Logic is the beginning of wisdom ... not the end

Captain Spock (Meyer et al., 1991)

---

Preconceptions about using technology to teach mathematics.

I have been using technology to teach mathematics for as long as I can remember. When I was a peer tutor in high school, I taught my classmates how to use their graphing calculators and how to use Excel. This was the majority of my math-technology exposure until sometime in the very early 2010s. I was tutoring a friend's high schooler as a favor, over the phone, and would email links to them from an up-and-coming YouTube channel called Khan Academy.

This setup worked well for my student: they could see what was happening in the videos instead of remaining frustrated while I tried to explain something over the phone. When we would talk on the phone after they watched the videos, they would ask me questions and tell me what they had learned. Khan Academy worked me out of a job with them, but they got much better results - and I was always a phone call away if they had more questions.

The inclusion of Khan Academy was my only additional tool for math technology until I returned to school in 2016 to finish my bachelor's in mathematics. This is when I discovered the Desmos Graphing Calculator.

Until I embarked upon this journey, I thought these tools were what was meant by using technology to teach mathematics. This is consistent with Dick & Hollebrands, "the mathematics classroom itself has been slow to take full advantage of using technology" (2011, p. xi).

In retrospect, my Calculus I course at Birmingham-Southern College (Summer term 2000) included a "calculator lab" where we learned how to use and program our TI-82/83 calculators in the context of the course. I didn't take Calculus II until Spring 2006 and remember being surprised that no calculators were allowed. At some point, the college math curriculum was preparing to embrace this technology but abandoned it. I suspect that it was seen as a crutch as opposed to a supplement, also consistent with Dick & Hollebrands (2011).

When good apples begin to rot. Or, bolstering the healthy elements before they manifest blight.

While reading about relational and instrumental understanding, the concept of faux amis stood out to me. I had a difficult time working on my master's thesis, and the underlying link among all my troubles was that of vocabulary and communication. My advisor and I were not speaking the same language. We were both speaking in English, but words and phrases I would use didn't mean the same thing to him, and vice versa. The language of mathematics may be universal, but the expression of it in spoken language is not.

Once I realized this, I started asking more questions, attempting to get him to explain to me what he meant when he said something. Unfortunately, this resulted in his having a significant amount of frustration with me, and eventually not listening to me at all. He made it clear to me that if it weren't for departmental politics and personal pride, he would have dropped me as a student. It was a very painful and heartbreaking process, and if I weren't stubborn, empowered, and privileged, I would have given up the pursuit. The thought of quitting never came up, but the anger of knowing that I wouldn't give up, the resulting stress (on my body, physical health, mental health, and to relationships with my children and partner), plus the continued pain inflicted by my advisor left me wondering if it was worth it. However, the dream of earning my Ph.D. in mathematics was destroyed, leaving me to wonder if I might ever achieve a Ph.D. in anything at all.

The specific example where I realized that we were not communicating is the concept of "research." In one of our weekly meetings early in the project, I told him that I had been to the library and gotten books on my thesis topic, and that I was doing research by reading them. He informed me that what I was doing was not "research" - yet he couldn't tell me what it was, and he was unable to express to me what it means when you say "research" among mathematicians. To be quite honest, I still don't know what exactly it means.

I continued to make vocabulary mistakes for the duration of my thesis project despite my best efforts to the contrary. The responses I received from my advisor whenever we miscommunicated left deep wounds and have made me more conscious about how I speak to people, especially students, when explaining a term or concept. I endeavor to instill a relational understanding to my students, the reasoning behind the mathematics, instead of instrumental understanding, rote memorization.

While reading through the chapter "Relational Understanding and Instrumental Understanding" (Skemp, 1978), the concept of faux amis and the pain I've endured prepossessed reading the remainder of the article. I have believed for some time now that a student's perception of themself impacts their ability to learn, long before I learned the term "fixed mindset" properly used to describe this phenomenon. Aside from teaching the required content, my primary goal as a tutor and a teacher is to empower my students to believe that they can "do math."

Combining fixed mindset with faux amis, I now wonder how many students we as educators are failing to reach because they have a fixed mindset, and we aren't aware that we aren't communicating effectively with them. How many students have been told by their teachers (and truly, a single one is too many) that they can't do math, when in reality the teacher and the student aren't speaking the same language? If that is what's happening, how do we overcome it? How did I overcome it?

My privileged background provided a strong foundation for me. I was always praised for my "high aptitude in math and science." I was never told that I couldn't do math, and by extension was never told I couldn't do math because of my gender. Quite the contrary: at my private, boarding high school, I enrolled in BC Calculus my senior year, and I was the only girl in the class with 7 boys.

I decided to drop the course around Thanksgiving, and my teacher, Mr. Casazza, called me into a meeting. He was concerned that I was dropping because I was in some way uncomfortable or harassed because I was the only girl, situations he took very seriously. It took a solid 30 minutes of discussion to convince him that I was dropping because I didn't have a grasp on the material, and because I already had college credit for Calculus I (I had taken it at a 4-year college the preceding summer), in order to make the class "worth my time" I needed a 4 or 5 on the AP Exam to get the Calculus II credit: a 3 wasn't good enough in my situation. He was still hesitant to sign my drop form, though, so I added that there was a programming class being offered during the spring term that was a scheduling conflict with BC Calculus and that I wanted to take it. This was finally enough for Mr. Casazza to sign my form - he may not have any girls in BC Calculus the rest of the year, but there would be a girl in JavaScript Programming. Incidentally, I still love programming.

I didn't realize back in 2000 that having girls in math and programming classes was a big deal; the boy-girl ratio at my school was 3:2, and as the previously-all-boys school had been co-educational for only 9 years at that point, I didn't find it odd that there had been just 3 girls in BC Calculus before me and only 1 in a programming class. The available population hadn't existed for very long and these particular classes were small even by my high school's standards.

I had another math teacher in high school, Mr. Stubbs, who made a lasting impact on me. Truth be told, I despised him when I was in geometry my freshman year. We, again, were not speaking the same language. Now I describe his speech as that of a pure mathematician, but at the age of 14 I thought him arrogant and rude; in retrospect, I was the one who was arrogant and rude. At the insistence of my mother, I went to daily tutoring with Mr. Stubbs and slowly began to grasp the concepts, eventually catching up with the course material. This was the first time in my life I had to study mathematics - something I hadn't yet learned to do, or that needed to be done. Having Mr. Stubbs as a teacher, someone who spoke plainly and logically (sometimes logical to a fault - he would have made an excellent Vulcan) was a crucial factor of my finishing and successfully defending my master's thesis in mathematics.

These memories and feelings - plus many more - came flooding back while reading Skemp's article. I had truly wonderful teachers, supportive family, and financial resources to keep me ahead throughout my entire education. I rarely encountered a math teacher who had an instrumental approach to the material, past elementary school at least; I have less than fond memories of Mrs. Portera's 3rd grade timed multiplication quizzes, and a special place in my heart for the classmate who would mimic my handwriting to pencil in a few extra answers for me as we peer-graded for the teacher. She was the only person - for decades - who recognized my struggle with regurgitation of abstract facts, my difficulty with speed-quizzes, and my anxiety surrounding them. How it is that this 8-year-old friend recognized these things about me, which my parents and all of my teachers failed to see, is baffling. I didn't get my learning disability diagnoses until I was 38 years old. If I hadn't been struggling, significantly, in graduate school I may never have gotten those diagnoses.

Understanding that you don't know jack.

I joined a "math mamas" Facebook group about a year ago. The women there share tips, pedagogy, and lesson plans - as well as discuss nonacademic topics. From them, I became aware of other technology, namely Geogebra, that can be used to teach mathematics. I didn't have the need or opportunity to use Geogebra until I began mathematics secondary education courses and I wonder how I ever lived without it.

I have also learned that technology is far broader than I previously considered. I transitioned, along with another intern, the precalculus courses at Huntingdon College from WebAssign to MyOpenMath, coupled with an LTI (Learning Tools Interoperability) integration within the Canvas LMS (Learning Management System).

When I was teaching precalculus 100% virtually, I used the Zoom whiteboard feature to write and draw for my students the same as I would on a classroom white/black board. I uploaded these pages to h5p and embedded the course presentations into Canvas pages for my students. I used Kahoot quizzes to liven up our 8am class. It never occurred to me that these items were mathematics teaching technology. Even the handouts I created in LaTeX (there's that programming love surfacing again) are, in some respects, technology. I've been converted to a believer in GradeScope and have been using it for about a year and a half, with tremendous success with grading transparency and student satisfaction.

I came into this genuinely perplexed as to what more was out there, besides Desmos, Geogebra, Excel, and a fancy handheld graphing calculator. I am more excited after every class to explore ways I can implement technology into my teaching and discover new apps to engage my students.

References

Meyer, N., Winter, R., Nimoy, L., Nimoy, L., & Narita, H. (1991). Star Trek Vi-- The Undiscovered Country. United States; Paramount Pictures.

CBS Studios, Inc. (2019). Captain Spock and Lieutenant Valeris discussing logic and wisdom in the captain's ready room. Iconic Trek Motivation. CBS Studios Inc. Retrieved July 4, 2022, from https://www.startrek.com/sites/default/files/styles/content_full/public/images/inline/2019-01/3a89966234620ddfca12d88f97f188d0.jpg?itok=MMHoBdII.

Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school Mathematics: Technology to support reasoning and sense making. National Council of Teachers of Mathematics.

Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15.