Elements of Difficulty Level in Mathematics* - Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing - page 82

overestimated. Current test has only resolution that divides about several stages.”
(Shojima 2009).
The way of giving questions is one of the methods of measuring how learners
understand. Leaders are making and choosing for measuring learners understand-
ing, but leaders make and choose questions by implicit knowledge because the
degree of difficulty for each question is not given by number.
Therefore, if it is possible to make the degree of difficulty of the problems in
mathematics, it will be used in a wide range of areas.
One of the ways of making a decision about the degree of difficulty for each
question is percentage of questions answered correctly (Maeda and Nishio 2000).
But, it has no mean to compare the degrees of difficulty between deferent catego-
ries of semantic structure (Riley and Greeno 1988). So, at least it has no mean to
decide the degrees of difficulty only percentage of questions answered correctly.
As another way of making a decision about the degree of difficulty for each
question, there is IRT (Item Response Theory) (Baker 1992). IRT is proposed as
the way to be able to account ability fairly and used TOEFL and TOEIC
(Tsukihara et al. 2008). But, in mathematics, there is less of a chance of used case
(Tsukihara et al. 2008).
Further, studies identified using the methods of the data mining difficulty is al-
so, it is still under development presently believed to be necessary and a study on
the learning history data (Baker 1992).
Therefore, in this research, we collected answers of each learner for knowing
element of difficulty. As well, “Difficulty” defines intractableness for solving each
question.

2 Methods of Making a Decision of Difficulty Level

For making a decision of difficulty level in mathematics, we were examined 3
types of method, “statistical approach”, “subjective method”, and “discovery of
the elements”.

2.1 Statistical Approach

Statistical approach can find out whether the number is correct or not because we
can do additional test.
For example, Hirashima defined difficulty by percentage of questions answered
correctly (Hirashima et al. 1955), IRC defined difficulty by ability of each learner
and percentage of questions answered correctly(Riley and Greeno 1988), and Neu-
ral Test Theory(NTT), the method for standardizing test, defined difficulty by
self-organizing map (Kohonen 2000)(Shojima 2009).
But, in statistical approach, we can know conclusive difficulty level after exam-
ination designers make math problems, as exemplified by IRC.