Limerick has been interested in different base systems for several months now; he’s enjoyed playing place value games in bases other than 10, converting between bases, counting in alternate bases, and so on.
The other day he came up to me and told me that he’d figured out a new way to count on his fingers. Holding up the pinky finger on one hand, he told me that was 1. Just the ring finger extended was 2; the ring and pinky together was 3. The middle finger on its own was 4, middle and pinky together was 5, middle and ring together was 6, and middle, ring and pinky all at once was 7.
It may be easier to see his logic with 0’s and 1’s, where a finger curled down represents a 0 and a finger extended represents a 1.
00000 00001 (one pinky extended) 00000 00010 (ring extended) 00000 00011 (pinky and ring extended) 00000 00100 (yeah, this one looks wrong, but he’s six so he has no clue) 00000 00101 (middle and pinky) 00000 00111 (middle, ring, and pinky)
And so on.
It’s binary! Assigning each finger one place in the binary place value system, he figured out how to count on his fingers to 1023 (which would be all fingers extended). In addition to showing how natural this aspect of mathematics is to him, it also shows a good foundational understanding of computer science – because electric signals are either on or off, just like Limerick’s fingers could either be extended or curled down, and therefore represent data in this same bitwise manner. It is so amazing what kids can think of and create when they are given the chance to deeply explore something they love.
(We tried developing a base 3 counting system but found it was too difficult to keep some fingers folded only halfway down when the fingers next to them needed to be extended or completely curled – someone with more fine motor control might have more luck though!)
We’ve been working on place value for a while. Rondel unfortunately decided that my default place value game was his least favorite thing ever, probably primarily because Limerick utterly loves it and finds it intuitive and easy while Rondel has struggled more with the concept. Fortunately, however, we were able to adapt it using place value blocks (wooden blocks in denominations of one’s, ten’s, hundred’s, and a huge thousand cube) into a game that let each kid operate on their own level of mathematical ability while working together to earn chocolate chips!
Our goal as a team was to reach 1000, rolling dice to add to our total on each turn. Along the way, we could get chocolate chips: one for each person every time we added a new hundred square, and ten for each person if we made it to the thousand cube. On Aubade’s turn, she would roll just one die and practice counting the dots to find how many she had rolled, then practice counting again as she put the right number of cubes onto our combined tower in the center.
On Limerick’s turn, he rolled two dice, multiplied them together, and then added his total to the combined tower. (Yes, this is easy for him. Next time I’ll have to come up with something more challenging for him to do! He also tends to supervise everyone else, however.)
On Rondel’s turn, he rolled four dice and added them all together (which was perfect for him! Adding two dice is easy for him at this point, but four lined up with the addition we’d been encountering in Life of Fred and he remembered and mentioned that.) Seeing how his one’s cubes lined up to form a group of ten, and how his ten’s lines added up to form a hundred square, the concepts of place value finally started to make sense to him! These blocks are such a nice visual/tactile representation of that 🙂
By working together, we eliminated both the stress of competition and the need for everyone’s individual rolls to come out to, on average, comparable amounts. Because we were working together, it didn’t matter if Aubade was rolling much smaller numbers than Limerick or Rondel, or that Limerick’s highest possible total was higher than Rondel’s – everyone just contributed towards our shared goal in their own way. It also didn’t matter who was fastest or reached a goal first, and the shared celebration every time an intermediate goal was met (i.e., the chocolate chip for each hundred) prevented anyone from becoming jealous or discouraged. And finally, because none of those things were important to the game, we could tailor it to each participant’s math level, allowing all three kids to play together despite ranging from counting to multiplication/division with their math skills (which I’ve found surprisingly difficult, mostly when it comes to including Aubade in the game.)
He counts everything he can, comparing amounts and sizes: he counted all the straps on the plastic pool chairs at the public pool Saturday afternoon, at least ten times in a row; he makes long chains of flakes and counts them over and over again as he builds to find out how long they are and which color is the longest; he counts the number of bites or slices on his plate and recounts each time he eats one. Essentially, he is spending time with the numbers and quantities, becoming friends with them, getting to know how they interact with each other and discovering their unique qualities.
Limerick figuring out how many sticks could fit on the length of our big wooden cutting board – I think it was somewhere around 40.
He will break numbers down into their component groups: if he builds a tower with fourteen wooden blocks, he will tell me proudly that it is two groups of seven. When he expanded his tower to eighteen blocks, I asked if it could split into two equal groups and he lined them up into two equal lines and counted each one to make sure that they both contained nine blocks. When I then asked if it could split into three groups, he made many little groups of three blocks and found out there were six of them – so, six groups of three and three groups of six. Since he’s been doing this on his own anyway, I introduced the vocabulary of “even” and “odd” to him.
“See this group of seven? I can’t break it into two groups of the same size – one of them is always bigger than the other. Numbers like that are called odd.”
Limerick pondered, then declared, “But eight is two groups of four!”
“Exactly! Numbers that can be split into two groups of the same size are called even!”
“Nine would be five and four,” Limerick told me with a slight frown, “but ten would be five and five!”
“So nine is odd, and ten is even!”
We’ll see if he remembers – he tends to hold a new word to himself for a week or so after he learns it, before he brings it out for everyday use.
Emboldened by all the math talk going on, I pulled out one of my favorite math tools from my own childhood (one that I believe my mom created, and that she now uses in her job as a remedial math professor at the community college):
It’s a physical representation of place value – the numbers on the cards in front show what the number would look like written down, while the buckets hold the appropriate number of craft sticks. In this picture, there are three single craft sticks in the box on the right, six bundles of ten craft sticks each in the middle box (the tens place), and nothing in the box on the left (the hundreds place).
We pulled out some huge foam dice and decided to take turns rolling and adding the number we rolled to the number already in the buckets:
Sometimes the boys knew right away what the answer would be; other times they would line up all the sticks to count them to make sure, and to bundle up the new ten-pack if needed.
Before this we hadn’t done much with written numbers – the boys know all the digits, but they didn’t understand double digit numbers. So it was a bit confusing at first for them, but by the time we stopped I think they were beginning to understand what the numbers meant and looked like, which is really cool!
Math tools and games like this aren’t necessary for learning math – Limerick has certainly been picking up on concepts ranging from quantitative comparison to division (with even a touch of fractions) just from everyday conversations about the numbers around us – but they are definitely helpful for illustrating a specific concept, challenging the mind to use a concept in a new way, or just having fun together with numbers! And since all you need for this particular tool are buckets, sticks, and paper, it doesn’t get much easier 🙂