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On a recent walk through my neighborhood I came across a father teaching his young son how to ride a bicycle. The father was holding on to the bike as the boy sat atop the seat, gripping the handlebars tightly. It appeared that the father was much more confident in the boy’s ability to learn to ride on his own than his son, as the boy clearly was not ready for his dad to let go of the bike. As I walked by them I heard the father say, “Don’t worry, I won’t let go until you’re ready to ride on your own.” Panicked, the boy responded, “OK, but promise you won’t let go. I CAN’T do it by myself!”
The panic and frustration were evident in the boy’s voice, and dad appeared to be getting a bit discouraged as well. It was all I could do to hold myself back from walking over and whispering one little word in the boy’s ear – “Yet.” The boy was absolutely right. He couldn’t ride the bike by himself – yet. I yearned to let him know that just because he couldn’t do it yet, that didn’t mean that he’d never be able to. I wanted to tell him that none of us could ride a bike before we could, and then once we could we were on our way (literally and figuratively). I also wanted to tell him that he might not be very good at it at first but with enough practice and desire he’d be flying on his own before he knew it. I was a little worried that he would get discouraged if he wasn’t riding like an expert right away. I think we, meaning parents, teachers, and in Vygotsky’s terms the “more capable peers” (Vygotsky 1978) often think that telling someone who is trying to learn to do something we’re already accomplished at is easy. This trivializes both the efforts of the person who is trying to learn the new thing, and the “thing” being learned. A former student of mine shared this sentiment with me years ago, “Anything worth doing well is worth doing poorly at first.” I’ve never forgotten this phrase and it’s reminded me through the years that lots of wonderful things that we learn to do throughout our lives may first seem insurmountable, but as long as we keep plugging along, sometimes through trial and error, often with a lot of help and maybe a few tears, we can do most of what we set our minds to. In addition, if we limit ourselves to only those things that are really easy to do right off the bat, we seriously limit our possibilities.
As I continued on my walk my thoughts, as they often do, turned to the teaching and learning of mathematics. Mathematics instruction in the U.S. has been characterized as “a mile wide and an inch deep.” Because of this, teachers are often pushed to cover topics at a breakneck speed, leaving students with the impression that mathematics is more about speed and application of the correct procedure than reasoning and sense-‐making. Years ago Alan Schoenfeld, a professor in both the Mathematics Department and Graduate School of Education at UC Berkeley, found that high school students generally think that if they can’t solve a math problem within ten minutes, they won’t be able to solve it at all (Schoenfeld 1985). To this day a typical math class in the U.S. starts with the teacher showing students exactly how to solve a particular type of problem, followed by students applying the procedure they just learned to similar problems, and then checking their answers and doing more of the same types of problems for homework. Nowhere in this scenario do students learn to approach mathematics in terms of “What I can do now” and “What I can’t do yet.” The idea of trying different approaches and sticking with a problem over time, maybe coming back to it after some time has passed, is absent in most American math classes.
We all need to shift our perspective of what it means to be a good math teacher from one who carefully provides students step-‐by-‐step procedures to one who provides students with opportunities to engage with rich mathematical tasks and a bit of “constructive struggling.”
The first of the eight Mathematical Practices from the Common Core State Standards says that students “Make sense of problems and persevere in solving them” (CCSSI 2010). In order for students to be able to do this, students and teachers alike need to begin to think in terms of “yet.“ This means that students have to be prepared to engage in a fair amount of what Cathy Seeley calls “constructive struggling” (Seeley 2009) and teachers have to be prepared to allow them to do so. As Seeley writes, “we sometimes spoon-‐feed our students too much information and ask too little of them in return,” (p. 89). In essence, we don’t allow them to struggle or have the opportunity to push through something challenging and feel the satisfaction of accomplishing (or being on the way to accomplishing) something difficult.
During my last year as an elementary school teacher I taught a class of fourth graders in Berkeley, California. One day, after I had introduced a math task for the students to tackle, one of my students, Olivia, said something that has stayed with me to this day. As she began to jump into the task, I overheard her say, “I don’t really get this right now, but I know I will.” I’ve thought about this statement many times over the years and the multiple messages it implied about Olivia’s understanding of mathematics, as well as the confidence she had in herself as a learner. Olivia was very comfortable thinking in terms of what she got right now and what she didn’t get yet. She embodied the first Standard for Mathematical Practice in that she knew that even if she wasn’t sure how to approach the problem right away, given time and effort, she was capable of solving it. Olivia knew she first had to make sense of the problem, and then she had to persevere to solve it. Her understanding that she had to make sense of the problem also implied that she knew there was something to make sense of. Unlike many of her peers, Olivia expected mathematics to make sense, and she believed she was perfectly capable of doing so.
What does this mean for American teachers and students? It means that we all need to shift our perspective of what it means to be a good math teacher from one who carefully provides students step-‐by-‐step procedures to one who provides students with opportunities to engage with rich mathematical tasks and a bit of “constructive struggling.” It also means that we have to shift our perspective of what it means to be a good math learner from one who can memorize rules and apply algorithms without thinking to one who expects mathematics to make sense and is willing to stick with a task until it is understood. This may seem like a daunting endeavor… it is! As with many things that are worth doing, it’s going to take time, effort, trial and error, a lot of help, and maybe even a few tears. But please don’t despair if you feel like you can’t do it. Just remember, all that means is you can’t do it – yet.
References
Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/the-‐standards.
Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando, FL: Academic Press.
Seeley, C. L. (2009). Faster Isn’t Smarter: Messages about Math, Teaching, and Learning in the 21st century. Sausalito, CA: Math Solutions.
Vygotsky, L. S. (1978). Mind in Society. Cambridge, MA: Harvard University Press.
Julie McNamara is co-author of Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense, 2010, as well as Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fractions Sense, Second Edition, 2015. In addition, Julie has written a follow up fraction book, Beyond Invert & Multiply: Making Sense of Fraction Computation, Grades 3-6, 2015, all published by Math Solutions.
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