Final Reflection: A New Perspective on Teaching Math

Exciting. Thought-provoking. Challenging. Insightful. Encouraging. These are just some of the words that come to mind when reflecting on my experiences in this math course. Over the past twelve weeks, my understanding of how to effectively teach math has been completely transformed for the better. Before this course, I was apprehensive about teaching math. Now, I feel energized and excited to take what I have learned in this course and to apply it in the classroom by creating a positive and engaging learning environment where my students will have the opportunity to develop a meaningful understanding of math!

The Importance of a Growth Mindset 

One of the biggest takeaways from this course is the importance of a growth mindset. Before this math course, I held the common misconception that some people weren't good at math. This is simply not true! Research shows that every child has the ability to excel in math. As a teacher, it is critical that I do not foster a belief in my students that they are not good at math but rather that I create a positive learning experience where every student in my class truly believes they have the ability to succeed. I need to establish a learning environment which promotes a growth mindset where my students understand the power of yet. I want my students to understand that mistakes and struggles are an important and valued part of learning which can be used as stepping stones leading to growth and understanding. Mistakes should not be stigmatized but rather should be embraced as a powerful learning opportunity! Rather than focusing on speed and efficiency, I need to reward hard work. As Carol Dweck discusses in her TED Talk entitled "The power of believing that you can improve", it is important for me to to praise the process the students are engaging in: their efforts, their strategies, their focus, their perseverance, and their improvement.

TED [2014, December 17] "The power of believing you can improve."
Retrieved from https://www.youtube.com/watch?v=_X0mgOOSpLU

Making Math Meaningful 

The Importance of Relational Understanding

This course has also taught me the need for students to develop a relational understanding of math. After completing this course, I now have a much better understanding and awareness of how intimately connected mathematical concepts and ideas are. As a teacher, I need to ensure that I develop lessons which help students to explore and discover how various mathematical concepts and ideas are connected. Rather than compartmentalizing different mathematical ideas and learning how to do specific tasks quickly using given steps or formulas, students need to explore ideas and learn about processes so that they are able to  develop an overall understanding and build a conceptual structure where mathematical ideas are linked. By exploring and making connections, students will develop a deeper, more meaningful understanding of math and will be able to adapt and apply their knowledge and understanding to new and diverse tasks. 

The Value of Manipulatives 

This course has also taught me the value of manipulatives in helping students to develop a more meaningful understanding of mathematical concepts. Before this course, I held the misconception that manipulatives were only helpful for students who struggled with math and thus would be kept at the side of the classroom as an option for students who need it. I now realize that manipulatives are valuable for all students as they make abstract concepts concrete, allow students to actively explore math concepts, and encourage students to prove their knowledge and understanding in a meaningful, concrete, and visual way.

Using chocolate bar pieces to explore fractions.
Olij, B. © 2016

Marian Small's discussion of the Van Hiele Taxonomy of Geometric Thought in Chapter 17 of Making Math Meaningful really drove home the value of manipulatives. In order for students to develop their geometric thinking, they need to have geometric spatial experience. For students to develop their spatial awareness and their understanding of the qualities and properties of shapes, they need to have opportunities where they can explore and discover these concepts through physical interaction with concrete materials. As a math teacher, facilitating opportunities where students work with a variety of manipulatives to explore ideas and demonstrate their understanding will be an integral part of my lesson plans.

Using tangrams to explore shapes.
Olij, B. © 2016

The Creation of Effective, Open-Ended Math Problems 

Another important takeaway from this math course is how to create effective, open-ended math problems which help my students to develop a more meaningful understanding of math. Effective, open-ended math problems help students to see math as sensible, useful, and doable! These problems should be grounded in engaging and relevant scenarios which help students to understand how math is useful and applicable to their own lives. Effective, open-ended math problems should also have a wide base and a high ceiling. This means that students at all levels of understanding should be able to engage with the problem. Students who are not as comfortable with the mathematical concept involved in the problem should still see the problem as challenging but doable and should be able to get started while students who are performing at a higher level of math should have opportunities to extend the problem in order to further challenge themselves. Effective, open-ended math problems should also involve more than one possible answer and more than one method of solving the problem. As a math teacher, I need to respect the diversity of thinking that occurs in my classroom. It is not fair or right for me to expect my students to all solve a problem using the same algorithm. Rather, I should create problems where students have the opportunity to use different algorithms to find the solution so that they understand that there is no single correct answer or single correct method of solving math problems. 

The Role of Facilitating 

This course has also taught me that direct instruction is not a very effective method of teaching mathematics. As a math teacher, my role is not to teach students through direct instruction what the equations or steps are for solving math problems. This will not create a positive learning environment or help my students to be engaged learners who develop a meaningful understanding of math. Rather, my role is to act as a facilitator in an environment where my students are actively involved in their learning by exploring, questioning, taking risks, and discovering as they build and share their understanding of mathematical concepts and ideas. This will create a much more positive and engaging learning environment where my students feel confident, empowered, and excited to learn about math! Rather than me directly instructing my students, my students need to have the opportunity to take ownership of their own learning! My role is to guide and support my students by asking effective questions which promote inquiry and thinking and encourage students to build new understandings and connections and to communicate their thoughts and ideas. The math congress which Marian Small discusses in Chapter 4 of Making Math Meaningful and which we experienced in class is a great example of how teachers can facilitate discussions and empower students to take ownership of their learning as students explain their work, ask questions, and share feedback. 

Overall, this math course has been an incredibly insightful experience which has taught me to see math and the instruction of math in a new, much more positive light. Over the past twelve weeks, my fear of teaching math has been transformed into eagerness and excitement as I now feel like I have a much better understanding of how to effectively teach math and how to create a positive experience for my students when they learn math. I have learned to embrace math with an open mind. I have learned to let go of the one-method, one-answer mentality of solving math problems and to instead embrace alternative algorithms and open-ended math problems. I have learned to let go of worksheets that involve rote questions and to instead embrace engaging, relevant math problems and creative math activities that encourage students to explore, take risks, and discover as they actively build their knowledge and understanding. I am eager to share this newfound enthusiasm and open-mindedness with my students and to continue to develop and build on my understanding of how to effectively teach math in an engaging and meaningful way!