The idea of problem posing in Mathematics is not new.
Past studies on Mathematics problem posing has shown that problem posing strategy could be perceived in various ways. It could be (1) a means to improve students’ problem solving; (2) a feature of creative activity; (3) as a window into students’ Mathematical understanding; (4) a means of improving students’ disposition towards Mathematics; (5) a means to increase students’ confidence in raising questions (Silver, 1994).
If teachers could vary and pose problems more creatively, by modelling these thinking processes to students and introducing the process of problem posing and problem variation to the students, we made our questioning and thinking explicit. We also model how we reformulate complex problems explicitly by variation to simplify the problem or break it down into smaller parts and vice versa, such that we are able to develop the novice or developing problem poser into an expert problem poser.
In a recent lesson, I introduced the idea of problem variation to Year 3 students in the lesson on solving exponential equations. The students had already covered the laws of logarithms, solving the logarithmic equations and are beginning to solve exponential equations.
We covered three types of exponential equations (focusing on the second and third type):
(a) ax = b (where a and b share same base)
(b) ax = b (where a and b do not share a common base)
(c) pa2x +qax + r = 0 (to be solved by substitution)
For type (b), the examples given in the worksheet are
Solve the following equations:
Solve the following equations:
As these are rather direct questions, they are a good place to introduce problem variation using “what if we change something in the question?” idea to see if the same approach still work and how much more complex can the question become.
With some prompting, students were able to propose variations of the above as follows:
where the index can be linear, quadratic or cubic which will involve solving quadratic equations and cubic equations.
For example 3(b), they were able to propose variations like
and so on by changing the base or the index
These provided them with more questions to work on. A more complex extension was proposed where the bases on both sides were unknown constants and they need to solve for x in terms of the bases.
This little exercise also allows them to contrast the question type with a previous type in the form pa2x + qax + r = 0.
Although these problem variations appear simple, it may boost the confidence of the less confident students. It also helps the majority to identify the basic characteristics and traits of the different exponential equations. It prompts the students to ask what types of questions can be solved with this particular approach which indirectly improves their problem solving. As I told them in class, this helps them to think like a teacher when she is setting a test paper.
These examples were simple enough to introduce the idea of problem variation, which can be further extended in more complex tasks in the future.
Education Development Centre, Inc. (2002). Problem Posing. Making Mathematics. Retrieved from http://www2.edc.org/makingmath/handbook/teacher/problemposing/problemposing.asp on 30 May 2017.
Lavy, I & Shriki, A. (2007). Problem Posing as a means of developing Mathematical knowledge of prospective teachers. In Woo, J.H., Lew, H.C., Park, K.S. & Seo, D. Y. (Eds.). Proceedings of the 31st conference of the International Group for the Psychology of Mathematical Education, (3), 129 – 136. Seoul: PME.
National Council of Teachers of Mathematics (NCTM) (2000). Principles and Standards for School Mathematics
Silver, E. A. (1994). On Mathematical problem posing. For the learning of Mathematics, 14, 19 – 28.
Posted by: Ms Goh Li Meng
Raffles Girls’ School