I have never heard about "Arbitrary and Necessary" in mathematics education. Digging into my memories, most of my math education was me listening to a teacher's lecture and receiving their wisdom from the teacher. The only times I had a more profound understanding was when I understood the reasoning behind a particular principle rather than memorizing it. While many facts in math are not arbitrary, you can find most of the arbitrary curricular content in elementary-level mathematics. This curricular content includes multiplication, fractions, division, addition, subtractions and many more. At that level, many students are introduced to these mathematical conventions for the first time and, most of the time, are asked to memorize them. In contrast, secondary math consists of much curricular content that falls under the "necessary." However, this requires students to have a good foundation of arbitrary content introduced in earlier years.
When teaching students at the secondary level, it is essential to realize that many students will come from elementary and previous years at varying levels of math abilities. To cater to a broad range of students, asking students to develop their understanding of everything would be manageable. As a result, some knowledge must be given to students as received wisdom as a guiding hand in hopes that they can better understand and develop their understanding. As shown in Figure 5 of the reading, arbitrary content generally consists of "words, symbols, notations and conventions," while necessary content consists of "properties and relationships." When doing lesson plans, content detailed as arbitrary will likely be given to students as received wisdom, along with a sprinkle of properties and relationships to guide students.
An excellent way that I will try to implement is to start a unit by presenting the students with a problem that they can solve at the end of the unit. Asking the students to attempt it using their given knowledge first and then start the lessons to build the skill necessary will allow students to develop their understanding. Thus, by using these lessons, I hope that it will build up supplementary skills and understanding to solve the main unit question.
Thanks Jacky. But I disagree with you quite strongly about elementary math education being mostly about learning arbitrary facts! Young children are quite capable of thinking logically and reasoning for themselves, and processes like adding, subtracting, multiplying, dividing and working with shape, probability, etc. have plenty of necessary properties that elementary school students can work out for themselves. You may want to visit some exemplary elementary math classes to see for yourself how capable the kids are! I recently taught a sequence of three lessons on binary numbers to an elementary class and was MOST impressed at the ways the students could reason and understand this new material!
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