ACADEMIC PERFORMANCE IN MATHEMATICS
The Mathematics tests were designed to take five subareas into account:
- numbers and operations;
- measurement;
- geometry;
- data analysis, statistics and probability;
- algebra and functions;
Within the scope of "numbers and operations", an attempt was made at evaluating the competence of students in use of different types of numbers and operations and in handling proportionality along with estimation and rounding up. The block "measurement" considered the skills in use of different conventional measurement procedures and systems and in estimation of lengths, surfaces and volumes. "Geometry" included the spatial skills of the students and their application to solving every day problems. The block "data analysis, statistics and probability" estimated the students’ skills to interpret and represent sets of statistical data and information in a simple way, and to predict probability results. Lastly, the subarea "algebra and s" aimed to evaluate student skills in understanding and use of algebraic language, as well as interpretation and construction of functions.
The questions were designed within these blocks to allow estimation of the knowledge and skills of the students and their ability to apply procedures and solve problems.
Table 8 shows the average percentage of correct answers in the area of Mathematics and relevant subareas, for students aged 14 and 16.
The subareas in which there seems to have been better mastery by the 14 year old students are numbers and operations, geometry and data analysis, while those of measurement and algebra provide worse results. At 16, the questions most easily solved are those in numbers in operations, and in algebra and functions, while in average the have a fairly low result. It is thus clear that most students have greater difficulties in dealing with measurement related questions. It would thus be convenient to reflect on what causes there may be for such low performance and carry out detailed analysis of the time spend on real practice in class on this area of the subject, as well as the space assigned to it in the text books.
The national average obtained after tests administered to students aged 14 was 226.1 points, this lying between the anchor points of 200 and 250. This means that the average 14 year old student is able to perform the tasks linked to level 200 and still has not reached the skills of point 250. As to the students aged 16, the national average was 263.1, that is to say, somewhat less than 37 points above that for 14 years, thus lying between 250 and 300.
The definitions of the performance levels in Mathematics are shown on Table 9, specifying, for each one of the performance levels, the skills and knowledge the students are able to master. It also shows the percentages of students in each one of the levels and those who pass their respective level and, thus, the previous ones.
The students aged 14 had difficulties in answering questions involving performance of operations with fractions, calculation of areas, memorisation of certain basic formulas, handling units of volume and representing functions. They also had difficulties in solving problems involving relations of proportionality or percentages, application of knowledge to triangle geometry and in solving simple linear equations. As to students aged 16,the most difficult tasks seemed to be performance of operations with powers and those requiring a spatial view of figures, such as representing functions, and handling conventional measurement units to calculate areas and volumes.
At the age of 14, the global result showed that 72% of the students passed level 200, which means they are able to solve simple problems in daily life through simple algebraic operations, estimations and rounding up, and intuitive concepts of statistics, interpret simple graphs, express and recognise simple problems in algebraic language. However, 27% only knows how to handle basic algebraic operations with simple fractionary numbers and 1% is not able to achieve the most basic skills specified in the performance levels.
Likewise, it was also noted that 28% of the students achieve the relevant skills for level 250, that is to say, are able to solve simple problems in daily life that include relations of numerical proportionality and percentages, to know flat bodies, have notions of probability and geometry of the triangle and similarity of figures, to solve simple linear equations,estimate the probability of simple situations (the Laplace Law), construct simple graphs and interpret tables of frequencies. However, when the contrary is considered, this also means that 72% of the students aged 14 do not reach that level.
At the highest end of the performance scale, 4% of the students exceed the 300 cut-off point, so in addition to the aforementioned, they are able to used the algebraic language to solve practical problems, perform operations with fractionary numbers in problems in daily life, fluently handle the concept of numerical proportionality and apply it to practical situations, to knowledge the lengths and surfaces of spaces and objects, to handle the measurement systems for lengths, space and object surfaces, over or underestimation and have knowledge of rounding up.
At the age of 16, 10% of the students is only able to perform basic algebraic operations with simple numbers. Moreover, 28% is only able, in addition to the above, to correctly solve the matters described above at level 200. This result is obviously unsatisfactory, as more than a third of the sample of students aged 16 only masters the four most basic skills defined in the performance levels.
If the 300 level on the scale is taken, scarcely 23% of the students aged 16 are skilled enough to solve the questions included. There are few students (3%) who reach level 350 and, thus, the aspects defined.
Overall, the results in Mathematics may not be considered satisfactory. The performance levels achieved at 14 and 16 years old are under no circumstances near to the desirable ones. On the other hand, although there is progress of 37 points between the two ages, it seems to be insufficient progress.