Assessment: Improving Student Learning

What is the primary purpose of assessment? A common response, and misconception, might be that the purpose of assessment is to collect marks so that the teacher can decide on a final grade that the student will receive. This week we had the opportunity to delve into the topic of assessment in order to develop a better understanding of how we can effectively use assessment for its main purpose: to improve student learning. 

Assessment AS Learning: Engaged and Empowered Students

As the graphic below illustrates, there are three different types or purposes of assessment: for learning, as learning, and of learning. As teachers, it is important that our assessment practices reflect this and that we integrate assessment for and as learning in our classroom in a balanced way. This week, as I reflected on my assessment practices during my last placement, I realized that my approach was not very balanced as I tended to focus more heavily on assessment for learning and less on assessment as learning. 

© TES. Retrieved from http://bit.ly/2hm6dpt. 

As Growing Success discusses, assessment as learning is critical for improving student learning as it helps students to "develop their capacity to be independent, autonomous learners who are able to set individual goals, monitor their own progress, determine next steps, and reflect on their thinking and learning" (Ministry of Education, p. 28). It is time to move away from the traditional practices of assessment where the teacher is the only active agent engaged in assessing student's work and to instead promote a more collaborative relationship where our students are actively engaged in the assessment process. When

Retrieved from http://bit.ly/2yW6wRW. 

our students are active participants in assessment, they are more likely to be interested in their learning, to be committed to their learning, and to be confident in their ability to learn. 

One of my main goals for my upcoming placements, therefore, is to incorporate more opportunities for my students to engage in self- and peer-assessment in order to empower them to take ownership for their learning. One key way to do this is by sharing the learning goals and co-creating the success criteria before a task so that my students and I have a common and shared understanding of the goals and criteria. If my students do not know what success looks like for a given task, how can I expect them to succeed? Rubrics and checkbrics that are based on the success criteria and rating scales are another great tool for students to use to monitor their progress and to identify their strengths, needs, and next steps for improvement. 

Descriptive Feedback: The Most Powerful Assessment Tool

Our readings for this week and our class discussion both emphasized that effective descriptive feedback is far more valuable and powerful for improving student learning than giving grades. By continuously giving our students specific, timely, and meaningful feedback, we can help our students to identify their strengths and needs and to develop a concrete plan for how to improve those steps. 

Retrieved from http://bit.ly/2ycVHex. 

With this knowledge of the power of descriptive feedback in mind, my webinar on report card writing focused largely on how to write effective report card comments as a means of effectively communicating with and collaborating with parents and students in order to improve student learning. In our webinar, we explored three criteria for effective report card comments that were discussed in Reporting Student Learning: Guidelines for Effective Teacher-Parent-Student Communication: 

• Precise: The comments should focus on specific knowledge, skills, and criteria that are found in the curriculum that are most relevant to the student's achievement or development. It is important for the comments to provide concrete next steps for learning and concrete suggestions as to how parents can support their child's learning at home. 
• Personalized: The comments should be tailored to the individual student, referring to their interests, learning preferences, and level of readiness, using personal examples gathered from conversations, observations, and student products.
• Meaningful: The comments should be written in parent-friendly and student-friendly language and should be written in a positive tone. 

Technology in the Classroom: Assessing Apps

The Math Mindset module for this week also emphasized the value of assessment, albeit in a different context, as one of the videos focused  on the importance of assessing or evaluating the value of math apps before integrating them in the classroom. As teachers, we have a responsibility to ensure that the technology that we are incorporating in our classroom is being used for the purpose of improving or enhancing student learning rather than just for the sake of having it present in the classroom. Some key questions to keep in mind when we are assessing whether or not to include a math app in our classroom are:

• Does it help students to develop an understanding of key mathematical ideas? 
• Does it promote a positive math mindset where students do not equate time with intelligence?
• Is it an app that students will find challenging, interesting, and engaging?
• Does it encourage students to develop a deeper or more meaningful understanding of math concepts through the use of visual representations? 
• Does it encourage students to explore, play, and interact with mathematical ideas?

Blended Learning: Embracing Technology in the Math Classroom

As a student in elementary or secondary student, what role did technology play in your educational experiences in the math classroom? If I reflect on my own experiences as an elementary and secondary students, the vast majority of my math classes consisted of pencils, papers, and textbooks. Besides the occasional use of an overhead projector to display a word problem or an image on the board, the use of technology was minimal. This week, therefore, was an exciting and eye-opening week for me as I was able to explore how to integrate technology in engaging, meaningful, and powerful ways through blended learning in order to enhance teaching and learning in the math classroom.

What Is Blended Learning? 

Blended learning combines traditional face-to-face classroom instruction with online learning. Before this class, I thought that blended learning simply involved using technology in the classroom. As the graphic below shows, however, blended learning seeks to mix digital and face-to-face learning to create a dynamic learning experience where students have the opportunity to take more responsibility for the time, pace, and place of their learning. 

© Adams 12 Five Star Schools

How Can We Implement Blended Learning in the Classroom? 

The idea of creating a dynamic learning experience that fuses face-to-face instruction and digital learning was somewhat intimidating at first. Some of the initial questions that ran through my head were: Where do I begin? How do I know whether the technology that I am integrating is creating a rich and dynamic learning experience for my students? During our in-class exploration of blended learning, I found the SAMR model to be a valuable resource in helping me to overcome my initial trepidation and to understand what blended learning looks like in a classroom. 

Imagine Easy Solutions. 2014, May 29. "The SAMR Model Explained"
Retrieved from http://bit.ly/2xbLomw.

© Dr. Ruben Puentedura. Retrieved from http://bit.ly/JaoAcN.

As the video above the image on the right illustrate, tech integration happens on a continuum or a scale. I can use technology to enhance my students' learning experiences by replacing traditional tools with technology as in the case of substitution or augmentation. I can also use technology to transform students' learning experiences in rich and meaningful ways by reconstructing tasks or creating new tasks that would not previously be possible without the use of technology. While my ultimate goal is to transform students' learning experiences through modification and redefinition, I think it is important to recognize that, in reality, all four parts of the SAMR model can be present when I am integrating technology in my classroom. Some lesson plans may simply involve using technology to increase the functionality of the task such as doing an online quiz with the app Kahoot! which provides students with instant feedback for self-assessment. Other lesson plans may involve tasks that would not be possible without the use of technology such as creating a video tutorial that contains videos, audio, diagrams, and photographs. 

If I reflect on my math lesson plans from placement last year, the majority of the technology that I integrated into my lessons served the purpose of substitution or augmentation. In my future classrooms, my goal is to make a more conscious effort to find ways to integrate technology to modify and redesign tasks for the twenty-first century learners in my classroom. The Padagogy Wheel is a fantastic resource for developing ideas about how to integrate technology to create rich, meaningful, and engaging tasks for students. 

© Allan Carrington. Retrieved from http://bit.ly/2gWj0yX. 

Why Integrate Technology into the Classroom?

Improving Students' Mathitudes

In Making Math Children Will Love, Dr. Lynda Colgan emphasizes the importance incorporating non-traditional and attention-grabbing resources into the math classroom in order to "spark curiosity about mathematics, improve appreciation for and interest in mathematics, and contribute to understanding and relevance of mathematics in everyday life" (p. 1). Integrating interactive digital tools into the math classroom is a great way to create positive and engaging learning opportunities where students are excited to play, explore, and learn more about math. For example, during our webinar, we talked about the importance of integrating financial literacy into our lesson plans. Digital resources such as budgeting tools, games that involve financial decision-making, and videos about setting SMART goals can help our students to develop the knowledge and skills to become informed and engaged citizens in the global economy in a way that is personal, fun, and engaging. In our unit plan, we have also made an effort to include interactive digital activities and games such as the Tangram Game and the Cube Nets activity from Illuminations to spark students' curiosity about geometry.

Enhancing Student Learning

As Dr. Catherine Bruce discusses in Technology in the Mathematics Classroom, interactive whiteboards have tremendous potential to enhance student learning when they are used as a dynamic thinking tool. Interactive whiteboards promote collaboration and conversation as they allow students and teachers to show and interact with websites, diagrams, images, videos, and records of students' work. Our Math Mindset module this week discussed the value of representing mathematical concepts with drawings and manipulatives in order to develop a deeper and more meaningful understanding of mathematical concepts. Interactive whiteboards provide this visual support by making allowing students to use visuals to explore mathematical ideas and to illustrate and explain their thinking. As a teacher, I would definitely integrate the use of a SMART Board or an app like Explain Everything in my classroom as a means of encouraging students to take risks, illustrate ideas, and justify their reasoning as they collaborate with their peers.

Explain Everything™ Interactive Whiteboard from Explain Everything™ on Vimeo.

The Value of Rich Math Tasks

Retrieved from http://wodb.ca/index.html.

If I showed you the image on the right and told you that any of the four quadrants could be a correct answer to the question of which one doesn't belong, how would you feel? Would you challenge yourself to find more than one answer or would you wish that I had told you that there was only one correct answer to find?

The "Which One Doesn't Belong?" activity from this week's class emphasized the importance of breaking the stereotype that math involves only one right answer. As the article What is a Mathematically Rich Task? emphasizes, when we integrate more open, rich, and meaningful math problems into our math classrooms, it creates a "far more interesting, engaging, and powerful" learning experience for our students.

Rich Math Tasks: What Are They?

In order to integrate rich math tasks into the classroom, it is important to understand what they are. The articles Rich Tasks and Content and 6 Characteristics of Rich Math Tasks were valuable in helping me to understand the following key components of rich math tasks so that I had a better understanding and awareness of how to create and include them in my unit plan and future classroom.

1) Accessible to a Wide Range of Learners
The struggling student, the advanced student, and every student in between should see the task as challenging yet doable and should feel that they have the opportunity to contribute to the task with confidence.

2) Situated in a Real-Life Context
The task needs to be authentic and relevant to our students' lives so that they develop an interest in and understanding of how math applies to the real world around them.

3) Allows for a Variety of Approaches and Representations
As we create rich tasks, it is important to recognize, respect, and embrace students' diverse learning styles and approaches to problem solving.

4) Promotes Collaboration and Discussion
Rich tasks encourage students to share and compare their strategies, reasoning, and responses in order gain new insights and perspectives as they develop their mathematical understanding.

5) Fosters Students' Creativity
Rich tasks promote creativity and imaginative thinking as students use a variety of strategies and approaches to work through challenging problems.

6) Includes Opportunities for Challenges and Extensions
It is also important for there to be a potential to extend the task for more advanced students so that the other students can continue to work through the problem without feeling pressured to finish quickly.

© Pat McEachren. 2017.

Rich Math Tasks: Why Include Them?

While there are many benefits to integrating rich math tasks into the classroom, I would like to focus on two key benefits that really stood out to me throughout this week's content.

Students Develop a Deeper Understanding of Math

Retrieved from http://bit.ly/2x6UJuZ. 

As the Math Mindset modules this week emphasized, if we want our students to become successful mathematicians, we need to give them opportunities to collaborate, reason, and make connections so that they can develop a deeper understanding of math. Through rich math tasks, students have the opportunity to share and compare their strategies, ideas, and answers with their peers, to explain and justify their methods and choices, and to make meaningful connections to other math concepts and the real world. Rich math tasks help to prevent students from thinking that math is a frustrating subject that deals with a series of isolated concepts and that always involves using one right method to quickly find one right answer. As a teacher, it is easy to feel pressured to rush through tasks and activities in order to get through all the curriculum content, but developing rich math tasks and giving students time to collaborate, reason, and build connections is key to building a rich and positive learning environment where students build deep mathematical understanding.


Students Feel Empowered and Engaged
Rich math tasks are also valuable in helping students to become empowered and engaged learners. Our webinar this week focused on Differentiated Instruction and the importance of ensuring that all of the diverse learners in our classroom feel valued, included, and capable of achieving success. Rich tasks respect and promote diversity as students are encouraged to use a variety of methods and approaches to solve the problem and to share their strategies and ideas with their peers, ensuring that students of all levels feel that they are valued contributors to the learning taking place in their classroom.

© RSW. Retrieved from http://bit.ly/2xTgYcb.

Rich tasks can also be key in building a sense of community where students feel that the teacher knows them and cares about them. When rich tasks incorporate students' names and interests and aspects of the teacher's life, the tasks become much more personal and students feel a sense of belonging. During my placement, I created a problem that involved my brother who owned a sailboat. The students loved the fact that I shared this information and a story about my own experience sailing for the first time with them and I could even see the level of engagement increase. In my future lesson plans, I will definitely make the rich tasks personal to our classroom so that students feel valued and see math as fun and meaningful.

© Belinda Olij. 2016.

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? 

© Cathy Tomlinson. Retrieved from http://bit.ly/2hqu0nq.

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. 

© Pat McEachren. 2016.

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.

© Sylvia Duckworth. Retrieved from http://bit.ly/2fj4m3i.

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! 

© Oskar Cymerman. Retrieved from http://bit.ly/2hmUGJy.

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.

© Education Week. Retrieved from http://bit.ly/1Kk1qJD.

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.

© 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!

© Sylvia Duckworth. Retrieved from http://bit.ly/1VOXD0C.

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. 

© QuoteFancy. Retrieved from http://bit.ly/2wSheFy

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. 

©  Reid Wilson. Retrieved from http://bit.ly/1nzUoMC.

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. 

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!