"Even a rock is changing on a molecular level. Fractals can be designed to simulate almost any shape you can imagine."
This is the fourth page of the section "Fractal Worlds." The above passage tells us that real world objects which change dynamically are better represented by fractals than by Euclidean figures.
Here are a few more excerpts from this page:
"Fractals are not necessarily confined to one rule, but a series of rules and stipulations can be the rule. Pick a simple object and design a rule to apply to it."
There is only one picture on this page. Its picture reads:
"The first five stages of a computer generated geometric fractal."
Of course, I won't be able to find this exact fractal online, but here's a link to some software that does generate such fractals:
Today is the third and last day of finals week. It has been my tradition on the blog on the third finals day to post a review of the old semester and preview of the new semester.
Yesterday, when I was searching for MTBoS blogs, I did find the following blog:
https://thegeometryteacher.wordpress.com/
Well, this teacher does include "The Geometry Teacher" in his URL, so surely his blog must have something to do with Geometry, right?
The blog belongs to Andrew Shauver, a Michigan teacher and instructional coach. In fact, Shauver has a link to a complete Geometry course here:
https://thegeometryteacher.wordpress.com/courses-i-teach/the-geometry-course/
And so for our review, let's compare our U of Chicago course to Shauver's to look for any key similarities and differences.
Let's start with the first semester plan from the U of Chicago text:
1. Points and Lines
2. Definitions and If-then Statements
3. Angles and Lines
4. Reflections
5. Polygons
6. Transformations and Congruence
7. Triangle Congruence
This corresponds roughly to the first three units of Shauver:
1. Introduction to Transformations
2.Rigid Motions and Congruence
3. Angles, Triangles, and Parallelograms
Most members of the MTBoS cite as an influence Dan Meyer -- even though he's no longer the King of the MTBoS -- and Shauver is no exception. Meyer's famous "3-Act" lessons permeate Shauver's pacing plan. Indeed, even his First Day of School activity is one of Meyer's, "Best Circle." (Note: This is a different project from "Circle Square" mentioned in yesterday's post.)
The new Queen of the MTBoS, Fawn Nguyen, also appears on Shauver's pacing plan. Unfortunately, the link Shauver gives is dead. Apparently, Nguyen's original post was dated May 7th, 2013, but I found the correct post on Nguyen's blog dated May 8th instead:
http://fawnnguyen.com/let-problem/
Her activity is a miniature golf problem and is thus similar to Lesson 6-4 of the U of Chicago text.
Shauver writes:
A word about PROOF: When I arrived in my math classroom in 2008, Geometry was one of the most universally-hated classes at the high school where I taught. Proofs were a huge reason why. I set out to change that and we made some significant progress. You’ll see our philosophy on proof permeate lots and lots of the activities that appear below, largely because we believe that good proofs don’t start getting written until the structure and purpose are really internalized by the students. But, it would be unfair of me to not give you a chance to further explore our thoughts during the transition.
Of course, it's possible that Geometry was hated because it's math, and many students already hate math from their experience in Algebra I. Then again, we know that many students don't necessarily like proofs either.
Shauver gives several links to old blog posts, including one where he allows students to measure triangles before giving the proof:
https://thegeometryteacher.wordpress.com/2012/12/07/when-measuring-is-okay/
Shauver writes:
I was raised in the math field to believe that measuring with ruler and protractor have no business in deductive reasoning. “Don’t trust the picture,” they’d say. I once sat at a PD session instructed by a man who told me that he would purposely give his students ridiculous pictures that had no bearing in reality so they, “learned to only trust the given information.”
This bothers me.
It bothers me because it excludes students whose logical sense is still developing.
This year, I scrapped that. I started with what it means to “prove” something first. Let them write in their own language. Lots of words; paragraphs, arguments like that.
Then, when they get tired of so many words, I teach them the shorthand and the notation.
Then, when they get tired of making the same arguments over and over and over again, then I teach them about theorem and postulate.
But, it required me letting them use them at first and still being okay calling it proof.
Compare this to what mathematician David Joyce writes about proof:https://mathcs.clarku.edu/~djoyce/java/elements/geotfacw.html
[Note: I've linked to David Joyce several times on this blog before, but apparently Clark University in Massachusetts changed the URL.]
Chapter 1 introduces postulates on page 14 as accepted statements of facts. The four postulates stated there involve points, lines, and planes. Unfortunately, the first two are redundant. Postulate 1-1 says 'through any two points there is exactly one line,' and postulate 1-2 says 'if two lines intersect, then they intersect in exactly one point.' The second one should not be a postulate, but a theorem, since it easily follows from the first. And what better time to introduce logic than at the beginning of the course. No statement should be taken as a postulate when it can be proved, especially when it can be easily proved.
Final conclusion. Much more emphasis should be placed on the logical structure of geometry. Postulates should be carefully selected, and clearly distinguished from theorems. Every theorem should be proved, or left as an exercise, or noted as having a proof beyond the scope of the course. Very few theorems, or none at all, should be stated with proofs forthcoming in future chapters.
Indeed, if the text Joyce critiques here were to contain something similar to Shauver's lessons, that text would probably call it a "Work Together." Here's what Joyce says about "Work Togethers":
It should be emphasized that "work togethers" do not substitute for proofs. They can lead to an understanding of the statement of the theorem, but few of them lead to proofs of the theorem. It must be emphasized that examples do not justify a theorem.
To Joyce, if an argument isn't rigorous, it's not a proof -- it's not mathematics. To Shauver, an argument can be less rigorous and still count as a proof and mathematics.
Shauver explains that his method gives results:
You know what? Engagement is soaring! Students aren’t afraid. They are willing to try. They are willing to listen to the feedback. They are willing to learn.
The big question is, would engagement be soaring, and would students be as willing to try, if the course were taught the way Joyce would like it to be?
The U of Chicago text takes an approach between those of Shauver and Joyce. Proofs are introduced in Chapter 3, and only in a few introductory lessons (like Lesson 7-1) are students encouraged to make Shauver-style triangle measurements. On the other hand, proofs are still not as developed as Joyce would prefer (such as proving the slope properties -- a Common Core Standard). In past years, I used Joyce's principles to guide my modifications to the text, but this year I'm restoring the U of Chicago order and dropping many of Joyce's suggestions.
(And of course, we must not leave out John Chase from yesterday's post -- not only does he appreciate statements justified in Geometry, but he extends this idea to other courses, and rejects Statistics because it makes unproven claims. Clearly Chase leans closer to the Joyce philosophy.)
Here is our second semester plan. We'll begin with Chapter 8:
8. Measurement Formulas (January 8th-17th)
9. Three-Dimensional Figures (January 18th-31st)
10. Surface Areas and Volumes (February 1st-15th)
11. Coordinate Geometry (February 16th-March 2nd)
12. Similarity (March 5th-16th)
13. Logic and Indirect Reasoning (March 19th-April 9th)
14. Trigonometry and Vectors (April 10th-23rd)
15. Further Work With Circles (April 24th-May 9th)
Since the new semester begins on Day 84, we start with Lesson 8-4, "Areas of Irregular Figures." It means that Lessons 8-1 (perimeter), 8-2 (tessellations), and 8-3 (rectangle area) are omitted. But 8-1 and 8-3 are easy to squeeze in -- "perimeter" just means "add up all the sides."
On the other hand, we see that Shauver skips almost all of Chapter 8 on area. Our second semester corresponds roughly to his Units 4-8:
4. Triangles and Similarity
5. Trig or Treat
6. 3D Geometry
7. Circles
8. Year-End Wrap-Up
We see that he begins with similarity, which is very much like how I used to begin the second semester with Chapters 12 and 14 before jumping back to 10. He covers circles later than I (Unit 7), meaning that he doesn't reach it until after Pi Day.
Meyer's lessons continue to appear liberally in Shauver's pacing plan. Where a Meyer lesson isn't available, Shauver often uses a Desmos lesson instead. (Recall that the Desmos online calculator is the official calculator of the California SBAC.)
Let's look at a sample Shauver second semester lesson:
https://thegeometryteacher.files.wordpress.com/2012/12/g-penn-1-intro-and-explore-public.pdf
Shauver's Unit 5 is very much like Chapter 14 of the U of Chicago text, and indeed both his Unit 5 and our Chapter 14 begin with special right triangles. But the U of Chicago goes straight into proofs of the 45-45-90 and 30-60-90 theorems, while Shauver opens with an activity instead. To this end, Shauver's lesson is not unlike Michael Serra's Discovering Geometry text -- Lesson 10.4 in my (old, of course) version of Serra starts a bit like Shauver's.
In a way, Shauver's proofs have a level of rigor somewhere between those of the U of Chicago text and the Serra text. Shauver's second semester is very similar to Serra's, but even in the first semester, Shauver slowly builds up formal proofs. In particular, Shauver tells us that he uses paragraph proofs, while Serra either uses flowchart proofs or no proofs at all in the first semester.
Unfortunately, there won't be much time for me to link to Shauver second semester activities. As you already know, these U of Chicago chapters in the second half of the book are longer than the chapters in the first half. In particular, Chapters 8, 10, and 15 have nine sections each, while Chapter 12 on similarity has ten sections. In order to maintain our digit pacing plan, some review worksheets will have to occur on lesson days. Last year, after I left my old classroom, I began with Chapter 12, and so you can check how I accommodated the ten sections of Chapter 12 in last year's posts.
Well, that does it -- this is the end of the first semester, and tomorrow starts winter break. I plan to post four times during winter break, with the first post on Christmas Eve -- a tradition I established the first two years of this blog (though not last year). That will feature yet another computer music post, with some Christmas songs and another new scale.
The first day of the second semester will be Monday, January 8th, 2018. As I wrote earlier, this will be Day 84 and Lesson 8-4.
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