In his book, Developing Number Knowledge, Wright defines the term (p15) this way;
Mathematization means bringing a more mathematical approach to some activity.
For example, when a student pushes some counters aside and solves an addition
task without them, we say they are mathematizing, since it is mathematically
important to reason about relations independent of concrete materials.
Others define it as "reduction to mathematical form" (Merriam Webster), "to treat or regard mathematically" (The Free Dictionary) and "explaining mathematically" the Collins dictionary.
The really, really interesting thing about all these definitions is the idea of reduction or movement from real life, concrete situations such as that described by Wright, to the symbolic form of symbols and algorithms typically used in math. This is completely opposite to the way math has traditionally been taught and how it is sadly still taught in poorly taught math classes.
A classic example occurred in my math class yesterday when I asked student what 1/2 ÷ 1/4 meant. No one knew. My hunch is that if you randomly asked 100 people on the street only a handful would be able to tell you that this meant how many quarters are in a half. What really makes this intriguing is that most people would tell you to change the sign, flip the second fraction, multiply and get the answer 2.
In other words people have not gone through the process of mathematization when they have learned this procedure. A real world, concrete idea has not been "reduced to a mathematical form". They learned the mathematical procedure without any sense of what it meant or connection to any concept or relationship. There was no derivation, if you like from, of a mathematical relationship from an idea or concept. This happens all the time in math.
Students are taught a square number is the result of "a number times itself" instead of a number that makes a square.
They are taught a prime number is "a number divisible only by 1 and itself" instead of a number that can only make one rectangle (e.g 1 x 7 or 1 x 13).
Students are taught the symbolic mathematics first and not the idea so they cannot be mathematized. This is what the Common Core State Standards for Mathematics is trying to achieve.