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     “If you can’t solve a problem, then there is an easier problem you can solve: find it.”

~ G. Polya

Why a “different” problem-solving model?

This desire came from my many attempts to make students adopt a model of problem solving that they can use independently and which can realistically describe the steps and the skills of a problem-solver. I have used the standard one introduced by Polya in 1945 in his book “How to Solve It”: understand the problem, make a plan, carry out the plan, judge the solution.

There is nothing wrong with this only that I felt that certain steps needed more detail and a lot of direct modelling. For instance: “Understand the problem.” Easily said but how hard is that? It is particularly because this is a reading comprehension skill and just like we use a variety of reading strategies: summarizing, making inferences, finding the main idea, visualizing, to name a few. Students need a lot of practice with this first step.

I don’t see the problem solving like a linear process but mostly like a circular model, almost like a spiral, if I have to use a metaphor to describe it. The more you understand it, the greater your ability to solve the problem. One has to revisit the context of the problem several times during the process, will re-read the problem even while carrying a strategy. We return to check the solution within the initial context. We adopt a plan and we test our understanding of the problem. More and more in mathematics I reach the same conclusion in my class: the students are solving a different problem. This is a very nice way of saying that they misunderstood the problem.

So, I started my quest. I dropped all the formulas I knew from before. All the KWC ( Know-Want to find-Conditions) and KWCD ( Know-Want to find-Conditions-Do) seem completely useless. Why? There is so much work that needs to be done by a teacher to show the students that KNOW means more than what is “given” or stated. “Know” includes one’s prior knowledge and the ability to infer . Ultimately, KNOW can be the knowledge about the context, a similar context or real life context that the problem reminds us of. I hope I am in agreement with whoever reads this blog entry that simply telling students to write what they know and expecting them to do a good job just because of a straightforward visual organizer won’t help. If you are like me, then surely you encountered the same problem. The students need to practice how to “KNOW”.

HENCE, THE METACOGNITION CONNECTION

Luckily I discovered, like many others before me, Singaporean mathematics. Metacognition is part of their curriculum. Visualization is part of their problems-solving model. I initially used Dr. Lee Ngan Hoe’s problem solving model.

First, I felt  so vindicated because of the “circular” model he proposed (which he had borrowed and simplified from Richard Paul’s Reasoning Wheel) and secondly, I found out that he adopted this model as he felt dissatisfied with the results of usin Polya’s model as well.

The Problem-Solving Wheel includes the following steps:

1. The GIVEN ( main idea, stated and inferred information)

2. FIND ( what do I have to find out?)

3. PICTURE ( this helps students visualize the problem and understand it better as well as making the first connections betewen its components)

4. TOPIC ( identify the topics and strands that this problem might fit in and thus it provides students with opportunities to make connections/apply knowledge)

5. FORMULAE ( formulae that might have been used to solve problems that fit the topics)
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Now, this is a great model but as I teach in an Ontario school it needs more work on components number 3 and 5.

“PICTURES” because in Singapore, whose mathematics education is stellar and so well recognized internationally ( see PISA) as one of the strongest in the world, the “Model Drawing” is taught since kindergarten. Students are used to using rectangular diagrams to represent the contexts of problems and thus, their ability to generalize and use algebraic thinking is stronger from a younger age.

“FORMULAE” because of what I mentioned above. The connections between topics are so easy to make. Think: representing numbers as rectangles helps with understanding rational numbers, measurement, number lines and leads to algebraic generalizations.

These two are my current goals: teaching the model drawing and encouraging algebraic thinking ( which is based on making connections and generalizations, observing relationships between concepts, patterns) in order to make the above model of problem solving applicable in my Ontario mathematics classroom. I know from what I read and from my elementary schooling experience that this will lead to success.