Category: learning together

Posted in family life, learning together

learning together: counting in binary

Posted on by kristinabean

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!)

Posted in learning together, musings

learning together: from history to current events

Posted on by kristinabean

Rondel and I have been slowly making our way through Joy Hakim’s wonderful series A History of US, an American history narrative that manages to be both honest about our nation’s flaws and proud of her successes at a level that children can understand. We’re on the third book now, watching the 13 colonies gradually coalesce into a single nation, and so far we’ve encountered quite a few striking dualities: religious persecution and the pursuit of religious freedom; the desire for liberty and the acceptance of slavery; the mingling of cultures and traditions combined with vitriolic racism. We’ve seen people leave England to be able to practice their faith freely (as Rondel commented, when will those kings ever learn that laws won’t change what people believe?) – and then create governments equally as intolerant of dissent. We’ve seen how European settlers as a whole used and abused the Native Americans they encountered, and how they built their economic success on the backs of unpaid labor.

He’s six, it’s his first time sailing through the choppy waters of history, and a lot of it is going to go over his head or be remembered in only the most general of ways – but the concept of slavery has been probably the most jarring and concerning element to him. He asked me why one group of people would think it was ok to treat another group of people that way, and all I could say was that people do a lot of horrible things because of greed and the love of power – that people did, and still do, attempt to convince themselves that another group of people is less than human or deserving of less dignity and justice if doing so will make their lives easier or more profitable in some way. We talked about how the consequences of those horrible parts of history echo down to the present time: how difficult it is to ever fully eradicate that toxic way of thinking, and how generational disadvantages persist unless deliberate work is done to counter the wrongs of the past. We talked about the privilege that he will have as a white man, not because there is anything innately superior about being one, but because of those historical roads our nation has traveled – and how that privilege comes with the responsibility to seek justice and equity for those to whom the trajectory of history has not been so kind. Which all sounds pretty intense for a six year old, but it flowed naturally from what we were reading (especially since he is very sensitive to injustice against others) and I think it’s one of those conversations that has to be had throughout life in age-appropriate ways.

And with all of this fresh in my mind, watching Hamilton for the first time thanks to DisneyPlus, I was able to see the diversity of the performers in a way I don’t know if I would have before. The characters’ races are not historically accurate, but I honestly think it is better that way – more thought-provoking, more eye-opening. We are so used to seeing our history in the color white – which is in our present culture the color of privilege. But those revolutionaries, while still racially privileged at the time, were looked upon with scorn and contempt by the British government for being colonists and “provincials.” Many of them were poor, working their way up the ladder of opportunity in a way that wouldn’t have been possible in England; many of them were the unwanted refuse of London seeking a place to thrive in a new world; many of them professed a faith that differed from the official Church of England and had fled from persecution there. They were a motley crew, to use the expression: their government saw them as a source of profit, as second-class, rather than as full citizens to whom full rights ought to be given. In our modern culture, showing us the revolutionaries as black and brown helps remind us of those historical truths. And it beckons to its audience with a call of hope: if the second-class citizens of Britain, the outcast and oppressed, could fight for liberty and justice against a “global superpower” and succeed, then just maybe the oppressed peoples in our nation today (most prominently people of color) have a chance to establish more perfect justice and liberty for themselves as well.

So study history well. Notice the parallels between the present and the past; follow the pathways that led from then to now. Whether you’re six like Rondel, thirty-one like me, or any other age entirely, there are stories to learn, connections to make, and hope and wisdom to be found for shaping a more perfect future.

Posted in family life, learning together

learning together: infinite series

Posted on by kristinabean

I asked Limerick the other day what was, in his opinion, one of the neatest things he knew about math and numbers.

In response he told me he thought the neatest thing was that if you start with the powers of 2 (say, 128 for example) and kept dividing by 2 (32, 16, 8, 4, 2, 1, 1/2, 1/4, and so on, he said), you would go on forever and never actually reach 0.

Basically my five year old uncovered on his own the concept of an infinite series approaching a limit and (very naturally!) decided it was just about the coolest thing numbers do. I love how his brain processes numbers and analyzes the world in their light!

Posted in learning together

learning together: a multi-level cooperative place value game

Posted on by kristinabean

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!

757 represented by a stack of place value blocks, with three dice and a pile of one’s cubes on the side

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.

Aubade carefully placing her one’s cubes on top of our combined tower.

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.)

Limerick straightening out the blocks on the combined tower after adding his total

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 🙂

Rondel adding a one’s cube to the top of the tower

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.)

Now I suppose I just need to come up with a name!

Posted in learning together

learning together: erupting citrus!

Posted on by kristinabean

Inspired by one of Limerick’s class experiments at our co-op, we spent an afternoon watching baking soda and citric acid react explosively in our kitchen 🙂

Limerick applying the baking soda to the lemon! (and his shirt, lol)

One thing I really liked about the way his co-op teacher presented the baking soda/lemon juice reaction was that she asked a lot of questions designed to help the kids come up with hypotheses and logically critique those hypotheses. Each lemon volcano had three components: the lemon, the food coloring (to make it look more like lava!), and the baking soda. So she asked them what they thought made the eruption happen, for example, and when a lot of the kids said the baking soda, she pointed out that the baking soda wasn’t making fizzy bubbles when it was all by itself in the bowl!

Limerick at co-op, right after adding the baking soda to his lemons.

So when we replicated and expanded upon the experiment at home (Rowan was so jealous that Limerick got to do it at co-op and he didn’t!), I tried to ask similar leading questions. We also decided to test other citrus fruits with the same reaction, so I had the kids think about the differences between the fruits and guess which would make the biggest reaction beforehand, so we could compare our hypotheses with our results.

To my surprise, the two types of oranges we tested had drastically different results. The big navel orange was even less reactive than I’d expected, while the small juicing orange was almost as explosive as the lemon! The grapefruits were also quite dramatic, being overripe and thus extremely juicy and very fun to squeeze everywhere to create great “lava flows” of fizzy reactive liquid. I do think the lemons were still the most reactive, although the results were not anything like quantitative 😛

While we didn’t draw chemical diagrams and get into the atomic reason acids and bases react, we did have a lot of fun exploring the reaction itself! It’s such an easy and exciting way to see how different types of substances can interact.

Posted in learning together

learning together: biomes

Posted on by kristinabean

While the kids were playing together, I set up an activity for the next lull in their imagination. Pulling out two of our giant whiteboards, I quartered them and placed a biome card (from our Waseca materials) into each of the eight sections: Oceans, Wetlands, Tropical Forests, Temperate Forests, Grasslands, Desert, Mountains, Polar Regions. Pulling out the box of toy animals, I began sorting them into the biomes: zebras in Grasslands, tigers in Tropical forests, dolphins in Oceans.

It wasn’t long before Rondel came out and was instantly engaged, asking if he could help sort the animals. So we sorted together, talking about which biomes would make the most sense for animals which are domesticated, for example, or adapted for a range of habitats. When we had at least a few animals in each biome, I brought out the biome question and answer cards.

There were six questions altogether (asking about temperature, moisture, soil, plants, animals, and humans), and each one had an answer for each biome, so Rondel’s task was to match each answer to the correct biome after I read the card. He only needed help on one or two of the cards, and showed a good conceptual understanding of how the environment differs between biomes, how plants and animals have adapted to different biomes, and how humans have interacted with biomes in different ways (both positive and negative).

The pictures show the animal question and answer cards; I thought that matched best with the animals in their biome quadrants! You may notice Rondel added some people to the biomes also, and split the polar animals into Arctic and Antarctic subsets 🙂

I noticed that our animal representation was heavily skewed towards African grasslands and oceans; the Waseca teacher’s guide recommended using animals cutout from magazines, which would increase the diversity, but I didn’t have any that I was willing to cut up. I may just need to buy a big batch of old National Geographics or ZooBooks off of Ebay – old magazines can be really useful craft and learning supplies!

This activity was a good summary of the information we’ve learned about biomes as well as a good overview before diving into more detail on any one area; I think we’ll explore animal adaptations next but I have a lot of ideas.

Posted in learning together

learning together: a “3R’s” treasure hunt

Posted on by kristinabean

Rondel really enjoys looking for treasure (thing-finding in the tradition of Pippi Longstocking, where almost anything can be considered treasure), and as I pondered what to do with our morning (unexpectedly open since Aubade had too bad of a cold to handle the hike we’d planned), I thought he might enjoy following a series of clues to find a treasure at the end.

Because there is always some way to incorporate math, reading, and handwriting into life’s activities (please take that with some humor!), I decided that each clue would be a numerical cipher but that the numbers in the encoded message would have to be determined by calculating a series of math problems. I made short messages like “under the desk” and “lego box”, converted them to series of numbers using a key, and then came up with arithmetic problems at Rondel’s level. He started out with the key and the first clue, which led him to the second clue, then to the third, and all the way to the treasure after six clues in total (probably about 50 math problems all together, which is a lot more than he’d normally do otherwise!). In addition to the math, he got a lot of handwriting practice in from writing down the numerical answer to each problem as well as the corresponding letter value from the cipher, and then even got to do a bit of reading to put all those letters together for each clue.

Above are two examples of keys and clues – the orange set was for Limerick and the red set was for Rondel. Changing the key values to larger numbers would let you create even more difficult math problems without needing to alter the method of encryption. Limerick kept commenting on how the problems corresponding to the letter “A” – where the answer was 1 – were too easy, and a different key value would have eliminated that issue.

Rondel did a whole treasure hunt, despite the difficulty of focusing with two younger siblings running around at high volume and also being very interested in everything he was doing and not giving him any quiet or space! I was really impressed with his determination and motivation, because nothing about this was easy for him but he didn’t give up.

Rondel sitting at the kitchen table with sound-reduction headphones, writing out the answer to one of his clues.
The sound-reduction headphones were a huge help! I could see him relax and focus more easily the instant he put them on.

Limerick wasn’t interested at first, because he isn’t all that into finding treasure, but he like the idea of following a path of math clues so I made a set for him later and he finished his as well! He does not like to write or draw often so pulling him into an activity where he writes this much is a rare and pleasant thing (I think he couldn’t resist the math).

Limerick standing by the kitchen table writing on his clues

One thing I did notice from the activity was that both boys have legibility issues, and I’m going to have to find a way to work with them on pen grip and letter formation that hopefully doesn’t result in daily fights. Rondel’s letters in particular are like people, each with their own personality and opinions, and they dance around the page and swing by their toes and jump on each other’s heads and sometimes sword fight – and they are highly offended by the idea that they should arrange themselves in a neat orderly line! So if you have any ideas or suggestions I would love to hear them.