Algebra can be taught in self-discovery projects starting from the Peano Axioms. In this approach, the students are given the job of coming up with algebraic definitions of successor, addition, multiplication of first natural numbers, then rationals. Then they do signed integers and then signed rationals.
Each step is given as a project or task. They have to develop an object oriented definition using data and then functions or operations that use that data.
Each new step is given as a project. So instead of lecturing from Peano Axioms to the properties of rational numbers, students are given each step to do as a project. They have to come up with an algebraic answer, not an analogy to area.
This is close to doing up through the rational number part of the book by Edmund Landau, Foundations of Analysis 1930 as a series of projects. However, Landau’s start with Peano Axioms and addition is too complicated because of the complicated definition of addition. That is why I wrote the book Pre Algebra New Math Done Right Peano Axioms using an easier definition of addition based on the recursion theorem explicitly, as Dedekind did in 1888.
This approach is the opposite of applications. Here the students have to retrace arithmetic using and creating algebra as a symbolic and mathematically meaningful object. This teaches them the concepts of algebra as linked to the procedures of each object oriented number type.
Successor and predecessor functions before addition are an example of chunking. Foundations of math is chunking.
Multiplication of natural numbers is repeated addition. Multiplication of a rational number by a natural number can also be defined as repeated addition.
Division of a rational number by a natural number has to be done in a formal way, however.
(m,n)/q = (m,n*q)
This concept of division may be motivated by pizza slicing and refinement, but it is not defined by pizza slicing. Area is not the definition of rational number.
Area for simple point sets of rational numbers can be a separate set of projects. That will help teach the difference between a foundation subject such as rational numbers that are defined algebraically and the application of numbers to area.
Current teaching confuses number with area as concepts. So fractions are justified as existing by an invocation of geometry. This is what Euler tried in 1765 in his Elements of Algebra. But that failed. Euler ignores his own remarks and simply defines the operations with fractions using letters and algebra ignoring geometry completely.
That should teach us that using area to teach fractions is a mistake. If Euler could not use area to justify the rules for fractions, then we should give it up as well. Euler would have done it if he could have but he failed and gave up on it. So should we.
The project approach to algebra starting from Peano Axioms also teaches or incubates algebraic ingenuity and mastery. Students will learn to be maestros of notation. Students will self discover or re-invent bits of algebraic notation. This will give them confidence in abstraction and in symbolic manipulation skills.
This is very different than confidence from applications much less drills. Having the feeling of inventing a bit of notation or of how to combine notations to define the next step in the progression from Peano Axioms to rational numbers is powerful.
Retracing the construction of arithmetic and algebra from the Peano Axioms, reinventing bits along the way gives a feeling of understanding and mastering the insides. This is learning to build the music instrument, not just play it.
The project is distinguished from the lecture approach by asking the students to try to find an algebraic definition of each step from the road from Peano Axioms to rational numbers.
If we have ordered pairs for fractions already (m,n), then we ask students to define subdivision by p parts. So they are asked it this way. They basically are being led to define
(m,n)/p = (m,n*p)
This can be done for m=1 first.
Pizza can be used for analogous reasoning, but they should distinguish the algebraic answer from the area justification. They should know that ultimately, they will be asked to do projects defining area in terms of sets of rational numbers and develop properties of area based on the properties of rational numbers, not the other way around.
This might also be done in parallel. So simple properties of area can be made either axioms of area or requirements on a definition of area in terms of a function from a point set of rational numbers to the rational numbers. This definition will eventually run up against problems such as some point sets of rational numbers don’t have an area as a rational number if we maintain nice properties for an area function.