Development and application of a two-tier diagnostic instrument to assess middle-years students’ proportional reasoning

Abstract

Proportional reasoning involves the use of ratios in the comparison of quantities. While it is a key aspect of numeracy, particularly in the middle years of schooling, students do not always develop proportional reasoning naturally. Research suggests that many students do not apply proportional methods appropriately and that they often erroneously apply both multiplicative and additive thinking. Further, students cannot always distinguish non-proportional situations from those that are proportional. Understanding the situations in which students mistakenly use additive or multiplicative thinking and the nature of the proportional reasoning that students apply to different problem types is important for teachers seeking to support their students to develop proportional reasoning in the classroom. This paper describes the development and use of a two-tier diagnostic instrument to identify situations in which students could and could not apply proportional reasoning and the types of reasoning they used. It presents data from an Australian study involving over 2000 middle-years students (Years 5 to 9) as a means of illustrating the use of the instrument for diagnosing students' reasoning in different situations. The findings showed that the instrument was useful for identifying problem types in which students of different ages were able to apply correct reasoning. It also allowed identification of the types of incorrect reasoning used by students. The paper also describes useful applications of the instrument, including its use as a diagnostic instrument by classroom teachers and its use in the design of classroom activities included in teacher professional learning workshops.

Multilevel Rasch modeling of two-tier multiple choice test: A case study using Lawson’s classroom test of scientific reasoning

“…To improve the assessment features of MC items, researchers have explored different designs including two-tier multiple choice (TTMC) [14], ordered multiple choice (OMC) [15], multiple-tier multiple choice (3 or more tiers) [16][17][18][19][20], and test combination analysis [21]. This study will focus on the TTMC design, which was first proposed by Treagust [14] and has gained popularity in recent years [12,[22][23][24]. TTMC items are constructed to jointly measure students' ability to correctly answer tier 1 questions and their explanatory reasoning with tier 2 questions [14,25].…”

“…Probing reasoning and explanation in tier 2 provides a way to assess more complex understanding and can also serve to diagnose misconceptions [14]. In recent studies, researchers have made substantial progress in developing TTMC items as effective diagnostic tests [12,22,24,[26][27][28][29][30][31].…”

An Evaluation of Proportional Reasoning of Middle School 5th , 6th and 7th Grade Students according to a Two-Tier Diagnostic Test

“…Buradan hareketle, öğrencilerin uygun toplamsal ve çarpımsal akıl yürütme stratejileri geliştirmelerini desteklemek için öğrencilerin orantısal akıl yürütmede yaşadıkları zorlukların teşhisinin öğretmenler açısından yararlı bir ilk adım olduğu vurgulanmıştır (Bright, Joyner and Wallis, 2003;Misailidou and Williams 2003;Van Dooren, De Bock, Hessels, Janssens and Verschaffel, 2005). Öğrencilerin orantısal akıl yürütmede yaşadıkları zorlukları belirlemek için kullanılan orantısal akıl yürütme teşhis aracının, öğretmenlerin, öğrencilerin belirli orantısal akıl yürütme öğrenme ihtiyaçlarını belirlemek için hangi öğretim faaliyetlerini seçmeleri gerektiği konusunda yararlı olacağı düşünülmektedir (Hilton, Hilton, Dole and Goos, 2013).…”

Full issue

“…These are of particular relevance to us because such designs sometimes indicate fairly detailed models of the mathematical components of the topic. For instance, to measure student difficulties with the different proportion type of tasks, (Hilton, Hilton, Dole, & Goos, 2013) designed a two-tier diagnostic instrument to measure the degree to which students' master "proportional reasoning". However, the underlying reference model remains implicit in this and many similar studies: it seems that the authors take for granted that readers share the same idea about proportion or proportional reasoning; instead of definitions, the reader is left with the test instrument which, evidently, consists of examples of tasks, rather than explicit types of tasks described theoretically in terms of techniques.…”