3D Printer and more Technology in the Classroom (plus the last Step II lesson of the semester)

 The week before Thanksgiving was pretty busy as it usually does as the semester draws to a close with all of its final presentations and final exams. I think that the first thing I want to talk about this week is my final lesson for Step II since I will be writing 5E lesson plans to close out the semester. Unlike the two previous lessons, my time management was poor. I made the strategic decision in the middle of the lesson to forego the evaluation I had prepared since our statistical examination that closes Step II will look at one evaluation in particular. The lesson was productive for my own growth. Upon reflection, I realized why my time management was so bad. On the surface, it looked like I had just allowed students too much time in the exploration section of the lesson. Looking more closely, I realized that I should have stopped the exploration section. Many students gave answers that indicated that either the lesson section's instructions were unclear or that the students ...
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Technology in the Classroom (Part II) and a Growth Mindset

 This week we looked at some of the real world applications that are available to on the TI INspire calculator. We looked at 4 and each group got to do one of them. My group got the pendulum lab.

We got a motion sensor like this:
https://www.amazon.com/Texas-Instruments-PWB-1L1-Motion/dp/B00LGNUGUE/ref=asc_df_B00LGNUGUE/?tag=hyprod-20&linkCode=df0&hvadid=693276021741&hvpos=&hvnetw=g&hvrand=11727828872040461247&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9011420&hvtargid=pla-570086422041&psc=1&mcid=671d50a168613a4a99f920a50c762d16&tag=hyprod-20&linkCode=df0&hvadid=693276021741&hvpos=&hvnetw=g&hvrand=11727828872040461247&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9011420&hvtargid=pla-570086422041&psc=1


The device measures the distance and speed of an object relative to its position. It plugs into the TI calculator and the data is plotted. Overall the lab was pretty disappointing because the sensor is not very accurate. We got something that resembled an EKG rather than a sinusoidal function. However, the lab was interesting in another way. It got me thinking about how the simplified form that we were supposed to observe on our calculator is derived. We were supposed to get our sensor to present something that looked like the graphs below. What I learned from the exercise is that physicists use the approximations, $$ \sin \theta = \theta$$ and $$ \cos \theta = 1 $$. In the last step, they simplify the distance that the sensor measures into its sinusoidal form by taking Taylor approximation to remove the square root from the distance function. In any case, the results are striking. Without any approximation (other than avoiding elliptic integration in $$ \frac{d^2 \theta}{dt^2} = -\frac{g}{l} \sin \theta \text{ by solving } \frac{d^2 \theta}{dt^2} = -\frac{g}{l} \theta \text{ instead to get: } \theta(t) = \theta_0 \cos\left( t \sqrt{\frac{g}{l}} \right) \, \mathbf{(1)} $$


I got these results:


Actual value of distance with time using the correct distance function and approximation (1):


using the worksheet's approximation (the distance function is replaced by a Taylor approximation and cosine of the theta of t function is assumed to be 1):


They are almost an exact match; their vertical deviations: $$ \int_0^{10} |d_1(t) - d_2(t)| \, dt \approx 0.0097 $$.

$$ \tiny \text{Conversely, } x \in \mathbb{Z}, \; \lambda \in \mathbb{R} \setminus \mathbb{Q} \implies \overline{\lbrace \lambda x \rbrace} = [0,1] \text{ and } 2\pi \sqrt{\frac{l}{g}} \diagup 2\pi \mathbb{Z} \text{ is finite.} $$



 On a final note with this experiment, I found that the elliptic integral and method 1 are numerically so close that a difference is outside the precision of a computer. The approximation using the second method is detectable but small.

I don't know how much application tools like this will have in a mathematics classroom but it was neat to see the kind of portable gizmos in production. Later this week we are completing an assignment that will examine the type of learning software that TI has. 

On a more pedagogical note, in Step II, we watched a TED Talk by a psychologist, Carol Dweck. Her discussion of fixed and growth mindsets coincided with a random math post I saw and there I found a cool connection between the ideas she discussed a mathematical pedagogy. I made me contrast two approaches to what seemed like a pretty easy math question on Reddit. I found that there was a strong connection between what Dweck's Growth Mindset and the Redditor's question asking why 6 times 8 is less than 7 times 7. In reading through the comments, I saw several connections with the algebra tiles that we have used this semester. I also saw connections with less tactile means that were equally compelling. I think that I saw a connection with the idea that these algebra tiles are a tool in a student's tool kit just like saying squares are no less than zero. I think that there are some connection worth exploring between the visual connections that students make with other equally compelling ways.
The ideas of growth, algebra tiles, some yarn, a simplified version of Dido's problem and a trivial inequality came to mind; its seems like there is an obvious 5E lesson here and I am glad that I watched the TED Talk to see the connection.

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