Creating Realistic Mathematics Tasks Involving Authenticity, Cognitive Domains, and Openness Characteristics: A Study with Pre-Service Teachers
Abstract
:1. Introduction
2. Theoretical Framework
2.1. Mathematics Tasks
- Real-life contexts (i.e., the non-mathematics aspects present in the task formulation) can be used as a didactic tool to support mathematics learning [31]. [32] establish three types of tasks based on their connection to the real world: intra-mathematics tasks, disguised tasks, and real-life application tasks. Intra-mathematics tasks have no connection to reality and require only mathematical procedures to be solved. Disguised tasks are those whose real context is given—i.e., all needed data to find the solution are provided and the outcome is achieved just by verifying the mathematics part. Real-life application tasks are often named authentic tasks; these are real-life situations where the solvers really feel the need to use mathematics [33]. Although National Council of Teachers of Mathematics (NCTM) claims that teachers should use such real-life tasks to facilitate the construction of student knowledge [20], these are rarely present in school lessons [34].The authenticity of mathematics tasks has also been analyzed in both textbooks and assessments. For example, [35] analyzed the authenticity of mathematics tasks in Finnish and Swedish national evaluations. Their analysis included five dimensions: the chance to find the event of the task in everyday situations, the adequacy of the question (emerging from the task) for the event, the adequacy of the data offered in an everyday situation, the explicit presence of the task purpose, and the specificity of the information about the event. The results showed that the event was simulated in more than 90% of the tasks, while the others were simulated in a range from 25% (the presence of the purpose) to 60% (task adequacy to the event). Based on the aforementioned dimensions, [36] analyzed, over a seven-month period, the mathematics tasks of two primary school teachers of 6th-grade pupils in Flanders. Their results showed that the context of the tasks as well as the data and specificity of the information were well simulated. [37] established levels of authenticity for 8373 mathematics tasks presented in the textbooks of a Spanish publisher across the six primary education courses. The results revealed that only 2% were authentic mathematics tasks, although about 26% may be easily convertible into authentic tasks.
- Concerning the variety of responses to a task, we talk of open-ended or closed-ended tasks. The former has only one correct answer, such as the procedural activities appearing in textbooks for practicing a specific skill [38,39]. The latter allows more than one correct answer; that is, they are flexible enough to take into account the solver thinking, reasoning, and creativity [29]. Open-ended tasks are also divided into well- and ill-defined answers: the former occurs when the answer is clearly defined to be either correct or incorrect, and the latter takes place when the answer is subjective in the sense that there is no right or wrong answer. For example, an exploratory task such as “Make up a story where the answer is the result of 2.4 × 5.3” relates to a well-defined open answer [40], whereas a real-life task (e.g., “Design a playground for the school”) relates to an open ill-defined answer [13]. Referring to the realism of the tasks, realistic tasks could involve real-life situations. To solve them, mathematics calculations are not enough but also require the interpretation of when and how mathematics and non-mathematics knowledge should be applied [18,41,42]. An example of this would be: “John runs the 100 m in 17 s. How long will it take to run 1 km?” [43]. To solve this task, the use of proportionality is unsuitable because it is unusual for a person to run 1 km at the same constant speed, as when running 100 m.
- Cognitive domains are understood as student thinking skills, which include aspects based on general processes such as working memory, attention, or language [47]. According to cognitive domains, mathematics tasks can be classified on three levels [48]: knowing, applying, and reasoning. Knowing tasks imply the evocation and repetition of knowledge that has already been taught. Applying tasks requires the integration and relationship of diverse mathematics knowledge, based on knowing and framed in non-routine situations related to familiar settings. Finally, reasoning tasks entail complex situations that, based on applying, demand reasoning and reflection to achieve the solution.Several authors have analyzed the cognitive domain involved in the mathematics tasks of secondary school textbooks, with a special focus on plane geometry [44] and linear equation tasks [45]. Such analyses have revealed that textbooks encompass mostly mathematics tasks involving routine activities related to the knowing or applying domain. Likewise, [46] showed that most tasks that appear in primary mathematics textbooks only aim to evoke knowledge. Similarly, in an analysis of tasks proposed by primary teachers, [49] found that about 81% and 19% of the mathematics tasks related to the knowing and applying domains, respectively.
2.2. Creating Mathematics Tasks by Pre-Service Teachers
3. Materials and Methods
3.1. Context and Participants
3.2. Procedure and Data
- Selecting three contexts appropriated for elementary school education. Step carried out in groups for 5 min.
- Choosing a context and creating three realistic mathematics tasks for an elementary school lesson. Step carried out in groups for 15 min. Delivery 1: initial Created Tasks 0 (CT0).
- Orally presenting the CT0s to the whole class. Proposing changes and improvements to other groups. Step carried out with the whole class for 15 min.
- Modifying the CT0s according to the suggestions received in step 3 for attaining realistic mathematics tasks. Step carried out in groups for 15 min. Delivery 2: refined Created Tasks 1 (CT1).
- Solving six CT1s: the three proposed tasks and another three from a different group. Step carried out in groups for 15 min.
- Modifying the CT1s according to the outcomes of step 5 to better achieve realistic mathematics tasks. Step carried out in groups for 30 min. Delivery 3: final Created Tasks 2 (CT2).
3.3. Categorization System for Analysis
3.3.1. Realism
3.3.2. Cognitive Domains
3.3.3. Authenticity
3.3.4. Openness
4. Results
4.1. Realism
4.2. Cognitive Domains
4.3. Authenticity
4.4. Openness
5. Discussion
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Students | N | Group | CT0 | CT2 |
---|---|---|---|---|
Master | 19 | 9 | 27 | 27 |
Degree | 43 | 16 | 48 | 48 |
TOTAL | 62 | 25 | 75 | 75 |
Levels | Indicators |
---|---|
Knowing: facts, concepts, and procedures students need to know | Recalling: definitions, terminology, number properties, units of measurement, geometric properties, and notation (e.g., a × b = ab, a + a + a = 3a). |
Recognizing: numbers, expressions, quantities, shapes, and equivalent entities (e.g., equivalent familiar fractions, different orientations of simple geometric figures, etc.). | |
Classifying/sorting: numbers, expressions, quantities, and shapes. | |
Computing: algorithms +, −, ×, ÷, or a combination of these with whole numbers, fractions, decimals, and integers. Undertaken straightforward algebraic procedures. | |
Retrieving: information from graphs, tables, texts, or other sources. | |
Measuring: using instruments; choosing appropriate units of measurement. | |
Applying: knowledge and conceptual understanding to solve problems or answer questions | Determining: efficient/appropriate operations, strategies, and tools for solving problems for which common methods exist. |
Representing/modeling: data in tables or graphs; equations, inequalities, geometric figures, or diagrams. | |
Implementing: strategies and operations to solve problems involving familiar mathematical concepts and procedures. | |
Reasoning: about unfamiliar situations, complex contexts, and multi-step problems | Analyzing: relationships among numbers, expressions, quantities, and shapes. |
Integrating/synthesizing: different components of knowledge, representations, and procedures to solve problems. | |
Evaluating: alternative problem-solving strategies and solutions. | |
Drawing conclusions: valid inferences on the basis of information and evidence. | |
Generalizing: statements representing relationships in more general and more widely applicable terms. | |
Justifying: mathematical arguments to support a strategy or solution. |
Components | Values | Indicators |
---|---|---|
Event | 1 | The proposed situation is feasible in real life. |
0 | The proposed situation is imaginary, although it could be related to a real situation (e.g., odometers measuring in different units according to the time of day). It could happen in real life but in an unusual way (e.g., farmers with large greenhouses watered with domestic cans) or purely mathematics (e.g., children who draw the reflection of musical notes in a mirror). | |
Question | 1 | The question is formulated on a regular basis for the described event, and its answer has a practical value or is of interest to others who are not attracted by mathematics. |
0 | The question would not be formulated in the real world, or it does not correspond to the event described. | |
Data | 1 | The data correspond with the real ones. |
0 | The data does not correspond with the real ones or this information is only accessible through competencies that are required in a simulated situation (e.g., means or standard deviations). | |
Purpose | 1 | The purpose is explicitly mentioned and it is in accordance with that of the actual situation. |
0 | The purpose is not clear enough or the task is described without referring to any specific situation. Thus, the task could be adjusted to many situations and purposes. | |
Specificity of information | 1 | The characters of the problem have proper names, the objects are defined or familiar, and the places are specific. The problem is formulated in the 1st or 2nd person or the origin of the graphics is mentioned. If the situation is not specific, the elements undergoing mathematical treatment provide, at least, their specific role. |
0 | The situation is general without specifying objects and subjects, or the names of the characters are provided but not their role, which means that other aspects such as the realism of the data cannot be assessed. For example, it is not the same to say that “Andrew picked up 100 kg of potatoes” when he is farmer compared to when he is not. |
Authenticity | Values of the Matrix (Event, Question, Data, Purpose, Specificity of Information) |
---|---|
Authentic | (1,1,1,1,1) |
Believable | (1,1,1,0,1) (1,1,1,1,0) (1,1,1,0,0) |
Fictitious | (1,1,0,-,-) (1,0,1,-,-) (0,1,1,-,-) (1,0,0,-,-) (0,1,0,-,-) (0,0,1,-,-) (0,0,0,-,-) |
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Paredes, S.; Cáceres, M.J.; Diego-Mantecón, J.-M.; Blanco, T.F.; Chamoso, J.M. Creating Realistic Mathematics Tasks Involving Authenticity, Cognitive Domains, and Openness Characteristics: A Study with Pre-Service Teachers. Sustainability 2020, 12, 9656. https://doi.org/10.3390/su12229656
Paredes S, Cáceres MJ, Diego-Mantecón J-M, Blanco TF, Chamoso JM. Creating Realistic Mathematics Tasks Involving Authenticity, Cognitive Domains, and Openness Characteristics: A Study with Pre-Service Teachers. Sustainability. 2020; 12(22):9656. https://doi.org/10.3390/su12229656
Chicago/Turabian StyleParedes, Sara, María José Cáceres, José-Manuel Diego-Mantecón, Teresa F. Blanco, and José María Chamoso. 2020. "Creating Realistic Mathematics Tasks Involving Authenticity, Cognitive Domains, and Openness Characteristics: A Study with Pre-Service Teachers" Sustainability 12, no. 22: 9656. https://doi.org/10.3390/su12229656