There is math and mathematical thinking in K-12 math. K-12 math is the good stuff in math. It isn’t some preliminary that isn’t used or good for anything. K-12 is used and it is useful. It is math.

When you do K-12 math and you are self aware of what you are doing, then you are engaged in mathematical thinking.

Mathematical thinking is not something that happens only at Stanford, Harvard and Cambridge and a few similar places. Math thinking happens whenever you use math or do math and are aware of what you are doing.

Dedekind’s 1888 book on the natural numbers is math thinking. It is the core of New Math. It is a major development in the use of set theory. The idea of closed sets as David Joyce calls them in Dedekind is a forerunner of concepts in topology, as in effect David Joyce points out in his notes on Dedekind.

When we add the use of the Dedekind prime notation for successor, the concepts in K-6 are even more exposed. A notation that expresses a concept should be taught precisely because it is a mechanical way to teach conceptual thinking.

n’ means the successor. So 2′ =3. 0′ = 1. x’ = 0 has no solution (before signs are introduced).

i+0 = i

i+j’ = (i+j)’

Is one way to define addition. This is from Grassmann 1861.

The other way that I invented is pitch line addition.

i+0 = i

i’ + ‘j = i+j.

Here ‘j is the number before j. So ‘3 = 2.

The successor is a function. (i,i’) is an ordered pair in the graph of the successor function. No ordered pair (n,0) exists. If (i,j) and (m,j) are two pairs then i=m. We also have (i,j) , (i,k) then j=k.

These are concepts. The Dedekind 1888 book is a conceptual book. Dedekind 1888 is mathematical thinking.

We can put all of that into K-8 teaching. This leads naturally to algebra. It also teaches proofs.

Lemma (i+j)+0 = i + (j+0)

Lemma (i+j)+k’ = i + (j+k’)

These two lemmas teach the associative law of addition of natural numbers. They can be taught in K-8.

They may be in the other Coursera course, introduction to logic or perhaps in Devlin’s Mathematical Thinking or both.

The Babylonians had mathematical thinking. Mathematical thinking did not start with the Greeks.

The Egyptians had mathematical thinking to build the pyramids. One pyramid fell down because the slope was too steep. They lowered the slope.

The Greeks tunneled a water tunnel from two sides of a mountain. This was mathematical thinking.

Tom M. Apostol writes on this.

http://www.mamikon.com/TunnelSamos.pdf

Apostol wrote a calculus book widely admired as more conceptual than others. He also wrote a mathematical analysis text that is considered better than Rudin by many.

http://en.wikipedia.org/wiki/Tunnel_of_Eupalinos

http://www.projectmathematics.com/samos.htm

http://science.discovery.com/videos/what-the-ancients-knew-eupalinos-tunnel-like-water-f.html

http://en.wikipedia.org/wiki/List_of_ancient_architectural_records#Tunnels

Mathematical thinking was not invented in the 19th century. It is not restricted to algebraic topology. You can do things other than algebraic topology and it is mathematical thinking. If algebraic topology was the only thing that counted as mathematical thinking, we would call it algebraic topology thinking.

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