Tool 4: Assessing Mathematical Understanding Through Formative Means
Using good Socratic questioning techniques, teachers can determine if the students actually understand the concepts in class. However, if time is a constraint, it is then important to have some form of formative assessment where students are able to check for their understanding in their own time and pace.
Existing methods of testing of mathematical knowledge have been limited to mainly procedural skills. This is evident in the ‘O’ and ‘A’ levels where questions expect students to apply formulae to obtain solutions as the final outcome. While it may be difficult to overhaul the summative assessment such as the national examinations, formative assessment modes can be used as a tool to promote deeper conceptual understanding in learners. This can be done by phrasing questions differently in homework assignments where more open-ended questions can be posed. Below shows two different types of questioning:
- (a) Write 3.20449 to 3 significant figures.
- (b) What is the difference between writing 3 and 3.0 (2 s.f.)?
It can be seen that (a) only requires the learner to use a mathematical fact (or rather, a skill) to obtain the answer, while (b) requires the learner to fully understand the concept of numbers and estimation before being able to answer the question fully.
Enrichment exercises could be provided for the high ability learners to deepen their conceptual understanding and explore a wider range of applications. These exercises, in the form of challenging problem-solving questions, keep them excited about their learning and prevent boredom in the subject. Teachers can also encourage these students to take part in a range of mathematics contests where they can pit their knowledge against the most challenging questions in, for example, the Mathematical Olympiad.
The exposure to non-routine questions often found in these contests will widen their comprehension of the mathematical concepts, and the students will also get to see how concepts can be applied in complicated and complex situations. Some of these non-routine questions can also be incorporated in the classroom when time permits, where students can be split into groups, and the teacher can use problem- based learning strategies to get the students to work together towards solving real- world problems.