When I was in the fourth grade, I created my own interpretation of Vincent Van Gogh’s Sunflowers painting. Our assignment was to choose a famous work of art, select a medium, and turn an 11×17 sheet of white paper into our own beautiful creation. Prior to unleashing my “art skills” on this blank sheet of paper, I had no significant connection to the original painting, nor did I know much about the artist except that Van Gogh went mad and sliced off his own ear—a fact shocking and fascinating enough to convince me that Van Gogh was the artist I would emulate. (I, however, had every intention of keeping both ears!) I don’t recall how long our class spent working on the art pieces or what we did before or after the project. What I do recall is that before this project, I had convinced myself I was not an “art person,” that I had no capacity to create something beautiful. What also remains perfectly vivid in my mind, fifteen years later, is the process of creation that I experienced and how it has influenced me as a first year teacher.
Six months have gone by since I started teaching 11th grade math. I’ve already learned so much from my classic mistakes and other, more successful classroom experiences that I feel I could write one heck of a first-year teacher book. To be fair, though, it would likely be found on a shelf in the Self-Help section. Chapter 1: Don’t lose hope…you’ll soon feel like a human being again! Of all the things I’ve learned thus far, one fact that sticks out is that the majority of my students hate math. They think it’s boring. I think they’re onto something.
When you hear a beautiful piece of music or read a great piece of writing, the synapses in your brain connect, you feel moved, your mind wanders, and you are reminded of past experiences. Completing math exercises #1-23 odd doesn’t generally have the same effect. Just as a musician or writer needs inspiration to create beautiful work, students need to feel encouraged and inspired to learn math. They need an outlet through which to be creative, make attempts, make mistakes, and grow in the process. Part of the problem resides in the fact that most math “exercises” aren’t working the right muscles. If I ask all of my students to complete the exact same 12 math problems, all of which are identical in nature and are to be completed by following the protocol laid out during class lecture, well then it’s no wonder they don’t feel inspired! Students in this case are directed to be robotic performers, following what Paul Lockhart (2009) refers to as shampoo bottle directions: listen, take notes, practice, test, forget, repeat. Key word: forget. What real learning is going on here? Likely none. If this is the approach we’re taking, then our students are right—math is boring!
During the second month of school I presented my students with an open-ended word problem about cookies in a cookie jar, allowing them time to work individually, to think. Students were to read the problem, fully grasp what it was communicating and gather their thoughts: This is what I know, that is what I need to figure out. Now, how am I going to get there? I remember looking around the room as my students began sorting and applying their knowledge in a way that would hopefully allow each of them to reach a solution. One student organized the information in charts of what was already known and what was needed. Some assigned variables to the ideas that were to be solved. Others used pictures or small circles to visually represent the elements of the problem. The paths taken to reach the end goal were all unique, yet with little instruction and no specific guidelines or rules to follow, each student arrived at a similar answer.
As my students got into small groups to share their chosen methods, it dawned on me that they were actually enjoying talking about math! That makes sense—they were given time in class to talk about themselves, to share their thoughts, to explain what went through their minds, what they tried, what worked and what didn’t. One of my students explained to her group, “I had no clue where to start so I just began trying different things.” With each failed attempt, figuring out what she did wrong helped guide her towards the process needed to reach her goal. Trial and error at its finest!
After sharing methods and experiences in small groups, students took time to reflect on the entire process and share their thoughts on paper. What struck me about their responses was that they reflected, not only on the activity, but also on how the activity related to their experiences with math itself—why it often frustrates them, why they don’t enjoy it. One student wrote, “Usually math is really hard for me. This activity wasn’t too challenging but I think it’s because I had to force myself to stop and figure out what was going on in the problem. I don’t usually do this with homework problems. Since I had to figure out what I was solving for, I understood what was happening in the problem and why I was doing the math process that I was doing. I think if I could understand more why I am doing the math processes I’m doing for homework on a daily basis, it would help me to understand the math better.”
I am amazed at how essential writing is in the learning process, even (and, arguably, especially) in a math class. Writing has become an integral part of my students’ math experiences, from brief end-of-class reflections to more elaborate math blogs. It’s a way to stop and give credit and value to what just happened. It’s also a way for me to catch a glimpse into the minds of my students, and what and how they are learning.
I think back to my experience as a 4th grader recreating Van Gogh’s Sunflower painting. I remember how my final piece turned out, what the pastel chalk smelled like, what colors and shades of yellow I used. I remember getting frustrated when the chalk smeared from the side of my hand pressing on the paper, and my repeated attempts to give the vase perfect symmetry. I never truly achieved that symmetry, but that certainly didn’t have any effect on the value of the process! I remember where I was sitting in the classroom and how proud I felt at the end of the project because I knew I had accomplished something. The learning that took place through this art project occurred naturally, with very little guidance, no step-by-step instructions, no teacher whispering to me that my flower vase wasn’t perfect and was therefore incomplete.
At the beginning of the school year, I thought I was appropriately following the sage advice that an astute colleague once shared with me during my pre-teacher days as an academic coach: Be firm and correct. Well, I soon realized that I had perfected the “firm” part, but needed some revision in the “correct” department. By putting so much importance on correct answers to math problems (those awful, robotic math problems!) even to the point of giving no credit for an answer that merely lacked a forgotten negative sign, I encouraged my students to lose sight of the journey of mathematics and become solely grade-driven. I felt like a failure, and my students felt like failures when they didn’t reach correct answers. It was right around this time that I began to hear those heartbreaking statements like “I’m not a math person” or “I’m not good at math” or “Math just isn’t my thing.”
Now, similar to my struggle with the symmetry of the flower vase, I’m learning not to put too much focus on reaching the right answer. I’ve changed my grading policy to emphasize mathematical thinking over mere correctness. I’ve also created a “no erasers” rule. Why? Our students are encouraged to keep drafts of essays, preliminary sketches and models. Why then should we encourage students to “throw away” first, second, or even third attempts at solving a math problem? Mistakes may be temporary roadblocks, but just as my students discovered during the cookie jar word problem, sometimes mistakes help point you in the direction you want to go. I hope to encourage and even celebrate mistakes with my students. With challenging open-ended questions, students can take their learning in various directions, get creative, get creatively frustrated, struggle, make mistakes, make more mistakes, and have a real learning experience. In the process, students write about all attempts, thoughts, and questions, creating a written record of their experience, and reflect on their experience at the end.
I still have the final product of my interpretation of Van Gogh’s Sunflowers. Though it will never look quite like Van Gogh’s original masterpiece, it serves a more important purpose as a continual reminder to myself that it doesn’t take the perfect symmetry of a flower vase to be an “art person.” Similarly, it doesn’t take perfectly correct math answers to be a “math person.” As this first year plays out, I’m striving to steer clear of monotony, to help my students engage with open-ended math problems, to create opportunities free of “shampoo bottle” approaches to learning and, instead, to allow my students to find inspiration as they write freely about their experiences as “math people.”
Lockhart, P. (2009). A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. New York: Bellevue Literary Press.
When I was in the fourth grade, I created my own interpretation of Vincent Van Gogh’s Sunflowers painting. Our assignment was to choose a famous work of art, select a medium, and turn an 11×17 sheet of white paper into our own beautiful creation. Prior to unleashing my “art skills” on this blank sheet of paper, I had no significant connection to the original painting, nor did I know much about the artist except that Van Gogh went mad and sliced off his own ear—a fact shocking and fascinating enough to convince me that Van Gogh was the artist I would emulate. (I, however, had every intention of keeping both ears!) I don’t recall how long our class spent working on the art pieces or what we did before or after the project. What I do recall is that before this project, I had convinced myself I was not an “art person,” that I had no capacity to create something beautiful. What also remains perfectly vivid in my mind, fifteen years later, is the process of creation that I experienced and how it has influenced me as a first year teacher.
Six months have gone by since I started teaching 11th grade math. I’ve already learned so much from my classic mistakes and other, more successful classroom experiences that I feel I could write one heck of a first-year teacher book. To be fair, though, it would likely be found on a shelf in the Self-Help section. Chapter 1: Don’t lose hope…you’ll soon feel like a human being again! Of all the things I’ve learned thus far, one fact that sticks out is that the majority of my students hate math. They think it’s boring. I think they’re onto something.
When you hear a beautiful piece of music or read a great piece of writing, the synapses in your brain connect, you feel moved, your mind wanders, and you are reminded of past experiences. Completing math exercises #1-23 odd doesn’t generally have the same effect. Just as a musician or writer needs inspiration to create beautiful work, students need to feel encouraged and inspired to learn math. They need an outlet through which to be creative, make attempts, make mistakes, and grow in the process. Part of the problem resides in the fact that most math “exercises” aren’t working the right muscles. If I ask all of my students to complete the exact same 12 math problems, all of which are identical in nature and are to be completed by following the protocol laid out during class lecture, well then it’s no wonder they don’t feel inspired! Students in this case are directed to be robotic performers, following what Paul Lockhart (2009) refers to as shampoo bottle directions: listen, take notes, practice, test, forget, repeat. Key word: forget. What real learning is going on here? Likely none. If this is the approach we’re taking, then our students are right—math is boring!
During the second month of school I presented my students with an open-ended word problem about cookies in a cookie jar, allowing them time to work individually, to think. Students were to read the problem, fully grasp what it was communicating and gather their thoughts: This is what I know, that is what I need to figure out. Now, how am I going to get there? I remember looking around the room as my students began sorting and applying their knowledge in a way that would hopefully allow each of them to reach a solution. One student organized the information in charts of what was already known and what was needed. Some assigned variables to the ideas that were to be solved. Others used pictures or small circles to visually represent the elements of the problem. The paths taken to reach the end goal were all unique, yet with little instruction and no specific guidelines or rules to follow, each student arrived at a similar answer.
As my students got into small groups to share their chosen methods, it dawned on me that they were actually enjoying talking about math! That makes sense—they were given time in class to talk about themselves, to share their thoughts, to explain what went through their minds, what they tried, what worked and what didn’t. One of my students explained to her group, “I had no clue where to start so I just began trying different things.” With each failed attempt, figuring out what she did wrong helped guide her towards the process needed to reach her goal. Trial and error at its finest!
After sharing methods and experiences in small groups, students took time to reflect on the entire process and share their thoughts on paper. What struck me about their responses was that they reflected, not only on the activity, but also on how the activity related to their experiences with math itself—why it often frustrates them, why they don’t enjoy it. One student wrote, “Usually math is really hard for me. This activity wasn’t too challenging but I think it’s because I had to force myself to stop and figure out what was going on in the problem. I don’t usually do this with homework problems. Since I had to figure out what I was solving for, I understood what was happening in the problem and why I was doing the math process that I was doing. I think if I could understand more why I am doing the math processes I’m doing for homework on a daily basis, it would help me to understand the math better.”
I am amazed at how essential writing is in the learning process, even (and, arguably, especially) in a math class. Writing has become an integral part of my students’ math experiences, from brief end-of-class reflections to more elaborate math blogs. It’s a way to stop and give credit and value to what just happened. It’s also a way for me to catch a glimpse into the minds of my students, and what and how they are learning.
I think back to my experience as a 4th grader recreating Van Gogh’s Sunflower painting. I remember how my final piece turned out, what the pastel chalk smelled like, what colors and shades of yellow I used. I remember getting frustrated when the chalk smeared from the side of my hand pressing on the paper, and my repeated attempts to give the vase perfect symmetry. I never truly achieved that symmetry, but that certainly didn’t have any effect on the value of the process! I remember where I was sitting in the classroom and how proud I felt at the end of the project because I knew I had accomplished something. The learning that took place through this art project occurred naturally, with very little guidance, no step-by-step instructions, no teacher whispering to me that my flower vase wasn’t perfect and was therefore incomplete.
At the beginning of the school year, I thought I was appropriately following the sage advice that an astute colleague once shared with me during my pre-teacher days as an academic coach: Be firm and correct. Well, I soon realized that I had perfected the “firm” part, but needed some revision in the “correct” department. By putting so much importance on correct answers to math problems (those awful, robotic math problems!) even to the point of giving no credit for an answer that merely lacked a forgotten negative sign, I encouraged my students to lose sight of the journey of mathematics and become solely grade-driven. I felt like a failure, and my students felt like failures when they didn’t reach correct answers. It was right around this time that I began to hear those heartbreaking statements like “I’m not a math person” or “I’m not good at math” or “Math just isn’t my thing.”
Now, similar to my struggle with the symmetry of the flower vase, I’m learning not to put too much focus on reaching the right answer. I’ve changed my grading policy to emphasize mathematical thinking over mere correctness. I’ve also created a “no erasers” rule. Why? Our students are encouraged to keep drafts of essays, preliminary sketches and models. Why then should we encourage students to “throw away” first, second, or even third attempts at solving a math problem? Mistakes may be temporary roadblocks, but just as my students discovered during the cookie jar word problem, sometimes mistakes help point you in the direction you want to go. I hope to encourage and even celebrate mistakes with my students. With challenging open-ended questions, students can take their learning in various directions, get creative, get creatively frustrated, struggle, make mistakes, make more mistakes, and have a real learning experience. In the process, students write about all attempts, thoughts, and questions, creating a written record of their experience, and reflect on their experience at the end.
I still have the final product of my interpretation of Van Gogh’s Sunflowers. Though it will never look quite like Van Gogh’s original masterpiece, it serves a more important purpose as a continual reminder to myself that it doesn’t take the perfect symmetry of a flower vase to be an “art person.” Similarly, it doesn’t take perfectly correct math answers to be a “math person.” As this first year plays out, I’m striving to steer clear of monotony, to help my students engage with open-ended math problems, to create opportunities free of “shampoo bottle” approaches to learning and, instead, to allow my students to find inspiration as they write freely about their experiences as “math people.”
Lockhart, P. (2009). A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. New York: Bellevue Literary Press.
