Among teachers and within educator groups, discussions about fractions abound and frequently revolve around concerns that so many of our students don’t have a solid foundation in fractions or the fact that fractions cause anxiety for so many.
Fractions are anxiety-producing because we often send a message to students that fractions are “all new territory.” We teach fractions as if we are starting afresh and we fail to connect our teaching around fractions to what we’ve taught prior. This causes anxiety for students, as they feel the uneasiness of learning that is not integrated and has little relevance to prior understandings.
Additionally, we teach a series of operations on fractions rather than teach for deep understanding of those operations and the mathematical base from which these operations are generated. This, combined with too much of an emphasis on the digits within the fraction, causes students to approach fractions as something to manipulate, rather than something to understand. Simply put, we teach our students what we ourselves learned, which for most of us was quite limited, very rote, and unnecessarily disconnected.
In this posting we will explore two starting points for altering this pattern.
We can begin by ensuring that the use of the terms “numerator” and “denominator” do not limit students’ conceptualization of fractions. As we begin the process of teaching fractions and most certainly as we embark on the teaching of addition of fractions, we quickly begin to talk of the “numerators” and “denominators.” It is of course incredibly important that we use explicit mathematical terms in our classrooms; however, we often find that students have been moved too quickly to thinking in terms of these labels of “numerator” and “denominator” and fail to appreciate the fraction as a number: a symbol that conveys a concept or specific quantitative idea.
That is to say, we move too quickly from helping students illustrate the idea of ¾ with an area model or number line, as an example, to making sure students remember key vocabulary related to the abstract representation and can reproduce the operational steps for finding the sum of two common fractions. “Add the numerators but not the denominators,” we often hear teachers repeat. Many assume that once students have knowledge of the vocabulary for the abstract representation, they have access to the idea.
The difficulty is this: most students start thinking of the numerals of the fraction – the numerator and denominator – as two distinct numbers. Every algorithm we teach to our students encourages this same separation. We say “add the numerators” or “multiply the numerator and the denominator by the same number.”
Such as often happens with students who have been in classrooms in which there is an over-emphasis on the traditional algorithms for addition and subtraction of whole numbers, students start to see those whole numbers just as digits. They ignore the stated value, they miss the big picture, and they just manipulate the digits. They sometimes fail to notice that they are just rotely completing an algorithm, rather that seeing the values of the numbers themselves and recognizing alternative strategies that demonstrate conceptualization. Students fail to use number sense, they fail to use elegant strategies, and they fail to keep the concept of number foremost in their minds as they manipulate expressions. We see this clearly when a child employs the borrowing algorithm for a simple problem such as this: 200 – 198.
The same happens with fractions. To many kids, the fraction 3/4, is merely a numerator of 3 and a denominator of 4. Yes, these kids can shade 3/4 on an area model and most likely can find it on a number line, but they manipulate it as if it is just digits – just a numerator and a denominator. We can change this.
A helpful idea to bear in mind is the meaning of the word “numerator” and its connection to its root verb, “numerate.” When we numerate, we are articulating the units in terms of a particular number. With the example ¾, we are actually numerating or specifying the number of fourths. The ¼ is the units digit, and we are numerating or stating that there is a quantity of 3 of the unit. We are saying “there are 3 of one-fourth.” That is, 3( ¼ ) or ¼ and ¼ and ¼. We are numerating the unit of ¼, and we are numerating it by 3. ¾ is just shorthand for 3 (1/4).
Give your students an opportunity to understand this clearly. Consider giving them the chance to think in these terms before you begin to speak too much about numerators or manipulating numerators using traditional algorithms for operations on fractions. Strive to help your students keep the concept of the unit intact, as well as the process of numerating that unit.
When students are given time to understand this, they often make the cognitive leap to recognizing that, for example, 3/4 + 13/4 is the same as 3(1/4) + 13 (1/4), which is equivalent to 16(1/4) or 16/4. They recognize that they are “counting one-fourths” when they begin to calculate this sum. They should understand that this is the same concept as addition of whole numbers. When combining whole numbers, students are dealing with like units, the unit of one. Thus, 3 + 13 is 3(1) and 13(1), which is equivalent to 16(1) or 16. They are not just manipulating numerators and leaving the denominator alone. Instead, they are combining fourths: “three one-fourths and thirteen one-fourths.” They are counting one fourths, and the total is 16 (1/4) sixteen one-fourths.
Let’s explore a second idea that we can bear in mind as we strive to teach the concept of fractions more deeply:
Focus less on the operation and more on the properties that guide the operations of number.
The same properties guide our manipulation of all number types. And this includes fractions. Fractions are just a different unit. Kids get used to working with whole numbers, and numerating in terms of whole numbers (ex. 3 is really 3(1), that is “three ones”). But they can easily and with confidence transition to fractions, if they see they are just numerating units that are less than one. It’s all the same. And then when they want to manipulate those fractions via an operation – be it addition, multiplication, etc. – they find that the same properties they learned with whole numbers guide those operations.
To emphasize again: no need to set students up to believe that encountering fractions is a big deal, or overwhelming, or implicitly difficult and “all new.” As you approach fractions, emphasize to students that they already know what they need to know, they will just be extending their knowledge from whole numbers to fractions. Teach for connections, and show the beauty of our number system. The understanding of the operation shouldn’t be different just because the types of numbers you use in the operation are different. What a relief, and how empowering! How fascinating to realize that what you have learned thus far continues along and just deepens as you extend your knowledge to a new type of number, as you move from whole numbers to fractions. This idea can empowers our students, and reduces anxiety.
Let’s look at adding fractions and think in terms of extending the idea of addition to fractions using key properties.
Addition, at its base, is the combining of like terms. This doesn’t change when fractions are involved. In starting with this premise, then all you are doing when you add two common fractions with different denominators is substituting the values to make like terms.
Let’s walk this back to whole numbers first.
With whole numbers, we might think of adding in this way:
3 trucks + 2 trucks = 5 trucks
or 3 + 2 = 5
With units explicitly stated, we might write: 3(1) + 2(1) = 5(1)
An expression could also have units other than one, and be written such as:
3 (2) + 2(2) = 5(2)
Regardless, in all of these whole number examples, we are adding like terms. The like terms are either “trucks” or “ones” (either explicitly written or not) or even “twos.” We can combine the like terms to find a sum, expressed as the same term or unit.
Thus, when we move from whole numbers to fractions, we see we can numerate 3/4 + 13/4 as 3(1/4) + 13 (1/4). We can then recognize we have like terms (in both terms ¼ is numerated), and thus we know we can find the sum without any other manipulations. (Helpful hint: “terms” are separated by a + or –.)
3 (1/4) + 13 (1/4) = 16 (1/4)
It’s not a big step when we move to a little more complexity. When the units aren’t “like,” we know we first have to create like units or “like terms.” In finding the sum of 3(2) and 4(1/2), we will have to either express both terms as ones or both as halves. A little number sense (i.e. having been a student in classrooms in which the prevailing thought was “think before you manipulate” and “use relationships and patterns” or “use strategies flexibly),” one might quickly notice a simple way to find like terms that does not involve finding the lowest common multiple and multiplying numerators and denominators. One might use the relationships, which in this case are pretty simple.
What we are essentially doing when we make the terms “like” is using the principle of substitution. Since like terms are necessary for addition, we always evaluate the terms of an expression, look for an equivalent expression for one or all terms that can replace the original terms in order that each numerates the same unit. In the case of ⅔ + ⅚ , there are units of ⅓ and ⅙. These are unlike terms. However, each one-third is composed of two of one-sixth. ⅔, then, can be replaced or substituted with 4/6: 2(⅓) = 2(2/6) = 2/6 + 2/6.
As we determined the sum of these two terms, we were guided by the Substitution Principle and the Property of Addition. We didn’t have to alter our understanding of how addition works, as this principle and this property provided the foundation of our understanding of addition. Furthermore, the understanding of the fraction as a numeration of a unit and the seeking of relationships in order to implore the Substitution Principle meant that we kept our conceptualization of fractions intact.
With a deeper understanding of fractions and the sense-making that underlies an integrated curriculum, students inherently feel less anxious and are able to approach fractions with confidence.