Saturday, July 18, 2015

The Perniciousness of Negative Numbers: Are Our Children Safe?

What follows is a paper I just wrote for a class called History of Mathematics for Middle School Teachers. The actual name of the paper was "The History of Negative Numbers in Mathematics and Education," but that's not exactly click bait, so I livened it up here. Most of the paper is about how mathematicians sort of pretended they were ignoring negative numbers for a while, then publicly freaked out about how ridiculous they were, then finally came to love them. I'm sure many of us have had an in-law relationship like that, so maybe you can relate. The end of the paper is an appendix with some thoughts about activities for students that reinforce the methods and philosophies described in the rest of the paper.


Why do we believe –5 is a number? To adult math teachers, the question seems silly or startling. We were told about negative numbers before our teenage years, and we’ve seen them used in budgets or economic reports, weather reports of temperatures, elevations below sea level, and, of course, math textbooks. We’ve added, subtracted, multiplied, and divided them just as we have with positive numbers. Why wouldn’t they be numbers? To our middle school students, however, understanding negative numbers and performing arithmetic with them are often far from natural, and in their confusion and skepticism they have much in common with mathematicians from previous centuries, who believed negative numbers were ridiculous or impossible. Considering the history of how mathematicians came to accept and embrace negative numbers provides some guidance for how we can help our students to understand them and work confidently with them.

In past centuries, many mathematicians, like our students, accepted the idea of unitless or abstract positive numbers, although with some limited exceptions, they avoided or openly scorned negative numbers in their methods and solutions. Philip E. B. Jourdain described abstract numbers in The Nature of Mathematics in 1913 [Newman p. 24]:
In arithmetic we use symbols of number. A symbol is any sign for a quantity, which is not the quantity itself. […] When we shake off all idea of “1,” “2,” &c., meaning one, two, &c., of anything in particular […] then the numbers are called abstract numbers.
Early Egyptian, Mesopotamian, and Greek mathematicians came to use abstract positive numbers to varying degrees, but they did not use negative numbers. In about 300 A.D., Diophantus, says Herbert Westren Turnbull in The Great Mathematicians, referred to “the impossible solution of the absurd equation 4 = 4x + 20” [Newman p. 115], although he accepted fractional solutions without difficulty.
Early Chinese and Indian mathematicians began using negative numbers in a limited way, often as tools for intermediate steps of problem solving. Early Chinese mathematicians used negative numbers as coefficients in intermediate steps of solving systems of equations. [Berlinghoff p. 81-82] In the 600s, Brahmagupta of India used and calculated with negative numbers to represent debts, and other Indian mathematicians later continued to use them and develop arithmetic rules for them. In the twelfth century, in his text the Lilavati, Bhaskara II found a negative distance for the position of a triangle’s altitude from a vertex, and he correctly interpreted it as “in the contrary direction,” producing an obtuse triangle. [Mumford, p. 126-127] Still, the Indian mathematicians were somewhat dubious about negative numbers, rejecting them as solutions to quadratic equations, for instance. [Berlinghoff p. 82]
In any case, Chinese and Indian mathematicians’ work with negative numbers did not end up being passed to other cultures. Arabic mathematicians, like Muhammad Ibn Musa Al-Khwarizmi in the 800s and Umar Al-Khayammi (Omar Khayyam) in about 1100 AD, avoided negative numbers in their work on complicated algebraic equations. Al-Khwarizmi expressed algebraic problems in words; Al-Khayammi placed each term on the side of the equation for which the coefficient would be positive. The limitations in how they wrote equations hindered them from seeing all quadratics or cubics as particular examples of a single type of equation. [Berlinghoff p. 82, 109-110]
In Europe from the time of the Renaissance onward, many types of mathematics flourished, including methods to solve increasingly complicated algebraic equations and the development of the coordinate plane. The focus on higher-order equations seems to have been part of why negative numbers seemed so unreasonable to European mathematicians, because if they accepted that negative numbers existed, they also felt obliged to accept taking their roots, and complex and imaginary numbers seemed completely over the top to them. (As David Mumford puts it, “It was because of negatives that square roots had a problem, so maybe it was best to consider them both as second class citizens of the world of numbers. […] The fate of –1 and i were inseparable.” [Mumford p. 140 & 142]) European mathematicians were finding negative numbers increasingly useful, yet they still shied away from embracing them fully. In his work Arithmetica integra in 1544, Michael Stifel represented all quadratic equations as a general form by using negative coefficients, but called them numeri absurdi and would not accept them as solutions. [Hollingdale p. 109] Turnbull says that in the 1500s, Girolamo Cardan (sometimes known as Cardano), “surmised the need” for negative, imaginary, or complex roots to cubic and quartic equations in accord with his ideas of how many roots these equations should have [Newman p. 119], but he seems to have avoided such solutions in his publications.
Before the late 1600s, many mathematicians were also confused about the size of negative numbers and their physical meaning. Mumford attributes European discomfort with negative numbers largely to “the overwhelming importance of Euclid,” with his focus on geometry and positive quantities, in the development of math in Europe. [Mumford p. 140-143] Antoine Arnauld felt, not unreasonably, that if -1 were truly smaller than 1, then it was not reasonable for their ratio to be the same no matter which came first (since this would never happen with positive numbers of different sizes). John Wallis believed that dividing by a negative number was essentially an even more extreme case of dividing by zero, and that therefore all such quotients would yield infinity. [Berlinghoff p. 84] Wallis was, however, a pioneer in the 1600s in using negative coordinates in the coordinate plane, which had been originally developed by Descartes with only positive coordinates. [Berlinghoff p. 140] This alteration allowed Europeans to think about positions of points more flexibly, in a way similar to that used by Bhaskara II centuries earlier in his writing about the obtuse triangle.
In his Treatise on Algebra in 1685, Wallis developed clear explanations of the meaning and laws of arithmetic of negative numbers, despite his confusion about division with them. He explained that multiplying by a negative number could mean taking away that many times. He also described multiplying a negative number by a positive number, and even explained the produce of two negative numbers: “[T]here may well be a Double Deļ¬cit as a Double Magnitude; and−2A is as much the Double of –A as+2A is the Double of A. . . But to Multiply –A by −2 is twice to take away a Defect or Negative. Now to take away a Defect is the same as to supply it; and twice to take away the Defect of A is the same as twice to add A or to put 2A .” [Mumford p. 137]

In another breakthrough, Wallis explained and illustrated a number line with negatives on the left and positives on the right, explaining that
 [W]hen it comes to a Physical Application, [a negative number] denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense. […] [If a man] having Advanced 5 Yards […] thence retreat 8 Yards […] and it then be asked, How much is he Advanced […]: I say –3 Yards […] That is to say, he is advanced 3 Yards less than nothing […] (Which) is but what we should say (in ordinary form of Speech), he is Retreated 3 Yards […] [Mumford p. 138]
Very soon thereafter, Isaac Newton was thinking about negative numbers in a similar way:
[I]n local motion, progression may be called affirmative motion, and regression negative motion; because the first augments, and the other diminishes the length of the way made. And after the same manner in geometry, if a line drawn in a certain way be reckoned for affirmative, then a line drawn the contrary way may be taken for negative. [Mumford p. 138]
It is easy to picture that Newton’s acceptance of negative numbers helped pave the way for his tremendous advances in calculus and physics.
Other scientists of the Enlightenment era in Europe were probably less comfortable than Newton was with negative numbers. The development of temperature scales gives some interesting evidence of a pronounced desire to avoid them. Early scales of the 1700s, including Fahrenheit’s, set 0° temperatures low enough that lab scientists of the time could describe lab temperatures with non-negative numbers; cold weather temperatures were not yet an area of interest. Anders Celsius developed a scale using the endpoints that are still familiar to us, of water’s freezing and boiling temperatures, which he determined precisely through experiments. Negative temperatures on the Celsius scale are, of course, much more likely to occur than with the Fahrenheit and related scales; for instance, the freezing temperature of brine used to set 0° Fahrenheit is negative on the modern Celsius scale. But Celsius decided to avoid negative temperatures for lab scientists by reversing the direction of the scale, making his 100° temperature water’s freezing point, and 0° the boiling point! Scientists used this reversed direction for a few decades before settling on the modern Celsius scale in the mid-1700s. [Beckman]

As for the post-Enlightenment European mathematicians, even after advances like Wallis’s and Newton’s, many remained skeptical of negative numbers for more than a century. In 1843, Augustus De Morgan wrote in his article Negative and Impossible Quantities, “These creations of algebra retained their existence, in the face of the obvious deficiency of rational explanation which characterized every attempt at their theory.” [Mumford p. 113] Philip Jourdain, the twentieth century mathematician mentioned earlier, perceived that negative numbers were useful; he acknowledged that “‘generalisations of number’ and transference of methods to analogous cases” were useful tricks of the trade that had led mathematicians to have “arrived at the truth by a sort of instinct.” Nevertheless, he did not feel that the generalization of numbers to negative numbers had been on a sound logical footing historically:
For centuries mathematicians used “negative” and “positive” numbers […] without any scruple, just as they used fractionary and “irrational” numbers. And when logically-minded men objected to these wrong statements, mathematicians simply ignored them or said: “Go on; faith will come to you.” And the mathematicians were right, and merely could not give correct reasons—or at least always gave wrong ones—for what they did. [Newman p. 25-27]
Jourdain reconciled himself to the logic of negative numbers by concluding that a negative number is the “operation which is the fulfillment of the order: ‘Subtract,’” adding, “Mathematicians call it a ‘number’ […] because of analogy: the same rules for calculation hold […] when ‘addition,’ ‘subtraction,’ &c., are redefined for these operations.” [p. 27-28] He goes on to describe negative numbers on a number line in a way that would be familiar to any of us in 2015, including the idea of +a and –a being placed symmetrically about 0. Later, he summarizes “the essence of algebra” as describing generally the “exceedingly complicated relations in which abstract things stand to one another. The motive for studying such relations,” he continues, “was originally, and still is in many cases, the close analogy of relations between certain things we see, hear, and touch in the world of actuality round us,” but in general, he notes approvingly, “we have reduced the definitions of all ‘numbers’ to logical terms” [p. 65], which is obviously a relief to him. Modern algebraists took similar ideas much further, defining our number system as a ring in ways that require negative numbers to be included as inverse elements for addition. At long last, we have come to a time when mathematicians consider negative numbers on an equal footing with their positive counterparts, and teaching children about them is not only accepted, but encouraged.

Activities for Students

As mathematicians came to accept negative numbers, they were used more freely in various contexts, so they are more familiar to today’s middle school students than they would have been to math students before the 1900s. Debts are still frequently described as a negative amount of money. Now that outdoor temperatures are frequently measured, negative temperatures are common contexts. Negative elevations (locations below sea level) are rare for land, but make sense to students as a context for problems. Students might be familiar with scales that go from a negative to a positive number (for instance, rating mood on a scale of -10 to 10). Finally, and perhaps most importantly for their flexible use of math as they grow up, students are now exposed to a variety of changes described with negative numbers, such as stock market drops or weight loss, and in middle school science they start describing physical changes (in position, for example) in terms of negative numbers as well. These examples provide a rich source of contexts for teachers to help students make sense of negative numbers, and our cultural acceptance and familiarity with negative numbers give our students advantages in learning about them that mathematicians in past centuries did not have. Seeing the contrast between historical and current use of negative numbers in “regular life” helps me better appreciate how essential it is to build upon this familiarity in order to help my students achieve a greater degree of comfort with negative numbers than mathematicians of the past ever experienced. The Illustrative Mathematics tasks “Above and below sea level,” “Comparing temperatures,” and “Bookstore account” would each help students explore negative numbers in a meaningful and familiar context.
Nevertheless, negative numbers will always represent another level of abstraction for our students beyond what they have experienced with positive numbers or even zero, which are easier to “see”. Mathematical advances of the past can guide how we help students with this abstract thinking. Multiplication and division, in particular, often are difficult to understand, and don’t have a clear meaning for many contexts, such as temperatures in degrees Fahrenheit or Celsius. Reading about historical explanations of negative numbers and how mathematicians made sense of them has given me a renewed appreciation for the number line developed by Wallis and currently promoted within the Common Core Mathematical Content Standards. The Illustrative Mathematics tasks “Integers on the number line 1,” “Integers on the number line 2,” “Fractions on the number line,” and “Distances between homes” are representative student work I will consider to help them understand and use number lines.


References

  1. Olaf Beckman, “History of the Celsius temperature scale,” 2001.
  2. William P. Berlinghoff and Fernando Q. Gouvea, Math Through the Ages: A Gentle History for Teachers and Others, Oxton House Publishers, Farmington, Maine, 2002 [1st edition].
  3. Common Core State Standards Initiative: Mathematics Standards. 
  4. Stuart Hollingdale, Makers of Mathematics, Penguin Books, London, 1989.
  5. David Mumford, “What’s So Baffling About Negative Numbers? — a Cross-Cultural Comparison.”
  6. James R. Newman, The World of Mathematics (vol. 1), Simon & Schuster, New York, 1956.


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