When students are asked to mathematize situations (recognize and extract the mathematics embedded in the situation and use mathematics to solve the problem). They must analyze, interpret, develop their own models and strategies, and make mathematical arguments including proofs and generalizations. These competencies include a critical component and analysis of the model and reflection on the process. Students should not only be able to solve problems but also to pose problems.
These competencies function well only if the students are able to communicate properly in different ways (e.g., orally, in written form, using visualizations). Communication is meant to be a two-way process: students should also be able to understand communication with a mathematical component by others. Finally we would like to stress that students also need insight competencies—insight into the nature of mathematics as a science (including the cultural and historical aspect) and understanding of the use of mathematics in other subjects as brought about through mathematical modeling.
As is evident, the competencies quite often incorporate skills and competencies usually associated with the other two levels. We note that the whole exercise of defining the three levels is a somewhat arbitrary activity: There is no clear distinction between different levels, and both higher- and lower-level skills and competencies often play out at different levels. In the TIMSS framework, this level relates best to the mathematical reasoning performance expectation: developing notation and vocabulary, developing algorithms, generalizing, and conjecturing.
This level, which goes to the heart of mathematics and mathematical literacy, is difficult to test. Multiple-choice is definitely not the format of choice. Extended response questions with multiple answers (with [super-] items or without increasing level of complexity) are more likely to be promising formats. But both the design and the judgment of student answers are very, if not extremely, difficult. Because at the heart of our study, however, we should try, as much as practice permits, to operationalize these competencies in appropriate test items. The three levels can be visually represented in a pyramid (Figure 1; de Lange, 1995). This pyramid has three dimensions or aspects: (a) the content or domains of mathematics, (b) the three levels of mathematical thinking and understanding (along the lines just defined), and (c) the level of difficulty of the questions posed (ranging from simple to complex). The dimensions are not meant to be orthogonal, and the pyramid is meant to give a fair visual image of the relative number of items required to represent a student’s understanding of mathematics. Because we need only simple items for the lower levels, we can use more of them in a short amount of time. For the higher levels we need only a few items because it will take some time for the students to solve the problems at this level.
The easy to difficult dimension can be interchanged with a dimension that ranges from informal to formal. All assessment questions can be located in the pyramid according to (a) the level of thinking
called for, (b) mathematical content or big ideas domain, and (c) degree of difficulty. Because assessment needs to measure and describe a student’s growth in all domains of mathematics and at all three levels of thinking, questions in a complete assessment program should fill the pyramid. There should be questions at all levels of thinking, of varying degrees of difficulty, and in all content domains.
Essential to mathematical literacy is the ability to mathematize a problem. This process of mathematization will therefore be described in a little more detail:
Defining mathematization: Mathematization, as it is being dealt with here, is organizing reality using mathematical ideas and concepts. It is the organizing activity according to which students used acquired knowledge and skills to discover unknown regularities, relations and structures (Treffers & Goffree, 1985). This process is sometimes called horizontal mathematization (Treffers, 1987) and requires activities such as—
• Identifying the specific mathematics in a general context.
• Schematizing.
• Formulating and visualizing the problem.
• Discovering relations and regularities.
• Recognizing similarities in different problems (de Lange, 1987).
As soon as the problem has been transformed to a more-or-less mathematical problem, it can be attacked and treated with mathematical tools. That is, mathematical tools can be applied to manipulate and refine the mathematically modeled real-world problem. This is the process of vertical mathematization and can be recognised in the following activities:
• Representing a relation in a formula.
• Proving regularities.
• Refining and adjusting models.
• Combining and integrating models.
• Generalizing.
Thus the process of mathematization plays out in two different phases. The first is horizontal mathematization, the process of going from the real world to the mathematical world. The second, vertical mathematization is working on the problem within the mathematical world (developing mathematical tools in order to solve the problem). Reflecting on the solution with respect to the original problem is an essential step in the process of mathematization that quite often does not receive proper attention.
One can argue that mathematization plays out in all competency classes because in any contextualized problem one has to identify the relevant mathematics. The varying complexity of mathematization is reflected in the two examples below. Both are meant for students of 13–15 years of age and both draw upon similar mathematical concepts. The first requires simple mathematization whereas the second requires more complex mathematization.
Example 1. A class has 28 students. The ratio of girls to boys is 4:3. How many girls are in the class?
Source: TIMSS Mathematics Achievement in the Middle Years, p.98
Example 2. (Level 3) In a certain country, the national defence budget is $30 million for 1980. The total budget for that year is $500 million. The following year the defence budget is $35 million, while the total budget is $605 million. Inflation during the period covered by the two budgets amounted to 10 percent.
a. You are invited to give a lecture for a pacifist society. You intend to explain that
the defence budget decreased over this period. Explain how you could do this.
b. You are invited to lecture to a military academy. You intend to explain that the
defence budget increased over this period. Explain how you would do this.
Source: de Lange (1987)
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