Flipped Learning
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ABOUT I am passionate about the teaching and learning of mathematics through inquiry and problem solving. Driven by sound pedagogy, the effective incorporation of technology and ongoing feedback to students has allowed me to further engage students to deepen their understanding and ability to solve problems in mathematics. To my students, I hope that this site helps to serve you well in your learning. For parents and educators, I would encourage you to explore this site, alongside students, to determine how you can best support them in their learning journey. To gain further insight into my teaching philosophies and practice, I would like to encourage you to consider the materials provided in the sections, below, about creating a classroom culture for learning mathematics.
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*As for item D (above), assessment is a social, human endeavour. That means that it includes students, teachers and parents. Thus this section of the website will continue to be 'in development' throughout the course of the year. That is, I'm envisioning assessment defined as a co-construction--taking into account students' and teachers' experiences and parent feedback.
If you have questions, comments or feedback about this and other postings, please feel free to contact me.
Sincerely Yours, Chris Stewart (OCT)
If you have questions, comments or feedback about this and other postings, please feel free to contact me.
Sincerely Yours, Chris Stewart (OCT)
CREATING A CLASSROOM CULTURE FOR LEARNING (MATHEMATICS)
A) Seven Positive Norms Professor-Researcher at Stanford University, Jo Boaler, describes several classroom norms that are critical to creating a classroom environment that allows students to learn (mathematics) to the highest degree possible (see below).
These are standards that we will do our very best--students, myself--to building a classroom culture that values learning, putting the following at the front-end of everything that we do:
I would encourage all visitors to LT@NGDHS to read and reflect upon the seven, classroom norms. Discuss them with someone...students, share them with your teachers...if these norms aren't reflecting in your experience, what will you/we do? |
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B) Math Talk
Another critical component to successful mathematics programs is the communication of student thinking. Professor-Researcher at Trent University, Cathy Bruce, describes math talk as an effective way of helping students to deepen their understanding and suggests some guidelines for creating math talk norms in the classroom.
Communication needs to be more than just dialogue (or presentation of ideas); it must produce rich, meaningful discussion that is, over time, guided by students (i.e., student-to-student) as they conjecture, reason, and defend their mathematical arguments. Careful listening is required by all in order to produce rich discussions. For the listener, there is a great benefit: as your own ideas are filtered against those of others, you might gain a clearer understanding and new ideas to share with others. For the speaker, significant gains can be made to one's understanding: as you search to make your thinking clear to others, you actively engage in a process that not only conveys information but simultaneously puts you in a place where you're reflecting and clarifying your thinking. For more information concerning math talk, use the links (video) below and/or visit the Curriculum Services Canada website. |
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C) Open, Low-floor/High-ceiling Tasks in Mathematics
1-What are they and why use them?
There are many different problems that we can choose to explore in a mathematics class, but one type that educators might find particularly useful are those that are referred to as "Low-floor, High-ceiling" (or "Low Threshold-High Ceiling, LTHC") tasks.
Imagine yourself engaging in the following youcubed favourite LTHC task (Try it out! Discuss your thinking with a partner).
Now that you've tried out the task, take a moment to consider each of the following. In fact, take both polls below to share your thoughts with the LT@NGDHS community!
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Based on your feedback (see left) and what has been published on the topic of LTHC tasks, the following can be seen and experienced in classrooms when students are working collaboratively to solve LTHC tasks:
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2-How do these tasks relate to the mathematics curriculum?
These types of problems, generally, lend themselves to creating the conditions where students see mathematics as a meaningful whole--much different than the perception that can be created by a unit-by-unit program. And this is important when it comes to helping all students to engage in learning mathematics. Sometimes, this is referred to by educators as "spiralling" through a curriculum.
When "spiralling," often additional time can be created for students to work at mastering concepts. The tasks that are used can be quite useful for both teachers and students for a number of other reasons. With respect to curriculum, the following are important to teaching and learning:
These types of problems, generally, lend themselves to creating the conditions where students see mathematics as a meaningful whole--much different than the perception that can be created by a unit-by-unit program. And this is important when it comes to helping all students to engage in learning mathematics. Sometimes, this is referred to by educators as "spiralling" through a curriculum.
When "spiralling," often additional time can be created for students to work at mastering concepts. The tasks that are used can be quite useful for both teachers and students for a number of other reasons. With respect to curriculum, the following are important to teaching and learning:
- Assessment for and as Student Learning
Opportunities can also be created where students are challenged to think about how they've tackled problems and how their arguments make sense to them. Chapter 4, in the following document (Growing Success) describes current assessment theory and policy.
Secondary school teacher, Alex Overwijk (Ottawa-Carleton DSB), has been exploring how to improve student learning in his courses over the last five years through this type of planning. Some educators might describe this process as 'spiralling' or 'interleaving'. |
Spaced Practice or Interleaving
Spaced practice (or 'interleaving') refers to a different approach to the teaching of mathematics. That is, as opposed to teaching specific types of mathematics in blocks or units (i.e., massed practice), various types of mathematics are introduced, developed, and applied in the context of a variety of activities. And with each subsequent activity, some of the previous mathematical expectations and processes studied are further developed and used in new contexts.
According to researcher, John Hattie, as per some of the top teaching methodologies that have significant, positive impacts on student learning, spaced practice ranks in the top 12. |
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So ... why is spaced practice an effective means of positively impacting student learning? As identified by Overwijk (see video, above right), spaced practice has the effect of increasing the effort required to retrieve items, producing a deeper form of processing of the item(s). Massed practice, on the other hand, requires less effort to retrieve information since there is usually less time between the first and following times that you're required to use the information.
3-Another Example of a LTHC Task
Let's wrap up this section with another example of a LTHC task that one could experience in Grades 10 through 12. This is something that I've tried in the past and have seen publicized on Twitter (below; btw...can be a great learning tool).
Let's wrap up this section with another example of a LTHC task that one could experience in Grades 10 through 12. This is something that I've tried in the past and have seen publicized on Twitter (below; btw...can be a great learning tool).
@MathManAnusic has created a fantastic blog post that not only describes the task, but he has also captured photos of his students' work and posted them. In short, students (throughout their secondary school experience) spend a significant amount of time learning about a set of equations called functions. With each passing year, more functions and more elaborate means of describing them are studied--including the graphing of functions. The task, which typically can be used from Grades 10 to 12 (as a student's repertoire of functions starts to go beyond linear functions in Grade 10), provides students an opportunity to |
create their own artistic design incorporating the functions they've learned and the elaborate ways of describing (or defining) what it is that the viewer is seeing. If you haven't yet gone to @MathManAnusic's blog post to view these works of art, I would like to encourage you to take a few moments to do so. As an incentive, I've screencaptured a few samples from his post.
As you peruse the 'gallery', think about some of the defining features of low-threshold, high-ceiling tasks:
And as part of the planning process, the task can serve both grade-to-grade level and cluster planning. Below, I've pulled out a different grade-level, overall curriculum expectations, from Grade 10 to Grade 12, to highlight some of the goals that students and teachers can target through the process of working through such a task (BUT...there are so many more...again, I've only shown a few expectations from each grade-level).
- There is a place for each student to begin.
- Students can show what they know.
- There is room for students to explore and challenge themselves--raising the ceiling on this task.
- The level of thinking can increase in complexity.
And as part of the planning process, the task can serve both grade-to-grade level and cluster planning. Below, I've pulled out a different grade-level, overall curriculum expectations, from Grade 10 to Grade 12, to highlight some of the goals that students and teachers can target through the process of working through such a task (BUT...there are so many more...again, I've only shown a few expectations from each grade-level).
D) Assessment: Ongoing, Narrative Feedback (*in development)
*As for item D (above), assessment is a social, human endeavour. That means that it includes students, teachers and parents. Thus this section of the website will continue to be 'in development' throughout the course of the year. That is, I'm envisioning assessment defined as a co-construction--taking into account students' and teachers' experiences and parent feedback. |