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Sunday / November 19

Understanding Abstraction: Math Practice #4

The Japanese Footbridge, by Claude Monet — modeling

The Japanese Footbridge, by Claude Monet. Public domain, image obtained from Wikimedia Commons at

Modeling with Mathematics

Claude Monet was one of the founders of the art movement known as Impressionism. Artwork prior to this period in the late 19th century was primarily representational or narrative; artists were attempting to faithfully record what something actually looked like, or to capture a moment in a story. Impressionists, by contrast, were seeking to capture other things in their visual artwork: the character of light and color, for instance, or conveying a sense of movement in a still painting.

The painting above is not easy to identify at first. It seems like a mass of random colors and brushstrokes. But when you know the title, “The Japanese Footbridge,” you can easily see the shape, and you recognize the play of sunlight on foliage behind it, and the shimmer of water beneath.

Monet clearly was not trying to capture a photorealistic image. Instead, his work gives us a different perspective of his home and how he perceived it. This painting, and the others he painted of the same bridge, evoke a sense of sitting by the side of the pond without precisely capturing every edge and detail.

It helps to know more of the history as well. This is one of more than two dozen paintings Monet created of the same subject. This bridge was located at his home in Giverny, France. The home and its gardens at Giverny were not merely Monet’s residence. They were something of an obsession. He poured his energy and finances into it, tirelessly giving detailed instructions to his gardeners for constructing and maintaining it. He insisted, for example, that the water lilies be cleaned every morning of the soot from passing trains the day before. He also developed cataracts around 1912. Monet’s earliest paintings of the bridge from 1899 were fairly representational, but this work from around 1920 is much more abstract; as his eyesight failed, Monet’s later paintings were based more on memories and emotions than on the scene in front of him.

Now, when you look at this painting, you probably are able to see much more than just random blobs of color. The abstract work captures important insights about Monet and his relationship with his environment.

What, if anything, does this have to do with mathematics? One important purpose of using math is creating models of the world, hence the inclusion of Mathematical Practice #4 in the standards. These models give us insights into how the world works, help us see complex patterns that are difficult to perceive, and allow us to make meaningful predictions. But mathematical models are highly abstract. Just like viewing the painting without knowing the context, students will have a difficult time understanding a model without relating it to the underlying situation. It’s even better when students develop the model themselves by analyzing that situation and translating their observations into mathematical language.

According to Joseph Malkevitch, “A mathematical model is a simplified version of the real world that employs the tools of mathematics—algebraic equations, probability, statistics….” In other words, an abstraction of that real world painted with mathematical brushstrokes. Dan Meyer puts it more succinctly: “The process of turning the world into math and then turning math back into the world.” The model is not intended to capture the entire phenomenon; instead it attempts to describe a specific relationship or pattern that occurs in it, and the model can then be used to answer questions about that situation.

At its simplest, a mathematical model might just be an addition equation written to describe this scene: “Martina has eleven pretzels in her lunch bag. Reshard gives her four more pretzels from his lunch. How many pretzels does Martina have now?” The model (11 + 4 = 15) ignores much from the real world scenario that would truly matter to the two students involved: What kinds of pretzels were they? Why did Reshard decide to give his pretzels to Martina? Does she even want his pretzels? But it does tell us something useful (though admittedly trivial to an adult) about the exchange of resources.

Dan Meyer’s Three Act Tasks provide a robust and adaptable model for using problems to prompt student curiosity, then leveraging that curiosity to develop deep understanding of math concepts through reasoning and modeling. This is the essence of Math Practice #4.

This week we will explore three additional strategies for incorporating more mathematical modeling in your instruction.


Interpreting and creating infographics is a type of mathematical modeling, though a fairly straightforward one. Start by having your students study infographics related to topics you are studying or that relate to other classes they are taking. Besides learning to understand the data and analyze the patterns and relationships they see, you can use this as a tool to study bias and misuse of statistics, as well as what features make a good infographic. Have students think about how the images help the reader understand the connections between the numbers and the real world, and how the data represents one model of those relationships.

Some of the best sources for infographics are listed below. Be aware that some of these are not necessarily aimed at a student audience and may have content inappropriate for children, so it’s best to review the graphics yourself and select the specific ones you want to share with your class.

When you start having students create their own infographics, they can use data they collect themselves, or peruse one of many online data sources for interesting information. And while the best and most interesting student infographics are often those students create by hand with paper and pencil and markers, there are some great tools out there if you want to dive into the digital world, including Piktochart,, Canva,, and Venngage.

Engineering Challenges

Scientists and engineers use mathematics to model phenomena that happen in the real world. Research work in those fields is in part about refining those mathematical models to make them better predictors of what will happen when a new bridge is built or a genetically modified organism is introduced into a region.

I taught a three week course this summer for middle school students about the physics of potential and kinetic energy. I could have gone the traditional route and begun with pages of equations and days of computation, but instead I challenged them to build a roller coaster.

I gave students a few simple materials—foam tubing split to make flexible track, painter’s tape, and a glass marble—and a challenge: use these materials to make a roller coaster with at least one loop and two other turns. During the run, the marble must remain in the groove of the track at all times. After all, we don’t want passengers flying off the roller coaster mid-course!

They set to work building their creations. Some of them worked, some didn’t, and they kept at it until they were able to create something functional.

After discussing this design process, which included prototyping, testing, and lots of failure, we talked about how engineers who design real structures don’t have the same luxuries. They can’t build a real roller coaster or skyscraper just to find out if it will stay standing. They need to know before they build it that it will work exactly as planned. How do they do it? With a mathematical model.

This provided me the perfect opportunity to introduce the equations that describe the various forces and energy involved in running the roller coaster. Some of them we derived ourselves from data we collected during various experiments. Some of them I simply gave them, but the students had the real life experiences to understand what every part of the equation represented in the system.

Challenges like this one can be time consuming and a bit complicated to execute, but the payoff in depth of understanding is huge, and you can actually save time in the end because you will have less re-teaching and remediation to do.


Coding is hot right now. President Obama has called for an enormous investment in computer science in K-12 education. The International Society for Technology in Education (ISTE) includes computational thinking, of which coding is a part, in their standards for students. But it isn’t just trendy, it’s extremely valuable for supporting Math Practice #4.

You don’t need to wait for programming classes to come to your school or district to take advantage of its benefits for your students. Learning to code is directly connected with learning to model things mathematically. Every coding problem involves taking a situation, breaking it down into steps, determining how to represent those steps algorithmically, and translating those algorithms into a computer language. Students who become facile at coding will also get better at thinking mathematically, and specifically with thinking about mathematical models.

Two excellent sites that provide a large number of resources to get started with coding your own classroom are Teaching Kids to Code at Edsurge and Coding in the Classroom from Edutopia. Coming in November is a book by Jane Kraus and Kiki Prottsman titled Computational Thinking and Coding for Every Student.

From Abstract Back to Concrete

As students get more comfortable with the idea that math is not just a series of computations but is a tool for making sense of things going on in their own worlds, they will begin to see mathematical connections in unexpected places. They will also have many more concrete experiences to which you can help them connect the abstract meaning behind the equations and algorithms they are learning in your math class.

Just as Monet’s painting becomes more understandable with background information, so the math will stop being disconnected procedures and formulas that students must pull out of the air when completing a textbook exercise and start being real tools and powerful strategies for solving complex and challenging problems.

Written by

Gerald Aungst has more than 50 years of personal and professional experience fostering curiosity in learning through play and making stuff. During the last 25 of those years, in his various roles as a classroom teacher, gifted support specialist, administrator, curriculum designer, and professional developer, Gerald has worked to create a rich, vibrant, and equitable learning culture in schools. Gerald is currently a gifted support teacher in the Cheltenham School District in suburban Philadelphia. Gerald is also one of the organizers of Edcamp Philly and a founder of His book, 5 Principles of the Modern Mathematics Classroom, is available from Corwin. You can also connect with Gerald on Twitter at @geraldaungst.


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