Conjecture. Sure, my students know what it is. They are even asked to make a conjecture of their own in a homework problem here or there. But should that be the extent at which my students develop their own ideas in a geometry class? Should I really be devoting little to no class time for students to explore mathematical concepts? My graduate professor, John Golden, is challenging me this semester to answer these difficult questions.
I remember one of my least favorite undergraduate math professors, and I remember him well. What was so frustrating about my professor was that he would never just answer my questions! Rather he would follow it up with another question, or give me a “hint” that didn’t seem to help at all. Many times I sat in his office hours only for him to tell me that I just needed to “think about it some more.” Wasn’t it his job to help me?! That’s what all my math teachers did previously. As much as I hate to admit it even today, I have probably never worked harder or learned more in another math class. It was a humbling experience, but also quite satisfying to know what I had accomplished. I left that class feeling much more confident in my abilities.
The reason I bring up my undergraduate math professor in this post about conjecture is that I think I need to follow his lead a bit more with my geometry students. Rather than simply provide definitions and theorems (as is often the case), I need to get my students thinking. After all, spoon feeding leads to helplessness. Just as I learned more and developed a deeper understanding in my undergraduate math course, my students will have a similar experience if I allow them to make conjectures and explore whether or not they are correct. It is the struggle that will cause them to develop into more mature math students. As I continue to teach my current unit on triangles, I will strive to find opportunities in which students can discuss, experiment, and play. I will encourage them to make conjectures before I tell them the desired result. My hope is that they will feel a sense of accomplishment, in addition to the cognitive benefits that will surely surface.
One unit in particular that I am excited about is our construction unit, which will take place during second semester. In the math department, we are revamping this unit to provide more of a challenge for our students. In the past, students were simply given a list of steps for various constructions. After performing those steps, they moved on to the next construction. This was a rote process that required very low-level thinking. By giving our students time to experiment by hand and then on Geometer’s Sketchpad, they will get a taste of what famous mathematicians went through as they created those constructions for the very first time.
Professor Golden: The balance activity we did in class would be great to do with high school students! I would appreciate feedback for any other areas that you think a hands-on activity would fit in well. Thanks!
I remember one of my least favorite undergraduate math professors, and I remember him well. What was so frustrating about my professor was that he would never just answer my questions! Rather he would follow it up with another question, or give me a “hint” that didn’t seem to help at all. Many times I sat in his office hours only for him to tell me that I just needed to “think about it some more.” Wasn’t it his job to help me?! That’s what all my math teachers did previously. As much as I hate to admit it even today, I have probably never worked harder or learned more in another math class. It was a humbling experience, but also quite satisfying to know what I had accomplished. I left that class feeling much more confident in my abilities.
The reason I bring up my undergraduate math professor in this post about conjecture is that I think I need to follow his lead a bit more with my geometry students. Rather than simply provide definitions and theorems (as is often the case), I need to get my students thinking. After all, spoon feeding leads to helplessness. Just as I learned more and developed a deeper understanding in my undergraduate math course, my students will have a similar experience if I allow them to make conjectures and explore whether or not they are correct. It is the struggle that will cause them to develop into more mature math students. As I continue to teach my current unit on triangles, I will strive to find opportunities in which students can discuss, experiment, and play. I will encourage them to make conjectures before I tell them the desired result. My hope is that they will feel a sense of accomplishment, in addition to the cognitive benefits that will surely surface.
One unit in particular that I am excited about is our construction unit, which will take place during second semester. In the math department, we are revamping this unit to provide more of a challenge for our students. In the past, students were simply given a list of steps for various constructions. After performing those steps, they moved on to the next construction. This was a rote process that required very low-level thinking. By giving our students time to experiment by hand and then on Geometer’s Sketchpad, they will get a taste of what famous mathematicians went through as they created those constructions for the very first time.
Professor Golden: The balance activity we did in class would be great to do with high school students! I would appreciate feedback for any other areas that you think a hands-on activity would fit in well. Thanks!
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