Encouraging Meaningful Quantitative Problem Solving - Journal of

In the Classroom

Encouraging Meaningful Quantitative Problem Solving

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Jeff Cohen Fenway High School, 174 Ipswich St., Boston, MA 02215 Meghan Kennedy-Justice Brookline High School, 115 Greenough St., Brookline, MA 02445 Sunny Pai and Carmen Torres Boston Arts Academy, 174 Ipswich St., Boston, MA 02215 Rick Toomey Department of Chemistry and Physics, Northwest Missouri State University, Maryville, MO 64468 Ed DePierro and Fred Garafalo* Massachusetts College of Pharmacy and Health Sciences, Boston, MA 02115; *[email protected]

Quantitative problem-solving is a challenging aspect of any physical science course. Traditionally, students have been encouraged to pursue various techniques in an effort to provide structure to this task. While these techniques may help to generate numerical answers, they can become exercises in symbol manipulation that leave the student without a clear picture of the physical situation associated with the problem. In recent years, various authors have stressed the need for approaches that emphasize qualitative understanding and better equation writing on the part of students (1–5). Several of us have been working to address these issues in the freshman chemistry curriculum at the Massachusetts College of Pharmacy and Health Sciences (MCPHS). We use an approach suggested by Arons (6 ), that stresses interpretation of ratios, coupling it with the interactive problem-solving strategies of Lochhead and Whimbey (7). During the 1997– 1998 academic year, the corresponding author spent part of a sabbatical at Fenway High School in Boston, introducing these methods in an effort to help high school science students improve their problem-solving skills. This paper begins with a discussion of traditional problem-solving methods, and then explores the ideas of Arons (6 ) and how they can be used to introduce quantitative problem solving. It next describes the evolution of our approach over the past eight years at MCPHS and our experiences introducing the methods at the high school level, and concludes with suggestions for further work.

Box 1 Problem Set 1: How would you solve the following problems? 1. A car travels 270 miles in 5 hours. How many hours will it take the car to go 150 miles? (Assume it travels at constant velocity.) 2. Convert 3000 meters into kilometers. 3. How many grams of solid sodium chloride are needed to prepare 2.5 L of a 0.5 M solution of sodium chloride? The molar mass of NaCl is 58.5 g. 4. The density of a particular material is 2.3 g/cm3. What is the volume of 800 g of this material? 5. Let x = number of meters and y = number of kilometers. An equation using the symbols x, y, and 1000, which expresses the relationship between number of meters and number of kilometers is (a) 1000x = y (b) 1000y = x (c) x + y = 1000 (d) xy =1000 ________________ Solutions to Problem Set 1 Although each of these problems can be solved in several ways, here is a set of solutions to the first four problems that represents a range of traditional approaches often suggested to students.

270 mi = 150 mi xh 5h The student is told to set up a proportion, cross multiply, and solve for x: 270x = 150(5); x = 2.78 h 1. Ratio and proportion:

2. Unitary conversion (or unit-factor method):

3000 m

How Do You Solve Quantitative Problems? In 1990, the corresponding author was introduced to interactive learning at a Chautauqua Short Course taught by Jack Lochhead (8). A valuable lesson learned from that experience is that active participation encourages reflection. It is in this spirit that the reader is now invited to participate by solving the problems in Problem Set 1, listed in Box 1. These and the problems in Box 2 are intended to elucidate ideas and to involve the reader in the process of reflecting upon his or her own problem-solving strategies. Some of these problems have been discussed in recent talks on teaching and learning (9, 10) and used in interactive workshops on quantitative problem solving (11, 12).

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1 km = 3 km 1000 m

The student is told that since 1 km = 1000 m, 1 km/1000 m is equal to 1, and so multiplying 3000 m by this factor does not change the distance. It merely changes the unit. 3. Dimensional (unit) analysis or factor-label method:

0.5 mol × 2.5 L × 58.5 g = 73.1 g L mol The student is told to set up an expression in which the units cancel to give the desired unit. 4. Plug and chug calculation:

d = m/V; 2.3 g/cm3 = 800 g/x cm3; x = 348 cm3 The student is told to plug known quantities into an equation and solve for the unknown quantity.

Journal of Chemical Education • Vol. 77 No. 9 September 2000 • JChemEd.chem.wisc.edu

In the Classroom

While any of these approaches is fine in the hands of an individual who has a clear picture of the physical situation associated with the problem, each of them has the potential to become an exercise in mere symbol manipulation when used by novice problem solvers. In addition, each has unique problems associated with its use. For example, few students can express why it is all right to cross multiply in problem 1. Interpreting ratios as being equal to one (used in problem 2) is mathematically incorrect in many instances and encourages the concept of equivalence, which the chemical community in this country has abandoned in freshman chemistry (1). Dimensional analysis, ratio and proportion, and plug-in type calculations may lead to correct numerical answers without a clear picture of the associated physical situation and to the inability to write correct equations (2, 3). Lack of skill in equation writing leads most students to choose the incorrect answer a in problem 5, which uses x and y as labels instead of variables. The correct answer is b. Is there any feature common to all these problems that we can draw on in an effort to guide students toward becoming better problem solvers? The reader is invited to solve Problem Set 2, in Box 2a, before we consider this question. Arons (6 ) suggests that students struggle with quantitative problem-solving because they are unable to articulate the meaning of division and various ratios. With this in mind, let’s look at the answers to problems 6–11 (Box 2b). Box 2a Problem Set 2: How would you solve the following problems? 6. Using only the following set of circles, show how you would represent the operation, 6 ÷ 2. O O O O O O Explain in words what you are doing, without using the word division. 7. Based on the definition of division, draw a picture that justifies the answer to the problem, 2 ÷ 1/2 = 4. 8. Consider a group of 40 apples and a group of 5 apples. Evaluate the following ratio and state what your answer means: 40 apples/5 apples. 9. Consider the ratio, $40/5 gallons. When we divide 40 by 5 we get 8. What does 8 represent in this case? Now consider the ratio 5 gallons/$40. 5/40 = 0.125. What does 0.125 represent in this case? 10. A certain liquid costs $5 per gallon. What does the following expression represent: $40/($5/1 gallon)? When you perform the division, what is the label on the number that you obtain? Explain why. How could you get the same result by a different method (i. e. do not divide 40 by 5)? 11. Consider the reaction: MnO4 + 5Fe2+ + 8H+ → Mn2+ + 4H2O + 5Fe3+. Does the following analysis correctly explain how many moles of MnO4 would react with 3 moles of Fe2+? 1 mole MnO4 = 5 moles Fe2+. Dividing both sides by 5 moles Fe2+, we get 1mole MnO4/5 moles Fe2+ = 1. To find the number of moles of MnO4 that reacts with 3 moles of Fe2+ we write:

1 mole MnO4

2+

× 3 moles Fe = 0.6 moles MnO4 (equation 1) 2+ 5 moles Fe Since 1 mole MnO4/5 mole Fe2+ = 1, we see that 0.6 mole MnO4 is equal to 3 moles of Fe2+.

Box 2b Solutions to Problem Set 2 6. Separate the circles this way: OO / OO / OO The word quotient comes from Latin and means “how many times?” In this case, the quotient three refers to the fact that we can remove two circles a total of three times from six circles. Many freshman chemistry students draw only one line, dividing the group of six circles in half. In this case two would represent the number of groups, and three would represent the number of circles in each group. 7. When we divide a number by a fraction, the rule is to invert the fraction and multiply the number by it: 2 ÷ 1/2 = 2 × (2/1) = 4 We can see why this works when we know the meaning of division. The problem is asking how many times can 1/2 be removed from 2? While most students can produce the answer symbolically by inverting and then multiplying, no student in the past several years of freshman chemistry class has been able to give a qualitative description in which two identical objects are cut in half, allowing half an object to be removed from the group a total of four times. 8. We divide to evaluate a ratio: 40 apples/5 apples = 8. The number 8 indicates that a group of 40 apples is 8 times larger than a group of 5 apples. To reinforce this idea, a proportion can be helpful:

40 apples 8 apples = 5 apples 1 apple We can say that for each apple in the denominator group, there are 8 apples in the numerator group. Students often incorrectly indicate that the label on 8 is apples. 9. Eight represents the fact that $8 will purchase 1 gallon. 0.125 represents the number of gallons that can be purchased for $1. Many college freshmen struggle when they are asked to express the meaning of the number obtained when the quantities in a ratio are divided. Often their response involves repeating the original information, or saying that they do not know. It helps students to see the proportions

$40 = $8 ; 5 gal 1 gal

5 gal 0.125 gal = $40 $1

10. Here, the idea that division entails repeated subtraction is useful. Each time we give the store owner $5, we get 1 gallon. With $40, we can do this a total of eight times and receive 8 gallons. To evaluate the set of labels $/($/gal), we do the same thing that is done when dividing by a fraction, we invert the denominator and multiply by the numerator: $ x (gal/$) = gal. The unit “$” cancels and we are left with the unit “gallon”. Here is another way to represent the number of gallons that can be purchased for $40: Invert the ratio $5/1 gal: 1 gal/$5 = 0.2 gal/$1. This is the number of gallons that can be purchased for one dollar. For $40 we can get 40 times this amount. Keeping track of the units we write: $40 x (0.2 gal/$1) = 8 gallons. Once again, we have found that it is initially extremely challenging for freshman chemistry students to express these ideas in words. 11. When writing an equation, the label on the left side of the equal sign should be the same as that on the right side of the equal sign. So 1 mole of MnO4 is not equal to 5 moles of Fe2+ . The ratio in equation 1 should be interpreted as 0.2 mole MnO4/1 mole Fe2+, and not equal to 1. In this way the label on each side of equation 1 works out to be moles of MnO4, and we avoid the conclusion that 3 moles of Fe2+ equals 0.6 moles of MnO4.

JChemEd.chem.wisc.edu • Vol. 77 No. 9 September 2000 • Journal of Chemical Education

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Revisiting Problem Set 1 Since ratios are fundamental to so many aspects of chemistry, it is worthwhile taking time to help students learn to articulate their meaning. Revisiting problems 1–5, we can see that articulating the meaning of ratios can serve as the foundation for expressing solutions to all of these problems. Problem 1 Arons recommends asking students to interpret a ratio and its inverse (6 ). In this case we have: 270 mi/5 h = 54 mi/1 h; 5 h/270 mi = 0.0185 h/1 mi The second ratio gives the time it takes to go one mile. If the car travels at constant velocity, it will take 150 times as long to travel 150 miles. Once the student can express this fact, then she or he can write a statement that formally keeps track of the units: 0.0185 h × 150 mi = 2.78 h 1 mi 5 h × 150 mi = 2.78 h and can270 mi celling the units does not guarantee that the student can express the reasoning. Setting up the correct proportion, as was done earlier, also does not guarantee that the student can express the reasoning. A second way to solve this problem would require the student to interpret the expression 150 mi/(54 mi/1 h). Here the student should be encouraged to state that each time the car travels 54 miles, one hour will elapse. For a distance of 150 miles, 2.78 h will elapse if the car travels at constant velocity. After the student can express this in words, she or he can be guided to the formal statement in which the units work out to be hours. A third approach to this problem involves interpreting the ratio 150 mi/270 mi and, using this along with the time to go 270 miles, to reason toward the answer.

Note that merely writing

Problem 2 There is 0.001 kilometer in one meter: 1 km/1000 m = 0.001 km/1 m In 3000 meters we will have 3000 times this many kilometers: 3000 m × (0.001 km/1 m) = 3 km

1 km × 3000 m = 3 km , as 1000 m long as 1 km/1000 m is interpreted as the number of km in 1 m, this reinforces the qualitative meaning. A second way to solve this problem would have the student interpret the expression 3000 m/(1000 m/1 km) in a fashion similar to that used in problem 1. Problem 3 Break the problem down into a series of simpler ones and state the meaning of the ratios: Step 1. 0.5 M means that one liter of solution must contain 0.5 moles of NaCl. Therefore, 2.5 liters at this concentration will contain 2.5 times as many moles: 0.5 moles of NaCl × 2.5 L of solution = 1.25 moles of NaCl 1 L of solution Even if one just writes

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One can allude to the fact that the units on the left work out to be those on the right, but the reason why one multiplies should be stressed explicitly. Step 2. 58.5 g of NaCl has 1 mole of NaCl units. We need 1.25 times this much NaCl to get 1.25 moles: 58.5 g × 1.25 moles = 73.12 g 1 mole Problem 4 800 g/(2.3 g/1 cm3) = 348 cm3. Each time we remove a 2.3-g clump of this material, we get 1 cm3. From 800 g we can do this 348 times, so we have 348 cm3. Another way: Invert 2.3 g/1 cm3: 1 cm3/2.3 g = 0.435 cm3/1 g. This is the volume of 1 gram of the material. The volume of 800 g will be 800 times this amount. Keeping track of the units, we write: (0.435 cm3/1 g) × 800 g = 348 cm3. Problem 5 The answer is b: The ratio x/y is fixed at 1000/1. Writing x/y = 1000, we can rearrange this equation and write 1000 y = x. In words, the equation states that 1000 times the number of kilometers is equal to the number of meters. Discussion Note that there is more than one way to solve each of these problems using ratios. The important point is that when the student can express the meaning of ratios in words, qualitative understanding of the physical situation is enhanced. This is an important first step in becoming a successful problem solver. Hayes indicates that lucid explanations from teachers and instruction in expert problem-solving techniques have proved to be of little value in helping students to become better problem solvers (13). Unless students are required to extract explanations and interpretations of the underlying phenomena in their own words, they memorize formulas or patterns for performing calculations (14 ). In addition, the difficulties associated with proportional reasoning do not diminish with additional years of science training, indicating that they must be explicitly confronted (15). Any of the other techniques that were initially used in Problem Set 1 could be applied consistently to all problems in that set. All five could be solved by the methods of ratio and proportion, dimensional analysis, or even unitary conversion (if we embrace the concept of equivalence). Nevertheless, an approach in which ratios are created and explicitly interpreted, as in problems 8–10, offers advantages over the other methods. The ability to interpret ratios reduces the chances that students will develop misconceptions about the material. Such misconceptions inhibit learning, appear regardless of the student’s level of reasoning, and persist even in students who do well in courses that stress quantitative problem solving (16, 17). When students are encouraged to create and interpret ratios, this also sets the stage for the correct setup and interpretation of equations. Equations similar to that in problem 5 could be written for problems 1–4 as well, but note that creating such equations requires the correct interpretation of ratios. For example, in problem 1, if we let x equal number of hours and y equal number of miles and use the ratio 0.0185 h/1 mi, we can write an equation to calculate the time it would take to go any number of miles under these conditions: 0.0185y = x. Research indicates that the difficulty students have with set-


In the Classroom

ting up and interpreting equations stems from a difficulty with translating words into equations and vice versa, rather than a difficulty with simple algebraic manipulations (18). The method avoids the tedium of ratio and proportion when a problem has more than one step, such as problem 3. It is also less mechanical than the dimensional analysis method, since it focuses on the relationship between physical quantities,1 not the relationship between units (2). The method of unitary conversion, while harmless when used in converting units, encourages students to embrace the idea of equivalence when applied in other situations (1). The student has to know when to interpret a ratio as being equal to one, and when not. For concrete reasoners this adds a complicating factor to an already difficult task. In problem 11, when the ratio is interpreted as being equal to one, this harks back to the idea of chemical equivalence. When the ratio is interpreted as 0.2 mole MnO4/1 mol Fe2+, this sets the stage for writing a general equation to express the number of moles of MnO4 that react with any given number of moles of Fe2+: 0.2(x moles Fe2+) = y moles MnO4 . Interpreting the ratio as the amount of numerator for one unit of denominator ensures that the units are the same on both sides of the equal sign. Interpreting the ratio as being equal to one makes the jump to a general equation more difficult. This suggests that ratios like 1 km/1000 m in problem 2 should always be interpreted as 0.001 km/1 m, and should not be set equal to 1. Ratios equal to 1 can be created if we say something like this:

moles of electrons gained by 1 mole MnO4 moles of electrons lost by 5 moles Fe2+

= 5 =1 5

the distance represented by the symbols 1 km =1 the distance represented by the symbols 1000 m Problem Solving in Freshman Chemistry The corresponding author has been working in the area of curriculum development at MCPHS since the mid-1980s (19–21). Initial work focused on integrating freshman chemistry with freshman biology in an effort to elucidate connections between the sciences. Instructor-generated handouts were used to supplement the chemistry text and drive classroom discussion. Progressively more handouts were added each year until the text was eliminated in academic year 1990–1991. Even though the integrated curriculum was discontinued in 1992, efforts throughout the 1990s have centered on using educational research findings and classroom feedback to guide further curriculum development in freshman chemistry. Topics are introduced through experimental observations or operational definitions, the presentations are coupled to the development of specific thinking skills, including reasoning using ratios, and an effort is made to cycle back to ideas throughout the year, developing them in progressively richer contexts. Traditional lecturing is downplayed and each class session usually contains one or more think-aloud problem-solving sessions, in which students work in pairs or small groups (7). In recent years, class size has been in the vicinity of 110 to 140 students. The instructor (Garafalo or DePierro) and a

learning facilitator (DePierro or Toomey) circulate among the students during the problem-solving session, listening to their approaches to solutions. Immediately after the 3–10-minute session, the instructor provides rapid feedback to the entire class in the form of a question-and-answer period (22). Students also experience active learning during the instructor’s daily office hour, which is conducted in a classroom and usually attended by 10–15 students. Delaying the presentation of homework answer keys encourages student interaction and attendance at these help sessions. A typical session contains Socratic dialogs between the instructor and individual students, small discussion groups comprising two or three students, or an open discussion driven by the students themselves. The most valuable source of feedback on how students are progressing in their understanding of material and how the instructors can improve future presentations continues to be that from interactive learning sessions in the classroom, office hours, and the laboratory. Other sources are surveys and performance on examinations. One of the major challenges in curriculum development at MCPHS has been in the area of quantitative problem solving. Initial efforts in the integrated curriculum of the late 1980s centered on postponing quantitative treatments one term until weaker students had the chance to complete a basic mathematics course. A modified version of dimensional analysis was then used in chemistry, but this overall approach proved to be of little help. Since the 1991–1992 academic year, the corresponding author has stressed the importance of interpreting ratios, introducing the topic at the beginning of the first semester. Problems similar to those in problem set 2 above are used in think-aloud sessions during the first week of class to stress fundamental ideas about division and ratios. Using familiar concepts (dollars, gallons, miles, hours, etc.) in the early part of the course helps students learn to interpret ratios correctly and encourages them to develop their own algorithms for problem solving. This approach is continued as the students encounter the many ratios that appear throughout the year. Our progress in using this approach has only been possible through the feedback obtained primarily in interactive learning sessions and from examinations. Our experience indicates that most students come to freshman chemistry schooled in the method of ratio and proportion or dimensional analysis. Many are unable to interpret the meaning of ratios, and they fall victim to the difficulties with problem solving that are described earlier in this paper. Over the last few years, consistently poor results on the first test in semester one have led to a steady decrease in the number of concepts that are introduced in the first few weeks of the curriculum. Some have been eliminated entirely, while others have been moved to more appropriate places. For example, the law of constant composition is used to give students practice in working with mass ratios, but chemical formulas and mole ratios are not studied until later in the term. An effort is made in think-aloud sessions to strike a balance between providing problems that allow students to demonstrate that they have command of an idea and providing problems that place students in a position of conflict as a step toward deepening their understanding. Surveys conducted from 1992 to 1995 indicated that about 20% of the students (including many who were doing well in the course) felt that the think-aloud problems used in class were too difficult. This


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has resulted in the use of a greater number of simple problems placed earlier in the term. For example, several problems using ratios lead to the concept of percent composition, and several examples require the student to convert percent composition data into ratios before limiting reagent problems are considered. The fact that some students are unfamiliar with active learning strategies and initially resist their use adds to the difficulties in the first few weeks of semester one. In recent years, students have been introduced to the difficulties and potential benefits associated with active learning through a letter sent to them in the summer before their freshman year and an orientation session in the week prior to the beginning of classes. Throughout the year, students are encouraged to reinterpret problems in their own words, to recognize that fully expressing a thought allows one to study it, and to realize that analyzing the path to an answer is more important than getting a correct answer (7, 8). In academic year 1998–1999, the handouts on quantitative problem solving were expanded into a formal unit on basic mathematics skills, which was referred to throughout the semester. The unit and classroom presentations encourage students to link new information to what they already know, through progressively more complicated questions followed by feedback. Stress is placed on connecting qualitative descriptions of phenomena with quantitative symbolic representations. Socratic lines of questioning are used to guide students through conflict resolution. The presentation anticipates student misconceptions that have been presented in the educational literature or unearthed in interactive sessions. For example, introducing the idea that division entails repeated subtractions led one student to question why 2 ÷ 4 should not be equal to 2 instead of 1/2. Discussion led to reinforcing the idea of finding the number of times the denominator could be subtracted from the numerator, and adding this question to a problem set. In spite of these efforts, many students rely on previously learned methods, particularly dimensional analysis or ratio and proportion, when solving problems. While students are free to choose any method they want, feedback during in-class question-and-answer periods and help sessions encourages them to proceed in a problem first by interpreting ratios. Students who favor dimensional analysis are encouraged to use that method to check their setup, and those who favor ratio and proportion are encouraged to see the algebraic and logical connections between that method and explicitly interpreting ratios. In this way, students can still develop qualitative insights while retaining some connection with a familiar approach. It has not been possible to measure the effects of these interventions on student performance in a controlled environment. Several factors have changed from year to year. These include class size, average SAT scores, and the number and sequence of topics presented. During this period we also changed from a trimester system with no lab in trimester one to a semester system with lab in both semesters, and different learning facilitators of varying ability were involved. Nevertheless, extensive feedback from students during the past eight years has enabled instructors to respond more effectively to their difficulties in quantitative problem solving, primarily by slowing the presentation so that students have time to develop skill in interpreting and using ratios. 1170

Engaging High School Students in the Method During the 1997–1998 academic year, the corresponding author spent part of a sabbatical at Fenway High School in Boston, in an effort to help science students improve their problem-solving skills. Fenway is a member of the Coalition of Essential Schools based at Brown University and has a significant African-American and Hispanic population. Over the course of several months, the corresponding author rotated through three classes of 11th and 10th graders, each with about 20 students, teaching them with the help of the high school authors. These classes were interspersed among other activities and other classes conducted by the high school teachers. The major focus was constructing and using ratios to solve problems, but other topics included review of manipulation of fractions and scientific notation. The general approach taken was similar to that used in the college freshman course: an interactive classroom served as an environment in which students and instructors could receive immediate feedback. Past experiences with college freshmen (including a former Fenway student), prior classroom observations at Fenway, and a diagnostic examination helped determine the appropriate level of engagement. Questions were presented to the students from a unit on math skills, which stressed the meaning of division and interpretation of ratios. The unit was created from handouts used at MCPHS in the first weeks of freshman chemistry, but with more elaboration. These questions were presented to the students either one at a time or as a group of three or four questions. The students worked alone or in groups trying to come up with answers. Two to five minutes were allowed per question, depending on the difficulty. During this time, the instructors circulated among the students, listening to their discussions and guiding them with the help of Socratic dialogs through the nontrivial tasks of interpreting and creating various symbolic representations in their own words. The students were encouraged to express their thoughts out loud so that they could reflect upon them and not to worry about whether they were right or wrong (7, 8). An instructor then engaged the entire class in discussion, during which consensus was reached on the answers. Research suggests that this type of approach is very powerful, even if the student does not come up with the correct answer. Engaging students in the struggle to find a solution makes the solution presentation more meaningful, provided it comes immediately after they have tried the problem (8, 22). The students were constantly reminded that the clumsiness and frustration that initially accompany efforts to express things in one’s own words are not an indication of an inability to learn, but rather an indication that one is engaging in real learning. At times, the instructor only coordinated the discussion of a problem, leaving the students to reach agreement among themselves on the final answer. The Fenway teachers used and created their own activities based on the MCPHS material, which helped them gain familiarity with this approach and provide multiple ways of engaging the students. These included problem sets, a quiz game patterned after the Jeopardy television show, studentgraded mock quizzes in which the students exchange papers and grade them using specific grading criteria, and handson measurement activities. A set of evaluative criteria were


In the Classroom

developed for assessing student performance. Some written exams were given, primarily in the classroom of C. Torres. Since Fenway evaluates students primarily by portfolio, performance on written exams was used at the discretion of the Fenway instructors in assigning final grades during the term that this activity took place. The visiting instructor did not participate in the process of assigning grades. The classes at Fenway provided an interesting contrast to those at MCPHS. While many college freshmen find it difficult to engage in interactive learning, most Fenway students were comfortable with this process, since it is encouraged throughout their curriculum. A number of students learned to articulate the meaning of ratios with less difficulty than many college freshmen. While a few chose to work alone, they would often participate during group discussions. The students’ willingness to interact enabled instructors to clarify points and uncover misconceptions that might have gone undetected. The in-class activities created by the Fenway teachers added variety to the presentations and usually helped students to stay engaged. However, the diversity of the students’ math backgrounds contributed occasionally to displays of frustration or boredom. Although it was frustrating to have students disengage, the interactive format provided instructors with valuable feedback that enabled them to modify the pace and focus of presentations and identify students in need of extensive remediation. For example, one conversation led instructors to discover that a new student thought the fraction 1/2 was represented as .12 in decimal format. Our work indicates that both the ability to describe the relationship between physical quantities and the ability to write such relationships in symbolic representations demand explicit attention in problem solving using ratios. After a couple of weeks of work with the first class, students were able to articulate the meaning of ratios and use them to reason verbally through simple problems. Up to this point, writing symbolic representations had been downplayed in order to stress correct verbal descriptions. As a result, the students were less skilled at producing correctly labeled symbolic representations of their reasoning on paper. For example, students who were capable of using the fact that $18 purchases 15 gallons of gas could reason toward the cost of one gallon, and then use this ratio to predict the cost of seven gallons of gas. However, many of these students expressed this fact symbolically with an equation in which the units were incorrect: $1.20 × 7 gal = $8.40. In retrospect, the visiting instructor underemphasized written symbolic representations. This is consistent with the environment at MCPHS, where large class size leads to more verbal than written interaction during help sessions, and the multiple-choice exam format can discourage attention to detail in written setups. In the second and third classes, an effort was made to coordinate the verbal descriptions with correctly labeled written symbolic representations. The solutions to problems 6–10 and the second set of solutions to problems 1–5 presented earlier are based on this approach. While this seems to have merit, care must be taken to stress the description of the physical situation, so that students do not slip into meaningless symbol manipulation. This was particularly evident with some students who were having difficulty verbalizing the physical situation associated with problems. In such cases it was clear that the students wanted to know what they

should do with the symbols in order to obtain a correct numerical answer, and that they were neglecting the practice of first creating a picture of what was physically happening before using symbols to represent it. The different skill levels in math aggravated this situation. During discussions, students with stronger math backgrounds were sometimes eager to express themselves in terms of symbols and their manipulation, whether they could articulate a clear picture of the physical situation or not. (For example, a student who could use the ratio 1 s/1000 ms in a calculation struggled to articulate the fact that the ratio indicates that there is one one-thousandth of a second in one millisecond.) This may have encouraged those with weaker backgrounds to focus on symbolic representations rather than physical descriptions. Our work also indicates that different approaches to quantitative problem solving are not easily interchangeable, at least not at first. This became evident as the visiting instructor was finishing with the first class. At that time, the permanent instructor planned to introduce dimensional analysis. When we attempted to make the transition to this method, the students did not easily embrace it. The shift from interpreting ratios to manipulating units was a new challenge for them. We gave them a written quiz shortly thereafter and their performance on problems that could have been solved by interpreting ratios or dimensional analysis was very poor. This was in spite of the fact that they had performed well a week before on an activity that required them verbally to construct and interpret ratios in order to solve problems related to preparing snacks. They had not had enough time to master the dimensional analysis approach, but their exposure to it was apparently enough to draw them away from the method of interpreting ratios, which we had been using for several weeks prior to this. These experiences lead to an interesting dilemma. While interpreting ratios leads to qualitative understanding and the development of formal reasoning (6, 17), dimensional analysis has been advocated by some for students who are concrete reasoners (23, 24 ), and it may have merit when students are involved in timed tests in which speed in obtaining a result is important. Conditions did not lend themselves to controlled experiments (e.g., the number of consecutive days of classroom visits varied with each group owing to other required activities, and some student attendance was erratic), but our experiences suggest that a combination of approaches may work to help students develop problem-solving skills. In the latter two classes that the visiting instructor attended, the extra attention paid to writing symbolic representations included demonstrating how units cancel, but this was always done after the problem had been set up by interpreting ratios, not as a primary means of obtaining a correct answer. Mock quizzes indicated that these students were learning the material in this way, but time constraints prevented extensive development of this approach. It is likely that multiple-step procedures such as converting hours into milliseconds can be introduced after students learn to interpret the ratios used in the individual steps, then pointing out how the units cancel in a correctly constructed multiple-step problem. Once again, the instructor must guard against the student’s slipping into meaningless symbol manipulation. In more complex problems, where a student can set up several different ratios and proceed in several possible ways,


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dimensional analysis can be used to verify that individual steps make sense. For students who face timed tests or the need to quickly gain some familiarity with manipulating quantities in the physical sciences, dimensional analysis can be of help, but it does not foster conceptual understanding (1, 2, 5, 6, 14 ). The fact that technical vocabulary can be a deterrent to creating correct symbolic representations, even for students who can create such representations using familiar concepts, suggests that helping students to achieve qualitative understanding of the many ratios used in the physical sciences is essential (25). Our experience also indicates that when physical pictures are stressed, students must be shown when they can let go of the picture and trust symbol manipulation when that is faster. For example, once a student understands that division of a number by a fraction means repeated subtraction of that fraction from the number, she or he can then mechanically perform the operation of inverting the fraction and multiplying this by the number. We found that several students had to be taken explicitly through a problem in which it was not easy to make a picture (e.g., 1/7 ÷ 2/3) and made to realize that they could trust that the manipulation was consistent with producing a correct answer. Concluding Remarks How will an instructor recognize “meaningful quantitative problem solving” when she or he sees it? In introductory physical science courses, if students are capable of engaging in meaningful quantitative problem solving, this suggests that they can maintain some sort of picture of the relevant physical or analogical objects (dollar bills, gallons of milk), properties of objects (mass, volume), or general concepts (distance, time interval) and the numerical interrelationships among these as they create, interpret, and manipulate various ratios (engage in proportional reasoning) in an effort to draw conclusions about the situation described in the problem.2 Clearly, in order to know whether a student is capable of meaningful problem solving, the instructor must probe for understanding of physical concepts, ratios, and symbol manipulation as well as evaluate setups and numerical answers. Our experience indicates that high school students and college freshmen struggle with proportional reasoning, but that tenth graders are capable of learning to articulate the meaning of various ratios and to express relationships between physical quantities in symbolic representations using ratios. The diversity of math backgrounds in the students that come to Fenway and the need for instructors to experiment with presentations contributed to the process’s taking longer than it might have. The former point argues for students being exposed to qualitative descriptions of mathematical operations, particularly with fractions, as early as possible. Interactive activities that help students to interpret and use ratios can and should be used in small and repeated doses throughout their science instruction, starting at least as early as the ninth grade (and probably earlier). Small homework assignments can be a powerful method of gauging the progress of the class, even for instruction that is primarily project-based. Just one or two problems at a time can be assigned to determine how well students are progressing in learning the necessary skills. This can be coupled to interactive problem-solving sessions 1172

and mock quizzes in class so that students get substantial feedback without inundating the instructor with grading. Interactive sessions can be interspersed among projects or formal classroom presentations to reinforce ideas. This may be particularly important for curricula that introduce concepts through projects or themes. While it is valuable for students to learn science within some practical context, they also need the opportunity to internalize the key linguistic elements that will allow them to master various types of reasoning (14 ) (e.g., working comfortably with ratios). Perkins indicates that transfer of thinking skills from one type of problem to another can be achieved when general reasoning principles (such as interpreting ratios) are taught with self-monitoring practices and potential applications in varied contexts (28). Clearly, the physical sciences provide a rich source of varied contexts for learning proportional reasoning. Bitner has found that proportional reasoning, one of several formal operational reasoning modes tested for in the Group Assessment of Logical Thinking, is a significant predictor of critical thinking ability (as measured by the Watson–Glaser Critical Thinking Appraisal) and grades assigned by high school science teachers (29). This suggests that instructors should take the time to help their students gain a deeper understanding of the quantitative problems we assign to them and that the education environment should support further work in this area. One reviewer of this manuscript indicated that currently the time involved in achieving the former goal is more than most educators are prepared to commit. However, the number of students who struggle with college freshman science courses because of poorly developed proportional reasoning skills would likely be reduced if more time were invested at the secondary school level to help them acquire these skills. Research indicates that short-term memory is easily overloaded when students are forced to work with too many new ideas (30), including those ideas associated with learning proportional reasoning (31). Such overloading can inhibit the processing activities necessary for preparing information for long-term memory. If students came to college with proportional reasoning skills firmly in place, they would be able to devote more time to issues surrounding content in the disciplines and less time to working on the process skills that make acquiring that content possible. Finally, we concur with Beasley that teacher development is more central to the quality of education than curriculum development (32) and with Feldman, who suggests that teachers’ knowledge about teaching and their educational situations grow when they are engaged collaboratively with other teachers in inquiry on their own practice (33). A copy of the math skills unit is available from F. Garafalo and copies of activities used at Fenway are available from J. Cohen and S. Pai. Acknowledgments We thank the reviewers for many helpful comments. FG acknowledges the generous support of the CVS Corporation for funding part of his sabbatical, and FG and RT acknowledge the generous support of the National Science Foundation for curriculum development at MCPHS, under grant #DUE9155849.


In the Classroom W

Supplemental Material

Supplemental material for this article is available in this issue of JCE Online.

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Notes

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1. Relationships can be between physical quantities, but one reviewer pointed out that relationships between analogical bridge objects, such as the circles in problem 6, are also possible. 2. Operational definitions often allow one to avoid difficult philosophical questions. For example, defining the gravitational mass of objects by comparing their relative attractions to the earth (6 ) avoids questions about why objects are attracted to each other in the first place. Similarly, the attempt at an operational definition of meaningful quantitative problem-solving avoids the question of why humans have been able to invent mathematical operations that relate to their activities and scientific descriptions of the world. References 26 and 27 make some interesting philosophical comments about this.

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Literature Cited

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1. Ochiai, E. J. Chem. Educ. 1993, 70, 44–46. 2. Canagaratna, S. G. J. Chem. Educ. 1993, 70, 40–43. 3. Niaz, M.; Herron, J. D.; Phelps, A. J. J. Chem. Educ. 1991, 68, 306–309. 4. Dierks, W.; Weninger, J.; Herron, J. D. J. Chem. Educ. 1985, 62, 839–841. 5. DeLorenzo, R. J. Chem. Educ. 1994, 71, 789–791. 6. Arons, A. A Guide to Introductory Physics Teaching; Wiley: New York, 1990; Chapter 1. 7. Lochhead, J.; Whimbey, A. In New Directions in Teaching and Learning, No. 30; Stice, E. Ed.; Jossey-Bass: San Francisco, Summer 1987; pp 73–92. 8. Lochhead, J. Chautauqua Short Course for College Teachers; State University of California, San Francisco, Feb 1–3, 1990. 9. Garafalo, F. Constructing Concepts in the Natural Sciences; Presented at 7th National QEM/MSE Network Conference. Washington, DC, February 1998. 10. Garafalo, F. Presenting Traditional Topics in Nontraditional

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