Chapter 4: Analytical Framework - Notion of OTDMP and Descriptor table 30
4.4. Table of Descriptors of OTDMP
The descriptors of each rating of the OTDMP strands to a large extent draw on the larger body of work in mathematics education, though implicitly. It should be noted, however, that a particular system of values could possibly be detected in the choice of descriptors – yet this is also true for the focus on mathematical proficiencies including adaptive reasoning, productive disposition and conceptual understanding in the first place. Furthermore, the descriptors are potentially more related to more teacher-centered or expository forms of teaching, and thus may need further development to be more inclusive. However, since teacher-centered methods interspersed with individual seat work mostly focused on practising taught work dominated all the videos, the current set of descriptors appeared adequate for the data collected in this study.
Table 1: Descriptors per opportunity to develop each strand per numerical rating.
Numerical rating→ 1 2 3
Components of OTDMP↓
Opportunity to develop Conceptual Understanding – OTDCU
Few opportunities are provided to build understanding of concepts.
Explanations or developments of concepts are not linked to other concepts in explicit ways (low discursive saturation of links).
No attempt is made
Opportunities are provided which clearly clarify the concept with some explicit links made to other concepts, horizontally or vertically. Only mathematically key links are engaged – in other words,
‘structural overload’
Clear explanations when stating or developing concepts, are provided. In addition, two of the following three opportunities to develop conceptual understanding are evident: (i) How and why specific concepts are used as well as
Noor Ally Promotion of Mathematical Proficiency
to explain the relevance or significance of the concept.
Representations are not linked to the concept in explicit ways or are irrelevant, in the sense that they do not capture essential aspects of the concept.
is avoided. Any representation is linked to the concept in explicit ways.
their significance is formulated and demonstrated.
(ii) Connections to other concepts are indicated, as per rating .
(iii) More than one representation are explicitly compared, discussed and connected, but only mathematically relevant
representations are included, and not so many of these that the key characteristics become difficult to discern
Opportunity to develop Productive Disposition – OTDPD
Opportunities to encourage, persevere and instill confidence (eg. encouraging learners to persist, praising effort and performance, explain strategies, adhering to tasks) occurs at times, but is not consistent (as when learners are told to collaborate, then scolded for talking to each other; or the teacher has a
negative disposition a lot of the time, e.g., displays anger, aloofness, sarcasm).
Real world situations are described but opportunities to relate these to the mathematics are not made explicit to the learners.
Opportunities to reinforce effort, comment on learners’
performance, encourage interest is developed, but only on occasion. Making sense of maths is identified and recognized but not fully exploited. Out of class situations are mentioned and used to motivate the mathematics, but the connections are only partially made explicit.
Opportunities to regularly reinforce effort as well as create enthusiasm in mathematics are developed and exploited. A positive approach to maths showing sensitivity, respect and interest in learners’ responses and questions is evident. Learners are regularly encouraged to persevere -‘keep trying’. Opportunities to create an
environment
conducive to fostering confidence is
observable (eg.
conveying that mistakes are okay, providing
explanations to learners in difficulty, organizing content so that it is personally meaningful to learners).
Opportunities to
develop links between out-of-class situations and the mathematics are made explicit.
Opportunity to Develop Procedural Fluency – OTDPF
Development of procedures is inefficient and inappropriate.
Opportunities to provide explanations of what and why a procedure is used are non-existent.
Opportunities to develop reasons for doing a procedure in a particular way are not forthcoming and alternate procedures are not explored.
Procedures may not be performed fluently.
Opportunities to perform procedures appropriately and fluently are developed. One procedure is used and an explanation of what procedure is used is
communicated.
Opportunities to provide reasons for doing the procedure are only partially made explicit.
Alternate procedures are not explored.
Opportunities to develop appropriate, efficient and fluent procedures are maximised. What, when and how a procedure is applied is explicitly and
competently communicated.
Opportunities to break down a procedure into its components are developed. It is coherent, orderly and sequenced. Multiple procedures are used and opportunities to provide reasons for and differences of these are made explicit.
Opportunity to develop Adaptive Reasoning – OTDAR
Opportunities to develop reasons underlying explanations are incoherent and inconsistent. Formal or informal proof or justification is not observed.
Justification may be simply with reference to authority
(textbook, teacher,
‘rule’).
Reasoning is explicit and valid.
Opportunity to develop informal proof or justification is sometimes used.
Learners are not encouraged to neither justify nor prove their work.
Opportunities to develop explicit reasoning are validated.
Explanations of procedures or concepts are
immediately followed with informal proof, justification and/or deductive reasoning.
Learners are encouraged to consistently justify their answers or claims.
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Opportunity to Develop Strategic Competence – OTDSC
Opportunities to develop and use a heuristic (pictures, lists, flow chart, etc.) are inappropriately selected for the mathematical problem at hand.
The opportunity to develop a single heuristic (pictures, lists, flow chart, etc.) that is appropriate to the mathematical problem at hand is seized.
Opportunities to develop multiple heuristics to solve problems are evident.
Opportunities to choose flexibly among these are explicitly linked to the
mathematics at hand.
In this chapter I introduced the notion of OTDMP which will now permeate the remainder of the study becoming the mainstay of the research. It informs the final design of the instrument and will be an integral part of the results and analysis. The table is the core around which the theory of OTDMP was first formulated. The table went through multiple changes mainly in the belief that opportunities that develop mathematical content and quality play a key role in mathematical understanding.
The overall quality of the lessons is seen in terms of “the opportunities that the lesson provided for students to construct important mathematical understandings”
(Hiebertet al, 2003, p. 199). This vision, framed by the five strands of mathematical proficiency, eventually led to the ratings which merge the mathematical content with the quality. A key component of the formulation was the generic use of the table in all areas of mathematics learning viz. number concepts and operations, geometry, measurement and data handling. The table could be applied equally well in any of these domains.
The acronyms introduced in this chapter will be used substantially in the remainder of this study.