1. What did I learn?
Today I learned the most incredible way to multiply binomials (x-1)(x+3). It didn't involve the standard approach: FOIL, although in actuality, the steps we took with the manipulatives had a direct relationship to the FOIL process. It fascinates me how we can use symbols or abstractions to create meaning. FOIL always felt like a "trick" to me. But using manipulatives to do the same work cements my understanding because I used a much more visual format. I solidified my understanding of how I am a highly visual learner (why I was able to learn to dance so quickly) and I would much prefer to be in the math than to be "lectured" about the math.
2. What do I have questions about?
I cannot figure out why my polyhedron does not follow Euler's Theorem. I like to describe my polyhedron as looking like the very top of the Luvre (one pyramid pointing upwards, offset by a pyramid pointing directly downward. The two are stacked). After picturing this, picture in your mind 3 Luvre(esque) shapes situated side by side. I've counted, recounted, and tried to think differently about what I created. I was only playing at the time that I created my 3-D structure. I wanted to first create the structure and then use it to test Euler's theorem (I didn't know it was his, but I had solved for a pattern: F+V=E+2...or...F+V-E=2. Reworking the problem allowed me to see how I was counting structural edges, not actual edges, yet this reworking leads me down a path where the outcome is jsut as perplexing. There are: 10 vertices + 24 faces = 30 edges + 2. 34 does not equal 32. I can't tell if I'm doing anything wrong. I'm open to suggestions. I wonder if my model doesn't fit the definition of a polyhedron. I love that it has created confusion!
3. What are the implications for classroom practice?
With respect to my baffling Euler problem, the implication is definitely that with investigation comes puzzlement, and there will be students who absolutely chomp on the bit for the challenge, and there will be others who will give up long before the challenge ensues. Expect diversity. With respect to the use of manipulatives, I can pull directly from the article, Group Worthy Tasks, "Different resources and hands-on materials attract more students and entice them to participate, thus opening additional avenues for students to understand the learning" (Lotan, 2003, p. 74). I responded to the manipulatives because they helped me to understand a concept that I had only ever been able to understand using FOIL.
Today I learned the most incredible way to multiply binomials (x-1)(x+3). It didn't involve the standard approach: FOIL, although in actuality, the steps we took with the manipulatives had a direct relationship to the FOIL process. It fascinates me how we can use symbols or abstractions to create meaning. FOIL always felt like a "trick" to me. But using manipulatives to do the same work cements my understanding because I used a much more visual format. I solidified my understanding of how I am a highly visual learner (why I was able to learn to dance so quickly) and I would much prefer to be in the math than to be "lectured" about the math.
2. What do I have questions about?
I cannot figure out why my polyhedron does not follow Euler's Theorem. I like to describe my polyhedron as looking like the very top of the Luvre (one pyramid pointing upwards, offset by a pyramid pointing directly downward. The two are stacked). After picturing this, picture in your mind 3 Luvre(esque) shapes situated side by side. I've counted, recounted, and tried to think differently about what I created. I was only playing at the time that I created my 3-D structure. I wanted to first create the structure and then use it to test Euler's theorem (I didn't know it was his, but I had solved for a pattern: F+V=E+2...or...F+V-E=2. Reworking the problem allowed me to see how I was counting structural edges, not actual edges, yet this reworking leads me down a path where the outcome is jsut as perplexing. There are: 10 vertices + 24 faces = 30 edges + 2. 34 does not equal 32. I can't tell if I'm doing anything wrong. I'm open to suggestions. I wonder if my model doesn't fit the definition of a polyhedron. I love that it has created confusion!
3. What are the implications for classroom practice?
With respect to my baffling Euler problem, the implication is definitely that with investigation comes puzzlement, and there will be students who absolutely chomp on the bit for the challenge, and there will be others who will give up long before the challenge ensues. Expect diversity. With respect to the use of manipulatives, I can pull directly from the article, Group Worthy Tasks, "Different resources and hands-on materials attract more students and entice them to participate, thus opening additional avenues for students to understand the learning" (Lotan, 2003, p. 74). I responded to the manipulatives because they helped me to understand a concept that I had only ever been able to understand using FOIL.
No comments:
Post a Comment