Introduction
Numbers and data are so pervasive in media and culture that our students need to be prepared to interpret and interact with them. Most students going through a physics classroom won't become physicists, so what should they get out of the course? The key offerings of a physics education are an intuition about the physical world and a critical eye that can make observations, ask questions and answer them. The foundational skills for this are a strong number sense and the ability to decompose complex problems into smaller ones.
The fact of the matter is students arrive in my classroom without experience in those skills and with habits that act as a direct barrier to their acquisition. Their issues with number sense are hindered by not being able to relate numbers to each other. What would happen if you asked your students, "What's the difference between 1017 and 10-17?" If they're like mine, the best answers you'll get involve a long string of nines. This question succinctly highlights two conceptual deficiencies. The first is that students are not yet fluent in techniques for relating numbers of different size. Math educators would say, they think additively, not multiplicatively. That is, when you say “difference”, they think of subtracting, rather than taking a ratio, i.e., dividing. The second is that they don’t yet understand precision, its implications and the limitations it imposes on the world. This ends up with students writing down all of the digits that show up in their calculator, all of the time. Not all of those digits are created equal, and the place values they hold are far more telling than the digits themselves.
Students also arrive to my class with very limited experience with problem decomposition and perseverance. The way word problems are frequently taught in lower grades leaves students in the habit of taking all of the given information and multiplying it together as a starting point, without considering the narrative of the problem and charting the steps that need to be followed to arrive at whatever is being asked of them. This problem compounds when the given information starts to disappear and they're left to find or produce it on their own. It's part of what makes problem solving a difficult skill for them to translate to the real world. In the work force, tasks aren’t assigned like traditional textbook problems; problems are phrased generally and need to be solved independently.
As a final observation, my students have trouble communicating about their work. They fail to attend to units in a way that preserves the meaning of the quantities involved, losing the computational narrative that runs through the problem and its solution. Without this ongoing understanding of what they're doing and why, students frequently can't articulate the justifications for the steps they're taking and the trajectory of their solution. This unit seeks to increase student understanding of number sense, problem decomposition and communication through the use of a style of estimation questions known as Fermi Problems.
Demographics
This unit is designed to be the introduction for my 11th grade General Physics and AP Physics 1 classes. The majority of students who are taking it are concurrently enrolled in Algebra 2, though a handful of the AP students will be in Precalculus. The unit doesn't depend on any specific physics content knowledge. It would, therefore, be immediately applicable as an introductory unit in a chemistry or environmental science class. The unit, with little to no modification, would also fit well into an Algebra 1, Algebra 2 or Precalculus class offered at the high school level. The complexity of questions and techniques can be adjusted depending on the age or ability level of the students.
I work at a small magnet high school where the students themselves are incredibly diverse. Despite being a magnet, it's not atypical for students to come to me below grade level in reading and/or math. This contrasted with the incredibly high abilities of some of them, means that every classroom contains the full range of abilities. In addition, students come from a variety of backgrounds that reflects the demographics of Philadelphia as a whole. There are a large number of recent immigrants and English language learners, both from Southeast Asia and Latin America. This enormous variety of backgrounds and skill levels is at the front of my mind as I design this unit.
The final demographic consideration is that I teach in the Philadelphia School District, an enormous and diverse urban district that serves the entire city. The financial state of the district is such that schools function with operating budgets so slim that basic needs go unfilled on a daily basis. Realizing this as my own reality and the potential reality for other teachers, the activities and strategies are designed to be free. No special equipment, manipulatives or physical resources are required.
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