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 mathematics. 
One view of algebra : quadratic equations
The quadratic formula A number x satisfies x^{2} + bx + c = 0 if and only if 
Factoring If r + s = b and rs = c, then x^{2} + bx + c = (x  r )(x  s) and x^{2} + 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
month.
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?
AlKhwarismi, Hisab aljabr w’almuqabala, 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 onehalf 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. x^{2} + 10x = 39 x^{2} + 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
If
x^{2} + bx + c = 0
then let
Exercise Give an explanation, purely in terms of the structure of the expressions, of why these two numbers satisfy r + s = b and rs = c: 
Answer When you add r and s, the plus and minus signs cancel. 
If
x^{2} + bx + c = 0
then let
Answer When you add r and s, the plus and minus signs cancel. 
If
x^{2} + bx + c = 0
then let
Answer 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 Ca^{x}
An example from biology: the MichaelisMenten
equation
initial velocity of reaction
is initial concentration of substrate
vmax, are constants
How do you know if a reaction follows the MichaelisMenten
equation?
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
n?
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