Solving Chemistry Problems without Utilizing the Factor-Label Approach Frank Cardulla Niles North High School, Skokie, IL 60077 I t is almost certainly safe to say that of all possible approaches to teaching prohlem solving in introductory chemistry classes, the factor-label approach is by far the most popular. It is the cornerstone upon which most of us have n ~ & ~ t e r lto huild problem-sol~ingskills in our studcntu. Illdeed, this method has arhieved such widespread popularity that it might seem blasphemous even to suggest that another method could constitute a viable alternative. Perhaps it is true that the factor-label (F-L) approach is the monarch of problem solving in introductory chemistry, but as I look at His Majesty, he appears to he without clothes. There is no question that motivated, competent problem solvers can be oroduced without wine the F-L method. M y uwn studmts provide testimonials to this statement. As to uhether the auorouch I ut~lizeis "suorrior" to a F-L approach is a matte.rLfor debate. ~learly,'motivated, competent prohlem solvers are beina produced hv numerous appioaches; and it would he extremely difficuit to do the type of research needed to choose amona them. Rather. I will simply present my own ideas for your j;dgment. I suspect that most of us have had the experience very early in-our teaching careers of immense frustration in trying to teach our students to solve chemistry problems. Let's take the relatively simple problems that are involved in couverting among units like grams, moles, molecules, and liters of a gas a t STP. T o us, these prohlems are very simple and obvious. Yet to a significant percentage of our students, these prohlems appear to present almost insurmountable obstacles. They multiply when they should divide, divide with the wrong number on top, and continually make errors that, from our perspective, no functioning human mind could possibly make. T o us, they seem incredibly inept. Yet we objectively know that this is not true. We are dealing with reasonably intelligent, reasonably motivated groups of students, yet they appear to transform into mental microns when faced with these simplest of prohlems. Let's take, as a simple example, the prohlem of converting 4.25 g of Ne gas at S T P into liters. We divide the 4.25 g by 20.2 glmol, and then multiply by 22.4 Llmol. It's elementary. But to our students, it's a mystery. "Why did you divide the first time. but m u l t i ~ l vthe second?" We trv to explain our logic. Thky do not understand. We become Gustrated. We explain aeain. Thev still do not understand. Now we begin to get angry with &em, and we feel guilty for it, because we know that they are trying. In desperation, we search for a "better way," thus leading us to the F-L approach.
There are only two reasons that justify utilizing the F-L method, in my opinion. First, as a teacher, you deem the obtaining of the correct answer to he the only goal of prohlem solvine. or, second.. vou . think that althoueh it would be wurthwlde for students to undtmrund what they are duing, unf~lrtunatelv,most uf them are incapable of achie\,inr t h ~ s understanding, so you shoot for the next best thing anduhope that understanding mav come with the vassaae of time. I accept neitheFof these premises. if ana&zing units to obtain a correct answer is our aoal, then the problems might just as well he stated in nonsense terms. he-student solving prohlems this way is taking a course in algebra, not chemistry. Further, I a m convinced that virtually all of our students are capable of correctly engaging in the type of quantitative thinking required to solve correctly most of the problems that we give them. Let me try to illustrate what I mean. The first prohlem I give my students is the most important problem of the year. It goes as follows. There are 3000 students in a high school. There are 30 students per roam. How many rooms are there in the high school? In my 23 years of teaching, no student has missed this problem because thev multiplied when tbev should have divided or divided withtbe wrong number ontop, and this includes a class of special education students I taught two years ago. "One hundred rooms", they immediately shout out. I ask them how they got the answer. They of course simply reply that they divided 3000 by 30. No, I say, I mean how did you decide what to do with the numbers? How did you know to divide 3000 bv 30? Whv didn't vou multivlv 3000 bv. 30.. or diuide 30 by 3000? They will literally laugh a t this suggestion. Students of humor would probably tell us that they are laughing because they sense an outrageous incongruity in these two suggestions. What is interesting, of course, is that they rarely can give a coherent justification for why they divided like thev did. Thev iust knew that i t was the wav to work the problem. If you-multiply, you get 90,000 ro'ms. They usually can explain that this just doesn't make any sense! If you had only one student per room, 90,000 rooms would give you 90,000 students, and you have 30 students in every room. How in the world could you end up with only 3000 students altogether? Dividing the wrong way would give you 0.01 rooms. That's ridiculous also. A whole room would have only 30 students in it. How can a tiny part of a room have 3000 students?
Volume 64 Number 6 June 1967
519
This problem is so simple that i t may appear to have absolutely nothing to do with teaching chemistry. But it has euerything to do with teaching chemistry, because the vast majority of the problems we ask our students to solve in our introductory courses involve exactly the same sort of logical thought orocesses. F; example, consider the problem of converting 7.56 g of HN03 into moles. This problem involves quantitative logic that is completely identical to that involved in the students and rooms problem. Students divided by students per room is rooms, and grams divided by grams per mole is moles. Yet we all know that when presented with a problem such as this, many of our students will work it incorrectly. Why? Interestingly enough, my experience has taught me that most students do immediatelv see that the room and the mole problems are, indeed, thesame problem. For some, this realization. bv itself. uroduces a dramatic imurovement in For others, however, it doesn't their probl'em"-solvin~~bility. seem to urovide much assistance at all. OK, so we know that they are fully capable of engaging in the tvpe of auantitative thinkina reauired to solve most of the Grbb~emHthat we give them.-~heiwhy do they have so much trouble when we exuress the uroblems in terms of moles, grams, liters, moles/liter, lit&lmole, joules/mole, meterslsecond, or any of the other multitudinous ways these quantities can go together? There are two answers to this question. First, the kinds of numhers that are involved in our calculations throw students. The numbers rarely are nice and even, like 3000 and 30. Instead, they involve decimals and exponents, and it is quite natural for students to have much more difficulty trying to work problems when the numbers aren't "nice". My experience, however, is that this problem isn't a terribly major one. I t takes work. but students auuear to he capable ofdivorc.ing the logic from the numhers u%h a lot of practice. They still, of courae, find urohlems with even numhers simpler; but that's only natural. We probably do too, but while they may have to think more slowly and carefully when the numbers are not "nice" integers, they can learn to divorce the numbers from the logic given time, constant instruction, and practice. Second, and probably most importantly, they donot know what we mean by terms likegrams, moles, liters, molesfliter. etc. Oh. thev mav be cauable to eivine a definition of theie things; bu
