...and teaching them is even more fun! Teaching them so students truly understand can also be quite difficult, so it is important that we as teachers approach this topic carefully. I learned today how much of an impact language can have on student understanding of math concepts.
I have often used the words "out of" to explain fractions: 1/4 is one out of four, 3/8 is three out of eight, 6/10 is six out of ten. This leaves students thinking that 1/4 is greater than 3/4 because 1/4 has just one piece taken out of it (leaving 3 pieces), while 3/4 has three pieces taken out of it (leaving one piece). You "take more out" with 3/4, so it most be smaller, students reason. The teachers language leaves them with the opposite answer. Fancy that! I never would have realized this on my own - what an awesome opportunity to get into the minds of my future students. Understanding a student's thought processes when making an error is the first step toward helping him or her correct that error.
As a substitute teacher, I have helped several students with problems involving fractions. Just last week, a student told me to talk in "kid language" while explaining fractions. This was humbling, as I felt I was using very clear, easy-to-understand terms. Looking back on this experience, I realize I used "out of" in my explanation. Oops.
I was trying to explain how mixed numbers and improper fractions can be used interchangeably. This student was puzzled by mixed numbers. We worked with 1 1/4. He pointed to the first 1 and then the second, very confused. "How can we have more than a whole? We have one already." I made some silly putty pies to demonstrate. By the end of our conversation, he recognized we had one whole pie and that the 1/4 was part of a second pie. This was definitely an "a-ha!" moment.
Next, I partitioned the first pie into four pieces (imagine in my graphic - I'm new to Canva and am still struggling some, though it's awesome), so we could count five total pieces to get to the improper fraction 5/4. I didn't get too far on this explanation. I could tell the student was drained from all of the hard thinking we had done. Plus, I had been so totally "in the zone" talking to this student that I had neglected the numerous raised hands and "Miss Harrison's" throughout the rest of the classroom. I decided to call it a day and went to answer the other student questions. I faced a teacher's dilemma that day: I could either focus on truly and fully explaining concepts to this one student who was struggling so, or I could brush the topic's surface while assuring everyone in the class got his or her questions at least partially answered.
Khan Academy has some amazing videos explaining fractions in a straightforward, easy-to-understand way. In the documentary Teach, a math teacher implements a model of blended learning in which students work on Khan Academy on individual computers. I think this is an awesome way for students to work at their own pace and develop a solid understanding of topics before moving on. It would, however, be hard as a teacher to relinquish this "control," as this model isn't generally what we picture when we think of teaching. As great as this program seems, I would still be sure to pull struggling students aside in order to enhance their understanding with manipulatives, especially for a fundamental topic like fractions.
I am happy my struggling student understood at least a fraction (punny, I know) of what he needed to know the other day, but I am frustrated that I couldn't help him more. Has anyone else been in this boat? Please share!
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