| Aspects to maintain | Suggestions | Interactivity |
|---|---|---|
| - Walking around to help students | - Raise volume | - Encouraged group interaction |
| - Pace | - More clear definition of slope at beginning or end of lesson | - Asked good guiding questions |
| - Good intro | - Have the app available for students to practice | - Advanced group got a chance to discuss more |
| - Good applet and technology use | - Fun/relevant | |
| - Addressed confusion | - If more time had been available, drawing more lines for practice would have been helpful | |
| - Eye-contact | ||
| - Visuals | ||
| - Group work | ||
| - Positive words/ feedback/ answered questions effectively | ||
| - Relating topic to “real life” |
I felt that the lesson went well, overall. I was really excited by the hook, because there was immediately a connection between slopes and skateboarding/snowboarding/skiing. There were several things which I would like to improve, in particular. Firstly, I would like to improve my teaching for intuition. Although this is a very broad goal, I will narrow it below. It is very difficult to wander from traditional teaching styles of stating definitions, demonstrating problems, then practicing problems of definitions. I think that our group may have chosen a topic which was a bit too ambitious to cover in its entirety in 15 minutes, and given the applications of a slope in calculus, it would not be wise to rush the lesson. I would stick by my choice to leave negative slopes until the end of class, however, because it has been decided, by a combination of the definition of a slope as well as the selection of which axes are positive or negative, that a line falling from the second quadrant to the fourth quadrant has a negative slope. If horizontal axes increased to the left and decreased to the right, the sign on a slope would be reversed. The magnitude of slope is really crucial here, the sign of the slope can be discussed later in the lesson. I think a first start to developing inquiry teaching would be emphasizing what is necessary and what is arbitrary in mathematics, such as the sign on a slope; I feel that this is a good guide as to what needs to be learned and what can be (self-)taught in an activity.
My second point for improvement would be the understanding of when to introduce definitions. After all the articles we've read in class, especially the Hoffman/Brahier article, I feel as though introducing a rote definition is almost a faux pas in mathematics, because if it is used incorrectly, a definition becomes not a summative statement but an introduction of something arbitrary being passed off as necessary. After all, why did we actually choose for slope to be a ratio of the change in vertical to the change in horizontal, rather than the other way around? Rather than a concept of "more steep", we would likely instead use the terminology "less flat". However, everyone who has ever encountered any wheels in their life will have an intuitive sense of what slope means, even as a matter of survival. Finding a mathematical definition is a fine balance between over-complicating an intuitive definition and over-simplifying a (often) physical concept. When is a place for definitions during a math lesson? I have always encountered definitions at the beginning of a math lesson, especially in calculus. However, could calculus also be taught intuitively, without starting from the definition? After all, this obsession with definition (I would argue) originates with the formation of proofs. The definition of a concept being proved is one of the most useful tools for verifying a mathematical statement around that concept. When, if ever, does mathematics become really intuitive in an upper level? What is the evolution of the necessity of a formal mathematical definition as a function of time in a math classroom?
My second point for improvement would be the understanding of when to introduce definitions. After all the articles we've read in class, especially the Hoffman/Brahier article, I feel as though introducing a rote definition is almost a faux pas in mathematics, because if it is used incorrectly, a definition becomes not a summative statement but an introduction of something arbitrary being passed off as necessary. After all, why did we actually choose for slope to be a ratio of the change in vertical to the change in horizontal, rather than the other way around? Rather than a concept of "more steep", we would likely instead use the terminology "less flat". However, everyone who has ever encountered any wheels in their life will have an intuitive sense of what slope means, even as a matter of survival. Finding a mathematical definition is a fine balance between over-complicating an intuitive definition and over-simplifying a (often) physical concept. When is a place for definitions during a math lesson? I have always encountered definitions at the beginning of a math lesson, especially in calculus. However, could calculus also be taught intuitively, without starting from the definition? After all, this obsession with definition (I would argue) originates with the formation of proofs. The definition of a concept being proved is one of the most useful tools for verifying a mathematical statement around that concept. When, if ever, does mathematics become really intuitive in an upper level? What is the evolution of the necessity of a formal mathematical definition as a function of time in a math classroom?
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