Place value. Remember it? The foundation for borrowing in subtraction, and carrying in addition, place value pretty much represents the sink-or-swim factor in mental and paper-pencil calculation. While math is so much more than calculation, imagine your confidence level in anything having to do with math, including handling money, without it
Most of us function more out of habit than understanding when it comes to making calculations, and we often defer to the excuse "I can't do mental math" at times when the skill would come in handy. When I think back to my own experience of learning 2-digit addition and subtraction, my heart pounds - I think at 4th grade I began to get the rules of the game, executing the routines solely out of wrote memory. Yet, if we all understood it to its fullest extent, we'd be that much more empowered to execute math in everyday life. We might even be able to do mental math!
The concept of place-value is obscured by the rules of borrowing and carrying, story problem contexts, and the absence of the all-important critical competitor. Direct instruction of what place-value is, and how it works, seems complex because it can involve using base number systems other than 10. Children need critical competitors to create understanding.
What is a critical competitor? Think about a dog, any kind will do. Now compare it to any other breed of dog. How might you recognize their common canine features? The tail? The bark? The ears? The snout? Even though they can look very different, there are some things about most dogs that make them recognizable as such. The same goes for batteries. They come in all shapes and sizes, yet because their function is to provide power and they have positive and negative sides, you can recognize most of them when you see them.
In literacy, we use something called Nonsense Words as critical competitors to test phonics understanding. It might look like this: The letters "ph" make the "f" sound, as in "phonics", "phone", and philanthropist". In order to assess whether a reader completely understands the letter-sound pattern of the digraph "ph", we might put it in a nonsense word, such as "pheld". This way, when the student says "feld", we can feel confident he or she recognized the digraph, even in a brand new, all-be-it nonsense, context.
I propose, and maybe confirmed, we can take that nonsense-word theory into math, using base number systems. In other words, teach place-value with the base number system of 5. 3. 9. Anything but 10. here's how it worked for my 1st-graders yesterday and today:
We reviewed a few basic operating principals of our base 10 number system:
1 - there is no single-digit number representation higher than 9
2 - noticing only the one's column, the pattern 0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0 repeats
3 - we show whole sets of numbers from the ones column, in tens column, in other words, a 1 in the tens column represents 10 ones that would be in the ones column, except for the fact that we only go up to the number 9.
Then we kept basic place-value rules, and applied them in the base 5 number system. Some noticings:
1 - there is no single-digit number representation higher than 4
2 - noticing only the one's column, the pattern 0,1,2,3,4,0,1,2,3,4 repeats
3 - we show whole sets of numbers in the ones column, in the place where the tens column would be, only we named it the "zag" column, just for fun.
Result: After two days of instruction on base-number-systems, about 1/2 the class is really getting it. The rest are very close, even my lowest performers. A few of my higher-performing students, after some significant cognitive dissonance, now have such a clear understanding of place-value that they've ventured out to explore other base-number-systems like 3 and 4. Tomorrow they'll have an opportunity to explore others of their choosing.
The look of pure conquest on the faces of 4 of my students, having established a clear understanding of place-value was...indescribable. Literally, it's bringing tears to my eyes as I type. Those children know, beyond a shadow of a doubt, when and how to move numbers into just the right columns, at just the right times, and WHY! They have calculation down pat, because they've built for themselves a crystal-clear understanding of place-value. I hope they feel forever confident as mathematicians.
What I want to know is why this type of lesson is reserved for post-graduate education? Base number systems are the keys to unlock all sorts of mathematical understanding. Talk about repeating patterns! Grouping! Number sense! Leaving base-numbers out of primary math education is like leaving phonics out of reading, or color theory out of art, or technique out of music, or the periodic table out of chemistry.
It felt extremely satisfying to give my young mathematicians access to this level of understanding. I only hope to have the privilege of watching their learning progress in years to come. I very much hope to see deep understanding, confidence, and joy of learning.
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