There is no x and y in the real world

Five months ago, when we were studying systems of linear equations in Algebra II, I gave my kids the following problem on a test:

x + y = 20

5x + 3y = 70

I remember distinctly that most students – over 80% – got the correct answer. I was pleased – to my mind they understood what I had been teaching and were able to apply it successfully to a problem they had not seen before.

Fast forward to last week, when in preparation for the finals, I had just concluded a week of review of linear functions, equations and systems.  Thursday was the test on the review material. As part of the “free response” part, I put the following problem on the exam.

“In a room there are four-legged chairs and three legged stools. There is a total of 40 seats and 145 legs. How many chairs and how many stools are there in the room? (SHOW WORK FOR FULL CREDIT).”

Again, about 80% of the students got the correct answer, but what a difference! Most got the answer by trial and error – I could clearly see on their papers how they scribbled  different combinations of stools and chairs filling up the answer space. What is going on here?

When I asked my kids what was so difficult about that problem, what I got from their various answers is that many (a majority?) are so unused to word problems that they practically shut down when they see one.  My “big” exams always do have two or three free response problems, but usually they are more involved than this one. The student do more poorly on the free response part than they do on the multiple choice, but I had blamed that difference in performance on the caliber of the problems.

Not so – or not completely so. It is the intrinsic fact of having to deal with a word problem that causes the poor performance. Perhaps the kids are not to blame. The whole California Algebra II State Test is multiple choice. Teachers who want to prepare the kids for the test – and who is not under pressure to do so? – tend to also give simple multiple choice exams that mimic the State test.  Plus, let’s face it – multiple choice tests are so much easier to grade! As far as I know, I am the only Algebra II teacher in our department who gives free response problems.

Unfortunately, there is no x and y in the real world. In the real world we deal with variables that have names, represent quantities and have units associated with them.  Heck – in the “real real” world,  half the battle is just formulating the problem.  No wonder the kids ask “When am I ever going to use this?” Honestly, the answer is “never”. Given all the abstraction and recipe math we are throwing at them, no wonder so many of our students are bored. We need to resolve the tension between teaching algorithmic methods and teaching how to set up problems.


2 responses to “There is no x and y in the real world

  1. Many thoughts early in the morning here. Would you feel that you were compromising the goal of a question like the stools and chairs by explicitly pointing them to write equations? I find my self SO frustrated with the ‘guess and check’ strategy that I either give numbers they will not reasonably think of or explicitly note that reasonable ALGEBRAIC support is necessary for credit. But I usually end up feeling that I am not being particularly reasonable in those cases, I don’t want grades to ever feel arbitrary or punitive. The degree to which our students compartmentalize their skills is a depressing thing at times. They can absorb the algebra but don’t look to apply it elsewhere. An example I keep seeing is in my Calculus class with rational functions. I ask for oblique asymptotes so they divide and rewrite the function correctly. But then I ask for the derivative and the students go back to the complex rational function to use a quotient rule. I have to believe that in their mind the rewrite of the function is not actually the same function as before. Watching the extra effort expended is exhausting to me.

  2. Jim – excellent point. I need to be more strategic (a) in having a longer “track record” of asking for word problems and (b) in instilling in the kids that guess-and-check is not what I am after and that instead they should model with equations. Thanks for putting my laments in a clearer perspective.

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