During last summer when I planned out what I wanted to discuss in this series of “Are tests necessary” this topic was not included, so to me this feels like a small spin-off post. However, in part 1 of this series I said that I will focus on things “what not to do” and this current topic falls under that category.

During last years a new trend has come into schools. A complement, or a substitute for tests, could be matrices for grading students. For the purpose of this article I will invent two new names for matrices in order to separate two completely different types: “Matrices for subject knowledge” and “matrices for student abilities”.

So what is a “matrix for subject knowledge”? A matrix for subject knowledge is a matrix that takes a chapter, for example equations in grade seven, and explains for the students what kind of equations they need to solve in order to get a specific grade. Since different countries use different scales for grading I will just use level one, two and three for the purpose of this article. A very simple example to demonstrate this type of matrix could look like this, but in reality with more details:

Equations |
Level 1 | Level 2 | Level 3 |

Mechanical solving | 2x-2=10 | 2x-3=3(x-1) | (2x-3)/3=0,1 |

Problem solving | Easy | Moderate | Hard |

The idea is to visualize the content for the students in order to support the learning process. However, the idea that this matrix will help students is questionable. If you tell a student during the introduction of equations that now we are going to learn how to solve equations and the goal is to learn how to solve “(2x-3)/3=0,1” I am pretty sure that the student will have no idea what you are talking about, or for what purpose you will need that equation. So if you would introduce equations in this way I am sure you would have a classroom filled with confused students, so the idea of this being a good visual aid for students falls apart really fast. However, after a while when you have been working on equations, you could show this kind of matrix for the students in order to explain what is expected of them to get a specific grade. So could it be that this matrix then becomes a good tool for teachers in order to grade students?

Let us pause for a moment and think about how this matrix is different from a normal test. A test also has easy, moderate and hard tasks in order to grade students. So could you not as a teacher just as well show an old test for your students in order to demonstrate what is expected of them? Is there a significant difference between the matrix and the tests? The answer is; no. It is the same old idea in a different package, but could it be that it is worse? There is one simple reason why I would choose a standard test over this matrix if I had to choose between the two.

For anyone having worked with teenagers will know that some of them will always try to find the easiest way out (until they become interested in the subject), and if you present them with the idea of solving the equation“3x+2=11” and you will pass the course, they will stop there and do nothing more. If that happens, and it will, this will lead to poor understanding of equations. With a poor understanding in equations you will have a hard time applying them in future chapters, resulting in a spiral of superficial knowledge. This is obviously not the way to go if you want students to gain interest or deeper knowledge in mathematics.

If someone is thinking “what about the average or skilled student, does not this matrix help them to know what to learn?” My answer to that is that many students striving for high grades are good at learning what the teacher expects of them. However, it has been shown in studies that just learning the necessary information to achieve a good grade does not result in any deeper knowledge of the subject, strange is it not?

These kind of matrices obviously will not work as intended but I have not even started discussing its most fundamental flaw. If after you have been practicing equations for a significant amount of time and a teacher notices that a student is still on “level 1” it is not enough to document it and tell the student to practice more of the same. Instead you should focus on what skills or abilities the student need to gain in order to learn in the first place. This is where we need to start talking about the “matrix for student abilities”.

When students learn mathematics it is not enough to have a teacher telling them everything what to do. This is not how problem solving works. Students, no matter their level, need to develop different abilities in order to research mathematics and become independent problem solvers such as gathering information, critical thinking, applying their knowledge in new situations, presenting results, skills in technology and collaboration to name a few. In order for students to learn all of this they need to practise it. That is why practising these abilities should always be an integrated part of the lessons, and then of course be of major importance in their grade. Basically all modern national curriculums even states that the students need to develop these kind of abilities and the matrix for subject knowledge does not even take this into account!

Why I wanted to discuss this topic is because schools have been stagnating for decades and I find it good that it is now under development. However, it is easy being stuck in old ideas and just change details without making any major progress, which has happened in this case of matrices for subject knowledge. But since there is a will to reform schools to better suit modern students, maybe now is the time to take a few steps back and start thinking in a whole new direction.

Learn more about matrices for student abilities in “Are test necessery? Part 2 – Grading without tests”, but please read “Are tests necessery? Part 1 – Introduction” before reading part 2 if you have not done so already.