Chapter 4 of Morris Kline's Mathematics and the Physical World has the title of "The Deeper Waters of Arithmetic." In this chapter Kline describes the discovery of irrational and negative numbers.
"...where ignorance is bliss, 'Tis folly to be wise." -- Thomas Gray, 18th century English poet
Kline begins Chapter 4 as follows:
"Mathematics could have been a rather simple and innocuous subject if the concept of number had been limited to the whole numbers and fractions."
This chapter reminds us that for centuries after their discoveries, many mathematicians would rather not learn about irrationals or negatives at all, rather than accept them as numbers. And in many ways, our students in the modern middle school classroom feel the same way.
Kline starts his chapter out with some Geometry -- the Pythagorean Theorem, in fact. But he only brings up Pythagoras in order to introduce his discovery that sqrt(2) is irrational. He mentions the proof of the irrationality of sqrt(2) -- the same one I mentioned on the blog a few weeks ago right after Square Root Day:
"There is a legend that the discovery of sqrt(2) was made by a member while the entire group of Pythagoreans was at sea. The member was thrown overboard and the rest of the group pledged to secrecy."
While Pythagoras and his followers were struggling to accept irrational numbers, negative numbers were being discovered, not in classical Greece, but in India, 3000 miles away and 1000 years later. As Kline writes:
"The Hindus saw that when the usual, positive numbers were used to represent assets, it was helpful to have other numbers represent debts."
But, as Kline continues, it took centuries for numberhood to be granted to both of these new, strange types of number. He writes that it was not until the 17th century when mathematicians fully embraced irrationals, and not until the 19th century before they accepted negative numbers! To many mathematicians, the idea that there could be numbers that aren't ratios or are less than zero was too undesirable for irrationals and negatives to be granted numberhood.
What did it take for mathematicians to accept these new numbers at long last? Kline writes:
"These new numbers, suggested by physical uses such as the representation of lengths in the case of irrational numbers or debts in the case of negative numbers, were as legitimate as the whole numbers and fractions. They recognized further that the axioms about number -- that is, the premises on which reasoning about number is based -- applied as well to the new numbers as to the old ones."
Kline goes on to describe the commutative, associative, and distributive axioms -- that is, the postulates, or properties, from Lesson 1-7 of the U of Chicago text. The new numbers, the irrationals and negatives, satisfy these properties as well as the whole numbers and fractions. But it's additional properties that ultimately forced the acceptance of irrationals and negatives.
In particular, the Additive Inverse Property tells us that every number has an opposite. The idea that there can be numbers without additive inverses is more undesirable than the idea that some numbers can be less than zero. And more advanced properties from Calculus, such as the least upper bound property, force the existence of irrationals. The idea that there can be a bounded sequence without a least upper bound is even more undesirable than the idea that some numbers aren't irrational.
Kline compares the distinction between definitions and axioms of mathematical terms to that between the definition and axioms of citizenship. The definition of citizenship tells us exactly who is a citizen (sufficient condition), but the axioms tell us what rights it bestows. Just as every real number has the right to an additive inverse, citizens have the right to vote and defend self and country.
As I reflect upon this chapter, I think about the five named sets of numbers -- natural numbers, integers, rational numbers, real numbers, and complex numbers. These five sets are so important that they are labeled with bold letters:
-- N, Z, Q, R, and C.
This chain approximately represents the order in which these sets are discovered -- people were working with natural numbers for many millennia before they considered real or complex numbers.
Likewise, students learn about natural numbers in kindergarten or even earlier, but they don't see complex numbers until Algebra II. Even if we ignore the complex numbers (since we haven't reached them in Kline yet), the real numbers aren't fully studied until eighth grade under the Common Core.
But this isn't exactly the order in which the set were discovered. As Kline tells us, not only did the Egyptians and Babylonians use fractions well before the use of Z, but the Greeks even discovered irrational numbers well before the Hindus started using integers. So the order in which the sets were discovered looks more like:
-- N, Q+ (fractions), R+ (unsigned reals), R, C
The proof website Metamath builds up complex numbers from sets, and this is the order in which the sets are built up -- naturals, fractions, unsigned reals, signed reals, complex numbers.
On the other hand, this is the order in which students learn the sets under Common Core:
-- N, Q+, Q, R, C
This explains why the seventh grade Common Core Standards mention "rational numbers" so often -- this is the year when students learn about positive and negative numbers, but since they already learned about fractions, they can now work with all of Q.
I've also mentioned the Classical Curriculum here on the blog. It is based on the idea that students, in four-year intervals -- learn all of history, and align English and science to the historical period about which they are learning.
Math is left out of this Classical Curriculum pattern, but there's no reason we can't include it. Here in California, for example, both Ancient Greece and India are studied in sixth grade history. So we could introduce both irrationals and negatives to sixth graders, right around the same time of year that they are learning about the cultures. Then as irrationals and negatives weren't fully accepted until around the time of the American Revolution, these concepts can be revealed until eighth grade, when the students are taking U.S. history.
As interesting as this idea is, it may be troublesome because it would require linking Common Core national standards to history state standards. And as controversial as Common Core is, the idea of national history standards are somewhat dangerous, since it's too easy to teach history with a specific ideological bias -- and we wouldn't want biases to be enshrined in national standards.
Kline ends the chapter thusly:
"One might expect that the power of mathematics to tackle and explore deeper and more significant physical problems would be considerably increased thereby. We shall soon see whether the game is worth the candle."
Question 4 of the PARCC Practice Test is on the equation of a circle:
4. The equation x^2 - 10x + 17 = -y^2 - 2y describes a circle in the coordinate plane. Find the radius of the circle and the coordinates of the center.
This is one of those questions that requires completing the square. We begin by setting up the equation for completing the square, with the variable terms on the left and the constant on the right:
x^2 - 10x + 17 = -y^2 - 2y
x^2 - 10x + y^2 + 2y = -17
x^2 - 10x + 25 + y^2 + 2y + 1 = -17 + 25 + 1
(x - 5)^2 + (y + 1)^2 = 9
And so the radius is 3 and the center is (5, -1). There are all sorts of points in this problem where students might make mistakes:
-- They might complete the square on -y^2 - 2y, instead of transposing these terms to the left side.
-- They might forget to take half of -10 and 2 before squaring.
-- They might miss the negative sign on 17 and obtain a wrong value on the right side.
-- They might forget that the right side is r^2, not r, and claim a radius of 9.
-- They might forget (x - h) and (y - k), not (x + h) or (y + k), and claim a center of (5, -1).
To me, this problem belongs in Algebra II, not Geometry. The U of Chicago text has a lesson on Equations for Circles, Lesson 11-3. In this lesson, all the equations are already in the correct form, and so only the last two student errors in the above list are possible. But PARCC requires the students to complete the square to put the equation in the correct form. Now we see that there are five common mistakes students might make, rather than just two.
Furthermore, we know that completing the square isn't taught in many Algebra I classes anymore. The U of Chicago Algebra I text doesn't mention completing the square at all. And in many Algebra I classes whose texts do contain completing the square, teachers end up skipping the lesson in the rush to finish the unit on Quadratic Equations before the PARCC or other end-of-year test.
In some ways, the U of Chicago text represents a compromise between the idea that students should see circle equations in Geometry and the opposite idea that they should wait until Algebra II. In our text, the students at least see that there's a connection between algebra and geometry as they are exposed to the derivation of the circle equation. They learn that not only do lines have equations -- as they've seen in Algebra I -- but so do circles. And they are now aware that they must be careful not to confuse r with r^2 or h and k with -h and -k. This will prepare them for completing the square, which they will see the following year in Algebra II. But this compromise is impossible. As long as equations of circles and completing the square appear in the Common Core Geometry Standards, we are obligated to teach it in Geometry, period.
By the way, I decided to look up the conic section unit in an actual Algebra II text. In the text I chose, the first lesson reviews the Distance and Midpoint Formulas. This is followed by one lesson on each of the major conic sections (parabolas first, then circles, ellipses, and hyperbolas), but in these lessons, the vertex (of the parabola) or the center is always at the origin. Only in the following section are the conics translated and completing the square required. So surely if we are to each it in Geometry, we ought to do the same as the Algebra II text, and devote the first day to circles centered at the origin and the second to translated circles and completing the square. But of course, I didn't have time to teach circle equations that way on the blog.
PARCC Practice EOY Question 4
U of Chicago Correspondence: Lesson 11-3, Equations for Circles
Key Theorem: Theorem (Equation for a Circle)
The circle with center (h, k) and radius r is the set of points (x, y) satisfying
(x - h)^2 + (y - k)^2 = r^2.
Common Core Standard:
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Commentary: This standard is a bit difficult for students at this level, nevertheless the students are expected to learn it. When we covered circle equations a few weeks ago on the blog, I made sure to include completing the square as I was aware of the presence of this sort of question on the PARCC. The worksheet gradually introduces equations with more and more algebraic manipulation required to put them in center-radius form.