In Brief
The restriction to problems that can be quickly solved has biased the school math curriculum in ways that cause difficulty in later study. Are there ways to include long problems in the programs of at least high-achieving students?
The Situation
School math problems (in the US) are carefully contrived to be short and most are single-step. The justification is that most material at this level can be learned this way: a student who can do short problems can also, with care and persistence, do long problems. This approach has substantial advantages:
- more problems can be given;
- less sensitive to errors since damage is limited to a single problem;
- lower demands on attention span and organizational skills; and
- the teaching of short tasks is less vulnerable to classroom disruption.
There are obvious disadvantages:
- students learn to tolerate error rates that almost guarantee failure with long problems;
- they do not develop the attention span and organizational skills needed to carry through long or multi-step problems; and
- genuine applications are not accessible: the shortness constraint leads to word problems so contrived that they become jokes (a train leaves Chicago … ).
There are disadvantages that are less obvious but no less serious. For instance symbolic work is slower than numerical, especially if calculators are available. Symbolic work has therefore been de-emphasized, leading to weak skills.
The worst problem is that at the algebra II level the justification is wrong: there are important points that arise only in long problems. The need for brevity leads to a focus on special cases that miss these points. Teachers sometimes compound this problem by using methods limited to these special cases. Some examples:
- Teaching and testing of polynomial multiplication focuses on products of binomials. Teachers introduce the FOIL nemonic that leads to good success with this case. Unfortunately it obscures underlying structure so trinomials seem qualitatively different, and many students have difficulty with them. (See the teaching note No FOIL).
- Teaching and testing of roots of quadratic polynomials focuses on quadratics with integer roots. Teachers introduce factoring tricks that work well in this setting but fail or encourage errors in general. When (or if) the quadratic formula is introduced there are still often enough integer-root problems on tests for students to get by without the formula. Unfortunately mastery of the quadratic formula is vital for university science and engineering math courses.
Perspective
Most students will not use the analytical tools developed in school math courses. They benefit from the medium (development of formal manipulation and logical skills in a modestly complex setting) rather than the message. The disadvantages described above do not seriously effect these students.
The advantages of the current system must be taken very seriously. Powerful factors (television, video games … ) seem to be driving reductions in attention span, decreased willingness to develop effective skills, and increases in disruptive behavior. Schools have no choice but to adapt to this. Calls for higher standards across-the-board are unrealistic.
The minority of students interested in high-level technical careers do need analytical tools and the described disadvantages do seriously effect them. The best (say top 5%) of US high-school graduates are significantly less well prepared than the top 5% 30 years ago, or the top 5% in many other countries. Low US representation at top levels in technical careers indicates that most of these students are unable to overcome their deficits.
Questions
The overall question is: how can long problems be incorporated in coursework and (this is vital) in testing of at least high-achieving students interested in technical careers?
Is it feasible to have a small amount of long-problem work in early education, say by fifth grade? This would have to be nearly universal since it is too early for students to be tracked into special classes.
Advanced Placement courses might seem to be a natural home for this material. Unfortunately these courses are essentially defined by the AP Calculus test and this has short-problem defects in spades. Learning in these courses is often so defective that students in university science and engineering calculus courses have to junk it and start over. Is there any way to work around this short of a complete about-face by AP test designers?