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URBAN ACADEMY INTRO
l APPLYING LOGIC IN MATH AND SCIENCE
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URBAN ACADEMY INTRO
l APPLYING LOGIC IN MATH AND SCIENCE
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Below we take a closer look at teaching and learning in two classes: trigonometry and a science course called “Chemical Puzzles.”
When Urban Academy math teacher Terry Weber says in his course description that students will solve problems outside the classroom, he really means it. “In trig, they go outside eight times,” he says. “We ask them to figure out how far is it across the East River, and how tall the smokestacks are on the other side. Or they go to the Flatiron Building, which is a triangular solid, and estimate its volume. They figure out the distance between Manhattan and Staten Island, knowing the height of the Statue of Liberty—if they don’t know its height they have to figure it out.”
But using familiar reference points to gain practical skills isn’t all that Weber is after. “My philosophy is that people have to understand why they do stuff, not just how to do it,” he says.
Weber wants his students to be able to estimate whether their answer is in the ballpark, within the realm of possibility. So, when they estimate the height of the street lamp using trigonometric functions, for example, if their estimate doesn’t make sense after they do a scale drawing, they have to figure out why.
Urban Academy Course Catalog
This class will cover identities, proofs, triangles, equations, patterns, and graphing. You will plan solutions to practical problems and will use a hands-on approach to problems outside the classroom. You will work mainly in small groups and will have regular homework and periodic quizzes and tests.
Prerequisite: Teacher recommendation
What Weber aims to show, he says, “is that in math there is room for students to put their own opinions into it.” For whatever problem he throws their way, he wants his students thinking, “Could I do this another way?”
CLASSROOM NOTES| 10.15.02
On an autumn morning, Lashawn and Andy are computing the width of the East River as it skirts mid-Manhattan, using the distance between a couple of garbage cans and a lamppost they can see on the other side. Creating a simple sketch to scale on graph paper, they explain their procedure in writing:
Lashawn measured 55 feet by using ruler walking from one garbage can to another. Andy measured 80 degrees from garbage can to top of lamppost across the river by using a protractor horizontally.
After working out the equations and re-checking their figures, Lashawn and Andy find that their solution for x, the distance across the river, equals 312 feet. “Uncommon answer,” they note skeptically and add:
The only difficulty we had was maintaining a precise measurement of the angles. In addition, the distance between the two garbage cans was small, compared to the large distance across the East River. This added to our inaccuracy.
By the end of class, Lashawn and Andy are onto an alternative way to solve the problem,
using a tangent method.
PROCEDURES AND PROFICIENCIES
1.0 “Practical procedure”
Note: Anyone who reads your practical procedure should be able to understand exactly what you did and why you did it.
Page 1:
Page 2:
Page 3: (Graph paper)
Page 4:
2.0 Sample pre-requisites and proficiencies
Students in Urban math classes routinely complete a step-by-step series of “pre-requisite” assignments and short proficiency tests that ask them to demonstrate and apply what they are learning. Below are two examples. Click here for sample student answers to these examples.
1) 8 students live in the houses A B C D E F G H. Some are
connected by phone lines (direct lines between the oval houses).
It’s good when a teacher allows you to fail and accepts something even though it’s not right, but you’re on the road to something right. There’s a lot of pressure on high schoolers to get it right or else not hand it in, because you think it’s stupid. Lots of teachers will give you every step leading up to the answer. But you’re not really learning anything, you’re just reciting it.—Vance
Terri Gross’s enthusiasm for discovery and that of her students find common ground in her semester-long chemistry course. “Using puzzles works because they really want to know the answers,” explains Gross. “It’s compelling, so by the end of the semester students get in the habit of asking questions. You can put something down in front of them—a liquid solution, for example—and you don’t have to say anything. They automatically start asking questions.”
She starts the semester with “Mystery Powders” (which range from baking soda to Plaster of Paris) and three liquids (water, dilute vinegar, and dilute iodine) which students group, test, heat as they work to identify them. Answers in hand, they then create a logic puzzle.
Next Gross presents students with eight clear, colorless liquid solutions, along with a list of names and chemical formulas. “First they group the solutions as to how they react—do they change colors? Turn cloudy?” she says. “Then they do Internet research about the chemicals, learning the chemical formulae, elements, atomic structure.”
Urban Academy Course Catalog
Labels have fallen off several bottles each containing different solutions of chemicals. How do you find out which solution is in which bottle?
You’re given a test tube containing a solution. How do you find out what it is? How do you find out if it’s a solution of a single chemical or a mixture of several chemicals?
These are some of the puzzles students will try to solve by designing and carrying out appropriate experiments. Each student will be required to record observations and data collected in the laboratory notebooks. Class time will also be spent on analyzing experimental results. Students will be expected to become familiar with the chemists’ shorthand of writing formulas and reactions.
Students will be evaluated according to the following criteria:
One major project will be to identify a series of “unknowns.”
In this unit’s final puzzle, students make up their own experimental question about a rusting nail in an agar solution with potassium ferrous cyanide. “A good question has to be something they can test within the constraints of the classroom,” Terri explains. Students have learned from the succession of puzzles that it could be quantifiable (like the third puzzle) or qualitative (like the first and the fourth). “It has to have a narrow focus, but be broad enough to be interesting, and I prefer it to have some importance,” she says.
“Sometimes you can’t tell until the very end if it was a good question,” Gross adds. One foolproof indicator: “If it raises a lot of other questions, it was a good question.”
Students in “Chemical Puzzles” wrestle with weekly assignments and experiments that build on each other and that ask students to make new connections in what they’ve been learning and to think like a scientist. The final exam has two parts: a written section and a practical application. Gross gives students, in advance of the exam, an outline of what to expect. Below is a sample homework assignment and final exam explanation.
PART ONE: Solving a “chemical” puzzle
PART TWO: Prussian (or Turnbull’s) Blue
We will be generally using the process described in the attached reading.
By Monday, you and your partner need to have chosen a question, developed an experimental design and written that design out carefully. One design per group is fine.
Chemical Puzzles:
This is the explanation of your final examination — Identification of an Unknown.
You will be given two unknown solutions. You will get 30 ml of each and no more! If you spill it or use it all up, you will not get any more! Each will be a compound from this list:
(NOTE: If your scheme requires a chemical that is not on this list, you must adjust your scheme. We will not have access to any other chemicals during the class period! I suggest that you adjust the scheme before class!)
You will then choose one of the unknowns and determine the molarity (moles per liter) of the solution that you have been given.
You will need to work efficiently in order to complete this assignment on time. There will be no extensions. The write-up will be due on the last day of class. At that time, you will take the written part of the final examination.
The Write-Up
1. Title: Identification of an unknown chemical.
The Written Exam
1. The structure of the atom. Given an element symbol, you should be able to tell me how many protons, neutrons and electrons are in a single atom.
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APPLYING LOGIC IN
He tied two students together with a string, and that was the ten-minute puzzle of the day, to think about with logic. All we had to do was reverse hands. But he wouldn’t let me get out, not even after ten minutes. I was so mad, but when I finally got it, I would have felt like such an idiot if I had let him tell me the answer. — Mika
HOMEWORK AND FINAL EXAM
Homework Assignment #12