“Problem solve” Dept

A recent article announced that the National Science Foundation (NSF) funded a grant for West Virginia University College of Education and Human Services The grant is to help educate math teachers on a new way of teaching math to teachers. For those of you new to all this, NSF spent billions of dollars in grant money in the early 90’s to fund (in my opinion and the opinion of many others) ineffective and damaging math programs including Investigations in Number, Data and Space; Everyday Math; Connected Math Program; Core Plus; and Interactive Math Program.

Of particular interest to me was this sentence: “The hope was for math teachers to find ways to teach students how to problem-solve.”

It used to be that students solved problems.  But now in today’s era of math reform, they “problem-solve”. Popular use of this rather irritating verb form harkens back to NCTM’s 1989 standards which downplayed the importance of procedural skills, and replaced those with students achieving “deeper understanding” and being able to problem-solve.

The core belief behind the current math-reformers’ use of the term “problem-solving” is that it is a core competency that can be taught independent of the domain in which a problem appears. Little to no importance is given to mastery of procedural skills, instruction on how to solve particular types of problems, nor sufficient practice solving such problems.

The typical problems of the past (distance/rate, mixture, number, coin) are being replaced with what reformers believe are problems that students are interested in wanting to solve. These are typically one-off problems that don’t generalize and for which little to no prior problem solving procedure has been taught. 

One math reform approach has been to present students with a steady diet of “challenging problems” that neither connect with the students’ lessons and instruction nor develop any identifiable or transferable skills. The following problem from Hjalmarson and Diefes-Dux (2008) is one example: How many boxes would be needed to pack and ship one million books collected in a school-based book drive? In this problem the size of the books is unknown and varied, and the size of the boxes is not stated.

While some teachers consider the open-ended nature of the problem to be deep, rich, and unique, students will generally lack the skills required to
solve such a problem, skills such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification.  The belief is that just as students develop problem solving habits for routine problems, a similar “habit of mind” or problem-solving schema occurs for solving non-routine problems.

Based on my experiences as both student and teacher, as well as the experiences of veteran math teachers, I submit that a substantial education in mathematics should steer a middle course between the proliferation of routine problems and reliance upon unique, complex projects. Students should learn to apply basic principles in a much wider variety of situations than typically presented in texts. Such problems, however, should not be as
complex or as time consuming as the example above. A math problem is not necessarily useful just because it requires outside-of-the-box insight and/or inspiration and will generally not result in a problem-solving “habit of mind” or schema.

Problem solving techniques taught independent of the domain in which they occur include such things as “work backwards”, and “find a simpler but similar problem”. But without experience, practice and mastery of domain-specific problems, asking a student to find a simpler but similar problem is as useful as telling a novice bike rider to “be careful” when taking a ride on their own.

Sweller et al. (2010) state that problem solving cannot be taught independently of basic tools and basic thinking. Over time, students build up a repertoire of problem-solving techniques. Ultimately, the difference between someone who is good and someone who is bad at solving nonroutine problems is not that the good problem solver has
learned to solve novel, previously unseen problems. It is more the case that, as students increase their expertise, more nonroutine problems appear
to them as routine.

Looks like the idea of problem solving as a core competency will be taught to a bunch of lucky teachers in West Virginia thanks again to the misguided largess of the National Science Foundation.

References

Margret A. Hjalmarson and Heidi Diefes-Dux (2008), Teacher as designer: A framework for teacher analysis of mathematical model-eliciting
activities, Interdisciplinary Journal of Problem based Learning, Vol. 2, Iss. 1, Article 5. Available at http://dx.doi.org/10.7771/1541–
5015.1051.

John Sweller, R. Clark, and P. Kirschner (2010), Teaching general problem-solving skills is not a substitute for, or a viable addition to, teaching mathematics, Notices of the American Mathematical Society, Vol. 57, No. 10, November.