Talk About Teaching (Math Literacy)


Talk About Teaching

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                  2    2

      2     2   -x  - y

z = (x  + 2y ) e

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(A graph of the equation above accompanied this piece.)



By Dennis DeTurck





   What is mathematical literacy? How is it achieved? Why bother with 

advanced mathematical education? The immediacy of these questions is 

emphasized by the nature of the debate over the book The Bell Curve, by 

Murray and Herrnstein. The book brims with data, statistical analyses 

and interpretation. Its conclusions concerning race and IQ have been 

criticized widely. Unfortunately, much of the commentary is emotional 

rather than rational, even though there is plenty of room for analytical 

criticism of the work. 

   There is a story about the great eighteenth-century Swiss 

mathematician Leonhard Euler. He was summoned to the court to debate an 

esteemed but nameless philosopher about the existence of God. The 

philosopher offered a long, eloquently-worded argument to refute the 

existence of a deity. Then, Euler stepped up to a blackboard and wrote 

some complicated mathematical equation (perhaps the title of this 

essay), stepped back, and intoned, "Therefore, God exists." The 

philosopher was speechless in the face of the mathematics because he was 

not mathematically literate enough to recognize its irrelevance. 

   Similarly, when we hear that "Housing starts were up 3% in 

October," how many of us know what this statistic means? How is it 

measured? Up 3% from what? From September housing starts? From October 

1993 housing starts? Is this good news?

   How does all this relate to freshman Calculus? It all comes back 

to the issue of why one learns mathematics. Probably, if a student 

intends to be a physician, an engineer, a lawyer, a businesswoman or a 

humanist, she will never need to calculate derivatives and integrals. 

But she must understand that mathematics provides useful tools and 

language for describing, measuring and predicting all manner of natural 

and human phenomena. Moreover, she should know how problems particular 

to her discipline get translated into the language of mathematics, to 

what extent realism is lost in the translation, and how accurate, 

reliable and relevant the results of mathematical analyses are to the 

original problems. 

   College students arrive having taken mathematics for many years. 

Most Penn students who need Calculus for their majors have studied some 

in high school. They say they "know Calculus." But anyone who has taught 

Calculus has heard the student lament: "I understand the math, it's 

those word problems!" These students have missed the whole point of 

studying mathematics in general, and Calculus in particular. Their 

mathematical education consisted primarily of drill on problems that 

were stated in mathematical terms ("Solve the equation...") and ended 

with "the answer in the back of the book." It's as though one learned 

the conjugations of all the regular and irregular French verbs, but 

didn't bother to understand the meaning of any of them. 

   Developing facility with symbols and getting correct answers to 

drill problems is important, but is only a small part of learning 

Calculus (or any mathematics) for two reasons. One is described above: 

for all but a few of us, mathematics is pointless unless it is done in a 

non-mathematical context. We must be using it to solve real problems. 

Thus, we must be able to translate something from the real world into 

mathematical language, solve the resulting mathematics problem, and 

interpret the answer in the language of the real world. We might do this 

to compare mortgages or investments, to decide to elect a risky surgical 

procedure to avoid not-quite-certain dire consequences, or to form 

opinions about public policy (should medical professionals be tested 

randomly for AIDS?).

   The other reason that "symbol-pushing" is such a small aspect of 

mathematics education is that we can now relegate most of it to a 

machine. This causes controversy at every level. In elementary school, 

one wonders whether students should use calculators (although, in the 

words of Peter Drucker, "Life is too short for long division"). For 

Calculus students, the issue is the use of computers and programs like 

Maple.

   Computing has evoked many essays that begin like this one: What is 

mathematical literacy? etc. This question is only half-answered in much 

that is said and written around the University. Many students come to 

Penn having used word-processing and publishing programs, e-mail, 

perhaps a spreadsheet, and games. They say they "know computers". 

Similarly, many faculty (see the Provost's essay "The electronic 

environment" in October's Penn Printout) view the computer primarily as 

a tool for communication: we use it to "talk" to each other more, and to 

gain access to more information. This is a limited view, which ignores 

the fact that computers are also used for  computing. What do we do with 

information once we have it? What kinds of thought take place before we 

post the message to the newsgroup? What are the tools for organizing, 

analyzing and interpreting textual, graphical and numerical data?

   Against this background, the use of Maple in Calculus courses can 

be discussed. Maple is a programming language that can solve most of the 

"rote" problems in a typical Calculus book: Without much thought, the 

user types in the problems with only minor notational adjustments and a 

few decorative commas and semi-colons, and the computer types back the 

answers. Maple can draw two and three-dimensional graphs (the one on 

this page [in the print version of Almanac] is the graph of the title), 

and can solve problems in many parts of advanced mathematics. Like any 

programming language, Maple has a fairly rigid syntax, and demands of 

the user a high degree of precision of thought. So using Maple comes 

with a cost. What is the return for paying this cost? 

   Many students will attest that the return is not that Maple makes 

Calculus easier to learn. Calculus is a profound achievement of human 

thought, and takes a great effort to master. Maple's graphics can help 

students visualize some difficult concepts, but making Calculus "easier" 

or helping students finish their homework faster is not the point. 

Rather, we return again to the theme of this essay: why we learn 

mathematics. Tedious details of mathematical calculations should be 

relegated to the computer, so that humans can focus on the intellectual 

and difficult part of the enterprise: the translation of the problem 

into mathematical language and the interpretation of the results of the 

mathematical analysis. In fact, this has been one of the biggest sources 

of student frustration. Since the machine can do the routine 

calculations, courses now place more emphasis precisely where the 

students are the least comfortable: translating word problems into math 

problems. The return for using the computer comes as students develop a 

deeper appreciation for the power and usefulness of the mathematics they 

are studying, and a greater ability to apply mathematics and 

quantitative thought when it is appropriate. This is evidenced as 

students and faculty in Biology, Chemistry, Physics, Engineering, and 

Economics classes begin to use Maple (or one of its cousins) to do 

mathematical work. 

   Another effect of the use of Maple in Calculus is the experience 

students gain in dealing with the software. Although most of our 

students will not be using this software in the future (indeed, it will 

undoubtedly be obsolete by the time this year's freshmen graduate), many 

will be faced with computer languages of one form or another (whether 

for the operating system of a machine, or statistical analysis, or 

specialized no-tation for music, etc.). Basic programming skills learned 

in one environment transfer to others, and are becoming more and more 

valuable.

   Penn's Mathematics Department continues to look to the future. 

Mathematicians, together with faculty from the natural sciences, 

engineering, medicine, and the social sciences are working to develop a 

collection of classroom modules and interdisciplinary courses that 

combine the teaching of mathematical techniques with applications drawn 

from the research of the Penn faculty and from the work of users of 

mathematics in industry. The Society for Industrial and Applied 

Mathematics (whose national headquarters is in Philadelphia) is 

assisting with industry links, and is part of a Penn-led consortium of 

institutions pursuing this "holistic" approach to mathematics and its 

applications throughout the curriculum. Our new administration has 

committed itself to a significant new initiative to position Penn as a 

national leader in undergraduate education. The goal of this 

interdisciplinary effort is to make Penn a national leader in innovative 

undergraduate mathematics education. 





Dr. DeTurck is professor and undergraduate chair of mathematics. 

His column is the second in a new Almanac series developed by 

the Lindback Society and the College of Arts and Sciences.


Almanac

Volume 41, Number 11
November 8, 1994

Return to Almanac's homepage.