler notes, “In my different research studies, I have asked hundreds of children, taught traditionally, to tell me what math is. They will typically say such things as ‘numbers’ or ‘lots of rules.’ Ask a mathematician what math is and they will more typically tell you that it is ‘the study of patterns’ or ‘a set of connected ideas.’” Discovering patterns, clearly communicating ideas, making connections between different material, and applying concepts to novel situations are important skills to help our students be successful.
Finally, to cultivate the ability to be numerate, we begin problems embedded in real-world contexts that children can relate to and understand. When toddlers are learning to count, parents don’t write down digits on a piece of paper and ask the children to repeat the names of the numbers. We start with stories and songs in which children can visualize “Three Blind Mice” or “Ten Green Bottles” (the UK version of “99 Bottles of Beer”). And just like that, when they get to school, we teach them to play games like the one I saw in the second-grade classroom last week so that there is a purpose to the need to manipulate the numbers, as well as a real-world application.
At our math coffees, parents to come to ECFS to hear about all the ways math is taught as a dynamic, integrated, vibrant, and relevant subject in their child’s grade.
They may just learn a new way to subtract 19 from 25.