Assessing fraction knowledge: Does the format of the assessment task make any difference?

An Australian research project (Pearn, 2019) investigated the links between fractional competence and algebraic thinking as middle-years students solved mathematical tasks using whole numbers, fractions, decimals and pronumerals. Fractional competence included understanding fraction size and relationships, demonstrating the understanding of fraction concepts and basic arithmetic competence with simple fractions. Algebraic thinking included students’ understanding of equivalence, transformation using equivalence, and the use of generalizable methods (Stephens & Ribeiro, 2012). Siegler et al.’s analysis of longitudinal data from both the United States and United Kingdom (2012) showed that competence with fractions and division in fifth or sixth grade is a uniquely accurate predictor of their attainment in algebra and overall mathematics performance five or six years later. Empson, Levi and Carpenter (2010) also argued that the key to learning algebra meaningfully is to help students: "to see the continuities among whole numbers, fractions and algebra" (p. 411). Demosthenous, Christou, and Pitta-Pantazi (2021) stated that students tended to solve tasks that share critical properties with textbooks’ tasks by recalling facts and procedures, while they use creative reasoning for those tasks that do not share those critical properties. The tasks described in this paper were constructed to enable students to demonstrate their understanding and reasoning rather than the instant recall of facts and procedures. More than 600 Australian students from Years 5 – 9 (10 – 16 years) completed two paper and pencil assessments: The Fraction Screening Test and the Algebraic Thinking Questionnaire (Pearn, 2019). This paper discusses the results from two tasks using the fraction two-thirds. Students needed to determine the number of objects representing the whole group if given a quantity of objects representing two-thirds of that group. Fraction Task A (Fraction Screening Test) was a worded task with a diagram. For example, students were shown a diagram of a collection of ten counters and asked: This collection of 10 counters is 2/3 of the number of counters I started with. How many counters did I start with? The second task, Fraction Task B (Algebraic Thinking Questionnaire) included symbols only (no diagram) similar to those given in school texts. For example, students were given the task: Write a number in the box to make a true statement. 2/3 × _ = 18 Explain your working briefly. After a detailed analysis of the solutions used to solve the three reverse fraction tasks from the Fraction Screening Test a Classification Framework for Reverse Fraction Tasks was created to classify the strategies (Pearn, 2019). Five strategies were identified: Advanced Multiplicative, Fully Multiplicative, Partially Multiplicative, Additive/Subtractive and Diagram Dependent. For example, students using additive strategies after the explicit partitioning of diagrams were deemed to be Diagram Dependent. Students who used both multiplicative and additive methods were deemed to be using Partially Multiplicative methods. The responses to Fraction Task A were analysed according to the Classification Framework for Reverse Fraction Tasks. More than 80% of all students solved Fraction Task A using one of the five solution strategies. Fraction Task B is in the form that Demosthenous, Christou, and Pitta-Pantazi (2021) would refer to as a textbook task. The majority of students did not attempt to answer this task, or gave an incorrect answer, even though they had successfully solved Fraction Task A. Many students giving an incorrect answer found two-thirds of 18 (12) rather than finding the whole given that two-thirds was 18. The correct response for the task is 27. Researchers such as Knuth et al. (2006) believe that students’ dependence on the operational conception of the equals sign hinders both arithmetic and algebraic calculations so the missing response for Fraction B was deliberately placed before and not after the equal sign in the Algebraic Thinking Questionnaire. Expecting students to show all their working for the written tasks and elaborate their responses during interviews meant that their strategies were able to be analysed using the Classification Framework for Reverse Fraction Tasks (Pearn, 2019). The common misconception that the equal sign indicates that an answer will follow was demonstrated by the large number of students who gave an incorrect response for Fraction Task B. This analysis confirmed the work of Demosthenous, Christou, and Pitta-Pantazi (2021) who stated that students tended to solve tasks that share critical properties with textbooks’ tasks by recalling facts and procedures. When attempting to solve Fraction Task B students attempted to perform the standard procedure for finding two-thirds of a number. However, when responding to Fraction A they used reasoning as this task was not presented in textbook task form. The findings from this Australian research project have implications world-wide particularly for countries undertaking international testing. References Demosthenous E, Christou C, Pitta-Pantazi D. (2021) Mathematics classroom assessment: A framework for designing assessment tasks and interpreting students’ responses. European Journal of Investigation in Health, Psychology and Education. 11(3):1088-1106. Empson, S. B., Levi, L., & Carpenter, T. P. (2010). 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