The first step in personalizing a problem-solving framework is to analyze your students’ written solutions to practice and exam problems to determine the problem-solving difficulties of your students. Your experience in working with students during your office hours or other discussions will also help. The Checklist of Symptoms of Student Difficulties in Figure 14.1 and the table in Chapter 3, page //, might be helpful. Note that a symptom may have more than one cause.
It is tempting, at first glance, to ascribe many student difficulties to mathematical weaknesses because students’ solutions tend to be mostly mathematics.
Considerable research indicates, however, that mathematics is only a minor element of students’ difficulties solving real problems.i For example, Figures 14.2 – 14.4 give the results of our analysis of student solutions to three final examination problems for a calculus-based physics course (two problems from the first semester and one from the second semester). These results were used to develop our problem-solving framework for that course.
The problems were selected because they required students to apply their knowledge to slightly different situations than they had encountered in their course and
textbook. They were also standard problems for a calculus-based course. We began by analyzing the students’ solutions to the Modified Atwood-machine Problem (Figure 14.2). At first, the major student errors seemed to in the mathematics. However, we adopted the following procedure:
Look first at the student’s diagrams (if any).
Then look at the first equation written down (or the first equation in each sub-part). Does the equation match the diagram? Is the equation an application of a fundamental concept or principle, or something else?
Based on this procedure, we classified over 250 student solutions to each problem for major errors (i.e., the error that prevented students from arriving at a correct solution).ii Approximately 40% of the students solved the three final examination problems correctly (or with minor errors). As expected, only about 10% of our students had major
difficulties that could be directly ascribed to mathematics, as shown in Figure 14.2 and Figure 14.4.
In contrast, approximately 50% of our students failed to solve each problem because of incorrect approaches to the problems. These students either drew no pictures or diagrams or drew incomplete diagrams. This is a symptom of the problem-solving difficulty of visualizing a physical situation (Difficulty 1 in Figure 14.1). Approximately one-half of our students could not apply their knowledge consistently (primarily Difficulties 3 and 4) . They did not appear to have integrated their knowledge into a coherent conceptual framework (Difficulty 5 in Figure 14.1).
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Part 4: Personalize a Problem Solving Framework and ProblemsFigure 14.2. Major errors in students’ solutions to Modified Atwood-machine Problem
Modified Atwood-machine Problem. In the diagram shown at right, block 1 of mass 1.5 kg and block 2 of mass 4 kg are connected by a light taut rope that passes over a frictionless pulley. Block 2 is just over the edge of the ramp inclined at an angel of 30o, and the blocks have a coefficient of sliding friction of 0.21 with the surface. At time t = 0, the system is given an initial speed of 11 m/s that starts block 2 down the ramp. Find the tension in the rope.
Major Type of Error in Students’ Solutions % (N=272)
1. Correct or minor errors Total: 29
2. Careless, many omissions, no sense of order 9
3. Incorrect Physics Approaches
a. Funknown = ma. The unknown force, in this case the tension in the rope, is mass times acceleration (ma). Usually no force diagram (or very minimal diagram) is drawn. These students appear to have no idea what the “tension” force is.
T = F = ma, where the acceleration could be g, gsin, or vo b. Funknown = Fknown. The unknown force (tension) is the sum of all
the known forces acting on the two blocks. Usually only a minimal force diagram is drawn, often without the tension force.
T = F1 + F2 - F3 . . .
c. Tension = Friction. The unknown force (tension) is the frictional force on m1 or the sum of the frictional forces on m1 and m2. Usually minimal force diagrams are drawn, often without the tension force shown. (They may be setting F = 0 in their heads).
T = m1gor T = f1 +f2 = m1g + m2gcos d. Incomplete, can’t tell.
52 6
22
11
13
4. Mathematical Difficulties
a. Can’t solve simultaneous equations b. Trigonometry or algebra errors
9 6 3
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Figure 14.3. Incorrect physics approaches in students’ solutions to the Enterprise Problem
Enterprise Problem. The “Enterprise, a ride at the Valley Fair Amusement Park, consists of a vertical wheel if radius 9 meters rotating about a fixed horizontal axis with seats for the occupants around its outer edge. The wheel rotates so that the occupants are moving at 11 m/s. The seats pivot so the occupants’ heads are towards the center of the wheel. When a 56-kg woman is upside down at the top of the wheel, what is the force she exerts on the seat?
(N=289)%
Incorrect Physics Approaches in Students’ Solutions
a. Funknown = ma. The unknown force (in this case the force of the woman on the seat, usually written as “F”) is mass times acceleration (ma). Usually no force diagram is drawn.
b. Funknown = Fknown. The unknown force (force of the woman on the seat) is the sum of all the known forces in the problem. The force of the seat on the woman is not drawn (or given a symbol). FC or mv2/r is drawn as a force vector.
c. F = 0. The sum of the forces acting on the object (woman) is zero. In this case, the unknown force (force of the seat on the woman) is shown in the force diagram. FC or mv2/r is drawn as a force vector.
d. N in wrong direction. The normal force is drawn in wrong direction, but the application of Newton’s second law is correct.
46 9
18
11
8
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Part 4: Personalize a Problem Solving Framework and ProblemsFigure 14.4. Major errors in students’ solutions to the Wave Problem
Wave Problem. A violin String 55 cm in length and placed near a loudspeaker is observed to respond strongly when the speaker is driven at a frequency of 1320 Hz exhibiting two nodes between endpoints. What is the tension in the spring if it has a mass of 0.5 grams?
Major Type of Error in Students’ Solutions % (N=217)
1. Correct (T = v2, v = f)
a. No mistakes
b. Math Mistake (units, calculator, forgot to square v)
44 35
9
2. Incorrect Approaches 49
Incorrect Approach for Determining String Density ( = m or = mg)
Incorrect Approach for Determining Wave Velocity (v).
(No useful diagram drawn for determining ) a. Used v = 2Lfn/n, with n = # of nodes.
b. Used formula for constructive interference ∆L = n
c. Used velocity of sound
d. Used other relationships (pendulum, harmonic oscillator) e. No idea how to calculate v, so quit.
Student solutions to both mechanics problems (the Modified Atwood-machine Problem and Enterprise Problem) indicated similar incorrect approaches:
Funknown = ma. The unknown force is always the mass times the acceleration (ma).
Funknown = Fknown. The unknown force is always the sum of all the known forces in the problem.
F = 0. The sum of the forces acting on an object is always zero.
These three approaches could indicate students may have misconceptions about acceleration, the nature of forces, or about the meaning of Newton’s second law (Difficulties 4 and 5 in Figure 14.1). This view is supported by our results for written conceptual questions (see Chapter 10, pages //-//). At the end of the quarter on mechanics, 57% of students still confused acceleration and velocity, 39%
did not understand the nature of forces (e.g., included the force acceleration or the force of momentum), and 75% did not understand that acceleration is caused by the sum of the forces acting on a object.
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The three incorrect approaches could also be due to the pattern-matching strategy for solving quantitative problems – memorizing the series of equations needed to solve different types of problems (Difficulties 5 and 7). That is, students use a different network of knowledge to answer conceptual questions than to answer quantitative questions. This view is supported in a study by Mel Sabello and Edward Redish.iii When administering a problem with several parts, they found that 17% of students set the sum of the forces equal to zero when solving the quantitative part of a problem, despite the fact that they drew a non-zero acceleration vector in an earlier qualitative part of the problem. This indicates a clear disconnect between their qualitative knowledge and their quantitative knowledge. The reality is probably somewhere between these two extremes, with students having misconceptions and exhibiting an over reliance on pattern matching.
Additional analysis of similar problems and open-ended conceptual problems indicated that the primary difference between the students in the algebra-based and calculus-based courses was their mathematical dexterity. About 90% of our calculus-based students had sufficient mathematical dexterity that, together with their over-reliance on pattern matching, they could arrive at a numerical answer. In our algebra-based course, students made more mathematical mistakes, some
students would give up half-way through the algebra, and more students used the novice plug-and-chug strategy – randomly plugging numbers into memorized formulas until they arrive at a numerical answer (Chapter 2, pages // - //).
Pattern-matching Novice Strategy Plug-and-chug Novice Strategy
Our initial problem-solving framework was based on the observation that the majority of our students in both the algebra-based or the calculus-based physics course exhibited the problem-solving difficulties in Figure 14.1 on the final examination. Subsequent analysis of students’ problem solutions led to the refinement and elaboration of slightly different frameworks for each course, as discussed in Chapter 5.