Transition from “I don’t know it”

Poster at 2006 AAPT Summer Meeting

Syracuse University, Syracuse, NY

July, 24, 2006

Transition from

“I don’t know it”

to “I know it”

is memorizing

Transition from

“I can’t do it”

to “I do it”

is training

Transition from

“I don’t understand it”

to “I get it”

is thinking

Constructing Learning Aids

for Teaching

Algebra Based Physics

Dr. Valentin Voroshilov, Physics Department

Boston University, valbu@bu.edu

Physics teachers experience occasional difficulties in helping students understand the reasons behind selecting formulas used in constructing a solution to a particular problem in Physics.

A set of specific exercises is a resourceful tool helping teachers develop better explanatory skills and be able to identify students’ setbacks during the actual process of solving a physics problem.

The same exercises can be used by teachers as a learning aid (teaching tool) to help students master their problem solving skills.

The presented below learning aids are applicable for solving problems in Kinematics; however, after the specialised training a teacher becomes capable of developing similar aids for any part of Physics.

All exercises are based on the fact that, when a student gets into a problematic situation, the brain starts constructing a solution from a recognition (i.e., it tries to recognize the situation first).

To help teachers develop a technique that an experienced physicist applies when developing a solution to a given problem, the set of learning aids are being constructed by teachers as an exercise (with the help of a facilitator).

1. A terminological dictionary that connects an everyday lexicon to physics terminology.

2. A classification table of typical physical models matching a situation described in a problem.

3. A correspondence table between models and physical quantities needed for qualitative description of the models.

4. A correspondence table between the models and formulas needed for quantitative description of the models.

5. A schemata of logical and procedural connections between categories involved in the analysis of the physical situation described in a problem.

Two main obstacles need to be overcome by students in order to recognize a specific physical situation described in a given problem.

The first obstacle represents a lack of ability of a student to convert written text of a problem from an everyday language to a specific physical language. For example, a situation “a car starts from rest” and a situation “a stone is dropped from a height” are two different situations for students. Students do not recognize that both of these real world situations describe the same physical situation, i.e.,” an object accelerates from rest and starts a linear motion”.

A terminological dictionary or a table of a correspondence between an everyday lexicon and a subject terminology can be used to help students perform a necessary interpretation of the problem.

1. Terminological Dictionary

Empirical Term

(Everyday Word)

Theoretical Term;

Category

Physical Quantities Describing the Category

(and the common notations)

A car; a stone; a rock;

an arrow; a plane; a rocket;

a box; a man

A body; An object

coordinates (x, y, z);

mass (m);

volume (V);

density (D)

Goes; drops; flies;

rolling; sliding;

pushed; pulled;

Moving; at a motion

displacement (S); distance (L);

velocity (v); acceleration (a);

time taken for the motion (t)

Getting at rest; moving from rest; starts; stops;

making a turn

Changing the velocity; Accelerating

displacement (S); distance (L); average velocity (v_av); initial velocity (v_i); final velocity (v_f); time taken for the motion (t);

acceleration (a)

Lies; hangs; sits; stands

At rest; does not move

the speed is 0; v = 0; no acceleration

The second obstacle is that students cannot recognize the physical model (models) needed to investigate a situation described in a given problem.

The process of recognition is always based on some classification parameters and their values.

In Kinematics, to identify the model needed to solve a problem, we deal usually with the following parameters and their values (within the framework of a physics school curriculum):

The form of the trajectory:

a) STRAIGHT LINE; b) CIRCLE.

The behavior of the speed:

a) DOES NOT VARY (constant); b) VARIES.

Four main kinematical models can be used in relation to values of these parameters:

2. Classification Table of

Typical Physical Models

The Form of a

Trajectory

The

Behavior of

a Speed

STRAIGHT LINE

CIRCLE

DOES NOT VARY

Linear motion with constant speed

Uniform circular motion

VARIES

Linear motion with constant acceleration

(Remember, it is not an exact case, but for 99 % of problems it is true!)

Circular motion with constant acceleration

(Remember, it is not exact case, but for 99 % of problems it is true!)

We cannot use this table to solve

every problem in Kinematics,

but we can use the principle!

For some problems

a combination of models

should be used.

When the two main steps are completed and the necessary models are identified; then, the set of the most important physical quantities needed to investigate the problem can be useful (Refer to Table 3. “Physical Quantities” (school Kinematics)).

Finally, a set of formulae needed to analyze the model can be constructed. The table of the correspondence between the models and the formulae can be used for this step (Refer to Table 4.“Main Equations”).

At this point, it is important to emphasize that this step – choosing the equations – is the last step of the analysis of a physics problem. After this step, mainly the mathematical calculations are left.

3. Physical Quantities

MODEL	MAIN PHYSICAL QUANTITIES
Linear motion with constant speed	Displacement (initial and final points), distance, trajectory, velocity, speed, time taken
Linear motion with constant acceleration	Displacement, distance, trajectory, time taken, initial velocity, final/terminal velocity, (initial and final instant), acceleration.
Uniform circular motion	Displacement, distance, velocity, time, angle, angular displacement, number of revolutions, frequency, angular velocity, period, centripetal acceleration, the radius of the circle.
Uniformly accelerated circular motion	Displacement (initial point, final point), distance, velocity, time, angle, angular displacement, angular velocity, angular acceleration, centripetal acceleration, tangential acceleration, the radius of the circle.
Mixed model	Concepts of parent models; intervals of motion, average velocity, average speed; average acceleration.

4. Main Equations

Model

Formulas

Linear motion with constant speed

v = s/t; s = x – x_o,

a = 0

Linear motion with constant acceleration

v = v_o + at; s = x – x_o

s = v_ot + at²/2

Uniform circular

motion

w = j/t; wT = 2p

n = N/t; v = wR

n = 1/T; a_c = v²/R; j = s/R

Uniformly

accelerated circular

motion

j = S/R; w = w_o + εt;

j = w_ot + ε t²/2

v = wR; a_c = v²/R;

a_t = ε R

In addition, the exercise on constructing the System of Operationally Connected Categories (SOCC) is a very effective training tool to help physics teachers develop specific skills an experienced physicist has developed.

We can think of the SOCC as a knowledge system, as schemata, as a mental model or as a concept map. The main idea of the SOCC is based on the fact that information becomes knowledge as a result of the connections between different parts of that information.

For example, without sensible connections, all of the words that our memory retains are just names of objects, which we can point at: “desk”, “chair”. A generalization does not exist without sensible connections. There is no “furniture”.

When constructing a solution to a physical problem, we have to expose and employ connections among terms, categories, names of objects, names of processes, physical idealizations, symbols, etc.

The SOCC helps us visualize all of the important connections. The SOCC can be drawn for both: a single given problem or for a specific part of Physics.

The SOCC itself is just a web. Any vertex of the web represents an important physical category. Any edge connecting to vertexes represents an important direct connection between the two corresponded categories. In Physics, a direct connection usually means that there is an equation involving both of the categories (or a verbal statement, like a definition).

The elementary part of a SOCC looks like this:

Here represents the term “an angular speed”; represents the term “a radius of a circle”; the line between them represents the equation v = wR.

When all the important terms and equations are being represented, we have a SOCC build up.

5. SOCC

Each vertex answers the question: “What kind of physical quantity can be used to describe the analysed physical processes?”

Each line answers the question “Is there a direct connection between two given physical quantities?” We may, also, ask: “Does the value of this quantity affect the value of that one”?

Whenever necessary, a brief explanation of main vertexes and links can be provided. For example:

2. A point mass is a body which size is not essential for the given problem.

…

5. A circular motion is a motion when all points of a body lie on circles which centres belong to one straight line (axis), and the planes of the circles are mutually parallel.

…

7. Distance is the length of a trajectory. To find a distance we can put a thread along a trajectory, then to stretch this thread into a strait line and to apply it to a ruler and read the number.

33. An angular speed is a physical quantity describing the rate of changing of angular displacement during the time. The angular speed is defined by the formula w = j/t (lines 16 - 33 and 31 – 33). Also the angular speed is connected to the linear speed by the formula v = w·R, which is represented by a line 19 - 33.

The main question the SOCC helps to answer is “What is connected to what?”

The actual connections, i.e. equations, become natural consequences of the analysis of the physical processes involved in the analysed problem.

References:

1. Erich Mazur, “Peer Instruction”, Prentice Hall, Inc., 1997).

2. George Polya, “How to Solve It: a new aspect of mathematical method”, (Princeton University Press, Expanded Princeton Science Library Edition, 2004).

3. Mark Vondracek, “Improving Student Comprehension by Thinking about a Topic in Multiple Ways”, (The Physics Teacher, November 2005, Volume 43, Number 8).

4. Arnold Arons, “A Guide to Introductory Physics Teaching”, (Wiley, New York, 1990).

5. Donald Scarl, “How to Solve Problems: For Success in Freshman Physics, Engineering and Beyond”, (Dosoris Press, 6^th Edition, 2003).

6. Carl Wieman, Katherine Perkins, “Transforming Physics Education”, Physics Today, November 2005.

7. David Hestenes, “Modeling Methodology for Physics Teachers”, Proceedings of the International Conference on Undergraduate Physics Education (College Park, August 1996) (located at www.modeling.asu.edu).

8. Richard P. Feynman, “Six Easy Pieces”, Helix Books, 1994).

9. F.K.L. Chit Hlaing (F. K. Lehman), “Cultural models (and Schemata) and Generative Knowledge Domains: How are they related?”, Paper for the panel on Cultural Models and Schema Theory, American Anthropological Association Annual Meeting, October, 2000, San Francisco (located at real.anthropology.ac.uk/AAA2000SF).

10. Valentin Voroshilov, “Universal Algorithm for Solving Problems in School Physics”, (in the book “Problems in Applied Mathematics and Mechanics”, Perm, Russia, 1998).

11. Valentin Voroshilov, “Quantitative Indicators for the Learning Difficulty of Physics Problems”, (in the book “Problems of Education, Scientific and Technical Development and Economy of Ural Region”, Berezniki, Russia, 1996).

12. Valentin Voroshilov, “Application of the System of Operationally-Interconnect Categories for Diagnosing the Level of Student Understanding of Physics”, (in the book “Artificial Intelligence in Education”, part 1. - Kazan, Russia, 1996).