What distinguishes an expert problem solver from a novice? By Au Kelly (Based on the research lesson by Prof A Takashashi at Monfort Secondary School

When I viewed the video vignettes, my first thought was, “Hey! I’ve used this in my class before!” A colleague had sent me the link to a website and having figured out how it works, I used it as a class filler after a lesson on Algebra with my secondary 2 students back in 2006. Being the naïve teacher that I was, I was quick to show how Algebra could be used in such a puzzle and did not exploit the potential of the puzzle the way Prof A Takashashi did.

It was interesting to see the way he questioned the students and elicited responses from them to see how they would go about figuring out the puzzle. As I watched the various responses from the students, it occurred to me that there were distinct differences in the way the students tackled the problem, which led me to the research problem: What distinguishes an expert problem solver from a novice?

From the video vignettes, I observed the following:

• One student only used random numbers and stopped after 3 tries without much attempt at drawing any form of conclusion although he knew the numbers ended up with the same symbol. He also found it difficult to articulate his thoughts and the professor presented the student’s answer on the board and another classmate attempted (to some level of success) to explain it on his behalf.
  
• Some tried more numbers and found that they all ended up with the same symbols when they compared it to the list on the screen but did not progress into much detail.
  
• Some were able to see the link in that the numbers obtained were multiples of 9. One boy went into more details, such as observing that numbers in the 90s would yield 81, numbers in the 80s would yield 72, numbers in the 70s would yield 63, right down to numbers in the 20s which would yield 18 and that all these were multiples of 9.

The students seem to be from the same class, which means that the instruction received during classroom lessons should be the same, and yet from the video vignettes, it can be seen that some were more able to see patterns emerging and were able to make more detailed observations and make conclusions as compared to the others.

As it is impossible to open up the heads of expert problem solvers to see what lies inside, the next best thing to do would be to adopt a qualitative approach to the research, as this method will yield richer data. Based on the video vignettes, I would categorise students into three main categories: Novice, Average and Expert problem solvers. I would conduct the research with one student from each category or if time and resources permit, two from each, for better corroboration. Students would be asked to think aloud while attempting to problem solve a specified number of questions and a video recording would be made to capture what the student does and says during the problem solving process.

This method of research would follow quite closely to research that was previously conducted by Muir, Beswick & Williamson (2008) who studied the strategies used by a selection of grade 6 students and detailed three individual cases involving naïve, routine and sophisticated problem solvers, and that of Montague & Applegate (1993) who examined the verbalizations of learning disabled, average achieving and gifted middle school students as they thought aloud while solving three mathematical word problems.

From observations made of these students, results would be able to show some differences between each category of students. For mathematics educators, we can help to leverage the learning of novice problem solvers by teaching them some strategies to employ that were used successfully by expert problem solvers to help them in problem solving.

I think rather than take the stance that some students just have it (problem solving abilities) and some don’t, educators should always try to bridge the gap between novice and expert problem solvers by providing the necessary facilitation and guidance. I believe this research can shed some light on how expert problem solvers think and act and how we educators can help less able problem solvers emulate this.

If I were to conduct such a research, it would be based on convenience sampling, as I would probably use the students I am teaching as research participants. As students in different schools may differ in terms of abilities, instruction received during lessons and strategies exposed to, the use of blogs in researching this problem can extend to a wider reach as other teachers may wish to replicate the study on their students and discussions on the result findings can be made online. The use of blogs as a mode of communication can most certainly be tapped into as it is available 24/7 and is cheap with a wide reach.

References
Montague, M., & Applegate, B. (1993). Middle school students' mathematical problem solving: An analysis of think-aloud protocols. Learning Disability Quarterly, 16, 19-32.
Muir, T., Beswick, K., & Williamson, J. (2008). “I’m not very good at solving problems”: An exploration of students’ problem solving behaviours. The Journal of Mathematical Behavior. 27, 228-241.