Obstacles and Affordances for Integer Reasoning: An Analysis of Children's Thinking and the History of Mathematics

Abstract

We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understanding or knowledge that may, initially, interfere with one's learning integers (which we call cognitive obstacles) and understanding or knowledge that may afford progress in understanding and operating with integers (which we call cognitive affordances). We analyzed historical mathematical writings related to integers as well as clinical interviews with children ages 6-10 to identify critical, persistent cognitive obstacles and powerful ways of thinking that may help learners to overcome obstacles.

Conceptual models for integer addition and subtraction

“…Noticing the complicated nature of integers and contexts, we turned our attention to the literature on student thinking about integers. This literature (e.g., Bishop, Lamb, Philipp, Whitacre, Schappelle, & Lewis, 2014;Bishop, Lamb, Philipp, Whitacre, & Schappelle, 2016;Bofferding, 2014) has demonstrated that students are capable of robust reasoning about integers, but is mostly situated in how students make sense of integers in symbolic settings (e.g., how they solve problems like -2 --5 = ☐) or pre-determined contexts.…”

“…They may pose unrealistic stories (Mukhopadhyay, 1997) and change the structure of the number sentences (Rowell & Norwood, 1999). When working with integer operations in contexts, students may not think about the integers as adults do (e.g., Bishop, Lamb, Philipp, Whitacre, Schappelle, & Lewis, 2014;Bishop et al, 2016;Bofferding, 2014). Students may be unable to transfer integer understanding in one context to another context (Mukhopadyhyay et al, 1990).…”

Kemampuan Bernalar Siswa Kelas V Sekolah Dasar Pada Soal Cerita Perbandingan Bilangan Bulat

“…Tidak hanya bilangan bulat positif, siswa juga mengalami kesulitan dalam memahami bilangan bulat negatif. Ketika siswa diminta untuk membandingkan bilangan tersebut, banyak diantara siswa yang menjawab salah (Bishop, Lamb, Philipp, Whitacre, Schappelle, et al, 2014;. Kemampuan bernalar siswa dapat diketahi melalui pengerjaan soal-soal, terutama soal yang membutuhkan alasan logis.…”

Investigating Patterns of Pre-service Teachers’ Written Feedback on Procedure-based Mathematics Assessment Items

“…In this vein, researchers also recognize the importance of students' perceptions of the feedback they receive as a way to glimpse the teacher's point of view (Bansilal, James, & Naidoo, 2009). Because how students view feedback is critical to their learning, much emphasis has been placed on the quality of teachers' feedback (Sadler, 1998) as well as a way for them to understand the state of each student's knowledge by employing strategies to elicit student understanding and methods to interpret students' products (Bishop et al, 2014;Leatham et al, 2015). Thus, formative assessment feedback is related to the effectiveness of teachers' questioning and more generally to the teacher's role as a facilitator of learning, which depends on his/her ability to observe students' learning and attend to their thinking (Lee, 2018;Lee & Cross Francis, 2018;Peterson & Leatham, 2009;Stockero, Rupnow, & Pascoe, 2015).…”