How would you integrate the mathematical concept of properties of integers in Social Studies, Science or Music?
How would you integrate the mathematical concept of properties of integers in Social Studies, Science or Music?
How would you integrate the mathematical concept of properties of integers in Social Studies, Science or Music?
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Mathematics, and Science
The preceding chapter explored implications of research on learning for general issues relevant to the design of effective learning environments. We now move to a more detailed exploration of teaching and learning in three disciplines: history, mathematics, and science. We chose these three areas in order to focus on the similarities and differences of disciplines that use different methods of inquiry and analysis.
A major goal of our discussion is to explore the knowledge required to teach effectively in a diversity of disciplines.
We noted in Chapter 2 that expertise in particular areas involves more than a set of general problem-solving skills; it also requires well-organized knowledge of concepts and inquiry procedures. Different disciplines are organized differently and have different approaches to inquiry.
For example, the evidence needed to support a set of historical claims is different from the evidence needed to prove a mathematical conjecture, and both of these differ from the evidence needed to test a scientific theory. Discussion in Chapter 2 also differentiated between expertise in a discipline and the ability to help others learn about that discipline. To use Shulman’s (1987) language, effective teachers need pedagogical content knowledge (knowledge about how to teach in particular disciplines) rather than only knowledge of a particular subject matter.
Pedagogical content knowledge is different from knowledge of general teaching methods. Expert teachers know the structure of their disciplines, and this knowledge provides them with cognitive roadmaps that guide the assignments they give students, the assessments they use to gauge students’ progress, and the questions they ask in the give and take of classroom life.
In short, their knowledge of the discipline and their knowledge of pedagogy interact. But knowledge of the discipline structure does not in itself guide the teacher. For example, expert teachers are sensitive to those aspects of the discipline that are especially hard or easy for new students to master.
National Academies of Sciences, Engineering, and Medicine. 2000. How People Learn: Brain, Mind, Experience, and School: Expanded Edition. Washington, DC: The National Academies Press. https://doi.org/10.17226/9853.
This means that new teachers must develop the ability to “understand in a pedagogically reflective way; they must not only know their own way around a discipline, but must know the ‘conceptual barriers’ likely to hinder others” (McDonald and Naso, 1986:8). These conceptual barriers differ from discipline to discipline.
An emphasis on interactions between disciplinary knowledge and pedagogical knowledge directly contradicts common misconceptions about what teachers need to know in order to design effective learning environments for their students. The misconceptions are that teaching consists only of a set of general methods, that a good teacher can teach any subject, or that content knowledge alone is sufficient.
National Academies of Sciences, Engineering, and Medicine. 2000. How People Learn: Brain, Mind, Experience, and School: Expanded Edition. Washington, DC: The National Academies Press. https://doi.org/10.17226/9853.
Some teachers are able to teach in ways that involve a variety of disciplines. However, their ability to do so requires more than a set of general teaching skills. Consider the case of Barb Johnson, who has been a sixth-grade teacher for 12 years at Monroe Middle School.
By conventional standards Monroe is a good school. Standardized test scores are about average, class size is small, the building facilities are well maintained, the administrator is a strong instructional leader, and there is little faculty and staff turnover. However, every year parents sending their fifth-grade students from the local elementary schools to Monroe jockey to get their children assigned to Barb Johnson’s classes. What happens in her classroom that gives it the reputation of being the best of the best?
National Academies of Sciences, Engineering, and Medicine. 2000. How People Learn: Brain, Mind, Experience, and School: Expanded Edition. Washington, DC: The National Academies Press. https://doi.org/10.17226/9853.
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