# It's not enough for Johnny to add

A little-noticed but critical battle is taking place in Congress at this moment. The nominal issue is money for a small research program administered by the National Science Foundation. The real issue, however, is the future of mathematics and science teaching in this country.

Everyone recognizes that American scientific and technical education is in trouble. Engineering jobs are going begging -- even in a recession economy. The military is worried about the poor technical skills of its recruits. We have a shortage of science and math teachers in our schools.

Clearly we need more and better mathematics and science teaching at all levels of the educational system. Yet in its 1983 budget for the National Science Foundation, the administration proposed to eliminate all support for research on science and mathematics education.

A few sensible legislators have been fighting to reinstate this small research program that could be crucial to the national effort to improve scientific and technical education. In the next few weeks a vote on appropriations for the National Science Foundation will determine whether this research will survive.

All of this is happening at a time when the evidence of a variety of achievement tests is clearly showing exactly what it is that needs to be improved in science and math teaching. Consider the National Assessment of Educational Progress tests in mathematics, for example: in the two most recent surveys, the results show that, although skills in calculation are improving, students are doing very poorly in their ability to solve mathematical problems. This means that our children know how to do arithmetic, but they are not good at knowing when to do it -- or, more accurately, which calculations to use to actually solve a problem involving numbers.

In other words, many children are functionally illiterate in mathematics even though they can perform computation fairly well. Much the same can be said for science: our children have learned a smattering of scientific ''facts,'' but their ability to reason about even simple scientific problems is badly underdeveloped.

This grave deficiency in problem-solving ability seems hard to understand at first. After all, every school math program includes problem solving among its goals, and every math course includes practice on solving various kinds of problems that simulate real-life situations. So why should school instruction on calculation be getting results while instruction in problem solving is failing? It is because the teaching of calculation skills is based on an accurate and knowable instructional theory, but we still do not have a well-developed theory of how to teach mathematics problem solving.

This situation reflects our past history of research on mathematics learning. Much of the teaching of calculation in school is based on an analysis of arithmetic learning done more than 60 years ago by Edward L. Thorndike, one of the giants in experimental psychology. On the basis of his analysis, Thorndike developed a theory of ''drill and practice'' in arithmetic. His theory has been modified and refined over the years, but it has not been fundamentally changed.

Thorndike's theory works well whenever the things to be learned can be thought of as collections of simple associations and rules. Drill and practice can produce very good computational skills as long as the practice exercises are organized so that children can make correct calculations often enough to learn the mathematical rules of procedure and the proper associations between numbers.

But drill and practice alone cannot succeed in teaching problem solving. This is because the process of real-life mathematical problem solving is much more complex than simply responding to a set of rules or associations. I can make this statement with assurance because of a body of very recent research on how children and adults solve real-life science and math problems. This research has made it clear that there is no reliable direct association between the words in which a problem is stated and the kinds of mathematical calculation necessary to solve it. Successful problem solvers, therefore, do not translate work problems directly into calculations to be performed. Instead, they use the words to build a mental model of the problem. Then they use this model to think through the situation and arrive at an answer.

The complexity of this process is surprising, because people who are good at this kind of reasoning are usually unaware that they are doing it. But a great deal of recent research on the psychology of problem solving has revealed that expert problem solvers in any field are not usually conscious of all their thinking processes. It takes patient work by psychologists and other scientists to figure out what is really going on as people think. The results of this kind of research can then be used in designing instruction that can be accurately focused on developing specific problem-solving skills.

We cannot succeed in teaching sophisticated science and mathematical problem-solving skills with the scattershot approaches we have been forced to use in the absence of a research-based understanding of how problem solving occurs.

We are on the verge of developing a strong theory of instruction for problem solving. In the face of this country's urgent need for better scientific and technical skills, a withdrawal of support for research on improving math and science instruction is particularly ill-timed.