Hung-Hsi Wu writes that the transition from arithmetic to algebra in 5th to 7th grade is depressed. It is too much play.
From page 7
However, the current 5–7 curriculum chooses to dumb down the
mathematics and replace precise reasoning and abstraction with
hands-on activities, picture-drawings, analogies, and metaphors,
with emphasis on “replace”. (More on this later).
This is an artificially depressed curriculum.
As math ed psychologists Asmuth and Rips pointed out, proofs for elementary school math are based on mathematical induction. (See the previous post.)
Pre-Algebra New Math Done Right Peano Axioms is an approximately 390 page book on the start of natural number addition. This has numerous examples and exercises. Most text books cover natural number addition in 10 or 20 pages. Some just put up a number line and then go on to signed numbers. That is not good enough.
A major part of the problem is that school teachers in the US have never seen Peano Axioms or mathematical induction taught without heavy algebraic formulas. This is done in PANMDRPA.
Around page 33, Hu criticizes the pre-New Math talk of what a variable means. A variable is really one of the elements in an ordered pair that makes up a function. Variable is an earlier way of thinking that has a powerful persistence.
The old idea was a variable was a thing itself. A variable was like a register in a computer memory that can change in value.
The modern concept is on a function. This change in emphasis is made in Pre-Algebra New Math Done Right in the way Wu is talking about. This is not done in any other text at this level.
Wu makes the point that modern math really deals with numbers and functions as opposed to variables. Getting people to stop using the word variable is close to hopeless. Variable comes with so much baggage that we need a positive example to counter-act the sloppiness it brings. This positive example is an extensive work with functions in a meaningful way that is different in kind than solving algebraic equations for “the variable”.
Pre Algebra New Math Done Right provides this contrasting positive example and antidote to fuzzy thinking about what the word variable means. It uses functions as the basic object of understanding arithmetic. The successor function has its own notations. S(0) = 1. Or 0′ = 1. This notation and the emphasis on meaningful functions such as successor, predecessor, the pitch lines introduced for the first time in the book, and ultimately addition itself provide the needed counter agent to the fuzzy thinking on variable that students are immersed in.