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Written problems tend to be hard for many students. I will describe the method and rationale I have used for several years now teaching word problems in college algebra.
There is a very general approach to solving problems that is applicable and used in many areas of math. You work through some examples and then assign some problems. Textbooks do this, and so do teachers. Is this sufficient for the type of word problems we traditionally get in algebra? I would say that it has to be. It’s about all we have. So we might then ask how many and what types of examples should we work through? How many problems should we assign for homework? How many word problems should we put on a test? How many class periods should we devote to word problems?
The answer to all these questions are given implicitly, to some degree at least, by whatever text book we are using. In the text I am currently using Chapter two is on equations. The chapter is divided into sections, as are most math textbooks. Section one covers simple equation solving, and is therefore, or should be, review. Section two is titled "An Introduction To Problem Solving", which of course means word problems. Since this topic is only one section long we have some basis for saying we should devote only one class period to it. For many textbooks it is approximately true that the number of sections that should be covered and the number of class periods in the semester are roughly equal, hence we have to average something like a section per class period. Is this sufficient?
I have concluded over the years that this is not sufficient. A fifty minute class period is time enough to thoroughly cover a small number of examples, perhaps five, but often less. In my customary way of doing things this would be split into two parts. On any given day I expect to spend a little class time going over and answering questions on the homework the students have completed. So one class period for written problems would translate to perhaps 35 minutes one day introducing the topic, and 15 minutes the next day going over a few of them. Then we have to move ahead to whatever the next topic is. However I spend four class periods on written problems. That might be considered excessive by some. My basis for saying that at least this much time is needed is simply my experience. It seems to take that much time to be effective.
In any class there will be a few students who have successfully done word problems before, but they seem to be quite a minority. Much more common are the students who have been exposed to them before, but were frustrated and unsuccessful. (Of course some of those students who had no trouble with word problems are not in my class. They're taking calculus.) I can't claim that I ever approached anything like getting 100% of the students in a class to really understand them. However I think I generally manage to get a majority of the class to understand them, at least when we are working them in class. When we come to the test that understanding, for many students has fragmented enough that they make mistakes. I often find good reason to give partial credit, but often not full credit. Word problems remain difficult for many students.
The author of our text gives a six step guide for solving word problems. Many other texts give something very similar, but varying in the wording and the number of steps presented. I have not found this helpful. I mention to my students that these steps are in the book, and suggest they read through them carefully and try to apply them. But that is all I do. I don't disparage these steps, but I don't promote them either.
Step number one, in our author's enumeration of the problem solving steps, is very simple - "Understand the problem". It is very simple, but to me it is utterly inane. To me that is not where you start, it's where you end up. If you understood the problem you could do it. You wouldn't need any more steps. Why does the author start this way?
The answer to that, unfortunately, is very simple. That was George Polya's first step, and Polya "wrote the book" on problem solving. Polya, I understand, was a capable and well respected mathematician. But all I know about him, and what he is apparently most famous for, is his book on problem solving. He enumerated four steps for solving problems. His first step was "Understand the problem". I don't remember when I first heard of Polya and his four steps, but right from the start it struck me as unhelpful. Understanding the problem is the final step, not the first. To put it first, it has always seemed to me, must be discouraging to many students. They have had a few years to become painfully aware that they do not understand the problem. To make that the first step seems to me would suggest to such students that they are dummies.
Our text book next presents four examples and shows the solution, attempting to show how the six steps are to be applied. When I first started using this text I decided those four examples are not at all sufficient, and I do not think they are even well chosen examples. Therefore I made out my own homework assignments as "handouts". These are available as a separate file here. I have also put a link to these assignments at the end of this article. These are referred to as Homework 2, Homework 3, Homework 4, and Homework 5.
I don't have enumerated steps for problem solving. But I do have a few ideas that I will try to develop.
I start out on day one of word problems with the idea of translating words to symbols. I give a few examples like:
- Write an algebraic expression for five less than six times a number. Answer: 6x - 5
- Write an algebraic expression that represents what you get if you take a number, divide it by five, and then add thirteen. Answer: x/5 + 13
- Write an algebraic expression that expresses four times the sum of a number and 9. Answer: 4(x + 9)
This doesn't take long. I put some problems like these on homework, and they are usually pretty easy for students to do correctly. Still in day one I go on to the next step, which is to take complete sentences and translate them into equations. Example:
Five more than three times a number is equal to seven less than the number. Find the number.
I explain about declaring the variable. "Let x = the number". I talk them through the equation. 3x + 5 = x - 7. Most of this seems quite manageable. Students seem to understand and can do problems like these. So next, still on the first day of written problems, I introduce consecutive integer problems, such as:
The sum of three consecutive integers is 52. Find the integers.
I explain about declaring variables again, but it's just a bit more complicated this time. If we let x = the first consecutive integer then x + 1 must be the second, and x + 2 must be the third. Then we translate to an equation. x + (x + 1) + (x + 2) = 52. I talk about the parentheses. They are not needed mathematically but as punctuation they help to make it plain what we are doing. I once again mention that any two numbers can be added, but sometimes we can do nothing more than put a plus sign between them. In this case putting a plus sign between the numbers is a way to begin. At this point we have translated a problem in words to an equation, which is the basic idea behind solving any written problem in algebra.
I mention that when I grade homework I will take off points if they don't declare the variable, explaining once again that their job is to present a line of mathematical reasoning. Show an algebraic solution. Don't expect any credit for just getting a right answer. A correct answer may often be found with a bit of trial and error, but that is not building a structure of mathematical knowledge than can be used as a basis to learn more math.
The next day's work on written problems brings in another step, and it is an important step. I use coin problems. Example:
There are thirteen coins in a pile. Some of them are dimes and some are quarters. The total value of this pile of coins is $2.80. How many dimes and how many quarters are there?
Once again I mention that the student's job is to present a chain of mathematical steps, not to get an answer by trial and error. I explain about declaring a variable again. Then I explain that there is an important difference between today's coin problems and yesterday's number problems. There has to be an expression for the number of dimes, and there also has to be an expression for the value of that number of dimes. I presume to many in the class this is obvious. However apparently it is not obvious to all. Then I work the problem.
At this point I explain something that I consider very important about written problems. Every written problem assumes you have some background knowledge, knowledge that is not explicitly stated in the words of the problem. In these coin problems that important background knowledge is simply how to figure, and express, the value of the coins. I mention that in the consecutive integer problems of homework 2 the background knowledge is the meaning of "integer", and the meaning of "consecutive". I return to this idea of background knowledge not expressed in the problem several more times in the course of the next few days.
Should this be hard for college students? Didn't they learn how to figure the value of mixed coins back in elementary school? For most of their lives haven't they been able to find the value of two nickels and six dimes, or the value of eleven dimes and 14 quarters? Isn't that simply whole number addition and multiplication. Should I have to repeat it here?
I really don't know the answer to those questions. Perhaps the answers are that of course they can add up the value of a mixed pile of coins, but the problem is in fitting that into algebra. But I'm not sure. Could it be that some students get to college and really cannot do simple problems like this?
I would think the problem is understanding the language, the notation. One would think that if you have eight objects worth two dollars each, every one would know to multiply eight times two to get 16 dollars as the total value. And one would think that if you have eight objects worth x dollars each then you would multiply eight by x to get their value, 8x, or that if you have x objects worth eight dollars each then you would multiply x by eight to get 8x as their total value again. Or is that the case? Is there a conceptual gap between those two ideas?
To continue, one would think that if you have 8 objects worth two dollars each and fourteen objects worth six dollars each, then the total value would be 8*2 + 14*6. So if you have x objects worth two dollars each and y objects worth six dollars each then the total value would be 2x + 6y. Or is that not so obvious? If it is not obvious then where is the glitch?
One glitch is that at this stage we don't want equations with x and y. We want equations only with x. In my original coin problem in declaring the variable we could let x = the number of dimes. Then 13 - x would be the number of quarters. Is this harder to understand than x dimes and y quarters? Perhaps. Maybe .1x is easily understandable as the value of the dimes, but perhaps .25(13 - x) is less understandable as the value of the quarters. Perhaps .1x + .25(13 - x) is even less understandable as the value of all the coins. And perhaps the equation .1x + .25(13 - x) = 2.80 is even less understandable yet.
How can we make these algebraic expressions more understandable? I have few things I mention in class that I hope will make the algebra more understandable. I talk about appearances and expectations. "Does this expression, .25(13 - x) look like a number of dollars?" I will ask the class. And I'll give the answer, "No, not really." I'll continue, "But what should a number of dollars look like? Should we put a dollar sign in front of it. Generally we don't. We don't want to clutter up our work." I then proceed to explain that an algebraic symbol will not generally look like any particular quantity. The expression, 2x - 5, might represent a number of gallons of gas. It might represent the value of a diamond. It might represent a number of miles, or a number of apples, or a number of inches, or a hat size, or a zillion other things. What it represents comes from the problem, and we must know not only what it represents, and why it represents that.
Often there are several steps in building up an algebraic expression. If we say that x is the number of dimes we are already a little bit abstract. "x" does not look like a number of dimes. But we say that it is anyway. We have agreed that for the moment at least, that is what we mean by x. Then we come to ".1x". That doesn't look like a value. It doesn't look like money. Should it? We can remind students that if we had 3 dimes and each dime was worth .1 dollars then we'd multiply 3 x .1 to get the total value. I say "remind" students, because if that bit of math is missing from the students minds then the prospects of going further seem grim. But surely that is understandable to college students. Still it is one step more abstract than simply saying that x is the number of dimes.
Should we talk about why we use .1x instead of 10x? I think we should. It only takes a minute.
Then we repeat the process with quarters. "13 - x" does not look like a number of quarters. How could it? Yet it is, anyway. Does it have to look like a number of quarters? So we press on. Each step does not seem abstract to the teacher, but I must conclude that the abstractions mount up for many students.
Having just said that you have to know what each expression in an equation means, I risk seeming contradictory when I also tell the class that once the equation is set up, then you no longer care what each individual part of the equation means. Solving the equation should be done in a mechanical fashion, blindly, by rote. Equation solving should be a skill, not an exercise in logic. We use logic to set up the rules and procedures by which equations are solved, and that is very important, but we need to go a step farther than that. We need to solve enough equations so that we can forget the logic. We need to be able to solve equations (simple equations I'm talking about here) quickly, automatically, and efficiently. This automaticity needs to have been developed previously (which gets into levels of fluency, a very important concept in education).
I have said that every written problem assumes you know something that is not explicitly stated in the problem. For coin problems this background knowledge is simply how to add up the value of mixed coins. Next we go into distance problems. Here the background knowledge is not quite so elementary.
Does everyone in the world know that distance is the product of rate times time? Or should at least anyone who gets into college know that? Should this be considered background knowledge too obvious to need mentioning? Well, perhaps it should be, but I do not treat it as such. I think it needs to be explicitly presented as knowledge that is needed to do simple distance problems. So I do. It doesn't take long.
"If you go fifty miles an hour for three hours, can you figure how far you have gone?" I ask the class. "Then will it always be true that distance will be the product of rate times time?" I develop this idea with another example or two, and with the formula d = rt. My impression is that this is familiar to most students, if not all, but that does not mean that I can omit it. I also mention that we are talking only about a constant speed. Some students in the class, I presume, have had high school physics and has studied accelerated motion.
I next develop the formulas for rate and time. I like to do this two ways. First from a intuitive way - if you travel 150 miles and it took you three hours, then you must have gone 50 miles an hour, so it appears that you can always compute speed by dividing distance by time. And second in an algebraic way - if the formula d = rt is true, then we can solve that formula for r by dividing both sides of the equation by t, thus r = d/t. And of course I do the same for the formula for t, t = d/r.
The homework page for distance problems contains a few exercises in writing algebraic expressions for distance, rate, and time. This is easy for many students, but certainly not all, at least not in my classes. I point out that if you have any two of those three quantities then you can get the third. More concretely if two boxes in a row in the chart are filled in, then the third box can be filled in, with either a number or an expression. How come? Well, look back at the formulas.
Then I proceed to the sample problem. Fred and Joe start 270 miles apart and drive directly toward each other until they meet. Fred travels at 40 mph and Joe travels at 50 mph. In how many hours will they meet?
The method to do this problem is pretty straightforward, of course. Start with declaring a variable. At this point I talk about using a chart to declare a variable. I tell them that a chart works very well for some written problems, but not for others. A chart works very well for distance problems. I point out, or at least try to, that we don't have to use a chart. We could declare the variables just like we did with the number problems and the coin problems. "Let x = Fred's time", and the second step could be "Let x = Joe's time". Then the next step could be "40x = Fred's distance," followed by "50x = Joe's distance", and then the equation 40x + 50x = 270. And similarly we could set up a chart for the coin problems.
Can students understand that using a chart is just a way of declaring variables? And do students understand that "declaring a variable" just means to decide what our unknown will be. Or do students just memorize a recipe? I don't know. I suspect many students don't have the understanding I would like. I suspect many students just try to memorize the recipe.
Here is something I would like to do, and I have done it to some extent. I would start the chart with x in the box for Fred's time. Then I would ask the class if x has to be time. "What would happen if we put x in for the distance that Fred travels? Could we do that? Is there any rule that says x has to be time. Where would such a rule come from?" Then I would leave the problem I had started on the board, move over the other side of the board, draw a new chart, and put in x for Fred's distance. "Will this work?", I ask the class. "Should we follow up on this? Over there on the other side of the board we started one way. Over here on this side of the board we're starting another way. Over there we said x will be a number of hours. Over here we are saying we'll let x be a number of miles. We won't try to let x be the speed, because speeds are given in the problem. But should x be miles, or hours? Would it work either way? Does it matter what you choose to let x be?" I would then proceed to work the same problem both ways. I would point out that, "over there we came out with x = 3, but over here we came out with x = 120. So what is x? Did we make a mistake. Then of course I go back to how we declared the variables. In one method x is 120, because we said that x is the number of miles. In the other method x is 3 because we said that x is the number of hours.
I would like to do this because I think it would be well worthwhile. It would give a perspective, an understanding, that would be very valuable So why not do it? Time is why. To explain the problem starting with x for time takes a while. Going through the same problem by a different method also takes a while. Is that "while" available? The clock is ticking. At the beginning of the hour we spent some time going over the coin problems on today's homework. Then I spent some time explaining the relationships between distance, rate, and time. At this point maybe there is 15 minutes left in the hour. Is that enough time to give a good explanation of what I am talking about. Or is that just enough time to get everyone confused?
How much time is needed to thoroughly explain everything that needs to be explained for this topic? What can I leave out? The explanation of the relation of distance, rate, and time? Maybe. Did they all get that pretty good in the eighth grade? Or is it obvious? My conclusion based on my experience is that I do need to explain and develop the relation of distance, rate and tine. Could we leave out the examples that apply this relation? Could I leave out the few minutes I spent pointing out that if you have two of those quantities in a row then you can fill in either a number of an expression for the third quantity. Could I leave out the intuitive explanation of why time is the quotient of distance divided by rate? Could I leave out the algebraic explanation of why time is the quotient of distance divided by rate? Could I leave out the lecture that starts out "40x, does that look like a number of miles? What should a number of miles look like? . . . . ." and goes through "okay 40x and 50x are both expressions for a number of miles, so does that mean that the expression 40x + 50x is also a number of miles? or maybe if you add miles to miles you get apples . . . . ."
Can I just talk faster?
The last type of written problem I try to cover is mixture problems. An example (the first example on homework 5) is:
“How many pounds of beans worth $1.20 a pound must be mixed with 30 pounds of beans worth $1.50 a pound to make a mixture worth $1.40 a pound?” How should the problem be explained? How should mixture problems in general be approached.
I start out by reminding the students of the general algebraic approach that we are using. We declare a variable, or variables, and try to come up with an equation. Then we solve the equation. By this time I have hopefully pointed out that in general, but not invariably, a good guide for declaring the variable is to find the question in the problem. In this problem the question is “how many pounds of beans . . .”, so can we start out “Let x = the number of pounds of beans.” Hopefully students will realize that there are two types of beans, the cheap ones and the expensive ones. Therefore we need to pin it down a bit more. Should x be the number of pounds of cheap beans or expensive beans? Again we can look for the question in the problem a bit more closely. The question is “is “how many pounds of beans worth $1.20 a pound . . .”, so can we start out “Let x = the number of pounds of beans at $1.20 a pound.”
But where do we go from there? Somewhere around this point I remind students that for distance problems it worked very well to declare variables in a chart, so perhaps that would be a good thing to do in mixture problems also. Indeed I make it a general rule. Use a chart to declare your variables for both distance problems and for mixture problems.
Mixture problems provide another opportunity to stress that declaring variables is just a way of thinking precisely. If we don’t know what we mean for x, then why bother to solve for x?
And again I will emphasize to the students that every written problem assumes some background knowledge. The background knowledge here is a little more vague than the background knowledge of time distance, and rate. The background knowledge is the meaning of concentration, though the word “concentration” may not actually be used. Do the students really know what we are talking about when we say “a 30% solution of alcohol and water”? Do they understand what we mean by “a 30-70 mixture of red beans and brown beans”? Do they know what we mean when we say “30% of the beans are kidney beans”? I’m not sure. I assume that for many students all of those statements are just everyday language. But perhaps for a significant part of the class those phrases do not have the meaning we would expect.
And, perhaps even more importantly, various ideas like these might be more language dependant that we realize. Perhaps when studying science students made sense out of a phrase like “a 30% solution of alcohol and water”, but don’t get much meaning out of “My portfolio is a mix, about 30% in common stocks”. Or perhaps they can understand “30% of all my retirement investments are common stocks”, but don’t quite interpret “a 30% mix of common stocks” in the same way.
And in mixture problems we have a concept similar to value in coin problems, but a little more obscure. When we say “two liters of a 30% alcohol solution and five liters of a 42% alcohol solution . . .” that is very much like two stamps worth 30 cents each and five stamps worth 42 cents each. But is that understandable to students? And if it is not, what do we do?
It seems to me that in this sort of thing back ground knowledge and understanding the written problem has a pretty nebulous border between them. In distance problems, in contrast, it seems much easier to separate the background knowledge - the relations between rate, time, and distance - from applying that knowledge in an algebra way. Indeed for mixture problems it may be that “background mathematics” may be what is needed more than “background knowledge”. Either way, if that background is weak, the written problems we are trying to teach will be difficult.
I put investments in with mixture problems. I start out with this problem:
A sum of $8000 is invested in bonds, part at 8% and part at 14%. The total return on the investment is $840. How much is invested at each rate? (Hint: This is very similar to a mixture problem. Start by filling in the chart.)
Unfortunately I also end up with this problem. This is problem five on the mixture problem assignment. But is it a mixture problem? Well, yes and no. Is it understandable at this point, assuming that students first see this problem when doing mixture problem homework, and then we go over it in class the next day? My experience has been that it is about as understandable as the mixture problems. Do they have the background knowledge, what we mean by interest, and "return" on an investment? I'm not sure. But this is the fourth assignment on written problems. There is simply no time to do any more.
Next comes work problems, such as:
John can paint a fence in three hours. Mary can paint the fence in two hours. If they work together, how many hours would it take them to paint the fence?
I don't teach work problems. There is no time. I already spent four class periods on written problems. We did them carefully and thoroughly. I gave thousands of words of explanation. At this point many students, those who are both conscientious and capable, understand them in the way that I want them to. Many other students, unfortunately, skipped class and/or didn't do their homework. And some, I presume, have made little progress in spite of a lot of effort. I wish we could do work problems. For those who understand, work problems could help to cement that understanding in place. For a few work problems might make that understanding attainable. But I don't feel another day on word problems is justified, so in recent years I have not tried to do work problems.
But I am going to discuss work problems anyway.
In explaining this problem we should first consider what we should let x be. Following the general rule of letting x be whatever the question asks for we could say, "Let x = time to paint the fence together". But where do we go from there?
The next step is to say that 1/3 is the amount of the job that John can do in one hour, and 1/2 is the amount of the job that Mary can do in one hour. Then we can say that the amount that John can do in one hour and the amount that Mary can do in one hour can be combined to get the amount that they can do together in one hour. Therefore the equation is 1/3 + 1/2 = 1/x. Solve for x and we have the answer.
Is this understandable to students? My experience (though I have not tried to teach this recently) is that no, it is not understandable to students. The more capable and conscientious students may understand it enough to be able to do some more work problems, but it has never been the sort of thing that can "click" into place in the students' minds the way we would like it to. Is this the best we can do?
My conclusion is that work problems would be understandable (in the way we would like students to understand them) only be a rather long process. That process is to treat as background knowledge the idea of rate in a general sense, to present enough examples and problems with rate to gain some fluency with the concept, and then to consider reciprocal rates, and then to consider and develop the idea of "parallel addition" as a mathematical topic. It seems to me that to do all this would require several days, even at the college level.
To understand what I mean by "parallel addition" consider the formula for combining resistors in parallel. I have seen this presented in a number of books, but never with an intuitive explanation of why it works. Rather we are simply given the formula , and are expected to learn to use it. But the reason it works (and surely some books explain this) is that 1/R is conductance, measured in mohs. If a resistor has a resistance of 20 ohms, then it has a conductance of 0.05 mohs. If two resistors are connected in series, then their resistances are combined. But if two resistors are connected in parallel, then their conductances are combined. That idea produces the familiar formula above.
Are there other examples of "parallel addition"? I presume there are many, but they don't necessarily make up a part of everyday life for most people. Here is an example:
A car gets 19 miles per gallon of gas. A truck gets 13 miles per gallon. What is their combined gas mileage?
Could we treat this is a typical written problem - declare a variable and come up with an equation? Can we start with "Let x be the combined gas mileage", and then say 1/19 + 1/13 = 1/x? We can, but is it understandable? My argument is that, at least for normal people (math geniuses need not apply), it is not. But it can be made understandable to college students (perhaps to eighth graders) if we develop the ideas carefully enough. A car that gets 19 miles per gallon has a "gas mileage" of 19, but a "fuel consumption" of 1/19. Gas mileage would be measured in miles per gallon, but fuel consumption would be measured in gallons per mile. We have to make fuel consumption concrete enough to work with. Then it is quite understandable that if two vehicles are both operating at the same time, their combined fuel consumption is the sum of their individual consumptions. To get both vehicles one mile down the road we have to supply the car with 1/19 of a gallon of gas and the truck with 1/13 of a gallon of gas. Thus the combined fuel consumption is 1/19 + 1/13.
Can we come up with a situation in which it makes sense to add two gas mileages? Can we make a problem in which the solution comes by adding 19 miles per gallon and 13 miles per gallon to get 32 miles per gallon? I’m not sure how. Would it be worthwhile to spend class time thinking about this? I’m not sure of that either.
What is the net speed of a fly that flies forward at 1.6 miles per hour in an empty truck that is traveling forward at 50 miles per hour? Here we combine speeds. Could we think up a problem in which gallons per hour of water are added? Could we think up a problem in which students per week are added? Rates, and reciprocal rates, and how rates may be combined, is not an easy subject. Can we fit this into the curriculum somewhere? How about eighth grade?
Should we try to teach work problems without first developing the idea of rates and how to work with them first? I'm not sure.
Many algebra books try to deal with written problems with a generalized approach. They present and explain a distance problem. Then they present and explain an investment problem. Then they do another problem or two of the usual categories. Is this successful? Can it be successful? I'm not sure. My conclusion from my experience with my students is that the method I have described is needed, and it won't fit into one class period A generalized problem will not transfer to distance problems. You have to teach distance problems if you want students to be able to do distance problems. A generalized problem will not transfer to mixture problems, or coin problems, or consecutive integer problems, and certainly not to work problems.
Since I settled into this four day method I have felt it is one of the important things I do in college algebra. I have felt that every semester a few students understand word problems for the first time, or in a way that they have not understood them before. And I feel this understanding, imperfect though it may be, comes across to students as part of the beauty and intrinsic value of math. Of course this is purely subjective opinion.
Written problems compete in the curriculum with many other topics. In my present situation we have "course guidelines" that list the topics we should cover. I feel pressure to cover all those topics as best I can. Some topics, in my view, are not worthwhile in terms of the time and effort they take. I think we would do well to throw out absolute value inequalities, and have more time available for more important topics, such as written problems. But that is just my opinion. I may have to cut back on the four days I devote to written problems in order to squeeze in all these other topics. I hate to do this. To my mind four days are about what is required for the "minimum teachable structure". That means three days might as well be zero days. No lasting benefit comes from going below that minimum. Maybe I should just spend one day on written problems, just so I could say I covered the topic. I think it would be a waste of one day, but maybe that's better than taking four days to do it right. Trade offs must be made. I’m not sure what the optimum set of trade offs might be in something like this. But I do feel I have a realistic understanding of what it takes to teach written problems to the students in my classes.
Click here for the handouts I use in teaching written problems.