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 As a starting point, I nominate “imposter solution” to replace the outdated “extraneous solution”.
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 As a starting point, I nominate “imposter solution” to replace the outdated “extraneous solution”.
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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
“All of our lives are enhanced if [a student] get[s] that good education.”  Governor Tom Wolf Pennsylvania Public School State Funding It may not shock you to learn that the quality of public education varies widely across the commonwealth. Availability of funding is certainly one of many factors determining the quality of education a child receives in their school district. In 2016, Governor Wolf signed Act 35, which provides a new formula for the allocation of state resources to public schools, to be phased in slowly. The act itself describes a complex formula with over fifteen variables for each district. In an attempt to make the law more accessible to the people it most directly affects  the students  I have created three hypothetical funding formulas and applied them to ten representative districts from across the state. These formulas are much simpler than the formula provided in Act 35 and are intended to give some insight into the considerations lawmakers valued when crafting the law. The three formulas given are only a sample of the wide variety of formulas on might create.
When teaching these formulas to a high school audience, there are several ways to have students engage with the material. Broadly, students can investigate funding from a mathematical or from a civics viewpoint. Some starting points for student engagement are Mathematics
Civics
Resources
Sources
I had just introduced the class to the AA Similarity Postulate  the idea that if triangles have two pairs of congruent angles, then one is a scaledup version of the other, i.e. they are similar. I posed my class the following question. Is there a corresponding AAA Similarity Theorem for quadrilaterals? I was impressed by how clearly my students argued two different views on the question.
Who do you believe?

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
September 2017
