When math education is student focused, it has a tendency to end up being about skills, drills and tests.
This is in contrast to math theory oriented math ed.
The reason is that math theory oriented math ed is about structure, explanations, logic and meaning.
When the math ed is student focused it starts from the student instead of the theory. But the student doesn’t know math theory, and doesn’t know the starting point of math theory.
Starting from the student, they will then have to ask how one lesson compares to another. Compares how? Since they don’t teach the meaning, which is math theory, they have to teach a skill.
Then the math ed compares one teaching method to another based on how well the student learns the skill.
So the decision of what lesson to teach depends on which lesson teaches the skill.
To determine which lesson teaches the skill, they have to test the skill with a test.
The lesson plan with the higher score on the test is the better lesson plan and it wins.
So they pick the lesson plan that does better on the test. I.e. they teach the test. I.e. they drill the skill.
The students then complain math is meaningless, and is one random lesson after another.
This is because math ed decided to be student focused instead of theory focused.
When you start with theory, you say the beginning is set theory, truth tables, and the Peano Axioms. Then you define addition as a function and prove its properties. You continue on until you get to place value notation and construct it and prove its properties. Then you continue to fractions.
Theory oriented is New Math. In theory oriented math ed, you first figure out what it is you have to teach the student. Then you try to figure out what works.
In student oriented math ed, you figure out what lesson plan produces a testable result that is better than another. So you end up at skills, drills and teaching the test. You forget completely what meaning is.
If we look at 1950 to present we see that. New Math started in the 1950s and 60s and then got a bad name. So they switched to student oriented ways of thinking.
New Math got a bad name in part because parents and teachers were not prepared for it. But the big reason was New Math did not have a program to start with the Peano Axioms and then go to place value notation step by step and then fractions. So New Math had no payoff in terms of covering the standard arithmetic structures and methods. So no one saw any point to it. That is what killed New Math.
What we have now is the ultimate result of abandoning theory and being student oriented. We have a high stakes testing regime that is skills oriented and the decider on what lesson plan to use is how well the student does on a skills assessment test.
And everyone is complaining it is one random skill after another with no pattern or structure and seems meaningless.
They also miss the chance to teach recursion. Recursion is the basis of much of economics, finance, probability and decision theory. So by leaving out recursion, which is theory and not skills, they miss the chance to teach the applications of elementary math that are more advanced but actually used now. One example is the binomial lattice for option pricing. This is recursive but with two dimensions instead of the linear recursion on the number line.
Addition and multiplication are linear recursions as typically defined. If you learn those well, including proofs, then going to the two dimensional recursion of a binomial lattice is much easier. That sets up many real world applications that are actually used in business today and which seem powerful and relevant.
The problem with math ed today is it is student focused. This produces a skills, drills, high stakes testing regime that turns off students and teachers alike. We can call this the Paradox of Student Focused Math Ed. It produces a curriculum students don’t like. This is because the students are not inspired.
Vision and Works have to go together in any great endeavor. When it is all works, i.e. applications, it lacks the power to inspire.
Burnham was quoted after his death as saying, “Make no little plans. They have no magic to stir men’s blood and probably will not themselves be realized.”
The version I misremember is
Make no little plans, they have no power to move men’s minds.
Students are not stirred or moved by one seemingly random skill lesson after another, followed by homework drills and high stakes testing. They want off that roller coaster. It leaves them disoriented.
The structure of math starting from Peano Axioms and then constructing place notation and proving its properties including the standard algorithms and other ones is a vision that has the power to move student minds. It is not a sequence of meaningless lesson plans whose only goal is skill drilling for high stakes testing.
It is Peano Axioms or high stakes testing. This is because it is known from the 19th century that arithmetic has a logical starting point consisting of the Peano Axioms. If you don’t start with the Peano Axioms then you are off the meaningful path and that means you are in the woods of skills and high stakes testing.
Only the proof path is meaningful. Once you go off the straight and narrow proof path, they you are doomed to the shoals and shallows of skills and drills. Eventually, a high stakes test carries you off and you are finished.
Keith Devlin’s Mathematical Thinking Coursera adventure is exciting because it is not just skills, but is getting back on the path of math. The proof path of math is where the meaning lies. Where the meaning lies is where your heart lies also.
The heart has its reasons that the high stakes testing knows not of. Those reasons are found on the proof path not the high stakes testing sheet.