Keywords

1 Introduction

Teachers of students in general mathematics classrooms accept and welcome the learners they are assigned to teach. Students’ characteristics influence the planning teachers undertake, the learning activities they provide, and the learning outcomes achieved by their students as a result. This chapter explores the impact of student characteristics that are beyond the control of teachers, and yet are within their powers through their actions to make a considerable difference to the mathematics learning outcomes of their students (Manizade et al., 2019).

In recent times, two significant developments—the recognition of streaming (the practice of grouping students “within-grade-level on the basis of perceived ability” (Forgasz, 2010, p. 57)) as harmful and the recognition of inclusive education as beneficial—have changed the nature of general mathematics classrooms. The research on assigning students to mathematics classes based on their achievement has long shown that this is a harmful practice (Hunter et al., 2020; Wilkinson & Penney, 2014; Zevenbergen, 2005). Streaming is detrimental to not just low and average groups but also has limited benefits along with possible risks for high achieving students (Linchevski & Kutscher, 1998; Parsons & Hallam, 2014).

In a separate development, the inclusive education movement, where all learners are welcomed and supported in general learning environments, has also become prominent. General Comment Number 4 of the United Nations’ Committee on the Rights of Persons with Disabilities (2016) distinguishes inclusive education from integration, segregation and exclusion. Inclusive education requires the learning support needs of all learners to be met within the general classroom where all learners are working towards the learning outcomes of their class program. Again the research evidence is unequivocal (Hehir et al., 2016)—all learners do better in inclusive education environments. Through international charters and conventions, the provision of inclusive education has become a requirement of States Parties to these agreements. In the case of the United Nations Convention on the Rights of Persons with Disabilities (United Nations, 2006), all but seven countries in the world have signed the convention, thereby signalling their intention to provide inclusive education for all learners with disabilities.

For mathematics education, these two evidence-based movements—heterogeneous rather than streamed groupings and inclusive education—combine to provide strong impetus for the need to teach all learners in general classrooms. In practice, this means that teachers can expect to teach students across the breadth of human variation. This might seem overwhelming and indeed it would be if the approach were to attempt to teach individual lessons to each student in the class. Research in this field has led to alternative approaches that indicate approaches that teachers might use to plan effective mathematics lessons for all learners, including those with intellectual disabilities in the one classroom. These will be canvassed in this chapter. In this way, it is argued that teachers use to advantage the characteristics learners bring to the classroom, leading to improved mathematics learning outcomes for all.

2 Literature Review

Individual student characteristics are beyond the teacher’s control. If we assume inclusive classrooms, the students they teach are also beyond the teacher’s control. Teachers do not exclude students who have particular attributes, such as intellectual disability. The example of intellectual disability is taken here because it is possibly the area of student diversity that presents as most instructive for understanding the impact of learners’ characteristics on mathematics learning outcomes. The impact of learner attributes on the presage, process, and products (PPP) of mathematics teaching must be understood for the best learning outcomes to be achieved.

One challenge encompassing both theoretical and methodological aspects of this area relates to the historical approach to the education of low attaining students in mathematics. The emergence of two education systems—special education and mainstream—has meant the mathematics education of students with intellectual disabilities and learning difficulties mostly had been undertaken by special education teachers. In recent times, inclusive education has been adopted around the world with the aim of educating all students in mainstream classrooms (Florian, 2012) and therefore the mathematics education of learners with intellectual disabilities has become the task of general mathematics teachers.

Mathematics education research typically takes a different approach to research from that of special education research in the field of mathematics (Xin & Tzur, 2016). It is not a happy marriage and reconciliation is not easy. Behaviourist approaches to mathematics learning, with a focus on remediating deficits and explicit teaching of procedures (often with mnemonics to aid recall of the procedures, see, for example, Flores, 2010), have been favoured by many special education researchers in mathematics (for a review, see Tan et al., 2019), though not all (see, for example, Browder et al., 2012). Special education research focuses on interventions and understanding learner deficiencies. Mathematics education research focuses more on pedagogy appropriate for learning mathematics. The systematic review by Tan and colleagues, sampled literature published between 2006 and 2017 related to mathematics education of students with intellectual disabilities. From their review, they distilled three categories: Deficit-oriented, Discursively-aligned and Socio-political. 48% of reviewed articles fell within the sub-category “Behaviorism” of the “Deficit-oriented” category, where they note, “In these studies, mathematics education is characterized as reproducing or memorization of prescribed facts or procedures. … Students in these studies are shown a particular sequence of solving mathematics problems, followed by assessing their ability to exactly follow each step in the pre-defined procedure.” (Tan et al., p. 6). The goal of the mathematics taught was found to be functional, “research focused on the importance of mathematics fact recall as they connected their work to functional life skills and the importance of mathematics for everyday tasks such as shopping” (p. 6).

By contrast, general mathematics education research emphasizes the value of mathematics as a discipline and favours less utilitarian views of the purpose of learning mathematics, for all students including those with disabilities (Scherer et al., 2016). In the Tan et al. (2019) review, a category of research was identified that aligned with this view of mathematics education research.

In contrast to the dominant forms of research in mathematics education involving students with intellectual disabilities that focuses on direct forms of instruction and basic mathematics skills development …, undermining conceptual construction and understanding of mathematics, the studies in this category presume, to a greater extent, that students with an intellectual disability are mathematics thinkers and doers, capable of a range of mathematics engagement. (p. 8)

For the purposes of this chapter, providing an overview of current understandings in this field, three areas of diverse student characteristics are reviewed: mathematics learning disabilities, which are inherent to the student; learned difficulties, which are not; and other learner characteristics, not related to intellectual development, that have an impact on mathematics learning leading to learning difficulties.

2.1 Current Understandings of Mathematics Learning Difficulties, Disabilities and Dyscalculia

Some students struggle to learn mathematics. This is hardly a revelation. No matter whether the assessment is norm-referenced (with a proportion of the population automatically performing worse than expected for their age) or criterion-referenced (with a possibly similar subset of the population not meeting learning targets for their grade level), teachers, parents, and the students themselves have known that learning mathematics does not come easily for all. The source of these difficulties is of particular interest because that affects approaches teachers might take to improve learning outcomes.

It would seem that the source of some mathematics learning disabilities is neurobiological in origin leading to differences in cognition. Developmental Dyscalculia (DD), possibly with sub-types (Skagerlund & Träff, 2016), is a condition caused by atypical development of the parts of the brain that support the understanding of quantity such as numerical magnitude processing, the development of a mental number line, and the ability to calculate using known facts (Kaufmann & von Aster, 2012; Peters & De Smedt, 2018; Skagerlund & Träff, 2016). Diagnosis is not undertaken based on brain imaging studies, however. Instead, diagnosis is usually made based on the determination of “a serious impairment of the learning of basic numerical-arithmetical skills in a child whose intellectual capacity and schooling are otherwise adequate” (Kaufmann & von Aster, 2012, p. 769). This comparison with progress in other areas of schooling or in comparison with other intelligence measures leads to the challenge of diagnosing DD in learners with intellectual disabilities who have relatively more difficulty in acquiring a sense of number than other areas of learning (Cuskelly & Faragher, 2019). The significant challenges of defining mathematics learning disabilities by discrepancy from “normal” achievement are well articulated by Scherer and colleagues (Scherer et al., 2016). Lewis (2014) draws attention to the challenges of diagnosis based on an arbitrary cut-off score and instead advocates for analysis of ‘persistent understandings’ that are divergent from correct mathematical understanding as an indication of mathematics learning disability.

Some aspects of difficulties learning mathematics overlap with language difficulties. Dyslexia, a learning disorder characterized by difficulties with language and reading, has a documented overlap with difficulties retrieving arithmetic facts (Peters & De Smedt, 2018) and is also neurobiological in origin. The area of the brain affected is not the same as that proposed for DD. Aspects of mathematics affected by dyscalculia also differ from that of dyslexia. Some aspects of arithmetic, such as retrieval of number facts, appear to be more language based (Dehaene, 2011). The impact of limitations of remembering arithmetical facts could arguably have been overstated, being a carry over from a time when the inability to calculate by written methods severely hampered further work in parts of mathematics reliant on these processes. With the ready availability of alternative methods of calculation, it might be the case that number and quantity need no longer be seen as core aspects of mathematics upon which the rest of the discipline relies. More research is needed in this area (Verschaffel et al., 2016).

Beyond those learners with mathematics learning disabilities such as DD, there are those with learning difficulties. The proportion of individuals who meet diagnostic criteria for DD is thought to be in the range of 5–7% (Butterworth et al., 2011). However, “a much larger number of children and adults experience less severe or less specific difficulties with mathematics which are nevertheless sufficient to cause significant educational and occupational difficulties” (Dowker & Kaufmann, 2009, p. 339). These learning difficulties with mathematics are the result of factors separate from the neurobiological functioning of the learner and reflect low achievement due to other factors. These factors include poor teaching, environmental factors, affective factors (Lewis, 2014), minority status (Hunter et al., 2020), previous academic attainment, gender, age, health, family socio-economic characteristics (such as parental education, income and health), and school characteristics (Parsons & Hallan, 2014).

Understanding the impact of factors that lead to mathematics learning difficulties has been the focus of mathematics research endeavors (Faragher et al., 2016; Scherer et al., 2016; Vale et al., 2016) and the concern of teachers. Lindenskov and Lindhardt noted in their study (2020, p. 65) the concern of teachers with the paucity of mathematics learning experiences they felt compelled to offer students who had difficulties learning mathematics.

The distinction between the two groups—disabilities and difficulties—is critical for intervention. Underpinning both, though, is the need for, and value of, good teaching with the right support.

2.2 Current Understandings of Learned Difficulties

Low attainment in mathematics, as the argument is building, can be the result of learning disability leading to different development, or learning difficulty due to factors in the ecosystem of the learner. Low attainment can be the result of a mismatch of mathematics education approaches with the needs of the learner (Lewis, 2014; Lindenskov & Lindhardt, 2020).

The third group of low attaining learners is comprised of a group who have acquired their difficulties with mathematics while they have been learners at school. These are those students who did not commence school with a disability or difficulty but through engagement with the school mathematics environment, they acquired difficulties with mathematics. This third group can be designated as those with learned difficulties. A group of learners in this category that has received research attention in recent times are learners with mathematics anxiety (Dowker et al., 2016). This group of students can become so fearful of mathematics that they will actively avoid even those mathematics tasks that are easily within their capability (Wilson, 2018).

Learned difficulties with mathematics can be pervasive, leading to limitations on the mathematics individuals are willing to undertake in the contexts of their lives. This important research finding, documented in many government reports including the influential Cockroft Report from the UK (1982), has led to impacts on conceptualization of numeracy—the use of mathematics in life contexts. These models acknowledge that it is insufficient to know mathematics and make sense of contexts; affective attributes such as willingness to undertake mathematics are also essential to consider (Goos et al., 2015).

While the three groups proposed (learning disabilities, learning difficulties and learned difficulties) have been presented as disjoint, intersection between groups may well exist. Students with learning disabilities could quite conceivably gain more pronounced difficulties via the same mechanisms  as other students e.g. development of mathematics anxiety.

2.3 Interventions

Having outlined three main groups of low attaining students in mathematics—those with learning disabilities, learning difficulties, and learned difficulties—attention is now turned towards what research indicates might be done to improve the attainment of those students.

Overcoming a neurological developmental difference requires specific attention to the cause, with brain-based interventions implicated. Some studies have tested the impact of computer interventions based on repeated task training aimed at developing new neural pathways (Räsänena et al., 2009) with some localized benefits. Unfortunately, transfer to areas of mathematics learning beyond the immediate training tasks was not found. It would seem that treatment approaches that correct neurological impairments to mathematics learning are yet to be uncovered. Instead, or in the meantime, teachers must take other approaches. It is not acceptable to do nothing—ways around the barriers to learning must be found if students are to be successful at learning mathematics in inclusive classrooms. Lewis’ work (2014) indicates the need for teachers to deeply comprehend the way a student is understanding or making sense of a mathematical concept and basing teaching to build from that conceptualization towards a correct mathematical understanding.

For students with learning difficulties, Lindenskov and Lindhardt (2020) indicated teachers in their study held a commonly accepted view that low attaining students required training and task repetition. These teachers also noted, however, “the low motivation of students who are vulnerable partly due to the monotonous and tedious tasks they were offered” (p. 65). After engagement with the research project, these teachers subsequently noted: “that the early use of calculators may prevent students’ low calculating skills to slow down processes towards conceptual understanding” (p. 66). This is an important example because it moves beyond a focus on number facts and arithmetic, characteristics used by much of the research literature to identify mathematics learning disabilities and difficulties. The calculator is used to move beyond those early aspects of arithmetic to a much more productive mathematical focus on understanding.

With respect to learned difficulties, overcoming detrimental thinking patterns acquired in the school years can be exceedingly difficult to change (Dowker et al., 2016). Some approaches have been trialed based on techniques used in psychology to treat other forms of anxiety. From a mathematics education perspective, this is an area of low attainment where the intervention should surely be to prevent anxiety from developing in the first place.

In this section of the chapter, a brief review of research understandings of types of student difficulties with learning mathematics has been given. It is now time to turn attention to the research investigating work that teachers might undertake to account for these student attributes in lesson planning and teaching.

3 Impact of Students’ Attributes on Their Learning Outcomes

Where do teachers need to take account of these diverse attributes? In the Medley (1987)/Manizade et al. (2023) framework, we are looking at the impact of Type G (student attributes) on the relationship between Types B (learning activities) and A (learning outcomes). It is not possible to teach separate lessons to each student in the class, nor desirable, and yet the current focus on learning trajectories and more traditional approaches of teaching from where the student is at might suggest that is required. The implicit assumption is that mathematics is inherently hierarchical and rarely questioned (Forgasz & Cheeseman, 2015) and therefore, learners follow the same path, though perhaps at different rates. There is an alternative view, that there are many paths to mathematical achievement.

If we were to make the assumption, or perhaps working hypothesis, that alternative pathways to mathematics attainment are possible, we could then consider some alternative approaches to curriculum design and the planning of mathematics learning activities by teachers. In this section, three will be considered: Universal Design for Learning, the use of digital solutions, and year level adjusted curriculum.

3.1 Universal Design for Learning

Universal Design (UD) was first used in architecture where it was suggested that new buildings and spaces could be made accessible by their design from the beginning. The underpinning idea required consideration of access for individuals with disabilities, and all others, as a key principle in the design phase of architectural planning. In brief, UD involves the design of products and environments to be useable by all people to the greatest extent possible without the need for adaptation or specialized design. The impact of UD in public building design is obvious, once noticed. The frustration of lack of access or the expense of adding facilities to provide access for an individual after a building is completed is avoided. In countries where UD is written into building design codes, we come to expect that there will be ramps or lifts, easily accessible light switches and power points, good lighting and accessible toilet facilities. If we need them, they are there, if we do not, they do not impede our use.

The idea of UD was expanded into other areas and by the late 1990s, it was applied to education. Meyers and Rose, researchers at the Center for Applied Special Technology (CAST) developed an application of UD to learning situations and coined the phrase Universal Design for Learning (UDL) (CAST, 2020). In an analogy to universal building design, UDL emphasizes meeting as many learning support needs as possible in the one lesson plan. The key feature of the designed curricula is the promotion of access, participation, and progress in general education for all learners. UDL becomes a way of thinking: planning (presage) always to provide multiple ways of presenting information, engaging with content, and demonstrating accomplishment. Diversity is expected, planned for, and valued for adding richness and alternative ways of thinking about the topic.

The use of rich problem solving tasks, an established practice in mathematics pedagogy (Chan & Clarke, 2017), is an example of how teachers might engage and challenge all learners in the one classroom with the one task (Lindenskov & Lindhardt, 2020; Sullivan, 2017). There are many sources of tasks of this nature and in the collection of Downton et al., (2006) they also provide work samples from students demonstrating different ranges of performance and accomplishment on each task. For example, Mason et al., (2010, p. 184) offers the following task: “What numbers have an odd number of divisors?” In order to plan to use this in an inclusive classroom, a teacher would consider what students would need to know to be able to make a start. What “enabling prompts” (Sullivan et al., 2006) would be needed? Understanding the question is likely to be needed, including definitions of keywords, such as “divisor”. Ways to find divisors (factors) of numbers would be needed. Demonstrating an example with blocks would be one way. For example, a teacher might show how to find the factors of 12 by taking 12 blocks and arranging them in rectangles. The side lengths are the factors. A teacher would also need to prepare ways to extend the problem for learners who have solved the original task. Mason et al. suggest “Is there a number with exactly 13 divisors?” as one option.

Just as universal building design will not meet the needs of some users with very specific needs, some learners require very specific adjustments. These adjustments can be added into the planning stage when those requirements are known. For example, one learner with Down syndrome in a senior secondary mathematics class, who was involved in the research project discussed in the later section (see Faragher, et al., 2019, for details), required step-by-step instructions for using his graphics calculator. His teacher prepared these adjustments as part of her lesson planning for specific topics. Any additional adjustment she made, she also provided for other students. In the case of the graphics calculator instructions, she made and laminated two copies—one for the student with the individual plan, and the other was placed on the table at the front of the class for use by any other student (and many did).

3.2 Digital Solutions

It is undeniable that technological advances have made astonishing possibilities for adjustments in mathematics classrooms. They have also fundamentally changed the nature of numeracy. Numeracy is the use of mathematics in life contexts and therefore, how we engage with these contexts and the technology available for our use, changes our numeracy needs. Implications for learners with intellectual disabilities have been discussed previously (Faragher, 2019). When we consider the impact of Type B activities (Student Mathematics Learning Activities) on learning outcomes, it can be argued that there is a fundamental change needed here: in what students are required to do, and what they need to be taught for numeracy development.

Digital solutions also affect how we know what students are able to do and understand as a result of learning. It is possible to go far beyond the time-honored techniques of tests and examinations to gather evidence of students’ mathematics learning. For students who find writing difficult, we can record them demonstrating techniques or presenting their work to peers. For students who have limited expressive language, we can observe them making choices or undertaking adjusted tasks. Video records can capture the ‘aha’ moments. In a recent research study discussed in Sect. 5.2 below (see Faragher et al., 2019, for background), such a moment was captured on video. The researcher and teacher were working in a secondary mathematics classroom with a student with Down syndrome. The student had limited expressive language. He was working on trigonometry with other students in the class, including his friend without a disability who was helping him learn to distinguish right-angled triangles from other triangles. The moment when the student reached for a right-angled triangle from a collection of possibilities, rather than testing at random, was observed by the researcher and the teacher, and captured on the video recording. In this way, the video recording is an example of a digital solution where a record of learning is made that can be analyzed and used to confirm learning. Of course, a camera has to be on at the time and except in the context of research studies, it is unlikely to be the case in routine classroom activities. However, the use of video can be used strategically. For example, in the concluding phase of a lesson, where a teacher wishes to gather evidence of learning, particularly from a student with communication limitations, a video clip could be taken of the student demonstrating the performance of a task.

Individual learner characteristics (Type G) affect the types of activities they are given (Type B) that allow them to demonstrate their learning (Type A). Digital options allow many more valid approaches to the assessment of learning with the likelihood of uncovering Type A outcomes that may never have been imagined. A study of the use of technology for formative assessment of mathematics was undertaken by Dalby and Swan (2019). They considered “the potential for iPad technology to facilitate and enhance formative assessment processes by contributing to the construction of richer and more efficient processes, that bring benefits to student learning” (p. 835). In the research, six lessons were co-designed with teachers and researchers, and these were then trialled in two secondary schools in the UK. The research used “a cyclical process of design, testing, feedback, reflection and redesign” (p. 836). Data analysis of the process of formative assessment used coding and categorization from which five categories emerged. The analysis indicated the potential for the use of technology for assessing mathematics using existing pedagogies while cautioning that “the greatest challenge for teachers in using technology in the classroom is not the technology but an understanding of the process by which it can enhance student learning.” (p. 843). More research into the use of digital solutions for assessing learning, particularly of students with intellectual disabilities, is needed to understand its full potential.

Sometimes, the promise of digital solutions is promoted as the panacea for improving the learning of low attaining students, and particularly those with disabilities. It is clear that while these tools do hold great promise, they are not sufficient in themselves and the other variables around teacher classroom practice need greater exploration. In this chapter, the possibilities the digital context affords to teachers are recognized in conjunction with other aspects of their work.

3.3 Adding Adjustments to Year Level Curriculum

The learning theorist Bruner argued that it is possible to teach any topic to a school aged child in an intellectually honest manner (Bruner, 1960, 1977). Over the years since then, examples from mathematics have emerged where students have indeed been taught seemingly more sophisticated mathematics than their years or curriculum attainment would suggest would be possible. Intriguing examples have emerged from Italy, a country that has not had a special education system for more than 50 years and so has had opportunity to explore possibilities of curriculum innovations. In a paper by Monari Martinez and Bennetti (2011), we see examples of students with significant intellectual disabilities being taught and achieving learning outcomes in areas of mathematics such as algebra and coordinate geometry. Perhaps the most astonishing is a student who learned to use the distance formula to find the distance between two points and then graphed these on centimetre graph paper. Subsequently, the student came to understand measuring with a ruler as she learned the ruler could be used to obtain the same answer as she had already calculated.

Learning how to measure with a ruler through co-ordinate geometry is beyond intriguing to be completely counter-intuitive. Replication studies are needed to determine if the specifics of that particular study can be repeated. Further research with similar results from other areas of mathematics is already emerging, however. Studies from the United States also indicate what is possible (Browder et al., 2012; Creech-Galloway & Collins, 2013). In the United States, the “No Child Left Behind” legislation encouraged the development of teaching approaches to support the requirement that all students would be assessed on the curriculum aligned with their grade level (Browder & Spooner, 2014).

Known variously as “age-appropriate”, “grade-aligned” or “year-level” curriculum, teachers have devised innovative approaches to learning design (Browder & Spooner, 2014). I choose to use the terminology of year-level adjusted curriculum (YLAC) because some students are older or younger than their class peers due to a number of possible factors including transfer across school districts, ill-health, and delayed entry. The purpose of adjusting the year-level curriculum is to begin with the curriculum being planned for the class and then meet specific learning needs by planning adjustments.

In the YLAC approach, teachers begin with the curriculum for the year level they are teaching; that is, they start with the lesson as they intend to plan for their assigned class. This approach is discussed in more depth elsewhere (Faragher, 2017; Faragher et al., 2019), and outlined here. Using principles of UDL or other planning methods, teachers plan multiple approaches for each aspect of their lesson, ensuring the learning support needs of as many students as possible are provided for in the standard plan. This means that enabling prompts to assist learners to enter a learning task are provided as well as extending prompts to ensure all learners, and particularly gifted and talented students, are challenged in the lesson (Sullivan et al., 2006). Similarly, provisions for students with language, social-emotional, physical, and sensory needs are planned.

Once the lesson has been planned, teachers then consider specific additional adjustments that may be required by some learners. In research studies exploring the practices of teachers who were including students with Down syndrome in regular primary and secondary mathematics classes (Faragher & Clarke, 2020; Faragher et al., 2019), teachers considered each stage of the lesson and thought about where the student might face barriers. At this point, teachers would look for ways to work around the barriers. These situations arose where students’ impairments were having an impact on their work, such as difficulties hand-writing or using the layout of a calculator. On occasion, the barriers were to do with intellectual disability, though these were mostly attended to in the general plan.

In these three approaches to lesson planning (UDL, digital solutions, and YLAC), the impact of the characteristics of learners (Type G), is clear. Diverse classrooms bring a diversity of Type G variables that directly affect the work of teachers. In the decades since inclusive education has become policy around the world, the nature of teachers’ work has fundamentally changed as well. This work, and the impact of Type G variables on learner outcomes, needs much greater exploration. In the following section, a more detailed analysis of examples from two recent studies into YLAC provide an illustration of the interaction between Type G factors and Type B (student learning activities) and Type A (learning outcomes). Through this analysis, an overt investigation of the interaction of these variables can be considered.

4 Learning Year Level Curriculum

At the turn of this century, a significant research project in Australia was undertaken to track the early numeracy development of children (Clarke et al., 2002). The project developed task-based interviews for teachers to use with their students to track their mathematical development. In a subsequent study, the interview was adapted to explore the mathematical development of young children with Down syndrome. While the interview had been used with children enrolled in special schools (Clarke & Faragher, 2004), to our knowledge, it had not been used with children with Down syndrome. Down syndrome is a genetic condition leading to varying degrees of intellectual disability. Difficulties with number have been documented in research literature for decades (and often incorrectly generalised to difficulties with mathematics in general). Other areas of mathematics attainment have rarely been studied until recent times (Faragher & Clarke, 2014).

In our research project (Clarke & Faragher, 2014; Faragher & Clarke, 2014), we became aware that some children seemed to be making greater progress than others and that there appeared to be a teaching effect. We wished to know more about the classroom environments where these children were developing their early mathematics knowledge.

4.1 Learning Year Level Mathematics Curriculum in Primary Schools

So began a study of learning year-level mathematics curriculum in primary classrooms by children with Down syndrome. We were focused on the work of teachers, rather than the students themselves as we observed their teaching practice over one school year. In our project, professional learning workshops were interspersed with classroom lesson observations and interviews with teachers about their work. As reported in a recent paper (Faragher & Clarke, 2020), there were indeed practices that teachers adopted, in response to the learning characteristics of their students (Type G), that had an impact on the learning outcomes of students with Down syndrome (Type A) within inclusive classrooms. In that study, it was clear that effective teachers expected all students to think mathematically and were focused on ways to engage their students with Down syndrome in cognitively challenging mathematics. They made judgments in lessons about when to withhold from telling a student the answer, and instead, encouraged them to persist. Teachers also had a clear focus on the mathematics of the year-level and they communicated that focus to teaching assistants (adult helpers without teaching qualifications) assigned for the lesson. The mathematical focus of the lesson for the student with Down syndrome was the same as for the rest of the class and teachers made adjustments to enable that to occur. Consideration of the learning needs of the student occurred at the planning and lesson implementation stages. A key finding was that “the provision of reasonable adjustments in mathematics is highly skilled work exemplifying high-quality mathematics teaching. This involves knowledge of the learner, the mathematics and how to teach it” (Faragher & Clarke, 2020, p. 141). In the context of education research variables, here is the interplay between Type G variables (individual student characteristics, abilities, and personal qualities) and resulting learning outcomes (Type A). The provision of reasonable adjustments to allow students with disabilities to access the curriculum is a requirement in law in many countries, and specified in the UNCRPD. Determining and planning reasonable adjustments requires a teacher to carefully consider the learning characteristics of students at all stages of the presage, process, product model of classroom learning.

In our work with primary school teachers, it was evident that good teaching and the right support had an important impact on the mathematics learning outcomes of their students with Down syndrome. In the Medley/Manizade et al. model (2023), we can see the interaction of research types at play here and these will be discussed in the implications section below.

4.2 Learning Year Level Secondary Mathematics

The primary mathematics research led naturally to consider possibilities in secondary mathematics. In particular, a parent had learned of early findings of the primary project and was keen to have her school consider implications for her son (called “Brian” in the project) in year nine. At the time, he was struggling with single digit addition and was being given worksheets with simple examples, such as 5 + 3, written vertically. Brian was acutely aware that this work was childish and what is worse, he found it difficult. His teacher asked for advice and following a short conversation, planned an adapted worksheet on the topic being taught to his class—linear functions. A clever aspect of the planned adjustment was that Brian ended up doing lots of single digit addition, but now in the context of algebra where he was substituting for variables. He now enjoyed doing this work. As a result of this initial success, the school became instigators of a broader research project that involved the teaching teams of five students with Down syndrome in three Australian states over two years.

Background to the research study, including the methodology and methods involved have been discussed previously (Faragher et al., 2019). In this chapter, the mathematics learning of three of the students is studied through the analytical lens of the Medley/Manizade et al. (2023) framework. In analyzing the case studies, the focus will be on how individual student characteristics (Type G) mediate the connection between student mathematics learning activities (Type B) and the resulting learning outcomes (Type A).

In the sections below, three of the students will be introduced—Brian, Jay, and Mary. I have used pseudonyms for each. First, I present a brief description of their learning context before moving to give a short overview of key aspects of the different variable types.

4.2.1 Brian

Brian has Down syndrome and attended a mainstream Catholic boys’ school. At the time of the study, Brian was in his final two years of secondary school, studying Prevocational Mathematics. This is a subject designed for students who require mathematics beyond school in areas such as employment and trades. It is not designed for students intending to study at university.

Brian’s teacher had a genuine expectation that Brian could be successful at learning the mathematics content of his course. She focused her planning on considering barriers Brian might face and then working out what adjustments might be needed to assist him. In an interview, when asked about additional planning load, she agreed there was extra work involved but she discounted this as a problem. Her point was that while she was spending time making resources and undertaking task analysis for various procedures, this was an investment in future work because “there will always be students of mine who need this support”. She made a particular point of making two copies of any resource—one for Brian, and one for the rest of the class. Students in one observation lesson were seen to access the instruction guide for calculating means, medians, and modes.

Learner characteristics affected the teacher’s planning of assessment activities that would be supported by the teacher aide. She felt that a page of exercises would be daunting and so she took the required exercises and reprinted them in a more engaging format. The advantage of electronic texts was clear here, as the reformatting was relatively straightforward. She often used color coding so that Brian would be able to have a visual cue to the type of task involved. She had developed this strategy through her experience of working with Brian over time.

Most of the assessment for this Prevocational course was undertaken by assignments involving rich tasks that students completed during class time. One example was a task where students were to design a car park. Another was to investigate various data representation approaches in the quantity control of matches. Each required mathematics techniques from different branches of mathematics, in these cases, geometry and statistics. Planning the work for Brian involved breaking the assessment activities into smaller sections, considering the mathematics required and then undertaking a task analysis to break the learning into small steps. A further consideration was the work to be undertaken by the teaching assistant. At the start of each lesson, the teacher would spend a short time with the teacher aide (not necessarily the same person each lesson) explaining the mathematics and indicating what she required Brian to do. It was not her expectation that the teacher aide would be solely responsible for teaching Brian.

During class, Brian worked on the assigned tasks, supported by his teacher aide who assisted with understanding the task instructions. The teacher would move around the room assisting students as they needed. She would also routinely move to check on Brian’s progress. In the observation lessons, the interaction between the teacher, the teacher aide and Brian was noted. On one occasion, Brian needed assistance with a particular part of the carpark assignment. He was incorrectly using his calculator to find the perimeter. When his teacher approached, she sat down in the chair beside Brian to assist. The teacher aide listened briefly then moved away to assist other students. This seamless interaction was a way the two adults supported each other and the learning in the classroom.

Type G Individual Student Characteristics, Abilities, and Personal Qualities

Brian was eager to learn. This was apparent throughout the observations. More specifically, he was eager to learn mathematics that was being taught to the other students in the class. His enthusiasm was described in his exclamation “I just love it” which exemplified the finding on affect reported in a previous paper (Faragher et al., 2019). At the final observation visit, the class was in the last few weeks of secondary school and their assessments were largely complete. The other students were keen to leave the secondary mathematics lessons behind and talk about post-school parties. Not Brian! His teacher explained that a visit would be productive because she was still preparing mathematics lessons for Brian (and any other student who wanted—though there were no other takers!).

Mathematically, Brian continued to have difficulty with arithmetic and used a calculator to do any calculations required. Most lessons he was supported by a teacher assistant.

Type B Student Mathematics Learning Activities

The mathematics learning activities that Brian was engaged in during observed lessons were significant to inclusive practice and the resulting learning outcomes. Brian in his early years in secondary school indicated the desire to study mathematics that was like his peers’ program. Success with adjusted activities in the junior school led to the expectation of participation in and the possibility of success with learning year-level adjusted mathematics in the senior school. In the senior school, exit assessment is required. The assessment tasks developed by the mathematics department were designed to meet the requirements of the state syllabus. In preparing adjustments for these tasks, his teacher had to present work with the right level of challenge. This was not always easy to judge and the teacher talked in her interviews about the need to make further adjustments sometimes during a lesson. She relied on the teacher assistant and her interactions with Brian to discern when the level of challenge of the tasks was too little or too much.

Type A Student Mathematics Learning Outcomes

Brian’s learning outcomes fell into two types—mathematical and non-mathematical. As Brian was studying a senior subject, his work was assessed to state standards. He received a passing grade for Prevocational Mathematics with the culmination being the award of the Queensland Certificate of Education. Brian became adept at the use of mathematical equipment, including graphics calculators and spreadsheets, and indeed, he required these to remove the calculation load enabling him to engage in the mathematical thinking and processes of the subject.

Beyond the achievement of mathematics learning outcomes, other learning outcomes were evident. Brian enjoyed his engagement in the senior mathematics classes and spoke with pride of his work. His teacher reported that his learning behaviour had improved in other subject areas in the school that were not part of the research study. This transfer is reminiscent of findings in the area of quality of life, a framework for understanding disability. Researchers in that field have found that interventions aimed at improving quality of life in one domain have led to unintended benefits to other domains of life (Brown et al., 1989).

Impact of Type G on B and A

Brian’s learning characteristics could not be ignored by his teacher and had a significant impact on her work. The success Brian experienced in lessons (as evidenced through his obvious enjoyment in lessons, engagement with tasks, and through wanting to keep learning at the end of the school year) and in his exit assessment on completion of the course, are success measures for the activities planned by Brian’s teacher. These activities were carefully constructed based on his teacher’s deep knowledge of him as a learner. She explicitly considered his learning characteristics to develop activities that would build his mathematical understanding.

An intriguing, and rarely considered side benefit in the PPP analysis of mathematics education research, is the impact on the learning activities offered to the other students in the class. Because the teacher had to take account of Brian’s significant learning support needs, she provided activities that were supportive of the learning needs of other struggling students in the class.

4.2.2 Jay

Like all students in the study, Jay has Down syndrome and an intellectual disability. In the first year of the study, Jay was in his second year of secondary school, attending a mainstream, co-educational Catholic school. The school has a tradition of educating students from diverse backgrounds, with a variety of learning needs and accomplishments. Jay’s class included students from a range of nationalities, many recently arrived in the country.

His teacher had many years’ experience and Jay’s class was the bottom stream of mathematics for the year level (students were assigned to classes based on achievement, with four classes—one for high, two for moderate and one for low achievement). The bottom stream class still had the expectations of meeting the year level curriculum learning outcomes. In preparing for Jay’s class, his teacher, made few adjustments. This was a bottom stream class and Jay was able to undertake activities along with other students. The teacher used whole-class teaching of mathematics techniques with worksheets to allow students to practise.

Assessment of the junior secondary mathematics involved tests and assignments. Jay undertook the same tests as the other students in his class, which were designed for the bottom stream class. In the second year of the project, Jay was observed working on an assignment where he had the same task as others in the year level and was required to undertake exploratory data analysis of various data sets, including constructing back to back Stem and Leaf plots. He was observed using his laptop to complete the questions that were not modified.

Each observed lesson there was at least one teacher aide assisting the teacher and on occasions an additional cultural liaison assistant. Jay required little support from the teacher aides. For example, in one observed lesson, his teacher aide, who used a wheel chair, positioned his chair beside but a little behind Jay. From time to time he drew Jay’s attention away from his worksheet, to the teacher giving instructions at the whiteboard at the front of the class. The teacher aide assisted students nearby when they required support.

Following the practice on the worksheet tasks, the teacher returned the focus to the whiteboard, calling for responses from students. During this time, he asked direct questions of Jay, based on what he had noticed Jay was able to do on his sheet.

One observed lesson was taught by a casual teacher, replacing the teacher on sick leave. This relief teacher assumed Jay would need easier work than the other students and had prepared a simpler level worksheet on the topic ratio, the same topic as the rest of the class. When the worksheet phase of the lesson was commenced, the teacher immediately gave Jay the easier worksheet. I asked the teacher if Jay might also have the worksheet being given to the rest of the class. She immediately agreed and gave the second sheet to Jay who was initially concerned. He was unsure what sheet he should do and his desire to complete work required reassurance from his teacher that he only needed to complete one of the sheets. Jay completed the unmodified worksheet without error.

Type G Individual Student Characteristics, Abilities, and Personal Qualities

Although in year 8, Jay had been assessed by the school as being at a year 3 level in mathematics. Jay is a serious student, dedicated to his work. He is strongly motivated to complete assigned tasks. In observed lessons, he was never seen to be off-task. Amid exuberant adolescents, he sat quietly working away on his mathematics. For each of the observed lessons, Jay was prepared for his class, fastidious in his attention to having the right equipment, including his laptop, textbook, and notebooks. It was perhaps remarkable to note that Jay was arguably the most dedicated student in his class. Some of the adolescent behaviours expected in bottom stream classes, including disengagement, off-task distractions, and lack of motivation were exhibited by other students, but not Jay. In every observed lesson, Jay diligently completed the tasks assigned to the class, mostly without adjustments. An observable aspect of Jay’s work was his attention to the steps in a process. His dedication and emphasis on task completion were attributes that supported his work in this class, where his enjoyment was evident in following the steps in mathematical procedures and exercises that were required to be successful in that class.

These Type G variables are quite different from those usually reported for students with intellectual disabilities. Those reports, such as in research and professional literature more commonly focus on learning deficits. School documentation that indicated year 3 level in mathematics further emphasize deficit and when considering Type B variables, have a direct impact on tasks offered to students.

Type B Student Mathematics Learning Activities

By being included in the general mathematics class, Jay experienced the same learning activities as his classmates in the whole class instruction phase. In the phases that focused on consolidation and practice, usually through worksheets or computer based exercises, differentiation of learning activities occurred. Being the bottom stream class of the year level, the level of mathematical challenge had already been reduced for the class and Jay did not indicate he needed further adjustment during the observed lessons. However, as was noted earlier, on at least one occasion, the learning activity he was initially provided was made simpler than it needed to be.

Type A Student Mathematics Learning Outcomes

For Jay, the learning outcomes were measured through the use of worksheets and assignment tasks. As part of the data collection for the project, photographs of these completed sheets were taken. One lesson culminated in the completion of exercises on index notation where he completed each exercise without error. The lesson observation and the learning artefacts can only provide evidence of the completion of procedures—following the steps of the process. We can know little of Jay’s understanding of the structures of mathematics underpinning these steps. However, it must also be noted that such evidence was not obtained from other members of the class, either. All were asked to demonstrate that they could complete the assigned exercises and the requirement was met if the steps were completed without error.

Impact of Type G on B and A

The Type G variables—the learner characteristics—that Jay brought to his mathematics lessons might reasonably be presumed to offer expectations of mathematical success. Those Type A outcomes were initially at risk, though, through presumptions of more pervasive mathematics learning difficulties than actually existed. The relief teacher did not know the student and made the assumption, perhaps based on school documentation, that he would need a simpler sheet. She automatically gave the easier worksheet to Jay, without checking, and without offering the sheet to any other learner. This indicates the danger of offering learning adjustments before a student demonstrates the adjustments are required.

High expectations and presumed competence are needed to counteract the pervasive deficit discourse about learners with intellectual disabilities.

4.2.3 Mary

Mary lives in a small town in a farming district. She attends her local school and at the time of the project was in her first year of secondary school. Mary seemed to enjoy her time at school and liked interacting playfully with her teacher and teacher assistants. In the first observation, she was being taught alongside two other students with significant, but very different learning support needs arising from other disabilities than Down syndrome. The class had 30 students and was taught by the teacher with two assistants working with the students with disabilities to the side of the class. In the first year of the project, Mary’s teacher prepared different activities for Mary from the main lesson she planned for the class. This approach changed as the research progressed. At the first observation, Mary was being taught in a segregated approach and over the two years, she gradually became more included in the general class activities. This mirrored Mary’s developing social inclusion (Koller et al., 2018). For example, her teacher told of her walking to school each day, with a mobile phone should she need assistance when originally she had been driven to school. Similarly, on the first observation day, Mary had been having lunch on her own in the school library. On later visits, she spent lunchtime in the school playground.

In this section, the teacher’s changing inclusive practice will be presented and then, using the Medley/Manizade et al. (2023) model, these data will be analyzed to suggest how teacher reflection on the impact of Type G variables (individual student characteristics, abilities, and personal qualities) can lead to profound changes in their inclusive mathematics teaching practice.

The first lesson observed was from a year 7 statistics unit where students were learning about Stem and Leaf plots. The lesson was adjusted for Mary. She collected data about screen time usage, as the other students did, however, instead of using the data to construct a Stem and Leaf plot, she was given another task of putting numbers in order. In the post-lesson conversation, the researchers and the teacher discussed the possibility of teaching Mary to construct a Stem and Leaf plot. After the next lesson, the teacher sent the team a photo of the student’s work sample where she had completed a Stem and Leaf plot. This led the teacher to reflect on the structure of her teaching and in the following lessons, she adopted inclusive teaching practice, planning the one lesson with adjustments for Mary based on her learning characteristics.

In the next year, the now year 8 class was learning about adding and subtracting positive and negative integers. The teacher found a lesson from a website that focused on rich tasks for gifted and talented learners. In a pre-lesson conversation, she discussed how she had thought about how she would engage Mary in the context of the lesson. The task was a game that was to be played in pairs where each player had a hot air balloon with hot air and sandbags affecting the height. She found a video clip of hot air balloons to introduce the lesson to ensure that Mary would understand the context of the game. Because the lesson involved a game, she chose the student who she would assign to play with Mary. The activity game cards were prepared for each group in the class and were made by the teacher before the lesson.

At the start of the lesson, the video was played and Mary moved her chair so she could be directly in front of the screen. When the class broke into groups to play the game, the teacher assistant sat with Mary and her classmate and helped them to learn the rules of the game. Once the students were settled to the task, the teacher aide gradually withdrew. The teacher then asked the assistant to complete some administrative work on the computer at the teacher’s desk. The students continued to play the game without further teacher assistance.

Type G Individual Student Characteristics, Abilities, and Personal Qualities

Mary’s learner characteristics had a significant impact on the planning that her teacher undertook. In addition, she also considered the learner characteristics of the other students in the class, specifically when she selected a student to partner with Mary in the mathematics game. She chose a student who liked working with Mary and who would not over support her.

Type B Student Mathematics Learning Activities

Mary had the opportunity to learn year-level mathematics through engaging in learning activities with her class peers. Her teacher had embraced the idea of an inclusive year-level curriculum and developed her planning whereby she no longer prepared separate lessons for learners with intellectual disabilities. Instead, she made adjustments to ensure Mary could engage with lessons. Mary, in turn, responded by expressing her enjoyment of working in the mathematics lessons. She also enjoyed working with her classmates.

Type A Student Mathematics Learning Outcomes

Achievement of learning outcomes was indicated by completed tasks. Lesson artefacts were collected, such as photographs of completed activities. Other indications of Mary’s learning were also obtained by observing her responses in the lessons. In the lesson on operations with integers, Mary quickly learned the rules of the game. The winner was the balloon that reached the top of the vertical axis first. Each turn of the cards involved turning over a positive or negative sign and then a positive or negative number. Mary loved to win and when she turned her cards and saw that she had to move the balloon down, she refused to move the balloon. It was obvious that not only did she understand how to operate with integers, but also, she could do so in her head. Furthermore, she insisted on playing until she won, gaining much practice in the process.

Impact of Student Characteristics on Learning Activities and Outcomes (Type G on B and A)

As for Jay, Type G variables initially led to restrictions in the mathematics offered to Mary. There was an assumption that her intellectual disability would require different activities for the lesson (Type B), leading to detrimental impacts on the learning outcomes (Type A). What also was evident was how quickly the teacher was able to make profound changes to her practice through reflection on the lesson and engagement in a conversation with researchers about the lesson. Here is an indication of the role of researchers in a teacher’s professional ecosystem. Teachers do not work in isolation and having a colleague to support reflection and stimulate professional growth has clear benefits for students’ learning outcomes.

The teacher already had the required pedagogical skills and practices in her repertoire. Her initial approach appeared to be governed by what she believed was a necessary response to the Type G learning support needs of her student. As her knowledge of the student grew, and encouragement to try adjusting the year level curriculum had an impact on the learning outcomes (she could see what the student could achieve) she fundamentally changed her teaching approach to a genuinely inclusive mathematics classroom.

A striking development was the use the teacher made of rich tasks from a collection of resources to challenge gifted and talented learners. She saw value in the use of these tasks in meeting the diverse learning needs of her students.

4.3 Learner Characteristics, the Mathematics They Engage in and the Learning Outcomes They Achieve

In times past (and not that long ago), students with learning disabilities were rarely given the opportunity to learn mathematics at their year-level. Exploration of examples where teachers have adjusted the year-level curriculum for their students at primary and secondary level are instructive for considering how learning characteristics attenuate relationships between mathematics learning activities (Type B) and the resultant learning outcomes (Type A). A focus on students who struggle with learning mathematics allows us to examine what might be possible for all learners.

What is clear is that these learner attributes in no way predetermine the learning outcomes. Learners need the opportunity to engage with year level mathematics, adjusted to be at the level of productive challenge (Gilmore & Cuskelly, 2014). Before and during lessons, teachers make adjustments in dynamic ways where they bring their understanding of mathematics, the learning outcomes intended, and their knowledge of the learner to the decisions they make. Dynamic judgments about the level of challenge and required support allowed teachers to respond where initial over-support was being made to students’ learning activities. Over-support has the potential harm of reducing the mathematics learning outcomes, and it was important that teachers responded and corrected this at the point when it occurred.

In all the lessons observed in both the primary and secondary projects reported earlier in this section of the chapter, the students were engaged in learning mathematics for their year level. In contradiction to what is commonly portrayed or anticipated by guides to teaching students with mathematics learning difficulties, the students in our project were rarely off task, when the work was the same topic as their peers. Indeed, as we saw with both Jay and Brian, these boys were at times more focussed than their peers without disabilities. The impact on learners, particularly those in the secondary years, was pronounced with benefits not only to their mathematics learning outcomes but also to their self-concept as learners of mathematics and learners in general.

4.4 Implications from These Studies

In the examples from YLAC mathematics research projects, we see students with significant individual characteristics and abilities likely to have an impact on their mathematics learning. These attributes are outside the control of the teacher and yet as has been seen, teacher actions can ameliorate student attributes and lead to productive learning outcomes. Rather than these attributes determining the learning outcomes, a case can be made that while student attributes might affect the work of teachers and the learners themselves, mathematics learning outcomes at the year level are indeed possible.

Teachers in our studies underpinned their work with the expectation that their students could be successful at year-level mathematics. If there were barriers, they worked to find ways around through adjustments to learning materials or approaches to the topics. A common feature was that small suggestions regarding possibilities for changing approaches to be more inclusive were taken up by the teachers in creative and reflective ways, leading to new approaches in their classrooms or different responses from students.

These studies of the practices of effective inclusive mathematics teachers indicate the dynamic nature of teaching and responding to the learning of students due to the characteristics they bring with them to the learning process. Teachers’ planning is affected before, and during, the lesson as teachers think and problem solve and make decisions in the moment. It is clear that individual student characteristics affect learning outcomes but in surprising ways. Creative, reflexive teachers respond to these characteristics by changing, adjusting, and developing the mathematics learning activities offered not just to the students with learning disabilities and difficulties, but to all learners in the class. Learning outcomes are likely to be improved for all. As a way out of low attainment, this is a critical aspect of mathematics education and an imperative of mathematics education research to further explore its possibilities.

This research also calls into further question the persistent, detrimental practice of streaming based on previous attainment. By planning for all learners, anticipating diversity and providing learning adjustments as required to year level curriculum, teachers have an alternative to separating students into coarse class groupings with known detrimental impacts.

5 Implications

Learning characteristics of students with intellectual disabilities cannot be ignored—they have too great an impact on teachers’ work. When embracing the teaching of diverse learners, teachers commence planning with learners’ characteristics central to their thinking.

Medley (1987) defined Type G variables as “individual student characteristics, abilities and personal qualities which determine outcomes of any specific learning experience”. Recent research, as outlined in this chapter, raises questions about the determinism of outcomes. Similar questions also emerge from a greater understanding of the impact of disability on learning, a field that continues to evolve and with greater opportunities for learning afforded by inclusive practice around the world, continues to surprise teachers and researchers alike. Perhaps a more accurate definition for Type G variables might be those learner variables that “affect outcomes”, rather than determine outcomes. These learner variables do not determine outcomes, at least in a predictable way. Indeed, by continuing to explore ways to make learning accessible for students who are variable in their characteristics, abilities, and qualities (Type G), there is the capacity for teachers to be surprised (Russo et al, 2020). This surprise is likely to affect preactive teacher activities (Type D) with flow on implications right through to student learning outcomes (Type A).

In addition to modification to the definition of Type G variables, there are implications for the placement of these variables in the framework. Currently, Type G variables are shown to act between Type B and Type A. New ways of thinking about learning for students with learning difficulties and disabilities, challenges us to think about new ways of representing where Type G variables affect the PPP process. Teachers do not plan a lesson and teach it to the class only to find that learner characteristics are affecting how that information is received. When working with diverse learners, teachers factor in learner characteristics much earlier on, and throughout the PPP process.

In this chapter exploring the impact of learner variables on student outcomes, a focus has been taken on learners who struggle with mathematics. These students present a considerable challenge to teachers as they work to improve learning outcomes. In most countries around the world, it has only been in recent times that students with significant mathematics learning difficulties and disabilities have been included in mainstream classrooms with the expectation that they can achieve learning outcomes. A deep understanding of the practice of teachers in these contexts is still emerging (Tan et al., 2019). It would seem that there is much more to learn about Type G variables and their effect on student learning outcomes and especially so in inclusive mathematics classrooms. Furthermore, there is much to learn about how teachers grow professionally through reflecting on the interplay between learner variables, student activities, and the resulting learning outcomes.