Would 7-year olds really use calculus? Wouldn’t they forget it all by the time they encounter calculus in school, if ever? We frequently revisit these questions as we continue to plan and lead *Inspired by Calculus* local math circles for 7-11 year olds. The issues of knowledge retention, use, and transference came up again during the Week 3 meeting.

Do try these activities at home by yourself (good), with your child (better), or invite some friends over (the best). If you do, please share your experience with us. As always, we welcome your questions and comments.

Week 1 activities | Week 2 activities

**Week 3 – Pyramids and Integration**

We spent our third meeting delving deeper into integration – making objects and shapes out of infinitely many slices or lines. We also built beautiful LEGO towers and, while working on them, talked about slopes. For homework we asked kids and adults to look for examples of pyramids and stepped structures, perhaps even to create a model or two.

But first, we asked the kids to make paper squares and gave them standard 8 ½ by 11 sheets, markers and scissors to complete the task. Making a square out of a rectangular piece of paper seems simple – fold and cut. The trick, of course, is to fold on a diagonal. Interestingly, while the kids were folding and cutting, they were folding paper horizontally or vertically, sometimes folding a couple of times before trying to cut a square. Each time they ended up with quadrilaterals that were somewhat squarish. And each time we asked them to refine their models and make them more precise.

All the kids had previous experience folding diagonally to make a square out of a rectangle. They did it at the previous math circle, after all. Did the knowledge not stick? Why? Perhaps a couple of factors played a role:

- Up until now the kids have not had the need for precision. In this activity, precision was not an internalized need, but an imposed request. The super-compelling “why do it” was simply not there.
- At the same time, a diagonal fold is a pretty challenging one for young kids and, when done without any assistance from adults, the fold itself is not nearly as precise as the easier horizontal and vertical folds.

At the same time, one of the children suggested starting with a circle and “cutting away at the edges” to make a square. This led to some unsuccessful attempts and a lovely discussion about commutativity: making squares out of circles and circles out of squares.

Next came Photospiralysis, a free software that you can download or use online to apply the Droste effect to your photos. You can read about the Droste effect here or just play a while with Photospiralysis. Here’s an example of a nested fractal made with the software:

You might be wondering what do Droste effect has to do with pyramids and why spend time with Photospiralysis. The answer is “collapsible cups” – like this one:

Imagine taking a flat Photospiralysized image and pushing it up and out, extending it into the 3D space. Or imagine taking a pyramid and squishing it flat. Or look at a LEGO pyramid directly from above. What would you get?

We will do more of this work next time, hoping the connection between the Droste effect and stairs and pyramids clicks for the kids.

Next up, the kids looked at the two pyramids built out of interlocking blocks. One of the kids pointed out that only one of these objects was a pyramid. The other one was “just stairs” since it wasn’t filled in with blocks. After a little “mathematician vs. engineer” discussion (engineers build, mathematicians imagine), we asked the kids if they would build their own LEGO pyramids. An easy task, for sure! Except, there was a catch – their pyramids’ slopes had to be different from the slopes of the original two pyramids.

The kids did awesome! They had a pyramid with a variable slope, with a slope twice as steep as the original pyramid, three times as steep, half as steep, with a zero slope, with a vertical slope, and with a negative slope. More negative slopes were made by turning pyramids upside-down. Way to go!

The last task was building multiplication towers. And they had to follow certain “crazy” rules that, as we promised, “would make sense in the end”. The towers came out beautifully and the kids spent some time exploring them, pretending to be climbing different routes to the summit, exploring slopes and rates of change along the way.

**Collecting Math – More Than Happy Familiarity**

Remember how the kids forgot how to fold paper to make a square? It wasn’t just the lack of folding skills: they could’ve asked adults for help. Their learning experiences from last week and from other times they folded paper or observed folding did not *transfer* to this new situation.

If this happened to a relatively simple geometry concept, won’t the same happen to more abstract and complex calculus ideas? If so, why not wait until high school or college to introduce calculus?

Let’s think beyond learning this or that skill. How can we help kids develop logical thinking and problem-solving in new situations? One of the best tools for developing these skills is using analogies and metaphors. We use metaphors a lot in our activities. We also encourage children to define a mathematical object or process by comparing it to something else. For example, when asked about shapes, kids in our groups usually reply with iconic objects (a CD, a plate, a pizza) rather than math words (a circle, a disk).

What is going on and why? First, the kids use analogies and metaphors. Second, they engage in mini-scavenger hunts for collections of everyday examples. Once children collect enough examples, their minds start organizing examples into categories. Categories are more universal, thus more transferable to new situations than individual examples. Once the categories are in place, we encourage kids to use math words and more precise methods. Then particular skills and overall ideas stick better!

Building an extensive collection of usable mathematical imagery takes time. That’s why we encourage to start early. It gives kids *happy familiarity* with the math they find, make, and explore. Children build a multitude of connections, and notice patterns that are transferable to other contexts.

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