Paul Ernest proposed, in a tentative and incomplete way, some primary and secondary objects of study of research in mathematics education. While the secondary objects come after the primary objects, Ernest claimed that they are "important too, and should not be neglected." These primary objects of study are • The nature of mathematics and school mathematical knowledge • The learning of mathematics • The aims and goals of mathematics teaching and schooling • The teaching of mathematics, including the methods and approaches involved • The full range of texts, materials, aids and electronic resources employed • The human and social contexts of mathematics learning/teaching in all their complexity • The interaction and relationships between all of the above factors. The secondary objects of study are • The nature of mathematics education knowledge: its concepts, theories, results, literature, aims and function • The nature of mathematics education research: its epistemology, theoretical bases, criteria, methodology, methods, outcomes and goals • Mathematics education teaching and learning in teacher education, including practice, technique, theory and research • The social institutions of mathematics education: the persons, locations, institutions (universities, colleges, research centers), conferences, organizations, networks, journals, etc. and their relationships with its overall social or societal contexts. According to Hiebert and Grouws, "Robust, useful theories of classroom teaching do not yet exist." However, there are useful theories on how children learn mathematics, and much research has been conducted in recent decades to explore how these theories can be applied to teaching. The following results are examples of some of the current findings in the field of mathematics education.
Important results :One of the strongest results in recent research is that the most important feature of effective teaching is giving students "the opportunity to learn". Teachers can set expectations, times, kinds of tasks, questions, acceptable answers, and types of discussions that will influence students' opportunities to learn. This must involve both skill efficiency and conceptual understanding.) These connections can be made through explanation of the meaning of a procedure, questions comparing strategies and solutions of problems, noticing how one problem is a special case of another, reminding students of the main point, discussing how lessons connect, and so on. :Deliberate, productive struggle with mathematical ideas refers to the fact that when students exert effort with important mathematical ideas, even if this struggle initially involves confusion and errors, the result is greater learning. This is true whether the struggle is due to intentionally challenging, well-implemented teaching, or unintentionally confusing, faulty teaching.
Formative assessment :
Formative assessment is both the best and cheapest way to boost student achievement, student engagement, and teacher professional satisfaction. Results surpass those of reducing class size or increasing teachers' content knowledge. Effective assessment is based on clarifying what students should know, creating appropriate activities to obtain the evidence needed, giving good feedback, encouraging students to take control of their learning and letting students be resources for one another.
Homework :
Homework assignments which lead students to practice past lessons or prepare for future lessons are more effective than those going over the current lesson. Students benefit from feedback. Students with learning disabilities or low motivation may profit from rewards. For younger children, homework helps simple skills, but not broader measures of achievement.
Students with difficulties Source:
Algebraic reasoning :Elementary school children need to spend a long time learning to express algebraic properties without symbols before learning algebraic notation. When learning symbols, many students believe letters always represent unknowns and struggle with the concept of
variable. They prefer arithmetic reasoning to
algebraic equations for solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns. Students often have trouble with the
minus sign and understand the
equals sign to mean "the answer is...". suggests that effective mathematics teaching of culturally diverse students requires a
culturally relevant pedagogy that considers students' cultural backgrounds and experiences. The three criteria for culturally relevant pedagogy are academic success, cultural competence, and critical consciousness. More recent research proposes that culturally sustaining pedagogy explicitly aims to perpetuate and foster cultural and linguistic pluralism within the educational system, ensuring that students can thrive while retaining their cultural identities.
Mathematics Teacher Education :
Student teaching is a crucial part of a teacher candidate's path to becoming a teacher. Recommended reform in mathematics teacher education includes a focus on learning to anticipate, elicit, and use students’ mathematical thinking as the primary goal, as opposed to models with an over-emphasis on classroom management and survival.
Methodology 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 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" Many studies are "mixed", simultaneously combining aspects of both quantitative and qualitative research, as appropriate.
Randomized trials There has been some controversy over the relative strengths of different types of research. Because of an opinion that randomized trials provide clear, objective evidence on "what works", policymakers often consider only those studies. Some scholars have pushed for more random experiments in which teaching methods are randomly assigned to classes. In other disciplines concerned with human subjects—like
biomedicine,
psychology, and policy evaluation—controlled, randomized experiments remain the preferred method of evaluating treatments. Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods. or the difficulty of assuring rigid control of the independent variable in fluid, real school settings. In the United States, the
National Mathematics Advisory Panel (NMAP) published a report in 2008 based on studies, some of which used randomized assignment of treatments to
experimental units, such as classrooms or students. The NMAP report's preference for randomized experiments received criticism from some scholars. In 2010, the
What Works Clearinghouse (a division of the
Institute of Education Sciences, which is the research arm of the
Department of Education) responded to ongoing controversy by extending its research base to include non-experimental studies, including
regression discontinuity designs and
single-case studies. ==History==