The Thomas Jefferson High School admissions test is not hard enough and doesn’t qualify students good enough or advanced enough in intro algebra for the work at TJ. This results in the need for remedial teaching this summer at TJ in algebra.
My comment at WaPo.
The concept of intro algebra as teaching only problem solving should be revised. This makes algebra lack its core.
Every activity or human enterprise has 1) faith 2) works. For the secular, vision and works. Works alone are never enough.
The vision/faith part of math is axioms, definitions, theorems, and proofs. If you leave that out, then you leave out the vision part.
What happens is 1) works alone don’t make sense 2) problem solving seems contrived and pointless 3) there is no inspiration 4) there is no sense of progress to achieve a meaningful goal.
Algebra must be taught from the Peano Axioms of the basic properties of natural numbers and the laws of associativity, commutativity of addition and multiplication proven as theorems. This is the cathedral of intro algebra. If you don’t show them the cathedral, just the work shed, then they are not inspired.
My web page New Math Done Right has numerous entries on teaching Peano Axioms at below collegel level. As an example I give the Peano Axioms for a number line as a way to teach Peano Axioms at an intro level.
==end of WaPo comment
The errors that students make in intro algebra are ones where they don’t get what it means. This is predictable since they are taught problem solving immediately without any attempt to teach the structure of algebra.
To teach the structure of algebra and of numbers, you have to start with the 5 Peano Axioms and teach those. Then you teach the definitions of addition and multiplication. Then you teach the proofs of their properties. Once you have done this, you have taught the meaning of algebra.
In this approach, you use the methods of algebra such as letters to stand for numbers for a purpose. Proving the algebraic laws can’t be done using just numerical symbols. You are led naturally to using letters for numbers to teach the laws and their proofs. By the time you get through teaching the proofs of associativity and commutativity and the distributive law, you have taught the meaning of intro algebra inside and out.
Having learning the proofs of algebraic laws, students will not be so easily confused into making simple errors with order of operations and parentheses. This is because they have a structure to put these techniques into. That structure is the definitions of addition and multiplication as functions and the proofs of their properties.
==Summer Sale on e-book
Pre-Algebra New Math Done Right Peano Axioms
is on sale for 2.99 at e-book vendors.