Why are there 10 symbols in our numbering system?
Why?
- Start a discussion on biases and assumptions implicit in the way we look at math
- Introduce programming and logical reasoning into math classes
There is a lot in math that tends to be taken for granted. For example: our number system is in base 10. In other words, we have 10 different symbols: 0,1,2,3,4,5,6,7,8,9. When we add one more, we end up reusing the digits 1 and 0 to make 10 and continuing from there: 11,12,13 etc. Why 10? Essentially, its completely arbitrary. The most commonly cited reason is that we have 10 fingers for counting on. The inspiration for this post is Alex Bellos' book Here's Looking at Euclid. In addition to many other topics, Bellos discusses the concept of different bases and how they were used in different cultures throughout history (bases 20, 60 and 12 are the most common). Below is a screenshot from the app Number Base Conversion.
It blows my mind that I didn't really hear about number system bases until Grade 11 Computer Science, where it tends to be a common intro to how computers work. I find it strange that it never appears in the math curriculum because it is fundamental to how we think about numbers. The base 10 bias is never really explored or explained in school.
Programming in Math Class...
Conrad Wolfram (brother of Stephen Wolfram, founder of the awesome math/everything tool Wolfram Alpha) is a big proponent of utilizing programming as a tool for helping students understand concepts in math classes.
He believes the act of writing a computer program requires a much more in-depth understanding of the concept. Once the program is written, students can use it to solve problems for them. This way, they don't have to waste a lot of time doing arithmetic and other mundane tasks that aren't part of the curriculum. Thats what we have computers for after all. This is highly representative of how problems are approached in the real world as well. We aren't trying to teach students to be human calculators, were trying to teach them to be problem solvers.
The topic of number bases and base conversion would be an excellent example for using programming in a math class. Students could write programs that convert base 10 number to other bases. They could then use their program to explore strange bases like base 1, negative bases, irrational bases (like pi) and fractional bases. This is a perfect example of the 21st Competency Use of ICT for Learning. Students would be able to create a usable artifact.
Another interesting investigation is looking at how arithmetic works in other bases.
Curriculum connections...
Understanding number bases requires the use of exponents, logic, and a deep understanding of place value and how numbers work on a fundamental level. Though the topic is not directly in the curriculum, I see it as a way for a teacher to formatively asses students' comfort with basic math concepts and their problem solving ability.
Critical Math...
It is important to be able to step back and examine our biases. Examining why we use base 10 is a good way to bring to light the fact that a lot of math is actually rather arbitrary as a product of history and culture as opposed to a given fact of nature. It is an opportunity to challenge our basic assumptions with math and opens the door to discussion and the critical examination of other things that we may take for granted (example: why are there 12 semitones in an octave in our music, why does the major scale have 7 notes?). We can ask questions such as: If we primarily used a base 12 system for math, would we see things in different ways? Would we have made certain discoveries that we have not made yet or would we have missed certain discoveries? Would our world be different?
21C Continuum...
2. Knowledge Construction: entry - adoption - adaptation - infusion - transformation
6. Use of ICT for Learning: entry - adoption - adaptation - infusion - transformation
Future Lesson Ideas
- Examine the music/mathematical relationships in the 12 Tone Equal-Temperament system
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