Thursday, July 28, 2016

Mathematical Mindsets: Chapter 4

Welcome back! For the past four weeks, I have been reading and participating in a collaborative book study focused on the book Mathematical Mindsets by Jo Boaler. Chapter 3 described how the natural creativity and beauty of mathematics connects to the real world, but a disconnect exists between it and what is typically taught as part of the school mathematics program. (Read Chapter 3's post here.) Chapter 4 discusses ways to promote a mathematical mindset by helping students develop flexibility with numbers. 

Chapter 4 Summary

Jo Boaler states, "when students see mathematics as a set of ideas and relationships and their role as one of thinking about the ideas, and making sense of them, they have a mathematical mindset" (p. 34). Boaler also describes a study conducted by two British researchers which discovered that when compared to a group of low-performing counterparts, who had been exposed to the same amount of content, the high-performing students were able to employ number sense, a flexible and conceptual interaction with numbers, to solve problems successfully. The low-performing students, however, employed a standard algorithm which is often difficult to recall and execute successfully. The chapter also includes a robust discussion about purposeful math fact review and homework assignments. 

My Big Takeaway

Jo Boaler states, "number sense and mindsets develop together, and learning about ways to develop one helps the development of the other" (p. 36). I've always believed in the importance of helping students develop number sense at all grade levels, but this chapter really helped me solidify this belief. However, the piece that has not been as strong a belief for me is how mindsets support this goal. I now know that they both work together.

Ways to Use this New Knowledge to Support Our Students in the Classroom

Helping students develop number sense must be a regular classroom routine that includes conceptual activities focused on numbers and number facts. I describe my two favorite routines below. 
  • Number Talks: Jo Boaler describes them as "appreciating the connected, conceptual nature of mathematics" Number talks require students to describe how they solved an abstract problem mentally. After collecting multiple students' methods, the class then explores how and why the strategies work. I wrote an entire series about using number talks in the classroom when I participated in a book study focused on the book, Making Number Talks Matter. Read the series here
  • Greg Tang: His collection of books is fantastic for helping students build numeracy skills. Through his picture books, students learn how to group numbers to flexibly count the number of items in his colorful pictures. My favorite book is The Best of Times. It helps students discover strategies for mastering basic multiplication facts. Explore the online version here. I also wrote a post about how I used his books to help my students develop computational fluency skills. Read the post here

Developing strong number sense is an important way to help students become more successful math students; however, in order for students to be truly successful, it must be accompanied by a mathematical mindset.   

       Sound Off! How do you develop number sense in the classroom?

  • Boaler, J. (2016). Mathematical Mindsets. San Francisco, CA: Jossey-Bass

Thursday, July 21, 2016

Mathematical Mindsets: Chapter 3

Welcome back! For the past two weeks, I have been reading and participating in a collaborative book study focused on the book Mathematical Mindsets by Jo Boaler. Chapter 2 described the importance of mistakes in the learning of mathematics. (Read Chapter 2's post here.) Chapter 3 takes an in-depth look at how people view the world of mathematics and how school mathematics differs from real-mathematics. 

Chapter 3 Summary

In this chapter, Jo Boaler discusses how our view of mathematics affects how effectively we are able to learn and understand the subject. She states that people see mathematics as different from other subjects, a series of rules and procedures, because they do not understand the complex nature and beauty of mathematics. Because it is often viewed as a series of right or wrong calculations, Boaler describes school mathematics as a disconnect between "the mathematics that mathematicians use and the mathematics of life"-- "a creative, visual, connected, and living subject" (p. 27). 

My Big Takeaway

Mathematics is mathematics. There is no difference between school mathematics and the mathematics that mathematicians study every day. Jo Boaler states, "when we teach mathematics-- real mathematics, a subject of depth and connections-- the opportunities for a growth mindset increase, the opportunities for learning increase, and classrooms become filled with happy, excited, and engaged students" (p. 32). In addition, being good at math is not about being fast or first, it's about being a powerful thinker.

Ways to Use this New Knowledge to Support Our Students in the Classroom

1. Investigate the role of mathematicians. Post a chart entitled, "How to Think Like a Mathematician" with verbs that describe what mathematicians do, such as wonder, guess and check, ask questions, make conjectures, make connections, solve problems, etc. Continue adding to the chart throughout the year as students find other verbs to describe the work of mathematicians. When solving problems, refer to the chart and ask students to reflect on something that they did that made them "Think Like a Mathematician."  

2. Ask students to wonder! I love to use Dan Meyer's 3-Act Mathematical Stories for this. For example, show the students the video portion of the Girl Scout Cookies 3-Act with no leading information. Just show the video. After the video is over, go back to the 17-second mark and pose the question, "What are you wondering now?" Allow students to contribute their ideas during a class discussion before focusing on what question to tackle. This 3-Act is set-up to answer the question how many boxes fit in the trunk; however, there are many other questions to explore. Once a question is determined, discuss how students will investigate the question and find a solution. The beauty of this task-- it's tough to find an exact answer, so the focus is on the process, not the final answer. Give it a try!  

3. Look for math in nature and have students describe how what they see relates to mathematics. Collecting and posting pictures of math in nature is a good reminder to students that math stretches beyond the walls of the classroom.

Changing how we view the role of and purpose for mathematics in the classroom will help our students see it as more than a series of rules, procedures, and calculations. By showing them what real mathematics is and what mathematicians do, we may encourage more positive feelings about and increased success in the subject.   

       Sound Off! How do you describe mathematics?

  • Boaler, J. (2016). Mathematical Mindsets. San Francisco, CA: Jossey-Bass

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Thursday, July 14, 2016

Mathematical Mindsets: Chapter 2

Last week, I began participating in a collaborative book study with 12 other bloggers focusing on the book Mathematical Mindsets by Jo Boaler. Chapter 1 described what happens in the brain when we learn new things and illustrated the difference between a fixed mindset and a growth mindset. (Read Chapter 1's post here.) Chapter 2 discusses the power of mistakes in mathematics. 

Chapter 2 Summary

Making a mistake is good! It lights up our brains causing it to "spark and grow" (p. 12) even when we don't know a mistake was made. Coupled with a growth mindset, when an individual believes he or she has the ability to learn and that mistakes are just part of the learning process, mistakes will cause our brains to grow, especially when we participate in challenging tasks.  

My Big Takeaway

Mistakes should be valued, celebrated even. When I read this chapter, I immediately began thinking about my years as a classroom teacher and how many times I may have given the impression that making a mistake at a certain point in the year with curriculum that we had learned months earlier was wrong. I know that my intention was to build proficiency, but I'm not sure that is the message my students received. 

After reading this chapter, I understand the power that was in the mistakes my students made with curriculum we had learned sometime earlier and how taking the time to analyze them carefully may have led us down a new path. A path that may have led to a deep misunderstanding that the students still had-- an opportunity I missed to help them become more successful learners.  

Ways to Use this New Knowledge to Support Our Students in the Classroom

1. When we are reviewing assignments, move from analyzing assignments for correctness and begin to analyze them for mistakes. We should ask ourselves, "What do the mistakes reveal about where my students are in the learning process?" Then create an appropriate plan of action to address the mistakes.  

2. Use Jean Piaget's concept of "disequilibrium" to promote a growth mindset. Helping students understand this concept will send two important messages: 1.) It's okay not to know the answer right away and 2.) It's okay to struggle (Carter, 2008). Providing tasks that put students in a state of disequilibrium will help them develop and understand the role of productive struggle in the successful learning of mathematics.  

3. Celebrate and highlight mistakes! "My Favorite No" is a teaching strategy that does just that. Click the video link below to see the strategy in action in a real math classroom. Don't see the video? Click here

As we prepare to return to school for the fall semester, it's important to consider the role mistakes will play in our classrooms. Will our students hide and shyly respond to questions of which they are unsure of the answer or will they be proud of their thinking, even when it is wrong? While this way of thinking about the role of mistakes may be foreign to many of us, it's imperative to helping our students find success in mathematics.  

       Sound Off! How will you celebrate mistakes this year?

  • Boaler, J. (2016). Mathematical Mindsets. San Francisco, CA: Jossey-Bass 
  • Carter, S. (2008). Disequilibrium and questioning in the primary classroom: Establishing routines that help students learn. Teaching Children Mathematics, 15(3), pp. 134-37. 

Thursday, July 7, 2016

Mathematical Mindsets: Chapter 1

Today marks the start of a collaborative book study I am participating in with 12 other bloggers focusing on the book Mathematical Mindsets by Jo Boaler. This idea of mindset has sparked a lot of discussion in the world of mathematics education since the book's publication earlier this year. 

This work began with Carol Dweck's research detailed in Mindset: The New Psychology of Success published in 2006. Dweck's work revealed that we all have a mindset-- a core belief about how we learn. (Boaler, 2016). People with a growth mindset believe that smartness increases with hard work, whereas those with a fixed mindset believe that you can learn things but you can't change your basic level of intelligence" (Boaler, 2016, p. x). This idea of mindsets, then, becomes incredibly important for educators because the research conducted by Dweck and Boaler reveal that different mindsets "lead to different learning behaviors" and, in turn, "different learning outcomes for students" (p. x). 

The driving force behind this book pushes us to understand the power of mindsets and how we, as educators, can use them to change our students' learning pathways and allow them the opportunity to achieve higher levels of success. Join me each week as I, and my fellow bloggers, explore each new chapter of the book. And, if you're really interested, grab a copy and read along with us! Happy Reading!

Chapter 1 Summary

As we encounter new ideas, electric currents begin to fire in our brains making connections between the various areas and regions-- the more complex and intense the new learning, the more lasting the connections will be. This lays the foundation for our work with students; if we provide them with the tools they need to successfully accomplish more complex tasks, they will foster a growth mindset and believe that they can be successful mathematicians. 

My Big Takeaway

There are two types of mindsets: fixed and growth. Students who shut down when they reach their frustration point because they foster a belief that being good at math is an intelligence which you obtain at birth have a "fixed mindset." On the other hand, students with a "growth mindset" believe that the intelligence needed to be successful in mathematics can be gained through effort.

Ways to Use this New Knowledge to Support Our Students in the Classroom

1. Provide short, prolonged opportunities to interact with new skills so that students' brains have the opportunity to make connections and change structurally. 

2. Foster a belief that no one is born "good at math." Rather, emphasize the idea that making mistakes is part of the learning process and how we growth as mathematical thinkers.

3. Provide feedback that is related to actions and not student characteristics. For example, say, "Your work shows that you thought a great deal about how to approach this problem" instead of, "You are so good at this!"  

As teachers, understanding mindsets is a powerful way to begin a new year with our students. Not only will it help us understand our own mindset about mathematics, it will help us understand how the way we think about and teach mathematics can play a deeper role in what our students take away with them at the end of our time together. 

       Sound Off! How will this idea of a fixed/growth mindset change your classroom for this next school year?
  • Boaler, J. (2016). Mathematical Mindsets. San Francisco, CA: Jossey-Bass 
  • Dweck, C. (2006). Mindset: The new psychology of success. New York: Random House.