Several years ago I had a profound moment that led me to completely rethink what it meant to understand mathematics. I was still in the classroom and had been working with 6th graders on adding and subtracting mixed numbers. My formative assessments and observations showed that most students were proficient, and I felt pleased.
To end the unit I gave students an application of subtracting fractions using the context of a freeway sign with fractional distances. Specifically I gave students the picture below (which is the first picture in this lesson) and asked them “How far apart are the exits for Junction 90 and Jefferson Blvd?”
I clearly expected students to do well with this problem, but as I walked around checking students’ progress I realized that something strange was going on. I saw answers like:
- 1/3
- 1 1/3
- 3 3/4
Relatively few students got 1/4. When I asked a student why she got 1 1/3, she said, “It is 1 1/3 because 1 1/3 is between 1 1/2 and 1 1/4.” I felt like it must have been April Fool’s day with the joke on me. I didn’t know what had happened. Was I wrong thinking that students were proficient… or worse… could this minor little context have thrown students off so significantly?
I needed to know for sure, so the next day I came to class and asked students what I considered to be the same problem with no context at all. I just wrote 1 1/2 – 1 1/4 on the board and asked them for the answer. Again, the results shocked me. The vast majority of my students got the correct answer of 1/4. I didn’t know how to reconcile the results of the two problems and this is when I started asking myself “What does it mean to understand mathematics?”
In the days that followed I reflected upon what happened and I decided that my students primarily had procedural skill and fluency but very limited conceptual understanding or the ability to apply mathematics. I realized that for my students to “understand mathematics” they would have to have a more balanced understanding that included all three. This experience provided the foundation for why I value using real-world applications whenever possible. They provide a context for building the conceptual understanding and procedural skill needed for rigorous mathematical understandings.
Now out of the classroom, I work alongside teachers and my goal is to help them realize why the Common Core State Standards state that “educators will need to pursue, with equal intensity, three aspects of rigor in the major work of each grade: conceptual understanding, procedural skill and fluency, and applications.“
To accomplish this I recently recorded myself working one-on-one with sixth graders completing the same problem that had been so meaningful to my professional growth. I wasn’t sure if I could duplicate the results I had experienced years earlier but my plan was to begin each interview by asking the student about the freeway sign and then, regardless of how he or she answered, ask him or her to do 1 1/2 – 1 1/4.
Watch the first video below and note that I sped up time when he was working to make the video shorter.
Is this student demonstrating a rigorous mathematical understanding? Does he have:
- Procedural skill and fluency
- Conceptual understanding
- The ability to apply mathematics
To me it appeared that he had none of these mathematical understandings. Now watch the follow up question with the same student. Again I sped up time when he was working to make the video shorter.
Like I experienced in my classroom, to my surprise he got it right and it appears that this student does have procedural skill but could not navigate around a minor context to actually apply what he knows. He has limited conceptual understanding to fall back on. Clearly this one student is not representative of all students; however it has been my experience that students with superficial mathematical understandings exist in most classes.
Here is another student’s experience with the two problems. Note that I did not speed up the video so you could see the time he spent thinking.
How do you reconcile what you just saw? On the one hand you have a student who found the freeway problem so challenging that he sat for over thirty seconds thinking about how to solve the problem before giving up and stating, “Dang. This is hard.” Then he proceeded to solve the same problem procedurally and explained his process in a reasonably thorough manner. If you had only seen him solve the fraction problem, would you think he could solve the freeway problem? Does he have the rigorous mathematical understanding required by the Common Core State Standards?
Something also worth considering is how subtracting mixed numbers has been and will be assessed. The problem below is from the California Standards Test released test questions. Would these two students get this problem correct? Will this question determine whether they have a rigorous mathematical understanding?
This problem is from the new Smarter Balanced Practice Test for Grade 5 (Question #2). Would these two students get this problem correct? Will this question determine whether they have a rigorous mathematical understanding?
It is critical that we give students opportunities to develop rigorous mathematical understandings. Procedural skill is still an essential piece but it is just as important as developing their conceptual understanding and the ability to apply mathematics. Often times we teach students how to do mathematics with the belief that they will be able to apply it when the moment comes. Clearly that is not always the case.