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But do you really understand?

In response to Richard Skemp's article on Relational Understanding and Instrumental Understanding (available here)

The Skemp article prompted me to consider my own experiences in learning and teaching Mathematics, particularly in my home country, New Zealand. New Zealand has national assessment/certification for all high school subjects, the National Certificate of Educational Achievement (NCEA). All Year 11, 12, and 13 students - the final three years of high school - studying the same subject and level, will take the same exam at the end of the year, at exactly the same time. Due the set type of questions, year after year, teachers have become accustomed to teaching for the exam, more often than not, for instrumental understanding. Although the NCEA administration have made small changes to the exams year to year, in hopes of creating space for teachers to teach for more relational understanding, this is not always the case. I, myself, have been guilty for tutoring certain students for instrumental understanding, particularly when the student is at risk of failing an upcoming exam. 

Skemp emphasizes the importance of understanding and aligning the goals of the teacher and the student. Most commonly, the teacher's goals are for higher rates of student achievement, and the student's goals are to pass their exams. In this situation, the larger issue is to differentiate between education, where the goal is to create more curiosity for critical thinking, regardless of the subject, and training, where the goal is to gain knowledge and skillset in a particular field. If both the teacher and student make it their priority to expand upon their ways of thinking that will apply across multiple disciplines, rather than teaching/learning a set of formulas to answer particular problems, they can create a basis for developing the schema where the student becomes a curious learner.

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