This past weekend, Tracy Zager tweeted a picture that 70% of 7th graders answered incorrectly:

This led to a great conversation which also included Christopher Danielson, Michael Pershan and Curmedgeon. It also sparked perplexity in me. I found I could devise many different variations to this problem to explore different student thinking, but instead I took this week to determine if students understood fractions as equal parts or if they fell back on a quick rule to find an answer. This was the results from day 1:

The results: 46.4% of my 8th grade students got it wrong, a much higher number than I anticipated. Looking at the reasoning given by my students, it appeared that they were following an old* rule* they partially remembers of counting shaded parts. I was not totally convinced this was the case or if it was because of the even partitioning that was done on the triangle. I wanted to check their understanding of a fraction so I gave them this for day 2:

When I looked at this, I expected small unit fractions and circle diagrams. I was not sure what they would write explaining what the fraction was or an example of it outside of the classroom. I was hoping they would talk about equal parts. This is how they answered:

One-half was the overwhelming fraction used, and they expressed that using a circle. All diagrams had pictures where students created equal parts, many times painstakingly so (evidenced with many eraser marks). Most students simply wrote it is one-half of the circle, none referenced equal parts. For the real life example, a majority of my students expressed the fraction as pertaining to a circular food; such as pizza, pie, cake or even a cheeseburger!

This last part really told me about how students have learned fractions. To gain student interest, teachers have fallen back on the standby of “the quickest way to student interest is food!” That students chose pizza or pie was no surprise, but that students see fractional parts in terms of circle graphs was. I need to make a point of expressing fractions with a different diagram- having students with such a narrow view of how fractions are used outside of the classroom limits them. It is no surprise that when fractions are introduced as slope in 8th grade, students have a total disconnect.

I still had a suspicion that students were only looking at the partitioning of the base instead of the area represented by them, I decided to use a diagram I thought of and that Curmudgeon reproduced and posted on his site:

I was hopeful that students would not only compare the widths of the fractional parts, but also the length. I was right and 77% of students answered the question correctly, here are the results.

I was very happy and proud of the results, and then another thing happened that blew that away. My students started questioning me as to why we were doing these warm-ups, what the correct answers were and were trying to anticipate what I was going to give them tomorrow! I had not gone over anything in class and had not told them what the correct answers were. They tried to pry information out of me, but I told them I had one more warm-up for them tomorrow and then we would discuss the week. They started a student-lead classroom discussion about the problems, what was correct and were sharing ideas about why people answered a certain way. I sat back and allowed them this time, it was great to see.

After class that day, I still had a nagging doubt that students only did so well because they were able to compare two diagrams, there was still an underlying issue of following this rule of counting three out of five pieces shaded. So the next day when they entered the room, this is what they found:

As I suspected, only 20% of them were able to answer it correctly. Those student noticed that one piece was 1/4th the size of the circle, the rest came back with the old standby of three out of five pieces shaded. I was crushed. I asked in each class whether this represented 3/5ths and the first and immediate response was not from the people who said yes, but those that said no. I said, “If this is true that the figure does not represent 3/5ths, why would that be? Take a minute to think about your answer.” I would call on someone who I knew answered incorrectly and when they contemplated the possibility of it being wrong, told me that one piece was bigger than the rest. We then went back and revisited the problems over the week, and students realized that the triangle did not have equal parts either. My students once again really dug into this discussion and other than a guiding question or two- took control and explored the problems and what the correct answers were. I was very happy with the outcome of that day, and I hope the students made some connections that will follow them throughout their mathematical careers. I will recheck their knowledge in two weeks and see if this experience will make a difference in their understanding of the concept.