Writing for Math: Developing Growth Mindsets in Young Mathematicians
The beauty of mathematics only shows itself to more patient followers. — Maryam Mirzakhani
If I am being honest, part of what drew me to math education is the idea that “you don’t have to write in math class.” Fantastic, I thought! I envisioned myself challenging my students to think analytically and critically about math, while seamlessly avoiding the struggle, ambiguity, and vulnerability that the writing process demands.
Fast-forward several years, and I find myself blogging about using writing to develop mathematical thinking. What happened? After years of working with students and developing them as mathematicians, I realized that the very thing that makes math beautiful and interesting is precisely the struggle, ambiguity and vulnerability that comes with engaging deeply with mathematics. Working together with my colleague Katelyn Hayes, we wondered if reflective writing could be the tool our students needed to process their math struggles.
Brain science that has recently emerged around neuroplasticity has made mathematics one of the most exciting subjects to study today. The idea that some are predestined to be “math people” while others are not has been proven false.
According to Jo Boaler, PhD, Stanford Professor of Math Education, anyone can learn math to the highest levels. To be successful at the highest levels of math, we must embrace the mistakes and struggles that build our skills and connections among concepts. Mistakes and struggle provide the foundation for the vital work that develops our mathematical thinking, and using the work of Carol Dweck, we hope to develop a growth mindset towards struggle, even our most reluctant math students.
As teachers in a math classroom, Katelyn and I began to preach this growth mindset to our students, but we were still giving traditional quizzes and tests, rewarding students for “right answers” and clearly following a prescribed set of steps towards a solution. Our students would smile and nod when we got on our soapbox about making mistakes and the value of struggle while we became increasingly frustrated by their fear of mistakes and lack of ownership of their learning.
What we realized after studying Boaler’s work was that our classroom had no true safe space for our students to grapple with the sometimes messy learning that we realized is vital to the study of math. Our traditional assessment system did not acknowledge or recognize the growth process that we value most about learning math. Because we were only assessing a student’s ability to get a single “right answer” in a prescribed amount of time, we were sending mixed messages about the complexity of mathematical development.
We asked ourselves the question:
How do we get our math students to use writing to engage with the learning process, using evaluation and feedback to identify their own level of understanding, driving deeper connections?
To explore this question, we developed a self-assessment and reflection system that gives students ownership of the learning process. In order to launch this self-assessment with our students, we shared the following meditation on learning:
Self-Assessment and Reflection is a vital part of the learning process. It empowers your brain to recognize mistakes and errors in thinking, so that it can perform the necessary adaptation to improve your thinking. In this process, your brain engages different perspectives of an idea, which deepens your understanding of that concept.
Think of your brain like a bonsai tree. Learning is a constant trimming and shaping process. Each leaf represents a concept learned, with each branch representing your understanding of that concept. When you focus on your understanding, you edit and improve it, and the leaf gets stronger and healthier. Each new concept that is learned is filed into your brain. The more that you revisit and engage that file, the stronger it becomes.
The Learning Targets
Students begin a unit by working towards mastery of a set of learning targets that are outlined below. Students use the “I can” statements for each target as a guiding light for working through the material, helping them identify when they are ready to assess.
To assess a learning target, students take a written assessment in pencil. Upon completion of the assessment, each student checks out a copy of the teacher’s solution and evaluates her own work by comparing it to that of the teacher. The student will find and analyze all errors to identify where and how her work diverges from the teacher’s solution. She will then thoughtfully explain her error in thinking, digging into the “why” behind the error. This writing exercise is also an opportunity to recognize unique ways of thinking—if a student believes in a solution different from the teacher’s, this is where she defends her thinking, explaining her thought process.
After assessing her work, each student returns to the “I can” statements to complete a written Self-Assessment Reflection guiding her through the important reflection process.
Part I: I am challenged by…
For each “I can” statement, the student identifies something from her assessment (or her learning from that target) that challenges her. This is an opportunity to focus on her struggle, reflecting on the process and strategies for working through it. Students answer questions such as: What is hard for you and why? Is there a concept that you struggle to visualize? Do you have ideas for how you can overcome this struggle in the future? Also included in the reflection is the following reminder:
Everyone struggles with something. That is what makes math fascinating—we will never figure it all out! No matter how well you did on this learning target, there is something that challenges you. Maybe an extension of this target would challenge you—write about that!
Part II: I am proud of…
For each “I can” statement, the student then identifies something from her assessment (or her learning from that target—not limited to the assessment) that she is proud of. This is an opportunity to celebrate and reflect on the connections that she made in this target. For example, she will answer questions such as: What strategies worked? What connections did you make? Tell me about your aha moments.
Sample student reflection:
As the teacher, I evaluate each student’s work with a focus on the self-assessment and reflection process. This process is much more subjective than traditional grading. I comment on her reflections, highlighting insightful observations and challenging her to dig deeper.
Reflecting on this process myself, I have found that it has transformed my classroom. It has challenged me to create assessments that produce worthwhile reflections by writing questions with multiple “right” answers, asking opinion questions about the mathematics involved, and pushing students to think outside the box. Over the last few years, I have noticed that this process deepens my relationship with each student as I get to know her learning style and watch her develop as a thinker. My goal is for her to see me as a coach as opposed to an evaluator.
As my classroom has shifted its focus towards written reflections that reveal a student’s growth and away from performance, I find that students are eager to push themselves, taking ownership of their learning. It is fun to watch a student’s light bulb moment--when she takes off with an idea. As students continue to engage in this reflective process of developing their thinking, they are slowly strengthening their relationship with the mathematical and analytical world, which will open doors for them in the future.
Boaler, Jo. Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching. Wiley, 2016.
Dweck, Carol. Mindset: The New Psychology of Success. Ballantine Books, 2006.
For more information on Jo Boaler's work and recent studies on the brain's neuroplasticity, see youcubed, Stanford's center for mathematical research that provides resources for teachers, students, and parents.
Claudia Mathison has taught high school mathematics since 2003. Through her years studying math education, Claudia has worked with many platforms for learning, developing her instruction around a student-led model. The focus of her work centers on building student agency in mathematics learning. She recognizes the power that intrinsic motivation and student agency have on student engagement. She uses discovery-based tasks and activities to inspire and build concepts, with Self-Assessment and Reflection as key components of the learning process. Claudia has presented her work at local and national conferences, such as the National Council of Teachers of Mathematics 100 Days of Professional Learning, and has a Bachelor of Science in Civil Engineering from LSU. Claudia spends her free time searching for the ultimate pi joke, especially on March 14th.