Problem solving and solving problems: Inquiry about inquiry

School Science and Mathematics, Mar 2000 by Lederman, Norman G, Niess, Margaret L

Nobel Laureate Richard Feynman would often sit in the corner of the room while his cousin received tutoring in algebra. After one of these tutoring sessions, the inquisitive Feynman had the following conversation with his cousin:

Feynman: What are you trying to do?

Cousin: 2x 7 =15. You're trying to find out what x is.

Feynman: You mean 4.

Cousin: Yea, but you did it by arithmetic. You have to do it by algebra.

Reflecting on the conversation (many years later), Feynman concluded, "The whole idea was to fmd out what x was. It doesn't make any difference how you did it. There's no such a thing as you don't do it by arithmetic, you do it by algebra."

After describing the algebraic solution to the problem (e.g., subtract 7 from each side of the equation, etc.), Feynman further elaborated about the procedure:

This was a lie that they had created in schools so that kids who had to take algebra could pass. They had created a set of rules that you could follow, without thinking, and get the right answer if you didn't know what you were trying to do.

Richard Feynman was a world-renowned physicist and never claimed to be a mathematics educator or expert teacher. However, he is well known for inventing numerous mathematics ideas and procedures to help solve problems in theoretical physics. Indeed, one of his mathematical inventions lead to his receipt of the Nobel Prize. In short, we believe he may have something valuable to say about mathematics and problem solving, not to mention mathematical reasoning.

The current reforms in science and mathematics stress science as inquiry, mathematics as problem solving, and mathematics as reasoning. The revised, and soon to be published NCTM Standards use slightly different wording, but the meaning of problem solving and reasoning remains the same. These emphases, perhaps more than anything else, distinguish current reform efforts from reforms in the history of science and mathematics education.

The current reform documents in science (i.e., National Science Education Standards, Project 2061 ) present inquiry from three different perspectives. Inquiry is viewed as a teaching approach, as process skills, and as content. As a teaching approach, inquiry is viewed as the desired instructional approach to facilitate students' understanding of the subject matter specified in the content standards. As process skills, the focus is on students' ability to successfully perform certain skills (e.g., observing, inferring, concluding, etc.) within the context of developing research questions of interest, designing investigations, completing investigations, and communicating results. Although many science teachers confuse these two meanings, neither is unique to the current reform efforts in science education. The third perspective of inquiry is knowledge about inquiry. That is, students are expected to learn about the inquiry they have performed. For example, science students are expected to know that there is no single set and sequence of steps (known as the "scientific method") that all scientific investigations follow. The stress on knowledge about inquiry (which overlaps with nature of science) is what distinguishes current science reform from previous reform efforts. Unfortunately, it is also the perspective of inquiry that is most misunderstood by science teachers and science educators, and it is the perspective that is rarely assessed.

The current mathematics reform documents (i.e., current and revised NCTM Standards) have standards analogous to the first two perspectives of inquiry in the science reforms. That is, the teaching standards clearly emphasize inquiry-- oriented teaching. It is such a teaching approach that is viewed, as in the science reforms, as being the best way to facilitate students' understanding of the mathematics content standards. The standards currently labeled as "mathematics as problem solving" and "mathematics as reasoning" are analogous to the process skills in science and heavily emphasize students' ability to solve problems, complete proofs, test conjectures, and assess the validity of arguments. There is really nothing in the "old" NCTM Standards or those that will be unveiled later this year that corresponds to the science standards' knowledge about inquiry and knowledge about the nature of science.

So, what does this all have to do with the conversation a young Richard Feynman had with his cousin? Feynman, in his reflections, was focusing on solving problems and the idea that there were various ways of solving problems. He also was quite incredulous about the idea that problems had to be solved in certain preferred ways, as opposed to other ways that would also yield the correct answer. Finally, Feynman was highlighting the algorithmic approach to solving problems that is promoted and rewarded in public schools. That is, being able to correctly solve a problem without having any conceptual understanding of the problem or the mathematics in the problem. You may also be happy to know that Feynman did not exonerate science instruction and curriculum from this problem. The conversation between Feynman and his cousin is discussed in a videotape called The Pleasure of Finding Things Out. Later in the tape, Feynman discussed the difference between science and pseudoscience. He described pseudoscience as following all the procedures of science and appearing to look like science, but never arriving at knowledge that is substantively connected to the knowledge base of science. He further inferred about those practicing pseudoscience: