NumberTheoryChapter2FollowUp

Place value is the root of all understanding when it comes to most K-12 math. And if a student doesn't truly understand the Base 10 numbering system, they're just memorizing their way through math without ever truly understanding it. The root of all understanding of place value comes down to counting and understanding (believing) the patterns involved in counting. There's not much you can do to introduce counting at the middle/secondary level that doesn't feel like a real insult, unless... you introduce them to another counting system.

STORY #1: Simone (EDU 361 Fall 2013) introduced me to the story of [|a woman who taught her third graders base 2 numbering system by using the Socratic method]. I have actually done this a number of times. The "how would aliens with only two fingers on each hand count" approach is exactly what I did and though I didn't do as much Socratic method (wish I'd known about it back then), I can assure you that 9 year olds pick up counting in other bases quite well. The beauty of that is that for students who already knew base 10, this was just a fun extension of their knowledge. For students who never understood place value in a decimal number system, they started to get it as they explored base 2. And for students who had faked their way through place value by only memorizing the expanded notation process or parroted how to read numbers, they actually started to understand place value in general and all those memorized processes started to make sense. I have taught many an adult friend how to count in other bases ... you'd be surprised how often it comes up in dinner conversation. Somehow, they are never insulted to be taught alien math but are very excited to learn it.

WARNING: When you are working in other bases, do not teach your students to convert to base 10 and then back to the original base. For example, if you are adding 101 and 11 in base 2, don't say, "Well 101 in base 2 is 5 in base 10. 11 in base 2 is a 3 in base 10. 5 plus 3 is 8 in base 10 and 8 in base 10 is 1000." That defeats the whole purpose and only further confuses the students who didn't understand place value in the first place. Have them get out their alien fingers, allow them to count, Start at 101 and count up +1, +1, +1. Or let them set up the addition problem on paper and start on the far right and say outloud 1 plus 1 is 2, so I put down a 0 and trade the 1 into the 2's place (because 10 is 2). 1 plus 0 plus 1 in the 2's place is 4 (because 100 is 4), so I put down a 0 and trade the 1 into the 4's place, etc.

EXTENSION: Do you have some students who really catch on to base 2 easily and some who don't? Challenge those who got it so quickly to try hexadecimal numbering system. It's a fun challenge and it's also used in computer code (as is base 2), so you're helping them learn a little about the science and math behind the technology they use every day.

STORY #2: As part of an integrated unit on Egypt, I introduced students to the [|Egyptian numbering system]which is a semi-decimal system but with very different characters to represent the digits. In other classes they were learning hieroglyphics, reading //[|The Gift of the Nile]//, studying the architecture of the pyramids (and learning about their mysteries), etc. No one had seen the numbering system before and it was interesting to see who caught on more quickly and why. It wasn't always those who were good at base 10 place value problems who "got it" first. Listening to the students help each other solve the problems was an amazing formative assessment for me as a teacher. It helped me not just understand their understanding of Egyptian numbers but it was even more revealing of their understanding, skills, vocabulary, and misconceptions related to base 10.

BONUS OUTCOME: I gave a pre-assessment on the Egyptian numbering system before the introduction. It was a simple worksheet with 5 numbers written in Egyptian numbers and they had to give the Base 10 equivalency, and they were given 5 numbers in Base 10 and they had to give the Egyptian equivalent. It drove the students crazy because they didn't know anything on the paper. The lesson didn't take long at all but was fun and powerful. In the same class period, after the lesson, I gave the post-assessment and everyone felt so much better. I sent them home that day with both their pre- and post-assessments so they could always remember that it's okay to not know something on the pre-assessment because I'm really telling the truth when I say I need to know what it is they don't know so I can figure out what I need to teach. Additionally, they ended up with a better comfort level with the decimal numbering system.