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Goodman & Etkina
Pedagogical approach of the 9th grade physics course
Teaching methods in Algebra-based Physics First emphasize active student
engagement, group problem solving, and cooperative learning [14, 15]. Concept
construction involves a variety of methods and settings so that students with a range of
learning styles can be successful. Methods include observations of physical phenomena,
discourse of the patterns and explanations, problem-solving, laboratory experiments and
frequent formative assessments. Settings include large and small group work in both the
classroom and the laboratory. Formative assessment is a critical element and is
performed in the large group settings via the Classroom Performance System (CPS) in
addition to the more traditional individual pencil and paper assessments.
Problem solving is a core activity of the course, and something students enjoy.
Once students acquire some basic understanding of a new idea, they apply that
understanding to solving problems. The teacher shows what type of questions can be
answered using this new concept and works through several examples. The teacher then
poses similar problems to the class and the class works through them as a large group,
with the teacher as moderator and recorder. This lasts for about ten to fifteen minutes,
with the students solving a few variations of the new type of problem. Students’
contributions to all the steps keep the class fast moving and all the students involved.
Especially important, due to the weakness of student algebra skills, is that every step of
the solution is discussed explicitly, for example dividing both sides of an equation by the
same variable in order to derive an expression for a physical quantity.
The students then spend the remaining fifteen to twenty minutes of the class
solving problems of increasing difficulty in groups of three or four sitting at round tables.
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They quickly reach problems that require them to consult together. The first part of the
class allowed them to construct some basic understanding. The latter part of the class
uses what was learned and builds on it to get the students into their zone of proximal
development [16, 17], a zone where they are challenged enough to stay interested but in
which they can be most successful with the assistance of others. The teacher’s primary
role during this process is to work with the students just enough to keep them challenged:
she/he offers just enough help and advice to keep them moving forward, but not so much
as to eliminate the challenge. This process requires making the judgment as to how much
help to provide, and when which is part of the art of being an effective teacher.
A crucial element is the creation of an open atmosphere where students are free to
ask questions without being embarrassed and all the steps to solving problems are
discussed explicitly. In that way, no student is singled out for not understanding even the
most basic steps. The class is fast paced, fun and non-critical.
Assessment
As the classes are held in an informal atmosphere of discussions between the
teacher and students and students with each other, ungraded formative assessments [18]
occur constantly during the class when a teacher poses a question and students respond.
Weekly, there are graded quizzes (either full period or shorter) which are at about the
same level as the homework. No grades are given for homework, class participation,
projects, etc. Only demonstrated ability to answer questions or solve problems in class is
graded.
Summative assessments (chapter tests, midterms and finals) all have the same
form as the AP exam: half multiple choice conceptual questions and half free response
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multi-step problems, often taken from prior AP exams. The identical summative
assessment is given to all the students in the course on the same day, regardless of their
teacher. The common assessments and rubrics are developed through a joint effort of all
of the teachers who currently teach the course.
Common school-wide assessments and dates enable and encourage students to
study together in groups, with or without a teacher, to advance their skill and
understanding: they help engender a culture in which physics is a common topic amongst
all students. This is the opposite of the effect of tracking, in that it bonds students rather
than separating them.
After school support
This atmosphere is supported by the availability of after-school opportunities to
study. In fact, a critical element of the course is that teachers take turns staying after
school for up to two hours twice a week throughout the year. During this time, students
are encouraged to work in groups, with or without the teacher, to advance their
understanding. Students who have taken the course in previous years often work with
those groups as informal tutors. This opportunity to study after school is a critical
element in reducing the amount of tracking in the school. Students who are not satisfied
with their grades on any assessment are also welcome to come after school to study with
others and, when ready, take a new version of that same assessment. This helps generate
a positive atmosphere in the school as students see that their teachers care about whether
they learn.
Curricular Materials
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Most physics textbooks are either conceptual, such as Hewitt’s Conceptual
Physics [19], or require the use of both algebra and trigonometry, such as Physics by
Giancoli [20]. It is assumed that students are either too young to employ mathematics
effectively or, are older and would know both algebra and trigonometry. The students in
this course fall into neither of those categories.
Since no textbooks support our approach, we had to choose whether to add the
algebra content to a conceptual text or subtract the trigonometry content from an
algebra/trigonometry based text. We decided to subtract rather than add, so we use
Giancoli’s text and omit topics requiring trigonometry. Aside from the third chapter, with
vector operations in two dimensions and projectile motion, the vast majority of the
problems in the book do not require trigonometry.
However, three difficulties remained. First, the algebra-based problems in the
text do not progress clearly from simple to difficult. Second, there are few complex
interesting problems that solely use algebra. Since an important aspect of this course is
the social constructivist group work, students need challenging problems. Third, the
book reading level is high; it is difficult for students to understand a topic by reading the
book.
We addressed the first difficulty by supplementing the book with sets of problems
of progressively increasing difficulty that students solve prior to attempting those in the
text. (See Appendix 1.)
The second difficulty required supplementing the book with a set of more
challenging algebra-based problems that students can solve working in groups and that
are consistent with the style of an AP free response problems (See Appendix 2).
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The third difficulty was addressed by putting in writing explanations that teachers
provide in class summarizing discussions on a particular topic. These written
explanations now serve as both supplementary readings for the students as well as a guide
for teachers who are new to teaching the course.
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Appendix 1
Introductory Problems
1. A 0.40 kg toy car moves at constant acceleration of 2.3 m/s2. Determine the net
force exerted on the car that is responsible for that acceleration.
2. If a net horizontal force of 175 N is exerted on a bike whose mass is 43 kg what
acceleration is produced?
3. A car travels at constant acceleration of 2.2 m/s2. Find the mass of the car if a 5.3
kN net force is exerted to produce this acceleration.
4. A wooden block is pulled at a constant acceleration of 1.4 m/s2. Find the net force
exerted on the block if its mass is 0.6 kg.
5. A 95 N net force is exerted on an ice block with a mass of 24 kg. Find the
acceleration of the block if it moves on a smooth horizontal surface.
6. A net force of 345 N accelerates a boy on a sled at 3.2 m/s2. What is the combined
mass of the sled and boy?
7. What average net force is required to stop an 8500 kg truck in 10 s if it’s initially
traveling at 26 m/s?
8. What average net force is required to accelerate a 9.5 g bullet from rest to 650 m/s
over a distance of 0.85 m along the barrel of a rifle?
9. A 7.5 kg cannon ball leaves a cannon with a speed of 185 m/s. Find the average
net force exerted on the ball if the cannon muzzle is 3.6 m long.
10. What average force is needed to stop a 15000 kg train in 5 s if it’s traveling at 20
m/s?
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Appendix 2
Free Response Problem
M
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1
1
1
m
The object, “M”, is being pulled across the table by a string which goes over the pulley
and is connected to a second object, “m”. The coefficient of kinetic friction (µk) between
the table and M is 0.25 and M is sliding across the table. Show all your work.
1. Draw a free body diagram, indicating all forces in approximate scale, for M. Show
the direction of acceleration and velocity next to the free body diagram.
2. Draw a free body diagram, indicating all forces in approximate scale, for m. Show the
direction of acceleration and velocity next to the free body diagram.
3. Use Newton’s Second Law to write an equation of motion for M in terms of T, g, m
and µk.
4. Use Newton’s Second Law to write an equation of motion for m in terms of T, g, and
m2 .
5. Solve that system of equations for a, the acceleration.
6. Solve for the value of the acceleration assuming that M = 20 kg and m = 40 kg.
7. Determine the tension in the string.
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