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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Models in Mathematics and Computer Modules

 As a future math teacher, one of the most valuable strategies in your teaching will be the use of models to explain mathematical ideas. For example, this week a UTeach faculty member had us work through an exercise on Celsius/Fahrenheit conversion. The activity is versatile; it could be adapted for science teacher by allowing students to make their own measurements of the temperatures of water at various states using both a Celsius and Fahrenheit thermometer. Students would then fill in tables of associated values and attempt to deduce a relationship. Likewise, this activity could be adapted into a statistics classroom where students use linear regression to account for measurement error in their attempt to create a best guess for the relationship between the two temperature scales.

An item that stood out for me was the incorporation of physical measurements into equations. That is, slopes contained the proper measure in the numerator and denominator as well as the constant term.

We then worked on estimating the number of M&Ms in a bag by finding the average weight for an M&M using scales. We didn't have time to finish the activity; I would have been curious to learn how closely our estimate is to the average number of M&Ms in a bag.

I think that there is a focus on model because they offer a way to represent abstract concepts in a more visual or tangible form and make these ideas easier for students to grasp. I am curious to know if the old-school geometry models are still built using a straight edge and compass. I know that when I took geometry I dreaded those exercises but now find their value.

I am surprised at the focus on models in mathematics classes though. It seems like the things we have learned are particularly relevant to students in science classrooms where experimentation and the scientific method are central themes.

There is a cool quote from Neil de Grasse Tyson that sums up this view: “Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.”

With the focus on models and understanding the applications of mathematics to the real world it is quixotic that there isn't a focus that tends in this direction. In exploring the uses of computer aids in learning we visited some of the options that are available at (Link to the PhET modules for math teachers: https://phet.colorado.edu/en/simulations/filter?subjects=math-and-statistics&type=html )

One that I chose to discuss in a Step III assignment is the module on parabolas. The last module allows students to interact with parabolas with a focus on the directrix, vertex symmetry and the focus. For me a student I remember one of the coolest facts I learned about the parabola: that the focus is where a beam that hits the parabola will end up regardless of the point on the parabola where the beam first touches. And that it is for this reason that satellite dishes have a cross-section that is a parabola.

In any case, after experimenting with their site, I learned that the interactive math modules provided by PhET simulations take this modeling concept to the next level. These simulations allow students to manipulate variables, observe changes, and experiment with mathematical ideas in a virtual environment. For example, in the "Graphing Quadratics" module, students can adjust the coefficients of a quadratic equation and immediately see how the graph changes. This type of exploration allows to visualize the equations they manipulate.

However, it seems like these modules are replacing homework as we once knew it. A term paper I wrote last year in English composition had us examine differing philosophies of education. One of the articles (published in the early 90s) remarked presciently how, even then, the course load of students had diminished markedly. I don't know how much freedom teachers get to teach but I know that I learned the most from teachers that gave really difficult assignments. I can clearly remember when returning from that level of rigor, standard things seemed much easier. I know my classmates at the time agreed that that was their experience too and I honestly wonder what the opinion of educators is toward that kind of classroom environment. Sometimes I feel like this real-world thing has gone a bit far. For example, in my college linear algebra class, we learned to balance chemical equations.

But, returning to the module on parabolas: I think that there is common ground. I think that the inquisitive classroom that we learned about from the Mathematics colloborative can benefit from such modules. I think that a good one would take time to conceive and create but that it would stimulate interest and foster the kind of passion that can make students want to learn the information.

In this light, I had toyed with the idea of a real-world class activity so students could preview Gauss' drawing of the heptagon.

For posterity, the radical is: 



So that students can draw:
Dummit, D. S., & Foote, R. M. (2004). ——— (3rd ed., pp. 602, 605). Wiley.



I think that, as a prospective teacher, integrating these interactive modules into your lessons can transform how students engage with math. Good uses of technology provide clear connections between algebraic ideas and graphical depictions. And I think that students can benefit from them.

These kinds of active learning modes can not only improve comprehension but also foster a lasting interest in the subject.

Link to the PhET modules for math teachers: https://phet.colorado.edu/en/simulations/filter?subjects=math-and-statistics&type=html

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