Curriculum

Flatland: A Project of Many Dimensions

joshwilkerson3 Comments

Last fall my advanced (freshmen) geometry classes completed a semester project on Flatland, a book by Edwin A. Abbott. In case you haven’t heard of this obscure yet significant work, let me start this post by providing a brief overview of the book and then I’ll describe the structure of the project.

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The Book (From Amazon.com):

Flatland is one of the very few novels about math and philosophy that can appeal to almost any layperson. Published in 1880, this short fantasy takes us to a completely flat world of two physical dimensions where all the inhabitants are geometric shapes, and who think the planar world of length and width that they know is all there is. But one inhabitant discovers the existence of a third physical dimension, enabling him to finally grasp the concept of a fourth dimension. Watching our Flatland narrator, we begin to get an idea of the limitations of our own assumptions about reality, and we start to learn how to think about the confusing problem of higher dimensions. The book is also quite a funny satire on society and class distinctions of Victorian England.

The Project:

The essence of this project was to have students interact with the subject matter of this book through weekly reading comprehension quizzes, two in depth class discussions, and a culminating extension project done in response their understanding of the book.

Reading Schedule (links are to the reading comprehension quizzes):

Class Discussion:

There were two major class discussions: one over the first half of the book that extended to a discussion of Flatland and society, and one over the second half of the book that extended to Flatland and theology. One week prior to our class discussion the students were given the following prompts and extra reading resources.

Discussion 1: Flatland and Society: In addition to being an interesting piece of Geometric fiction, the first half of Flatland serves as a satirical social commentary on Victorian England. Do a little research on what society was like at during that period. Be prepared to discuss this social culture and link it by analogy to the society of Flatland.

In your preparation I would like for you to wrestle with the following quote about mathematical advances in Victorian England (taken from Equations from God: Pure Mathematics and Victorian Faith, by Daniel Cohen) where we find “origins of the rebirth of abstract mathematics in the intellectual quest to rise above common existence and touch the mind of the deity” (emphasis added). Relate this comment to the notions presented in Flatland.

Here are some questions that you might think of answering prior to class discussion, and these concepts may help you begin to organize your thoughts. It should be noted that none of these questions have easy answers, but they do have plausible ones. I am not looking for you to give a “right” answer, but to wrestle through the concept and tell me what you think.

  1.  How does the society of Victorian England mirror that of Flatland?
  2. How are women depicted in Flatland? How were they treated in Victorian England? What point is the author trying to make in his depiction of women?
  3. Why might Flatlanders be interested in rising above common existence? Why would this be said of people in Victorian England?
  4. How does A. Square “touch the mind of deity?” How might we “touch the mind of deity” as we study mathematics?

Resources:

Discussion 2: Flatland and Theology: Edwin A. Abbott was not only trained in mathematics but theology as well. The second half of the book can be seen as a theological analogy: the visitor from Spaceland interacts with Flatlanders in a way that might mirror how God, who is not bound by the limits of three dimensions, interacts with our three-dimensional world. Draw out this analogy using specific examples from the book and from the Bible to answer the question: “How do we experience God?” Be sure to cite any references you make to Flatland and use biblical examples of God’s interactions with His creation to support your ideas.

Here are some questions that you might think of answering prior to class discussion, and these concepts may help you begin to organize your thoughts. It should be noted that none of these questions have easy answers, but they do have plausible ones. I am not looking for you to give a “right” answer, but to wrestle through the concept and tell me what you think.

  1. How might our universe appear to a being outside of it? How did Flatland appear to the Sphere?
  2. What does it mean for God to be “over all, through all, and in all” (Ephesians 4:4-6)? How might this be said of the Sphere in relation to Flatland?
  3. How might it be possible for God to be omniscient (know what is happening in all parts of the world at the same time)? How can this be said of the Sphere in relation to Flatland or of the Square in relation to Lineland or Pointland?
  4. What were A. Square’s limitations in fully understanding the Sphere? What are our limitations in fully understanding God?
  5. What would we expect to see if/when God entered the world? How might it be similar or different from when the Sphere entered Flatland?

Resources:

Project Options:

Choose from one of the following options as a project response to your reading of Flatland. The resources cited for each option are available on the course webpage under the “Projects” folder. You are not restricted to these resources, in fact you are encouraged to find additional ones, however the cited resources must be analyzed prior to beginning your project.

The project proposal needs to include your name, class period, and the project option you have selected. You also need to provide a preliminary outline or plan of attack. The big idea here is that you need to demonstrate that you have spent time considering your project and have a clear plan for moving forward.

All papers, while they may deal with a subject that is not explicitly mathematical, need to contain significant geometric content. Show me that you understand the geometry as it is presented in the book and then take that knowledge to analyze the subject matter of your paper. Papers will be graded more on mathematical understanding than on writing ability, though proper grammar and spelling need to be used (see rubric for more details).

Option 1: Book review, extra reading plus a 2 page review. Read one of the following “sequels” to Flatland and write a short review:

  • Sphereland: A Fantasy about Curved Spaces and an Expanding Universe, by Dionys Burger (ISBN: 0064635740)
  • Flatterland: Like Flatland, Only More So, by Ian Stewart (ISBN: 073820675X)

The book review should take 1 page to summarize the book and 1 page to critique it. In the critique you need to address what the author did well/poorly and why. You should also address what impact the book had on your understanding of Flatland. Be sure to comment on significant/interesting mathematical descriptions as they are presented. You will find several sample book reviews linked on the course website to give you a feel for what a book review looks and sounds like.

Resource: How_to_write_a_book_review

Option 2: Creative writing assignment, minimum of 4 pages, on one of the following topics :

  • Rewrite A. Square’s description of the visitor from Spaceland had that visitor been a Cube instead of a Sphere. Then rewrite the experience again if the visitor had been each of the other Platonic Solids (Tetrahedron, Octahedron, Dodecahedron, Icosahedron). This amounts to writing ¾ to 1 page on each solid.

Resource: Descriptions of Platonic Solids

  • Write an additional chapter for the book in which the Sphere is visited by a person/object from the 4th dimension. Write the Sphere’s description of the visitor in a way that mirrors A. Square’s description of his encounter with the Sphere.

Resources: Short story “The 4D Doodler” by Graph Waldeyer & “What is the 4th dimension?” by Eric Saltsman

Option 3: Explore 4 Dimensional Hypercubes: Complete questions 1-4 on the Hypercube Activity posted on the class website (Kuyers Institute Lesson on Hypercubes). This project will culminate with an activity in which you build a 3D model of a 4D hypercube.

Resources: Helpful websites are linked within the Hypercube Worksheet document & “What is the 4th dimension?” by Eric Saltsman

Project Rubric

Additional Resources:

Reclaiming Math

joshwilkerson1 Comment

By Katherine Loop

(Disclaimer: The views expressed by guest authors do not necessarily reflect those of GodandMath.com. Guest articles are sought after for the purpose of bringing more diverse viewpoints to the topics of mathematics and theology. The point is to foster discussion. To this end respectful and constructive comments are highly encouraged.)

How can “1 + 1 = 2” be anything but neutral? How can math facts be approached biblically?

If you’ve ever wondered how a biblical worldview applies to math, I’d invite you to join me on a little exploration. We will take a look at some biblical principles that give us a framework for understanding math, explore the worldview conflict in math, and examine how we can “reclaim” this vital subject for the Lord.

Applying a Biblical Worldview to Math

While the Bible does not specifically talk about math, it offers principles through which we can view and understand each aspect of life, including math. For example, the Bible teaches us that Jesus created and sustains all things (Colossians 1:16-17). Thus He created and sustains math! This does not mean He created the symbols on the piece of paper we have come to associate with math; it means He created the real-life principles those symbols represent.

Sit back and think for a moment about why math works. Why do equations we solve on paper apply in real life? They apply because they are ways of describing the consistent way God holds the universe together! One plus one consistently equals two because God keeps objects adding together consistently. We can only rely on math to work because a faithful, never-changing God consistently sustains this universe.

Biblical principles can also govern our use of math. The Bible teaches us that God has given each person work to do—work we are to work at as unto the Lord (Colossians 3:17). So as we learn math, we want to learn to use it to help us in our work—and we want to do it as unto the Lord!

Math proves a useful tool in many areas, both directly and indirectly. Math also helps us learn logical thinking, structured thought process, and problem-solving skills, thus indirectly serving as a tool at a deeper level.

The Secularization of Math

Now that we have an overview of a biblical worldview toward math, let’s take a look at what worldview is prevalent in most math teaching today. Pick up nearly any math textbook, and you’ll notice it does not mention where math came from or why it works. Math “facts” are taught and drilled into students day in and day out, but students are never given a foundation on which to understand those “facts.” They’re left looking at math as a neutral, self-existent structure.

While calling something neutral may not seem like a worldview, it is! It subtly takes God out of math. Neutral means indifferent or, “not engaged on either side; not aligned with a political or ideological grouping.”[1] When taught to view math as neutral, students are subtly being taught that there are certain absolutes and principles that exist independently from God—absolutes that are not engaged with a worldview.

This is a subtle but very dangerous twist on reality. Math should be reminding us to trust God, the One who day in and day out sustains this universe with such faithfulness that math facts can be used to describe how objects will operate! Instead, math facts end up encouraging students to place their faith in human reasoning and math itself. They are left viewing math as the source of truth.

Math works consistently. But if we view math facts as “neutral,” we are subtly giving math itself—or man’s reasoning—credit for the consistency around us. Instead of leaving math awed at God’s faithfulness, we’re learning to trust math and human reasoning. This sort of thinking lies at the root of naturalism and humanism.

Looking back, I can see how the things I learned in math for years subtly harmed my heart. When I began wrestling to make my faith my own, I approached the topic with the thought that I needed to see what math and “science” proved or disproved before I could commit myself to God. My understanding of math, which should have been reinforcing God’s faithfulness (after all, math could not exist without the biblical, faithful Creator!), was actually drawing me away from Him and encouraging me to trust math itself and human reasoning.

Reclaiming Math

So how do we reclaim math? How do we teach the next generation to base their thinking in every area, including math, on God’s Word?

When I first thought about teaching math Biblically, my first inclination was to add a Bible verse or analogy to a math lesson.

However, adding a Bible verse to a secular lesson does not fix the problem. It is similar to taping Bible verses over a bomb. We might succeed in hiding the danger, but not in diffusing it.

If we really want to reclaim math, we need to diffuse the bomb and begin looking at each concept, not as a neutral fact, but as a useful way of describing the consistency in our universe God created and sustains. As we do so, we will discover fascinating glimpses of God’s character revealed in math and begin seeing math as more than numbers on a piece of paper.

For example, when studying addition, rather than merely memorizing how to add two numbers, we can pause and realize that 12 + 4 only consistently equals 16 because day in and day out, God is holding all things together that consistently—by the power of His Word! Wow! What a powerful, faithful God we serve!

Then, rather than memorizing the steps to the addition problem by rote, we can realize that these steps are just one way men, using the ability God gave them, have come up with to describe the consistency around us. There are other ways to add—including ways that do not involve carrying digits at all. We might take a look at some of those other ways…or at various ways addition can help in our daily tasks or in exploring God’s creation.

Rather than looking at math as an independent fact, we are now seeing it as a useful tool to describe God’s creation. We are left marveling at God’s faithfulness and power, encouraged to trust Him in every area.

Yes, indeed, math can be—and needs to be—reclaimed. Its very existence serves as a testimony to God’s faithfulness. Let’s not miss out on seeing Him in math!

Katherine Loop, a homeschool graduate, is the author of various homeschool resources, including two books on teaching math from a biblical worldview. View more information on viewing and teaching math from a biblical perspective and sign up for her free e-newsletters on her website, www.christianperpsective.net.  

This article is based on one originally published in the Home School Enrichment Magazine. Reprinted with permission.

[1] Webster’s New Collegiate Dictionary, 1974 ed., s.v. “neutral.”