People give up trying to understand structure with real numbers

Infinite decimals make people give up thinking.  They don’t believe they can understand an infinite decimal, especially one that does not repeat.  Therefore, K-8 should avoid reals as much as possible.

It is natural to think you can understand finite sets of naturals perfectly.  Then rational numbers you believe are finite and constructive when taken one at a time or when the denominator is fixed or bounded.  When you have an infinite number of rationals in an interval, we take a step down in our confidence, but we still see it as feasible.  But an interval of real numbers and infinite decimals that don’t repeat convince us we can’t really understand it completely. So we don’t try.

Pedagogy should take this into account.  Do everything possible with the new idea on finite sets of naturals first. So length, area, slope, functions as ordered pairs, algebra formulas, and laws of algebra.  Work out where this runs into limitations.   This is where we need rationals.  So we set up the need for rationals by restricting to naturals.  We do this over and over grade after grade by doing the new concept first on naturals only.

Then do the rationals on the new concept.   If we have set up the need for rationals from the naturals, the rational case will be easy.  We then end up where the rationals are not sufficient.  This helps reinforce what the rationals are and how they work, as we struggle to make them work for the new concept or application such as length of the hypotenuse or area of a circle.

We can then do the real case, but this should leverage off of the transition from naturals to rationals.  We should reason by analogy how things change when we change the number concept.  We should spend the least time on the real case in K-10.  We should still emphasize the natural case with a greater emphasis on induction, recursion theorem, definition by induction-recursion, and proof by induction.  In higher grade levels, the distinction between these should be emphasized more.