Chapter 7
“But You Didn’t Give Me the Formula!” and Other Math Challenges in the Context of a Chemistry Course
It’s Just Math: Research on Students’ Understanding of Chemistry and Mathematics Downloaded from pubs.acs.org by UNIV OF ROCHESTER on 05/14/19. For personal use only.
Amy J. Phelps* Department of Chemistry, Middle Tennessee State University, Murfreesboro, Tennessee 37132, United States *E-mail:
[email protected] Chemists at the college level believe that the application of previously learned mathematical skills drives student success in chemistry. Although math is a central part of any chemistry course, the data demonstrating that success or failure of students is solely dependent on mathematical ability, at least for one university, is inconclusive. The value that has been placed on quantitative problem solving as a measure of rigor in a chemistry course has made it possible for some students to be successful because of their math skills, without necessarily learning basic chemistry concepts. Conceptually, students struggle with math and chemistry even when they show mastery of basic computational skills. Problem context influences how students approach problems and which prior knowledge they apply in the problem-solving process. Research done in an electrochemistry laboratory demonstrates the influence of the context of a question on how students approach the solution. Think-aloud interviews with general chemistry students were used to get a student perspective on the issues they face when translating between chemistry problems and mathematical relationships. Taken all together, this data reveals that a chemistry course is a series of word problems requiring recognition of skills that apply to a particular context and translation of chemical concepts into mathematical relationships. To address these issues, we should allow for the struggle in our classrooms and avoid the temptation to take shortcuts by asking students to give discrete answers that can be reached merely by plugging in values and computing.
It is a widely held belief of chemists at the college level that a student’s success in chemistry is highly dependent upon the application of previously learned mathematical skills. In fact, a recent ConfChem online conference titled “Mathematics in Undergraduate Chemistry Instruction” held in the fall of 2017 was dedicated to this topic. The eight papers presented offered a variety of approaches for dealing with mathematical deficiencies in students so that they might be more successful in © 2019 American Chemical Society
chemistry. Addressing deficiencies is a decidedly more proactive approach to the problem than merely concluding it is not our fault, which is the view of many chemistry professors who blame the struggles students have in chemistry on their insufficient precollege education. In the eyes of these professors, it is not our responsibility as chemists to fix it because it is not our job to teach math. Although some professors are comfortable to leave the argument right there, the idea of accountability, which has been a part of the K–12 system for years, eventually arrived to varying degrees at the postsecondary level (1). That movement led my university administration to begin to question whether the failures of a professor’s students were in any way attributable to the professor. In the chemistry department at my institution, we have been challenged over the last three years to address the lack of success of our students in some meaningful way. This mandate has been focused in particular on large-enrollment introductory courses, which includes General Chemistry I. In four sequential semesters, more than 2,300 students enrolled in the first course of the General Chemistry sequence. These students were taught in multiple sections by 14 different professors. Of that group, 37%, or about 860 students, did not successfully complete the course, meaning they earned a D or an F or withdrew from the course after the drop/add period in the first two weeks of the term (DFW). One way to explain this rather high rate of failure was that students were not properly prepared for the course. Since making students better prepared was not within our power, we decided to search for a way to better advise students based on their prior knowledge, including their mathematical knowledge.
Identifying Better-Prepared Students In the fall semester of 2017, we gave all the students enrolled in the first course of the twosemester General Chemistry sequence the 2009 Toledo Placement Exam distributed by the American Chemical Society Exams Institute. This exam has 60 questions: 20 on math skills, 20 on general chemistry knowledge, and 20 on specific chemistry knowledge. We hoped that this exam would provide insight into who was prepared for the first course in General Chemistry and who was not. The assumption was that if students were more prepared, they would be more successful. We gave the exam during the first week of laboratory and collected the final grades of the students at the end of the term. We plotted the students’ percentile scores on the Toledo exam and the final grades they received in the class (Figure 1). The course grade was represented by the numerical quality points traditionally assigned letter grades in which an A is a 4, a B is a 3, and so on, and both F and W grades were assigned a zero. For the nearly 700 students in this data set, there was no clear relationship between students’ precourse Toledo exam scores and their final grades in General Chemistry. In an effort to make these scores easier to interpret, the data were plotted again using a violin plot (Figure 2), which helped us visualize the data by adding a kernel density estimation of the data, especially where we had several overlapping student scores (2). Wider sections of the plot indicate a higher probability of finding a student, and the thinner sections represent a lower probability.
106
Figure 1. Toledo percentile score versus course grade.
Figure 2. Distribution of Toledo Exam scores by final grades. This plot makes it easier to see that students making A’s in the course are more likely to have Toledo Exam Scores above the 75th percentile, and students failing the course are more likely to have scores below the 25th percentile, but it is hard to say that all students at the 25th percentile should not take the course when you consider the rest of the possible grades. Grades of B, C, and D are equally possible if a student scored at the 25th percentile based on this distribution. Since we were really more interested in enhancing course success rather than focusing on any specific grade, we decided to look at the students using two categories, successful in General Chemistry (A, B, C) and unsuccessful in General Chemistry (D, F, W). We broke the data out into these categories and plotted it again (Figure 3).
107
Figure 3. Frequency of success or failure based on percentile score. It appeared that doing well on the diagnostic exam meant you were more likely to do well in the course; however, doing poorly did not mean you would not do well in the course. It still was not clear how to use this data to advise students. Since the overwhelming opinion of professors is that people fail in chemistry because of deficient math skills, we decided to look at their successes and failures in the course in relation to their performances on just the math portion of the placement test. We argued that the other sections were specifically taught in the general chemistry class, so a good student could get that information by participating in the course, but much of the math was not part of the explicit curriculum of the course. The math section was 20 questions, so we plotted the raw score on that section against the percent A, B, or C and the percent D, F, or W (Figure 4).
Figure 4. Frequency of success or failure compared with math raw score. While it does appear that scoring better than 50% on the math section does favor success in the course, the vast majority of students who did not succeed still had a math score over 50%. Even if we advised all the students who made less than 50% on the math portion to take a preparatory chemistry class, we would not make a big impact on our DFW rate. Although this work is ongoing, as we collect 108
more data and seek to analyze it in a more informative way, it seems clear that math, at least in terms of measurable computational skills, is not the only thing students find challenging about General Chemistry.
Conceptual Chemistry: More Than the Numbers One reason math skills are so important in chemistry is the value chemistry professors have placed on presenting highly quantitative courses. It is one of the ways chemists have traditionally defined rigor. It is easy to equate the two sets of skills, chemical and mathematical, and conclude that the struggles students have with chemistry are really thinly veiled struggles with mathematics. Easy, that is, until researchers in chemical education proposed that students who were good at math could succeed in chemistry without knowing very much chemistry. When first proposed by chemical education researchers, this was a radical idea. Students could have success in chemistry because of their math skills and still have basic gaps in their chemistry knowledge. Could it be that students’ abilities to do math allowed them to “get by” in chemistry without knowing much chemistry? Had we tailored our assessments to the quantitative skills so much that students who could do the math did not have to know the chemistry in order to succeed? Was problem solving more than successfully navigating computational challenges? It appeared that allowing easy-to-assess quantitative problems to be our primary evaluation of student competence in chemistry assumed an artificial equivalence between understanding chemistry concepts and doing mathematical computation. I first encountered this idea that conceptual understanding in chemistry could be more than computational skills as a young graduate student. Shortly after arriving on campus, having been broken down by a series of qualifying exams that made us question our decision to pursue graduate study in chemistry, our class of new graduate students were asked to take a test on common misconceptions in chemistry. The concepts on this test were from General Chemistry. The results of this endeavor were rather interesting. All of us had majored in chemistry prior to coming to graduate school, we had been accepted for graduate study in chemistry at a large Midwestern university, and we were competent math students, yet there were basic gaps in our chemistry knowledge (3). This test evolved into a more formal multiple choice exam called the Chemistry Concept Inventory (CCI) (4) and has been used many times to point out gaps in the traditional quantitative-focused approach to chemistry instruction. It has been a valuable tool for pointing out to graduate students and preservice teachers the failings of teaching as you were taught; noting that despite their success in chemistry, they still have basic misconceptions (5). Giving a misconception test to chemistry graduate students was a rather interesting endeavor decades later with my own set of shiny young graduate students. In a recent study of graduate teaching assistants’ (GTAs) identity, we used the CCI to attempt to inform the GTAs about the importance of their understanding of chemistry and how it impacted their roles as instructors of chemistry. During the first of six workshops conducted throughout the spring semester of 2017, the GTAs completed a 21-question version of the CCI. The GTAs’ scores were presented at the second workshop and ranged from 6 to 19 with a mean of 11 and a mode of 10. Clearly, even the most advanced doctoral candidates in our department held misconceptions about basic chemistry knowledge. Of course, these scores were not unexpected by the researchers. We hoped to convince the teaching assistants that professional pedagogical knowledge could increase their success as laboratory instructors and that if they saw that the methods used on them as students left them with gaps in their understanding, they might be more open to other approaches to instruction. A GTA we called Fluorine described his appreciation for the experience. 109
Something is like the misconception and the test, she took us, you know? That was interesting and that really strikes me. The thing is like, I know, it looks like a really easy question, a simple question [he smiles and laughs a little]. Right? And it happens to the student as well - the same thing I observed the next day. Next lab, when I was in the probably you noticed that - you were there for that, in my class. When I was doing solubility or something like that. Yeah, doing the solubility class. They have a concept like, if you have a certain solubility it can dissolve in almost everything, right? And that’s what my student was saying, I got this really less amount, and I said, what do you think? And he said, it should dissolve. That’s what the concept is, even I think, you do too, right? Yeah, so there are a lot of misconceptions about the very basic phenomenon. So, see that was really nice, sincerely. That was a very good part, like I learned, I need to [he taps a finger to his temple] here, think about misconceptions. And it is very important, like if you can help a student to come up with a good, I mean like correct a misconception, even one misconception - that is really a lot. (6) The GTAs were required, as part of their teaching assignment, to attend a workshop hour once every other week throughout the semester. Initially, many GTAs did not want to do this, but they were very engaged in the results of CCI. For some of the GTAs, the CCI served as a disequilibrating event with regard to how they should conduct themselves as teachers. The GTAs began to identify ways in which their students’ experiences differed from those they had been exposed to in their own undergraduate experiences, many of which were in schools outside the United States. During their interviews, GTAs explained that the instructional methods in their countries focused more on theory and mathematical problem solving, rather than on laboratory experiences that they called practicals. A GTA we called Chlorine (Cl) explained the difference in thinking shortly after taking the CCI. Cl: I memorize a lot and do the calculations a lot. Actually, over there (in his home country) they emphasize about how to solve a problem in chemistry, not about all the symbols, but basically right here, I can see it. I: So, what do you mean by solve a problem? Cl: Actually, it is very difficult for me to show you one example, but um right here for example we can take some symbol like S each multiple… This is a problem you can use only one or two formulas and that’s it, over there you use more formulas to solve the problem. Or they ask you many steps. So, you have to solve one by one, and one by one and all five steps before you can find the results. It’s not difficult, but it is complicated. But if you understand the problem like here it’s easy. Like here they ask you what’s A, B and C you solve, and you have the results for B & C. Over there they just ask you to explain C so you have to find A and B (on your own). (6) This GTA pointed out what many GTAs saw as a fundamental difference in the problem-solving expectations in the chemistry labs they teach when compared with the chemistry classes where they were taught. They saw the problem solving required of their students broken down into discrete steps that they viewed as simplified. Based on their experiences as students, quantitative problem solving and memorization were the keys to success in chemistry. This study demonstrated a disconnect between the way graduate students were prepared and the role they were expected to play in the education of their students (6). In the late 1980s and the 1990s, the chemical education literature was full of research demonstrating that students who could do the calculations did not necessarily understand the 110
chemistry involved. One of the first papers to propose that chemistry was more than the quantitative narrative that had become so popular was by Nurrenbern and Pickering (7) titled “Concept learning versus problem solving.” They looked at paired pictorial and quantitative gas laws problems and found that students were more successful with the mathematical problems than they were with the more conceptual questions. It was surprising to many to think that students were performing better on gas law calculation problems than problems that required no computation at all. Many studies followed up on this one trying to better understand what was happening and trying to be sure that the results were real, but time and again, the students were more perplexed by the questions that did not require a calculator (8–12). It was a difficult idea for chemists to accept because it challenged their perceptions of the role of mathematics in chemistry. Maybe strong skills in math could get a student through a course in chemistry without having to learn it. Some chemists started to think about problem solving in a different way, recognizing that problems could be conceptual as well as quantitative (13, 14). Certainly, computational skills are valuable, but computational competence should be only part of what it means to be a successful chemistry student. Since students quickly learn to focus on what we value, this led to a re-evaluation of how assessments in chemistry were designed. The American Chemical Society Exams Institute developed a conceptual exam and a paired questions exam that included both conceptual questions and quantitative questions on the same idea. These products contributed to the legitimacy of the idea that chemistry was more than calculations and those nonquantitative concepts could be assessed (15). Tests (and text books) began including questions based on diagrams of relationships and particulate illustrations of reactions. My mind was changed by much of this data. For example, I was convinced that having students spend hours memorizing formulas was not a good use of their time, so I started providing formulas to my students. Soon after I began this practice, I was approached by unhappy students after an exam on gas laws because I “didn’t give them all the formulas” as I had promised. I was confused—what could they mean? Their complaint concerned the ideal gas law formula. PV = nRT was the formula provided on the test, but the test question had asked the students to solve for pressure, and the students explained I had not given them the formula for pressure. How could it be that they could not see the connections between the formula provided and what they needed to solve the problem? My students did not see anything equal to P so, in their eyes, I had failed to provide the proper formula. This is not a failure of computation, and it suggests that the students have conceptual issues in mathematics as well, once mathematics is seen as more than computation. This concern about conceptual understanding led to a redesign in my courses beyond just a sheet of formulas. I moved chemistry concepts to the forefront when introducing new topics in hopes that a good grasp on conceptual understanding prior to focusing on quantitative problem solving would provide a better foundation for grappling with the math.
The Importance of Context: Lessons from Electrochemistry I began my career as a mathematics and science teacher in a high school. One of the more interesting opportunities I entertained in my first teaching assignment was the chance to teach an Algebra II class and a Chemistry class back to back. These two classes were inhabited by many of the same individuals, and it gave me the opportunity to observe firsthand how context could impact the students’ ability to DO the math. I had students who, in fourth period, could do the math without issue but, in fifth period, could not apply those same concepts to the chemistry problems successfully. Logarithms made sense in Algebra II when we were working problem after problem, but seemed a mystery when dealing with pH in chemistry. Math classes back in that time (and in many 111
places at this time) followed a very reliable pattern. A new skill was introduced, the teacher worked some examples to demonstrate that skill, the students were assigned some problems to practice that skill, and the next day, after checking those problems, we started again, until arriving at the end of a chapter where you found the “word problems,” hated by teacher and students alike because word problems disrupt the happy, agreed-upon social order. Now students would have to read and analyze a situation and apply these new skills developed in the chapter. I remember the struggle—I know colleagues who skipped those sections. This struggle continues to be played out almost daily in my chemistry class. Chemistry uses mathematics in a new context that is not as predictable or as structured as in a math class. In fact, the context could require applications of math learned in more than one course and rarely involves finding x. The difference between solving for x and understanding those same skills in the context of a chemistry class was illustrated by the struggles of students trying to balance oxidation-reduction reactions. A pair of students approached the instructor after an electrochemistry lab and expressed frustration with the balancing of oxidation-reduction reactions using the half-reaction method. Amanda: I still don’t get how you know how to do this. Instructor: Ok, what do you have? Amanda: 6H+ plus ClO3- goes to Cl- and 3H2O, right? Instructor: Ok, so now what do you do? Amanda: I don’t know. I think the charges need to be the same. Instructor: What do you need to do to make them equal? Amanda: Add 5? Add 7? I don’t know. Can I just use a formula to solve this? Why didn’t you just give us a formula? Instructor: What formula do you want to use? Amanda: Lets see maybe [writing now] x + 5 = -1. Can I just do that? Instructor: Ok, so what does that get you? Amanda: Well subtract 5 from both sides and you get x = -6 Instructor: Very good. What do you do with that x? Amanda: Subtract 6 from… No add 6 to … Instructor: 6 what? Amanda: Electrons?... Charges?... Ions? Instructor: What charges do electrons have? Amanda: Negative. Instructor: So what do you do with the 6 you solved for? Amanda: I’m just not sure. Instructor: Ok. Can we think about a number line where you are at +5 and need to get to -1 and every time you add an electron you move down. [Draws a number line vertically on the board and adds electrons moving toward -1.] How many electrons get us to -1? Amanda: 6 Instructor: So what do we need to do with the 6 electrons in our half reaction? Amanda: Give them to the +5 side? Maybe? Amanda and her friend wanted an equation they could rely on and in fact came up with an algebraic relationship that worked, but in the context of chemistry, the equation was not sufficient to solve the problem because they still needed the conceptual understanding of what it was they were trying to do in the first place. Amanda could add and subtract positive and negative numbers, but she could not 112
bridge the gap between solving for x and balancing the charges. It would have been easy to blame the issues these students were having on their inability to deal with negative numbers, but listening to the students made it clear that they could do the computation, but lacked understanding, perhaps of both the math and the chemistry, required to correctly apply their solution for x. Their skills with algebra did not solve the problem of balancing redox reactions in the context of chemistry class. We have seen in other situations how context can make a difference in how students look at problems. In a study of electrochemistry in the laboratory, we have demonstrated the power of context on the problem solving of students where the questions were much more closely related than when comparing math classes with chemistry classes. We taught an electrochemistry lab using two different methods in an effort to understand how the method of laboratory delivery impacted students. Initially, we noted that there was no measurable difference in how the students did on the electrochemistry questions on the tests after participating in one of the two different labs, so at face value, one method was no better than the other (16). We decided to repeat the study and focus on the student discourse in the laboratory to see if we could ascertain what impact the delivery method had on the students. We wanted to see what they were talking about and how those conversations changed based on the type of laboratory they were engaged in. A simulation of an electrochemistry lab (17) was used for one set of students and a traditional hands-on lab (18) was used for the other set of students. We analyzed the discourse using the three representations of chemistry proposed by Johnstone: particulate, symbolic, and macroscopic (19). When analyzing this data, we noticed that the traditional lab students were very focused on the macroscopic nature of electrochemistry but struggled with the symbolic and particulate questions, but students who were engaged in the simulation lab were much more focused on particulate explanations (20). Here are two students in the traditional hands-on lab talking about a strip of zinc placed in a copper solution. F1: So the metal definitely got darker. It went from like a silver color to orange. F2: It’s kind of a rust looking. F1: It does? F2: So observations…? F1: You can see it like—looks fuzzy. F2: Ok changing color and texture. F1: Yeah write that. It got orange and a little rusty. F2: I’m gonna take it out and see what it actually looks like. (20) As you can see, their conversation is very macroscopic, but the students using the simulation in lab while discussing a Daniell cell were focused on a very different representation. F1: Ok let’s see—at the molecular level—what happened on the zinc itself? Look at the surface, you see zinc is losing two electrons and becomes zinc two plus. Look at the surface of the copper. F2: So copper can actually gain electrons? F1: Yeah… (20) The context in which the lab was presented impacted the explanations the students gave, and this influence of context continued to be seen throughout various iterations of these electrochemistry 113
experiments. Students enrolled in the same class, studying the same material, and asked the same questions formulated answers consistent with the manner in which the material was presented. The way material is presented provides a context that cues students to engage certain modes of reasoning. This has been demonstrated in animation studies in which the subtle differences in representations in the animations influence the concepts students build (21). This relationship between the instructional context and how students connect the various levels of chemistry has been identified in advanced levels of chemistry as well (22). Students learn math in a very different context than in their chemistry class, and we need to find ways to cue their mathematical understanding while they are engaged in chemistry problem solving.
Translating Chemistry into Math: Students’ Voices By designing class time around cooperative group problem solving that focuses on studentgenerated solutions over mimicking instructor examples, instructors are provided with insights into how students conceptualize chemistry and math. Often, my students are asked to develop their own approaches to solving problems and to negotiate meaning with their peers and share these discussions with the class, which leads to interesting discussions about the math. When I do solve problems in class, I try to make explicit the translation of the chemistry into the necessary mathematical setups. Often, I set up problems while thinking aloud as we work through the chemistry, and then I say, “Now the chemistry is on pause for a minute while we solve the math,” or I might say, “Now the rest is algebra.” Not because the math is unimportant, but because I want them to be aware of what we are doing as we work through the various intertwined worlds of chemistry and math. I often tease that they have to know how to do it all, but they should be aware of what they like and don’t like about the process. I think if students can SEE that what they are doing is applied mathematics—mathematics they generally can do in math class—then perhaps they can engage those skills in the context of chemistry class. To better understand the issues student encounter while translating chemistry descriptions into mathematical setups, we talked to some students while they solved problems. We had students who were enrolled in a first-semester course solve stoichiometry problems in a think-aloud interview setting to see if we could get better insight into their perspectives on the application of math in the solving of chemistry problems. The interviews were done near the end of the first-term course, and the problems were similar to ones they had been working on in class. The preliminary results for this study provide some insight into the students’ perspectives on why the math in chemistry is difficult (23). At the beginning of the interviews, the students were asked which subjects were difficult for them, and math was on everyone’s list. One student (Roger) explained he didn’t apply himself very much in high school and was often in trouble, so in general, he did not do well in most subjects. He got to algebra in high school and then was moved into more “applied math,” like balancing a checkbook. He described his troubles with math this way: In my experience, I am more of the audio learner. I learn better from hearing and watching people do things out. You know, like work out problems. And I like skip steps and I have trouble keeping numbers straight so (I) get them kinda confused a little bit. So I have to spend a lot more time focusing, slowly working through things. It (math) just doesn’t come naturally to me. It is interesting that Roger assumes that people who are good at something do not struggle with it or do not need to stay focused and take their time. 114
Another student in the study (Sarah) described high school as not very challenging. This student got A’s in all her high school math courses, which included two years of algebra and a year of geometry. She was on a math track in high school designed to prepare her for postsecondary study, but her beliefs about her skills in math were similar to less successful students. My skill set is in English classes, in humanities, as far as math and science I fall short. I mean I’m still a smart student, like I got it from my parents. I have a very good work ethic and study habits but it’s a lot harder for me to comprehend math and science. I feel like for some people it’s easier because with math and science there are formulas and so there’s always going to be one answer but for some reason it’s very hard for me to comprehend the one answer and I’m not as good at memorizing formulas. And numbers are just a lot harder for me to put together and make sense of versus words. It’s a lot harder for me to get a hold of so like math especially. Science is easier for me than math is. Math is just very hard for me. Sarah’s understanding of math is really a view of doing computations correctly, usually using a formula. She is, not surprisingly, unaware of the more complex issues in math in which you can have multiple solutions to a question or various proofs for a theorem. Math for these students was about computation, which was consistent with the experiences they have had in math classes to this point. All the students were asked to solve a stoichiometry problem in which magnesium reacts with oxygen to form magnesium oxide. They were given 40 grams of oxygen to start with and asked how many grams of magnesium oxide would be formed. The students were encouraged to explain their approaches as they went. Roger solved the problem in the most straightforward fashion with the least prompting or hints from the interviewer. Roger: The first thing is trying to see what the question is asking–see what the final answer is and the information I have. So I believe I need to make sure this equation is balanced. So I got two Mgs over here and I got 2 Mgs over … and I’ve got 2 O2s and 2 oxygens. So it looks balanced to me. Roger: So now the next thing is to figure out how many grams. These are the products and these are the reactants. When I have 40 grams of O2 completely reacts – so you got 40.0 grams of oxygen and you can see in this one the conversion we have one gram or one equivalent of oxygen makes two of these so we have a 1 to 2 reaction. So 40—should be—so we’re in grams so we have to move—these would be moles I believe… [mumbling]-- I need the molar weight of oxygen. Interviewer: 32 Roger: It’s 32? Thanks. Interviewer: Let me ask you quickly—those coefficients in front of those compounds what are they? Roger: They represent moles—so we want grams—How many grams of Mg so we know that it’s 1 to 2 so that is the thing. So we don’t need to convert it. So I believe we have to go from grams to mole to moles to grams. So then—so we go from grams to moles and then from mole we would need how many of these we have—so we have 1 mole of O2 makes 2 moles of Mg O and then we need the grams in one mole. Interviewer: It’s 40.3 Roger: Thank you. 40.3 grams of MgO in one mole of MgO and then we would multiply. Interviewer: Don’t worry about finishing the calculation part. 115
Roger: Oh Ok. Interviewer: I just want you to explain something for me. What your logic is—what are you doing here? Roger: So the question wants it to be in grams and the information I have is in moles so I have to take it from… I’m also given grams in the beginning. So I have to go from grams to moles and then I have to do the conversion moles of O2 to moles of MgO and the question wants the final answer in grams so I have to convert it back to grams. He was able to set up the problem and think through the logic of how to solve the problem. Sarah had a different experience with the problem. Sarah: How many grams of magnesium oxide are produced when 40 grams of O2 react completely with magnesium? So first I would take—there is two MgO for one O2 so if there is 40 O2 and I double so it would be 80 grams. Interviewer: So that’s what you would do? Sarah: That’s how I would approach it. Yes. Interviewer: So if you were taking the test that would be your answer? Sarah: Yes. This represents the most common way the students solved the problem, but Sarah showed her persistence when told that she was incorrect. Sarah: I needed to change to moles. Don’t I? Interviewer: Well it’s not the same in terms of moles. Sarah: Yes that’s correct. Right. Okay, so then I’ll take 40 grams of O2 and I need to get that to moles so I would divide it by the molar mass of O2. Interviewer: 32. Sarah: So I will do 32 then… My calculator is over there, but I can use my phone calculator. Okay so then I have 40 and I divide it by 32 grams of O2 and it leaves me with 1.25 moles of O2 and now that I see that the ratio is 1 to 2 and I want to get MgO so if that is to this then I will multiply this times two and that would be my answer. 2.5 grams MgO. I believe this is my final answer. The students in this study all keyed in on the 1 to 2 ratio in the balanced equation but were challenged by the chemistry concept, 1 to 2 of what? All the students interviewed listed math as a subject they didn’t feel was their best or that they had trouble with regardless of past grades or experience. Yet some students were able to work through the logic of the problems and others were not even aware they were having issues. None of the students had computational issues per se. They were able to multiply and divide correctly, and they recognized the importance of the 1 to 2 ratio, but the conceptual underpinnings were not as evident to most students. I am often reminded of those algebra students so many years ago who would say, “I can do algebra just fine, except for the word problems.” Chemistry is one giant word problem, as is much of the world, so just as students struggled with word problems in a math class, they struggle with them in chemistry class. A student knows what is required in precalculus class on a particular day as they work to hone a particular skill, but in chemistry class, where those math skills are needed, students have to translate the chemistry into the computational forms that are required, solve them, and then
116
translate them back into chemistry to answer the question. As instructors of chemistry, we need to model these translation skills. Problem solving has long been an interest of mine, and I believed as a young teacher that if I could figure out THE STEPS to solving problems then I could justify my career. I was not convinced that knowing chemistry was necessary to live a full life, but problem solving was a skill everyone needed in all walks of life. I am not the only one to have looked for the secrets of problem solving in chemistry, and the literature is full of our successes and failures (24). We know that true problem solving in classrooms is rare, and that if you know the steps to solving it, then likely you are facing an exercise and not a real problem. We have often made chemistry too much about quantitative skills, and our colleagues in math are too boxed in by people believing that math is only about computation. The challenges for professors in these two disciplines are similar, and perhaps the solutions can be worked on together. Certainly, we cannot continue to do the same things and expect different results. We can start by posing problems and giving students a context in which to think about both chemistry and math. We can allow for the struggle in our classrooms that is required when solving real problems and avoid the temptation to take shortcuts to discrete computational answers that can be reached merely by plugging in values and crunching the numbers.
Acknowledgements There are people who need thanking: Graduate students Tasha M. Frick, Shaghyegh Fateh, and Vichuda (June) K. Hunter for all their hard work collecting and analyzing data; Scott Oswald for editing and discussion on the chapter in progress, particularly the data representation; Jeffrey LaPorte for reading and editing; and my colleagues who teach in the General Chemistry sequence, specifically, Michael J. Sanger, Gary White, and Greg Van Patten.
References 1. 2. 3. 4. 5. 6.
7. 8. 9.
King, A. F. The Changing Face of Accountability. J. High. Educ. 2000, 71, 411–431. Hintze, J. L.; Nelson, R. D. Violin Plots: A Box Plot-Density Trace Synergism. Am. Stat. 1998, 52, 181–184. Bodner, G. M. I Have Found You an Argument: The Conceptual Knowledge of Beginning Graduate Students. J. Chem. Educ. 1991, 68, 385. Mulford, D. R.; Robinson, W. R. An Inventory for Alternative Conceptions Among FirstSemester General Chemistry Students. J. Chem. Educ. 2002, 79, 739–744. Phelps, A. J.; Lee, C. The Power of Practice: What Students Learn from How We Teach. J. Chem. Educ. 2003, 80, 829–832. Frick, T. M. Explorations of Chemistry Graduate Students’ Professional Teaching Assistant Role-Identity: Connecting the Seen to the Unseen. Ph.D. Dissertation, Middle Tennessee State University, Murfreesboro, TN, May 2019. Nurrenbern, S. C.; Pickering, M. Concept Learning Versus Problem Solving: Is There a Difference? J. Chem. Educ. 1987, 64, 508. Nakhleh, M. B. Are Our Students Conceptual Thinkers or Algorithmic Problem Solvers? Identifying Conceptual Students in General Chemistry. J. Chem. Educ. 1993, 70, 52. Nakhleh, M. B.; Mitchell, R. C. Conceptual Learning Versus Problem Solving: There is a Difference. J. Chem. Educ. 1993, 70, 190.
117
10. Sanger, M. J.; Campbell, E.; Felker, J.; Spencer, C. “Concept Learning Versus Problem Solving”: Does Particle Motion Have an Effect? J. Chem. Educ. 2007, 84, 875–879. 11. Sanger, M. J.; Phelps, A. J. What are Students Thinking When They Pick Their Answer? A Content Analysis of Students’ Explanations of Gas Properties. J. Chem. Educ. 2007, 84, 870–874. 12. Sawrey, B. Concept Learning Versus Problem Solving: Revisited. J. Chem. Educ. 1990, 67, 253. 13. Phelps, A. J. Teaching to Enhance Problem Solving: It’s More than Just the Numbers. J. Chem. Educ. 1996, 73, 301. 14. Bruck, L. B.; Bruck, A. D.; Phelps, A. J. “Gone” into Solution: Assessing the Effect of HandsOn Activity on Students’ Comprehension of Solubility. J. Chem. Educ. 2010, 87, 107–112. 15. Holme, T.; Murphy, K. Assessing Conceptual and Algorithmic Knowledge in General Chemistry with ACS Exams. J. Chem. Educ. 2011, 88, 1217–1222. 16. Hawkins, I.; Phelps, A. J. Virtual Laboratory vs. Traditional Laboratory: Which is More Effective in Teaching Electrochemistry. Chem. Educ. Res. Pract. 2013, 14, 516–523. 17. Greenbowe, T. J. An Interactive Multimedia Software Program for Exploring Electrochemical Cells. J. Chem. Educ. 1994, 71, 555. 18. White, G. D. CHEM 1120 Laboratory Manual; Kendall Hunt: Dubuque, IA, 2005; pp 87–93. 19. Johnstone, A. H. Why is Science Difficult to Learn? Things are Seldom What They Seem. J. Comput. Assist. Learn. 1991, 7, 75–83. 20. Hunter, V. Encouraging Meaningful Learning in Electrochemistry Lab. Ph.D. Dissertion, Middle Tennessee State University, Murfreesboro, TN, December 2019. 21. Kelly, R.; Jones, L. L. Investigating Students’ Ability to Transfer Ideas Learned from Molecular Animations to the Dissolution Process. J. Chem. Educ. 2008, 85, 303–309. 22. Cole, R.; Becker, N. M.; Stanford, C.; Towns, M. Translating Across Macroscopic, Submicroscopic, and Symbolic Levels: The Role of Instructor Facilitation in an InquiryOriented Physical Chemistry Class. Chem. Educ. Res. Pract. 2015, 16, 758–768. 23. Hunter, V.; Phelps, A. J. Where Do They Go Wrong?: Listening to Students Solve Stoichiometry Problems. Middle Tennessee State University, Murfreesboro, TN. Unpublished work, 2018. 24. Bodner G. M.; Herron J. D. Problem–Solving in Chemistry. In Chemical Education: Towards Research-Based Practice; Gilbert, J. K., De Jong, O., Justi, R., Treagust, D. F., Van Driel, J. H., Eds.; Kluwer Academic Publishers: Dordrecht, 2002; pp 235–266.
118
