24. Mathematics Education Research
Poster
Centennial European Doctoral Disssertations in Mathematics Education. A Documentary Archology Exercise on the Origins of European University Research
Mónica Vallejo-Ruiz1, Natividad Adamuz-Povedano2, Manuel Torralbo-Rodríguez2, Antonio Fernández-Cano3
1Universidad de Murcia, Spain; 2Universidad de Córdoba, Spain; 3Universidad de Granada, Spain
Presenting Author: Adamuz-Povedano, Natividad
Scientists, in general, and researchers and university teachers, in particular, should have precise information and knowledge about the origins of their disciplinary field. This idea reminds us of Aristotle's argument that we understand a subject best when we see it grow from its origins (Metaphysics, Book I, Chapter I, p. 12). Specifically, he said: "seeking first to understand the causes of the entities that surround us .... in the search for their causes, they advanced to that one".
In the case of Mathematics Education, different authors (Kilpatrick, 1998, 2020; Artigue & Blomhøj, 2013; Fiorentini & Lorenzato, 2015) have described how, during the last two centuries, research in this field has focused on analysing what mathematics was developed in schools and how it was taught and what learning (usefulness) resulted from it. Parallel to this scientific development, Kilpatrick (1998) raised the problems inherent in this work; in particular, the constitution of a group of people who identified themselves as researchers in mathematics education and focused their efforts on defining them and instituting their own research methods. During this period, figures such as Augustin Cauchy, responsible for the reorganisation and foundation of the field of mathematical analysis at the beginning of the 19th century, Ulysse Dini, from the University of Pisa, who wrote the first modern treatise on functions of a real variable, or Emile Borel who, from his collection of works on the theory of functions, developed some twenty volumes that gave the international mathematical community access to the most recent developments in research in this field (Artigue, 2020). Belhoste (1998), for his part, states that the constitution and institutionalisation of the mathematical community in countries such as Germany, France and Italy is largely due to the development of educational institutions focused on the teaching of this discipline.
These and many other examples from the scientific literature describe in great detail the origins of the development of the scientific community of mathematics education; however, we have not found any bibliographical evidence on the origins of research in mathematics education through doctoral theses or dissertations in this scientific field.
As we know, doctoral theses have a long history from the Middle Ages to present-day universities, but the recovery of the first European theses on this field of study -their reading and analysis- is a hard and complex exercise in documentary archaeology. Institutional repositories and databases are sources in which to search for doctoral theses and master's theses; some have indexed dissertations (we speak of dissertations, as a general term, with two variants: doctoral theses and master's theses) from more than 200 years ago. Databases in this respect are: the Spanish database Catalog-CISNE, the British Library-Ethos: e-thesis, the German Deustsche Nationalbibliothek or the Belgian Dial. UCLovaine. Other European dissertation databases are more recent, dating from the mid-20th century, such as the French TEL-These en ligne or the Dutch NARCIS.
For all these reasons, the present study focuses on identifying and analysing the origins of university research in Mathematics Education. Following a documentary archaeology exercise, a total of 23 European doctoral and master's theses in the field of mathematics education, defended more than one hundred years ago (1854-1917), have been located. They are thus centuries-old dissertations and indicative of the origins of university research in this field. In this way, this work is a first step towards understanding and illuminating the field of mathematics education from its beginnings and, by extension, as a study of the history of science geographically limited to Europe.
Methodology, Methods, Research Instruments or Sources UsedThe study of documentary sources allows the use of different approaches, depending on the objective, the questions posed or the nature of the documentary sources. In this case, we have used what is known as "documentary archaeology", i.e., the analysis of documents extracted from historical records. In the field of archaeology, such studies aim to demonstrate to archaeologists that the historical record, far from being a finite body of specialised information, is in fact a rich treasure trove of new insights into the past (Beaudry, 1988; Saldanha, 2020; City of Alexandria, 2022).
It is also a descriptive-exploratory-retrospective study that analyses a total of 23 dissertations that could be called the pioneering European university works in Mathematics Education. These works have been located in four European databases: the Spanish database Catalog-CISNE, the British Library-Ethos: e-thesis, the German Deustsche Nationalbibliothek or the Belgian Dial.UCLovaine.
Through successive advanced computer searches in the various databases indicated, the time span 1800-1922 was established. The keywords entered interchangeably were: dissertation, thesis, education, mathematics, arithmetic, geometry. However, the validation of the retrieved document as a "pioneering European thesis in mathematics education" requires a detailed, quasi-craftsmanlike tracing. Since we have only been able to operate with secondary sources (the reference of the thesis contained in a database), access to the thesis itself (primary source) has only been possible in some cases where these had been digitised. The methodological procedure followed is that proposed by documentary archaeology in the following phases:
- Selecting the site: European Education.
- Conducting research: about Centennial dissertations in mathematics education.
- Excavating the site: in European databases on theses and masters. Search sequences. High concern about the validity of primary and secondary sources.
- Cleaning and cataloguing artifacts (dissertations): cleaning of retrieved documents, cataloguing dissertations in Excel database, analyzing data and interpreting.
- Reporting the results: Treatment of the information obtained (i. e. this poster).
The following indicators have been analysed: year of defence, conceptual analysis of the title, author, language, keywords and the producing university.
Conclusions, Expected Outcomes or FindingsThe documentary archaeology work carried out has allowed us to locate and recover 23 theses that fit the aforementioned characteristics; a meagre number, almost an appendix to the dissertations in the two main disciplines: mathematics and education.
Some general conclusions can be drawn from the analysis of these works:
- The conceptual analysis of the titles of the academic works determines that, except for four doctoral theses, the focus of university studies revolves around mathematics (its origin and history) or any of its disciplines (arithmetic, geometry, probability, calculus, and measurement).
- The author's genre according to his or her own name (Vallejo, Torralbo & Fernández-Cano, 2016). All the theses analysed have been written by men, except the one defended by Eva Sachs (1917), written within the area of Philological Studies, although with a notable connotation with mathematics education.
- Eleven of the doctoral dissertations are written in German, seven in Spanish, and the rest in French and English.
- The German producing universities are the Universität Erlangen and the Universität Munich, with three and two doctoral theses, respectively, and, in the case of Spain, production is centralised only at the Universidad Central de Madrid, with seven. The Université Catholique de Louvain is also worth mentioning, with a production of two doctoral theses.
Finally, it should be noted that given the current state of European databases as highly idiosyncratic, national, and not connected, a work of integration of European databases of doctoral dissertations is proposed as an alternative to the powerful American ProQuest Dissertations attempting to conform a specific and updated database for the field of mathematics education. Other educational disciplines and specialties would be able to develop their own databases of dissertations and other end-of-studies projects.
ReferencesARTIGUE, M. (2020). El desarrollo de la didáctica de las matemáticas, una mirada internacional. Revista Chilena de Educación Matemática, 12(3), 83-95.
ARTIGUE, M. & BLOMHØJ, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM, 45(6), 797-810.
BEAUDRY, M. C. (1988). Introduction. In: Documentary archaeology in the New World (pp. 1-3). Cambridge University Press.
BELHOSTE, B. (1998). Pour une réévaluation du rôle de l’enseignement dans l’histoire des mathématiques [For a re-evaluation of the role of teaching in the history of mathematics]. Revue d’Histoire des Mathématiques, 4, 289-304. Retrieved from: http://www.numdam.org/article/RhM_1998__4_2_289_0.pdf
CITY OF ALEXANDRIA, VA. (2022). Archaeological process. Retrieved from: https://www.alexandriava.gov/archaeology/archaeological-process
FIORENTINI, D., & LORENZATO, S. (2015). Investigación en Educación Matemática: recorridos históricos y metodológicos [Research in Mathematics Education: historical and methodological background]. Autores Asociados, LTDA.
KILPATRICK, J. (1998). Investigación en educación matemática: su historia y algunos temas de actualidad [Research in mathematics education: its history and some current issues]. In J. Kilpatrick, P. Gómez, & L. Rico, (Eds.), Educación Matemática: Errores y dificultades de los estudiantes. Resolución de problemas. Evaluación Historia [Mathematics Education: Students' errors and difficulties. Problem solving. Evaluation History] (p. 1-18). Una Empresa Docente e Universidad de los Andes.
KILPATRICK, J. (2020). History of research in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (p. 349-354). Springer.
SACHS, E. (1917). Die fiinf platonischen Korper. Zur Geschichte der Mathematik und der Elementenlehre Platons und der Pythagoreer. Berlin: Weidmann.
SALDANHA, G.S. (2020). Linha Cumeada: an archeology of the epistemological statements of Bibliography in the foundation of Information Science. Encontros Bibli-Revista Eletronica de Biblioteconomia e Ciencia da Informacao, 25, (Especial), 1-16. https://doi.org/10.5007/1518-2924.2020.e73443
VALLEJO, M., TORRALBO, M. & FERNÁNDEZ-CANO, A. (2016). Gender bias in higher education: Spanish doctoral dissertations in mathematics education. Journal of Hispanic Higher Education, 15(3), 205-220. https://doi.org/10.1177/153819271559
Infographics (links to databases consulted):
https://www.dnb.de/SharedDocs/Downloads/DE/Professionell/Netzpublikationen/anleitungSucheDissertationen.html
https://ucm.on.worldcat.org/advancedsearch?queryString=tesis&databaseList=
https://www.bl.uk/ethos-and-theses
https://dial.uclouvain.be/memoire/ucl/en/search/site
24. Mathematics Education Research
Poster
How Does the Compare and Contrast Strategy of Concept-Based Learning Affect the High-Order Thinking Skills of 7th Grade Students?
Almira Kaliyeva, Ardak Khaliyeva, Alibi Skakov
Nazarbayev Intellectual school, Kazakhstan
Presenting Author: Kaliyeva, Almira;
Khaliyeva, Ardak
This research considers the use of one of the strategies of the concept-based learning, which allows students to absorb knowledge in a universal way, on the subject of mathematics when teaching secondary school students with this method and its features.
According to the traditional two-dimensional model of the curriculum, the content of knowledge is presented within certain topics that contain data. The given data usually form the student's ability to know and understand. It is observed that students in this type of model usually study factual information mostly and are not able to answer questions of the high order thinking types for analysis and evaluation (Medwell, J. & Wray, D., 2020). The acquired knowledge remains in the state of individually isolated data without logical relation between facts and conclusions. This does not contribute to the revitalization and development of deep-level mental activity of an individual.
Meanwhile, conceptual learning assumes not a simple comprehension of knowledge through teacher delivering but full immersion and skills focused curriculum (Erickson L., 2017). According to Wiggins and McTighe (1998), "A concept is a principle or conception of a broad-minded, long-term character that transcends the context of origin, time period, and material content." Based on the given definitions, the acquired knowledge should not be limited to a certain level, the knowledge should be used in a variable manner, on a wide scale.
Unlike a two-dimensional curriculum based on facts and skills, concept-based learning is based on big ideas (Murphy, 2017), not just subject content. And the ideas that arise from knowledge are wide-ranging, interconnected and interdisciplinary. For example, through concept-based learning, students can explore the big idea of "change," from patterns in mathematics to civilizations in social studies to life cycles in science. They become critical thinkers, develop the ability to solve problems creatively.
According to the results of the monitoring of knowledge, conducted at the beginning of the academic year, it was found that students struggle in performing tasks that require higher order thinking skills. The rate of completion of tasks assigned to mathematical modeling, which requires analytical skills, was 47%, while the level of completion of tasks requiring data collection and processing was 68%. The results of monitoring showed that although students have a tendency to perform calculations based on theoretical knowledge but analysis, evaluation and drawing conclusions are difficult for them.
This made the relevance of this research to study and draw conclusions on how the use of comparison and contrast strategy contributes to the ability of middle school students to connect individual facts and the formation of inference skills.
In this context, teaching through concept-based approach a teacher needs effective strategies to develop thinking skills of students. One of such strategies is a compare and contrast method that is aimed at drawing conclusions on the case studies by distinguishing similar situations and contrasting phenomena between two or more concepts.
The purpose of the study is to determine the impact of the comparison and contrast strategy of teaching on the basis of concepts on the formation of high-order skills of middle school students.
We also aimed to identify the difficulties that arise when using this strategy for students of this age to further find the best solutions on them.
Research questions:
- To what extent does the compare and contrast strategy affect the development of students’ high-order thinking skills on Maths lessons?
- What are the possible difficulties students may face when learning through the strategy and method?
This research is qualitative as the main method was to study five focused groups of students of grade seven (84 students in total).
Methodology, Methods, Research Instruments or Sources UsedDocument analysis including evaluation of students’ works and results, lesson observations, survey and interview methods were used in data collection. A comparison was made based on the results of the two sections, which are high in importance, and the changes in the students' knowledge and skills were differentiated.
Before the study, the program of the 7th grade was analyzed and micro-concepts related to the subject were determined. The main concepts that are convenient for using the strategy have been selected. Model lesson plans were developed according to the "compare and contrast" strategy of concept-based teaching. The pattern of planned lessons corresponds to the learning objectives in the curriculum. Lesson plans included questions and tasks aimed at revealing the vital importance of basic concepts such as numbers, equation, dependences and patterns, functions. The tasks aimed at the student's analysis and collection of information. In addition, instructions were attached to guide the student to independently research and plan actions to complete complex assignments accordingly. These activities were organized in the form of individual work, pair work and work in small groups.
In order to determine the effectiveness of the method being used, a questionnaire was taken from the students. The personal interview, which included structured questions, helped to find out the opinion of the students, determine the priority directions, and make plans in a new direction accordingly.
According to the results of the survey after the lessons with the compare and contrast strategy, 70% of the students indicated that it was interesting to solve the tasks given to identify patterns, establish connections, and make conclusions by experimenting. They mentioned the need of using logic in these tasks and found them interesting. However, 32% of students reported that they have difficulty understanding the terms of tasks of practical importance.
18 out of 30 students who participated in the personal interview preferred to solve problems in class using only ready-made formulas and properties, but 12 students preferred to conduct research and draw conclusions on their own. 60% of the students emphasized that they liked to create definitions for new concepts and create rules on their own by describing and observing their features. According to them, it helped to remember the necessary definitions better. Two out of 14 students who liked to solve practical problems answered that it would be practical skill in the future.
Conclusions, Expected Outcomes or FindingsThe research made it possible to differentiate the effectiveness of the methods used in teaching based on the findings. To conclude, one of the most effective ways is to identify two or more related ideas and conclusions, and to give students tasks that combine these concepts or offer small research works. In this case, the educational goals of several subjects are covered by one task, and mutual communication is realized in a real way. The student uses the knowledge and data acquired to perform the task, determines and analyzes their connection, and accumulates on the basis of deceptions and reaches the level of assessment of thinking skills. Although integrated learning, which is usually used to transform knowledge, realizes interdisciplinary communication, in many cases it can lead to the creation of artificial communication. Therefore, in addition to small research works offered in class, giving creative tasks such as experimenting and designing outside of class or as homework allows students to think freely.
Thus, based on the findings, it is clear that the implementation of the strategy alone without the stage of learning facts and theories is not possible.
Using the compare-and-contrast strategy, students conducted analysis while completing small research tasks. Pupils distinguished common properties or features of the given concepts, classified them, and determined the interconnection. Based on the results of their analysis, they made a general conclusion. In this way, it was assumed that they would learn to perceive the main concepts in mathematics as a big idea related to the world, instead of taking them at the level of the topic as usual.
ReferencesBolter J., Burns G., Linsky J., (2011). Higher revision workbook. Pearson Company.
Bostock L., Chandler S., Shepherd A., Smith E., (1992). Mathematics to level 10 a full GCSE Course. Stanley Thornes Ltd.
Erickson, H., Lanning, L., & French, R. (2017). Concept-Based curriculum and instruction for the thinking classroom. Corwin, https://dx.doi.org/10.4135/9781506355382
Medwell, J. & Wray, D. (2020). CONCEPT-BASED TEACHING AND LEARNING: A REVIEW OF THE RESEARCH LITERATURE. 486-496. 10.21125/iceri.2020.0144.
Murphy, A. (2017). A Quick Guide to Concept-Based Learning and Curriculum - Atlas. Atlas Curriculum Mapping. Retrieved January 29, 2023, from https://www.onatlas.com/blog/concept-based-learning-curriculum
Strong S., & Associates (2017). Compare and contrast. Sample lessons. Thoughtful Education Press. www. ThoughtfulEd.com.
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