Making Sense of Math: The Shepherd Question
During our class this week, I had an "aha moment" while watching this video about the nonsensical question mentioned above that was given to a group of eighth grade students:
Robert Kaplinsky. 2013, Dec. 1. "How Old is the Shepherd?"
Retrieved from https://www.youtube.com/watch?v=kibaFBgaPx4
Of the 32 students interviewed, 75% gave numerical responses where they attempted to plug the given numbers into some sort of formula involving addition, subtraction, multiplication, or division. Watching the video, I could really empathize with these students. If I'm honest with myself, I probably would have done the exact same thing if I were in their shoes! As a student in elementary and secondary school, I was always the one who memorized the given formulas or algorithms but never understood why they worked or why they were important. I would plug numbers in without having a real understanding of what those numbers meant or represented. As I reflected on the video and my own past experiences, I was struck by the importance of ensuring that our students develop a deeper understanding of math rather than a superficial understanding that simply involves mechanically performing calculations without thinking about whether or not their process or answer makes sense. Our students are humans, not robots!
Knowing vs. Understanding: What's the Difference?
As an elementary or secondary student, did you prioritize speed in the math classroom? Did you believe that the students who were the fastest at math were always the smartest? I certainly did! Was I right to assume this? I certainly was not! Knowing how to use a formula quickly to get an answer is not the same as understanding why that formula is being used to solve the problem.
The Strands of Mathematical Proficiency highlights these two different aspects of learning mathematics. Knowing how to use a formula falls under procedural fluency, that is, the ability to "carry out procedures flexibly, accurately, efficiently, and appropriately" (p. 116). Understanding why the formula is important falls under conceptual understanding, that is, the comprehension of math concepts, operations, and relations (p. 116). Conceptual understanding involves a deeper understanding of mathematics as it goes beyond memorizing isolated facts or methods to understand why those mathematical concepts matter. It involves a more comprehensive understanding of math as students build connections between new ideas and concepts that they already know.
As teachers, it's important to understand that knowing and understanding do not have to be at war with one another in our classrooms. Quite the opposite! Procedural fluency and conceptual understanding are interconnected, and both are important aspects of building mathematical proficiency. As I work with my group to develop our unit plan, it is important to remember that we need to develop hands-on, engaging activities that encourage students to build connections to prior knowledge from the various math strands so that they develop a more meaningful and comprehensive understanding of the math that they are learning. It's time to let go of memorizing facts and formulas without understanding what they mean or why they matter! At the same time, it is also important for us to develop tasks in our unit plan that provide students with opportunities to develop and practice their procedural skills in a variety of rich and meaningful contexts.
The Strands of Mathematical Proficiency highlights these two different aspects of learning mathematics. Knowing how to use a formula falls under procedural fluency, that is, the ability to "carry out procedures flexibly, accurately, efficiently, and appropriately" (p. 116). Understanding why the formula is important falls under conceptual understanding, that is, the comprehension of math concepts, operations, and relations (p. 116). Conceptual understanding involves a deeper understanding of mathematics as it goes beyond memorizing isolated facts or methods to understand why those mathematical concepts matter. It involves a more comprehensive understanding of math as students build connections between new ideas and concepts that they already know.
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As teachers, it's important to understand that knowing and understanding do not have to be at war with one another in our classrooms. Quite the opposite! Procedural fluency and conceptual understanding are interconnected, and both are important aspects of building mathematical proficiency. As I work with my group to develop our unit plan, it is important to remember that we need to develop hands-on, engaging activities that encourage students to build connections to prior knowledge from the various math strands so that they develop a more meaningful and comprehensive understanding of the math that they are learning. It's time to let go of memorizing facts and formulas without understanding what they mean or why they matter! At the same time, it is also important for us to develop tasks in our unit plan that provide students with opportunities to develop and practice their procedural skills in a variety of rich and meaningful contexts.
Promoting Deeper Understanding: Math Daily 3
As we explored in class, Math Daily 3 is a valuable framework for developing a deeper conceptual understanding of math. By exploring mathematical concepts individually, with peers, and in writing, students have the opportunity to actively build and communicate their understanding of mathematical concepts and their connections to prior knowledge. Math Daily 3 also provides students with the opportunity to develop their procedural fluency as they practice different ways of applying methods in a variety of rich and meaningful concepts. I also love that the Math Daily 3 framework encourages students to take ownership of their learning as they engage in self-assessment to reflect on which areas of their understanding and skills need improvement. Throughout the different activities of Math Daily 3, I would encourage students to use a variety of strategies and approaches to reasoning about mathematics. For example, visualization strategies and the use of manipulatives are a great way to strengthen students' understanding of mathematical concepts and to develop their spatial reasoning.
As we explored in class, Math Daily 3 is a valuable framework for developing a deeper conceptual understanding of math. By exploring mathematical concepts individually, with peers, and in writing, students have the opportunity to actively build and communicate their understanding of mathematical concepts and their connections to prior knowledge. Math Daily 3 also provides students with the opportunity to develop their procedural fluency as they practice different ways of applying methods in a variety of rich and meaningful concepts. I also love that the Math Daily 3 framework encourages students to take ownership of their learning as they engage in self-assessment to reflect on which areas of their understanding and skills need improvement. Throughout the different activities of Math Daily 3, I would encourage students to use a variety of strategies and approaches to reasoning about mathematics. For example, visualization strategies and the use of manipulatives are a great way to strengthen students' understanding of mathematical concepts and to develop their spatial reasoning.
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| © Teaching in the Tropics. Retrieved from http://bit.ly/2x4jJVg. |
Mindset Matters Too!
While procedural fluency and conceptual understanding are integral aspects of being successful in mathematics, we cannot forget the importance of a positive mindset! As The Strands of Mathematical Proficiency discusses, another key aspect of mathematical proficiency is a student's belief that math is sensible, useful, and doable, and that, with hard work, they are capable of being successful in math (p. 131). This ties in well with the our online module's discussion of fostering a growth mindset in our classroom. As teachers, we need to promote a safe and positive learning environment where students do not fear mistakes or challenges but instead embrace them as an important and valuable part of training their brain to become stronger. We need to expect great things from every student, celebrate challenges, and praise hard work so that all of our students believe that they are capable of knowing, doing, and understanding math!
While procedural fluency and conceptual understanding are integral aspects of being successful in mathematics, we cannot forget the importance of a positive mindset! As The Strands of Mathematical Proficiency discusses, another key aspect of mathematical proficiency is a student's belief that math is sensible, useful, and doable, and that, with hard work, they are capable of being successful in math (p. 131). This ties in well with the our online module's discussion of fostering a growth mindset in our classroom. As teachers, we need to promote a safe and positive learning environment where students do not fear mistakes or challenges but instead embrace them as an important and valuable part of training their brain to become stronger. We need to expect great things from every student, celebrate challenges, and praise hard work so that all of our students believe that they are capable of knowing, doing, and understanding math!
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| © Sylvia Duckworth. Retrieved from http://bit.ly/1VOXD0C. |
Amazing post Belinda! I could not agree more with the points you have made about what we have learnt this week. I too, was a student that would want to get math questions done as fast as I could, because speed was directly linked to how smart you were...or that was what I thought and perceived being good at math was. I was also the student that would pick the easy questions because I knew that I was about to complete them without any problems, just because I was scared that if I did the harder question then I wouldn't be a math person. I like how you mention that we as teachers and that teachers in general need to aid math understanding rather than making it all about speed as that is what hinders most learners and dampens their confidence.
ReplyDeleteGreat work Belinda! You have brought up some very important points about the importance of knowing vs. understanding a concept. Despite thinking that the kids who completed their work faster were the "brighter" students in the class, I have realized that this is not correct. In most instances, it is better for the students to take their time while completing assignments in order to ensure they fully understand these mathematical concepts. In addition, I thoroughly enjoy your last image focusing on growth mindset praises. These phrases are a great way to motivate students; especially if these students are struggling with a new concept. These are definitely some phrases I will keep in my day book for future placements. Growth mindset phrases will help educators to create an active learning environment for students; maintaining their engagement in learning in any new concepts the students are learning.
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