By Judah L. Schwartz, Harvard University
From whence this blog
Nearly twenty years ago Paul Lockhart wrote a brilliant essay, A Mathematician’s Lament, on the parlous state of mathematics education. In it, Lockhart laments that mathematics education does not celebrate mathematics as an art and as an important part of human culture. I write this essay in the same spirit, lamenting that mathematics education does not do well in preparing our students to use their mathematical skills to model the world they encounter in the practical, economic, policy and social aspects of their lives.
I have spent many years trying to understand why so many people seem to have difficulty with mathematics. Many people have a distaste for the subject and will go a long way to avoid engaging any use of their mathematical knowledge.
Elementary and secondary schools, the social institution to which we entrust the education of our young, present the subject of mathematics as a “right answer” subject. What other inference is to be drawn from questions like “what is the sum of 34 and 28?” and then “what is the sum of 41 and 24?” Seldom do we see problems like “Make up an addition problem with two whole numbers the answer to which lies between the sum of 34 and 28 and the sum of 41 and 24.” One could follow that problem with “How many such problems are there and how do you know?” And follow that with “Suppose you could use integers rather than whole numbers, how many such problems are there and how do you know?” It is no surprise then that the public at large thinks of mathematics as a body of knowledge in which any question has a unique correct answer and an uncountable number of incorrect ones. On the other hand, I claim that there are important cognitive benefits to be derived from posing problems with multiple demonstrably correct answers and multiple demonstrably incorrect answers.
I have never thought of mathematics as a “right answer” subject. I have always believed that mathematics provides a way of approaching the world and making sense of what one sees, hears and feels there. Indeed this is precisely the reason most often given for the inclusion of mathematics in the curriculum of our K-12 schools. Mathematics offers the promise of allowing people to make reasoned decisions about their daily lives.
Mathematics provides a set of tools that can help to confront the extraordinary complexity of phenomena that surround us. Although the things that interest us and the relationships among those things are inherently complex, mathematics can clarify essential elements in the surround. Mathematical models can often provide great insight into the way our world works. That having been said, our ability to deal with the complexity of our world by using our mathematical models is complicated by the constant need for –
- the exercise of judgment about the conclusions we draw from our models, and
- the inherent imprecision of our senses [and their extensions, e.g., telescopes, microscopes, hearing aids, etc.]
Judgment and Imprecision
Counting and measuring are essential to all modeling. Even with these simplest acts of modeling, normally considered as an integral part of the K-12 mathematics curriculum, uncertainties about consistency and validity and the need for judgment are present.
Judgment is a serious matter in the act of counting. Does the head count in a ball park include only paying customers? People with complimentary tickets? Attendants, vendors, players, coaches? Clearly, purpose needs to be taken into account.
Imprecision and judgment are serious matters in the act of measuring. Consider the act of making simple measurements, even of such quantities as length or time. No measuring instrument can resolve the attribute it is being used to measure with infinite precision and as a consequence the best we can do is to assign a range of rational values as a magnitude. Additionally, judgment is required in deciding the extent of the attribute to be measured. Needless to say, no person has ever measured a quantity that resulted in an irrational number for a magnitude.
Further, judgment is required is assigning magnitude to the distance from Boston to San Francisco. An appropriate choice of unit might be the kilometer. It would hardly make sense to report the distance in millimeters, leaving aside all issues of from where in Boston to where in San Francisco. Reporting that distance in light-years would display a similar lack of judgment.
The need for judgment extends well beyond counting and measuring.
Suppose for example one needs to construct a cube of whose volume is as close as possible to 8 cm3. Deferring the issue of precision for the moment, our mathematical model of a cube tells us that a side length of 2 cm will work. But it also tells us that side lengths of –1+31/2i and –1–31/2i would do as well. The decision to discard the extraneous roots of the equation x1/3 = 1 is a judgment that is made based on the context. The craftsperson fashioning the cube from a block of steel will certainly be able to make the required physical choice and not either of the permissible mathematical choices.
Similarly, our mathematical model of how a ball thrown on a flat Earth moves tells us that there are two times that the ball is on the ground. Newton’s laws of motion have two correct mathematical solutions – the time in the future when the ball will hit the ground and the time in the past when the ball was launched [neglecting the fact that the ball was almost certainly not launched from ground level]. We are likely to be interested in only one of these times.
Some postulates about essences
At least for the sake of this discussion please allow me some postulates about essences –
The essence of
…learning is posing for oneself a provocative next question
…teaching is posing for students a provocative, engaging next question at the proper moment
…education is learning to ask of the world “what is this a case of?” and “what if not?”
…mathematics is making, exploring, proving and disproving conjectures
The reader will note that all these essences center on the posing of questions and the making of conjectures. This is particularly true of mathematics. None other than Georg Cantor once said, “In mathematics the art of proposing a question must be held in higher esteem than solving it.” By posing problems, David Hilbert, at the International Congress of Mathematicians in Paris in 1900, set the course of research for much of the mathematics of the twentieth century.
A physicist’s view of posing problems
Because as physicists we are engaged in modeling the world, albeit with mathematical tools, we do not always have the luxury of asking questions that have unique correct answers. Our problems, which we pose both to our students and to ourselves, are about models, their predictions and their consequences.
Models are never “correct” – at best they are “adequate.” Unlike an unsolved problem in mathematics, models in physics cannot ever be proven to be “correct.” This observation suggests, at least to this physicist, two guidelines about the posing of problems in mathematics. They are –
- think long and hard before posing a problem that has a single correct answer, and
- pose problems for which answers can be deemed to be “good enough.”
The reason underlying the first guideline is that a unique correct answer can only address the internal consistency of a model but not its validity or utility.
The reason underlying the second guideline speaks to the adequacy of a model—no model can take into account the full complexity of the phenomena being modeled. Judgment must be applied when deciding whether an answer to a problem is “good enough.” Perhaps the most widely known examples of the use of this guideline are Fermi estimation problems.
How might we think about mathematics when posing problems that follow these guidelines?
Objects and Actions – Measures and Models
The idea of mathematical objects and mathematical actions that can be performed on them or by them is central to the nature of mathematics. Simple mathematical objects and associated mathematical actions are often combined to form more complex objects which in turn have actions associated with them.
The power of this idea derives from the fact that essentially all of the languages spoken by people have utterances that are composed of a noun phrase and a verb phrase. A noun phrase tells us about objects and their properties. An associated verb phrase tells us about the actions that an object carries out or that is carried out on the object.
Measures are quantifications of attributes of nouns. Examples include the weight of a person or a gallon of milk, the circumference of a person’s waist or the height of a bookcase, the surface area of the body or that of a mountain lake. Other such measures are the volume of a human body or that of a bottle of wine, the length of a person’s lifetime or that of a movie, the number of a person’s red blood cells or the number of people in the US on any given day.
The measures cited here are basic ones. We are equipped by nature with sensory tools to assign some sort of magnitude to these measures or, at least, to order a collection of objects having that attribute.
These fundamental measures can be composed with one another to make composite measures. Composing a distance traveled with a time interval yields velocity as a composite measure. Composing velocity with a time interval yields acceleration as a further composite measure. Composing the number of people in a city with its surface area yields a measure we usually call population density.
Models are assertions of relationships among measures. Newton asserted a relationship between the measure of acceleration and the magnitude of the push or pull causing that acceleration. Mendel asserted a relationship between the color of the seeds of one generation of pea plants and the color of the seeds of cross-bred members of the next and subsequent generations.
The reason schools include mathematics in the K-12 curriculum is that the judicious use of mathematical models can help our students make sense of what they see and hear in the world around them. Such use offers the promise of allowing people to make reasoned decisions about their daily lives.
If, indeed, the end goals of our teaching mathematics and our students learning mathematics is to help them use mathematical models in their lives, we must put greater emphasis on modeling in our teaching—and not at the expense of reliably and correctly executed mathematical manipulations.
In a future blog, I hope to discuss one approach to doing just that.
 Reprinted in MAA Online, March 2008, Devlin’s angle
 Does one determine a person’s weight before or after a haircut? Does the surface area of a mountain lake include [all, part, none] of the runoff stream?
 For example, the π that appears in C = πd is defined as the ratio of a circumference to a diameter, but the magnitude of the number, itself, is not computed from measurements. A measurement of the circumference yield a value, C ± ΔC, and a measurement of the diameter yields a value, d ± Δd. The ratio of these two values (C ± ΔC)/( d ± Δd) is a bounded range of rational numbers. The definition C = πd expresses a model of the relationship between circumference and diameter, perhaps inferred initially from many measurements of circumferences and associated diameters but then determined by logical reasoning.
 I recall an incident in which a sixth grade student who fully understood the issue, displayed this understanding in an extraordinary way. I had asked him to formulate an estimation problem. With a huge grin on his face he said, “What is the average size of a postage stamp in square miles?”
 Proving and disproving each have a different meaning in mathematics than they do, for example, in law.
 A train is normally thought of as a rigid body. Thus, in this model, the caboose begins to moves the instant the locomotive does. Rigid bodies, inextensible strings, point masses are all instances of models that are valid enough and useful enough for many purposes. In the idealized world of such models, problems with single, correct answers can be, and often are, posed.
 For example, how long does it take a person to eat his/her own weight in food?
 Length, area, weight, volume, time, number.
 Measures can be defined within mathematics itself. Consider a collection of rectangles. People of all ages are willing to grant that a 5 x 6 rectangle is ‘squarer’ than a 2 x 15 rectangle. Thus one can pose the problem of defining a measure of ‘square-ness.’ Having done this with both elementary school and college students, I can attest to the fact that many reasonable answers are possible.
 I have written elsewhere on the topic of the reasons for the inclusion of mathematics in the school curriculum. See Can Technology Help Us Make the Mathematics Curriculum Intellectually Stimulating and Socially Responsible?, International Journal of Computers for Mathematics Learning, 1999, Vol. 4, Nos. 2-3, pp. 99-119.