Symbols

Symbolic expressions are typically much more powerful than numbers

• The formula for the area of a circle is simpler and more powerful than the approximation and far simpler and more useful than a table of numerical values.

There is a trend in K-12 to emphasize numbers over symbols. Factors driving this shift are suggested below, but whatever the reasons the trend puts students at a considerable disadvantage at the college level. It is vital that they be proficient in working with symbols in any context, not just as variables but in coefficients and other parameters. Problem sets reflect this by plentiful use of symbols.

Example

1. Find (a numerical approximation of) so that a triangle of height and width has the same area as a circle of radius .

1. Find (an expression for) so that a triangle of height and width has the same area as a circle of radius .

It is now common for students to see the first version as easy and solve it instantly using a calculator. They see the second version as qualitatively different and solve it more slowly. Some students are uncomfortable with any problem with non-numerical answers.

Contributing factors

Understanding root causes helps with developing strategies to fix a problem. The factors identified here seem reasonable but may not be complete or even primary. This does not change the fact that there is a problem.

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Calculators

Calculators are the most obvious driver of emphasis on numbers. Tactile learning of keystroke routines does not generalize to symbols in the same way that by-hand arithmetic seems to, so calculators do not support development of symbolic skills.

This does not mean calculators are evil, just that they should not drive the curriculum. It also may be possible to design calculators that do support symbolic skills. For instance it might be better to explicitly enter, and perhaps edit, a complete expression before evaluating it, rather than evaluating intermediate steps.

Multiple-choice tests

A perennial problem for test developers is that the structure of possible answers may enable identification of the correct one without working the problem. The easiest way to destroy structure is by numerical evaluation.

For example the numerical version of the example above has a numerical answer that is easy to hide among other numbers. The answer to the symbolic version has structure that is almost impossible to miss.

Another advantage is that simply changing numbers in the numerical version gives variations that students will see as different problems. Changing symbol names gives an obviously equivalent problem. Another way to phrase this is that numerical evaluation obscures commonality that is obvious on a symbolic level, and test designers exploit this.

The difficulty is that the test focus on numerical problems has made success with numerical problems the de facto objective of the curriculum. Decisions made for the convenience of test developers have driven instruction in exactly the wrong direction: away from symbols and abstractions that reveal commonality.

This does not mean that tests, even multiple-choice tests, are evil, just that they should be carefully designed to reenforce rather than undercut educational objectives. This is possible but difficult.

An Analogy

Symbols make expressions functional much like links make web pages functional. Plugging in numbers destroys this functionality like printing a web page kills the links. Calculators, in this analogy, are like printers. Yes everyone has one and should know how to use it, but printing a web page is rarely the best thing to do and makes little sense as a goal. Nor would generating printouts be a good way to test students' use of web pages.