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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.

Can you really teach calculus concepts to young children? Don’t they need to know algebra first, or at least arithmetic? We receive these questions a lot, and we do have our share of worries as we plan and lead “Inspired by Calculus” local math circles for 7-11 year olds. But let’s be brave together! 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.

The first post in the series is here.

Second Meeting – Bodies of Revolution, Part 2

Our second meeting had to be rescheduled because of the weather. So kids had some extra time to play with the concept of integration (making big things out of many little things) using String Spin. Their new scavenger hunt task was to search for edible bodies and surfaces of revolution.

We started by revisiting integration. But we also worked on the concept of differentiation. Which brings up the question: “How can we show something this complex to young children?”

We find activities that serve as metaphors for the concepts, for example, differentiation as slicing an object into many parts.

First, kids tried to make some circles: “Remember that circles are made of triangles?” Children remembered this folding and cutting method from the last meeting, but it was still hard (most ended up with squares!) – so they were motivated to look for an alternative.

So out came salt dough. For the first few minutes the children just sat in a circle on the floor, divvying up the dough, smelling it, rolling and crumbling it, pounding it and mixing colors, and talking about it. This play time helps children to become familiar with properties of a new manipulative, before using them for mathematics.

Logistics won’t allow making salt dough with the kids during the math circle. But do try it at home! Making manipulatives together with your child is a powerful way to explore mathematics. For another good example, read this post about math of henna art by Rodi over at the Talking Stick Learning Center.

After a few minutes of free play, most kids ended up with some salt dough circles, made by squishing spheres or sculpting by hand. It was time to show them a more precise way of making a circle out of paper triangles (integrating).

At this point, children had done a lot of free play, both with physical objects (paper, salt dough, bodies of revolution they found/noticed around them) and software (String Spin). We noticed that their guesses (i.e. what curve would produce what surface in String Spin) were getting more precise – a good indicator that the kids were starting to discover patterns.

So the kids made their circles out of triangles until the salt dough got way too soft to handle. Then we returned to the unanswered question from our first meeting.

“What if we had a thick book with very many pages and we opened it so that the covers would touch. What shape would it make?”

This time, we actually had a cylinder made out of an old paperback book. And the question was more concrete: “What shape should we draw in String Spin so that when it rotates, we get this cylinder?” Interestingly, while all the kids recognized the object as a book, many came up with answers that looked nothing like a rectangle. Here are a few of their suggestions.

What was happening? It took us a few days to come up with the hypotheses that kids were trying to integrate 2-dimensional solids (book pages) out of 1-dimensional lines. We will test the idea next time. We modeled all children’s ideas using the software, but the solutions looked more like explosions or slices than book art. We’ll seek more precise software next time to do what we think kids want with integration. This was a very cool experience for the adults as it reinforced the following idea.

When choosing a math activity, it is not enough for the activity to be a metaphor for whatever advanced concept you want to teach to a child. It should also provide a way to help children communicate and share their math discoveries. And the best way to do so for young kids is not through words or formal mathematical notations, but by making things.

It was time to move to the metaphor for differentiation (in the grown-up eyes) – slicing (in children’s eyes). We asked participants to bring fruits and vegetables that were bodies of revolution.

Off we were to find yet another solution to our “how to make a circle” problem. All the kids figured out how to slice their fruits to get circles. So we asked them what would happen if we stacked the circles on top of each other? Would we get a cylinder, as some suggested? What if we stacked circles of different diameters? And what shape would we end up with if we kept stacking smaller and smaller circles?

And here came an interesting twist. Most kids insisted, even after modelling in String Spin, that stacking smaller circles would get us a pyramid. We are pretty sure that all the kids in the Circle are familiar with the words “pyramid” and “cone”. But we are also pretty sure that for now they do not distinguish between the two. One reason is that there has been, up to now, very little practical need for them to distinguish the two. The other reason is that a pyramid and a cone share a lot of essential mathematical qualities!

Which brings us to the question:

Shouldn’t kids know the difference between a pyramid and a cone, before working on calculus concepts?

From the calculus point of view, the difference between the two is insignificant. The process of differentiating and integrating and the volume formulas are the same for pyramids and cones. Overall, the sequence of mathematical education, what comes first and what comes after, is largely a matter of the established tradition. And the importance of knowing particular math facts or having certain math skills is relative to the area of the mathematics you want to learn.