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Teaching Strategies

# This Is What Math Looks Like

Whether they’re in kindergarten or calculus, students can use simulations to see the math.
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Math Simulations are tools I’ve created to visualize math ideas. They let students write math and see what it looks like by representing it visually through, for example, lengths, rotations, and light.

One simulation lets students write any number and see it as a length. Learners can type in the number 3, for instance, and see it presented as a line. Then they can type in the expression 5 x 3 and see that the line gets five times as long.

Another simulation allows students to represent a number of objects. Typing the number 3 displays 3 dots. Adding 5 makes 8 dots. Adding 5 more makes 13. Continuing to add 5 provides an intuitive way to visualize the function 3 + 5x.

This video shows what the simulations look like:

The simulations I’ve made can be combined to create functions (the simulations at the site are free, but registration is required). For instance, think about a dimmer switch for the lights in a room. You rotate the dimmer and the lights get brighter. There is a simulation that turns rotations into an amount of light. Just write any function, let’s say f(x) = 3x, and then turn the rotator in the simulation. The amount of light in the simulation will adjust to be three times the number of degrees of rotation.

Here’s another example. The length of a metal rod changes with its temperature. There’s a simulation for that: Write any function—for example, f(x) = 0.1arctan(x - 20) + 5—and then change the temperature in the simulation. The length will visually adjust according to the function.

This video highlights some functions that can be visualized with Math Simulations in the context of a playground:

### Know Activities, Do Activities, and Make Activities

Know activities: What I call Know activities help learners know the definitions of math concepts. They expose learners to examples of a concept as well as non-examples. Let’s take the concept of proportional relationships. A mechanical pencil can serve as a tangible example. Clicking the end of the pencil one time causes the lead to stick out 2 millimeters. Clicking a second time causes it to stick out 4 millimeters, and a third click makes it stick out 6 millimeters. This, we can explain to learners, is an example of a proportional relationship for two reasons: 1) When the number of clicks is zero, the length of the lead is zero; and 2) each click produces a constant increase in lead length: 2 millimeters.

Once the concept has been defined, Math Simulations allow us to produce many more visual examples quickly. For instance, we can open the temperature and time simulation and type the function y = 12x. Then slide the time slider in the simulator. Is the temperature zero when the time is zero? Yes. Does the temperature increase at a constant rate as the time increases? Yes. This is proportional. Non-examples can be produced just as easily. Typing f(x) = 50sin(x) will produce an example that clearly does not change at a constant rate.

Do activities: The second type of exercise, Do activities, helps learners translate between three representations of a concept: symbolic, real, and simulation. For instance, learners can be presented with a simulation and asked to write a symbolic function that models it. Or you can present them with a symbolic function and ask them to generate a real-world example of that function.

Make activities: Finally, Make activities help learners be creative with the math they learn. They ask learners to use mathematical simulations to imagine new possibilities in their world. In these activities, learners are presented with a real-world context along with a matching simulation and symbolic function. For example, they might be given the context of a fishing reel. After being cast, the reel is rotated to bring the line back in. The length of the fishing line is a function of the rotation of the reel. A function for this might be f(x) = 600 - (1/90)x, where x is in degrees and f(x) is in inches.

After familiarizing themselves with the situation, students are asked to write new functions that represent different fishing pole behaviors. These can range from slight adjustments to wild redesigns. The function f(x) = 700 - (1/90)x, for instance, implies a pole design with a longer fishing line. The function f(x) = 600 + 10sin(x) implies a design where rotating the reel pushes and pulls the line in and out. This is a potentially innovative fishing pole feature that might help lure fish, and it was imagined using pure mathematics.

### Final Thoughts

I’ve used Math Simulations with students in kindergarten through calculus, in classrooms where students have devices and ones where they don’t. In classes with one-to-one student devices, there’s a lot of potential for student creativity and ownership, but even when students don’t have devices, the simulations can still be useful, in whole class prompts and demonstrations.

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L.L. Barkat
Author, Educator, Publisher

Love the simulations. But also especially love the imagination aspect of the "Make" section. For kids who are storytellers and dreamers, this could be quite a new and intriguing perspective to introduce. Math stories, math dreams--ones that find their unexpected beginnings in symbolic math.

Harry O'Malley
I design and teach math.

I am thrilled that the essence of the "Make" activities was communicated in the article. Your description is *exactly* what they are designed to do. That, I think, is the most important aspect of these simulations. They allow mathematics to become a tool for creativity, for dreaming about the world, and for making. The tools I am developing have that single aim: to create learners who naturally think about their ideas in symbolic mathematics; learner's whose natural world-view is anchored in that language the same way most people's worldview is anchored in verbal language. Thanks for your comment.

L.L. Barkat
Author, Educator, Publisher

Thinking further, then, if we want learners to have their ideas anchored in (or initiated from) that language, then more must be done along the "make" side.

Imagine, for a moment, if any of us tried to learn our native language simply from listening to others speak and parsing their speech and even directing speakers to speak to each other, but we ourselves never formed words, never mixed and matched and made phrases, sentences, whole arguments from within ourselves using the medium of language.

I think math education too often is like that kind of experience. It's like being handed other people's sentences and intentions and dreams and thoughts all the time in the form of pre-made equations, worksheets, tests, without ever really exercising one's own "math linguistic" capabilities. Granted, all language formation needs the outside pieces. Babies wouldn't learn to speak without that, nor would adults trying to grasp a new language learn without that. But, besides listening and parsing what's coming from the outside, the baby is driven to babble from within, driven to test-drive language from within and this leads to an interactive process.

Bottom line is that it could be fun to do a lot more with math play, math dreaming, right from the start, and build upwards so that most of what kids feel like they are seeing are little story and dream possibilities on the page. (Ha. I bet a few people would like their checkbooks to look like a string of different stories or dreams! ;-) )

Harry O'Malley
I design and teach math.

We dream in the language of experience, which is a curated combination of sights, sounds, smells, tastes and textures. Our verbal language refers to and is directly linked to those tangible experiences, and is therefore directly linked to our dreams. Until I began using these simulations I could not dream in symbolic mathematics. I can now. When I read an equation like y=3x+1 or y=3(x-2)^2+8 it is as vivid an image as the sentence "The dog ran slowly across a field" and just as instant.

I have not designed these simulations as a supplementary resource. I have designed them to serve a primary role in math education. A math education where learners can think about mathematics in ways almost no one understands because they have never experienced it. They are game changing in a fundamental way. Dreaming in mathematics is not possible as a world-wide goal without them.

James Gee said it so directly: "When human beings understand anything, whether it's a text or the world, they understand it not by abstract generalities, but by literally being able to run in their head a simulation of images, and actions, and experiences that the words refer to."

These simulations allow symbolic mathematics to be linked directly to those experiences.

Here's a link to the Jame Gee quote, which comes from an Edutopia video: