Capturing the Meaning of Words Using Mathematics

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

Question 1. Which is bigger - a baby giraffe or a baby elephant?

Question 2. Who was the most influential member of the Republican Party in 2012?

Question 3. Which were the countries with the five largest economies in 2013?

Addressing questions of this kind can benefit from using a mathematical approach. One might try to apply the ideas of mathematical modeling to study these questions. One aspect of teaching mathematical modeling is that it trains one to think about the "border" between things that at first glance may be grouped together but in fact are distinguishable. Presumably we would not want to compare a baby male giraffe with a baby female elephant. Our knowledge of the "world" teaches us that there is a different distribution by size and weight for the sex of human babies so might this not extend to giraffes and elephants even though we might not have much data at our disposal? Data might be limited if we were interested in the issue of giraffes and elephants in the "wild" as compared with those who were born in zoos. The original problem does not have many words but a lot of thinking goes in to trying to "answer" the question. Furthermore, part of what is involved in trying to get insight into this "real world situation" would be to try to understand what some of the words used in the situation description mean.

For example, in Question 1, one might, because one has some knowledge of animals ask, "What kind of giraffe and what kind of elephant are we talking about?" Certainly, even young children would realize that dogs come in very different sizes depending on what "breed" of dog one is talking about. Suppose for definiteness one decided that one was talking about Masai giraffes and African elephants (usually there are considered to be two species of elephant, African and Asian elephants). At one time it was thought there was only one species of giraffe but recently the view has changed and it is now believed, based on the analysis of mitochondrial DNA, that there are six species, one of which is the Masai giraffe. But one would still have the issue of what interpretation to give to the word "bigger." Bigger in weight, bigger in height off the ground when standing (baby giraffes and elephants typically can locomote on their four legs much, much earlier than human babies can walk on their two legs), bigger in surface area, etc. And there is also the issue of whether one is talking about "typical" baby giraffes and elephants or the extremes of what one might see. Thus, for human babies, Wikipedia (1/14/2014) reports: "In developed countries , the average birth weight of a full-term newborn is approximately 3.4 kg. (7 1/2 lbs.), and typically is in the range of 2.7-4.6 kg. (5.5-10 lbs..)" So, one needs to know which countries are considered developed, what the word "typically" covers, and what the word "approximately" involves. Similar ranges of numbers are given for length of babies at birth for "first-world" babies. Data for giraffes and elephants might be harder to come by!

While the mathematical world has many complexities and often grows because humans think of clever ways to distinguish between things that superficially look alike, in many ways we have more control in the mathematical world because we make definitions to expand distinctions. Euclid does not distinguish between convex and non-convex polygons but it is a common distinction now. What is less well known is that computational geometers have found other ideas for telling (simple) polygons apart. Thus, one sees in computational geometry these terms: monotone, rectilinear, equilateral, starshaped, spiral, etc. The "real world" seems to me rather more complex, like layers of an onion. The notation of a "baby giraffe" seems simple enough until one starts peeling the layers of the onion away. Which species of baby giraffe, one that lives in the wild or a zoo, a baby which was just born or is a month old?

Thinking about issues of this kind is very different from what is involved in carrying out complex algorithms, which is sometimes what is promoted as the essence of mathematics - at least as measured by what is often tested in high stakes testing. In lower grades students are required to carry out long division, learn to find roots of quadratic polynomials, and because some such polynomials have complex number roots, to learn the algebra of complex numbers. However, the number of occasions that students need to solve quadratic equations outside of STEM discipline classrooms is limited, and relatively few students major in STEM related fields.

You might wish to "brainstorm" the other two questions raised here from a mathematical modeling point of view, as well as trying to answer Question 1 by making suitable assumptions in the style of the discussion above.

References:

Blum, W. and P. Galbraith, H. Henn, M. Niss (eds.), Applications and Modelling in Mathematics Education, ICMI Study 14, Springer, New York, 2007.

COMAP, (multiple authors), For All Practical Purposes, 9th (and earlier editions), W.H. Freeman, New York, 2012.

Crisler, N.. and P. Fisher, P., G. Froelich, Discrete Mathematics Through Applications, Macmillan, NY, 1999. (3rd edition, Cristler and Froelich, 2005, W.H. Freeman, NY)

Malkevitch, J. and W. Meyer. Graphs, Models, and Finite Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1974.

Malkevitch, J., Discrete mathematics and public perceptions of mathematics, in Discrete mathematics in the Schools, J. Rosenstein and D. Franzblau, (eds.), American Mathematical Society, DIMACS Volume 36 (1997), pp. 89-97.

Matos, J. and W. Blum, S. Houston, S. Carreira (eds.), Modelling and Mathematics Education: ICTMA 9, Horwood, Chichester, 2001.

Swetz, F .and J. Hartzler, Mathematical Modeling in the Secondary School Curriculum, National Council of Teachers of Mathematics, Reston, VA, 1991.

Verschaffel, L.,and B. Greer, E. de Corte, eds.), Making Sense of Word Problems, Taylor & Francis, NY, 2000. (Also published by: Swets & Zeitlinger)

These thoughts were, in part, spurred by a comment of Rachel Levy, Harvey Mudd College.