What price are we willing to pay for ‘correct answers’? What gains (benefits) should students expect for dealing with applications in a math class?
In our beginning algebra class this week, we spent much of our time on applications. Many of these were the typical puzzle problems involving tickets and cars, integers and angles. As is normal for this course, students really wanted some magic — a rule that would help them get the correct answer for all of the problems. Some of the students remembered some magic from a prior math class; one piece of magic was the word ‘is’ … the other piece of magic was a triangle (for mixtures).
We often provide rules (whether perfect or not) that are meant to help students get more correct answers for applications (broadly stated as word problems involving a context). We tell students that “of” means multiply, and that “is” means equal; the prototype for both rules is the “a is n% of b” template (a worthless model, as normally taught). Students who have experienced this ‘correct answer’ driven course encounter many problems when faced with a narrative about an application, where ‘of’ is the normal preposition and ‘is’ is the normal verb connecting phrases. We train our students to surface-process language for the sake of correct answers, and wonder why students continue to have problems with applications.
One of the most challenging problems we did this week was this simply-stated problem:
A store claims that they markup books by 30%, and the selling price for one book is $79.95. Find the cost of the book to the store (before the markup was added).
Every student in this particular class was a graduate of our pre-algebra course, where this same problem was done as part of a longer chapter on percents and applications. Every student in this class wanted to either multiply by 30% or divide by 30%; a few students thought that there was a second step where they needed to add or subtract this result.
Quite a few of the students could do this problem:
A store sells a book that has a cost of $61.50, and they have a markup on books of 30%. Find the selling price.
Their success on this arithmetic problem was not based on understanding the words any better (the words are the same). Their success was based on the ‘magic’ rules we had given them that happen to work: multiply by the percent, add or subtract if needed.
The whole point of experiencing applications such as these is to build up the student’s mathematical reasoning. There might be magic in the world, but magic is not reasoning. Correct answers based on locally-working magic is worse than wrong answers based on weak reasoning. If our courses include applications, keep the magic of “is” out of the course … and all other magic.
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