Mathematics Curriculum Review: Learning Trajectories to Inform Curriculum Design and Pedagogy

Report Submitted to the Ontario Ministry of EducationAugust 2, 2013

University of Ottawa Research TeamResearchers:

Dr. Christine Suurtamm, Dr. Martha Koch

Research Assistants: Yasmine Abtahi, Ph.D. student

Jill Lazarus, Ph.D. studentJimmy Pai, M. A. student

Charlotte Pashley, M. A. studentLuisa Thuswaldner, M. Ed. studentRebecca Walton, M. Ed. student

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Table of Contents

1. Introduction and overview of the report (1/2 page)....................................................12. Overview/introduction to learning trajectories (2 - 3 pages)...................................13. Research in learning trajectories (12 pages)[work on consistency, length on Thursday]............................................................................................................................................. 4

3.1 Introduction.................................................................................................................................................43.2 Fosnot and Dolk’s Landscape of Learning......................................................................................43.3 Lawson’s Primary Math Trajectories in Inquiry Classrooms – [chris will revise by Thursday].............................................................................................................................................................63.4 Cross, Woods, & Schweingruber’s Teaching–Learning Paths................................................73.5 Clements and Sarama: Learning trajectories in mathematics for young children.....123.6 van den Heuvel-Panhuizen: Learning trajectories for calculation with whole numbers [Martha will revise for Thursday].......................................................................................133.7 Confrey: Hexagon Map of Learning Trajectories......................................................................16

4. Learning trajectory related resources................................................................................184.1 Mathematics in Context (max 1 page) [Chris – by Thursday].............................................184.2 Lawson’s resource (max 1 page) [chris – by Thursday]........................................................184.3 Building Blocks (max 1 page) [Martha – by Thursday].........................................................184.4 Turnoncc (max 1 page) [chris – by Thursday]...........................................................................184.5 PRIME (max 1 page)..............................................................................................................................184.6 First Steps in Mathematics (FSiM)..................................................................................................204.7 Observations [both - write notes by Thursday]........................................................................21

5. Learning trajectories in curriculum design (12 pages) – [Chris & Martha make notes to bring to discussion on Tuesday]................................................................................21

5.1 Overview.....................................................................................................................................................215.2 Quebec (2 pages).....................................................................................................................................22Comments..........................................................................................................................................................245.3 Australia......................................................................................................................................................245.4 United States (2 pages)........................................................................................................................285.5 Netherlands – [Martha – see what there is to bring to the table on Thursday]..........31

6. Summary (2 pages) – [make notes to bring to the table on Thursday]....................31cautions.............................................................................................................................................................. 32The ways in which trajectories might influence curriculum......................................................32Issues................................................................................................................................................................... 32Learning trajectories, instruction and assessment (influencing teacher practice)..........32Ways to support the curriculum..............................................................................................................32

7. Recommendations about how to proceed (2 pages) [bring notes to the table on Thursday]........................................................................................................................................... 33

Suggested conceptual map as to how to further integrate learning trajectories into the curriculum.........................................................................................................................................................33Suggested ways of supporting implementation...............................................................................33Criteria for assessing research and learning trajectory resources..........................................33Recognition of limitations of learning trajectory work.................................................................33Further work....................................................................................................................................................33

Preliminary Reference List – organized by topic.................................................................34Learning trajectories (in general) and their role in curriculum design.................................34

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Relevant examples of learning trajectories........................................................................................34Ways jurisdictions have approached curriculum design using learning trajectories......35Additional references related to recommendations (to come).................................................37

Appendices........................................................................................................................................ 37

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Mathematics Curriculum Review: Learning Trajectories to Inform Curriculum Design and Pedagogy

1. Introduction and overview of the report (1/2 page) This report provides a discussion of research in the development of elementary mathematics curriculum in an age appropriate manner. In so doing, we are presenting state-of-the-art approaches that inform both curriculum development and ways of teaching elementary mathematics that suit the age and development of elementary students at various stages. The report specifically addresses the intense research that is currently occurring in learning trajectories in elementary mathematics and the ways in which these learning trajectories are informing curriculum design as well as teaching approaches, instructional tasks, and assessment.

The report begins with an introduction to learning trajectories that helps to define learning trajectories and begins the discussion about their various aspects and purposes. It then moves into a discussion of current research in learning trajectories and provides samples of learning trajectory research that appears most salient with respect to this curriculum review. In order to discuss the implication on pedagogy we discuss a variety of teacher and professional development resources that have been created based on learning trajectories and present ways that pedagogy could be informed by learning trajectories. We will then present examples of ways in which this learning trajectory research is informing curriculum design in four jurisdictions: Quebec, Australia, the United States, and the Netherlands.

We conclude the report with a summary of the ideas presented in the report, specifically as they relate to the Ontario mathematics curriculum. The final section makes specific recommendations with regard to how developmentally appropriate approaches to the teaching of mathematics can be further integrated into the Ontario curriculum. A full reference list provides links to resources that informed each section of the report. Appendices provide examples of trajectories and other documentation that are referred to in the report.

2. Overview/introduction to learning trajectories (2 - 3 pages) Recently, the concept of learning trajectories has gained attention as a way to focus research on how students learn particular concepts. The focus on learning trajectories has also been influencing curriculum and assessment design, teacher resources, and, in the United States, funding priorities for research (Empson, 2011). The term ‘learning trajectory’ is often used interchangeably with ‘learning progression’ and the ideas behind these two terms are quite similar. Usually, the term ‘learning trajectory’ is used in research in mathematics education literature while ‘learning progression’ is used in more practitioner-based resources but this is not always the case (Daro et al., 2011). Other jurisdictions may use other terminology such as the use of the term developmental continuum in Australia.

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Learning progressions or trajectories helps to describe potential pathways by which the learning or understanding of key concepts may be learned. Learning trajectories are hypotheses, supported by empirical research, about the stages of thinking, knowledge, or skills that students are likely to go through as they develop an understanding of mathematical ideas. These trajectories address both the possible order and nature of points in the growth of students’ understanding. While the word trajectory gives us the image of a linear pathway, this is not necessarily the case. In fact, the words ‘network’ or ‘web’ might be more appropriate to emphasize the fact that learning is not linear but consists of building a web of interconnected ideas to develop an understanding of a particular mathematical concept. As such, learning trajectories take on a variety of representational forms such as webs, networks, pathways, connected hexagons, or a landscape. It should also be clearly noted that not all children go through the same sequence to connect ideas and develop an understanding of the concept, thus a network with a variety of pathways more closely resembles the diversity of learning and learners.

Martin Simon first used the term ‘learning trajectory’ in 1995. In his use of the term Simon proposed the phrase ‘hypothetical learning trajectory’ to reflect the notion that the path that learning takes may vary based on students’ experiences and context. Thus, the learning trajectory is a predication of the path by which learning might proceed that is informed by research, an understanding of mathematics, and experience. Simon provides the following analogy to explain the fluidity of this path:

The choice of the word "trajectory" is meant to refer to a path, the nature of which can perhaps be clarified by the following analogy. Consider that you have decided to sail around the world in order to visit places that you have never seen. One does not do this randomly (e.g., go to France, then Hawaii, then England), but neither is there one set itinerary to follow. Rather, you acquire as much knowledge relevant to planning your journey as possible. You then make a plan. You may initially plan the whole trip or only part of it. You set out sailing according to your plan. However, you must constantly adjust because of the conditions that you encounter. You continue to acquire knowledge about sailing, about the current conditions, and about the areas that you wish to visit. You change your plans with respect to the order of your destinations. You modify the length and nature of your visits as a result of interactions with people along the way. You add destinations that prior to your trip were unknown to you. The path that you travel is your "trajectory." The path that you anticipate at any point in time is your "hypothetical trajectory” (Simon, 1995, p. 136 – 137).

The notion of learning trajectories is influenced by Piaget’s developmental theories and Vygotsky’s concept of the zone of proximal development. It reflects Piagetian theory in the sense that the trajectories represent developmental stages. The use of a learning trajectory to determine the next steps for students relates to the zone of proximal development in so far as a teacher is really examining where a student is and what exploration might take him further. The original discussion by Simon (1995) was framed within a constructivist perspective that argued that if teachers are providing tasks to students that help to develop their mathematical thinking and construct mathematical meaning then the teacher must have some sense of where the student currently is and the potential path(s) that the students might take to develop and understanding of the concept(s) at hand. It is a recognition that teaching needs to take readiness into account in providing learning opportunities.

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Curriculum and learning trajectories

Learning trajectories can help to inform curriculum design so that concepts are sequenced in appropriate ways. In many cases, curricula have been previously linked to scope and sequence paths that could be thought of as being very similar to learning trajectories. Much like scope and sequence learning trajectories recognize that learning builds over time and what has come before and what will come after needs to be taken into account in curricular and instructional design. However, the scope and sequences were often developed based on what educators thought were a logical sequence of learning goals based on their understanding of mathematics and their own experience. The difference between scope and sequences and learning trajectories is that the learning trajectories focus on how students learn particular concepts and are generally empirically supported hypotheses about the thinking, knowledge, and skills that students experience as they develop their understanding of a related set of concepts. Thus, the trajectory focuses on how students learn concepts and they are supported by research that has examined that learning. The extent of the research that is connected to trajectories varies and this will be discussed later in the report as we examine some of the trajectories in more detail.

Pedagogy and learning trajectories

Knowledge of learning trajectories can also help to inform teachers so that they can select and sequence a series of tasks that help students to 1) connect new ideas to their prior knowledge and 2) move their learning forward. Understanding students’ mathematical thinking and the possible path it might take helps teachers in understanding students’ thinking and in choosing or creating particular tasks. Recent attention has been paid to the use of learning trajectories to inform teaching. For instance, learning trajectory-based instruction, or teaching that is grounded in empirical evidence as to how students learn, has gained attention as an area of research in mathematics education (Szatjn, Confrey, Wilson, & Edgington, 2012).

Assessment and learning trajectories

Formative assessment has been shown to have positive effects of student achievement (Black & Wiliam, 1989). Formative assessment involves a teacher examining student thinking during teaching through observing classroom activity; analyzing written work; conferencing; questioning, listening, and responding; working with tasks specifically designed for formative assessment purposes. These actions, however, are not enough as we also know that formative assessment works best when teachers have a sense of how students’ thinking progresses, what problems they might face, and how they can help to move their thinking forward. Learning trajectories hold promise in helping to inform teachers about where students are in their development of important mathematical ideas as they provide conceptions of students’ thinking, understanding and skills and how they develop. Thus, a learning trajectory can assist a teacher in determining where a student’s thinking is and where they might go next.

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3. Research in learning trajectories (12 pages)[work on consistency, length on Thursday]

3.1 Introductionjust notes: Why we are discussing these. These are occurring in the literature quite often or have influenced curriculum and/or teaching in a variety of ways. Influencing resources – dreambox, It should be noted that learning trajectories are context sensitive. How children develop concepts depends on the learning environment and the kinds of experiences they have had. Learning does not happen in a vacuum. A trajectory that is developed through empirical research with students in an inquiry-oriented classroom may look very different from research with students in a more teacher-directed ‘traditional’ classroom. Furthermore, current research that investigates learning trajectories takes into account the current mathematics classroom context that has moved away from merely using learning procedures and processes in a teacher-directed way to opening up spaces of inquiry in the classroom. These situations often provide students with space to explore ideas in a variety of ways and reinforces the notion that learning trajectories are not linear but are better represented by a network of ideas and students understanding can develop through diverse pathways. In other words, the sequence of developing understanding might be different for different children. However, recognizing the connections between key concepts, representations, and strategies helps the teacher to recognize the learning and to provide appropriate environments and tasks for exploration of key concepts.

3.2 Fosnot and Dolk’s Landscape of Learning

What they look like/view of learning trajectory

Catherine Fosnot is a US-based mathematics education researcher and Maarten Dolk is a Dutch mathematics education researcher who works with the Freudenthal Institute. Fosnot and Dolk use a landscape metaphor to present their ideas about how children develop an understanding of mathematical concepts. Their landscape representation is a strong reaction to views of learning progressions as linear trajectories. Rather, they recognize that children can take multiple paths in developing their understanding of concepts. In other words, not all children move along the same path.

Fosnot and Dolk have developed learning landscapes for a variety of concepts such as fractions, decimals and percents (2007), and multiplication and division (2001). Each learning landscape consists of 3 types of elements as landmarks: big ideas (depicted as ovals), strategies (depicted as rectangles), and models (depicted as triangles). Hence, for a particular learning landscape they have identified the big ideas that children must engage with, the types of strategies that children might use, and the types of models that are meaningful ways to represent the mathematical ideas. However, the paths that children take through the landscape and interacting with the landmarks are varied and not prescribed. In fact, Fosnot and Dolk have the view that children may revisit some of these landmarks and their paths may be retraced or crossed over as they tackle new mathematical problems and reinforce their understanding of key concepts.

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What mathematical areas/concepts/big ideas and grades do they cover

Fosnot and Dolk have developed learning landscapes for a variety of mathematical ideas. These include:

Number sense, addition and subtraction (Grades K – 3) (Fosnot & Dolk, 2001a) Multiplication and division (Grades 3 – 5) (Fosnot & Dolk, 2001b) Fractions, decimals, percents (Grades 4 – 6) (Fosnot & Dolk, Algebra (Grades K – 8) (Fosnot & Jacob, 2010)

Appendix XX provides an example of a learning landscape for multiplication and division. One can see that several models, represented by triangles, that students might use in multiplication might be multiplication as area, repeated addition, or arrays. It also demonstrates strategies that students might use such as skip counting, or grouping. We can also see that some of the big ideas that students explore are unitizing, the distribution and commutative properties, and place value.

Development and research to support the learning landscape

Fosnot began developing the learning landscapes by engaging teachers in the analysis of video clips of children and many samples of children's work as part of professional development activities. This work was collected in schools as children worked on the investigations that she and her team were designing and trying. This analysis of student work was augmented by the research of others. Initially they relied heavily on a group of Dutch researchers from the Freudenthal Institute who had done a substantial body of research on what they called "progressive schematization." The Dutch researchers had analyzed the historical development of mathematical ideas as a starting point and then they designed linear hypothetical trajectories and used them to test the progression of children's ideas and strategies. Based on this work, Fosnot initially had placed the ideas and strategies she was analyzing on a line as well and called them “learning lines” as the Dutch researchers had. Sequences of investigations for teachers to use were designed to support this progressive development. However, as Fosnot’s thinking and research into human development continued she was influenced by a variety of theorists and she questioned the use of linear models to represent learning. She experimented with a variety of models that looked more like networks of relations, and what emerged was the landscape metaphor.

Using this metaphor, Fosnot and Dolk decided to try and depict what we had seen with students and had been discussing with professional development participants as landscapes. They clustered ideas, models, and strategies together that they had seen emerge in many classrooms over many years and collapsed their learning lines and hypothesized how the ideas and strategies might be related. They also encouraged the view that multiple pathways could occur depending on culture and context, as well as different student experience. Their own work on analyzing student samples of videos of students working was augmented by empirical research of others that provided models of possible pathways. They did not try to describe something as a precursor to another idea unless there was empirical research to support that. In fact, they avoided prescriptive paths as they saw any specific path as context or student dependent.

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How are they used?

Fosnot and Dolk’s learning landscapes have been taken up in a variety of ways. One of the primary uses of the learning landscapes has been in professional development with teachers. This was initially how the landscapes were developed and teachers continue to use them to examine their own students’ mathematical thinking and to help to determine appropriate instructional tasks. A professional development resource was developed by Fosnot and colleagues called the Context for Learning Mathematics Series, published by Heinemann. This resource includes the landscapes for learning as well as instructional tasks.

These materials have been used in a variety of settings and in some jurisdictions has been adopted as a mathematics learning program (e.g. Washington State). Within Canada, Fosnot’s work has been used in professional development work with teachers in several provinces, including Ontario and the Central Algebra Regional Consortium.

3.3 Lawson’s Primary Math Trajectories in Inquiry Classrooms – [chris will revise by Thursday]

Alex Lawson is an Ontario-based mathematics education researcher who is interested in elementary students’ development of number sense and has done extensive work on documenting learning trajectories through her empirical research. In order to document these trajectories she followed a cohort of 60 children from the beginning of Grade 1 through to the end of Grade 5. These children are all in the same school and the teachers involved in the study are using inquiry-oriented teaching methods that are consistent with current thinking in mathematics education as well as with the Ontario mathematics curriculum. The teachers allowed students to develop their own methods of solving problems and did not teach the traditional algorithms until Grade 4 and at that point they were often alternative algorithms, based on students’ ideas. Teachers implemented these ideas to varying degrees depending upon their own comfort level. The teachers used Fosnot and Dolk's landscape of learning and context units to support the children (Fosnot & Dolk, 2001a, 2001b) and to note individual student’s placement on the landscape. Noting the context of the study is important as many studies do not take into account the teaching methods and in many previous studies the context was a more traditional teacher-directed setting. Lawson was interested in the ways that children’s understanding develops in a setting more oriented towards current thinking in mathematics education.

Lawson interviewed students twice yearly using the same items (with slight changes towards more complex as they progressed through the grades) to track the change in their thinking. Each interview was approximately 45 – 60 minutes long. Her findings included:

First, the children who came in well behind (25% could not either count reliably or add 5 + 7) were always behind but most, except a few who went into special remedial classes early, did not fall further behind. The fact that teachers could work with children's ideas and track them on the landscape knowing that they were progressing but had not started in the same spot was very useful for teachers. They relied as much on the landscape as the curriculum guidelines so

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that children who were not meeting grade level expectations were still being adequately tracked and one could see their progress.

Children have different paths once they move beyond counting on. Some paths are more beneficial than others. For example, in either addition and subtraction or multiplication and division doubling and either adding or subtracting 1 is a good stepping stone (or doubling and regrouping in multiplication & division) but it should not be the ‘go-to’ method in later grades as it is limited. Splitting for subtraction is also highly problematic and the teachers began working hard at the end of Grade 2 and into Grade 3 using the number line and taking leaps of 10 from any number. For example, with 41 - 23 if children split they often got into trouble with the single digit calculation. If they left the subtrahend whole and subtracted 20 and then 3 that worked. If they added up on the number line 23 to get to 41 that worked. (need to unpack this)

Certain strategies that are very important will only come up if they are promoted in the classroom. She notes that this would not be through direct instruction but through providing an appropriate context, problem, game, or mini-lesson.

Lawson’s study found that it is really important to have an overall landscape or continuum to have a sense of where things are headed and to know what to do for individual children. For example, if they are counting on to add what would be one of the next strategies you would want to promote? This does not mean to teach it directly but instead, to create a context that is likely to elicit it. She notes that this continuum is non-linear as there are multiple pathways and connections that a student.

This detailed work helps to understand students’ development of number and the ways in which teachers might use a landscape of learning (similar to a learning trajectory) to guide their instructional decisions such as providing appropriate tasks. Lawson’s work is soon to be published (by Pearson) and will include resources for teachers.

3.4 Cross, Woods, & Schweingruber’s Teaching–Learning Paths

Introduction

The teaching-learning paths described by Cross, Woods and Schweingruber (2009) emerged from their work on a committee funded by the US federal government and several other foundations to develop pre-school mathematics learning objectives and “evidence-based” suggestions for achieving those objectives (p.11). Cross et al. note several NCTM documents published between 1989 and 2006, that provide guidance and, in some cases, suggest a learning path for early mathematics. However, they saw a need for greater specificity with regard to the mathematics that should be learned at each grade level than what is provided in these publications (p.122).

Before presenting their teaching-learning paths, Cross et al. provide an extensive review of empirical studies of the mathematics knowledge of infants, toddlers and preschool children. Based on this review, they conclude that young children have a more sophisticated understanding of concepts such as number, space and measurement than was previously realized. Knowing that very young children have already begun to develop these concepts and will continue to do so if provided with appropriate

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opportunities and experiences was the starting place for the teaching-learning paths they present.

What they look like/view of learning trajectory

Cross et al. define a “teaching-learning path” as a “sequence of milestones” that consists of “the significant steps in learning in a particular topic; each new step in the learning path builds on the earlier steps.” (p. 121) The paths are based on “research that shows that young children generally follow particular paths when learning number-relations-operations and geometry-measurement” (p.121). Each teaching-learning path includes four steps organized by age and grade (i.e. Step 1 - from 2-3 yrs age, Step 2- to age 4/preK, Step 3 – age 5/K, Step 4 – age 6/Gr. 1). While the steps are described as “convenient groupings”, the authors note that they see development as continuous (p. 127). They acknowledge that there is considerable variability in when children can do particular mathematics tasks but they also point out that research suggests that much of this variability is due to the different opportunities children have had to learn, practice and receive feedback about their learning. Further, they note that variability can also be found within a given child who may use one strategy during one part of a session and another strategy a few moments later.

While each step in their teaching-learning paths assumes that the child has had sufficient experience with the previous step to learn the content, Cross et al. note that “many children can still learn the content at a level without having fully mastered the content at the lower level if they have sufficient time to learn and practice” (p. 131). Cross et al. maintain that most children are capable of learning the foundational mathematics content specified in the teaching-learning paths and they also acknowledge that children may exceed the specified content associated with any given step (p.131). In describing how the teaching-learning paths might be used in an instructional context, Cross et al. emphasize that while tasks in the path are sequenced, “teachers need to adapt for particular students’ conceptual development rather than rigidly follow a prescribed curriculum” (p.274). Thus, there is an explicit recognition of the need for flexibility in the use of these teaching-learning paths.

In addition, Cross et al. cite various studies demonstrating that some aspects of a teaching-learning path may be culturally specific (p. 129). Some examples include the role of the grammatical structure of a child’s first language in the development of understandings of cardinality and different conventions for counting on fingers across various cultures (p.132). Cross et al. describe how approaches to solutions may also differ culturally such as the “make-a-ten method” that is more commonly taught in East Asia than in the US (p. 165). At the same time, Cross et al. demonstrate some areas that are common across different languages such as the consistent order in which children acquire language for spatial relationships beginning with in, on, under, up and down terms, while language for proximity (e.g. beside etc.) and frames of reference (e.g. in front of etc.) are learned much later (p.181).

In each of the teaching-learning paths Cross et al. have developed, the steps for each component are explained both mathematically and developmentally. In addition to describing what children can typically do with regard to content at each of the four steps, the teaching-learning paths include descriptions of the need for practice, the importance of effort, common errors and misconceptions, and suggested instructional approaches to

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support each path. For instance, there is a full-page description of common counting errors in the Number, Relations and Operations teaching-learning path and a page and a half describing suggested approaches to teaching the concept of length consistent with their teaching-learning path.

With regard to assessment, the teaching-learning paths do not include specific assessment tasks for each content area as is the case for some learning trajectories. However, Cross et al. discuss the importance of formative assessment strategies in effective mathematics teaching and provide detailed descriptions of how the use of tasks, observation, and flexible interviews are necessary to get a sense of where children are on a given teaching-learning path (p.259). They cite a collection of task-based interviews developed by Clarke et al. (2001) based on a developmental trajectory for children beginning at age five (p. 263). The tasks are intended both as a professional development resource and as a formative assessment tool for teachers [could be worth mentioning in the section on assessment and LTs] A key message is that teachers must understand the development of mathematical thinking to interpret a child’s response to a task, during an observation or in an interview. They argue that familiarity with the teaching-learning path would help teachers better understand the development of mathematical thinking. They note that greater links between assessment and instruction are needed in early years mathematics and that the teaching-learning paths they have developed are helpful because “they keep the teacher situated in an organized set of goals with directionality both for individual children and for the class” (p.265).

What mathematical areas/concepts/big ideas and grades do they cover

Cross et al. suggest that giving equal time to all 5 math strands in pre-K to K spreads experiences too thinly across topics and may preclude a deep exploration of the foundational mathematics children need (p.124). They advocate for focusing on topics that are both mathematically foundational and ”consistent with children’s ways of thinking, developing, and learning when they have experience with mathematics ideas”. Previous NCTM documents focused on number and operations as well as geometry and measurement for pre-K to K students and these are also the focus of the teaching-learning paths developed by Cross et al. The two teaching-learning paths they have developed are: 1. Number, Relations, and Operations (ages 2 & 3 to Grade 1)

Number Coreo Cardinalityo Number Word Listo Counting 1-to-1 Correspondenceso Written Number Symbols

Relations Core: More Than/Less Than Operations Core: Addition/Subtraction

2. Geometry, Spatial Thinking and Measurement (ages 2 & 3 to kindergarten) Geometry and Spatial Thinking

o Shapeo Spatial Structureo Spatial Thinking

Measurement9

o Lengtho Area and Spatial Structuringo Volume

There are several core areas within each teaching-learning path, each of which is described in detail across the four steps. For instance, the “Counting 1-to-1 Correspondences” component of the Number Core is described at each step. For each component of each core area, the mathematical concepts are explained, and the research on the development of the concept in young children is reviewed. Summary boxes are provided for each core and component to indicate the “major types of new learning for each age/grade” (p. 146). [see example in Appendix n]. Connections across the mathematical ideas in the core areas within a teaching-learning path are also described. For instance, Cross et al. note that typically it is not until Step 3 (Kindergarten) that children integrate the four components of number. In addition, various important conceptual shifts are flagged as important. For instance, “being able to see ten ones as one ten is a crucial step on the learning path” (p. 146). Most aspects of the Number teaching-learning path are presented as a sequence of narrative descriptions whereas a table approach is used for the Geometry, Spatial Thinking and Measurement teaching-learning path.

What research supports them

Considerable research for these teaching-learning paths is presented with well over 200 articles cited. Much of the research cited is US research but some has been done in East Asia, Israel and various other countries. Many of the references are to empirical studies published in peer-reviewed journals as well as papers and proceedings from peer-reviewed mathematics education conferences.

For some topics a range of evidence is presented. For instance, for the five counting principles posited by Gelman & Gallistel (1978) several subsequent pieces of research that provide differing views of when each principle is understood by children are presented. In another example, Cross et al. note that children in the US tend to learn the pattern of the decade word followed by a number (1-9) before learning the order of the decade words. This claim is supported by references to four studies conducted in the 1980s (p. 143). For the Operations core, Cross et al. note there is considerable research from various countries that describe the three levels through which children progress with regard to numerical solution methods for addition and subtraction and 11 sources are cited as the basis for including these levels as part of the teaching-learning path. The authors also identify some specific areas within each teaching-learning path where more research is needed. For example, they note “there is not sufficient evidence to indicate the best time for teachers to start writing addition and subtraction problems in equations or for students to do so” (p. 162) and “there is insufficient evidence on the effects . . . of including such topics as congruence, similarity, transformations, and angles in curricula and teaching at specific age levels (p. 211). Cross et al. note that research on the development of geometry, spatial thinking and measurement is far less developed than it is for number but nonetheless they maintain that there is adequate research to provide guidelines for teaching-learning paths (p. 175). Cross et al. cite research demonstrating that most children in the US experience very limited geometry instruction, are less prepared than children of other countries and

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score at or near the bottom on geometry tasks in international studies (p.192). Their teaching-learning path for these topics is presented as a series of goals in a table that includes up to Step 3 (5 yrs/K). The table is a synthesis of a series of developmental studies that the authors claim constitute an empirically verified developmental progression for each concept they have included (p. 194). For some concepts, additional studies have demonstrated that interventions at the pre-school level result in significant gains in this ability – thus lending support for the teaching-learning progression.

How are they used?

We did not locate any specific evidence of the use of the teaching-learning paths developed by Cross et al. by any particular jurisdictions. We can describe how the authors suggest that the teaching-learning paths be used. Indeed, they maintain that these paths are “essential for high-quality teaching” (p. 269) and describe how they might be used to guide instructional practice, revise curriculum/standards, evaluate and revise commercial mathematics education resources, and as the basis for professional development programs in mathematics. Overall, they identify the use of teaching-learning paths as a key indicator of an effective pre-school mathematics program (p. 274). They also provide evidence to suggest that use of the teaching-learning paths they developed can reduce the gap in educational outcomes between children form low-SES backgrounds and those from higher SES families (p.344). [details of some of their suggestions are given below but some could be removed Cross et al. suggest areas that need to be revisited in some state standards or better ways of approaching some teaching and learning activities in light of their teaching-learning paths. For instance:

“many states require that kindergarten children understand some aspects of money, but sometimes they have goals that are not sensible for this age group” (p. 147).

research showing that early childhood curricula often introduce squares as distinct from rectangles making it more difficult for children to understand that squares are a special kind of rectangle (p.193). They recommend an approach where squares are included as examples of rectangles.

recent research has shed new light on Piaget’s theory of measurement (p.199) suggesting that beginning with non-standard units may not be as important as was once thought and may even be detrimental

Thus, one use of these teaching-learning paths may be to suggest specific ways to implement curriculum/standards and or to make changes to those curricula.

Another potential use of these teaching-learning paths is to suggest areas where additional research is needed.

The authors address the role of teaching-learning paths in standards. They note that many states have based their standards for early years mathematics on various NCTM documents such as Curriculum Focal Points (2006) but suggest that the greater detail of these teaching-learning paths may be more helpful. They include an analysis of the proportion of state standards in various states that focus on the content areas of their teaching-learning paths. They observe considerable variation across states and note

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that in general greater emphasis is placed on some areas of number than on other areas of number and even less emphasis given geometry and measurement. They recommend that states revise their standards to reflect the teaching-learning paths they have developed. Cross et al. also recommend that commercially published programs base their materials on these teaching-learning paths.

Cross et al. recommend that professional development activities be developed that help teachers recognize and better understand the developmental progressions in the teaching-learning paths (p. 249). They advocate for professional development that incorporates teaching-learning paths by matching a particular developmental progression with appropriate activities and they cite several studies where professional development focused on developmental trajectories increased teachers’ professional knowledge as well as students’ motivation and achievement.

3.5 Clements and Sarama: Learning trajectories in mathematics for young children

What they look like/view of learning trajectory

Clements and Sarama are well known for their work in the area of mathematics for young children and have been instrumental in developing learning pathways for several concepts in early mathematics. In this development, they believe it is important to 1) establish a clear picture of the big ideas of mathematics and 2) lay down paths for learning to help children develop those big ideas (Sarama & Clements, 2009). They assert that these pathways, or learning trajectories, help teachers to interpret what students are thinking and doing mathematically. This in turn helps to inform the teacher as to the next steps in terms of potential learning experiences. They define learning trajectories as

descriptions of children’s thinking as they learn to achieve specific goals in a mathematical domain, and a related, conjectured route through a set of instructional tasks designed to engender those mental processes or actions hypothesized to move children through a developmental progression of levels of thinking (Sarama & Clements, 2007, p. 17)

Clements and Sarama view children’s thinking and learning as going through a developmental sequence as they engage with big ideas. They see these trajectories as having both theoretical and pedagogical value. They suggest that learning trajectories can facilitate teaching and learning that is developmentally appropriate. and suggest that there needs to be correspondence between instructional experiences and the trajectory of students’ thinking. They also view teachers’ engagement with learning trajectories as outstanding professional development.

For Clements and Sarama, number and operations are seen as the most important area of mathematics learning in the early years (Sarama & Clements, 2009). They acknowledge that there has been a great deal of research on how children develop number sense and operation in the early childhood. Through their review of research literature they have developed hypothetical learning trajectories for the following areas. Number and quantitative thinking

o Recognition of number and subitizing (ages 0 – 8)o Counting (ages 1 – 7)o Comparing, ordering and estimating numbers (ages 0 – 8)

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o Addition and subtraction and counting strategies (ages 1 – 7)o Composing number and multi-digit addition and subtraction (ages 0 – 8)

Geometry and spatial thinkingo Shapes, composition, and decomposition of shapes (ages 0 – 8)o Spatial thinking (ages 0 – 8+)

Geometric measuremento Geometric measurement of length (ages 2 – 8)o Geometric measurement of area (ages 0 – 8)o Geometric measurement of volume (ages 0 – 9)o Geometric measurement of angle and term (ages 2 – 8+)

They present their learning trajectories in a chart form, with columns titled “age”, “developmental progression”, and “actions on objects”. Appendix XX provides a sample of a learning trajectory for addition and subtraction.

What research supports them

Sarama and Clements publication Early Childhood Mathematics Education Research: Learning Trajectories for Young Children (2009) presents a full chapter for each of their learning trajectories. Within these chapters, they outline the research that they drew on as they developed their trajectories. Within these chapters they also outline the ways that the experiences and education influence the choice of strategies and models that they might use for the concepts being explored.

How are they used?

As well as producing their detailed research volume, Clements and Sarama published a book for teachers, Learning and Teaching Early Math: The Learning Trajectories Approach (2009), that describes how children learn mathematics, and teachers can build this knowledge to realize more effective teaching practice. This work has also been incorporated in a commercial product, Building Blocks, marketed as a mathematical program for young children.

3.6 van den Heuvel-Panhuizen: Learning trajectories for calculation with whole numbers [Martha will revise for Thursday]A team of researchers from the Freudenthal Institute at Utrecht University and the Dutch National Institute for Curriculum Development collaborated to produce learning-teaching trajectories for calculation with whole numbers from kindergarten through Grade 6. The aim of the trajectories is “to offer clarity and insight into the educational route followed by primary school students learning mathematics” (p.10). The authors also note that learning-teaching trajectories will eventually be developed for all domains of primary school mathematics in the Netherlands.

What they look like/view of learning trajectory

Their view of learning trajectories draws on three interconnected elements: a learning trajectory that gives an overview of students’ learning process; a teaching trajectory that describes how teaching can most effectively stimulate the learning process; and a subject matter outline that indicates which of the core elements of the mathematics curriculum should be taught (p.13). The authors explicitly state that the trajectories are “not intended as a straightjacket” (p.11) and they acknowledge the complexity of the

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learning process where “students sometimes progress by leaps and bounds and at other times can appear to relapse” (p.13).

“A learning-teaching trajectory should not be seen as a strictly linear, step-by-step regime in which each step is necessarily and inexorably followed by the next. It should be seen as being broader than a single track.” (p.13)

They indicate that the trajectory should encompass the majority of students but will not “accommodate very weak or very able students” (p.11). They describe the trajectory as having a “bandwidth” that allows for individual differences among students in terms of how they read a target. They note that students are able to understand the same subject matter at different levels and can work on the same subject matter even though they are at different stages of their development. Given this, they describe this trajectory as a “communal learning-teaching trajectory” (p.14). The trajectory “sets out important signposts but is not a pre-ordained learning route” (get page number).

The learning-teaching trajectory is a narrative description of mathematics learning for the specific content area that includes a series of Intermediate attainment targets, which are described as “crucial steps that the students take” or “landmarks towards which the teaching can be oriented” (p. 9). They are also described as “benchmarks” and “calibration points” (p.15). They can be used a guidelines for assessment but are described as “more a source of inspiration than as a strict way of monitoring the students’ progress” (p.11). The authors caution that the intermediate attainment targets should not be used as a checklist and cannot “fully reflect the rich repertoire of students’ behaviour at a certain stage of development” (p.16). The levels for a particular attainment target should be seen as conceptual levels or levels of understanding rather than as achievement levels (p.17).

Each intermediate attainment target is presented in a box. Below each target a second box provides a teaching framework that summarizes how teaching can contribute to the target. (see appendix N for example) Each intermediate attainment target is prefaced by a description of the mathematics content as well as several practical examples of students’ behavior, often showing multiple approaches to reaching the target as well as examples of teaching activities related to a given target. There is a strong emphasis on the variety of strategies children may use. The authors emphasize that the learning process and the teaching approaches used do not proceed in the same way in each phase of the trajectory and that domain-related levels of understanding are more appropriate than a universal set of levels across the entire trajectory. They envision different sets of levels for different content areas such as counting or calculation. Thus, the number and nature of levels may change across content areas.

What mathematical areas/concepts/big ideas and grades do they cover?

This volume, edited by van den Heuvel-Panhuizen, presents two learning-teaching trajectories for calculation with whole numbers. The first trajectory covers pre-school to Grade 2 and 3 (ages 4-8) and the second trajectory covers Grades 3 to 6 (ages 8 -12). The learning-teaching trajectory and the intermediate attainment targets within it are described using overlapping grades (K1 & K2, Gr. 1&2, Gr 2&3) because the authors explain that it is difficult to “define unequivocally” targets for one specific year (p.10). Each learning-teaching trajectory gives only an approximate indication of the time

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course; the attainment targets are flexible in both length and breadth. The double grade approach makes explicit that some students will achieve the attainment target in the following year. No intermediate attainment targets are specified for the preschool years.

The description of the learning-teaching trajectory includes separate chapters for each topic or grade level as follows:

Calculation with whole numbers (primary - lower grades)o Kindergarten 1 and 2o Grade 1 and 2o Grade 2 and 3

Calculation with whole numbers (primary –upper grades)o Number and number relationso Mental arithmetico Column calculation and algorithmso Estimationo Calculator

In addition, a chapter explaining the links between the lower grades and upper grades primary learning-teaching trajectories is provided as well as a chapter describing how the learning-teaching trajectories for calculation with whole numbers connect with other topics in the Dutch primary mathematics curriculum such as measurement and geometry. The authors also note that there are important links between the learning-teaching trajectory for whole numbers mathematics and other school subjects though these may only be referred to in passing to keep the length of the volume reasonable.

What research supports them?

Acknowledge the contributions of teachers and schools who participated in the research conducted to develop the learning trajectories. The learning-teaching trajectories were based on classroom investigations and consultation with experts in teaching mathematics to primary grades children, pre-service and in-service teacher education, assessment, and information technology. More detail may be available in the complete volume.

How is it used?

We have seen reference to these learning trajectories in other trajectory research but we do not have specific evidence of their use in specific jurisdictions. However, a range of uses are suggested by the authors as follows.

Aim is to “be a support for teachers alongside mathematics textbook series” (p.7). Described as “a source of inspiration for transcending the textbook” (p. 11) and as a way for teachers to “rise beyond the limitations of the textbooks” (p.18). Teachers who have a good overview of the learning trajectories know when certain ideas or solutions are likely to be generated by students and they are in a better position to guide students’ learning and make instructional decisions. While modules for introducing teachers to the learning-teaching trajectories have been developed, the authors note that the way teachers use the trajectories will vary from school to school and teacher to teacher. That said, the authors suggest that mathematics coordinators can play a key role in the implementation

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of learning-teaching trajectories in schools and that the goal is for the learning-teaching trajectories “to become part of the teachers’ thinking” (p.19).

With regard to the connection to the curriculum, the intermediate attainment targets are described as a further development and supplement to the core goals for mathematics set by the Dutch government including the five general goals for mathematics and the specific core goals for the topic of calculation skills. Both the core goals and the intermediate attainment targets should be achievable for “virtually all students” (p.16). In some cases, the intermediate attainment targets will be the same as the core goals.

Further, the authors recommend using the learning-teaching trajectory as the basis for discussions about mathematics for the whole teaching team in a school. In particular, the trajectories may provide a useful frame of reference for conversations about the stumbling blocks children commonly encounter in particular aspects of mathematics learning. They also suggest that showing parts of the learning trajectories at parents’ evenings along with video clips of their classroom practice would be beneficial. Teachers can use the learning trajectories Work from the Freudenthal Institute and National (Netherlands) Institute for Curriculum Development to produce learning-teaching trajectories for the calculation with whole numbers in K – 6. This work has informed the Dutch curriculum.

3.7 Confrey: Hexagon Map of Learning Trajectories

What they look like/view of learning trajectory

Confrey and colleagues describe a learning trajectory as

a researcher-conjectured, empirically-supported description of the ordered network of constructs a student encounters through instruction (i.e. activities, tasks, tools, forms of interaction and methods of evaluation), in order to move from informal ideas, through successive refinements of representation, articulation, and reflection, towards increasingly complex concepts over time (Confrey et al., 2009, p. 2).

Confrey and colleagues have been doing extensive work in the area of learning trajectories. Much of their work on specific learning trajectories has focused on the concept of equipartitioning. [need more here and references] However, Confrey’s most recent work has been focused on drawing from the work of others to develop, organize and create representations of collections of empirically-based learning trajectories. Specifically, her research team at North Carolina State University has represented 18 learning trajectories. Specifically, this work has been focused on linking the learning trajectory work to the set of mathematical standards adopted by 45 of the 50 U.S. states called the Common Core State Standards.

In this work, each standard is represented by a hexagon and these hexagons are connected together in such a way as to show the progression that students might make in their development of concepts. A website, turnonccmath.net, provides an interactive map of the learning trajectories and common core standards. A screen shot of this mapping can be found in Appendix XX. As shown, the learning trajectories are represented within an inter-connected map. In this way, each outcome has a way of

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connecting in six different ways to other outcomes. This represents a representation of learning trajectories as non-linear and interconnected.

Confrey and colleague’s work provide the following learning trajectories:

1. Counting (K – Grade 2)2. Time and money (K – Grade 3)3. Addition and subtraction (K – Grade 4)4. Elementary data and modeling (K – Grade 5)5. Shapes and angles (K – Grade 6)6. Early equations and expressions (K – Grade 7)7. Length, area, and volume (K – Grade 8)8. Equipartitioning (Grades 1 – 5) 9. Division and multiplication (Grades 2 – 6)10. Fractions (Grades 3 – 5)11. Place value and decimals (K – Grade 8)12. Rational and irrational numbers(Grades 6 – 8)13. Linear and simultaneous functions (grades 6 – 8)14. Integers, number lines, and coordinate planes (Grades 5 – 8)15. Ratio and proportion, and percents (Grades 6 – 8)16. Triangles and transformations (Grades 7 – 8)17. Variation, distribution, and modeling (Grades 6 – 8)18. Chance and probability (Grade 7)

It should be noted that there is great variation in the level of development for these trajectories. It should also be noted that these trajectories were mapped on to these outcomes after the outcomes were determined. The grade level of each of the outcomes may have influenced the levels in the trajectories.

What research supports them

The learning trajectories are based on research evidence in a variety of ways – some more than others. In using the interactive map, by clicking on a learning trajectory, information about the trajectory becomes evident. This information includes a listing of the research that supports the trajectory. Some of the trajectories have a great deal of empirical research to support the trajectory. For instance, the learning trajectory for equipartitioning has a long list of research articles that are specifically focused on examining student learning with respect to equipartitioning. In some other cases, the trajectories are not as well supported by numerous specific empirical studies but rather by a few empirical studies and some more general theoretical or pedagogical discussion articles.

How are they being used?

These learning trajectories have been matched to the K-8 Common Core State Standards for Mathematics that are used in most U.S. states. The work connects research in mathematics learning trajectories to curriculum and also provides a resource to support teachers, teacher educators, professional development providers, and curriculum experts. It also provides directions for researchers through identifying areas that require further research. [more work to be done here]

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4. Learning trajectory related resources

4.1 Mathematics in Context (max 1 page) [Chris – by Thursday]

Description

Research base

How it is being used

4.2 Lawson’s resource (max 1 page) [chris – by Thursday]

Description

Research base

How it is being used

4.3 Building Blocks (max 1 page) [Martha – by Thursday]

Description

Research base

How it is being used

4.4 Turnoncc (max 1 page) [chris – by Thursday]

Description

Research base

How it is being used

4.5 PRIME (max 1 page)

Description

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PRIME, published by Nelson Canada, is described as a developmental program that provides assistance for teachers (Grades K-12) by helping them build their understanding of mathematics, helping them to track student learning using “Developmental Maps”, and providing strategies for student progression through five mathematics strands. The “developmental map” might be seen as somewhat of a learning trajectory. The PRIME resource is accompanied by professional development for teachers, facilitators, and administrators.

According to the information provided on the Nelson website, PRIME offers two to three-day courses for educators each of five mathematics strands (Number and Operations, Patterns and Algebra, Geometry, Data Management and Probability, Measurement). At the elementary level there are currently courses for grades K-8 teachers in Number and Operations, Patterns and Algebra, and Geometry. At the secondary level there are currently courses for grades 7-12 teachers Number and Operations, and Patterns and Algebra.

Research base

Information from the Nelson website suggests that the initial research, conducted by Marian Small and Douglas McDougall, was intended to answering the following two questions: “1. What are the phases of development that students “travel” through while learning key mathematical skills and concepts? and 2. What is the best way to identify a student’s placement in the phases of learning?” (PRIME, 2005, http://www.prime.nelson.com/facilitator/research.html).

Information provided states that data was gathered from more than 12,00 students Canada wide including western Canada, Ontario, Quebec and eastern Canada. This data provided the basis for PRIME as it indicated phases of student learning through key identifiers, which were used to create the Developmental Maps. This information comes from the Nelson website which references the following research document:

Small, M., McDougall, D.E., Ross, J.A., & Ben Jaafar, S. (2006). PRIME Developmental maps: Research study. (pp. 79). Toronto: Thomson Nelson (available with purchase of PRIME)

We were unable to find any peer-reviewed research on this resource or the developmental maps.

How it is being used

The PRIME website provides connections between PRIME strand materials and the curriculum expectations for CAMET (Council of Atlantic Ministers of Education and Training) Curriculum, The Ontario Curriculum, The Quebec Curriculum, and the WNCP (Western and Northern Canadian Protocol) Curriculum – though connections are not available for each strand for every curriculum.

Many school districts are currently using PRIME including (Kawartha Pine Ridge, Ontario; Dufferin-Peel, Ontario; Wolf Creek, Alberta; District 16, New Brunswick, Delta School District, British Colombia etc.). A number of studies have been done on the effectiveness of PRIME in various contexts (we could summarize and reference these).

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Diagnostic tool

4.6 First Steps in Mathematics (FSiM)

Description

First Steps, is a professional development program and series of teacher resource books published by Pearson that is intended to help teachers (Pre-school – Grade 9) diagnose, plan, implement and evaluate students’ mathematics learning. The program, developed in Australia through the collaboration of mathematics education researchers and elementary classroom teachers, includes four components: Diagnostic Maps, Diagnostic Tasks, Key Understandings and Sample Learning Activities. The Diagnostic Maps shows the progression of mathematics learning for students from pre-school to Grade 8. The maps present six phases in the development of mathematical thinking: Emergent, Matching, Quantifying, Partitioning, Factoring and Operating. Diagnostic maps show these six phases for each mathematics strand. Each phase for each strand includes a number of Diagnostic Tasks that are designed to help teachers uncover the critical misconceptions their students may have. First Steps teacher resources are available for four mathematics strands: Number, Measurement, Space (Geometry), and Chance and Data (Analysis and Probability). Facilitator professional development courses using the train-the-trainer model are available for the Number strand.

Research base

Information from the Pearson website indicates that development of First Steps began in 1995 when the Department of Education and Training in Western Australia commissioned Professor Sue Willis to develop professional development resources for K-8 mathematics. Willis began with an extensive literature review of empirical research that resulted in an 60-page bibliography, available from Pearson upon request. The findings from the studies in this literature review were placed on a wall sized continuum chart. Extensive analysis of this body of research demonstrated common misconceptions across countries, key conceptual phases and gaps (cite Yukon paper for development process). The development team combined Information from the literature review with analysis of more than 10,000 student work samples to develop the diagnostic map and tasks, the key understandings and the sample learning activities. An additional study of 2000 student work samples was used to validate the diagnostic tasks. Not clear if any of this was published in peer reviewed journals though some materials are available from Pearson. A Canadian edition of First Steps was created based on a study with 500 Canadian students that was conducted to confirm that the observations from the previous study were relevant in the Canadian context. Copyright seems to be held by Western Australian Minister of Education and Training, Canadian Edition, 2007 (see Yukon paper).

How it is being used

Pearson provides documents to show the links between the Key Understandings in First Steps and the curriculum expectations or standards for some jurisdictions. For instance correlations between Key Understandings and curriculum expectations from the Ontario curriculum for Grade 5 area available (add ref). Similar links are shown for the Western and Northern Canadian Protocol as well as the Common Core State Standards (add a

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ref for this). Some individual school boards in provinces such as Saskatchewan, BC and the Yukon have adopted First Steps as a mathematics program. First Steps has also been used by some school boards as the basis for creating their own mathematics program. For instance, Durham Catholic DSB created a continuum based mathematics (CBM) program complete with diagnostic tasks and tracking charts based on an amalgamation of materials from PRIME and First Steps (add ref). They trained 325 K-8 teachers from 20 elementary schools in the use of their CBM program and documented teachers use of the diagnostic tasks and tracking charts, common student misconceptions, and the impact of the program on student achievement. Similarly, PEI used First Steps as the basis of the Early Numeracy Intervention Program (ENIP) they developed in 2010-2011 (add ref). This program involved 255 students, 25 teachers in 14 schools.

While it will continue to be available in Australia, Pearson is closing the international offices as of July 2013. Seems like materials still available from Dept of Ed in Western Australia??

4.7 Observations [both - write notes by Thursday]

5. Learning trajectories in curriculum design (12 pages) – [Chris & Martha make notes to bring to discussion on Tuesday]

5.1 OverviewThis section will discuss the different ways that learning trajectories have been used in curriculum design as well as in informing instruction. The discussion will include cautions about how to look at learning trajectories as well as what they can and cannot tell us about the progression of student learning. It will also cover, in a general way, how jurisdictions have used learning trajectories to inform curriculum design or to verify certain aspects of their curriculum. This will help to set the stage for the next section that describes this process in detail for particular jurisdictions.

Trajectories take into account both where students are at as well as the desired learning outcome. For each jurisdiction, we will review curriculum and supporting documents to understand the implementation of age appropriate approaches to elementary mathematics, the process used to create developmental trajectories, and the strengths and limitations of those processes. We will also summarize the recommendations of each jurisdiction with regard to how student demonstration of mathematics knowledge and skills should be assessed, evaluated and reported.

this should be a discussion of the choice of jurisdictions, etc. see fullan’s discussion of how well Ontario does. list the other places we looked and why we chose those that we did – focus on work on learning trajectories. Note that higher performance may or may not have any relationship with the use of learning trajectories. Describe comparability as being considered in a variety of ways. Ontario does not have a need to raise scores.

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5.2 Quebec (2 pages)

Introduction

One reason we selected Quebec as a comparison jurisdiction is it’s high performance on recent international mathematics assessments. On the PISA 2009, Quebec’s overall mathematics achievement was the highest among Canadian provinces. Quebec scored statistically significantly higher than Ontario (543 vs. 526) and was the only Canadian province in the high performing group that includes China, Finland, Korea and Singapore (EQAO, 2010). Similarly, on the TIMSS 2011 Grade 4 and Grade 8 assessments Quebec scored statistically significantly higher than Ontario (Gr. 4 533 vs. 518; Gr. 8 532 vs. 512) (Mullis et al., 2012). A second reason we included Quebec as a comparison jurisdiction is that the Ministère de l’Éducation, Loisir et Sport (MELS) released learning progressions for elementary and secondary mathematics in 2009 and 2010 respectively.

In Quebec, the elementary curriculum is known as the Quebec Education Program (QEP). The document, published in 2001, is available in both English and French and includes nine cross-curricular competencies as well as subject-specific competencies for each subject area (MELS, 2001). The QEP includes competencies for the three primary cycles, which are equivalent to grades one through six in Ontario. The mathematics section of the elementary QEP (pp. 139-157) includes three subject-specific competencies: to solve a situational problem related to mathematics, to reason using mathematical concepts and processes, and to communicate by using mathematical language. In addition to these competencies, the QEP lists “essential knowledges” for mathematics (p.150) under six headings: arithmetic, geometry, measurement, statistics, probability and cultural references. An essential knowledge in the QEP is roughly equivalent to a specific curriculum expectation in the Ontario curriculum in terms of the scope of content included in each.

In addition to the QEP, MELS has more recently developed learning progressions for all elementary and secondary subject areas. The learning progressions are intended to help teachers with lesson planning and determining learning outcomes (MELS, 2009; MELS, n.d.). Each learning progression is organized as a series of content tables. The left column of the table lists the essential knowledges, which closely resemble the ones included in the QEP. The right side of the table indicates the progression of learning for each essential knowledge using arrows, stars and shaded boxes. An arrow indicates the grade level at which teachers should begin explicit teaching and learning activities related to the essential knowledge. A star indicates the grade level at which teachers should plan for the majority of students to have completed learning the essential knowledge by the end of the school year. A shaded box indicates grade(s) in which students “reinvest” the essential knowledge (MELS, 2009, p.5). To assist students in reinvesting the essential knowledge, teachers should plan teaching and learning activities that require the student to draw on the essential knowledge that has been learned. A sample page from the elementary mathematics learning progression is included as Appendix n. The mathematics learning progression also includes cognitive and metacognitive strategies to help students achieve the three mathematics competencies.

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Relationship between Curriculum and Learning Trajectories

The learning progressions document was written several years after the QEP was released and is described as a “supplement” and “compliment” to the QEP (MELS, n.d.). The two documents are closely aligned in that the essential knowledges included in each document are quite similar. For instance in the area of arithmetic operations involving numbers, fractions in curriculum there are five essential knowledges for grades 1-6, (MELS, 2001, p.152). One is “multiplying a natural number by a fraction, using objects and diagrams”. If we look in the learning progression document under arithmetic: operations involving numbers, fractions we see 4 essential knowledges (MELS, 2009, p.12) including “multiplies a natural number by a fraction”. However, across the document, several of the essential knowledges in the learning progression are broken down into smaller components and are more detailed than the essential knowledges in the QEP. In addition, the cognitive and metacognitive strategies are added to the learning progressions documents but not included in the QEP. In these senses, the learning progression document is almost an extension or revision of the previous curriculum document. In fact, on the website the learning progression documents for each subject area have been added to the table of contents for the QEP as part of the chapter for each subject area. So we can say that the learning progressions have augmented the curriculum in Quebec.

Research base

Unfortunately, little information is available on the process used to develop the learning progression. MELS acknowledges the participation of school-based educators (ie. teachers and subject specialists) as well as university researchers in the development of the learning progressions (MELS, n.d.), but we were not able to locate any research documents related to the development of the learning progression for mathematics.

Challenges & issues

We were unable to locate anything on the process of developing the learning progressions or any indication of challenges that were faced in designing or implementing the document.

Implications for Assessment

There is a suggestion that where the star is indicates when the teacher should evaluate the essential knowledge to determine if majority of students have completed the essential knowledge but no tools or strategies are provided.

The video introducing learning progressions makes a statement about the need for teachers to follow the Policy on the Evaluation of Learning but there is nothing specific about the learning progressions (MELS, n.d.). Teachers are required to use a new province wide report card for 2011-12 school year and an updated “Framework for the Evaluation of Learning” in reporting. The framework for math is organized by the three competencies. Under each competency are descriptions of the evaluation criteria and hyperlinks to the learning progression but no additional assessment or evaluation strategies, tools or tasks are evident. It is interesting to note that the third competency “to communicate by using mathematical language” is something that students must receive feedback on but it must not be considered when determining students’ marks throughout grades 1 to 6 (MELS, 2011). Thus, the evaluation framework connects to the

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learning progression and both supplement the curriculum. In fact, from the positioning of the documents on the web and the titles on each page it seems that the QEP is being expanded to include all three documents

Professional Resources for Learning Trajectory Use

Did not find any supports for teachers or evidence of professional development for use of learning trajectories.

CommentsWe see Quebec’s learning progressions as essentially a scope and sequence document rather than a learning trajectory. However, it is probably a useful document to teachers as it provides more detail than the original curriculum document. It appears as though they may have been developed to provide more guidance for the essential knowledges given that the curriculum document centers more on the competencies. It is interesting to note that the learning progressions address the essential knowledges but they do not explicitly relate to the three competencies.

We should note that the document provides only one learning progression for all students and is not effective in showing the connections across mathematics strands. Issues around coherence across 3 documents – curriculum, learning progressions, evaluation framework – this must be troublesome for teachers, in particular.

5.3 Australia

Introduction

In terms of international mathematics assessments, Australia’s achievement on the PISA 2009 assessment is slightly lower than Ontario (i.e. Australia - 514; Canada - 527, Ontario – 526) (EQAO, 2010; OECD, 2010). They also score somewhat lower on the TIMSS 2011 Grade 8 assessment (Australia - 505, Ontario - 512) though their achievement on the TIMSS 2011 Grade 4 assessment is very close to that of Ontario (i.e. Australia - 516; Ontario – 518) (Mullis et. al. 2012). Overall, Australia’s performance on international assessments can be described as comparable to Ontario, which is one factor that suggests Australia is a suitable comparison jurisdiction. However, the main reason we chose to include Australia as a comparison jurisdiction is that the states of states of New South Wales (NSW) and Victoria have both developed a type of learning progression or developmental continuum for mathematics in recent years.

New South Wales has a mathematics continuum for grades K through 10 that takes the form of a poster summarizing the key ideas in mathematics for “Number”, “Patterns and Algebra”, “Data”, “Measurement”, “Space and Geometry” as well as a category called “Working Mathematically” (NSW, 2010). The continuum is presented as a table detailing the mathematics expectations within each strand or sub-strand across the grades. The contents of the table are aligned with the NSW curriculum and the poster is intended to be displayed in classrooms and staffrooms so that teachers, students and parents can see “the big picture of mathematics learning” across the elementary and secondary grades (NSW, 2010). The NSW continuum is similar to the mathematics learning progression for Quebec in that it is essentially a scope and sequence document. For this report, we have chosen to focus on the “Mathematics Developmental Continuum P-10”

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for Victoria rather than the NSW continuum because Victoria’s continuum includes elements that go beyond scope and sequence.

Description of Victoria’s Mathematics Developmental Continuum

The Mathematics Developmental Continuum P-10 for Victoria is presented as a website of inter-related links that address five dimensions: “Number”; “Space”; “Measurement, Chance and Data”; “Structure”; and “Working Mathematically” (Victoria Department of Education and Early Childhood Development [VDEECD], n.d.) check ref. The continuum spans grades from “P” the “Preparation” year (age x years) through Year 10 (age y years). For each dimension, the continuum is presented as a two-column table. The first column lists the “Mathematics Standards and Progression Points”, which are aligned with the mathematics curriculum for Victoria. The second column provides “Indicators of Progress” which are points on the learning continuum that highlight the critical understandings students must have in order to progress in their mathematical learning. Indicators of Progress are intended to highlight common misconceptions of students and to support teachers in deepening their understanding of student learning in mathematics by providing “research-based descriptors of achievement” [need to find quote & ref]. The Indicators of Progress give teachers a sense of the progress students should be making and the types of learning and teaching experiences that are appropriate for further progress (cite users guide here). Table x shows the number of Indicators of Progress and the span of grade levels for each dimension in the continuum. Reviewing the table provides a sense of the grain size maybe mention grain size in intro of this developmental continuum. For instance, there are 19 Indicators of Progress associated with “Space” across the span of mathematics instruction from P to Grade 10.

Dimension # Indicators of Progress *Levels CoveredNumber 41 Level 1 (K & Grade 1) to

Level 6 (Grades 9 & 10)Space 19 Level 1 (K & Grade 1) to

Level 6 (Grades 9 & 10)Measurement, Chance and Data

25 Level 1 (K & Grade 1) to Level 6 (Grades 9 & 10)

Structure 16 Level 3 (Grade 3 &4) to Level 6 (Grades 9 & 10)

Working Mathematically 14 Level 1 (K & Grade 1) to Level 6 (Grades 9 & 10)

* each level in the curriculum (VELS) and the continuum includes two grades

For each Indicator of Progress links are provided to “Illustrations”, “Developmental Overviews” and “Teaching Strategies and Activities”. “Illustrations” are suggested focused observations and diagnostic tasks intended to assist the teacher in determining where students are in their mathematics thinking and to help them uncover misconceptions. Developmental Overviews map student progress in key concepts through all levels of learning. There are ten developmental overviews which cover various “big ideas” in mathematics including: “Proportional Reasoning and Multiplicative Thinking”, “Numbers and Operations” and “Methods of Calculation” add ref here. Teaching Strategies and Activities are suggested tasks teachers can use to support students in developing a conceptual understanding of the progression point. We have

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provided a few selected screen shots of the Indicators of Progress, Illustrations, Developmental Overviews, and Teaching Strategies and Activities in Appendix x) .

Another interesting feature of Victoria’s developmental continuum is the Mathematics Online Interview (https://www.eduweb.vic.gov.au/MathematicsOnline), a secure part of the website that can be accessed only by school staff. The Mathematics Online Interview is an assessment tool that can be used by teachers to determine students’ existing mathematical knowledge in relation to progression points. Analysis of the responses provided during a one-to-one interview process are intended to provide teachers with information to use when planning to meet student’s learning needs. While we could not access the interview process itself, the descriptions of the tool suggest that this may be a valuable resource for teachers.

Relationship between Curriculum and Learning Trajectories

Australia has a new national mathematics curriculum as well as a detailed scope and sequence document for mathematics which covers the foundation year to year 10 [ages for these years] (ACARA, 2013). Australia also has a “General Capabilities in the Australian Curriculum” document (2013) which is a document that details seven general capabilities similar to the 21st century learning skills documents that are appearing in many educational jurisdictions. One of the general capabilities in this document is “Numeracy” and there is a learning continua for numeracy included in the document. This learning continua is entirely separate from Victoria’s developmental continuum but it suggests that presenting information using a continua approach is viewed as worthwhile in Australia.

The state of Victoria has recently released their new curriculum document known as AusVELS (VCAA, 2013). AusVELS goes from Foundation to Year 10 and closely parallels the new national curriculum. For each year/grade level, the AusVELS curriculum presents content expectations for each mathematics strand and sub-strand along with achievement standards. Victoria also has a scope and sequence chart that is very similar to the national scope and sequence chart. AusVELS replaced the previous curriculum which was called VELS (Victorian Essential Learning Standards). AusVELS does not explicitly refer to the Mathematics Developmental Continuum P-10 perhaps because the continuum was developed when VELS was still in place (K. Stacey, personal communication, May 21, 2013). In fact, the “Mathematics Standards and Progression Points” in the developmental continuum are aligned with the VELS curriculum rather than the AusVELS document. Thus, the connection between the developmental continuum and AusVELS is a bit indirect. In any case, the developmental continuum is seen more as a teacher resource than as a curriculum document (K. Stacey, personal communication, May 21, 2013).

Research Base

As we reviewed each of the 115 Indicators of Progress in the developmental continuum, we noticed that for some indicators references are provided either at the bottom of the web-page for the indicator or through a “See more about” link. However, following the links and examining the references for each Indicator of Progress revealed that many of the references cite general resources for teaching primary mathematics or identify the source of a specific activity recommended for teaching the progression point. In fact, very few of the references associated with each Indicator of Progress are to empirical,

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peer-reviewed research. Moreover, in the few instances where an Indicator of Progress is supported by peer-reviewed research, there is typically only one paper cited.

We contacted Kaye Stacey, a mathematics education researcher and professor at the University of Melbourne, who was been involved in the development of the developmental continuum to learn more about the research the continuum is based on. She confirmed that some research on Number was gathered in New South Wales and later drawn on by the state of Victoria for that dimension of the continuum. However, the other areas of the continuum reflect the mathematics education researchers’ existing knowledge but the researchers involved did not conduct empirical studies of their own nor do they make explicit connections to studies done by others (K. Stacey, personal communication, May 21, 2013).

Challenges and Issues

One issue using learning trajectory research to inform the curriculum development process for Australia is that the curriculum is written in one or two year stages whereas developmental trajectory research is often more fine-grained than this (Stacy, personal communication, May 21, 2013). This difference in grain size may make it more difficult to bring learning trajectories research into the curriculum realm in any jurisdiction. We also suspect that it was not feasible for the state of Victoria to provide the funding that mathematics education researchers would require to conduct empirical studies as the basis of each Indicator of Progress in the developmental continuum (maybe cite the email?).

Implications for Assessment

In general, the Indicators of Progress in the developmental continuum are intended to be part of the ongoing assessment practice of teachers (cite users guide). In addition, as noted above, the developmental continuum includes two specific tools that are intended to support teachers in their formative assessment activities. The “Illustrations” provided for each Indicator of Progress suggest observations and diagnostic tasks to assist teachers with determining where students are in their mathematics thinking and to help them identify students’ misconceptions. The Mathematics Online Interview provides another tool teachers may use to determine students’ existing mathematical knowledge in relation to an Indicator of Progress. Providing these formative assessment tools within the developmental continuum may help to ensure that assessment activities are integrated with teaching and learning activities (assessment as and for learning) rather than assessment being seen as a separate activity conducted after learning has taken place (assessment of learning).

Professional Resources for Use of the Developmental Continuum

The website where the developmental continuum is made available also includes a “User’s Guide” to provide teachers with information about the design of the continuum as well as suggestions for effective use of the continuum for planning purposes. The User’s Guide includes sections on navigating the continuum and on learning and teaching using the continuum. We did not see any evidence of professional development sessions focused on the developmental continuum though these may be available.

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Comments

Victoria’s developmental continuum for mathematics goes a bit beyond a scope and sequence approach in that it provides Illustrations, Developmental Overviews, Teaching Strategies and an online interview tool. Both our review of online materials and literature and our personal communication with Kaye Stacey suggest that there is a limited research base for a few of the Indicators of Progress but no empirical studies for many of the Indicators of Progress. We acknowledge that the resources provided to teachers as part of the developmental curriculum are likely to be quite useful for teachers implementing the AusVELS curriculum despite their being based on the older VELS curriculum. We also see the emphasis on formative assessment as beneficial. However, we see a need for more empirical research for most of the Indicators of Progress before using this resource to inform the curriculum revision in other jurisdictions.

5.4 United States (2 pages)

Introduction

The United States was chosen as a jurisdiction to discuss in this report as it is has recently development national curriculum standards for mathematics, Common Core State Standards (CCSS), which are in the early stages of implementation. At the same time, there is a great deal of research in learning trajectories in the United States. In fact, much of the development of the CCSS was informed by some of this research. There is also ongoing research examining how well the learning trajectory research meshes with the Common Core State Standards. Some of this is led by the work of Jere Confrey and colleagues, but it is also informed by many of the researchers stated above who are working on learning trajectories. Examining the process of alignment and the learning trajectories that they lean out will help to inform Ontario’s work. As well, the current work in the U.S. highlights areas where more research needs to be done.

In 1989, the National Council of Teachers of Mathematics (NCTM) produced a seminal document called the Curriculum and Evaluation Standards for School Mathematics. This document, as well as its successor, Principles and Standards for School Mathematics. (NCTM, 2000) were research-based guidelines that, along with other resources and stakeholders, influenced the development of mathematics standards in most districts/states in the United States. This development occurred in a district-by-district or state-by-state fashion, resulting in a great deal of disparity among states as to which topics were taught, when they are being taught, and what order they are being taught in. However, particularly with the addition of high stakes national assessments in mathematics and the implications of No Child Left Behind, states were concerned about the variability in the mathematics standards from jurisdiction to jurisdiction. It was through the National Governors Association that work began on the Common Core State Standards. Many stakeholders, including NCTM, were involved in designing the mathematics standards and their approach involved examining learning trajectories.

Relationship between Curriculum and Learning Trajectories

One of the tenets of the work in developing the CCSS was that “a body of mathematics standards must also respect what is known about how students learn” (National Governors Association Center for Best Practices, Council of Chief State School Officers [NGA], http://www.corestandards.org/Math, 2010). Hence, the development of the

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standards heavily relied on research-based learning progressions that detail students’ mathematical knowledge, skill, and understanding and how these develop over time (NGA, 2010). As authors of the CCSS describe, three dimensions informed the sequencing of the standards: cognitive development, mathematical coherence, and the pragmatics of instruction (Daro et al., 2011). Examining cognitive development led the curriculum designers to examine learning trajectory research as a basis for the development of the standards. In fact, learning trajectory research was crucial to the design of the standards. The CCSS authors solicited and collected as much research as they could on how students learn mathematical topics. While much of the work leaned on this research, they also noted that for many of the topics there was insufficient research in how students develop their understanding of the topic. However, they noted that this way of thinking about how students develop mathematical ideas helped them think about sequence and coherence as they created the standards. It should also be noted that long with content standards, the CCSS also include mathematical practice standards [add a bit more here about these standards]

Research Base

It could be claimed that the CCSS have a strong research base as evidence-based learning trajectories form the basis of the development of the standards. There were also several researchers on the author team so that the selection of research used to inform the standards is presumed to be well counseled. In reviewing the development of the standards, research on learning trajectories such as that presented earlier in this report was consulted. However, it should be noted that there are many mathematical topics within the standards that have not been researched from a learning trajectory perspective. It should also be noted that K-8 standards were determined Elementary stands were more determined by research of learning progressions and secondary standards were guided more by the logical development of mathematics.by research of learning progressions and secondary standards were guided more by the logical development of mathematics. Research continues to determine what areas need more extensive research.

Challenges & issues

There were several challenges and issues in creating the CCSS documents. Curriculum standards have to take into account the full range of mathematics,

even if learning trajectories have not been developed. Learning trajectory researchers have not fully mapped the terrain of mathematical standards and hence, some of the sequencing of standards are not as research-based as others. Further to this, in putting together the standards document, the connections between mathematical topics, and thus learning trajectories needs to be considered. Connecting ideas across learning trajectories such as the ways measurement connects geometric and numerical ideas, needs to be carefully thought through as these connections are not always evident in fine-grained focused learning trajectory research on a particular mathematics concept. Each standard is not necessarily situated in a single trajectory as it might relate to several trajectories.

Learning trajectory research provides possible ways that students develop topics. However, there is great diversity in terms of what knowledge students bring to a situation. It may be difficult to map that diversity in learning trajectories and

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consequently in curriculum standards. Learning trajectory research establishes that learning trajectories are real for some students, a possibility for any student and a close approximation for most students. Learning trajectories and hence standards based on them should not be seen as a map that explicitly demonstrates the path that students take. Not only should learning trajectories be seen as networks of possible ideas but consideration needs to be given to the fact that learning trajectories have been developed in particular contexts and these may not be easily transferable to other contexts. Developing narrow, precise standards and consequently assessments based on this information

Implications for teaching [ look at confrey article, etc. – more to come]

Teacher knowledge has been shown to be a large challenge and much work is being done to create resources and professional development for teachers.

“With enough knowledge of relevant learning trajectories and enough understanding ofhow learning trajectories work, teachers will better anticipate and recognize the most common starting points they will find among their students (Murata & Fuson, 2006; Battista, 2010).” (Daro et al. 2011).

What knowledge do teachers have of the path a student has taken to be at their current point of understanding?

Supports for teachers and professional development for use of learning trajectories

Implications for Assessment

As previously described, learning trajectories have strong positive implications for formative assessment. As stated by CCSS authors:

The standards, based as much as possible on tested learning trajectories, identify what direction to lead the students from wherever they begin the lesson. A curriculum, based on the standards, with the diagnostic value of revealing how different students see the mathematics – how they think about it – and where they are along the learning trajectory would also help (Daro et al., p. 47).

To see the connection between learning trajectories, the CCSS and formative assessment, we describe a particular formative assessment project … Vermont to be inserted here [Chris – by Thursday]

Formative assessment work using frameworks – reference trajectories – distilled trajectories into frameworks; used to develop assessment materials, professional developmentFrameworks: fractions, proportional reasoning; multiplicative reasoningContinuous revisiting frameworks based on examining student work

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With respect to large-scale assessment, the development of the CCSS and the concern previously expressed in consistency between standards and large-scale assessments, development on large-scale assessments that align with the CCSS is currently underway by two assessment developers, Partnership for Assessment of Readiness for College and Careers (PARCC), and Smarter Balanced Assessment Consortium. Concerns have been expressed about the development of traditional large-scale assessments with respect to the CCSS. If the goal of the CCSS is to pay close attention to student thinking and the complex ways that students develop an understanding of mathematical ideas, traditional large-scale items may not be able to demonstrate how students move from one idea to the other. Furthermore, traditional large-scale items may not be able to provide sufficient evidence of students fluency with the CCSS mathematical practices. As a consequence,

5.5 Netherlands – [Martha – see what there is to bring to the table on Thursday]

6. Summary (2 pages) – [make notes to bring to the table on Thursday]Notes:

Trajectories are not a totally new idea nor are they a magic solution to all of the problems of mathematics education. They recognize that there are many paths to learning but there are particular pieces that most students experience as they develop mathematical ideas. They also provide information to teachers to help to recognize prior knowledge and to set appropriate instructional tasks so that students have the opportunity to experience key learnings to developing that understanding. Ideas for summary and recommendations:

Major shift in thinking about the mathematics that young children can do in terms of mathematical thinking; gaps are already there by kindergarten.

Reason for having more curriculum for young children, rethinking how young children learn; prior learning also means more gaps in terms of differences in students

What is age appropriate is being envisioned differently;

As we move from traditional classrooms to reform classrooms; we open spaces to see what young children are capable of; makes learning trajectories look differently; children can come at things in a variety of ways; opening up mathematics due to investigation, etc. (Cross et al. ) shift in mathematics pedagogy

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cautions

The ways in which trajectories might influence curriculumFocusing on identifying significant and recognizable clusters of concepts and connections in students’ thinking that represent key steps, trajectories form a stronger basis for guiding curriculum development, instruction, and assessment.

Most of the work on learning trajectories focuses on a very specific mathematics content area. Often these are treated in isolation from other learning trajectories that have been developed. Work in connecting these trajectories has just begun (e.g. see Confrey) and the interactions and cross over between trajectories will be very useful work to inform curriculum design. Theoretically, it should eventually be possible to string together the growing number of specific trajectories where careful empirical work is being done and to use these to design curricula.

Perhaps well-constructed and validated trajectories might provide better descriptions of how students’ mathematical understanding and skill should develop over time. These could e used to inform standards, curricula, assessments, and approaches to teacher professional development.

Issues While many things are called learning trajectories there needs to be some caution taken with how evidence-based the trajectories are. It should not be taken for granted that anything called a learning trajectory has been supported by empirical evidence. That being said, it would seem as though we should proceed with curriculum design recognizing that learning trajectories are worthwhile to be consulted but that in some cases, we will need to design curriculum based on a combination of the continuum of learning trajectories and what we have been calling scope and sequence.

Context is important

We don’t have a good description or understanding of the key steps in the development of mathematical knowledge and understanding, and we don’t have a codified, warranted body of knowledge about what to do for students who manifest particular problems or misunderstandings at particular points along the path (Daro et al., p.55).

No universal progression that applies to all students. Each progression is inherently embedded in a sociocultural context.

Learning trajectories, instruction and assessment (influencing teacher practice)Professional development connected to prime and prime as professional developmentTeachers’ careful use of trajectories – not recipes, etc.

Ways to support the curriculum

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7. Recommendations about how to proceed (2 pages) [bring notes to the table on Thursday]

Suggested conceptual map as to how to further integrate learning trajectories into the curriculum

Work by division with experts – leader in math ed and researcher in that division; consider trajectories and their alignment with the current curriculum; consider: the research base of the trajectory; the context; the grain size; flexibility, big ideas, implications for assessment, development, resources

Suggested ways of supporting implementation Find ways of sharing what we know about learning trajectories with teachers and develop resources that support classroom teaching, learning, and assessment

Criteria for assessing research and learning trajectory resourcesMuch of the work on developing learning trajectories draws on the research of others; it is a combination of empirical research, experience, and understanding of the structure of mathematics. The way that the research is synthesized seems to take a variety of forms. Teams synthesizing research – what research they synthesize and how they synthesize it varies based on their own understandings and experiences and contexts. We wonder – have they considered the context of the study – what types of classrooms was this research done in; what are the students’ prior experiences; what about the cultural differences?

Whereas some trajectories are more tied to specific studies

Need to examine what research supports the learning trajectory as well as the context where the research was done.

Recognition of limitations of learning trajectory workThe work of learning trajectories needs to be consolidated to inform curriculum design and teacher support. This would also include integrating the trajectories and finding connections across trajectories. At this point, the place where this works seems to be most promising is the work of Confrey and her team.

Further workExamination and development of learning trajectories in Ontario classrooms – based on understandings of mathematics educators who work in Ontario

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Preliminary Reference List – organized by topic

Learning trajectories (in general) and their role in curriculum design

Daro, P., Mosher, F.A. & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction (Report # RR-68). Retrieved from Consortium for Policy Research in Education website: http://www.cpre.org/learning-trajectories-mathematics-foundation-standards-curriculum-assessment-and-instruction

Duschl, R., Maeng, S. & Sezen, A. (2011). Learning progressions and teaching sequences: A review and analysis. Studies in Science Education, 47(2), 123-182.

Empson, S. B. (2011). On the idea of learning trajectories: Promises and pitfalls. The Mathematics Enthusiast, 8(3), 571-596.

Popham,W.J. (2007).The lowdown on learning progressions. Educational Leadership, 64 (7), 83‐84.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.

Stephens, M. & Armanto, D. (2010). How to build powerful learning trajectories for relational thinking in the primary school years. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the Future of Mathematics Education: Proceedings of the 33rd Annual Conference of the Mathematics Education Research Group of Australasia (pp. 523-530). Retrieved from http://www.merga.net.au/documents/MERGA33_Stephens&Armanto.pdf

Relevant examples of learning trajectoriesConfrey, J. (2008). A synthesis of the research on rational number reasoning: A learning

progressions approach to synthesis. Paper presented at The 11th International Congress of Mathematics Instruction, Monterrey, Mexico.

Confrey, J. , Maloney, A. , Nguyen, K ., Mojica, G. , & Myers, M. (2009). Equipartitioning/splitting as a foundation of rational number reasoning using learning trajectories. Paper presented at The 33rd Conference of the International Group for the Psychology of Mathematics Education, Thessaloniki, Greece.

Cross, C.T., Woods, T.A. & Schweingruber (Eds.) (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington DC: The National Academies Press. Retrieved from http://www.nap.edu/openbook.php?record_id=12519

Fosnot, C.T. (2013). Contexts for learning mathematics: Investigating fractions, decimals and percents [Landscape of Learning Graphic]. Retrieved from http://www.contextsforlearning.com/grades4_6/landscapeLearning.asp

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Fosnot, C.T. & Dolk, M. (2001a). Young mathematicians at work: Constructing number sense, addition and subtraction. Westport, CT: Heinemann.

Fosnot, C.T. & Dolk, M. (2001b). Young mathematicians at work: Constructing Multiplication and Division. Westport, CT: Heinemann.

Fosnot, C. T. & Dolk, M. (2002). Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents. Portsmouth, N.H.: Heinemann Press.

Fosnot, C. T. & Jacob, B. (2010). Young Mathematicians at Work: Constructing Algebra. Portsmouth, N.H.: Heinemann Press.

Generating Increased Science and Math Opportunities (GISMO) Research Team (n.d.). Learning trajectories for the K-8 Common Core Math Standards. Retrieved from http://www.turnonccmath.net/

Lawson, A (in press, 2013 Fall) What to look for: Understanding student’s mathematical thinking Gr 1-2. Toronto: Pearson (Book, DVD, ebook)  ISBN: 13: 978-0-321-88717-7  (virtual samples will be available soon)

Lawson, A. (2013) Documenting the trajectory of children’s invented methods of calculation. Paper presented at the annual meeting of the American Educational Research Association, San Francisco (April 27- May 4.)

Sarama, J. & Clements, D.H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.

Szilágyi, J. , Clements, D. H., Sarama, J. (2013). Young Children’s Understandings of Length Measurement: Evaluating a Learning Trajectory. Journal for Research in Mathematics Education, 44 (3), 581 – 620.

van den Heuvel-Panhuizen, M. (Ed.) (2008). Children learn mathematics: A learning-teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. Rotterdam, The Netherlands: Sense Publishers. Retrieved from https://www.sensepublishers.com/media/161-children-learn-mathematics.pdf

Ways jurisdictions have approached curriculum design using learning trajectories

Australia:

Australian council of Assssment and Reporting Authority (ACARA). (2013). The Australian curriculum: Mathematics. Retrieved from http://www.australiancurriculum.edu.au/Mathematics/Rationale 

Government of Victoria Department of Education and Early Childhood Development (n.d.). Matheamtics developmental continuum. Retrieved from

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http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/pages/mathcontin.aspx?Redirect=1

United States:

National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards: Mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers. Retrieved from http://www.corestandards.org/Math

National Research Council. (2001). Knowing what students know: The science and design of educational assessment. Pelligrino, J., Chudowsky, N., and Glaser, R., (Eds.). Washington, DC: National Academy Press. Retrieved from http://www.nap.edu/openbook.php?isbn=0309072727

The Vermont Institutes. (2013). Vermont Mathematics Partnerships Ongoing Assessment Project. Retrieved from http://www.vermontinstitutes.org/index.php/vmp/ogap

The Vermont Institutes. (2008). Vermont Mathematics Partnerships Ongoing Assessment Project Fractions Framework. Retrieved from http://www.vermontinstitutes.org/index.php/vmp/ogap

The Vermont Institutes. (2008). Vermont Mathematics Partnerships Ongoing Assessment Project Multiplicative Reasoning Framework. Retrieved from http://www.vermontinstitutes.org/index.php/vmp/ogap

The Vermont Institutes. (2008). Vermont Mathematics Partnerships Ongoing Assessment Project Proportional Reasoning Framework. Retrieved from http://www.vermontinstitutes.org/index.php/vmp/ogap

US Department of Education. (2008). The Final Report of the National Mathematics Advisory Panel. Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/reports.html

Quebec:

Ministère de l’Éducation, Loisier et Sport Québec. (2009). Progression of learning mathematics. Retrieved from http://www.mels.gouv.qc.ca/progression/mathematique/pdf/math_en_sectionCom.pdf

Ministère de l’Éducation, Loisier et Sport Québec. (2001). Quebec education program, preschool education, elementary education: Chapter 6 Mathematics, science and technology. Retrieved from http://www.mels.gouv.qc.ca/DGFJ/dp/programme_de_formation/primaire/pdf/educprg2001/educprg2001-061.pdf

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Additional references related to recommendations (to come)

AppendicesIn many cases, due to copyright laws, rather than providing hard copies of specific documents in appendices we will provide links to those documents.

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