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Using a Modified Form of Lesson Study to Develop Students’
Relational Thinking in Years 4, 5 & 6
Lei Bao
Alvie Consolidated school Victoria
<bao.lei.l@edumail.vic.gov.au>
Max Stephens
The University of Melbourne
<m.stephens@unimelb.edu.au>
This action research project aimed to develop students’ relational thinking and to
improve teachers’ knowledge of students’ thinking. The study used sentences involving
one unknown number with fifteen students in a multi-grade 4, 5 & 6 over a period of
two weeks. Students in all grades increased their understanding of equivalence and their
capacity to use relational thinking to solve number sentences. The three participating
teachers improved their knowledge about students’ relational thinking, and its relevance
to their future teaching.
Introduction
Research has shown that the transition from arithmetic to algebra is difficult for
many students. Carpenter, Franke and Levi (2003) found that many students perceive
arithmetic only as a series of calculations. They found that students often do not see the
relationships between numbers and operations when they carry out calculations, and
advocate a closer integration of number and algebra in the primary school curriculum.
The Australian Curriculum: Mathematics (ACARA, 2010) also aims to strengthen the
links between the teaching of Number and Algebra, especially to the middle and later
primary years.
This study used a modified form of Lesson Study as a research strategy to deliver
lessons to build students’ relational thinking and as a professional development platform
for developing teachers’ content knowledge by identifying the links between relational
thinking and teaching arithmetic. This paper investigates the performance of both
students and teachers on questionnaires before and after the Lesson Study and uses
student interview data. It aims to shed light on the following research questions:
What kind of thinking did students use to solve open number sentences before
the Lesson Study?
How can a modified form of Lesson Study to be used to move forward the
development of students’ relational thinking?
What advantages did the students see in using relational thinking? How
confident were they in thinking relationally across the four operations?
What did the teachers participating in the Lesson Study learn about their
students’ relational thinking and how will this inform their own teaching and
capacity to introduce relational thinking in the future?
Current Research on Relational Thinking
Stephens (2006) and Hunter (2007) explain that relational thinking depends on
treating the equal sign as an indicator of equivalence, whereas many primary school
students treat the equal sign simply as a command or direction to find an answer. This
was exemplified by Carpenter, Levi, Franke, and Zeringue (2005) in their study of
Grade 2 and 3 students, where they reported that some students, in response to the
missing number sentence 8 + 4 = __ + 5, gave 12 as answer by adding 8 + 4; and some
other students gave 17 as answer, by adding all the numbers 8 + 4 + 5. Hunter (2007)
along with Stephens (2006a), Molina and Ambrose (2008) also explain that students’
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inadequate understanding of the equality symbol leads to difficulties in solving
symbolic expressions and equations.
Stephens and Ribeiro (2012) and Irwin and Britt (2005) identify relational thinking
in the methods of equivalence and compensation that some students use in solving
number sentences; for example, by transforming 17 + 68 into 15 + 70 by adding 2 to 68
and subtracting 2 from 17. Researchers, such as Carpenter, Franke, and Levi (2003),
Hunter (2007), Molina and Ambrose (2008), and Stephens (2006a) suggest that
relational thinking can be fostered by posing true/false and open number sentences, and
assisting students to focus on the sentence as a whole rather than as a computation. For
example, Carpenter and Franke (2001) asked students if 78 – 49 + 49 = 78 was true or
false, and how did they decide; looking if students could refrain from computation and
affirm that it was true by attending to the mathematical structure of the sentence.
Investigating true/false number sentences played a key role in this action research study.
Jacobs, Franke, Carpenter, Levi, and Battey (2007) also used true/false and open
number sentences as contexts in which teachers could orchestrate conversations with
students in order to identify the kind of thinking embedded in students’ strategies in
deciding whether particular number sentences were true or false, and the methods they
used to solve missing number sentences involving the four operations.
Like the study of Jacobs et al. (2007), this study also aimed to enhance teachers’
content knowledge of students’ relational thinking; in particular, how that reasoning
supports students’ understanding of number and number operations. In helping to plan,
implement and evaluate the research lessons, participating teachers were able to focus
on using equivalence and number relations to simplify calculations. Participating
teachers interviewed students to elicit their strategies and to assess students’ relational
thinking. Teachers’ content knowledge of relational thinking was also assessed by
presenting them with students’ responses to various open number sentences.
Method
Four teachers and fifteen students from a rural primary school in western Victoria
participated in this research. The participants included the school principal, one teacher
from Prep/Grade 1, one teacher from Grades 2/3, the teacher researcher from Grades
4/5/6; and fifteen students from Grades 4/5/6, aged between 9 and 12.
Lesson Study as research
The usual aim of the Lesson Study is the professional development of the teachers:
how to improve teachers’ understanding of what students learn and how best to bring
that about (Fernandez and Yoshida, 2004). Teachers research and plan a lesson
collaboratively. The lesson is then taught by one of the teachers in the group with the
others members observing. In a debriefing session, the lesson design and its
implementation are analysed, with a special focus on how well the students were able to
demonstrate what they were intended to learn. Changes to the lesson are then made to
better achieve the intended learning goals. In the next cycle, the revised lesson is taught
to a different class by another teacher from the group, and that lesson is then followed
by further review and modification where necessary.
This study adopted the approach used by Pierce and Stacey (2009) where Lesson
Study is used for research involving students and teachers, as well as for teachers’
professional development. However, in their modified form of Lesson Study, the above
researchers planned the lessons and gave them to teachers to enact. In this case, the two
lessons were researched and delivered by the teacher researcher to a multi-grade 4, 5
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and 6 over two weeks. Evidence was collected before, during and after the lessons from
students and teachers as discussed below.
Space does not permit a detailed elaboration of the two lessons. Key goals of the
lessons were to have students become familiar with number sentences with more than
one number following the equal sign; to understand the equal sign means ‘same as’ or
‘equivalent’; and thus show how to balance equations involving the four operations by
using number facts and computations, thus opening up basic properties of relational
thinking. Students were asked to identify True or False number sentences; to solve
missing numbers sentences and to share the strategies they used; and to create some
sentences of their own where relational thinking can be used to simplify calculations. A
key question for the research was whether were students able to identify the variation
between numbers on both sides of equal sign, and to show in their solutions that the
direction of variation depends on the operation involved. A framework based on the
study of Molina and Ambrose (2008) was used to categorise those students who had
misconceptions in regard to the equal sign, and to categorise shifts in other students’
thinking from relying solely on computation, to using a mix of computational and
relational strategies, to relying only on relational thinking. Participating teachers
observed the lessons, recorded students’ mathematical thinking, and debriefed following
the lessons. The post-lesson debriefing was used to fine tune the delivery of the second
lesson. Interviews with students were conducted by the teacher researcher and by one of
the participating teachers.
Student and teacher questionnaires
Questionnaires before and after the Lesson Study were administered to assess
students’ capacity for relational thinking and to identify any improvement as a result of
the Lesson Study. Participating teachers used observation sheets consisting of the
problems given to students in order to record evidence of relational thinking based on
what particular students wrote during the lessons. Interviews with students were later
used to confirm the evidence obtained from these sources.
Two student questionnaires were administered before the Lesson Study. The Grade
4 questionnaire had six missing number questions involving all four operations, such as
33 + 19 = __ + 20; __ + 17 = 15 + 24; 78 – 39 = __ – 40; 14 × 5 = 7 × __; 12 ÷ 4 = __ ÷
2. Grade 5/6 students were given eight missing number questions with some involving
larger numbers, such as 199 + 271 = 200 + __; 137 – 98 = __ – 100. Students were
invited to explain how they had worked out the answer for each question. Each
questionnaire also included True/False questions. For Grade 4, these included 27 + 48 –
48 = 27 (T/F); 15 + 19 = 15 + 20 – 1 (T/F); 99 – 9 = 90 – 59 (T/F); 3 × 4 = 12 × 2 (T/F);
18 ÷ 6 = 6 ÷ 2. Students were asked in each case why they had chosen True or False.
Two similar questionnaires were given after the Lesson study. The one for Grade 4
and some Grade 5/6 students included open number sentences involving addition,
subtraction and multiplication. The one for Grades 5/6 involved all four operations with
relatively larger numbers that used in the Grade 4 version. The results of the
questionnaires and interviews are summarised in Table 1 below.
Before the first lesson, a questionnaire was conducted to identify teachers’
knowledge of relational thinking and their experience in teaching relational thinking.
Teachers were asked to solve similar missing number sentences as for Grades 5/6 and to
explain how they had solved each problem. They were also asked to identify possible
misconceptions of students in attempting to solve missing number sentences; and
whether they had previously used relational thinking in their teaching. After the Lesson
Study, teachers were given another set of missing number questions; and were asked
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whether they intended to incorporate relational thinking in their future teaching. The
questionnaire also re-examined teachers’ understanding of students’ possible
misconceptions of the equal sign.
Results
Table 1 shows students’ use of relational thinking before and after the Lesson Study.
The four levels are based on the categories used by Molina and Ambrose (2008). In
Grade 4, all three students moved from Level 1 to Level 3. In Grade 5, one student who
was absent for one of the lessons remained at Level 1, but the other three all moved to
Levels 3 and 4. Grade 6 students who were at Level 2 before the Lesson Study all
moved to Levels 3 and 4. One Grade 6 remained at Level 4.
Table 1
Relational thinking before and after the Lesson Study
Relational Thinking Level 1 Level 2 Level 3 Level 4
Students display
misconceptions in
relation to equal
sign.
Students use only
computation to solve
open number
sentences.
Students use a mix
of computation and
relational thinking to
solve problems.
Students use only
relational thinking to
solve open number
sentence problems.
Before After Before After Before After Before After
Grade 4 (N=3) 3 0 0 0 0 3 0 0
Grade 5 (N=4) 2 1 1 0 1 1 0 2
Grade 6 (N=8) 2 0 4 1 1 3 1 4
Students’ Understanding of the Equal Sign
Before the Lesson Study seven students had misconceptions in relation to the equal
sign; three from Grade 4, two from Grade 5 and two from Grade 6. Some students wrote
52 in answering the question 33 + 19 = ___ + 20, using 33 + 19 = 52 but disregarding
the 20 on the right side. Others wrote 72 using 33 + 19 + 20 = 72. None of the above
students could find the missing number for the question ___ + 17 = 15 + 24, as one
explained, “You cannot plus anything from 17 to make 15.” Most of these showed
misconceptions in true/false questions. For example, they circled True for the problem
99 – 9 = 90 – 59, giving a reason that 99 – 9 = 90. One student, in responding to the
true/false question 34 +28 = 30 + 20 + 4 + 8, gave 134 (should be 124) as answer,
evidently by adding 34 + 28 + 30 + 20 + 4 + 8. These students all appeared to treat the
equal sign as a command to give an answer to the operations expressed on the left side,
or even both sides, of the equal sign.
Interestingly, one Grade 4 student, Amanda, gave a correct response and reasoning
for the true/false question 99 – 9 = 90 – 59 even though she had misconceptions in the
first part of the questionnaire (see below). She worked out the left side 99 – 9 = 90 and
the right side 90 – 59 = 31 and chose False. Among most of the true/false questions,
Amanda started working out the value of left side and right side by using an algorithm.
During the interview, she said, “When I was working the question 34 + 28 = 30 + 20 +
4 + 8, I found 34 + 28 = 62 so is 30 + 20 + 4 + 8 = 62.” She added, “I also know my
times-tables very well, for the question 3 × 4 = 12 × 2, I know 3 × 4 =12 and 12 × 2 =
24 so it’s false.” Amanda’s grasp of number facts helped her to compute the value of
both sides of equal sign and develop her understanding of equivalence, which was
confirmed during the interview. Analysis of the interviews revealed that most students
understood that the equal sign means ‘the same as’ or “is equivalent to”. However, as
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mentioned above, one Grade 5 student who was absent in one of Lesson Study sessions
still displayed misconceptions in relation to equal sign.
Students’ Perceptions of Relational Thinking
In the pre-Lesson Study questionnaire, Amanda wrote 33 + 19 = 52 + 20
disregarding 20 on the right side. In another problem, __ + 17 = 15 + 24 she wrote, “ I
don’t know because you can’t plus 17 to make 15.” As mentioned earlier, Amanda
changed her understanding of the equal sign during the true/false questions, using
computational strategies to balance both sides. In the questionnaire after the Lesson
Study, Amanda used a mix of computational and relational thinking. In her answer to
the question __ + 16 = 15 + 24, Amanda explained, “Between 16 and 15 is +1 so
between 24 and 23 is –1 because 16 is a big number you need a small number to keep it
balanced.” However, she did not use relational thinking in the multiplication question
18 × 5 = 9 × __; instead, she did 18 × 5 = 90 and 90 ÷ 9 = 10.
Tom (Year 5) also displayed various misconceptions in relation to equal sign in the
questionnaire before the Lesson Study; writing incorrectly that 33 + 19 = 52 + 20 and
also 78 – 39 = 41 – 40. After the Lesson Study, Tom successfully demonstrated his
understanding of relational thinking in addition and multiplication equations. In
correctly solving 13 + 29 = 12 + 30 Tom explained saying
29 + 1 makes 30 and 13 – 1 makes 12”. In the multiplication question, 18 × 5 = 9 × 10 ,
he wrote that 18 ÷ 2 = 9 so 5 × 2 = 10. However, he was confused with the direction of
compensation in subtraction questions. For example, in giving 71 – 28 = 73 – 26, he
reasoned incorrectly that, since 71 + 2 = 73, so 28 – 2 = 26.
In the questionnaire before the Lesson Study, John (Year 6) used a mix of
computational and relational thinking strategies to solve problems. For example, in
responding to 18 + 29 = ___ + 30, John wrote that 18 + 29 = 47. He also used arrows
vertically to show 29 plus 1 makes 30 and 18 minus 1 makes 17, therefore 17 + 30 = 47.
In his response for the problem ___ – 38 = 75 – 40, John wrote that 75 – 40 = 35, then
he used arrows vertically to show 40 minus 2 makes 38 and 75 minus 2 makes 73,
therefore 73 – 38 = 35. For the multiplication problem 48 × 25 = ___ × 100, John
worked out the left side of equal sign 48 × 25 = 1200 then he calculated right side that
12 × 100 = 1200. For division problem, 24 ÷ 6 = ___ ÷ 3, John worked out the left side
of equal sign 24 ÷ 6 = 4, then right side 4 × 3 = 12 so 12 ÷ 3 = 4. Dealing with true/false
questions, for example, with 570 + 199 = 570 + 200 – 1, John circled True and
explained, “200 – 1 = 199.” After the Lesson Study, John used arrows from left side to
right side to show one direction and variation of compensation; then he used the correct
direction of variation between the uncalculated equations on each side of the equal sign
to solve the problem. He confidently used relational thinking and correctly solved four
operations without carrying out any computation to check the answer.
In the pre-Lesson Study questionnaire, Esther (Year 6) used arrows to show the
directions of compensation, consistently directing the arrow from left to right no matter
where the missing number was. For 18 + 29 = ___ + 30, she explained, “29 plus 1
makes 30 so 18 minus 1 makes 17 to keep it balanced.” In another question, ___ + 17 =
15 + 24, Esther explained, “17 minus 2 makes 15, so 24 minus 2 makes 22, therefore
you don’t have two big numbers on the same side.” In the true/false question, 570 + 199
= 570 + 200 – 1, Esther explained, “It’s true because it’s just split so it is easier.” In
another question, 12 × 6 = 72 × 6, she explained, “It’s false because 72 and 12 are very
different numbers.” Esther’s responses demonstrate that she looked at the equation as a
whole and simplified the calculation. In her post-Lesson Study questionnaire responses,
Esther not only worked out the variations between two numbers but also the directions
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of compensation. For example, in the question, ___ + 199 = 152 + 200 Esther used
arrows from 200 minus 1 to 199 and from 152 plus 1 to 153 and explained, “When you
plus one you have to take away one so it stays balanced.” In a later subtraction question,
12.8 – 3.2 = __ – 3, Esther used arrows from 3.2 minus 0.2 to 3 and from 12.8 minus 0.2
to 12.6 and she explained, “The difference of numbers has to stay the same.” In her
answer for the multiplication problem 36 × 25 = 9 × ___, Esther showed that 36 divided
by 4 makes 9, so 25 times by 4 makes 100, explaining “You can’t have 2 big numbers
on the same side.” In her answer for the division problem ___ ÷ 15 = 20 ÷ 5, Esther
showed that 5 times 3 makes 15, so 20 times 3 makes 60 and she explained, “The gap or
difference between two numbers has to stay the same.” Before and after the Lesson
Study, Esther applied clear relational thinking in open number sentence problems
involving addition, subtraction, multiplication and division; not needing to use any
algorithm to check her answers like many other students.
Teachers’ Understanding of Students’ Misconceptions of Equal Sign
In a questionnaire before the first lesson, teachers were asked to give all possible
student responses to the missing number sentence 15 + 8 = __ + 10. All teachers gave
13 as a possible student response for the question, reasoning that since 15 + 8 = 23, so
23 – 10 = 13. Only one teacher gave 33 as a possible student response, explaining that
students might think 15 + 8 + 10 = 33. No teacher identified 23 as a possible
misconception. Before the Lesson study, teachers’ awareness that students may treat the
equal sign as meaning ‘the answer comes next’ appeared to be limited.
After the Lesson Study, teachers demonstrated better understanding of students’
likely misconceptions in relation to the equal sign. In their questionnaire responses, all
three teachers identified 24 as the correct answer, and 34 as a likely incorrect answer for
the sentence 25 + 9 = ___ + 10. They explained that students might focus on 25 + 9 =
34 disregarding the number 10 on the right side of equal sign. Two teachers also pointed
out that 26 could be a possible answer for students who used relational thinking but did
not follow the correct direction of variation. Only one teacher recognised all the likely
misconceptions that students might make, identifying 44 as a further possible incorrect
answer for those students who added all the numbers up, 25 + 9 + 10 = 44.
Teachers’ Knowledge of Relational Thinking
None of participating teachers had explicitly taught relational thinking before the
Lesson Study, but they appeared to know that relational thinking relates to equivalence
problems. In solving missing number sentences involving addition and subtraction, they
did use relational thinking, but all used computational strategies to solve multiplication
and division problems, such as 48 × 25 = ___ × 100, and ___ ÷ 15 = 20 ÷ 5.
Throughout the two cycles of Lesson Study, participating teachers expanded their
understanding of relational thinking, particularly through discussing students’ responses
after the two lessons. During the lessons, teachers had opportunities to explore students’
strategies embedded in their solutions and to orchestrate discussion with some of
students in relation to equal sign, computation and the direction of variation they used.
In the post-Lesson Study questionnaire, teachers were all able to use relational
thinking to solve multiplication and division problems, and referred to the advantages of
relational thinking might have over a purely computational approach. They agreed that
students need to know their number facts, including multiplication facts, and what the
equal sign means before applying a relational approach.
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Teachers’ Intentions regarding Teaching Relational Thinking
Participating teachers studied students’ responses and found that it is easier to start
the direction of variation from the side where there is no missing number so students
could identify the correct direction of variation between the uncalculated equations.
After the Lesson Study, teachers all proposed to integrate relational thinking into their
mathematics lessons. For example, the Prep/One teacher wrote, “I will try to teach
students the near 10 and near double strategies so that students can see the links, for
example, if 6 + 10 = 16, then 6 + 9 = ___.” Another teacher wrote, “Establish a better
understanding of the equal sign, e.g. 10 + 3 = ___ + ___.” All teachers agreed that
relational thinking could be used to simplify calculations and to check answers; and
especially in simplifying calculations where addends or subtrahends can be rounded up,
or down, to the nearest ten or hundred. All agreed to focus more on developing students
thinking with regard to equivalence and compensation at whatever year level.
Discussion and Conclusions
This study provided evidence that Lesson Study focused on students’ mathematical
thinking in solving open number sentences was productive for both teachers and
students. It provided opportunities for teachers to discover students’ misconceptions in
relation to equal sign, and students’ needing to pay attention to the sentence as a whole.
Before the Lesson Study, many students relied completely on calculation.
According to Molina and Ambrose (2008), this behaviour is a result of the strong
orientation to computation which dominates arithmetic in early years. True/false and
open number sentences proved to be useful tools for seeding discussions about the equal
sign and developing students’ relational thinking. This modified form of Lesson Study,
was successful in moving forward the ability of students to use relational thinking
across all year levels. The most significant increase was evident among Grades 5 and 6
students. Many could solve number sentences using the four operations solely by using
relational thinking. Fluency with number facts played a vital role for these students.
As a result of the Lesson Study, almost all students were able to use equivalence and
the correct direction of compensation/variation between numbers to solve problems.
However, some students still failed to identify the correct direction of compensation or
variation. These responses confirmed findings by Irwin and Britt (2005) and Stephens
(2006) that students need help to distinguish between the direction of compensation for
addition and subtraction sentences. Students also need to know that direction of
compensation is also different between multiplication and division.
Throughout the two lessons, participating teachers had opportunities to investigate
relational thinking among Grades 4/5/6 students, and to observe the students’ movement
away from computational strategies to using relational thinking in solving open number
sentences. Focussing on the structure of number sentences, the operations involved, and
the key ideas of equivalence and compensation are all necessary to strengthen links
between number and algebra. Our study supports the findings of Carpenter et al. (2003),
Hunter (2007), Molina and Ambrose (2008), Stephens (2006a) that solving and
discussing true/false and open number sentence problems are effective ways to foster
relational thinking. It also showed that fluent recall of number knowledge is a pre-
condition of students’ use of relational thinking across the four operations; and that
relational thinking is influenced by the characteristics of the sentence, size and type of
numbers used, and the operations involved.
Participation in Lesson Study improved teachers’ personal understanding of
relational thinking strategies, and of children’s misconceptions that need to be addressed
7
directly. Teachers recognised the key role played by fluent recall of number facts to
support the twin ideas of equivalence and compensation. They were all able to point to
specific instances of how they could and would integrate relational thinking into their
teaching of number and number operations at whatever grade level. The study reported
here has implications for the use of True/False and open number sentences in
developing students’ thinking about equivalence and the structure of number sentences,
using all four operations. True/False and open number sentences were effective in
assisting students to move away from an exclusive reliance on calculation. These same
types of sentences can be used with teachers to extend their understanding of relational
thinking. Using them in a Lesson Study context allows teachers to observe students
closely as they begin to think relationally across the four operations. In this way,
teachers are more likely to use relational thinking effectively in their own teaching.
References
Australian Curriculum, Assessment and Reporting Authority (ACARA) (2010). The Australian
Curriculum. website. http://www.australiancurriculum.edu.au/ Accessed on 30/04/2012.
Carpenter, T. P., & Franke, M. L. (2001). Developing algebraic reasoning in the elementary school:
Generalisation and proof. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the
12th ICMI Study Conference. The Future of the Teaching and Learning of Algebra (pp. 155-162).
Melbourne: University of Melbourne.
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: integrating arithmetic and
algebra in elementary school. Portsmouth: Heinemann.
Carpenter, T. P., Levi, L., Franke, M. L., & Zeringue, J. K. (2005). Algebra in Elementary School:
Developing Relational Thinking. ZDM, 37(1), 53-59.
Fernandez, C., & Toshida, M. (2004). Lesson Study: A Japanese approach to improving teaching and
learning. New York: Lawrence Erlbaum Associates.
Hunter, J. (2007). Relational and calculational thinking: Students solving open number equivalence
problems. In J. Watson & K. Beswick (Eds.), Mathematics: Essential research, essential practice,
Proceedings of the 30th annual conference of the Mathematics Education Research Group of
Australasia, pp. 421-429. Hobart: MERGA.
Irwin, K. C., & Britt, M. S. (2005). The algebraic nature of students’ numerical manipulation in the New
Zealand Numeracy Project. Educational Studies in Mathematics 58, 169-188.
Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development
focused on children’s algebraic reasoning in elementary school. Journal of Research in Mathematics
Education, 38(3), 258-288.
Molina, M., & Ambrose, R. (2008). From an operational to a relational conception of the equal sign.
Third graders’ developing algebraic thinking. Focus on Learning Problems in Mathematics, 30(1),
61-80.
Pierce, R. & Stacey, K. (2009) Lesson study with a twist: Researching lesson design by studying
classroom implementation. In Tzekaki, M., Kaldrimidou, M. & Sakonidis, C. (Eds.). Proceedings of
the 33rd Conference of the International Group for the Psychology of Mathematics Education , Vol.
4, pp. 369-376. Thessaloniki, Greece: PME
Stephens, A. (2006a). Equivalence and relational thinking: Preservice elementary teachers’ awareness of
opportunities and misconceptions. Journal of Mathematics Teacher Education 9, 249-278.
Stephens, M. (2006). Describing and exploring the power of relational thinking. In P. Grootenboer, R.
Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures and learning spaces, Proceedings of the
29th annual conference of the Mathematics Education Research Group of Australasia, pp. 479-486.
Sydney: MERGA.
Stephens, M., & Ribeiro, A. (2012). Working towards algebra: the importance of relational thinking.
Revista Latinoamericana de Investigacion en Mathematica Educativa (Relime) 15 (3), 373-402.
Footnotes: The research reported in this paper was undertaken a partial fulfilment of the
Master of Numeracy degree at The University of Melbourne in 2012. All student names
cited in this paper are pseudonyms.
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