Through education, teachers influence change in their students' mindsets, which in turn can help students become successful individuals (Yeager & Dweck, 2012). We believe that the best teachers guide, motivate, and inspire their students. Teaching mathematics effectively is crucial to developing students who can solve problems and persevere. In AMLE's position paper (2010), This We Believe: Keys to Educating Young Adolescents, there is an emphasis on both Active Learning, in which students are engaged and are situated at the center of purposeful learning, and Challenging Curriculum, which promotes curriculum that incorporates students' ideas and questions. Here we consider how these characteristics can be employed in the mathematics classroom.
A Glimpse into a Mathematics Classroom
Students in the sixth grade mathematics classroom participated in a geometry unit in which the pedagogical focus was how we presented the lessons and promoted motivation through the use of growth mindset to help students feel confident and achieve success. Before starting with the lesson, we allowed time for students to express how they felt about the unit they were about to begin. Some students expressed feelings of being overwhelmed. Others, shared feelings of discomfort about geometry in general with statements such as "I feel ugh about it. Geometry isn't really my thing. It's a little confusing for me." This activity allowed us the opportunity to acknowledge students' feelings and assure them the goal was to create a positive mathematics learning experience.
Each lesson began with bell work in which students were to work independently for approximately two minutes before sharing their strategies with their shoulder partner. The task was to find the area of a shaded figure by applying knowledge about triangles and rectangles. Students needed to identify the information given to help them attempt the task and determine what was needed to be successful. They needed to develop a strategy to find the area of the rectangle and a triangle and then understand that the task was asking them to subtract both areas to find the shaded part. After working on the task and sharing with their partners, a whole-class discussion was conducted starting with the recording of all strategies used by the students. When the strategies were shared, students were asked to look around the classroom to observe how many of their classmates thought and attempted the problem in the same manner. This practice was used to ease student anxiety or doubts about the way they attempt mathematics problems. Then, students were asked to explain and try at least one of the strategies written on the board. Once a student demonstrated it, another student was encouraged to critique their peers' work. A different student was called to elaborate on what was shared and explain their rationale. Up to three students were called to critique their reasoning of others every time they worked on a problem.
This practice helped students make sense of their reasoning as well as that of others while deepening understanding of the skills presented. The following is an excerpt from the class using the mathematical practices of critiquing another students' reasoning:
Teacher: Let's see, Elijah can you explain what she did?
Elijah: Uhm…she did something wrong…she needed to divide the area of the triangle by two.
Elijah: She needed to divide the triangle in half.
Teacher: Do we all agree on that?
Teacher: So, you are telling me that you should divide the triangle by two? Is that it?
Teacher: Why is it that we need to divide by two?
Alanis: Because the triangle is half of a square.
Teacher: Because the triangle is half of a square. Kyra, what do you think of that? Agree or disagree?
Teacher: Why do you agree?
Kyra: Because if you take that triangle and you put it in one of the sides, it means you can have another one of it.
As the lesson progressed, students asked questions and shared their understanding of different strategies with their peers. By using effective questioning practices, students deepened their knowledge and articulated their own conclusions. Students who showed anxiety when trying to answer questions had the opportunity to gather their thoughts, listen to others before answering, and respond when they felt ready. This problem fostered a stress-free environment in which students felt at liberty not only to express their thoughts but also to accept their failures.
Before moving on with the lesson, the teacher assured the class how confident and proud she felt about their performance. She also used humor to express the need to challenge them more. This practice seemed to help students feel more confident and ready to do the mathematics at hand.
Teacher: All of you already know? WOW, then I should not even be teaching this lesson!
Before the challenge was given, students had the opportunity to reflect on their learning and the way they felt about the lesson. At all times, students demonstrated being engaged and responded enthusiastically to what the teacher proposed to them. An excerpt from the lesson is below.
Halie: I was kind of overwhelmed but now not that much, because I didn't know where really we were going to go to, but now that I know what I need to do it's kind of easy.
Emily B.: What reduced my overwhelmedness was how it was just like finding the area, but then just like adding another number to multiply to find the volume of the entire shape. It made things more simple to understand and see how to process it…it was just simpler.
Teacher: So, this is helping you? This strategy of seeing the floor first and then stacking it up is helping you?
Emily B.: Mhmm
Teacher: Ok, Kyra.
Kyra: I am not overwhelmed anymore because if you really just think about it…because with me and my brain it goes; ok, I see this problem…this problem looks hard, but then when you actually explain it, we see…and then you ask questions, show models…it's not that hard.
Teacher: Does that mean I can give you a challenge?
While students were describing how they felt about the lesson, they started sharing which strategies helped them the most. One strategy they liked the most when conceptually understanding how to find the volume of a rectangular prism was "the stacking strategy," because that is how they named it. This strategy was based on knowing that the volume of a shape can be given by finding the area of the base of the shape (a rectangle) and then "stacking" the same number of blocks or the same area given one on top of the other until the height had been reached. One of the students was able to connect this strategy of finding volume to a real world situation in which the first floor of a two-story house would represent the area of the base and the top floor would give the height. Another student, using this same idea, connected the same example to how the surface area of the house could be identified by how much paint or wallpaper the house would need.
During the challenge, students were encouraged to attempt to solve all problems given. Those who were able to get to the solution quickly were encouraged to use other strategies to support their findings. One significant thing noticed was how students persevered and showed strong effort when trying to solve the problem given. The prompt most students use is the use of positive language and the word "yet." The idea behind it is the belief that a student can change how they feel and the way they work on a problem by affirming that, even if they do not grasp the concept or strategy at that particular moment, they still believe they can understand it with a little bit more effort and perseverance. This strategy is called "The Power of Yet."
Through our experience, it is clear that incorporating more mathematical discourse and fostering a growth mindset belief in the middle school mathematics classroom can help students develop more confidence and achieve higher levels of mathematical understanding, which aligns with the Active Learning and Challenging Curriculum characteristics of This We Believe. Classroom discourse motivates students and promotes engagement in the classroom, and it encourages students to believe in their abilities, persevere into solving the problems given, and change the perceptions they might have of how they do mathematics.
National Middle School Association. (2010). This we believe: Keys to educating young adolescents. Westerville, OH: Author.
Yeager, D. S., & Dweck, C. S. (2012). Mindsets that promote resilience: when students believe that personal characteristics can be developed. Educational Psychologist, 47(4), 302-314.
Lynnette Sanchez-Gonzalez is a 5-9 mathematics High Impact Teacher and curriculum leader at South Seminole Academy, Seminole County, Florida.
Megan Nickels, Ph.D. is an assistant professor of STEM education in the College of Community Innovation and Education and College of Medicine at the University of Central Florida.
Published in AMLE Magazine
, August 2019.