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! 

Exploring Data & Probability

Have you ever thought about how often you make estimations or determine the probability of an event in your everyday life? Whether it's estimating how long it usually takes to drive to work in the morning or determining the likelihood of our favourite sports team beating their rival in an upcoming game, we are constantly involved with measurements and estimations of central tendencies and probability. Given the fact that data and probability are so relevant to our everyday lives, it is critical that we as educators create a learning environment where our students have the opportunity to explore and develop a more meaningful understanding of these topics.

Making the Measures of Central Tendency Meaningful

In her discussion of the measures of central tendency in Chapter 21 of Making Math Meaningful, Marian Small not only describes what mean, median, and more are but also when each of these measures are useful. When teaching my own students, I would definitely take Small's approach of teaching not only the meaning of the three measures of central tendency but also their usefulness with regards to different scenarios and different sets of data. 

In order to understand what the mean or average of a data set is, my students and I could talk about what it would mean to make everything fair or to level the bars. In class, we were given word problems where students received an unequal number of items. We were challenged to use Cube-A-Links to show how we could split these items evenly among the group of students who sharing them. 

Olij, B. © 2016

Olij, B. © 2016

Marian Small discusses how the mean is useful when the numbers in the data set are fairly close together but that the mean may not be the best indicator of the "average" group size when there are one or two extremely high or extremely low values. In order to demonstrate this concept to students, I could ask them to level bars where there was a considerable difference in the number of Cube-A-Links each bar contained. As students leveled the bars, they would be able to visually see and physically experience how the size of the leveled bars (the average) did not accurately represent the initial sizes of the various bars in the data set. 

I could then have my students explore the concept of a median in order to understand how the median is a valuable measurement when the data contains one or two extremely high or low values. Students could interact with several different data sets and compare the median and mean for each of these sets in order to see how the median can sometimes provide a more accurate reflection of the data. 

When discussing when the mode is a useful measure of central tendency, we could use the example of a glove factory. In order to know what size glove the factory should focus on manufacturing, the glove designers could look at which size glove is most frequently bought in order to determine the most common glove size. In order for our students to understand the measures of central tendency and when it is appropriate to use them, it is important to explore and discuss examples of these measures in real-life, engaging scenarios. 

Exploring Probability Using TinkerPlots

During class, we also had the opportunity to explore TinkerPlots which is a data analysis software where students can manipulate a set of data in order to create graphs and other representations which compare different properties of the data. In Chapter 22 of Making Math Meaningful, Marian Small discusses the importance of using a large set of data in order to more accurately determine the trends or the probability of an outcome or event. TinkerPlots is a particularly valuable resource for the classroom as it offers data cards which contain large sets of data for students to interact with. TinkerPlots is also very beneficial for visual and tactile learners as it is very easy for students to sort and manipulate the colour-coded data by clicking and dragging different points on the plot to explore different connections between various properties and to create colourful graphs. At the start of the activity, students could develop a number of "I wonder if..." statements related to the data set. Students could then explore whether their hypotheses were true by sorting and analyzing the data in order to determine what the trends and connections are among the various properties of the data set. TinkerPlots is a fantastic resource for encouraging students to develop a more meaningful understanding of data and probability as they make sense of real data and recognize trends in an interactive, visual, and tactile way. 

Learn Troop. 2014, December 27. "TinkerPlot Basics."
Retrieved from https://www.youtube.com/watch?v=wPFfIurEnUg