from Flatland by Edwin A. Abbott MS / Math Geometry, Idea, Mathematics, Perspective, Story Divide the class up into groups of three and have the groups draw the name of a three dimensional object at random. Provide each group with an example of its object (pyramid, cone, sphere, etc.). Provide each group with graph paper, rulers, etc. and ask them to draw an accurate two-dimensional representation of the assigned object. Discuss with the entire class what you gain and lose by moving back and forth between a two-dimensional and three-dimensional world. Share copies of the text with the students and ask that they read the first two paragraphs (ONLY the first two) and examine the rest of the text briefly without reading. Discuss as a class what they anticipate the whole text will be like. Discuss whether the text is fiction or non-fiction. Have the students number the paragraphs 1-8. Read the whole text aloud slowly and pause after each paragraph so that students can identify any unfamiliar words or phrases. 1
Share as appropriate: Edwin A. Abbott (1838-1926) was an English schoolmaster and theologian, best known as the author of the novella Flatland (1884). His primary interests were theology and mathematics, and he was an important scholar and teacher of both. Create a list of unfamiliar terms (include for example: privileged, luminous, forelands, etc.). List these on the board, separating the overall list into two sub-lists of geometrical and non-geometrical terms. Include on the geometrical list: Equilateral, Hexagon, Oval, Pentagon, Square, Triangle. Divide the entire class up into small groups and have each group define a term, using mathematical language to define the mathematical terms. Post all of the relevant vocabulary as posters in the classroom. Divide the class up into eight work groups. Assign one paragraph to each group and ask the groups to summarize their paragraphs in modern English. Work through the paragraphs in order, reading the original and then each group s summary in turn. 2
What sentence or part of a sentence from the text is the key to understanding the Nature of Flatland? (round-robin response) Why did you choose that passage? (spontaneous discussion) When does a Flatland circle appear as an oval? When as a straight line? Why? In paragraph 6, the narrator mentions three figures that are drawings in the original book. With a partner, draw each of the three figures on a piece of paper. Why did you and your partner draw the figures as you did? The narrator of Flatland is a square. How do you think that affects his perspective? Why in paragraph 8 does the narrator say that when our friend comes closer to us we see his line become larger; if he leaves us it becomes smaller? What would it mean if you lived in Flatland and couldn t tell a Triangle, Square, Pentagon, Hexagon, [or] Circle apart? In what way is our world like Flatland? Can you imagine dimensions to the universe that we can t perceive? 3
Revisit the original two-dimensional drawings from the Launch Activity. Discuss what they would look like in Flatland from the perspective of a Flatlander; from the perspective of someone floating in space above Flatland; and from someone who perceives three dimensions. How do three dimensional objects appear in two-dimensional space? After reading and discussing an excerpt from Flatland by Edwin A. Abbott, write an essay in which you describe a three-dimensional object (pyramid, cone, sphere, square box, etc) as it would appear within Flatland, and from above Flatland. Be sure to explain why it appears as it does. Support your discussion with drawings and with evidence from the text. (Informational or Explanatory/ Description) (LDC Task#: 14 ) 4
Invite participants to talk in pairs for two minutes to share thoughts about what the writing task is asking and how they might respond. Allow a few minutes for all to create designs (paragraph outlines) for their arguments. Have students draft their designs on paper along with preliminary sketches for their drawings and use the designs and sketches to refine their thinking. Challenge all to draft their explanations by writing the paragraphs defined by their outlines and refining their sketches. Refer to the original text in order to illustrate key points. Have participants work in pairs to read their first drafts aloud to each other with emphasis on reader as creator and editor. Share sketches as part of reading aloud. Listener says back one point heard clearly and asks one question for clarification. Switch roles. Give time for full revisions resulting in a second draft. Once the second draft is complete, have participants work in groups of three-four and this time take turns reading each other s second drafts slowly and silently, marking any spelling or grammar errors they find. (Have dictionaries and grammar handbooks available for reference.) Take this opportunity to clarify/reteach any specific grammar strategies you have identified your students needing. Give time for full revisions resulting in a third and final draft. Also, note that in producing the final drafts of their essays, students should also create final drawings to illustrate their points. 5
Publish the collection of argumentative essays both online (on the class website) and on paper. Display the finished essays in the classroom or other public space in your school along with samples of the three dimensional objects themselves. Invite other math classes to a gallery walk through the displays hosted by your students so that they can explain the thinking expressed in their papers. Terry Roberts National Paideia Center 6
From Flatland: A Romance of Many Dimensions Edwin A. Abbott PART 1 THIS WORLD Section 1 Of the Nature of Flatland I CALL our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows--only hard with luminous edges--and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said "my universe": but now my mind has been opened to higher views of things. In such a country, you will perceive at once that it is impossible that there should be anything of what you call a "solid" kind; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonstrate. Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle. But now, drawling back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view, and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line. The same thing would happen if you were to treat in the same way a Triangle, or a Square, or any other figure cut out from pasteboard. As soon as you look at it with your eye on the edge of the table, you will find that it ceases to appear to you as a figure, and that it becomes in appearance a straight line. Take for example an equilateral Triangle-- 7
who represents with us a Tradesman of the respectable class. Figure 1 represents the Tradesman as you would see him while you were bending over him from above; figures 2 and 3 represent the Tradesman, as you would see him if your eye were close to the level, or all but on the level of the table; and if your eye were quite on the level of the table (and that is how we see him in Flatland) you would see nothing but a straight line. When I was in Spaceland I heard that your sailors have very similar experiences while they traverse your seas and discern some distant island or coast lying on the horizon. The far-off land may have bays, forelands, angles in and out to any number and extent; yet at a distance you see none of these (unless indeed your sun shines bright upon them revealing the projections and retirements by means of light and shade), nothing but a grey unbroken line upon the water. Well, that is just what we see when one of our triangular or other acquaintances comes towards us in Flatland. As there is neither sun with us, nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes closer to us we see his line becomes larger; if he leaves us it becomes smaller; but still he looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will--a straight Line he looks and nothing else. By permission of Oxford University Press (Source - Pages 15-16 in Chapter One from Flatland: A Romance in Many Dimensions by Edwin A. Abbott, edited with an introduction by Rosemary Jann [Oxford World Classics, 2008]) 8