The Common Core State Standards have become a focus of controversy and I feel I must lend a voice to the discussion.
My most extensive experience is with a fraction of the Common Core, literally. I have examined the mathematics standards labeled Numbers and Operations—Fractions from the initial statement for grade 3, ‘Develop understanding of fractions as numbers’, to the final Standard of grade 5, ‘Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions’. These Standards for fractions set a goal for our schools, our teachers and our children that many are trying to resist, but it is a simple goal we should all strive to achieve: understand fractions as numbers.
Fractions are numbers. What else might fractions possibly be? Sadly, an amazing variety of ideas exist. The first problem I encounter when talking with teachers about fractions is this diversity of understanding.
Most frequently I hear the term ‘part-part-whole’, a widespread method of introducing children to fractions. The concept is ingrained and may seem reasonable, but it muddles future understanding because it uses portions of sets as models. For example: “I have four pencils. If I give you half of them, how many will I give you?” If I give you half of my set of 4 pencils, I give you 2. True, 2 is 1/2 of a set of 4, but this approach often leads children to confuse the number 2 with the number 1/2.
There are other problems with introducing fractions using sets and a part/whole explanation: first, 2 out of 4 is actually a ratio expressed in fractional terms; and more, as they progress, children can’t easily translate part-part-whole into an equation.
Another statement I hear from teachers relates to fraction equivalence: “For fractions to be equal, they must be parts of the same whole.”
Wait, what? This is like saying that for numbers to be equal, they have to count the same thing.
We count monkeys and we count elephants. Is 1 monkey equal to 1 elephant? Is 1 huge pizza equal to 1 small pizza? We don’t even waste time considering this absurdity; it’s intuitive. The only thing equal about 1 monkey and 1 elephant is the quantity we have: we have 1. Number only defines quantity. When we count we always define what we count; then number determines how much of it we have.
We might have 3 or 10 or 17. 17 what? Aha, that depends. ‘Let’s find out how many cups of water there are in this jug.’ ‘Let’s determine how many inches of string we have.’ ‘Let’s figure out how many apples we have left in the basket.’ We explain what we’re counting and then begin to count. If I have 1 monkey and you have 1 elephant, we have the same number, but you need a bigger apartment.
How much of the sandwich do you have? I have 1/2. How much of the yard did you mow? I mowed 1/2. Just like 1 or 4 or 7, 1/2 is simply a number, our way of expressing quantity.
Fractional equivalence is another idea entirely. It means expressing equal portions of the whole: 1/2 is the equivalent of 2/4 or 3/6 or 5/10. 1/5 is the equivalent of 2/10. Equivalent fractions define the same portion of a unit. If I cut my sandwich into 2 parts and eat 1, and you cut your cookie into 4 parts and eat 2, we have both eaten the same quantity of our snack. 1/2 is equal to 2/4: the portions are ‘equivalent’. The snacks are not.
Part/part/whole, equivalent fractions and other inconsistencies and misconceptions have wended their way through our education system since I was taught about fractions so many years ago. I worked through it; some didn’t. Quick, divide 24 by 1/3. How many of you answered 8? Raise your hands. Wrong.
We must teach children to count and compute with rational numbers. By the end of fifth grade, children should be able to tell you that a fraction is a number and that different fractions can be used to express equivalent portions. They should be able to compute with these numbers, too, just as they do with whole numbers. Following the guidelines of the Common Core strand Numbers and Operations—Fractions will instill this understanding.
The Mathematics Standards form a progression which builds number competence by establishing what children should understand. Let’s not dismiss these Standards until we devise something even better. Please, let’s not hold our children back by condemning the reasonable expectations of the math common core—at least not this fraction of them.