The Tremendous Value Of Solving Difficult Algebra "Story Problems"

Recently, my 14-year old son requested that I help him with the following beginning algebra problem:

The manager of a specialty food store combined almonds that cost $4.50 per pound with walnuts that cost $2.50 per pound. How many pounds of each were used to make a 100-pound mixture that costs $3.24  per pound?  From Algebra: Introductory and Intermediate by Aufmann, Barker, and Lockwood, 3rd Edition

My son's reaction?  "I don't get it."  Then after showing him some of the steps, I still heard "I don't get it".  I could imagine the frustration of a teacher with 30 or so students all reacting this way.  But, after having my son write down the steps similar to those shown below on about 4 or 5 problems, he did start to get it.

Always let X = what is being asked for.  In this case there are two things being asked for .  So we let x stand for one of them. We let X = pounds of almonds.

Form an equation by first sketching a diagram illustrating relationships. In this case, we have

And now, the equation must still be solved!

So What Is The Point Of Subjecting Students To This?
The value of students learning how to do this problem is not so much learning how to do mixture problems. Because truthfully, such mixture problems are rarely seen in day-to-day life.  The great value of this problem to the student is obtaining the confidence to solve multi-step problems in mathematics, science, and other subjects.  In this MAA Statement , they conclude

. . . the best mathematics preparation for college is continuous coursework throughout high school that fosters a strong background in algebra and geometry, and brings an ability to solve multi-step "word problems" and an open and positive attitude towards problem solving in general.

Until my son was assigned these problems, he only saw 1 or at the most 2 step problems that may have included some fairly simple algebraic operations. This problem, however required some real thought!  Ultimately, the goal should be that students be unafraid to tackle problems much more complex than the one above by chipping away at the problem, breaking it into comprehendible chunks, and persisting until a final solution is arrived at.

My Own Impossible Story Problem
When pursuing my degree in higher mathematics, the very first course I took was one called Abstract Algebra. To see what this course is about visit this Wikipedia page on Abstract Algebra.  On the very first day the seemingly insane instructor, writing at a seemingly insane pace, wrote something on the board similar to

 
Excerpt of Image From http://math.rice.edu/~rpreeti/math356/hw3-pg1.JPG 

Well, needless to say, I panicked.  I probably had that same sick feeling that 14 year olds have all over the planet as they tackle mixture story problems! And to top it off, none of the tutors in the math lab could help me since it was beyond their course work!  Here is what I did:

• I panicked some more and considered dropping out. After all, this math made NO sense to me. Heck, it's just a bunch of symbols.  It's not even math! Sound familiar?
   
• After deciding to at least try and tough it out, I obtained every book in the university library pertaining to Abstract Algebra that looked like it could be helpful.  I had a stack well over a foot high.
   
• With the help of the books, and through some great persistence on my part, I started breaking down the subject. First I learned what all these strange symbols stood for.  For example, the symbol | means "such that".  Also, I learned in detail what quantities like "groups" were.  I would devote as much as 6-8 hours in a single evening to this subject alone.

And it started to click! Toward the end of the course, it came to me.  This Abstract Algebra course is not about learning techniques or applications.  This course, in addition to teaching the general structure or our number systems,  teaches one how to tackle multi-step seemingly impossible abstract problems! In fact, I would say this is the best course I ever took in college.  After this course, I was no longer intimidated be very lengthy and abstract problems.  I knew that if I was persistent and tackled the problem a small piece at a time, I would eventually succeed. And I did.  In subsequent graduate level courses, when given 10-page assignments consisting entirely of very abstract and difficult proofs, I viewed these as challenges rather than obstacles, and I persisted and succeeded.  Later on, when I decided to learn PHP/MYSQL computer language in order to construct a data-base driven website,  I used this same persistence to learn the language by breaking down the tasks and concepts into manageable pieces.  And truthfully, learning PHP/MYSQL was much more difficult than learning Abstract Algebra - Yet I don't think I would have had the confidence or the symbolic reasoning ability to learn this computer language without my successes in math.

To my Abstract Algebra professor, I say "Thank you!"