Fly on the Math Teacher's Wall: Squashing Fraction Misconceptions

I am once again working with a group of math bloggers to join forces in ‘squashing’ some math misconceptions.  For the inaugural blog hop we tackled the big idea of Place Value.  This time around we are taking on FRACTIONS.  Read through my thoughts and then at the bottom is a link to another blogger and their thoughts on a fraction misconception.  Then keep following the links until you read all 16 of the posts or until you get tired of learning about fractions. 🙂

When you see the fraction 3/5, how many of you say “3 out of 5?”  I used to and sometimes I accidentally slip up and still say it, but I try my best to never use those words when talking about a fraction with elementary kids.  This blog post will give you two reasons why using “out of” damages students’ understandings of fractions PLUS I’ll tell you how we should be saying fractions, why that wording is so important, AND how it helps build fraction sense.

Two Reasons ‘Out Of’ Damages Kids’ Understandings of Fractions

1) In the early grades we only ever talk about fractions that are less than one…but when kids get into the upper elementary we ask them to think about fractions greater than 1, like 7/5. Think about the ‘out of’ term with fractions like that…how can you have ‘7 out of 5’????  This is a spot that fractions start making no sense whatsoever to kids and they just start following procedures without understanding.

2) The visual image ‘out of’ brings about does not help kids see fractions as part of a whole. Instead they see it as part of a set, which is important but….think about this; If I get 1 out of 2 that is not the same as getting 2 out of 4.  If I get 1 out of 2 brownies that definitely is NOT the same as getting 2 out of 4 brownies.

Kids struggle with fraction equivalence and my belief is it stems from this ‘out of’ terminology.  Another area this ‘out of’ image brings misconceptions to our students is when they go to add fractions.  How many times have you seen kids add 1/2 + 1/3 and say it is 2/5???  Again this ties back to the image ‘out of’ brings forth for the students.

This ‘out of’ view of fractions needs to come way later in our students’ mathematical careers, yet many textbooks use these kinds of visuals and the ‘out of’ understanding in early elementary to introduce fractions to children.  The Common Core standards intentionally has fractions mentioned in the early grades (1st-3rd) as part of 1 whole.  This image then helps kids see equivalence much better AND the need for same sized pieces along with same sized whole when adding.

So, if you shouldn’t say ‘out of,’ what’s the alternative??  Common Core Standard 3.NF.1 states “Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.”  And now we all shake our heads and say “WHAT???”  Let me give you an example to help explain this lovely standard.  The fraction 3/4 is formed by 3 parts of 1/4 size…three 1/4 pieces.  The fraction 4/5 is formed by 4 parts of 1/5 size….four 1/5 pieces.

There is a lot of number sense that needs to be built around fractions before we ask students to start operating with them…just like there is a lot of number sense we develop for whole numbers before we ask kids to operate with them.  I won’t address all the areas of numbers sense for fractions here because the post would become a novel.  Instead I’m going to focus on two of them that will help kids see fractions, like 3/5, as ‘three one-fifth pieces” instead of ‘3 out of 5.’

One/Two More or Less (aka Relating Counting to Addition & Subtraction)

Knowing the numbers just above and just below any given number is crucial for students to understand our number system.  We spend time helping kids count forwards and backwards with whole numbers, helping them develop the pattern of numbers and helping relate counting to addition.  Have you ever had students count by fractions?  Try counting by ⅔; starting at 0, go up to 6, and then back down to 0.  To help get you started, let’s try counting by 2s; 2, 4, 6, 8, 10, etc.  Now let’s count by ⅔s; ⅔, 4/3, 6/3, 8/3, 10/3, etc.  Doing activities like this helps students realize that fractions are just like whole numbers.  The numerator tells us the number (amount) of pieces we have or can count (Van de Walle, et al, 2014).  The only difference with fractions is that we have attached a denominator to that number.  The denominator tells us the denomination (size) of those pieces.  Just like the denomination of money helps us count our money; 1 ten, 2 tens, 3 tens…1 five, 2 fives, 3 fives, etc. There actually is a denominator with whole numbers, we just don’t write, or say, it; “two-wholes, four-wholes, 6-wholes…” but we do with fractions “two-thirds, four-thirds, six-thirds.” Once students have this understanding of what the numerator and denominator actually tell us, then it becomes easy for them to tell us what one/two units more or less is and to add and subtract with like denominators; what is ¼ less than 2/4?  What is ⅖ more than ⅗?  When we are finding one/two more or less (ie adding or subtracting), we are just counting forward or backward.  We can once again relate this to money denominations.  If I have 3 tens and then get 2 more tens, kids easily see that I have 5 tens.  When I have 3 tens and get 2 fives I can’t say that I have 5.  I do have 5 bills but they are different denominations so I can’t just add them together.  I have to get a common denomination to be able to just count the bills.

Part-Part-Whole

When children are working on their number sense of whole numbers, many of them see seven as just seven items.  They do not understand that seven can be broken into 6 and 1, 5 and 2, and so on.  Students also don’t know how to use that understanding to help them solve problems.  For instance when adding 9 + 7, a child might decompose the 7 into a 1 and 6 in order to make a friendlier problem.  But when they are adding 8 + 7, they might decompose the 7 into a 2 and 5.  Students need to know how to break numbers apart but more importantly how to use those decompositions to make problems easier.  The same is true with fractions.  It is not uncommon to see kids who when adding fractions just add the numerators AND the denominators.  Decomposing the numbers to get to a ‘friendly’ number is helpful when adding fractions, just like it is helpful when adding whole numbers.

This is just another reason why we need to pay close attention to how we SAY fractions “three one-fourths” instead of “3 out of 4.”  Saying it in that way helps us understand how we can then break 3/4 into ¼ + ¼ + ¼ which is hard to understand when we say it as “3 out of 4.”  This is extremely beneficial when students start dealing with fractions greater than 1, like 5/3.  It makes no sense to say it as ‘5 out of 3.’  If instead we think of it as ‘five 1/3 pieces,’ kids can take three one-thirds to get a whole and two one-thirds would be left.

“Children are bound to find fraction computations arbitrary, confusing, and easy to mix up unless they receive help understanding what fractions and fraction operations mean.” (Siebert & Gaskin,  2006, p. 394)

Don’t forget to continue on with the blog hop by visiting Teaching Math By Hart.