Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 61
Facilitating the design and enactment of mathematics curricula
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 62
curricula to convey both the mandated curriculum standards that specify the concepts and processes students are expected to learn in each grade, as well as the curricular materials educators use to enact these standards. Each researcher also provides examples of changes made to their map in response to user feedback and considers the ways their map functions as a shared resource across professional communities.
We see these projects as situated at the intersection of two MEDA 2020 themes:
mathematics teacher education and professional development in the digital age (Theme 1), and curriculum development and task design in the digital age (Theme 2). We look forward to critical engagement with conference participants on these topics.
THREE DIGITAL MAPPING PROJECTS Math-Mapper 6-8 (Jere Confrey)
Math Mapper 6-8 (MM6-8) is one component of a digital learning system which covers the content of middle grades mathematics in the United States (Siemens & Confrey, 2015). The map can be accessed by registering an account at sudds.co. The purpose of the map is two-fold. Firstly, the map creates a visual representation of the relationships among the big ideas and sub-constructs within middle school mathematics. We view big ideas as concepts that connect the content, processes, and forms of argumentation in mathematics. In doing so, big ideas can help avoid viewing mathematics as a set of fragmented topics and skills. Secondly, the map provides teachers with direct access to empirically-based learning trajectories (LTs) (Clements & Sarama, 2004; Confrey, Toutkoushian, & Shah, 2019) which can guide learner-centred instruction and ground the map’s related diagnostic assessments. Confrey (2019), wrote a synthesis of research on mathematics learning trajectories which summarizes the map’s theoretical foundation. MM6-8 was built and refined in a partnership among learning scientists and psychometricians in a “trading zone” that allows revision and refinement of the map (Confrey, 2019).
Figure 1: Components of Math-Mapper 6-8
The principal audience for the MM6-8 map is both students and teachers. The map replaces the linearity of a book’s table of contents in favour of multiple levels of visual illustration. MM6-8 uses a non-linear, hierarchical structure which includes nine big ideas, 25 relational learning clusters (RLCs), and 62 constructs, each of which is associated with a LT. Students who use the map can see how what they are learning connects to a small but powerful set of big ideas. Teachers who use the map gain access to empirically established ideas about learning using LTs. In addition, every level of LTs in MM6-8 has a related set of assessment tests and practice accompanied by
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 63
intuitive student reports to guide diagnostically-valid instructional moves. Thus, teachers can use MM6-8 to re-examine instructional materials and curriculum standards and diagnostically assess student progress along LTs (Confrey, Gianopulos, McGowan, Shah, & Belcher, 2017). Connections to other resources are also offered including the Common Core State Standards - Mathematics (CCS-M) and access to illustrative resources from the “Resource library”.
Early users of MM6-8 include partnerships with six middle schools with varied demographics. Over the last four years, students have taken over 75000 MM6-8 assessments enabling the research team to use item-response theory (IRT) to conduct on-going validation of those assessments. Annual interviews with teachers have led to modifications to the map based on the use of the diagnostics resulting in more explicit delineation of misconceptions in mathematics, revisions to the map, and shorter, more focused assessments. Student data show positive correlations of increased use with improved end-of-year growth on MM6-8 tests (Confrey, Toutkoushian & Shah, 2019).
Data also indicate that it takes time for teachers to learn to trust the learning trajectories and to see their relationship to instructional practices.
Dynamic Mathematics Curriculum Network (Martha Koch)
The DMC Network [1] is the result of a research project to represent the connections within school mathematics as perceived by individuals who are engaged in mathematics education (Koch et al., 2018). Theoretically rooted in complexity thinking (Davis & Simmt, 2003; Doll, 2008), the DMC Network was derived from analysis of the concepts, connections and related resources suggested by K-12 teachers, school division mathematics curriculum leaders, teacher educators, researchers, and graduate students from across Canada. In the first phase of data collection, participants engaged in video-recorded collaborative problem-solving sessions and created physical models of connections they perceived as they worked on the task. The task they were given is one that often prompts algebraic thinking. In subsequent phases, we invited mathematics educators to view the digital version of the DMC Network that we had developed through analysis of the models from the first phase, and to contribute their ideas through an online portal.
In the DMC Network, mathematics concepts and processes are represented as nodes connected to one another with curved lines. Clicking a node or connection line reveals definitions, explanations, examples, and links to research-informed resources. The position of any node is determined solely by connections between that node and other nodes. Readers are invited to view these features at dynamicmathcurriculum.ca. Based on our analysis of input from participants, some nodes connect to many others (e.g.
“Algebraic expressions” currently connects to 12 nodes) while other nodes have fewer connections (e.g. “Proportionality” currently connects to 4 nodes). Educators can create many paths through the concepts and processes that are shown. A central feature of the DMC Network is the “Add to the Network” tab which invites any user to suggest new nodes or connections or recommend related resources. Contributors are asked to
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 64
explain their thinking to assist the research team with evaluating their suggestions and deciding which changes should be made in the DMC Network.
We think of teachers as the main audience for this resource. The DMC Network can help teachers deepen their understanding of mathematics concepts, plan a sequence of lessons, discover a way to support a student struggling to understand a concept, or share ways of teaching mathematics that they have found effective. Teacher educators may use the DMC Network to help teachers become more aware of the connections that are called for but typically not made explicit in curriculum documents.
Participants who contributed to the first iteration of the DMC Network found the process of articulating and representing the connections they perceived as they engaged in collaborative problem solving to be challenging yet generative. Many noted that these activities deepened their understanding of mathematics concepts and processes in relation to the curriculum standards in their jurisdiction. Most saw mathematics as much more interconnected than they had realized and many sought to represent the connections they perceived by adding iterative elements to their models. Those who contributed their ideas through the “Add to the Network” feature more often suggested new connections rather than new nodes. The first iteration of the DMC Network had 10 nodes with 31 connections while the next iteration had 13 nodes and 59 connections.
Here again, we noted the tendency for participants to see mathematics as deeply interconnected. A few contributors suggested changes that reflect initiatives in their context such as one teacher educator who recommended including Indigenous views of mathematics in the DMC Network. In the most recent phase of the project, high school and college educators have been invited to envision a three-step path they might take within the DMC Network and to provide feedback on the nodes and connections that might be added.
Cambridge Mathematics Framework (Ellen Jameson)
In the Cambridge Mathematics Framework (CM) project a team of designers, teachers, and researchers are developing a tool to enable the dynamic generation of maps which highlight and describe connections between ideas and experiences in school mathematics (www.cambridgemaths.org). The maps, and associated content, are representations of knowledge about mathematics learning interpreted from reports of research and practice according to our design methodology (Jameson, McClure &
Gould, 2018). The purpose of the CM Framework is to support coherence in mathematics education by facilitating shared understanding of connections in mathematics learning within and between communities involved in curriculum design and enactment such as curriculum and resource designers, teachers, and teacher educators. Our purpose, theoretical influences and design methods are more fully elaborated in recent papers (Jameson, McClure & Gould, 2018; Jameson, 2019).
In order for the CM Framework to serve as a shared frame of reference, these mathematical ideas are not curriculum-specific but can be mapped to various sets of standards. Likewise, these ideas and relationships are expressed in ways which are
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recognisable and useful to audiences in multiple communities. Some people may be looking for a ‘way down’ to get a more detailed perspective, while others may need a
‘way up’ to see a bigger picture. Some might be looking back to see what ideas students may need to be working with at a particular point, while others may look forward to see what ideas students will need to be able to work with later on. Some may be working at a time scale of a few weeks, while others may be designing for learning over a few months, a few years, or a decade.
Mathematical content is expressed in the CM Framework maps as waypoints. Each waypoint contains a summary of the mathematical idea (the ‘what’) and its part in the wider narrative (the ‘why’), and lists examples of ‘student actions’ that would provide opportunities to experience the mathematics in meaningful ways. Waypoints are related to one another by themes. A theme is a way in which an idea develops into or is used when working with another idea. The CM Framework also includes Research Summaries which are documents that tell the story of a group of waypoints and themes.
They include a literature review, an interactive map of the waypoints and themes, and a section which describes how research has influenced the structure and content of the map. An example of a Research Summary is available on our website (Jameson et al., 2019). Connections to other resources are also managed within this layered structure.
These resources might be for designers (such as curriculum statements for curriculum comparison or revision), for teachers and teacher educators (such as professional development activities), or for both (such as glossary definitions of mathematical terms).
External reviewers evaluate our research summaries, and we conducted a Delphi study to evaluate our structure and theoretical influences (Jameson and McClure, 2020). We are piloting the use of the CM Framework for curriculum and resource design to identify core actions for the key uses of the CM Framework and to develop features, interfaces and training support. In one case, we used the CM Framework in the design of the UNICEF Learning Passport for Children on the Move (LPCM) mathematics curriculum framework, through which we were able to develop new tools and processes for mapping, analysing and revising curriculum statements, and for documenting the content and connections underlying the revised curriculum in order to provide a narrative for those who need to work with it (Jameson & Horsman, 2020).
We are also currently running a survey, CM DefineIt, to collect data on preferences and critiques of published definitions of mathematical terms relative to teachers’ contexts (Majewska, 2019). Our next step will be to trial the features and interfaces we are developing.
INSIGHTS FROM ACROSS THE PROJECTS Facilitating the design and enactment of curricula
Each of these digital maps is beginning to impact the design or enactment of mathematics curricula in distinct ways. Confrey identifies as a first order of impact of MM6-8, an increased awareness among teachers of their students’ thinking and a
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movement toward learner-centred approaches as teachers enact the curriculum standards in their setting. Koch notes that many participants who contributed to the DMC Network became more aware of the deeply connected nature of mathematics as they articulated their ideas. Insights from educators who describe their experiences planning a three-step sequence using the DMC Network will be shared at MEDA 2020.
In the CM Framework, Jameson describes the integration of important conceptual connections, highlighted by evidence from research and practice, into curriculum analysis and design in ways that improve coherence.
Fostering professional learning and knowledge sharing
Feedback from early users of each project illustrates the ways these maps foster professional learning and knowledge sharing within and across the groups engaged in mathematics education. For example, Confrey characterizes MM6-8 as a “trading zone” (Confrey, 2019) among different communities where practitioners, learning scientists, and measurement specialists can discuss the map as a shared resource. On a smaller scale, participants in the DMC Network project describe their learning in comments such as “I could see some connections quickly and then started to wonder about other nodes that might be appropriately connected . . . I wanted to break some nodes down into smaller chunks”. Others noted the value of the DMC Network for facilitating collaboration such as one participants’ comment “For its users and contributors it seems that this type of tool can emphasize the importance of researcher-practitioner relationships”. Returning to knowledge sharing at a larger scale, in the CM Framework LPCM pilot project, a curriculum development team used the CM Framework as a shared frame of reference driving discussions around strategies and trade-offs in the design of a curriculum with unusual constraints.
Digital maps as inherently dynamic tools
Each project includes processes for responding to feedback and for reflecting new research. These processes are essential for ensuring each map continues to support effective mathematics teaching and learning. The nodes, connections, definitions and resources in the DMC Network can be revised as researchers review contributions from mathematics educators. In the MM6-8 project, feedback from teachers has resulted in changes to the map to clarify mathematics misconceptions and facilitate ongoing development of assessment tools. Newly developed learning trajectories can also be added. The team developing the CM Framework has created a flexible format and structure to which modifications can be made. The CM team is currently using this flexibility to expand the range of content and to respond to feedback from external reviewers. As the CM Framework reaches a broader audience, opportunities to incorporate feedback will expand and new research can be included.
Dynamic tools can be used in ways which are creative rather than prescriptive; they leave room for choice and decision-making. A map or network can contain many overlapping paths, allowing users to focus on one or more paths relevant to their context while not losing the implications of the others. For example, the CM
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Framework contains more than a single curriculum could cover, but this is what gives it the power to explore the implications of choices for content selection and sequence when designing a curriculum or a textbook.
These projects provide evidence of the ways digital maps can foster vertical integration across K-12 mathematics curriculum standards, encourage informal collaboration or more formal partnerships between groups engaged in mathematics education, and lead to a better understanding of the processes and impact of instructional change on teachers’ knowledge and classroom practices.
NOTES
1. Social Sciences & Humanities Research Council of Canada partly funded the DMC Network.
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