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Symbols and Science

In Memoriam Andrew Gleason

Cardano, Sponge Cake, and Notation

Cardano’s famous book ”Ars Magna” (1545) marked the end of the
medieval period in mathematics and triggered the development of
modern mathematical notation. The famous formula for the cubic , as
presented in the ”Ars Magna”, resembles more a culinary recipe than
a modern formula. The switchover has implications for the teaching of

One view of algebra : quadratic equations

The quadratic formula

A number x satisfies

x2 + bx + c = 0

if and only if


If r + s = -b and rs = c, then

x2 + bx + c = (x - r )(x - s)


x2 + bx + c =0if and only if x = r or x = s:

Another view of algebra: understanding functions

H (daylight in hours)

Figure: Sunlight in Madrid

The graph shows the number of hours of daylight in Madrid for one
Why does the graph look linear ?
Estimate and interpret the slope of the line .
What month does the graph show?

What symbolic skills do modern students need?

Manipulative skills?
Graphical and numerical understanding ?
Symbol sense?

Al-Khwarismi, Hisab al-jabr w’al-muqabala, 830

what is the square which combined with ten of its
roots will give a sum total of 39? The manner of solving
this type of equation is to take one-half of the roots just
mentioned. . . . Therefore take 5, which multiplied by
itself gives 25, an amount which you add to 39 giving
64. Having taken then the square root of this which is
8, subtract from it half the roots, 5 leaving 3.

x2 + 10x = 39
x2 + 10x + 25 = 39 + 25 = 64
x + 5 = 8
x = 3


The quadratic formula in the 17th century

What is going on here?

Viete’s formulae and the quadratic formula


x2 + bx + c = 0

then let


Give an explanation, purely in terms of the structure of the
expressions, of why these two numbers satisfy
r + s = -b and rs = c:


When you add r and s, the plus and minus signs cancel .


x2 + bx + c = 0

then let


When you add r and s, the plus and minus signs cancel.


x2 + bx + c = 0

then let


When you add r and s, the plus and minus signs cancel.
When you multiply r and s, you get the difference of two squares in
the numerator ,

An example from economics

Figure: Cost price index for the last 100 years

Another view of the same data

Figure: Cost price index for the last 100 years

What skill is required to see the equivalence?

Manipulative skill.

Symbol sense. Understanding the structural similarity between

mx + b and Cax

An example from biology: the Michaelis-Menten

initial velocity of reaction
is initial concentration of substrate
vmax, are constants

How do you know if a reaction follows the Michaelis-Menten

An example from physics

What value of v makes this equal to ?
What value of v makes it equal to 0?

What are the differences between solving this equation for P, r , and

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