In taking this stance, we follow de Certeau in contending that an artifact such as an instructional sequence developed by one group is necessarily reshaped and transformed when others use it. This perspective is also useful for research on teacher professional development in that it orients us to explain why teachers adapt instructional sequences in particular ways by understanding their evolving instructional practices as they are situated in particular institutional settings.

If we had justified the instructional sequence solely with traditional experimental data, the teachers would know that this sequence had proved effective elsewhere but would not have access to the underlying rationale that would enable them to adapt it effectively to their own instructional settings. In contrast, the type of rationale that we developed constitutes a potentially important resource for teachers as they adapt, test, and modify the instructional sequence in their classrooms.

Our intent in the professional development experiment was therefore to engage teachers in activities that would enable them to reconstruct the rationale for the instructional sequence. As we noted when discussing the statistics design experiment, data generation discussions play a significant role in shaping the ways in which students interpret data.

Early in our collaboration, the teachers recognized that data generation discussions were an important aspect of instruction. However, it became apparent that, from their perspective, effective instructional activities involved a scenario that was immediately interesting and personally relevant to students. For example, the teachers considered that instructional activities that involve soft drinks or roller coasters were instructionally more promising than those that focused on issues of broader social significance e.

This approach was reasonably successful in that the teachers came to view a broader range of problem scenarios as potentially productive and saw it as their responsibility to develop the significance and relevance of problem situations with students. This development would have been unlikely had we not made the rationale for the instructional sequence an explicit focus of professional development activities. The products of teacher development research of this type might include prototypical sequences of activities and resources for teacher-researcher collaboration together with a rationale that is cast in terms of a the actual learning trajectory for a professional teaching community, and b means by which that learning can be supported.

We speculate that the rationales of this type will enable other researchers and teacher educators to adapt the prototypical sequences to the specific settings in which they are collaborating with teachers in a conjecture-driven manner. These heuristics have not been derived from a general background theory but instead they are empirically grounded in the activity of developing, testing, and revising specific designs in classrooms. The enduring contribution of RME resides in the solution that it proposes to the perennial question of how to induct students into established mathematical practices while simultaneously taking their current understandings and interests seriously.

These various means of support are strongly interrelated and can be viewed as aspects of an encompassing classroom activity system. The goal for instructional development is therefore to design classroom activity systems such that students develop significant mathematical ideas as they participate in them and contribute to their evolution.

The second adaptation emphasized the critical role of teachers as co-designers of classroom activity systems. We argued that the implementation of instructional sequences necessarily involves a process of adaptation. This orientation offers the prospect that research on teacher professional development might become a design science that involves developing, testing, and modifying designs for supporting the learning of professional teaching communities and the participating teachers. Planning for teaching statistics through problem solving.

National Council of Teachers of Mathematics. Learning to reason about distribution. With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal , 93 4 , Reform by the book: What is - or might be - the role of curriculum materials in teacher learning and instructional reform? Educational Researcher , 25 9 , , Developing practice, developing practitioners: Towards a practice-based theory of professional education.

Handbook of policy and practice pp.

- Mathematics education - Wikipedia.
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Software tools and mathematics education: The case of statistics. Mathematics education and technology pp. Creating Usable Innovations in Systemic Reform: Educational Psychologist , 35 3 , - Assisting teachers and students to reform the mathematics classroom. Educational Studies in Mathematics , 31 , Asking the right questions about teacher preparation: Contributions of research on teacher thinking.

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Educational Researcher , 17 2 , Overcoming misconceptions via analogical reasoning: Abstract transfer versus explanatory model construction. Instructional Science , 18 , Individual and collective mathematical learning: The case of statistical data analysis. Mathematical Thinking and Learning , 1 , Reasoning with tools and inscriptions. Journal of the Learning Sciences , 11 , Proposed design principles for the teaching and learning of elementary statistics.

Learning about statistical covariation. Cognition and Instruction , 21 , The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education , 14 , 83 — Participating in classroom mathematical practices. Journal of the Learning Sciences , 10 , Constructivist, emergent, and sociocultural perspectives in the context of developmental research.

Educational Psychologist , 31 , Belknap Press of Harvard University Press. What implementation research reveals about urban reform in mathematics.

## Learning from and Adapting the Theory of Realistic Mathematics education

Problems of developmental teaching Part I. Soviet Education , 30 8 , The influence of L. Vygotsky on education theory, research, and practice. Educational Researcher , 24 3. Relationships between macro and micro socio-cultural contexts: Implications for the study of interactions in the mathematics classroom. Educational Studies in Mathematics , 41 , The practice of everyday life. University of California Press. Learning and testing mathematics in context. The later works Vol. Southern Illinois University Press.

Individual and collective conflicts of centrations in cognitive development. European Journal of Psychology , 9 , Perspectives on discourse, tools, and instructional design pp. American Journal of Sociology , , Perspectives on learning to teach. American Educational Research Journal , 38 , Geometry between the devil and the deep sea. Educational Studies in Mathematics , 3 , Mathematics as an educational task.

Transforming teaching in math and science: How schools and districts can support change. The organizational context of teaching and learning: Context problems and realistic mathematics instruction. Developing realistic mathematics education. How emergent models may foster the constitution of formal mathematics. Symbolizing, modeling, and instructional design. Authentic inquiry with data: Critical barriers to classroom implementation.

Educational Pyschologist , 27 , Identity in cultural worlds. Highlights related to research. Collecting, representing, and analyzing data. Research of critical barriers. Teaching problems and the problems of teaching.

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Cognition and Instruction , 14 , Symbolizing, communicating, and mathematizing: Key components of models and modeling. Mathematics practice in carpet laying. Anthropology and Education Quarterly , 25 , The role of tools and inscriptions in supporting effective communication. Developing a conjectured learning trajectory. Journal of Mathematical Behavior , 16 , Building new knowledge by thinking: How administrators can learn what they need to know about mathematics education reform. Meanings of average and variation in nursing practice. Educational Studies in Mathematics , 40 1 , Street mathematics and school mathematics.

Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics , 26 , Reform, projects, and the language game. Educational Researcher , 24 7 , Reasoning with metaphors and metonygmies in mathematics learning. Analogies, metaphors, and images. Where is the context in contextual word problems? Alternative strategies for the organizational design of schools. Studying cognitive development in sociocultural context: The development of a practice-based approach.

Mind, Culture, and Activity , 1 , Issues in the transformation of mathematics teaching pp. Reflective reform in mathematics: The recursive nature of teacher change. Educational Studies in Mathematics , 37 , On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.

Educational Studies in Mathematics , 22 , The development of the concept of concept development: Relational understanding and instrumental understanding.

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Arithmetic Teacher , 77 , Efficacy and teaching mathematics by telling: A challenge for reform. Journal for Research in Mathematics Education , 27 , The teaching experiment methodology in a constructivist research program. Radical constructivism and mathematics education. Journal for Research in Mathematics Education , 25 , Underlying principles and essential elements. Teacher learning in a social context: Integrating collaborative and institutional processes with the study of teacher change. Fractions in realistic mathematics education.

A paradigm of developmental research. Rousing minds to life. Calculational and conceptual orientations in teaching mathematics. Talking about rates conceptually, part II: Mathematical knowledge for teaching. Notations, principles, and constraints: Contributions to the effective use of concrete manipulatives in elementary mathematics. Journal for Research in Mathematics Education , 23 , Epistemological analyses of mathematical ideas: A quest for synthesis. Learning mathematics as meaningful activity. The schools that America builds.

Problems of general psychology. John Dewey and American democracy. Association for Curriculum and Supervision. What is normal anyway? Elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece , the Roman Empire , Vedic society and ancient Egypt. In most cases, a formal education was only available to male children with a sufficiently high status, wealth or caste.

In Plato 's division of the liberal arts into the trivium and the quadrivium , the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclid 's Elements. Apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession.

In the Renaissance , the academic status of mathematics declined, because it was strongly associated with trade and commerce, and considered somewhat un-Christian. The first modern arithmetic curriculum starting with addition, then subtraction, multiplication, and division arose at reckoning schools in Italy in the s. They contrasted with Platonic math taught at universities, which was more philosophical and concerned numbers as concepts rather than calculating methods. For example, the division of a board into thirds can be accomplished with a piece of string, instead of measuring the length and using the arithmetic operation of division.

The first mathematics textbooks to be written in English and French were published by Robert Recorde , beginning with The Grounde of Artes in However, there are many different writings on mathematics and mathematics methodology that date back to BCE. These were mostly located in Mesopotamia where the Sumerians were practicing multiplication and division. There are also artifacts demonstrating their own methodology for solving equations like the quadratic equation. After the Sumerians some of the most famous ancient works on mathematics come from Egypt in the form of the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus.

The more famous Rhind Papyrus has been dated to approximately BCE but it is thought to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students. The social status of mathematical study was improving by the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in , followed by the Chair in Geometry being set up in University of Oxford in and the Lucasian Chair of Mathematics being established by the University of Cambridge in However, it was uncommon for mathematics to be taught outside of the universities.

In the 18th and 19th centuries, the Industrial Revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic , became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age. By the twentieth century, mathematics was part of the core curriculum in all developed countries. During the twentieth century, mathematics education was established as an independent field of research. Here are some of the main events in this development:.

In the 20th century, the cultural impact of the " electronic age " McLuhan was also taken up by educational theory and the teaching of mathematics.

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While previous approach focused on "working with specialized 'problems' in arithmetic ", the emerging structural approach to knowledge had "small children meditating about number theory and ' sets '. At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included:. The method or methods used in any particular context are largely determined by the objectives that the relevant educational system is trying to achieve.

Methods of teaching mathematics include the following:. Different levels of mathematics are taught at different ages and in somewhat different sequences in different countries. Sometimes a class may be taught at an earlier age than typical as a special or honors class. Elementary mathematics in most countries is taught in a similar fashion, though there are differences.

In the United States fractions are typically taught starting from 1st grade, whereas in other countries they are usually taught later, since the metric system does not require young children to be familiar with them. In most of the U. Mathematics in most other countries and in a few U. Students in science-oriented curricula typically study differential calculus and trigonometry at age 16—17 and integral calculus , complex numbers , analytic geometry , exponential and logarithmic functions , and infinite series in their final year of secondary school.

Probability and statistics may be taught in secondary education classes. Science and engineering students in colleges and universities may be required to take multivariable calculus , differential equations , linear algebra. Applied mathematics is also used in specific majors; for example, civil engineers may be required to study fluid mechanics , [12] while "math for computer science" might include graph theory , permutation , probability, and proofs.

Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to, realistic for, and considered socially appropriate for their pupils. In modern times, there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum.

In England , for example, standards for mathematics education are set as part of the National Curriculum for England, [14] while Scotland maintains its own educational system. Ma summarised the research of others who found, based on nationwide data, that students with higher scores on standardised mathematics tests had taken more mathematics courses in high school. This led some states to require three years of mathematics instead of two. In North America, the National Council of Teachers of Mathematics has published the Principles and Standards for School Mathematics , which boosted the trend towards reform mathematics.

In , they released Curriculum Focal Points , which recommend the most important mathematical topics for each grade level through grade 8. However, these standards are enforced as American states and Canadian provinces choose. A US state's adoption of the Common Core State Standards in mathematics is at the discretion of the state, and is not mandated by the Federal Government. The MCTM also offers membership opportunities to teachers and future teachers so they can stay up to date on the changes in math educational standards. The following results are examples of some of the current findings in the field of mathematics education:.

As with other educational research and the social sciences in general , mathematics education research depends on both quantitative and qualitative studies. Quantitative research includes studies that use inferential statistics to answer specific questions, such as whether a certain teaching method gives significantly better results than the status quo. The best quantitative studies involve randomized trials where students or classes are randomly assigned different methods in order to test their effects.

They depend on large samples to obtain statistically significant results. Qualitative research , such as case studies , action research , discourse analysis , and clinical interviews , depend on small but focused samples in an attempt to understand student learning and to look at how and why a given method gives the results it does.

Such studies cannot conclusively establish that one method is better than another, as randomized trials can, but unless it is understood why treatment X is better than treatment Y, application of results of quantitative studies will often lead to "lethal mutations" [19] of the finding in actual classrooms. Exploratory qualitative research is also useful for suggesting new hypotheses, which can eventually be tested by randomized experiments.

Both qualitative and quantitative studies therefore are considered essential in education—just as in the other social sciences. There has been some controversy over the relative strengths of different types of research. Some scholars have pushed for more random experiments in which teaching methods are randomly assigned to classes. In the United States, the National Mathematics Advisory Panel NMAP published a report in based on studies, some of which used randomized assignment of treatments to experimental units , such as classrooms or students.