## The Mathematical Education of Technicians

An interview with Kenneth Chapman, American Chemical Society

In science observations lead to hypotheses, whereas in mathematics applications seem to flow from decontextualized theory. How can mathematics and science teachers best reconcile these rather different approaches to the mathematical and scientific education of students?

Technology focuses on applying science and mathematics to solving real, contextual problems by applying science, mathematics, economics, and other tools as necessary. The mathematics needed in technology is fully contextualized and becomes a tool in the same sense that a carpenter uses a hammer to apply a force appropriate to the need at hand. The carpenter usually has several hammer designs to choose from and is concerned with a product, which is seldom a new hammer design. Teaching a prospective carpenter to design hammers may develop a better understanding of the factors important in selecting hammers, but it may be so distracting that other more important skills for carpentry are overlooked entirely. A carpenter who is taught primarily how to design hammers may never gain great skill in driving finish nails and leaving the wood unmarred.

Similarly, the science technician may need to develop a mathematical relationship to address the needs of a particular context, but the problem-solving skill required involves just the analysis of a specific chemical or physical system rather than understanding a mathematical relationship that is generalizable to all possible systems. Thus a technician is unlikely to use mathematical proofs or mathematical applications (as I interpret your term). Nevertheless, a technician needs to be just as rigorous as a scientist or mathematician in ensuring that quantitative results are correct and accurate.

On the science side, "hypothesis" may carry rather different connotations. For example, the use of mass spectrometry can lead a chemist to hypothesize a particular molecular structure for the components of a sample; the technician's hypothesis, in contrast, is likely to focus different issues: the specific treatment of the sample and the operation of the instrument in a manner that results in reliable and accurate data. Current teaching practices in science focus only on the structural interpretation for everyone. The new National Skill Standards for the Chemical Process Industries suggest this is wrong.

All this suggests that technician education is more difficult than the educaton of scientists and engineers. The technician is expected to know how to use mathematics to solve quantitative problems and how mass spectrometry data are interpreted to elucidate molecular structure. However, the technician is expected also to be highly skilled in acquiring data, caring for instruments, and producing error-free results. Unless technician education is designed very carefully so that real needs are determined and addressed, it will be packed with so much science and mathematics that technician skills will never be learned. These factors explain why so many technicians complain that their academic study was useless for their work.

This is a rather long way to suggest that maybe your question is wrong. Reconciliation of current instructional practices may be much less important than revolutionizing these practices so they match the needs of workers in the technological workplace rather than of scientists and mathematicians who operate at the frontiers of their discipline. For example, my high school geometry course consisted entirely of memorizing or developing proofs about geometrical relationships. What I have needed through the years is an ability to calculate lengths, angles, areas, and volumes. I have never been called upon to provide a geometry proof nor has my use of applied geometry in any way benefited from understanding proofs. In chemistry, everyone has learned the aufbau principle in general chemistry for the past thirty years. However, I have never encountered a chemical technician who has ever had reason to reference or use the principle. Both science and mathematics need to be taught very differently if they are to be of benefit to technicians.

Some educators have argued that departmental boundaries in education may set up artificial barriers to collaborations on curriculum development and improving instructional strategies. Based on your interest in the school-to-work movement and the fact that disciplinary boundaries are not relevant in the workplace, what do you think is the proper balance of disciplinary and interdisciplinary curricula in education, especially in grades 10-14?

Indeed, disciplinary boundaries are irrelevant in the workplace. They are convenient for organizing knowledge, skills, and academic administration, but they often confuse the development of problem-solving skills because useful knowledge and skill is often withheld, unavailable, or disconnected at the moment when it is needed.

I am not satisfied that interdisciplinary courses can accomplish the integration of knowledge and information needed. Most worthwhile technology problems are not pure chemistry or pure biology; but they also are not bio-chemical. They involve the total experience of science, mathematics, communications, computer applications, economics, politics, sociology, and other knowledge.

I would suggest reversing the typical teaching structure. Do away entirely with courses and with students changing classrooms every 45-60 minutes. Instead, let students work on complex projects under the tutelage of a master, who can call on specialists in mathematics, chemistry, biology, etc. to provide instruction and information as needed and as appropriate. Have the teachers change locations instead of the students.

The proper balance among the disciplines must be determined from the needs of the worker in the workplace and in society. The skill standards for chemical technicians that ACS has published are a starting point for identifying the needs, but more detailed work is necessary to determine the right balance, particularly for mathematics. The balance among the disciplines will need to be worked out as instructional materials are designed to match the needs we are starting to recognize. I hope the projects that work on curriculum and instructional materials in the future will use interdisciplinary teams. I do not believe the balance can be worked out in conferences or two-day workshops.

Mathematics teachers often worry that the context-rich environment of an interdisciplinary course will make it harder for students to sort out the mathematics principles from the surrounding context. What has been your experience as a chemist in this regard?

Your assumption is that it is useful to sort out mathematics principles. I have seen little concern for such sorting. Results count, even if terminology is wrong and understanding is absent. As an example, almost every science teacher introduces students to "dimensional analysis." As developed many years ago, dimensional analysis is a simple mathematical tool applied to complex systems to determine relationships among variables. The products of dimensional analysis are dimensionless numbers, some of which are used widely (e.g., Reynolds' Number). However, science teachers do not teach real dimensional analysis. Instead, they teach about unit cancellation as a way of bookkeeping in arranging quantitative information. They use the term incorrectly, and know nothing about the mathematical processes of dimensional analysis. Although it peeves me greatly (as a chemical engineer) to see this error propagated, I doubt that it causes much real harm. For the technician in particular, it is the result that counts and an appreciation for mathematical principles makes little contribution to a technician's success.

A context-rich environment offers students a better opportunity to recognize the necessity and value of mathematics. From that beginning, some teachers will awaken an interest in some students to become much more interested in mathematics for its beauty as well as its utility, thus leading to better understanding of mathematics principles than would have been achieved otherwise.

What do you see as the main opportunities and primary challenges in integrating mathematics and chemical or chemical technician education?

Contextual situations invariably must be addressed quantitatively. For the technician, quantification means mathematics. The necessity of understanding quantitative relationships offers an opportunity to demonstrate that a broad understanding of mathematics offers both a broader and deeper understanding of any context. Thus, contextual teaching should provide the motivation for understanding more mathematics than does a textbook or classroom without context-rich examples.

The primary challenge to mathematics teachers is to retain a position in the instructional programs for technicians. If the move toward contextual learning leads to the instructional environment I suggested earlier (a master leading students through large contextual problems), some will expect the master to do all the teaching and we will have returned to the one-teacher school house. I would oppose that. Rather, I would prefer looking at the education of the chemical technician as paralleling the medical education that takes place in a major teaching hospital. The master would be a generalist and would be competent at providing initial diagnoses. The discipline experts would be the specialists, who are called upon as appropriate to exercise their specialty knowledge and skill to benefit the patient.

To achieve the desired integration, mathematicians should be involved in each major technology education project for curriculum and instructional materials. However, NSF may have to make this a requirement since it is unlikley to happen otherwise. I find it difficult to move my colleagues in the direction of teamwork across discipline boundaries. Each one sincerely believes that they possess all the knowledge and skill that a technician needs to learn!

Given that many students change their minds about their career goals while still in school (either high school or college) and many workers change jobs five or six times during their working lives, how do we safeguard against having technological education become too narrowly focused on developing skills? What do you see as the right balance between theory and practice in technological education programs?

Perhaps we should discuss the balance of three strands--theory, application, and practice. Then we also should discuss learning styles. The combination really gets complicated and leads quickly to individualized instruction. However, I am not ready for that, and the expense would not be politically supportable.

I think the greatest emphasis for most students should be placed on applications. To me that would result in assessments that, for example, would determine whether or not all high school graduates could determine accurately the metric area of an irregular plot of land given a reasonable measuring tool marked in English units. Or for a student aiming to earn a livelihood in the practice of chemistry and having had the opportunity to study acid/base solutions, to calculate the amount of a strong acid required to produce a solution with a specified pH. Both examples require the ability to solve problems with multiple steps. The latter might be construed as requiring some theoretical knowledge--but not much. Both require applying mathematics by developing (not deriving) relationships in an equation format, and solving the equations using "practice."

I think students are ill-served when they are taught only practice. Anyone working in chemistry has not been educated properly if they cannot take the instrument manual of a complex new instrument and a brief written description of the chemical principles underlying the measurement and quickly understand what is required both to produce and to recognize good data--and to understand the implications of poor or poorly presented data. Chemists and chemical technicians who have to be retrained from the ground up for each new application will be of little value to their employer and advancement is unlikely.

Thus, I think all students need a strong foundation in applications that leads to an understanding of underlying theory, with enough practice so they recognize that additional practice can lead to skills needed to do a measurement accurately and with precision. If a current chemistry class for general students might be construed as 75% theory, 15% applications, and 10% practice, I probably would change the distribution perhaps to 60% theory, 25% applications, and 15% practice -- not very much. I would make substantial changes to the application and practice portions to reflect modern (not necessarily current) technology, and I would change the order of presentation. Typically, I would let the need for theoretical knowledge evolve from the applications and practice, and get theory to support application and practice rather than the reverse.

I hope this conveys my perspective that expectations and rigor should be increased, not decreased. However, I do not equate rigor and expectations with memorized facts and theoretical nomenclature. Rather, I equate it with processes that lead to understandable and useful ends.

For lack of sufficient master teachers who can work with students on long-term projects, how can we help faculty in different disciplines feel more comfortable working together? This seems especially difficult when such collaborations require teachers to reveal their lack of knowledge or understanding about topics in unfamiliar areas.

I think the short answer is: "work together until trust builds." In our alliance work, we find that a key requirement is that trust be developed among the alliance representatives and members. This typically takes several years of working together. I doubt that teachers in an institution are much different. For most, trust has to be earned and that takes time.

I think the problem is more sociological and political than technical. This is one place where observations from sports becomes useful. There are very few Michael Jordans who can do it all, but most of us science and technology educators seem to think we can do it all in our field. In most cases, a full team of dedicated enthusiasts can do a great deal more. An excellent example from my youth was Kentucky's "fiddling five" basketball team. The coach, Adolph Rupp, was highly surprised to find he had an NCAA national champion team. They lacked any stars, and none ever made a national names in sports; they just took their limited talents and found ways to make them work together perfectly. Every game was a crisis.

Teachers need to accept that they should be learners along with their students. We have to create environments where teachers work together over prolonged periods to reach the point where they begin to understand one another's vocabularies and working modalities. Maybe classes are too small. Perhaps economics will force us from one teacher per 20 students in a classroom to three teachers per 100 students in a single classroom (e.g., a chemist, a biologist, and a mathematician). Instead of one master teacher, go to three master teachers working together. Workshops of one day to four weeks are good for conveying some knowledge and skills, but they do not result in collaboration when classrooms are set up as inviolate single-teacher sanctuaries.

There may be some good models from the team teaching fad of a few years back. In particular, I like the Asian model of a bullpen office for teachers and large classes for students. Larger classes permit teachers to have fewer classes, so substantial time every day (compared to the U.S.) can be devoted to teacher collaboration and to enable experienced instructors to work with new teachers.

The quickest and easiest expedient for addressing one part of the need is to get teachers to spend significant periods in the technical workplace that reasonably matches their preparation. However, that addresses technical workplace knowledge needs only. Improving classroom management has to be addressed by enabling teachers to really work together across disciplines.

Some of the experimentation now taking place in a few selected schools may need much greater support to provide models than can be applied widely. Unfortunately, NSF seems too often to either limit funding below the needed threshold or to limit the scope of projects so that some important variables are not addressed adequately. Changing even a few schools is both difficult and costly.

Kenneth Chapman is Head of Technician Resources and Education at the American Chemical Society. He can be reached by e-mail at: kmc97@acs.org.