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An object for expressing knowledge which can be shared between communities to coordinate perspectives and actions (Stahl, 2006; Lee, 2007).
“Objects that coordinate the perspectives of various communities of practice (Wenger, 1998; Henderson, 1999)” (Lee, 2007, p. 308). Those objects might be concrete, like maps, catalogues or data entry forms, or abstract, like protocols or information types. Boundary objects “arise over time from durable cooperation” (Lee, 2005, p. 390). They have aspects which are more fixed in order to have recognizable meaning for all communities concerned and aspects which are more adaptable to be meaningful for specific communities.
A platform developed by the Cambridge Mathematics team, which is used for collaborative writing, querying and visualisation of the Cambridge Mathematics Framework. It acts as a multiuser interface for the Framework database and contains tools for searching, viewing, analysing and editing.
Process in which selected design features are mapped back to conjectures and theories that underlie them, and then mapped forward to possible user actions and outcomes. The conjecture map is an explicit model of the logic of the design and can help designers reflect on it as a whole (Sandoval, 2014).
Many different definitions of a curriculum exist as the word curriculum is used in a wide variety of ways with respect to the scope of educational content and the scale of space and time. Formally, a common definition from sociology is that a curriculum consists of designated knowledge to be taught and learned, organised in time as discrete chunks of content (Bernstein, 1971; Young, 1971). However, in some jurisdictions or fields the term can be used to mean something as general as a brief list of content, or as detailed as a sequence complete with plans for teaching. It can be a plan, a guiding document (national or localised), or a reality in classrooms, or all three at once.
The statements of content through which a curriculum is described in detail, also known as learning objectives, assessment objectives, standards, or learning goals. Examples of documents that contain curriculum statements include: the English National Curriculum in England; the Key Learning Area Curriculum Guide (Learning Content of Primary and Junior Secondary Mathematics) in Hong Kong; The New Zealand Curriculum Mathematics Standards for Years 1-8 or the National Curriculum Statements for Mathematics in South Africa.
A panel of experts which has repeated discussions about a complex issue for which published evidence is insufficient. The extent to which they agree or disagree supports decision-makers to make good choices and/or consider the most useful factors.
Delphi is a group survey method which identifies areas of consensus and disagreement among experts when existing evidence doesn’t point to one solution. Experts on a Delphi panel do not meet or know who is involved, which allows for anonymity and helps to reduce bias. They fill out a questionnaire, receive an anonymised report of the group’s opinions and statements, and then have a chance to react to the group in a new questionnaire. These rounds provide the mechanism for the panel to converge on specific points of agreement and disagreement through mediated discussion.
Research is dependable if the research methods used would allow someone else to carry out the same study. The methodology should therefore be clear enough for someone else to replicate the study (Jameson, 2019).
Targets or aims that a design is intending to meet, usually tied to what the design can be used to achieve. They are developed from an understanding of the problem context and the intended scope of the design project. They define the outcomes that the design is engineered to support. Initial design goals are usually set out at the start of a project, but may be adapted as experience with particular design increases the understanding of the context.
A set of interpretations of theory or background knowledge of the design context which lay out general perspectives on how to approach the design. Design principles guide the choices about the features a design should include and the functions that those features should support (McKenney & Reeves, 2012). They can also make a design process more transparent by providing justification for specific design decisions and features.
Features of the design which enable the outcomes the design is engineered to support.
The path or journey that a project or a design has followed from start to finish.
In graph theory, an edge is a line connecting two nodes in a graph. In the Mathematical Ideas layer of the Framework, an edge represents a relationship between two waypoints. This relationship may have a direction (e.g. from A to B) or be undirected (between A and B). Each edge is labelled according to a mathematical theme, and whether the connection between the waypoints is best described as a development of a concept, skill, procedure (conceptual progression) or progression in using the concept, skill, procedure.
An exploratory waypoint often comes at the beginning of a theme. It allows for exploration and play with ideas and may introduce students to a concept and provide essential foundation for understanding that concept intuitively.
The process of evaluating Research Summaries by international experts in different areas of mathematics education and research, to ensure the research underpinning the Framework is appropriate and has been interpreted reasonably.
Academic researchers, often with additional experience in teacher education, who have knowledge and understanding of mathematics education literature and can contribute their expertise by evaluating certain areas of the Framework.
Research carried out in the natural location relevant to the context of the study and the research topic rather than in a laboratory (Farrell, 2016).
Key mathematical terms or phrases are defined in glossary nodes. These allows us to access definitions while looking at the Framework content and find the content which is linked to a particular term.
Some response from members of the mathematics education community that we can interpret as feedback about their judgement of quality or value. This response may be direct as in an interview or a survey, or indirect in the form of community engagement and critique of project outputs.
The process of examining Research Summaries by the Cambridge Mathematics team, to ensure consistency in how the information is presented.
At landmark waypoints, ideas are brought together and synthesised so that the whole may seem greater than the sum of its parts.
A visual way of imagining how different Framework components and add-on modules are categorised in our database and can be linked together.
An industry-standard graph database management system which is used to store the Framework. It enables us to retrieve, edit and visualise connections between mathematical ideas more easily than if we used a spreadsheet or a relational database.
In graph theory, a node is a point in a graph, which can be connected to other points by edges. In the Mathematical Ideas layer of the Framework, the nodes are waypoints.
The guiding structure of the Framework, including what ideas can exist within it, how they can be expressed and related. It consists of a set of agreed guidelines for how mathematical ideas can be represented and related to one another in the Framework.
The process of evaluating scientific, academic or professional work by experts in the same field before it is published (Kelly et al., 2014).
Small-scale implementations which allow us to test the possible uses of the Framework.
A preliminary study with a small pool of participants conducted to test the feasibility, time, cost and possible barriers that the researcher may need to consider before carrying out the full-scale study (van Teijlingen & Hundley, 2001).
Research carried out directly by the investigators. It can include, but is not limited to: interviews, surveys and observations.
Data that is not numerical in nature and may be describing something, for instance, how a person feels about cats (Surendran, n.d.).
Data that is numerical in nature and expresses a quantity, amount or range (Economic Commission for Europe of the United Nations, 2000).
A way of examining our own thinking about the structure and content of the Framework. This involves following shared research processes and reflecting on our own perspectives, beliefs, and experiences and those of our collaborators when interpreting the literature and creating elements of the Framework.
The extent to which a process or a research method produces stable and consistent outcomes.
Research and evidence from a range of fields such as mathematics education, mathematics and psychology, which informs the mathematical layer of the Framework and the tools we design for using the Framework to create shared artifacts. The writing team carries out a semi-structured review process to identify relevant research literature which underpins the Framework.
Edges which connect research nodes to Research Summaries or directly to waypoints as appropriate. They also describe whether a source has been an important or a secondary influence on the waypoint or Research Summary to which it links.
Nodes which correspond to specific sources and contain metadata characterising those sources.
Concise reviews of literature in a given area of mathematics learning, together with their implications for the Framework. They include an interactive component of part of the Framework map.
An interactive, filtered selection of waypoints and the connections between them (Framework map), corresponding to an interpretation of reviewed literature on a particular mathematical topic.
Research which uses data that has already been collected through primary research. It can include, but is not limited to: reviewing existing research and evidence, statistically analysing existing data from single studies and analysing findings from multiple studies to come to a conclusion (Bhat, n.d).
Shared knowledge representation
A diagram, text, technical vocabulary, map or other representation of knowledge which has been discussed and agreed to have a particular meaning in a particular group.
Something that members of the mathematics education community can observe about our project – whether in the content, the design itself, our process, goals, influences etc. – which they might then interpret as a sign of quality or value.
The types of actions that students might do to help them build an understanding of a mathematical concept.
The Cambridge Mathematics Framework
A network of mathematical ideas which can be tied to teacher education and training, tasks and assessments. It is designed for the dynamic creation of knowledge maps which serve as representations of conceptual relationships in mathematics learning. Its structure and content are evidence and research-informed.
A connection between waypoints.
Research is transferable if others can use (apply) it in another field.
Research that aims to evaluate a product by testing it on users (Moran, 2019).
Places where learners acquire knowledge, familiarity or expertise about some form of mathematical idea. There are different types of waypoints and all of them can be found in the Framework. The two main types include exploratory and landmark waypoints.
The ability of learners to recognise, use and adapt a variety of definitions, representations, and strategies to solve problems in this/an area of mathematics.
Bhat, A. (n. d.). Secondary research – definition, methods and examples. QuestionPro. Retrieved December 10, 2019, from https://www.questionpro.com/blog/secondary-research/.
Bernstein, B. (1971). On the classification and framing of educational knowledge. In M. F. D. Young (Ed.), Knowledge and Control: New Directions for the Sociology of Education (pp. 47–69). London: Collier-Macmillan Publishers.
Economic Commission for Europe of the United Nations. (2000). Glossary of terms on Statistical Data Editing [Conference session]. Conference of European Statisticians Methodological Material, Geneva. https://ec.europa.eu/eurostat/ramon/statmanuals/files/UN_editing_glossary_2000.pdf
Farrell, S. (2016, October 23). Field Studies. Nielsen Normal Group. https://www.nngroup.com/articles/field-studies/
Jameson, E. (2019). Methodology: Building the Research Base. Cambridge, UK: Cambridge Mathematics. https://www.cambridgemaths.org/Images/methodology-building-the-research.pdf
Kelly, J., Sadeghieh, T., & Adeli, K. (2014). Peer Review in Scientific Publications: Benefits, Critiques, & A Survival Guide. The Journal of the International Federation of Clinical Chemistry and Laboratory Medicine, 25(3), 227–243.
Lee, C. P. (2005). Between chaos and routine: Boundary negotiating artifacts in collaboration. In ECSCW 2005 (pp. 387–406). Springer.
Lee, C. P. (2007). Boundary Negotiating Artifacts: Unbinding the Routine of Boundary Objects and Embracing Chaos in Collaborative Work. Computer Supported Cooperative Work (CSCW), 16(3), 307–339.
McKenney, S., & Reeves, T. C. (2012). Conducting Educational Design Research. Routledge.
Moran, K. (2019, December 1). Usability Testing 101. Nielsen Normal Group. https://www.nngroup.com/articles/usability-testing-101/
Sandoval, W. A. (2014). Conjecture Mapping: An Approach to Systematic Educational Design Research. Journal of the Learning Sciences, 23(1), 18–36.
Stahl, G. (2006). Group Cognition: Computer Support for Building Collaborative Knowledge. Cambridge, MA: MIT Press.
Surendran, A. (n.d.). Qualitative data - definitions, types, analysis and examples. QuestionPro. Retrieved December 10, 2019, from https://www.questionpro.com/blog/qualitative-data/
van Teijlingen, E., & Hundley, V. (2001). The Importance of pilot studies. Social Research Update, 35. Retrieved from http://sru.soc.surrey.ac.uk/SRU35.html
Young, M. F. D. (1971). An approach to the study of curricula as socially organized knowledge. In Knowledge and Control: New Directions for the Sociology of Education (pp. 19–46). London: Collier-Macmillan Publishers.