Word Problems and Reasoning … Rule 42

In my introductory algebra class, I gave a quiz today; this quiz included this question:

Some milk having 1% fat is being mixed with milk having 4% fat; the mixture will be 100 gallons, and have 2% fat. How much fat is in the mixture?

Now, I always review the quiz right after we complete … so sometimes I get to see some interesting reactions.  Please understand that all of the items on this quiz dealt with ‘puzzle’ word problems (not much real context), so students were not feeling mellow … many were feeling quite a bit of stress.  Besides the stated reactions about the question being ‘tricky’, I got to see students respond when they realized what they were supposed to do.

A typical wrong response to that question was to start working on solving the ‘problem’ they expected to see … how much of each type is supposed to be mixed?  Quite a few of these students wrote the correct equation including the “0.02(100)” for the milk fat in the mixture.  However, not many of these students realized that they had the answer to the question.  Our entire approach to these mixture problems in the class prior centered on “value = rate(quantity)” as a basic concept.

I was pleased that some students just wrote down the answer (2 gallons), perhaps with a note “0.02(100)”.  These students got that basic idea that value = rate(quantity).  I’d say that this 20% compares with the 40% who tried to ‘solve’ the problem but never realized that they had the answer … and the 40% who had no idea what to do.

One of the culprits for the difficulties is the inadequate way percents are done in math classes.  We focus so much on correct answers that we do not make it clear that percent is not ‘how much’ … and that every percent is a rate which is multiplied by a base.  For my question on the quiz, just knowing “percent times base” is sufficient to get the right answer (and show some understanding).

The other culprit is based on the high-anxiety suffered by students when faced with “word problems”.  I’d like to think that my class presents word problems as a reasonable use of language and algebra, even if the problems are either trivial or uninteresting.  Further, I’d like to think this positive approach helps students be more comfortable dealing with these problems.

Some readers might wonder “why do those puzzle problems at all” … perhaps we should “make the content relevant to the students”.  With all of the focus currently on ‘alignment’ and ‘context’, those are reasonable questions.  Based on my understanding of the learning process (along with some sociology), the question is not easily answered.  I am pretty sure that covering ONLY relevant applications is not a good idea for a mathematics course serving a general purpose; it might work in occupational math, or specialized math, but not so much when there is so much diversity among the students.  One student’s relevant problem is another student’s puzzle problem, and another student’s life survival issue; in addition, high context in problems can localize the learning and interfere with general reasoning and understanding.

So, I will continue to work with quite a few puzzle problems in our introductory algebra course — and keep a focus on the basic ideas that allow us to understand and solve them.  My goal is to help students develop a deeper understanding and develop connections.

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