Abstract
Our research is based on work with word problems. We define a word problem in accordance to the following definition: "verbal description of problematic situations that give rise to one or more questions whose answers can be obtained by applying mathematical operations to the numerical data present in the problem." (Verschaffel, Depaepe & Van Dooren, 2014). However, our type of word problem is atypical (Slezáková, Málková & Mottlová, 2021) as it does not meet the expectations solvers and teachers usually have.
In our research we study the phenomena that initiate the use of heuristic strategies (Polya, 2004). Such a phenomenon is a specific type of an intuitive argument with which a pupil confirms that their partial solution is correct. We have defined the following research question: Which phenomena triggered the need of heuristic strategies that pupils used in the solution of the specific word problem and become the basis of their justification? The additional question is: How will the approach to this type of word problem change in our sample of older pupils compared to our previous experience with younger pupils?
The tool in our experiment were three atypical word problems (of the type used in Singapore textbooks) consisting of two parts.
The first part is instructions: "Put the numbers back and check they make sense in the text." and the numbers that need to be filled in. This part replaces the question that is to be answered in usual word problems.
The second part of the assignment is a problem situation with missing numbers. Instead of numbers, there are gaps.
The order of the numbers in the first part does not have to correspond to the order of the gaps in the problem but it must make sense: a) mathematically (the numbers are in an additive or multiplicative relation), b) linguistically (the filled in numbers are in accordance with the rules of the language in which the problem is given), we reflect on its specificity and are aware of the limitations and advantages that this language system brings to both native and non-native speakers, c) with respect to real-life context (the problem situation makes sense in real life).
The experiment was conducted as a semi-structured interview with 5 pairs of pupils (11-13 years old). The pupils solved a series of three problems in the limit of 15 minutes. Pupils were continuously encouraged to think-aloud. Their comments on the solution were audio and video recorded. The recordings were transcribed and the transcript was subject to qualitative analysis.
The main results of the experiments are that pupils approach the word problem from three perspectives (mathematical, linguistic and real-life context). We focused on the phenomena that initiate the use of heuristic strategies in atypical word problems. We recorded five phenomena that are the basis of pupils' justification of a solution (The greatest or the smallest numbers, Declension, Signal words, Real-life experience, Benefit of calculation). These atypical word problems could provide potential opportunities for, among other, two key processual skills, mathematical reasoning and communication.
In our future research we would like to focus on an analysis of heuristic strategies used by pupils from different age groups. In a longer-term perspective, we want to monitor whether pupils who solve this type of word problems with heuristic strategies use them also in solving usual word problems.