For teaching students from K-?, we should have the following emphasis.
- Natural Numbers almost perfect. This includes mathematical induction and proofs.
- Limitations of natural numbers almost perfect.
- Rational numbers well. This includes ordered pairs.
- Limitations of rational numbers well.
- Real numbers leverage the transition from naturals to rationals to give them insight on rationals to reals.
Teaching algebra on the real numbers in 8th grade as the primary emphasis is a mistake. This is because the concept of function as ordered pair is poorly developed when dealing with the real number continuum. We can’t visualize the typical real number very well. It has an infinite decimal. It is a point on a line. These are vague notions compared to naturals and rationals.
With naturals, we understand them well. There is nothing between two succeeding naturals. This is taught as a defect of naturals instead of as a strength. When naturals are taught as indexing a count and keeping place, this not a defect but a feature.
Everything should be taught on naturals first. This reinforces the function as ordered pair concept. It reinforces the logic of numbers. It is easier to understand. It helps isolate a new concept like using letters for variables or the laws of algebra for letters. On a bounded set, there is a finite amount of information and we can list it all. This gives a valuable feeling of certainty and complete understanding. We can use these lists to prove special cases.
Next we go to using the rationals. Some of the benefits of naturals continue. For a fixed denominator, we have something very like the natural sequence. We can define successor on it, and from that mirror the definitions of addition of that rational and multiplication by natural numbers. Understanding this distinction of the two cases helps teach the concepts and distinguish naturals, rationals, and their properties and structure.
The rationals are concrete and constructive in a real analysis sense. They are computable. This means they help us focus on the lesson concept such as slope without having uneasiness from the difficulties of real numbers in a constructive and computable sense.
Real numbers are much harder than rationals. Functions on reals are even more harder than functions on rationals. In algebra we should teach functions and teach functions as ordered pairs. This is much easier when we consider functions restricted to the naturals and then progress from those to functions restricted to the rationals. This also leads into calculus and analytic geometry in a more structural way.
This way of thinking about pedagogy is only possible by reasoning from the foundations of math. Foundations of math is the meaning of elementary math. When we reason from those foundations, we can see where students are having problems can be fixed by stepping backwards to the easier cases first. The easier cases of functions are functions on naturals and then functions on rationals. To focus on functions, we should compare these two cases. This also emphasizes how the role of domain and range can change the procedures and methods we use and ultimately the logic we use to do proofs.
The conventional approach loses these chances to teach lessons where things are simple and finite such as bounded sets of naturals or simple and constructive such as rationals.
Restricting to such cases will let us isolate on the concept being focused on as well as reinforce naturals and rationals and their link to each other.
Rationals are a big step from naturals and this is not well appreciated in teaching algebra. The logical structure and methods change. The step to reals is even more radical and disconnected with finite calculations. We need to be mindful of this when designing lessons and curriculum.
Self discovery is much easier when we isolate out the naturals, rationals and reals and the step from naturals to rationals receives heavy focus. This teaches the structure of the progression from naturals to rationals. This is not emphasized well.
Keith Devlin in his discussion of repeated addition wants them all mashed together. He says the students can’t understand the difference or progression from naturals to reals, so just teach the field axioms. However, the reason they don’t get the difference is they are not taught it explicitly. Teaching this difference should be a major goal of K-8 education.
This is really teaching the structure of elementary math. This covers the concept of ordered pair and function much better. Rationals are much better understood. This helps focus on where many students become lost. The structure is also more interesting. It is more suitable to self discovery methods because it is conceptual.
This is part of the harmony of math foundations and pedagogic content knowledge. Math foundations is conceptual and is the right and consistent explanations of elementary math. The concepts and objects of elementary math are connected by math foundations. Pedagogical content knowledge then picks up on that explicit structure and makes sure the students learn it.