The following paper was published in The Tau Beta Pi Bulletin, LXXV, 3, March 2002, pp. 3-5. (Slight modifications appear in the final printed version.) See URL; click to select the March 2002 issue as a PDF file.

Training and Thinking

Allen Klinger

© 29 August 2001

This paper supports exposure and familiarization as well as challenge, as essential to educating professionals in engineering. Getting knowledge always involves learning the ways of the past. Engineering courses train students while exposing them to the fruits of past thought, often through laboratory experiments and design projects. Yet engineers often react with unease when given a new problem outside their experience. Thinking is a difficult task that we tend to avoid.

These statements describe what engineering college curricula give students:

1. Becoming familiar with technology, tools, and theories.

2. Gaining facility in use of mathematics, models, and computation.

But neither familiarization nor facility prepares individuals for work when there is radical change in the underlying technology or our model of nature.

Past Examples

Familiar terms and systems: radar, jet air-transport, computer- and satellite- communication, all came out of military research and development expenditures. Some developers were engineers, but ideas and innovations were by other kinds of professionals.

In attaining balance between familiarization/facility and basic knowledge in engineering, practical steps involve challenging work assignments. They can reflect changes about basic knowledge. Dynamism in radio engineering led power engineers into today's multifaceted Institute of Electrical and Electronics Engineers. Inexpensive active solid state devices, chips composed of large numbers of transistors, lessen needs to study passive circuits, resistors, capacitors and inductors. That subject was fundamentally important: now it is an aspect of engineering.

There are many different examples of change in established engineering disciplines. Widespread assumptions about Chemistry, and Calculus, have sometimes stayed the same over time. Engineering's first two-year college curriculum still has Calculus. It enables models using second-order ordinary differential equations useful in Dynamics/Electric-Circuits. Chemistry today has been dropped from some engineering college curricula although advances in Biology/Genetic requires engineers able to enter Biomedical Engineering, where it would be useful.

This paper's goal is to isolate qualities of engineering that enable and support engaging in different directions of work. We focus on ways to strengthen engineering education, enabling it to produce graduates possessing flexibility in dealing with new ideas.

Simple Questions

A multiple-choice format enables examining complex ideas. Consider the following statement and three possible completions.

1. In four births which is most probable:

a) Two male, two female.

b) Three of one sex, and one of the other.

c) All four the same in sex.

Fifty-fifty probability, i.e., male births being approximately as frequent as female, leads some to choose the incorrect answer a).

The three-answer multiple-choice format is best when there are correct, incorrect and possible-but-wrong completions to the initial statement. But some simple questions introduce other ideas. The next question goes beyond grade school Arithmetic.

2. In adding fractions which statement is true:

a) 1/3 plus 1/2 is 2/5.

b) 1/3 plus 1/2 is 5/6.

c) In Physics elements with values 1/3 and 1/2 can be connected to compose 1/5.

All three 2-completions hold for some real situation. There are valid alternative interpretations for numerical sums of rational fractions; e.g.,

a) Baseball batting averages [1];

b) Adding resistors in series [2], pieces of a whole fruit, etc.

c) Adding resistors in parallel [2].]

Consider a variation on fractions I call strange cancellation, and then two other puzzles that troubled several. (The section after this includes examples from more conventional mathematics.) Visual problems offer a rich source of challenging material: we leave them for a subsequent paper.

3. For a rational fraction composed of two "two-digit" numbers canceling a single common numeral:

a) Can't give the correct value.

b) Can give the correct value.

c) Reduces numerator and denominator numbers modulo the numeral.

4. To divide a circular pizza pie into eight exactly equal pieces by three straight-line cuts.

a) Is impossible.

b) The division can be done in one, and only one, way.

c) There certainly are three different ways to accomplish this.

5. Given: three "fives"; one "one"; and four arithmetic operations, plus, minus, times and divide; Find: twenty-four. I.e.: Given: 5, 5, 5, 1; and, + , - , * , / Find: 24

a) I can do this.

b) There's no way to do it.

c) The only way to get the result twenty-four is to not use one "five."

All five questions raise issues about training and thinking. They've raised the issue that there may be alternative solutions.

Anyone stumbling over 4 can view the following URL (the image includes brief text about alternatives):

Questions 1-5 possess non-obvious solutions to seemingly not difficult subjects: what is most probable; fraction addition; cancellation; division; and arithmetic operations, Let's now consider subjects engineers know about, that others may not.

Elementary and Fundamental

Some engineering courses train to redress sketchy secondary school coverage, as in algebra and trigonometry [3]. These five questions address secondary school mathematics. They show the failings in that sketchy coverage and the differences between engineers and non-technical people.

6. When two quantities, say a and b, are compared, we write a is greater than b (or a is more than b) as: a > b .

Which statement is true?

(a) 1/103 > 1/10-3 (b) 103 > 1/10-3 (c) 103 > 10-3

7. For the given pair of equations which is true about the variables x and y ?

2x + 3y = 2

6x + y = 12

(a) x and y have the same sign. (b) x and y are whole numbers. (c) x > y.


8. For the equation x2 + x - 1, solution values are:

(a) Odd. (b) Imaginary. (c) Irrational.

9. We write the logarithm of a number n to base 10 by "log n". Which statement is true?

(a) log ( n + m) = ( log n ) + ( log m )

(b) log ( n * m) = ( log n ) * ( log m )

(c) log ( n * m) = ( log n ) + ( log m )

10. There are exactly how many ways to arrange 5 things?

(a) 120 (b) 15 (c) 54

University students not interested in Mathematics had difficulty with 6. -10.; one asked "Where did you get those questions?" My response "They were just high school mathematics" brought the immediate reply "Not in my high school."

Play involves individuals and stimulates thought about quantity: see e.g., [4, and 5]. Training, specifically knowledge of terminology (imaginary, irrational) and procedures (elimination of variable in simultaneous linear equations) or the lack of it is what questions like 6-10 measure. Creativity is related to thinking: "Real problems don't come in compartmentalized form" [6]. Is training in past approaches a way to dodge or eliminate thinking? Perhaps - consider this quotation: "Students in college mathematics courses are unresponsive. They are afraid to speculate and afraid to reach into themselves for ideas [7, p. 844]."

Gaining Strength

By now I hope you've found a one-eighth slice of pizza, 24, and other solutions like removing sixes in 16/64 or nines in 19/95, yielding the correct answers 1/4 and 1/5, respectively to the strange cancellation question. [How many fractions under one hundred (one thousand, ...) can remain unchanged quantities when a single digit is eliminated from numerator and denominator? Anyone who thinks out how to compute the solution(s) for a given level learns the value of not accepting the established rules, and could become interested in cancellation-like procedures described in [8].] Technology can stimulate similar playful but thought-demanding efforts.

Computing. Binary, octal and hexadecimal number systems are important in computing. Here is a related version of the familiar "continue the sequence" problems posed to me by Merwyn Sommer:

What number follows 24 in 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, __?

For an answer review this URL:

Further explanation of the three question 2 completions follows in two parts. A visual covering some of the same ground is at the URL:

Circuits. Overall resistance (ohms) depends on interconnection method: series differs from parallel. Five-sixths results when two resistors of one-half and one-third ohms are in series, the same result as ordinary fraction addition. Parallel connection of the same resistors yields the overall result one-fifth ohm.

Baseball. Baseball batting averages are calculated as total hits divided by total at-bats. This gives yet another way to look at the addition of two fractions. Suppose a batter has one hit in two at bats on the first day. Let the second day involve one hit in three at bats. The common accepted wording, "batting two for five," goes along with the decimal fraction 0.400.

The thinking/training relevance of the above examples could be questioned since they are mostly based on quantitative ideas. But the above ideas are matched by the models and ferment involved in other practical fields. Physics has relativity or Einstein's expansion of Newtonian mechanics, Maxwell's equations, wave-particle duality, the uncertainty principle, the Schrodinger equation, quantum mechanics, quantum electrodynamics, and many other issues where thought independent of past knowledge or orthodoxy has prevailed. Philosophy, Astronomy, Cosmology ("Big-Bang," "continuous creation of matter" theories) also have evolved.

Exercise and Rest

The relationship between training and thinking is that they are of equal importance. Any curriculum should seek a balance with training in methods of the past taking a share of the time, but reserving an equal amount for thinking.

Professional schools, engineering in particular, are training individuals. Being able to function in a field requires special knowledge. That is independent of developing the ability to continue to learn. Reading widely and writing about ideas are basic exercises that support adaptability.

Thinking is necessary. It should be enshrined in the college and university curricula. Undergraduates and graduates who study engineering need the challenge of lucid writing about cultural, historical and technological issues [9-11].

The vitality of engineering as a profession depends on its responsiveness to real world issues. Education for engineers should prepare them to function in the world. Success in new areas can grow from appreciating thinking in different cultures [9].


Distinguishing between overview and in-depth presentation is artificial. There is a seamless nature between elementary and fundamental issues. In reality "simple" things constitute basic knowledge. One needs an overview or introduction that engages and provides some familiarity. Then more exercise is needed. Trying many things initially is the only practical way to stimulate ultimately gaining facility in some main area.

Every well-educated engineer has training plus. That person's thought processes have been stimulated, creating adaptable qualities, focussing life on "... discovering an underlying order in matter is man's basic concept for exploring nature" [11, p. 95], even if the matter is synthesized, and the order created by design.


[1] Kline, Morris, Mathematics for the Nonmathematician, NY: Dover, 1985, 1967.

[2] Staff, Dept. Electr. Eng., MIT, Electric Circuits, NY: Wiley, 1940, 1953, p. 94.

[3] Miller, Frederic, College Algebra and Trigonometry, NY: Wiley, 1945, 1952.

[4] Enzensberger, Hans, Tr. Heim, Michael, The Number Devil,

[5] Gardner, Martin, "It All Adds Up (Review of [4])," Los Angeles Times, 11/8 1998.

[6] Steen, Lynn, "Teaching Mathematics for Tomorrow's World," Educational Leadership, 9 1989.

[7] Thurston, William, "Mathematical Education," Notices of the American Mathematical Society, 37, 7, 844-850, 9 1990.

[8] Gillings, Richard J., Mathematics in the Time of the Pharaohs, Cambridge, MA: MIT Press, 1972; NY: Dover, 1982.

[9] Ascher, Marcia, Ethnomathematics, A Multicultural View of Mathematical Ideas, Pacific Grove, CA: Brooks/Cole -Wadsworth, Inc., 1991.

[10]Editors, National Geographic, The Engineers, 1997.

[11] Bronowski, J., The Ascent of Man, Boston MA: Little Brown, 1973.


Selected Web Bibliography ... pointers to articles. ... using the above in classes. ... reasoning about quantity. ... wisdom words about paths pointers to wisdom files ... course for non-technical people

The above appeared in the Tau Beta Pi Bulletin, Feb-Mar 2002, Vol. LXXV, No. 3. It was preceded by the editor's choice from these two items (the source was, Mathematically Speaking, A Dictionary of Quotations, Selected and Arranged by C C Gaither A E Cavazos-Gaither Illustrated by Andrew Slocombe, Institute of Physics Publishing, Bristol UK and Philadelphia PA ISBN 0 7503 0503 7 1998 IOP Publishing Ltd.):

von Neumann, John - "... in mathematics you don't understand things. You just get used to them." p. 249, Quoted in G. Zukav, The Dancing Wu Li Masters (p. 226, fn)

Wright, Frank Lloyd - "You can study mathematics all your life and never do a bit of thinking." p. 257, Quoted in Robert B. Heywood (Editor) The Works of the Mind, The Architect (pp. 57).