The What-If-Not strategy in action

This chapter uses several classroom examples to demonstrate the 'What-If-Not?' strategy with respect to problem posing and problem solving in mathematics teaching. The strategy is developed in 4 stages: 1. Attributes Listing, which brainstorms mathematical facts and previous knowledge to re-examine the given question and list the attributes of the question; 2. What-If-Noting, which encourages modification and adjustment of certain attributes to yield closely related questions; 3.Question Asking, generating new questions by modifying attributes of the original question; and then 4. Problem Analysing where students focus on interesting investigations. The strategy enables reducing rigid teaching while remaining focused on the topic.

Stop:

One of the obvious merits of the What-If-Not approach is that it makes students believe that there is more than one way to approach a question, rather than simply focusing on finding the “right answer”. We can expect that when the approach is actively implemented, students will have more opportunities to discuss their ideas and give more consideration to the meaning of the problem. However, some factors might affect the learning outcomes. Classroom culture and motivation are two of those factors. For example, how do teachers encourage less capable students to pose questions and maintain their focus on the activities? I have observed that some teachers do not prefer such students posing problems because they believe the approach is less helpful for these students than traditional learning. But if we want to our students to solve their problems, problem posing is a very important activity that encourages students to think creatively and critically. I will look into more research of the matter.

The author suggested that the value of the problem posing approach is not merely having the potential to engage students in creative activity, but for teachers to gain valuable knowledge during each stage of What-If-Not practice. Since transferring mathematical facts and computation is less stressed in the approach, teachers are able to pay attention to discoveries or errors made by the students, instead of giving the “right” answer. Students will make teachers aware of the significance of mathematical theorems and algorithms by witnessing the consequences of violating essential conditions of theorems and algorithms.

In the author's example below, we can see an instance of a student making a modification to the standard long division algorithm, yielding a negative number as a remainder. As teachers, we must understand the significance of having a negative remainder - what does it mean, and what does it teach us about the algorithm? In this case, some might argue that the remainder is by definition positive, but it ultimately depends on the lesson that the teacher is trying to convey.