Learning a new skill is a matter of repeating difficult, unfamiliar experiences so that new, similar experiences become less difficult.
When it comes to passing courses in math and science, the experiences in question involve successfully solving word problems, narratives that use common language to express a problem which is resolved through the use of mathematical relationships.
There is no shortage of web-sites which describe the the ‘best’ approach to mastering word problems, but generally speaking, most sites describe successful problem solving as a sequence of four basic steps:
- Understand the problem
- Devise a plan
- Implement the plan
- Evaluate the plan's effectiveness
This note will focus on the first two steps which may can be the most daunting obstacles to a successful exercise.
Understanding the problem is the first key step to a successful solution. Nerves or time pressure often result in the student skimming the narrative, looking for keywords that will trigger a reaction which kicks those problem-solving juices into high gear.
But too often, students start jumping to conclusions and implement a plan which may completely miss the target of the exercise’s objectives.
So after initially skimming the description of the problem, it is often best to just take a breath.
Once the hyperventilating stops, relax and re-read the problem. It may help to highlight key phrases, or even re-write the givens in tabular form to provide a good summary of the known facts behind the problem.
Depending on the problem (especially in courses like physics or mechanics), it is often a good idea to sketch out the actors, vectors, or any other symbols that help one better visualize a different perspective of the problem. Research shows that the more detailed and accurate the illustration, the better likelihood of success.
So let’s talk about the ‘devise a plan’ part.
Novices (students just getting their feet wet in a subject) may read a word problem and immediately search for an equation that seems to fit. More often, they try to use an approach that matches what was tried in the past to solve a similar problem. This method is indeed, part of the practice, practice, practice mantra that helps provide a good foundation for future problem solving.
But eventually, either in a homework assignment, or worse, on an exam, there is presented a problem that doesn’t quickly fall into any familiar pattern that can be used as a template for a solution.
Given that the student has invested the energy to really understand the nature of the problem (step 1), there sometimes comes the need to use what has been called the ‘expert’ approach.
The ‘expert’ approach (in this case dealing with physics problems) pushes the student to set aside known equations and formulas in favor of taking a step back to look at the bigger picture of the problem.
Sometimes, this involves the student re-writing the problem statement in his/her own terms, including notes which highlight known concepts the student thinks may be relevant (for an example of the ‘expert’ approach, see this document). Other times sketches of the elements in the problem may be simplified through simple boxes and arrows representing the actions described in the problem narrative.
The key here is that this phase involves less focus on finding the right equation, and more focus on identifying the physical laws and metrics in play. Once the relevant rules or relationships have been identified, the equations will eventually follow.
Next Month: Implementing and evaluating the plan.
References:
garyhall.org.uk - Mathematics Problem Solving Strategies
West Virginia University - Expert Approach to Solving Physics Problems
California State University - Physics Problem Solving Strategy
European Journal of Physics Do Students Benefit from Drawing Productive Diagrams