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Teaching for Skills vs. Understanding

Debate about the relation of skills to understanding dominated an October, 1995 Roundtable at the University of Wisconsin, Madison. Excerpts that follow are by John Janty, Mathematics Coordinator, Waunakee High School, Waunakee, Wisconsin; Sarah Krinke, senior, Waunakee High School, Waunakee, Wisconsin; Tom Kurtz, Department of Mathematics, University of Wisconsin; Robert Meyer, Harris Graduate School of Public Policy, University of Chicago; Margaret Ellibee, Wisconsin Center on Education and Work; Hal Schlais, University of Wisconsin Centers; and Don Chambers, Wisconsin Center for Educatonal Research.

John Janty: So what are the skills that I need to impart to students? I hear a lot of talk around this table about the need for higher order thinking skills, but believe me, we get a lot of messages that students had better know how to add fractions too. Should we spend our time teaching algebraic gymnastics or developing conceptual understanding? I'm getting very mixed messages.

We have very limited time for mathematics. How can we use it most productively--on conceptual understanding so that students can later understand how computer software functions, or on basic skills? What is it to be--problem solving or drill?

Students, I might add, are not buying into drill these days. Sarah is as likely to say "This is all very nice, Mr. Janty, but when am I going to use this in my life? When will I ever need to differentiate all these expressions?"

Sarah Krinke: That's right. I want to be a doctor, and they told me to take Calc I and II, so I'm taking Calc I and II. When it's over, I'll be done with math. But I still ask: "When will I ever use it?"

Tom Kurtz: There is no doubt, as John says, that the culture for mastering skills has changed. When I was in school, if I was told to factor ten polynomials, I did. But Sarah isn't going to factor ten polynomials--and she is a good student. So the question is not just whether these skills are needed, but is there any mechanism to engage students in learning these essential skills? Factoring polynomials is not so important by itself, but it is part of the language that one uses when discussing things that are important.

Robert Meyer: The message universities send is that if you master certain skills then you can come to the university and pass some entrance exams. But is there anything there? Do those skills mean anything? The typical skills on mathematics placement tests are limited to such a narrow repertoire that in a slightly different context these skills are virtually useless. So I'd say, even if students pass the test, that they don't really "know it."

Tom Kurtz: How important are skills, anyway? On one level I agree: they are not very important. But on another level, I know full well that when I am explaining something to a classroom of fifty engineering students, half of them won't have a clue as to what is going on because they are still trying to figure out what I did when I cancelled two factors two boards before. A failure to master those kinds of skills is a failure to understand a language. It is like being locked in a room where the course is taught in Spanish when you don't know any Spanish.

Margaret Ellibee: The NCTM Standards begin with process standards--math as communication, math as reasoning--before moving on to more math-specific areas such as algebra, geometry, and statistics. The [SCANS report] ("What Work Requires of Schools") looks at skills desired by employers--both basic skills like reading, writing and mathematics as well as higher order skills such as reasoning. In that way you have a nice match, in my estimation, between the NCTM Standards and some of these work force skills standards.

More important is that these skills--whether SCANS or NCTM--incorporate a student-centered approach that integrates work force skill and mathematics standards in a context that is relevant to students. School-to-work programs make learning relevant to the workplace in terms of students' career ambitions. It all has to be contextual--content, teaching, assessment--or it won't make any sense.

Hal Schlais: I often teach older students, who have a different need for skills. They have been away from school for a long time, and need to take courses to get ready for what they really want to study. Many of them succeed because they are motivated now to learn things that they never mastered--or maybe never studied-- when they were younger.

Don Chambers: I believe school mathematics is a cultural artifact that has been inherited from earlier generations. As such, its features are deeply embedded in the fabric of our society and are passed on, through school experiences, to each new generation. Many mathematicians, mathematics teachers, parents, and members of the public have developed expectations of school mathematics on the basis of their own school experience. Others are dissatisfied with this inherited school mathematics and are proposing changes. One vision of change is expressed in the NCTM Standards.

One important issue on which views differ has to do with the development of understanding. Some believe that conceptual understanding develops from a platform of procedural knowledge. Others believe that conceptual understanding and procedural knowledge develop simultaneously and support each other. There is some research support for this latter view, which is built into the NCTM Standards. More research, and more knowledge of outcomes of recent research, would be helpful in advancing conversations on school mathematics reform.

To add you voice to this discussion, e-mail comments, letters, and op-ed articles to: extend@stolaf.edu or click here if your Web browser is set up for e-mail.

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Last Update: 12/18/95