Analysing present day math education

Present situation

There are lot of problems with how mathematics is generally taught today in schools. The methods do not seem to be working too well, and a large proportion of the students are left behind, with only very rudimentary understanding of mathematics and no substantial skills in how to use math in everyday life.

We can see a number of issues with present day teaching practices:

The teaching is aimed at the average student. A certain amount of material has to be covered each term, irrespectively of how well the students will learn it. This means that the pace is too fast for the weaker students: who do not get enough time to practice and understand the material being taught. New topics build on previous topics, which they have not mastered properly, so learning becomes too difficult for them. With a slower pace, these students could have had a chance to stay on board. The pace is again too slow for the stronger students, who get bored and might loose interest for the topic alltogether.
Mathematics is a skill that has to be learnt, through practice. Students do get practice, through assignments that are solved in class or at home. But the students will usually not get individual feedback on their solutions. The teacher will show a model solution to an assignment, but the student usually has to figure out him/herself how this differs from own solution, and what were the errors there. Stronger students are usually motivated and will figure this out, but weaker students might find this too difficult of too cumbersome.
Students are not normally required to correct the errors in their solutions. The idea is that it is sufficient to show a correct solution, and that the student will remember this when solving a similar problem later on. However, the student learns best when finding and correcting errors in own solution. Hence, an important learning opportunity is missed.
The students are not usually taught in a very systematic way how to solve math problems. In particular, they are usually not required to justify and explain their solutions. The central idea of mathematics is that the results are achieved by a careful analysis of the problem, together with a sequence of correct derivation steps that lead to the required answer. In stead, the students are taught that the important thing is to get the right answer, with a combination of informal reasoning and calculations with paper and pen.
The teaching of mathematics in schools today does not correspond to the way mathematics is used in practice. A real life mathematical problem does not have a known answer. If it had, then there would be no need to solve the problem, we would just use the answer. In real life, it is very important that the answers we derive are correct. Very bad things can happen if we have a wrong answer to an important problem. The only way to guarantee that the answer is correct is to derive the answer in very careful steps, and check that each step is correct. And then also try to see whether the answer derived is reasonable.
A student gets points for a solution to an exam question based on what insights the student has demonstrated in the solution. This means that a student can get reasonably high points for solutions that give the wrong answer. In this way, the teaching de-emphasises the importance of getting the right answer, and emphasises the extent to which the solution follows a standard model solution.

Why is this the case

The best way to help both weak and strong students would have to provide an individual study paths for each student. But this is very difficult to organise in a class where the students are at very different skill levels in mathematics. The setup requires that the teacher goes through the material at a given speed, in order to fill the curriculum requirements. These again require that all students learn the same amount of material. Some learn this well, others not at all. So the fixed curriculum requirements are part of the problem.
We know that individual feedback for students would improve learning considerably. However, it is simply not cost effective for teachers to give individual feedback for each student. It takes too much time. To realise individual feedback would require that the number of math teachers is increased substantially. Parents can hire individual tutors for their children, but this is expensive, and not possible for all parents. Anyway, private tutors is rather a way to patch up a system that is not working correctly.
Students could correct the errors in their solutions for themselves. However, this is difficult when the model solution is too different from their own solution. So correcting would usually amount to just copying the model solution, even in cases where the students own solution could also lead to a correct solution. Partly this is due to the solutions written on paper, wihich makes correcting errors clumsy and messy,
Teachers are usually quite fluent in the mathematics needed to solve a specific problem. Hence, it is easy for them to just state the main steps in a solution, and outline a solution. Their solution is based on experience from solving a number of similar problems before. A student, however, may be solving a problem of this kind for the first time, and does not have the required expertise — this is what the student is trying to aquire. Hence, the presentation of the solution should be done much more carefully (which again could be quite boring for the teacher). Also, the justifications for each step in the solution is usually given verbally, only the algebraic manipulations are written out explicitly. When the students looks at the model solution later on, the verbal explanations are lost, and only the algebraic manipulations are shown. In other words, the logical structure of the solution is not shown explicitly in the solution.
The answer is given for a problem in school mathematics, so that the student can check whether the solution is correct or not. If the answer is wrong, the student can try find another solution. To this end, the correct answer serves a good purpose. However, the main method for making sure that the answer is correct in real life is then not taught, i.e., carefully checking that each step in the derivation is correct. Teachers often also prise short and insightful solutions, even if the reasoning can be quire sloppy.
Giving points for partially correct solutions is also a necessity for teachers. It shows the level of understanding that the student has about the theories needed to solve the problem at hand. The emphasis is on learning the underlying theory, not on getting the correct result. Partly this is because the students only have a limited amount of time in an exam, and have to provide a solution as far as they had time to work it out. But this then implicitly means that it is ok to have the wrong answer, as long as you had some good idea of how to derive a correct answer. In practice, this is not the case, a wrong answer is either useless or directly harmful.

eMath- a new method for teaching mathematics

We will here propose a new method for teaching mathematics that does not have the above listed weaknesses. This new method is made possible by the advances in digital technology, as it provides tools for the teacher and the student that have not been possible before.

The main ingrediants of this new method, eMath, are as follows:

A simple language for mathematical derivations, structured derivations, that is easy for student and teachers to use (after some training) and also easy for the computer to understand.
An editor for writing structured derivations, eMath editor, that makes it easy for students to write and edit their solutions.
An automatic system for checking the correctness of a students solution, the eMath checker. The checker goes through the students solutions step by step, and marks each step as proved correct, or not proved correct.
A learning environment, eMathStudio, that supports structured derivations and offers the eMath editor and eMath checker, The student can solve assignments in this environment using the editor and with the checker giving immediate feedback. The environment includes the course material needed, and allows the teacher and the student to follow in real time how the student is progressing in hid/her studies.

The digital environment allows us to update mathematics learning that can address the issues listed above, and make learning much more efficient.