Theory focused math tells the story. It has a beginning, set theory, truth tables, and Peano Axioms. Then it has an adventure. It makes sense. It has a progression. They end up constructing and proving the properties of place value notation and its algorithms.
Student focused math never tells this story or even thinks to tell it. Student focused math is about testing. Student focused math teaches the test.
Stories have internal meaning. Tests have external meaning. If you do badly on a test, then it is bad.
Theory focused math tells the story. It holds the interest of the teacher or textbook writer. This is a sign of a story.
Student focused math is about testing. It holds no one’s interest except for the external consequences for students and teachers.
Poor test performance results in raising the anxiety level of students, parents, teachers and administrators. This leads to even more tunnel vision focus on skills and drills to make up the past poor performance.
Student focused math leads to test cheating by students and even teachers. Student focused math leads to investigations of testing.
Student focused math leads to Bill Gates Foundation going around getting teachers fired, schools closed, and more testing for those left. Student focused math is one way to solve the teacher pension problem. Keep testing the students until their scores go down and fire the teachers.
Student focused math is about driving students and teachers all out with stress and anxiety to drill skills and teach the test.
Theory focused math, which we might call math focused math, instead tells the story of starting with basic concepts and defining addition and multiplication. We don’t just memorize the addition and multiplication tables, we define addition as a recursive function and build them.
When we use number lines, we use a pair of number lines to teach the right addition identities.
x+0 = x
x+y’ = (x+y)’
These correspond to the first number line having x, the second number line has its 0 over the x and the sum is over the y. If y is 0, we just get x.
If we go y on the second we are over the point x+y on the first. If we go one more on the second number line, to y’, then we go one more on the first number line to (x+y)’.
With math focused math we use the pair of number lines to teach the definition of addition of natural numbers, i.e. we teach the meaning of addition.
This is opposed to the current system of teaching a little number line counting and then memorizing the addition table and then testing. The link from counting on a single number line to the addition of two numbers as a function is never spelled out in the current methods. Not in books, videos, or lesson plans. It is ignored.
They just go onto memorizing the addition table and the test. That addition is a function with two inputs may be mentioned, but its meaning is not developed very much. This then becomes a key concept not explained that then hinders learning meaning thereafter. These blocks to understanding accumulate until skilling and drilling and testing are all that are left.