When I was an elementary student, fractions were by FAR my most difficult subject. I could not ever wrap my brain around them and embarrassingly enough, I still struggle to understand them. The other day I had trouble trying to figure what half of a 3/4 cup of butter would be! My cookies tasted a little more buttery than I would have liked. So today I am talking about misconceptions about fractions, because I’m really a pro!
This question was on a fourth grade test on fractions this week at one of my schools.
Amazingly, many students answered the question by saying they agreed with Molly. Their explanations said things like:
- “4/8 and 1/2 are the same number because they are equivalent.”
- “The diagram below each shows a half, so they are always the same.”
So somehow, somewhere we have a misconception here. Students are missing the idea that fractions can be different amounts if the whole is a different size. After all, one 8 inch pizza is not the same as one 16 inch pizza, right? I’d MUCH rather eat 1/2 of the 16 inch pizza!
My favorite way to clear up misconceptions is to relate it to real life…especially food. Food lends itself beautifully to math in so many ways. Once I brought in the skittles, suddenly light bulbs turned on. Equivalent fractions may be the same number, but they are not always the same amount.
I think we miss this step very often when we work with students. I think that real life connection is what helps them figure out what the symbols stand for. When we leave that out, students are unable to make sense of a problem.
If you want to see more examples of misconceptions that us math nerds have uncovered, check out The Math Spot:
Technically, Molly is correct. they ARE effectively the same number. Fractions are numbers, just as 2 and 5 are numbers. The problem arises right at the beginning, with “fractions are parts of a whole”. Nobody questions the action 5+2=7, but if numbers are “amounts” then what if it was 5 oranges and 2 apples? There is confusion between one half, the number, and half a pint, the amount. It is very sad indeed.
Great points! This is a huge misconception for my students.
… and compare with…
2 is not the same as 2
Well, 2 mountains “is” bigger than 2 molehills.
I like your idea for squashing this misconception. Very visual. I think my students will understand this now.
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I enjoyed your post! You discussed some of the common misconceptions students have about fractions that I have been learning about in my math methods course, at the University of Wisconsin Oshkosh. With that being said, there are some big ideas or concepts that students need to understand about fractions in order to be successful: One being that students need to understand that fractional parts are equal shares or parts (with equal measurements) of a whole or unit, and the whole must be specified. Another concept students need to understand, according to John A. Van del Walle (2014), (which relates to the 4th grade problem you discussed in your post) would be for the students to understand that when two fractions are equivalent, that means there are two ways of describing the same amount by using different-sized fractional parts. (pg. 202). Those are just some of the basic conceptual understandings that students MUST first understand about fractions before they can successfully solve problems using fractions. And yes, like you discussed in your blog, using problems that are in context helps students to connect their conceptual understanding of fractions to real-life!
This was a very interesting post because I don’t think that misconceptions, especially dealing with fractions, are usually addressed in math classes. I am currently a student at UW-Oshkosh and in my math methods course we just got done learning about fractions and how difficult they are for many students. I think most of the misconceptions stem from a lack in understanding when it comes to part-whole relationships. Daniel W. Freeman and Theresa A. Jorgensen, who wrote the article, “Moving Beyond Brownies and Pizza,” explain that, “one major conceptual hurdle that students must overcome is the idea that fractions are numbers in and of themselves, not a composition of two distinct, whole numbers,” (Freeman and Jorgensen, 2015), Once students see fractions in this light, I think it would be easier to manipulate them in word problems and simple operations. I don’t think enough teachers understand fractions themselves and that makes them afraid to allow their students to investigate that part-whole relationship. However, it is through the understanding of this relationship that students can truly learn what fractions are and how to use them.
Another way to squash this misconception is using candy bars. Ask students if they’d rather have a half or a whole candy bar. The “whole” turns out to be a whole miniature size bar where as a half is half of a king size. Shows that half can even be bigger than a whole sometimes depending on what you’re comparing. I also used this intro into finding common denominators- making sure we’re dealing with all the same sized pieces before adding and subtracting.