Noticing structural thinking through the CRIG framework of mathematical structure Mark Gronow Central Queensland University <[email protected]> Structural thinking skills should be developed as a prerequisite for a young person’s future mathematical understanding and a teachers’ understanding of mathematical structure is necessary to develop students’ structural thinking skills. In this study, three secondary mathematics pre-service teachers (PSTs) learned to notice structural thinking through the CRIG framework of mathematical structure. The framework consists of Connections, Recognising patterns, Identifying similarities and difference, and Generalising and reasoning. I report here on how the CRIG framework helped the PSTs’ notice structural thinking. To develop an ability to notice structural thinking, teachers must first of all be aware of mathematical structure. Mason et al. (2009) defined mathematical structure as “the identification of general properties which are instantiated in a particular situation as relationships between elements or subsets or elements of a set” (p. 10). Stephens (2008) described structural thinking as an awareness of how mathematical properties develop into generalisations. Furthermore, Mason et al. (2009) promoted structural thinking as understanding the concepts and knowing procedures to use and when solving mathematical problems. Varied theories exist about structure; as mathematical structure or structural thinking. Wertheimer (1945) proposed that mathematical structure is knowing how a formula is connected to a mathematical concept. Hiebert and Lefevre (1986) combined conceptual and procedural knowledge as ‘proceptual’ thinking across mathematical processes. Stephens (2008) defined ‘structure’ as synonymous with relational thinking (Skemp, 1976). Schwarz et al. (2009) proposed that structural thinking is knowing the relationships and connections between mathematical concepts. The concept of structural thinking in mathematics is not clearly understood by many teachers of mathematics (Richland et al., 2012). Mason et al. (2009) stated that teachers’ awareness of structural relationships transforms students’ thinking and disposition to engage, they believe that teachers need to focus on structure so students can think structurally. Research in teachers’ awareness of mathematical structure or structural thinking is limited. Gronow et al. (2020) explored secondary mathematics teachers’ understanding and use of mathematical structure. Their study investigated how teachers used mathematical structure and encouraged structural thinking through components of mathematical structure: Connections, Recognising patterns, Identifying similarities and differences, and Generalising and reasoning. The four components, known as CRIG pedagogical framework of mathematical structure developed during Gronow et al.’s (2020) study found teachers’ identified with structure but were not aware they were using it in their teaching. The CRIG framework, in this study, is introduced to PSTs as an effective mechanism for learning to notice structural thinking. The four components of the CRIG framework are detailed next. Connections. Vale et al. (2011) recognised connections between mathematical representations as fundamental to structural thinking. Making connections between contexts or concepts allows learners to develop mathematical understanding. Mathematics teachers 2021. In Y. H. Leong, B. Kaur, B. H. Choy, J. B. W. Yeo, & S. L. Chin (Eds.), Excellence in Mathematics Education: Foundations and Pathways (Proceedings of the 43rd annual conference of the Mathematics Education Research Group of Australasia), pp. 211-218. Singapore: MERGA. Gronow make connections between prior, present and future learning, and in real-world contexts of mathematical representations. Recognising patterns. Patterns are essential in children’s mathematical development which begin with their observations of the natural world. Children recognise, observe and generate patterns before reaching school and learn patterning in formalised learning situations that develop structural thinking processes that lead to a deeper understanding of abstract mathematical concepts. Mulligan and Mitchelmore (2009) found that children’s structural thinking, identified in patterning awareness, is essential for mathematical concept formation in future learning. Identifying similarities and differences. Learners develop structural thinking through noticing the differences in mathematical representations. Mason (1996) believed structural thinking is noticing similarities and differences in mathematical relationships. Mulligan and Mitchelmore (2009) discovered that children who found similarities and differences in patterns were involved in structural thinking. Generalising and reasoning. Mason (2008) described this as an activity that develops a more in-depth experience of mathematics. Mathematical thinking that eventuates into a generalised fact is structural thinking, it connects mathematical relationships from concrete representations to abstract ideas. Mason et al. (2009) wrote that appreciation of structure involves the experience of generality. Stephens (2008) applied structural thinking to designing arithmetic questions. He asserted that children who could articulate a generalised principle underlying a whole problem were thinking structurally. The framework of noticing also supports the process PSTs learning to notice structural thinking. Scheiner (2016) identified how noticing is not restricted to a single process. Mason (2002) asserted that “every act depends on noticing” (p. 7), he used the term “awareness” to characterise the ability to notice, referring to noticing as an awareness of what one is attending to. In this study, noticing structural thinking implies an awareness of understanding and use of mathematical structure. By adopting Mason’s (2002) approach to noticing, the development and use of mathematical structure has emerged as a form of directing PSTs’ attention to their mathematical thinking. Mason studied what he noticed when doing mathematics and called what he noticed the structures of attention of how one thinks mathematically. The aim of this study is for the PSTs to notice structural thinking through learning the components of the CRIG framework of mathematical structure. The PSTs use of the CRIG framework provides an opportunity to detect their awareness of structure, thus answering the research question: How does the CRIG framework help PSTs to notice structural thinking? Method Context and participants PSTs in their final year Bachelor of Education/Bachelor of Arts (secondary mathematics) degree at a Sydney university were invited to participate in this study. Three PSTs, referred to as Ms K, Ms M, and Mr T, volunteered to participate in the study during their professional experience placement. Each PST taught mathematics at a secondary school in metropolitan Sydney. Ms K taught an accelerated Year 9 class, Ms M taught a top streamed Year 8 class, and Mr T taught a mixed ability Year 7 class. The PSTs were familiar with the concept of 212 Gronow mathematical structure through the content of courses studied in their undergraduate degree; however, they had no prior knowledge of the CRIG framework. Study design, instruments, and data collection The study design comprised of three cycles of: professional learning workshops (PLWs), which were audio recorded. Video recordings of PSTs’ mathematics lessons and a noticing reflection audio recording of PSTs reviewing a recorded segment of their mathematics lessons. Analysis The audio recordings of the PLWs and noticing reflections were all transcribed to a word document and uploaded to NVivo (QSR International, 2017). The videos of the mathematics lessons were also uploaded to NVivo. NVivo was used to code the data from the PLWs, mathematics lessons and noticing reflections for PSTs’ utterances and comments that identified a CRIG component. The data were analysed for evidence of the PSTs’ noticing of structural thinking through the PSTs attending to the CRIG framework. The videos acted as the main source of evidence for identifying the PSTs noticing structural thinking through their use of the CRIG framework when teaching. The PLWs and the noticing reflections provide further evidence of the PSTs attention to the CRIG framework. Results This section presents a summary of the data collected for each PST from the three cycles of PLWs, mathematics lessons and noticing reflections. An outline of the results from the PLWs are given, followed by exemplars of each PSTs’ utterances from the mathematics lessons and comments made during the noticing reflections in Tables 1, 2 and 3, coded to a CRIG component. During the PLWs, the PSTs were taught to notice structural thinking through the CRIG components. The first PLW began with a presentation on the CRIG framework, followed by a viewing of a video titled Related Problems: Reasoning About Addition (Teaching Channel, 2017), where a teacher used the CRIG components to teach addition to a Year One primary class. Ms K Recognised patterns in the teacher’s instructions to students. Ms M also Recognised patterns as a teaching strategy to engage the students. Mr T noticed that the students used Similarities and differences to make generalisations. In PLW 2, the PSTs viewed a video recording of a child attempting several different arithmetic problems, they were asked to examine the child’s mathematical thinking when solving the problems. Ms K noted the child relied on calculations and did not Identify Similarities and Differences between the numbers. Ms M noticed the child was using Generalising and Reasoning in her structural thinking when she recognised that the problem could be solved another way. Mr T stated the child “Got it after the CRIG prompt, meaning she has structural understanding.” In PLW 3, the PSTs considered how the CRIG framework could be applied to teaching the expansion of binomial products. Ms K made Connections to the distributive law and expanding the expression using the FOIL method. Ms M was Identifying Similarities and Differences when changing numbers, pronumerals, signs and coefficients in the binomial expression. Mr T stated that Generalising and reasoning was identified as a way to summarise the process of expansion and apply it in other mathematical contexts. 213 Gronow Table 1 Exemplars of Ms K using the CRIG Framework to Notice Structural Thinking Cycle Mathematics lesson Noticing reflection 1 Topic: Simultaneous equations Connections between the equation Connections to the relationship between the and the graph. “I think to show how graphs’ intersection points and solving the the y2 and the x2 is giving us part of equations simultaneously. the circle, that relationship.” Recognising Patterns of the power of x to Identifying similarities and differences between graphs and determine the curve’s shape. equations: “So, they could see that Identifying similarities and differences all of them had a square except the “What is different about the line’s shape?” last one.” 2 Topic: Angle sum of polygons Recognising Patterns to develop Connections to prior learning “How did we the formula: “They understood it prove the angle sum of the quadrilateral?” better with the pattern.” Angle sum of a polygon formula: Identifying similarities and Recognising patterns: “Can you find the differences different patterns helped pattern of what is going on between the students’ thinking. “I had the number sides and triangles?” triangles meeting at a point. I adjusted it as I saw the pattern they Generalising and reasoning: “Calculate the were working out.” interior angle sum of any polygon.” 3 Topic: Quadratics Connections: “I was connecting it Connections “Quadratics and parabolas go to when we did the non-linear hand-in-hand. The visual representation of a simultaneous equations.” quadratic is a parabola.” Recognising Patterns, “Rather than Identifying Similarities and Differences of drawing random graphs, I’d link the x2 expression in an equation “This is not them to recognise any patterns from of degree two; it is a power of negative two. factorised quadratics.” So, this is not a quadratic.” Generalising and Reasoning “Generalising the solutions of when Generalising and Reasoning relationships between the equation and the graph. crossing the x-axis.” Table 2 Exemplars of Ms M using the CRIG Framework to Notice Structural Thinking Cycle Mathematics lesson Noticing reflection 1 Topic: Circumference of a circle Generalising and Reasoning through Connections to a real life example of a students’ discussion when dividing pizza as a sector of a circle. the circumference by the diameter. “I’m looking at what they just did. Recognising patterns in the ratio of a I’m asking them to contribute what circle’s circumference and diameter. they found and see what they Similarities and Differences comparing conclude from what they've done.” the circle’s radius and diameter. 214 Gronow 2 Topic: Area of composite shape Recognising patterns “asking them Identifying Similarities and differences to how to figure out the area. That explain the formula of the area of circles. could have been kind of recognising “Area equals 𝜋𝑟2 which is the same as patterns.” saying 𝜋×𝑟×𝑟.” Identifying Similarities and differences “How to write something Generalising and reasoning “How come in exact form and not exact form we have 𝜋 for every circle? Because the circumference divided by the diameter Generalising and reasoning “Asking was always equal to 𝜋.” them questions they can conclude.” 3 Topic: Volume of a cylinder Connections “How they could use Connections of a real-world problem: previous things they've learnt.” “This is a picture of the sinkhole. What Recognising patterns “By helping shape does it look like?”, them recognise patterns to work Generalising and reasoning “What do we mathematically.” need to know to solve this problem? What Generalising and reasoning are we trying to find in the end?” “Recognising the meaning and interpreting the information.” Table 3 Exemplars of Mr T using the CRIG Framework to Notice Structural Thinking Cycle Mathematics lesson Noticing reflection 1 Topic: Ordering fractions Connections “I should have Connections to a real example “What is reworded the question because this one-third of my chocolate bar.” was what we did last lesson.” Identifying Similarities and Differences in Recognising Patterns “What do you ordering fractions “When you look at this, notice I’m doing with these which one’s bigger? Or, which one’s numbers?” smaller?” Identifying similarities and Generalising and Reasoning defining a differences “Show the diagram of rule “The size of the parts needs to be the shaded fractions not symbolically.” same.” 2 Topic: Adding and subtracting fractions Recognising patterns: “I tried to set Identifying Similarities and Differences up some patterns and then asked “What do you notice about the them to recognise the patterns.” numerators?” Generalising and Reasoning “I’ve Generalising and Reasoning, using a tried to incorporate generalisation in whole number method to add fractions. terms of asking them, ‘What do you think would be the next pattern?’” “So, if 1+1 = 2, then, if I use the same 1 1 thing, for a + , is 1+1 = 2, and 2+ 2 2 1 2 = 4, so it’s over . Right?” 4 215 Gronow 3 Topic: Stem and leaf plot graphs Recognising patterns: “So I should Similarities and Differences between have put one number on so the graphs and stem-and-leaf plots. “Now students to see a pattern.” what were the things we compared. Identifying Similarities and What’s similar?” Differences “I should have asked Generalising and reasoning to analyse about the placement of these three stem-and-leaf plot data. “Take a look at numbers: “How are they different?” your graph and talk to the other person and tell them what the graph tells you?” Discussion During this study, the PSTs’ noticing of structural thinking developed through their learning of the CRIG pedagogical framework of mathematical structure. Noticing of structural thinking was evident in their references to the CRIG framework drawn from the statements made during the PLWs, utterances in their mathematics lessons, and noticing reflection comments. Exemplars given demonstrate the PSTs’ noticing structural thinking through the CRIG components. The PSTs use of the CRIG components were identified in varied pedagogical strategies. Ms K encouraged students to use a pattern to find the rule for the angle sum of a polygon, Ms M used real world examples for each of her lessons to connect students understanding to the mathematical concept and Mr T used the CRIG components in his questions. The PSTs’ teaching accommodated the CRIG framework and supported their understanding of the mathematical content. Ms K considered other patterning approaches to finding a rule for the angle sum of a polygon and Ms M noticed similarities and differences in binomial expansions. The PSTs’ pedagogy focused on a structural thinking learning environment, Ms K promoted students’ thinking by challenging them to connect the equation to a graph, Ms M connected mathematical concepts to real-world examples and Mr T asked questions so students would notice patterns, and similarities and difference. In their noticing reflections, the PSTs stated how the CRIG framework supported their teaching. Ms K, was thinking of her future teaching: “If I were to do this again, I’d teach the patterning way, and I would incorporate the CRIG more.” Ms M stated CRIG helped her understand student thinking “They're trying to understand the difference between volume and capacity.” Mr T reflected on how CRIG improved his explanations. “I should have made it more explicit, by connecting to their prior experience.” The CRIG framework in these cases supported the PSTs’ noticing of structural thinking. Prescott and Cavanagh (2007) found that secondary mathematics PSTs tended toward a traditional teaching pedagogy. Awareness of the CRIG framework encouraged the PSTs in this study to move beyond traditional teaching pedagogy. The PSTs were more inclusive of student learning, as noted when asking CRIG component focused questions. Mr T’s questions promoted students’ structural thinking. He challenged students’ thinking about why using a whole number method when adding fractions was incorrect. “So, if 1+1 = 2, 1 1 1 then, if I use the same thing, for a + , is 1+1 = 2, and 2+2 = 4, so it’s over . Right?” 2 2 4 The PSTs diverse pedagogical strategies also saw them use the CRIG components when instructing or communicating with students. In her second mathematics lesson, Ms K used Recognising patterns to help students develop the angle sum of a polygon formula. As the students had discovered a different pattern, one that was not considered by Ms K, she acted 216 Gronow in-the-moment and noticed the students’ new approach, she encouraged her students to continue with their strategy and asked one student to explain it to the class. Ms M promoted student involvement in her lessons by arranging students in groups to complete activities, many of which had a real-world experience, such as, here final lesson of finding the volume of a cylinder as a sink hole. The professional learning program to understand and use the CRIG framework helped the PSTs’ to notice structural thinking. Ivars et al. (2018) identified the need for a specific framework for PSTs to have effective noticing. The CRIG framework provided this focus. The ability of the PSTs to understand the CRIG framework and to use it demonstrated its simplicity as a practical and useful tool for teachers of mathematics. The PSTs’ content knowledge was established from their extensive mathematical background in their university studies. The CRIG framework, however, deepened the PSTs structural understandings of mathematical relationship, for example Ms K’s students finding an alternative approach to finding the angle sum of a polygon. The PSTs’ lack of professional experience before this study could have influenced their fundamental understanding of the CRIG framework and their ability to notice structural thinking. However, having more teaching experience in the future will provide continual opportunities notice structural thinking through the CRIG framework when doing mathematics and when teaching. The PSTs’ teaching experience was restricted to their university professional experience program. Researchers have identified how PSTs’ limited experiences influence what they attend to when teaching. Star and Strickland (2008) found that secondary mathematics PSTs were not good at noticing mathematical content. Mason (2002) also asserted that PSTs lack experience in recognising and using classroom interactions effectively to promote mathematical understanding. Contrary to the results of these studies, the PSTs in this study produced mathematics lessons that engaged students with activities, instructions and questions that focused on developing students’ structural thinking through using the components of the CRIG framework. The PSTs effectively demonstrated an ability to learn and apply the CRIG framework as a new pedagogical skill to mathematical content that they had not taught before. The introduction of structural thinking through the CRIG framework could be regarded as an extra burden for the PSTs to consider when teaching. Nevertheless, the evidence indicates that the PSTs were comfortable with identifying and including the components of the CRIG framework in their lessons and were able to notice structural thinking. The PSTs were able to articulate the benefits of the CRIG framework to notice structural thinking they indicated that the CRIG framework had shaped their noticing structural thinking and had changed their teaching. Ms K stated that thinking structurally helped her make sense and explain mathematical concepts. In the final PLW, Ms K stated, “You structure your practice to facilitate deeper thought as to what and how things made sense.” Conclusions and Further Research The CRIG framework proved to be useful for helping PSTs to notice structural thinking. The CRIG framework provided the PSTs with a foundation for teaching mathematics that helped them focus on developing their understanding of mathematical structure. Moreover, this provided PSTs opportunities to notice structural thinking. Mason (2002) introduced the concept of noticing into the lexicon of mathematics education, and with his colleagues (Mason et al. 2009) the notion of teachers’ noticing of structural thinking has emerged as a significant contribution to mathematics teaching. PSTs noticing of structural thinking as the focus of this study has demonstrated, as evident from 217 Gronow the results, that there is potential to advance the discourse of mathematics teaching in this area. The introduction of mathematical structure in the teaching and learning of mathematics and the noticing of structural thinking has implications for future research in mathematics teaching. Future research could consider how developing noticing structural thinking through the CRIG framework may benefit practicing teachers of mathematics (e.g., primary, secondary, pre-service, novice, experienced, and out-of-field teachers). References Gronow, M., Mulligan, J., & Cavanagh, M. (2020). 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