In How To Help Kids Learn Math, I described what a revelation it was to put myself into a young child’s shoes, and to remember what it was like to be introduced to math concepts. As I discussed, it can take awhile for young children to understand that numbers are symbols that represent different quantities.
A next step for kids as they learn mathematical sense is understanding how to use those symbols to make meaning. With only ten basic numerals, 0 through 9, the number we know as 10 isn’t really a number at all but two numbers being arranged together to create a third.
We have a tens based number system, which means that, in our mathematical way of doing things, ten objects make a whole group. Don’t assume this is the only possible way of making groups. Base-10 was not always the norm. Historically, all sort of values were used to decide groupings,. including ancient tribes that probably used their hands and toes in calculations as a base-20 system. In fact, if you live in or do business with the United States, you are probably using another system as well. Along with our base-10 calculating system, there is the imperial method of measurement, based on 12 inches in a foot, or groups of 12, and 16 ounces in a pound, or groups of 16. If you happen to work in computer programing, you may have used the “binary,” or base-2 system.
Our base-10 translates through place value. After 9, we have a whole group of ten objects, which is shown as 1 group and 0 singles, or 10. If you get nine more of this size group, you communicate that by writing 1 group of groups, 0 single groups, and 0 singles, or 100.
To be fair, by the time place value is a specific part of education, numbers and their values past 9 are very familiar to students. They are learning to understand just what the symbols mean. They need to know how to break that number down into its parts. You probably have no trouble understanding that 23 means two groups of ten and three single units. But what if we make it new again and take you back to when you were still learning about number groupings.
Let’s change the base grouping number so that you have to think about it again.
How much is 10 worth in:
Base-6? Base-12? Base-2?
The number 10 represents one whole group with zero extras, so each of these answers is it’s own base, 6, 12, and 2.
Let’s try a harder one and tackle that 23 that is so easy in our daily lives. How much is 23 worth in:
Base-6? Base-12? Base-4?
It probably took you a moment or two to calculate.
(ANSWER: In a Base-6 system, 23 means two full groups of 6 with three singles. 6+6+3= 18. Base-12- Two groups of twelve with three singles, 12+12+3=27. Base 4- Two groups of four with 3 singles. 4+4+3=11.)
Lastly, to approach the feeling your younger student has when tackling this new concept, try switching it around. If you have twenty three objects in front of you, how would you write that amount in:
Base-6? Base-12? Base-2?
(ANSWER: Base-6: You can make 3 whole groups of 6 with 5 leftover, so it would be written, 35. Base-12: You can make 1 whole group of 12 with 11 leftover, so it would be written 1*. In a Base-12 system, you would have to create 2 more symbols to represent ten and eleven as a single digit. I used * to represent eleven. Base-2: This one get complicated quickly. If you want to follow it closely, you can use 23 objects or a picture with 23 dots, just like your student would in class. Remember, two objects together makes a full group and moves it to the next place value. From 23, I can make 11 groups of two with one single, so there is a 1 in the single’s place. I can make 5 groups of 2 from those groups, which leaves me one single group in that column. I can make 4 groups of two of those with one single, which leaves me one in the next column. I can make 2 groups of these which leaves 0 singles in the next column. Finally, I can make 1 group of those two groups that leaves me with a 1 in the highest value column. 10111)
Did you brain start to spin a bit? Mine did. I’ve had years of practice switching bases in numbers and I still have to concentrate very hard to present Base-2 values. It’s easy to forget how challenging some of these new processes are when you’re learning them for the first time. Hopefully, this experience will help you have more patience when you see your student struggling over homework.