Semiotic Square: Easy-Complex-Simple-Hard
The semiotic square is a tool used for the structural analysis of relationships, developed by linguist Algirdas Greimas. We’re going to relate four different concepts in three different ways.
First we want to rule out the incompatibilities. The diagonal lines represent a contradictory relationship. Easy and hard live on opposite ends of one spectrum, so you can’t use them both in one sitting. Likewise, for something to be both complex and simple is a contradiction.
Next, the squiggly horizontal lines represent a contrary relationship. These are kind of like cousins. They might have similar properties, but they’re on different spectra, so they can coexist. When the stars align, something can be both easy and simple.
Lastly, the vertical connections represent a complementary relationship. This is a sort of double-negative. Complex is the anti-anti-easy. Simple is the anti-anti-hard.
If you want to go even further, you can add examples to your connections. My favorite way to do this is to embed the semiotic square in a rhombus, and label the points of the rhombus with examples that correspond to the connection. I think about pedagogy when I look at mine. The ideal activities are complex but easy, like building elaborate structures out of prefabricated LEGO blocks. Conversely, drilling long division in bulk is hard but simple.
(Click image to enlarge)
I first came across the concept in Kim Stanley Robinson’s Red Mars. The diagram above is the very same one that appears in the book. Red Mars chronicles the history of the first one hundred colonists of the red planet, and their successors. One of the characters, a psychologist and an aspiring philosopher Michel Duval, is responsible for choosing the first hundred, and later for their well-being when he joins their ranks. Michel uses the semiotic square to classify the other Mars colonists, as an attempt to extrapolate from the traditional introvert-extrovert spectrum.
I loved it immediately for how human it is. Once you figure out the semiotic square, you’ll start seeing applications everywhere. If you make your own, share them in the comments. I’d love to see.
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Math Mind Hack: imagine replacing one object with another
Asking “What if?..” always opens up mathematical possibilities. But this hack is about scale analysis, ratio and proportion, and other spatial reasoning skills. Yet it also helps with general problem-solving and imagination.
Photo sources: the skies; the hands; eye/mouth.
It’s a remake of my more general mini-poster. I like this version better, because it has clearer visual structure and a more doable call to action.
Posted in Grow
Inspired by Calculus Math Circle – Week 3
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.
Posted in Grow
Heart Collage
If you’re looking to teach kids about symmetry, a new app called Heart Collage is a great start!
Where there is love there is life’s Facebook Page
Available on Google Play and the iTunes Store, Heart Collage guides you through a series of photographs to create a composite image, using a system called integration. Integration is the mathematical concept that big objects are made of small objects. The idea itself is easy and versatile. Integration is one of the fundamentals of calculus.
The app can be used to create many shapes, as well as hearts. Heart Collage can create perfectly symmetrical images like the one above. The same mathematical skills and logical thinking are present throughout the creation process, no matter what shape the user creates. The possibilities are endless!
I did a mini-interview with Heart Collage creator Chiu-Ki Chan about the app and its relationship with math.
Do you like math? Are there any ways it influenced you to create Heart Collage?
Will you be disappointed if I tell you I did not think about math at all when I created Heart Collage? That’s the beauty of it though – math is everywhere, even when I am not thinking about it.
What kinds of mathematics did you have to take into account when creating the structure of the collages, to make sure the collection of images form a specific shape?
Symmetry plays a large part in structure of the collages. The heart shape is symmetric vertically, but the diamond shape is symmetric both horizontally and vertically. Things get really interesting in the alphabet series. Take a look at letter J:
The letter itself is not symmetric, but I want to create it with as few photos as possible, so I had to find symmetry within the letter. As you can see, I have mirroring for the top horizontal bar and the curve in the bottom, even though I am not mirroring the whole letter.
If you play Heart Collage with your kids, talk about symmetry and integration – and share what kids notice and what they say with us!
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