Exploring the Power of Relational T

Exploring the Power of Relational Thinking: Students’ Emerging Algebraic
Thinking in the Elementary and Middle School
Max Stephens
The University of Melbourne
[e-mail]
Masami Isoda
Tsukuba University, Japan
[e-mail]
Maitree Inprasitha
Khon Kaen University, Thailand
[e-mail]
Abstract
There is a strong case for arguing that the application of relational thinking to solve number sentences embodies
features of mathematical thinking that are centrally important to algebra. This study investigates how well students
in Years 5, 6 and 7 in three countries were able to use relational thinking to solve different types of number
sentences. Were there other students who appeared to rely solely on computational method to solve the same
number sentences? The study then examined whether those who had shown clear evidence of relational strategies
to solve the number sentences were better placed to solve symbolic sentences.
Keywords: algebra, relational thinking, number and symbolic sentences.
Relational thinking
In their study, The algebraic nature of students’ numerical manipulation in the New Zealand Numeracy
Project, Irwin and Britt (2005) argue that the methods of compensating and equivalence that some
students use in solving number sentences may provide a foundation for algebraic thinking (p. 169).
These authors give as an example the number sentence 47 + 25 which can be transformed into 50 + 22
by ‘adding 3’ to 47 and subtracting 3’ from 25. They claim “that when students apply this strategy to
sensibly solve different numerical problems they disclose an … understanding of the relationships of the
numbers involved. They show, without recourse to literal symbols, that the strategy is generalisable” (p.
171). Several authors, including Stephens (2006) and Carpenter and Franke (2001), refer to the thinking
underpinning this kind of strategy as relational thinking.
Solving number sentences successfully using relational thinking certainly calls on a deep understanding
of equivalence. Students need to know the direction in which compensation has to be carried out in
order to maintain equivalence (Stephens, 2006). Some children who correctly transform number
sentences involving addition reason incorrectly that a number sentence such as 87 – 48 can be
transformed to be equivalent to 90 – 45. These children do not understand the direction in which
compensation must take place when using subtraction or difference. These students fail to recognise
that the relationship of difference is fundamentally different to addition. Other children, however,
recognise this feature explaining that “in order for the difference to remain the same, the same number
has to be added to each number in the expression. These children write correctly 87 – 48 = 89 – 50. The
first part of this study was designed to probe children’s thinking with number sentences
The Study
Design of tasks, scoring procedures and results of the questionnaire
Three groups of number tasks shown in Figure 1 were given to students in Years 5, 6, and 7 using a
pencil-and-paper questionnaire administered in regular class time. In introducing the questionnaire,
classroom teachers told students that:
“This is not a test. It is a questionnaire prepared by researchers … looking at how students
read interpret and understand number sentences. For most of the questions there is more than
one way of giving a correct answer. Please write your thinking as clearly as you can in the
space provided after each question and don’t feel that you have to write a lot.”
The questionnaire and the teacher’s introduction were translated into Japanese and Thai. Each group of
problems was introduced with the words: “Write a number in each of the boxes to make a true statement.
Explain your working

ผลลัพธ์ (ไทย) 1:

พลังของความคิดเชิงสำรวจ: นักเรียนใหม่พีชคณิตการคิดในโรงเรียนระดับประถมศึกษา และกลางสตีเฟ่นส์สูงสุดมหาวิทยาลัยเมลเบิร์น [e-mail] มาซามิมหาวิทยาลัยอวกาศสึกุบะ ญี่ปุ่น[e-mail] Inprasithaมหาวิทยาลัยขอนแก่น ไทย[e-mail]บทคัดย่อกรณีแรงการโต้เถียงที่ใช้ความคิดเชิงแก้ประโยคเลขก็ มีลักษณะของคณิตศาสตร์ความคิดที่มีความสำคัญกลางพีชคณิต การศึกษานี้ตรวจสอบนักเรียนอย่างไรดีในปีที่ 5, 6 และ 7 ใน 3 ประเทศสามารถใช้ความคิดเชิงแก้แตกต่างกันของจำนวนประโยค ได้มีนักเรียนคนอื่น ๆ ที่ปรากฏให้ใช้วิธีคำนวณแก้ไขเหมือนกันหมายเลขประโยค การศึกษาที่ตรวจสอบแล้ว ว่าผู้ที่มีแสดงยกเลิกหลักฐานเชิงกลยุทธ์แก้ประโยคเลขได้ดีกว่าวางแก้ประโยคสัญลักษณ์คำสำคัญ: พีชคณิต คิดเชิง จำนวน และประโยคสัญลักษณ์ความคิดเชิงในการศึกษา ลักษณะของการจัดการแสดงของนักเรียนในการนิวซีแลนด์พีชคณิตโครงการ เชอร์ และ Britt (2005) โต้เถียงที่วิธีการชดเชยและเทียบเท่านั้นบางนักเรียนใช้ในการแก้เลขประโยคอาจให้รากฐานสำหรับคิดพีชคณิต (p. 169)ผู้เขียนเหล่านี้ให้เป็นตัวอย่างประโยคหมายเลข 47 + 25 ซึ่งสามารถเปลี่ยนเป็น 50 + 22by ‘adding 3’ to 47 and subtracting 3’ from 25. They claim “that when students apply this strategy tosensibly solve different numerical problems they disclose an … understanding of the relationships of thenumbers involved. They show, without recourse to literal symbols, that the strategy is generalisable” (p.171). Several authors, including Stephens (2006) and Carpenter and Franke (2001), refer to the thinkingunderpinning this kind of strategy as relational thinking.Solving number sentences successfully using relational thinking certainly calls on a deep understandingof equivalence. Students need to know the direction in which compensation has to be carried out inorder to maintain equivalence (Stephens, 2006). Some children who correctly transform numbersentences involving addition reason incorrectly that a number sentence such as 87 – 48 can betransformed to be equivalent to 90 – 45. These children do not understand the direction in whichcompensation must take place when using subtraction or difference. These students fail to recognisethat the relationship of difference is fundamentally different to addition. Other children, however,recognise this feature explaining that “in order for the difference to remain the same, the same numberhas to be added to each number in the expression. These children write correctly 87 – 48 = 89 – 50. Thefirst part of this study was designed to probe children’s thinking with number sentencesThe StudyDesign of tasks, scoring procedures and results of the questionnaireThree groups of number tasks shown in Figure 1 were given to students in Years 5, 6, and 7 using apencil-and-paper questionnaire administered in regular class time. In introducing the questionnaire,classroom teachers told students that: “This is not a test. It is a questionnaire prepared by researchers … looking at how studentsread interpret and understand number sentences. For most of the questions there is more thanone way of giving a correct answer. Please write your thinking as clearly as you can in thespace provided after each question and don’t feel that you have to write a lot.”The questionnaire and the teacher’s introduction were translated into Japanese and Thai. Each group ofproblems was introduced with the words: “Write a number in each of the boxes to make a true statement.Explain your working