Many Geometry textbooks instruct the teacher to distinguish between a line segment $\overline{DE}$ and its length $DE$ by writing a line over the letters and to distinguish between an angle $\angle BCD$ and its measure (in degrees) $m \angle BCD$ by including an $m$ for measure. However, I have gone the entire year not writing lines over my letters or extra $m$'s in front of my angle symbols, and not one student has asked me to clarify. It's the job of the teacher to sort through needed and needless distinctions and present the material predigested to the students. In our class, I have found this segment vs. length distinction is not needed.
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If you've been through a high school Algebra 2 course, you may remember polynomial long division. It's as much fun a regular long division, but now with algebra! Usually, after a few days of practicing long division, the teacher will introduce a faster way of completing polynomial division problems, called synthetic division. Essentially, this method allows the student to write less by keeping track of the variables in columns. Often students are resistant to this new method, simply because it is new. However, after practice, they usually concede that it is easier. Unfortunately, synthetic division only works when dividing by a monic, linear polynomial such as $x3$. Once we start dividing by quadratic polynomials such as $x^23x+5$ most textbooks do not provide any such shortcut... Today, I offered my students two options. Students could practice dividing by quadratic polynomials using long division, or, if they felt confident in that skill, they could opt to learn a synthetic division technique for quadratic polynomials that was not in the book. Four students had strong enough math skill and had sufficient interest to take me up on the offer to see if the synthetic shortcut for linear polynomials could be extended to quadratics. Below is the same problem worked out using both long division (left) and synthetic division (right). If you're familiar with synthetic division from high school, you might try to figure out how the procedure works. Overheard at the end of class, "This synthetic division is so much easier!"
Along the landscape of English vernacular, there are some words that we say because those are the words we say. Have you ever encountered an abode that was not “humble” or a “petard” that was doing something other than hoisting?
Mathematics is often criticized for having more than its fair share of these disconnected words. “Conjugate” and “discriminant” come to mind. The only connection we have to these words is from their usage in a very limited context. Why not change our terminology to allow our experience to influence our understanding of the mathematics? As a starting point, I nominate “imposter solution” to replace the outdated “extraneous solution”. There is no title that universally gives its bearer so much honor as “teacher.” Before I had posed a problem, before I had offered an explanation, before I could be judged on merit, in unison, the students rose from their seats and greeted me, “Good morning, teacher.”
Over the next week and a half, I had the privilege of teaching and, as always happens, learning from these students. In Ghana, as in the United States, I found students to be aware of their own learning preferences, in need of guidance to work through the difficulties of an unfamiliar problem, and eager to share something of themselves. After my initial warm greeting I began delivering my first lesson. After some time, students requested, “Please sir, exercises,” the British influence evident in their vocabulary. I reassured the class that I would explain the material again. It was not until the next day, when students repeated their request, that I understood what they were asking for. I was about to begin a new topic when the student from the day before raised his hand, “Please sir, exercises.” At that point another student chimed in, “And say that you will mark our work,” (another British choice of words). The students were selfaware enough to recognize that they needed practice and teacher feedback to feel comfortable with the topic. Perhaps my only defense for understanding the students’ request so slowly is their anglicized vocabulary! Later on in the week, the arrangement of several local landmarks along a single road provided the perfect setting for a math problem. Traveling inland from the coast, a driver will encounter, in order, the towns of Essiam, Ajumako, and Techiman. The school I worked at is located between Essiam and Ajumako. Teacher David begins his day at school. He walks 2 kilometers to Ajumako and then another 1 kilometer to Techiman. At the end of the day, he walks 4 kilometers from Techiman to Essiam. How far is he from school when he finishes walking? Students applied the skills they knew to provide me with a variety of possible solutions.  I added the numbers to get 7 km.  I subtracted the numbers. The answer is 1 km.  8 km. I multiplied. Of course each arithmetical calculation was performed correctly, calculation being the skill they had been taught and drilled on. However, mathematics only shows its full strength when students can discriminate among various situations and apply a fitting calculation to each. This skill of matching calculation to scenario was the skill my students lacked. So much so, that even an familiarity with local geography was unable to make up for this deficit. I was glad to be there to help them work through the difficulties of deciding which calculation best fits the scenario given. In another class I was assigned two students for an hourlong reading group. Our task was to take turns reading aloud from a children’s chapter book. Of my two students, one would read only after strenuous coaxing and the other refused to read at all. Given the students’ reticence, I wondered how we would manage to make it through the hour. After a few more minutes, the silent student spoke, “You read. Ask questions.” So that is what I did. Sentencebysentence I read the book and quizzed them on vocabulary. Students were eager to learn new words and volunteer translations in their language. In this manner we passed the hour quite happily. Where I thought the students were being difficult, they merely had limited experience reading English and needed someone to meet them at that level. In observing my Ghanaian students advocate for their own learning style, work through a challenging new problem, and eagerly share their own expertise, I saw much of what I see in my American students. While empowering students in all these areas is the daily struggle and joy of a teacher, I needed a different context to bring this into focus. Certainly the differences between my students in the two countries are many: material possessions, educational background, career aspirations, etc. Looking back on the experience, though, perhaps the biggest difference between teaching in Ghana and teaching in the United States is only a change of vocabulary. All my students call me Teacher David, but before I left Ghana all my students had changed that to Kyrekyrenyi David. A student was recently out of school for several days while traveling. She had the foresight to check with me about upcoming assignments that she would miss and to bring her Algebra textbook with her. However, she forgot to bring paper with her, and apparently was traveling somewhere with no paper. Being a resourceful student, she completed her work on a paper towel and submitted that to me. Santeri Viinamäki [CC BYSA 4.0 (http://creativecommons.org/licenses/bysa/4.0)], via Wikimedia Commons

About Me
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome. Archives
June 2017
