Mathematical problem solving in primary school

Abstract

A student is engaged in (non-routine) problem solving when there is no clear pathway to the solution. In contrast to routine problems, non-routine ones cannot be solved through the direct application of a standard procedure. Consider the following problem: In a quiz you get two points for each correct answer. If a question is not answered or the answer is wrong, one point is subtracted from your score. The quiz contains 10 questions. Tina received 8 points in total. How many questions did Tina answer correctly? The complexity of this problem lies neither in the size of the numbers nor in the execution of the appropriate calculations. What makes this problem perplexing for primary school students is that it requires keeping track of several interrelated values. Introducing such problems in primary school serves a two-fold role: firstly, to offer students opportunities to develop problem solving skills and reasoning, and secondly, to provide entry points to the development of algebraic thinking; therefore, we call these problems early algebra problems. In this thesis we investigated the ability of Dutch upper primary school students to solve early algebra problems and ways to support this ability. The results of the first study with 152 high achievers from grade 4 revealed that their performance in problem solving lagged behind their general mathematical performance and that they often lacked essential problem solving skills. To shed light on the students’ low performance we investigated the cognitive demand of the tasks in the mathematics textbook series for grade 4. This analysis showed that challenging non-routine tasks are rare and are mostly found outside the main book of the textbook series. Subsequently, we aimed to address the difficulties students encounter when solving non-routine problems by providing an environment for experiencing the interdependency of values. A dynamic interactive computer game called Hit the target was developed for this purpose. In this environment the students can observe how the score varies as they manipulate the values of the hits, the misses, or the game rule. After piloting the game with 24 high achievers from grade 4, we conducted a large-scale study with 785 students from grades 4, 5, and 6. The students of the experimental group were asked to solve a series of problems at home using an online version of the game. The analysis of the data for grade 6 showed that the intervention had a significant effect on students’ problem solving performance. Furthermore, although girls put more effort in the online activity than boys, their achievement gains were the same as for boys. A closer analysis of student activity in the online environment showed that they applied various strategies, ranging from trial-and-error to strategies implying exploring relations between variables. In conclusion, the interactive dynamic character of the computer environment and the variation in the series of problems had a significant role in supporting students’ reasoning when dealing with early algebra problems. The above-mentioned findings lead to suggestions for educational practice.