Apples, Hats, and Word Problems

In almost all math courses, an emphasis is placed on word problems … applications … real-world  … translation.  Is there a valid reason to include this work?  Or, should the mathematics in a course be restricted to that which is needed to deal with the contextual situations that students encounter?  Do these verbally-presented situations have a valid purpose in our math courses?

Here is the reason I am thinking about these issues — one of my ‘Facebook friends” (also a friend in ‘rl’ = ‘real life’) posted a link to a captioned picture at an online site.  The caption reads:

Everytime I see a math word problem, it looks like this:  If I have 10 ice cubes and you have 11 apples … How many pancakes will fit on the roof?
Answer: Purple because aliens don’t wear hats.

This ‘spoof’ takes its energy from the fact that we tend to have problems that are either obviously worthless (pancakes on the roof) or unreasonable (hats).  You are most likely responding the same way I did … “The problems in MY course are good and realistic problems!”  The criticism here is not what experts might see … the criticism is in what students (novices) see in this work.

First, here is a link to a short report I wrote a few years ago:  Ignore the Story  This report does not deal with how effective ‘word problems’ are — it deals more with qualitative studies.

Second, let us admit a basic fact:  The problems we can include in a course will not convince the majority of the students that those problems provide a justification for the mathematics covered.    Yes, I realize that some faculty will not agree, and hold the position that properly chosen contexts and applications will convince students.  We have a tendency to underestimate the complexity of going into a context to apply mathematics and coming back out of the context; this is hard work, and many students will seek any avenue possible to avoid dealing with the deeper relationships — sometimes working harder to avoid the process than it would be to complete it.  In other words, students will tend to map application processes to the procedures required at the shallowest level that ‘works’.

This minimalist tendency is not unique to word problems; nor is it unique to mathematics … our colleagues in other disciplines experience the same problem.  The difference is that we, in mathematics, expect students to deal with short (often cryptic) descriptions of situations in a variety of areas; students are expected to see mixing two levels of milk fat to be mathematically equivalent to mixing acid solutions of different percent concentrations even though the phrasing is often significantly different.

Elements of a course should support the instructional objectives of the course.  This implies that verbally stated problems should contribute to the mathematical outcomes for a student.  We should be using verbally stated problems to encourage and build a more complete understanding of basic mathematical ideas, and these verbal problems should also contribute to linguistic literacy for our students.  Achieving correct answers is not nearly as important as being able to paraphrase the situation, summarize it, state the known and unknowns, identify relationships between quantities that might be helpful, and write at least one mathematical statement that can be used to ‘solve’ the problem.  These abilities (which combine the linguistic and the mathematical) are much more related to a good prognosis for employment and other goals … more than basic skill accuracy, and more than algebraic manipulation by itself.

The emerging models for developmental mathematics (such as New Life, Pathways, and Mathways) tend to emphasize deeper processing of verbal problems, and de-emphasize repetitive ‘word problems’ which might look like the ‘apples, hats, and aliens’ spoof.  I encourage you to examine how verbally stated situations are used in your courses.  Do they contribute to both understanding basic mathematical concepts and linguistic abilities?

 
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1 Comment

• 

  By schremmer, May 24, 2012 @ 10:11 am

  What’s wrong with the kind of things at

  Regards

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