Inspired by Calculus Math Circle – Week 4

Even if five-year-olds can play with advanced ideas, can they do the next steps toward formal math? Can they actually integrate and differentiate? Are they capable of sufficient level of abstract thinking? These are the questions we are focusing on as our “Inspired by Calculus” local math circles for 7-11 year olds are getting closer to the end. 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 activities

Week 4 – Islands, Mountains and Steps

The pre-circle homework this time was to match bird’s eye views of mountains with their profiles. The kids seemed to like it and felt inspired by it. They brought pictures of islands and mountains and showed us how different mountains from the homework can be modeled using one’s hands and arms. Maria shared a neat papercraft, a hyperbolic paraboloid. In case you want to make one, here’s how.

The show-and-tell helped us get started with the theme of the day – the relationship between slopes, and slices or layers. Maria showed the kids a rolled-up strip of foam. What will this circle look like if we push it through the center? The kids’ answers included a “round pyramid”, a cone, a hat, Pilot Mountain (a North Carolina landmark), and an upside down tornado. The kids showed not just their understanding of the idea, but also their ability to seek similarities between objects, an important step on the way to developing abstract thinking (more on this – later).

Next we asked the kids to become map-makers, to draw their fantasy islands and add contour lines. Some of the islands the children designed included a spoon island, a heart island, a fish island, and a rooster island!

Once the islands were designed, the kids got felt, craft foam, and decorator foam. They started tracing and cutting out the layers of their islands. There was confusion over what needed to be done. We did have a pre-assembled model to show, but forgot to do that step. Usually our maker activities are very open-ended. We do not have to show what the end result needs to look like. But this time the island-making was not an end in itself, but a bridge to the next activity. And it needed to be done just so.

What happens to your math circle if you mess up the plan? It’s a measure of the robustness of the group. In this case, perseverance (kids stayed on task) and cooperation (parents helped) kicked in and all the children ended up with great islands. You need soft skills like perseverance and cooperation for most STEM professions.

Once the islands were ready, it was time to explore their mathematical properties. We started with a “can you imagine” game of wild questions. Can you imagine a mountain with an infinite slope? An infinitely high mountain? A mountain with a zero slope?

It can be risky to start a discussion with questions about “extreme cases.” Kids might lose the main concept altogether. After all, most slopes are neither zero nor infinite. But by now kids had many encounters with the idea of a slope. So we felt safe. Still, the questions threw them for a loop and made them look for very divergent ways to satisfy these extreme conditions. Some laid their pieces down flat, jigsaw puzzle-like, modeling a zero slope, others set the pieces on edge on top of each other for infinite slope.

One of the children chose thin material (craft foam) for her heart mountain. Others opted for the chunkier decorators foam or a mix of materials. As the kids were building vertical slopes (infinite slopes), Maria pointed out the fact that even though the Heart Mountain was not high, it could still be arranged to show infinite slope.

This Heart Mountain provided a perfect bridge to the next activity. Maria modeled three slopes – a vertical one (holding up an index card), a steep-angled one (folding that right angle in half) and a shallow-angled one (folding in half again). She asked the kids to rearrange the layers of their islands to match these slopes. How can we show ever smaller slopes?

A couple of kids suggested decreasing the height of the mountain by removing a few layers or squishing the foam. The chunky models, most with only 3-4 layers, were not very helpful. The kids struggled to imagine a smooth mountain. They needed smoother models with more, and thinner, layers. Using 3×5 cards to “smooth out” the slopes helped somewhat.

Why was it so hard for the kids to understand that a slope does not have to change if the height of a mountain (or pyramid) changes? Size is absolute, but slope depends on the ratio of two sizes – so the size is “in there somewhere,” but indirectly. Slope is a relationship OF sizes, so it’s a derivative concept in more ways than one.

Can we train the kids, at this point, to do formal differentiation? We can train them to do some, for sure. But that would not be interesting, probably not developmentally appropriate, nor helpful to the kids in reaching their current life goals.

So why bother? Are we making a mountain (pun intended) out of a molehill when we call it “early calculus”? After all, calculus requires a great deal of abstraction and, as we just said ourselves, the kids aren’t there yet.

Noticing Similarities  – Keys to Math Fluency

The goal of our math circles is to help children develop mathematical mastery. The part of mastery we focus on is not calculation, but the ability to recognize similarities and differences in ideas and abstractions.

During the Circle, children often describe something as something else. This week, for example, they described a shape as a silly hat or a “circle pyramid” or the Pilot Mountain. This shows us that they are developing their ability to move past superficial differences, to notice and celebrate similarities.

Noticing differences is an important skill, but it is also something we do well with very little practice. Noticing similarities is much harder, but it paves the way to abstract thinking. Introducing calculus concepts to young children helps them further develop this “similarities over differences” view. For more on the importance of noticing similarities, read this short Wiki entry .

The Inspired by Calculus series will wrap up next week when we post the notes from the math circle’s final meeting.