My class always welcomes conjectures. This is made explicit on the very first day of the semester. This goes for everything from grade nine to grade twelve. As the grades advance, the topics have us venturing into increasingly abstract concepts, but conjectures are always honoured.
Certain class structures promote conjecturing more than others. Students offer questions during lectures, but they are often of a surface variety. They notice a pattern that has occurred in three straight examples, or think they have discovered a short-cut. I don’t like using tricks, but if they are “discovered” or “re-invented” (to borrow a term from Piaget and genetic epistemology), then we use them.
Author: natbanting
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Garbage Can Task
The following task happened by accident:
I was about to introduce a problem to my Math 9 Enriched class that we were going to complete with group whiteboards. Before I could introduce, life got in the way. Students wanted to know about their most recent examination. As I launched into a speech on their performance, a student got up to sharpen their pencil. She walked right in front of me. I made a comment, and she replied that the garbage can should be in the back corner where it would be more convenient.
The other day, a future teacher asked what one piece of advice I would give to a soon-to-be mathematics teacher. I immediately had several. I settled on one that I felt encapsulated my belief both in and out of class:
Honour curiosity.
In class, this finds me wandering through student suggestions and constantly posing new problems that create relevant challenges. Curiosity (both student and teacher) keeps a vibrant ecology going, and I would argue that the intellectual tension so often provided through curiosity is necessary for a positive ecology to thrive.
I have students in an enriched class that demand for me to give them more practice. I tell them that we practice mathematics with daily class activities. They don’t want practice, they want repeated practice. They are accustomed to receiving repeatable drills to cement understandings.
I have learned to compromise with this demand. I do believe there is a place for basic skills training in mathematics, and would raise an eyebrow at anyone who claims these unnecessary. I do, however, also believe that the heart of mathematics is problem posing, problem framing, and problem solving.
I was given a section of enriched grade nine students this semester. I decided very early on that the proper way to enrich a group of gifted students is not through speed and fractions. They came to me almost done the entire course in half the allotted time. This essentially alleviated all issues of time pressure.
The beautiful thing about this is we are able to “while” on curiosities that come up during the class (Jardine, 2008). I am not afraid to stop and smell the mathematical roses–so to speak. In a recent tweet I explained it as the ability to stop and examine pockets of wonder. This has been a blessing because our curriculum has become far less of a path to be run and more of the process of running it.
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Road Building Task
The Pythagorean Theorem is often taught in isolation. It has connections to solving equations, but often appears in curriculum long before other equations involving radicals. It also has unique ties to both radicals as well as geometry.
Despite these connections, the theorem has developed the reputation of a surface skill. It involves the repetition of the rule alongside numerous iterations. Something so fundamental to geometry is reduced to a droning chorus of:
” ‘a’ squared plus ‘b’ squared equals ‘c’ squared “
This semester, I’ve been attempting to infuse my courses with more opportunities for students to collaborate while solving problems. This post is designed to examine the shift in student disposition throughout the process.
I have noticed an increased conceptual understanding almost across the board and this is reflected in the differing solutions on summative assessments. It is also nice to see their marks grow on these unit tests. I do not believe that paper-and-pencil tests are the best venues for displaying conceptual understanding, but it is awesome when the two intertwine.
I have talked about individual whiteboards on this blog before. My school bought me supplies and I was loving the various classroom activities. While the grouping questions facilitated good mathematical talk between peers, I was still searching for a method to encourage more collegiality where my role could diminish to interested onlooker or curious participant.
So I had this brilliant idea.
Why don’t we get group-sized whiteboards created where students could work collaboratively on tasks?
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Digitizing Exit Slips
I’ve tried many forms of student written reflection in my classroom. No matter the format, I have always phased them out due to the administrative details and increased time burden. I liked the idea of having students reflecting on their learning, and believe in the benefits of writing across all curricular areas. What I needed was an easy way to orchestrate the process.
It needed to be easy for me to access and for students to complete. Here is what I’ve come up with, and the results have been great:
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What Makes a Task “Rich”?
In my short career, I have seen the death of the lesson. I remember creating ‘lesson plans’ to the exact standards of my college of education, and then never looking at them when I began to teach. I was never really in tune with the rigidity of the plan, but knew that there were certain learning goals I needed to get to by the end of an hour.
The scene has shifted away from the harshness of a ‘lesson’ toward more student-action-centred words like project, problem, prompt, or task. I like these words because they accurately describe what I am trying to do as a teacher–make the students think.