Solution to Questions 1 and 2:

Students' Use of Rule-based Reasoning in the Context of Calorimetry and Thermal Phenomena* Ngoc-Loan P. Nguyen, Warren M. Christensen, and David E. Meltzer

Iowa State University*Supported in part by NSF Grant DUE-#9981140

Introduction

Calorimetry, the measurement of heat, is a basic part of most introductory science courses in physics and chemistry. The goal of this project is to investigate student learning difficulties in calorimetry, create focused worksheets and tutorials to improve learning, and test them in classes. The initial step involved identifying a baseline of student understanding using diagnostic questions and interviews. Initial testing was carried out in summer 2002. Preliminary versions of a worksheet were developed and tested, and further testing was carried out in spring 2003 in Physics 222, a second-semester calculus-based introductory physics course. One group (“Control”) received standard instruction, while a second group (“Intervention”) received instruction using the worksheet. Examination of the performance of the two groups indicates that further innovative development is needed to make improvements over standard instruction.

Follow-up Investigation 

Additional testing was done in a second-semester calculus-based physics course during spring 2003. The same type of pretest was administered to all recitation sections about a week after lecture instruction on calorimetry, on the day that homework involving calorimetry questions was due. Seven recitation sections had been randomly chosen using a random number generator to form the intervention group; after the pretest, they received instruction with our calorimetry worksheet during the normal recitation period. All other sections received standard instruction.

Here are two representative explanations offered to justify an incorrect answer to Pretest Question 2:

Student 1:

“A has a higher specific heat so [it] takes less time to reach the same temperature.”

Student 2:

“Since the specific heat of A is two times that of liquid B, and everything else is held constant (the initial temperature and mass and the heating rate), the liquid of solution A will heat up two times as fast as liquid B.”

A notable feature of the responses to Question 2 is that one category of erroneous explanations from Question 1 almost completely disappeared, despite the similarities between the problems. The proportion of students claiming that temperature changes would be equal because energy transfers were equal (the largest category in Question 1) fell from 9% to 1%, suggesting that application of this “rule-of-thumb” depends on the context in which the problem is presented.

Summer Results for Pretest Questions 1 and 2:

Approximately half of the students were able to give correct answers with correct explanations. Follow-up interviews were done with nine of the students, and were consistent with these results.

LSH = lower specific heat GSH = greater specific heat

Table 1: Pretest (Summer session 2002)

Question 1 N = 32 Question 2 N=32

Metal in Water Heating two liquids

Correct Correct

TLSH > TGSH 73% Slope B > A 66% With correct explanation 53% With correct explanation 56%

Incorrect Incorrect TLSH = TGSH 22% Slope A > B 22% TLSH < TGSH 6% Other 12%

First Pretest Explanations

Over half (55%) of the students gave correct explanations based on the definition of specific heat, its inverse relationship to changes in temperature, or explicit algebra. About 30% of the students provided brief explanations suggesting alternative conceptions based on several simple “rules-of-thumb.” The percentage of responses corresponding to each of these explanations is shown in Table 3.

Nearly half (49%) of all students who assigned the larger temperature change to the wrong material argued that the rate of change in temperature was directly proportional to the specific heat. This explanation indicated that these students were not merely confused about which specific heat was in fact larger, nor had they randomly selected the wrong answer.

Table 3 Pretest: Question 1

Explanations

Metal in liquid All students N=311

Correct (Tlower specific heat > Tgreater specific heat) With correct explanation 55%

Incorrect

(Tlower specific heat = Tgreater specific heat)

temperature changes are equal since energy transfers are equal 9%

temperature changes are equal since system goes to equilibrium 6%

temperature changes are equal because masses are the same 3%

temperature changes are equal because initial temperature is higher (or lower)

1%

Other 2%

(Tlower specific heat < Tgreater specific heat)

specific heat directly proportional to rate of temperature change 7%

temperature change is larger because initial temperature is higher (or lower)

3%

Other 5%

Results from Pretest Question 2:

Second Pretest Explanations

Exactly half of all students were able to give a correct answer with correct explanation for the second pretest question. The only common incorrect explanation offered was that the rate of temperature change should be directly proportional to the specific heat. This explanation accounted for 81% of all incorrect responses.

Table 4 Pretest: Question 2

Heating two liquids Overall N=311

Correct Slope B > A 70%

With correct slope ratio 56% With correct explanation 50%

Incorrect Slope A > B 28%

Other 2%

Of the students who said that the temperature changes of the two materials would be equal, 70% justified this conclusion either by the fact that the system was moving toward equilibrium, or with the argument that equality in energy transfers implied equality of temperature changes. For example, here is one student’s argument:

"Same. The system will reach an equilibrium since the copper will gain the heat that the water gives up they will both change the same amount of C."

A different justification was offered by this student:

“The temperature change of the copper and the water will be the same. Any heat lost by the copper will be gained by the water, or any heat gained by the copper will lost from the water. So T of both are the same.”

Here a student argues (incorrectly) that the temperature change is dependent on initial conditions:

“More than, since it has to go from a lower initial temperature to a higher system temperature. Q=mcT”

Students’ written explanations suggested that most of their answers were linked to certain specific rules (either correct or incorrect) which allowed rapid responses without requiring extensive reasoning.

Intervention vs. Control Group Pretest Analysis

A point of interest concerning the pretest results can be seen in the following table.

The substantial discrepancy between the intervention group and control group remains unexplained. Since the pretest was given before the intervention occurred and the groups had been randomly selected, no significant difference in pretest scores should have been anticipated.

Table 5 Pretest: Question 1

Metal in Liquid Intervention N=129

Control N=182

Correct TLSH > TGSH 55% 68%

With correct explanation 46% 59% Incorrect

TLSH = TGSH 24% 20% TLSH < TGSH 21% 13%

Table 6 Pretest: Question 2

Heating two liquids Intervention N=129

Control N=182

Correct Slope B > A 67% 72%

With correct slope ratio 50% 60% With correct explanation 43% 55%

Incorrect Slope A > B 29% 27%

Other 4% 1%

Posttest:

The posttest was administered as a two-part free-response test question on a midterm, whereas the pretest had been administered as a recitation quiz.

Suppose that a mass of aluminum (Al) is heated to a high initial temperature. At time t0 it is placed in

an insulated container of water that is at a lower initial temperature. The mass of the aluminum is the same as the mass of the water. (Note: The specific heat of water is higher than that of aluminum.)

a) On the grid below, graph the temperature as a function of time of the aluminum and water separately.

Make sure to clearly label your graphs. (Note: t1 represents a time shortly after the initial time, before equilibrium is reached.)

b) Please explain the reasoning that you used in drawing your graph.

Temperature

Time

Water

Al

Insulation

t1

t0

ASSUME THAT THE SPECIFIC HEAT OF WATER IS FOUR TIMES THAT OF ALUMINUM

The pretest and posttest questions are not completely equivalent; the posttest question requires a calculation of the slope ratio, whereas Pretest Question 1 does not. If one overlooks this difference and considers the gains in score from pretest to posttest, the intervention group seems to show a better performance than the control group. However, performance on other questions (see below) did not support this conclusion.

Table 7: Post-test Question: Free-response

Free-response question Intervention N=129

Control N=182

Correct Slope Al > Water 75% 78%

With correct slope ratio 49% 49% With correct explanation 49% 41%

Incorrect Slope of Water > Slope of Al 15% 8%

Both slopes same sign 5% 8% Other 5% 6%

Additional posttest questions: Two partitioned gases

The following multiple choice question was included both on the midterm and on the final exam.

Two kg of an ideal gas are placed in container A, and another four kg of the same ideal gas are placed in container B. The total kinetic energy of the gas molecules in A is exactly equal to the total kinetic energy of the gas molecules in B. The two containers are placed into thermal contact with each other, but insulated from all other objects in the surrounding environment. When the two containers come into contact with each other: A. The temperature of gas A will increase and the temperature of gas B will also

increase. B. The temperature of gas A will increase but the temperature of gas B will decrease. C. The temperature of gas A will decrease and the temperature of gas B will also

decrease. D. The temperature of gas A will decrease but the temperature of gas B will

increase.** (Correct response) E. The temperature of gas A will remain constant, and the temperature of gas B will also

remain constant.

Responses on Final exam: [Note: No significant difference between intervention and control groups]

Overall, 17% of all students chose options (A) or (B), both of which assert that the energy transfer between object and liquid is not equal. This indicates that even at the time of the final exam many students are confused about the concept of conservation of energy. In addition, 12% of all students selected option (D), which asserts that the energy transfer is equal and the temperature change is equal for object and liquid. As mentioned before, assertions of equal temperature change are likely to be closely associated with incorrect ideas about equilibrium, heat and temperature, and/or specific heat.

In option (C) the energy transfer is equal, but the relative magnitude of temperature change (between high- and low-specific heat substances) is reversed from the correct answer. As seen by analysis of free response questions, this type of error is likely to be closely associated with thinking that the temperature change is directly proportional to the specific heat.

Table 9 Final Results: Object Sealed in a Container

All Students N = 303

Choice A 4%

Choice B 13%

Choice C 13%

Choice D 12%

Choice E 57%

Solution to Free Response Posttest:

NOTE: These are approximate solutions. The actual temperature changes over time are exponential in nature. As long as graphs displayed the correct ratios over time the graph was scored as being correct with correct ratio.

Water

Al

InsulationTemperature

Time t1 t0

Water

Aluminum

Slope of H20 = 1/8

Slope ofAl = –1/2

Temperature

Time t1 t0

Water

Aluminum

Thermal Equilibrium

Slope ofAl = –1

Slope of H20 = 1/4

OR

Secondary Questions: Object Sealed in a Container

This was a qualitative calorimetry question fairly close in form to both the posttest and the pretest.

An object is immersed in a liquid within a sealed and insulated container. The mass of the object is the same as the mass of the liquid. The initial temperature of the object is lower than the initial temperature of the liquid, but the specific heat of the object is greater than that of the liquid. The calorimeter is left alone for several hours until it reaches equilibrium. Which of the following is true? Note: Here, “temperature change” means “number of degrees Kelvin increased or decreased.” A. The energy transfer to the object is not equal to the energy transfer away from the

liquid, and the temperature change of the object is greater than the temperature change of the liquid.

B. The energy transfer to the object is not equal to the energy transfer away from the liquid, and the temperature change of the object is less than the temperature change of the liquid.

C. The energy transfer to the object is equal to the energy transfer away from the liquid, but the temperature change of the object is greater than the temperature change of the liquid.

D. The energy transfer to the object is equal to the energy transfer away from the liquid, and the temperature change of the object is equal to the temperature change of the liquid.

E. The energy transfer to the object is equal to the energy transfer away from the liquid, but the temperature change of the object is less than the temperature change of the liquid.** (Correct Response)

t0 Time

TemperatureLiquid A Liquid B

The specific heat of A is greater than thespecific heat of B.

Heating Plate

Liquid A

Liquid B

WWCuCu TcTc

WCu

WCu T

ccT

WCuCuW TTcc

TmcQ WCu mm WCu QQ and

Notation: T absolute value of temperature change

Responses to “Two Partitioned Gases” Question on the Midterm and Final: [Note: No significant difference between intervention and control groups]

Between the time of the midterm and that of the final, students in both the intervention and control groups became less likely to believe that the temperatures of the two partitioned gases would remain the same (choice E); this seems to suggest a decrease in confusion between temperature and total internal energy. However, the numbers of students reversing the relative magnitude of temperature change (choice B) did not change in either group, implying stability in this particular misconception.

Table 8 Posttest and Final Results: Two Partitioned Gases

MIDTERM N = 303

FINAL N = 303

Choice A 1% 1%

Choice B 22% 19%

Choice C 1% 2%

Choice D 28% 46%

Choice E 48% 30%

Results from Pretest Question 1:

These results are consistent with the results from summer 2002.

Table 2 Pretest: Question 1

Metal in Liquid Overall N=311

Correct TLSH > TGSH 62%

With correct explanation 55% Incorrect

TLSH = TGSH 22% TLSH < TGSH 16%

Initial Assessment

Diagnostic questions were administered in a second-semester calculus-based course during the 2002 summer session. (See Figure below; similar questions from later course are shown.) The questions were administered after the students had a standard lecture on calorimetry but before recitation instruction; results are shown in Table 1.

Questions on Calorimetry: February 6, 2003

Name ______________________________________________ Section ____________________ 1. The specific heat of water is greater than that of copper.

A piece of copper metal is put into an insulated calorimeter which is nearly filled with water. The mass of the copper is the same as the mass of the water, but the initial temperature of the copper is higher than the initial temperature of the water. The calorimeter is left alone for several hours. During the time it takes for the system to reach equilibrium, will the temperature change (number of degrees Celsius) of the copper be more than, less than, or equal to the temperature change of the water? Please explain your answer.

2. Suppose we have two separate containers: One container holds Liquid A, and another contains Liquid

B. The mass and initial temperature of the two liquids are the same, but the specific heat of Liquid A is two times that of Liquid B. Each container is placed on a heating plate that delivers the same rate of heating in joules per second to each liquid beginning at initial time t0.

a) On the grid below, graph the temperature as a function of time for each liquid, A and B. Use a

separate line for each liquid, even if they overlap. Make sure to clearly label your lines, and use proper graphing techniques.

b) Please explain the reasoning that you used in drawing your graph. (Please continue on the back of the page.)

t0

Time

Temperature Liquid A Liquid B

The specific heat of A is greater than the specific heat of B.

Heating Plate