How Schools Fail to Teach Mathematics
Students are getting worse at math, and the educational system is the problem.
There is a global mathematics teaching problem. The US has regressed in recent PISA tests in mathematics, and is well below the OECD average which itself has been on a steady decline since it it started recording in 2003.
Young people are worse at mathematics now than 20 years ago, and are on a trend towards becoming worse still.
Students are taught mathematics, but they don't learn mathematics: "60 percent of U.S. students who enter community colleges are not qualified to take a college mathematics course, even though they have graduated high school" James Stigler UCLA
Note: While this article focuses on the US, many of the same observations apply to other countries as well, and the principles are universal.
Students don't like mathematics. Mathematics frequently shows up as the "most hated subject" by students.
So the teaching is ineffective, and fails to engage the students leading them to hate the entire field often for life. How did it come to this?
How most schools teach mathematics
Most schools use an archaic outdated teaching philosophy. There are at least 3 major roadblocks.
Lack of big picture
The curriculum leans heavily on memorization especially in the US which leads to poor adaptation to new scenarios. For example, lacking number sense may lead a student to compute $19\times6$ by rote instead of recognizing the easier variant $20\times6-6=114$.
Teachers have long lists of requirements and skills which they need to teach in a short amount of time, so they do not have time to properly relate the concepts to other concepts showing why they are important, and motivating the students. This leads to questions such as "why is this useful?" and "why are we learning this?".
The incentive structures in school are often aligned such that students will avoid making a mistake at all costs. Otherwise, it can have effects on their grades and their future prospects. This is not very productive for learning, and can lead to students failing to reach out for help. Mistakes are a natural and necessary part of learning.
These are all solvable by a proactive teacher, but there is an even more fundamental problem.
Lack of personalized education
Classrooms work at the abstraction level of the whole class. This is due to the expense associated with considering each student individually. Teachers are resource constrained, so they have to address many students, leading too little to distinguish different students in the class.
There are efforts to individualize education  , but it is limited, and can rarely extend much beyond the current classroom goals. You can give harder algebra problems, but you are still limited with how far you can go before you lose class cohesion, or become overburdened by designing personalized curriculums.
Moreover, classes are typically categorized based on age not academic level which leads a large variance in student ability making the need for individualized instruction even greater. And those classes are usually required to follow a set curriculum, so even if a student is two years (or ahead) they are forced to solve problems and tackle concepts that offer no instructional value. Either because they don't understand the prerequisites or have already mastered the concepts.
Mathematics is uniquely difficult to teach since every concept depends on previous concepts, so if you fail to learn one concept, it will have compounding effects as the class moves on without you. If you fall behind, it's very difficult to catch up.
This extends to homework as well which is usually given collectively to the whole class ignoring any individual differences. But there is also another limitation of homework. Usually you only get feedback later when you no longer remember what your thought process limiting how much value you get from it.
We can do better
I think we can do better.
While there are many facets you can improve, the fundamental problem with schools is that the teacher student ratio is too low, so you have to abstract the individual students away and handle the class as more or less one unit.
However, with big data analysis, we can personalize the learning experience in a way that's simply not available outside personal tutoring which is cost prohibitive for many.
But what are some of the properties that is required to create a good personalized learning experience?
The student should know why they're learning the concept, and how the concept relates to other concepts to properly contextualize the learning experience.
Principle of proximal learning
Perhaps the most important principle is that a personalized learning experience should meet the student at their level. This is formalized by the theory of the zone of proximal learning.
The learning material should tailored specifically for the student, and assist the student to reach beyond their current abilities.
Embracing mistakes & Immediate feedback
Mistakes are a natural part of learning, and should not be punished. Instead, we can use mistakes as learning opportunities with immediate targeted feedback.
By clustering common mistake types, we can leverage analysis techniques to find common errors, and help the students improve.
Putting it all together
Bringing all these principles together is a large endeavour, but I have been working on a project doing just that.
Njoror works by encoding the relationships between the concepts in a graph using that as a base for contextualization, and for the student-competency model. It analyses the answers from all students in order to recommend learning material that matches each student's level. Njoror also performs cluster analysis on the answers in order to identify common mistakes, and provides instant relevant feedback for the student to help them progress.
Njoror is still in pre-early access, but you can sign up for updates about the project, and an early invite.