Upon reading the framework, it is evident that the major idea of teaching mathematics in California public schools revolves around the development of a balanced instructional program that not only provides students with an enabling environment to become proficient in basic computational and procedural skills, but also to continuously develop conceptual understanding of the mathematical concepts and become proficient in problem-solving.
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As indicated in the framework, teachers should aim to achieve a balance between these three concepts (proficiency in computational and procedural skills, attainment of conceptual understanding, and proficiency in solving new or perplexing problems) if they expect their students to be competitive in mathematics (California Department of Education, 2006).
The concepts are interrelated; hence, teachers must strive to come up with methodologies and strategies to deliver them to mathematics students according to the standards of a particular grade level.
As indicated in the framework, “when students apply basic computational and procedural skills and understandings to solve new or perplexing problems, their basic skills are strengthened, the challenging problems they encounter can become routine, and their conceptual understanding deepens” (California Department of Education, 2006 p.5).
Consequently, it is suggested that the major idea of teaching mathematics to students in California public schools entails connecting their skills, conceptual understanding, and problem-solving capability to develop a network of mutually reinforcing components in the curriculum that are intrinsically aligned with the standards depending on grade level.
Examples of Teaching Strategies
From the framework, it is clear that no single strategy of instruction is the best or most appropriate in all contexts, and that teachers have a wide choice of instructional strategies including “direct instruction, investigation, classroom discussion and drill, small groups, individualized formats, and hands-on materials (California Department of Education, 2006 p.5).
As one of the approaches to teaching, direct instruction is not only skills-oriented, but the teaching practices it adopts are essentially teacher-directed.
In teaching algebra and functions to grade six students, for example, a teacher can use small-group, face-to-face instruction to demonstrate to students how to solve linear equations and develop algebraic reasoning at each step of the process by breaking down the instructions into small units, sequencing them deliberately, and teaching them in an explicit manner.
Although teachers can use this strategy to ensure that Grade six students are able to use their computational skills and conceptual understanding to solve problems in algebra through explicit, guided instructions, the strategy nevertheless limits student’s creativity and active exploration.
The other teaching strategy is investigation, whereby teachers play an active role in guiding students to identify a topic of interest, explore the current knowledge on the topic, frame the topic into manageable questions, gather appropriate information, analyze and synthesis the information, take action on the findings, and reflect on the outcomes found.
This strategy could be used to teach geometry to grade six students as it does not only facilitate an explicit understanding of geometric concepts, including raising students’ levels of geometric thinking, but also motivates students by presenting mathematical topics in an enjoyable and interesting manner that challenges their intellectual development.
Overall, upon reflection, I have learned that
- no single method of instruction is the best or most appropriate in all situations,
- it is important to balance the concepts of computational and procedural skills, conceptual understanding and problem-solving capability when teaching mathematics,
- in mathematics instruction, new skills are developed almost exclusively on previously learned skills,
- methods of assessing students for mathematics comprehension should be context-specific.
These learning outcomes can be implemented in real-life classroom situations by coming up with a well-formulated framework that does not necessarily follow a linear order to ensure students benefit from the mathematics lessons taught in class, and also by proactively aligning instruction with assessment.
California Department of Education. (2009). Mathematics framework for California public schools: Kindergarten through grade twelve. Web.