Math Matters: September 2017

Saturday, 23 September 2017

Differentiated Instruction: Reaching All Learners

This week we had the opportunity to explore how to make math meaningful and engaging for the diverse learners that will be in our classrooms!

Differentiated Instruction: What is It?

It is good to start off with a reminder of what Differentiated Instruction is. As the Capacity Building Series document "Differentiating Mathematics Instruction" discusses, differentiation involves "differentiating instruction by content, learning process and/or product to address the needs of students with varied experiences, learning readiness, learning styles, contextual interests, and learning interaction preferences" (p. 2). As an educator, I want to be proactive in understanding the individual and collective needs of my students and flexible in adjusting my teaching strategies and methods to those needs so that every child is able to thrive in my classroom!

An 'aha moment' for me during class was when we talked about how we could adjust the game "I have... Who has..." to support diverse learners. I love this game because it is a fun, engaging, and collaborative way for students to practice vocabulary and to reinforce or review key concepts! Even though I used this game in my placement last year, I had never thought about how a simple change like anonymously colour-coding the cards based on their level of difficulty and strategically giving them to students according to their readiness would be a great differentiation strategy to better support my students. I will definitely keep this strategy in mind when incorporating this game into my future lesson plans!

Differentiated Instruction: How Can I Achieve It?

Developing Parallel Tasks

As we explored in our reading and during our in-class activities, parallel tasks are a great way to meet a variety of needs in the classroom. By developing two problems that both focus on the same key concept yet differ in terms of their level of difficulty, teachers can address students with differing levels of readiness. I was amazed and excited to see how parallel tasks could create such a rich and inclusive learning opportunity as the common questions that are used to address the whole group encourage students to learn from one another, to make connections, and to develop a deeper understanding of the big ideas. This week's task for our unit plan where we created our concept map was great practice for learning to identify key concepts as sometimes I find that I can get so bogged down in the specific expectations that I lose sight of the big ideas. During my last placement, I tended to gravitate towards using open problems as a differentiation strategy. I'm excited to add parallel tasks to my differentiated instruction toolbox and will definitely make more of an effort to also include these types of problems in my future math lessons.

Encouraging Student Inquiry

This week's webinar focused on student inquiry. I thought this fit well with the topic of differentiation as student inquiry allows students to explore a topic in a way that fits their readiness and interests and to demonstrate their understanding in a way that suits their learning style. As we discussed in our webinar, educators can and should also differentiate for students' individual needs by using scaffolding questions to guide and support students throughout the inquiry process. As the image below mentions as well, I also love that student inquiry empowers students to take ownership of their learning and fosters their natural sense of curiosity.

Celebrating Mistakes

As the Math Mindset modules this week emphasized, another key aspect of reaching all learners is to foster a growth mindset where students learn to embrace mistakes and see them as a valuable and necessary part of their learning journey. If we want our students to challenge themselves when choosing a parallel task, to share seemingly wild questions or ideas during the inquiry process, and to persevere in the face of challenges, we need to create a positive and safe learning environment where students feel comfortable making mistakes and understand that mistakes and struggles lead to brain growth!

Friday, 15 September 2017

Knowing, Doing, and Understanding Math!

Let's pretend for a moment that you are a student in the eighth grade. Your teacher gives you this word problem: There are 125 sheep and 5 dogs in a flock. How old is the shepherd? How would you feel? What thoughts would be racing through your mind? What would you do? This nonsensical word problem highlights the importance of fostering a learning environment where students truly know, do, and understand math which was a common theme throughout this week's course materials.

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.

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.

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!

Saturday, 9 September 2017

The Discovering Continues...

One year ago, I walked into our math class at Brock feeling nervous, apprehensive, and insecure. There was a definite sense of fear and doubt when I thought about the possibility of teaching math to twenty to thirty children. Over the course of the last year, my anxiety and fears about teaching math have shrunk and my confidence and excitement have grown. This week, I walked into our math class feeling eager to discover and explore more about teaching math in a way that is meaningful, effective, and fun for my future students!

Putting Knowledge into Action in the Classroom

The highlight of last year was being able to take what I had learned during our math course and put it into practice during my placement. I had the opportunity to co-teach a unit on geometry and our class, both the teachers and the students, had a blast! As we learned in our course last year, manipulatives are particularly important for developing students' geometric thinking and spatial awareness. With this in mind, we pulled out the geoboards, toothpicks and marshmallows, tiles, and blocks and we filled our unit with open-ended problem-solving activities that encouraged our students to actively explore shapes and their properties. I quickly discovered that math can be a fun and rewarding subject to teach, and it was really encouraging and exciting to see hands-on, meaningful learning in action in the classroom!

Fostering Curiosity and a Positive Attitude

One of the big ideas that I took away from our first week back is the importance of curiosity in learning. As teachers, it would be easy to simply tell our students the information that they need to learn, but this does not lead to students are engaged in meaningful learning or who are developing a love of learning. As the video from class emphasized, we need to encourage children to wonder. In the language that we use to speak to our students and in the lessons that we develop, we need to foster curiosity by encouraging our students to ask questions, to share ideas, and to take risks.

Another big idea from this week is the importance of fostering a positive mindset both in myself and in my students. As the videos this week demonstrated, adults and children alike often hold the misconception that there are two kinds of people in the world: math people and 'not-math' people. It's time to stop believing that lie! The reality is that every single student has the potential to succeed at math. Their attitudes and experiences, however, play a key role in determining their success. It's time to instill in our students the belief and the confidence that they are capable of success and that mistakes and challenges are an important and valuable part of their success stories.

Looking Forward to More Growth

Knowing that students' experiences in the classroom have such a tremendous influence on their perceptions of math and their success in math, I am excited to continue to grow and evolve as a math teacher so that I can support my students in the best ways possible. As we participate in the in-class activities, webinars, and online modules and build our unit plans, I am looking forward to diving deeper into different strategies and approaches to teaching math in ways that help students to see math as fun, relevant, and valuable in their lives. I am eager to continue to discover how to promote curiosity, a love of learning, and meaningful math experiences in the classroom.