Lecture 14 of David Kung's Mind-Bending Math is called "Losing to Win, Strategizng to Survive." In this lecture, Dave Kung describes a few more paradoxes involving predictions and games.
The first one Kung discusses is called the Unexpected Exam -- actually, it's better known as the Unexpected Hanging paradox, but this is a school blog, so Unexpected Exam it is. A teacher tells the class that there will be an exam one day next week, but they won't be expecting it. The test can't be on Friday, since by process of elimination, if the test hasn't occurred by Friday, it must be on Friday and therefore no longer unexpected. So the latest the test can occur is Thursday -- which means that the test can't occur on Thursday itself, as it would no longer be unexpected. For the same reason, a Wednesday test is no longer unexpected, and so on via induction to Tuesday and Monday. The conclusion is, no unexpected test can ever be given. Then the teacher gives the test on Wednesday -- and no one's expecting it.
This is currently finals week. No test given this week should be unexpected -- teachers have been informing their students about finals week schedules ever since they returned from winter break. But there are everyday examples of the Unexpected Paradox that occur in the classroom -- and none of them involve tests.
We begin with the Unexpected Bell Paradox. Let's say that we are teaching a 50-minute period -- say it's first period from 8:00-8:50. Then around 8:45, the students stop paying attention. Sometimes I wonder whether these means the class is five minutes too long, and perhaps we should solve it by making the class five minutes shorter, to end at 8:40. But then students would stop paying attention at 8:40, and if we made class end at 8:40, students would stop paying attention at 8:35. So the class seems to be "five minutes too long" no matter how long it is -- hence the Unexpected Bell Paradox.
The other versions of this paradox are the same, but at a larger scale:
-- Unexpected Afternoon Paradox: During sixth period (think back to MTBoS Week 1), students stop paying attention. But if we shortened the day by one period, students would stop during fifth period.
-- Unexpected Weekend Paradox: Students are less attentive on Fridays. But if we shortened the week by one day, students would stop paying attention on Thursdays.
-- Unexpected Vacation Paradox: Students are less attentive the week before a long break (for example, the week of December 14th before the recent winter break). But if we shortened the term by one week, students would be restless the week of December 7th.
-- Unexpected Summer Paradox: Students are less attentive the whole month of June. But if we shortened the year by one month, students would be restless all of May.
All of these really mean the same thing -- it's hard to get students to work as hard during the last 10% of any unit of time -- a period, day, week, term, or year -- as it is during the first 90%.
Notice that unlike the original Unexpected Exam Paradox, these are not inexorable. If we shortened school to one day a week (think once-a-week music lessons), then students do not treat the entire day like a Friday, but other paradoxes such as Unexpected Bell still apply.
There are ways to fight the Unexpected Bell Paradox. I've seen some teachers block the clock on the wall with a large sign, "Time Is Passing, Are You?" If a student asks me how many minutes are left in the period, I always round up to the nearest five minutes.
Also, notice that there's a way that a weekend really can be unexpected. Students work hard all day Thursday (since Friday is the last day of the week), and then there's a snow day on Friday. Since this is unexpected, students really did work hard until the last day of the school week. (On the other hand, fire drills are not unexpected bells, since fire and other drills are announced well in advance and students end up already knowing when the drills are.)
Returning to Kung's lecture, we still haven't reached the paradoxes mentioned in the title. Here are links to describe these two paradoxes in more detail:
The Quick Conundrum for the lecture asks, how can a paperclip float on water? If we are careful, we can use surface tension to make it float. But adding alcohol to the water makes the paperclip sink.
Today is the last day of the semester. Like many other districts, there is a professional development day between the semesters for teachers, but not for students. So there are no students -- and therefore no subs -- tomorrow. The second semester will begin on Monday, January 25th, which is Day 89.
We notice that the first semester contains only 88 days, while the second has 92. This is more accurate than the district whose calendar I followed last year, with only 84 days first semester. In general, Early Start schools have less accurate semesters because they usually don't start early enough in August to have a full 90 days of school before Christmas. Indeed, LAUSD had only 79 days in the first "semester" -- actually 78, due to school being cancelled one day.
It's time for us to plan for the second semester. I've decided to follow more or less the same order of U of Chicago chapters that I used for the second semester last year:
Chapter 12. Similarity
Chapter 11. Coordinate Geometry
Chapter 14. Trigonometry and Vectors
Chapter 8. Measurement Formulas
Chapters 9-10. Surface Area and Volumes
Chapter 15. Further Work With Circles
The main difference between last year and this year is my elimination of Chapter 13. Actually, I've split up the material in Chapter 13 and spread it out into different units. I've covered most of Chapter 13 already, except for Lesson 13-5, "Tangents to Circles and Spheres," which I'll include with Chapter 15 on circles.
Now I want to discuss the first unit of the second semester (the seventh unit of the year, I suppose) in further detail. This will be a Dilations unit and cover Chapters 12 and 11. Last year I covered these two chapters in four weeks. But I believe that I can cover them in three weeks, because there are so many lessons that I'm leaving out. Last year I included Lesson 11-4, the Midpoint Formula, but we just covered the Midpoint Formula last week, so we don't need it again. For that matter, I also included a review of the coordinate plane, labeled as Lesson 1-3 -- but of course we've been graphing throughout the first semester.
Lesson 12-6 covers the Fundamental Theorem of Similarity. I'm dropping this lesson because it was based on a very complicated proof due to Dr. Hung-Hsi Wu. (Then again, that hard proof was based on induction, and now we can use the Blue-Eyed Islanders problem from yesterday to introduce induction to the class -- but I won't.)
Originally, my plan was to just to assume the key result of Wu's FTS -- that dilations with scale factor k multiply all distances by k -- as part of a new postulate, a Dilation Postulate. But I mentioned a problem with this postulate back in October -- and it all goes back to something posted on one of the traditionalists' websites. Of course I'm talking about Dr. Katharine Beals:
I discussed back in October that this PARCC problem is essentially asking students to use the SAS Similarity Postulate to prove the properties of dilations. But the Common Core Standards imply that students will be doing the reverse -- to use the properties of dilations to prove SAS Similarity!
So my plan is to rearrange the lessons in Chapter 12 by placing a SAS Similarity Postulate first:
First Week of Semester:
-- Lesson 12-9, The SAS Similarity Postulate
-- Lesson 12-2, Dilations Without Coordinates
-- Lesson 12-3, Properties of Dilations
-- Lesson 12-10, The Side-Splitting Theorem
Second Week of Semester:
-- Lesson 12-5, Similar Figures
-- Lesson 12-8, The AA and SSS Similarity Theorems
-- The Pythagorean Theorem and Distance Formula
-- The Slope Formula
Third Week of Semester
-- Slopes of Parallel and Perpendicular Lines
-- Equations of Lines
-- Review for Chapter 11/12 Test
-- Chapter 11/12 Test
But the traditionalists do have a point here. Common Core proponents may argue that the Core is a set of standards and not a curriculum. Yet right on this website, I keep changing the order of the lessons to match PARCC questions -- meaning that the Core is dictating my curriculum after all!
It's been a while since I linked to Beals's website. She still posts her "Math Problem of the Week" series in which she criticizes Common Core problems. I haven't linked there lately since I only want to focus on the Geometry problems. But now -- you guessed it -- this week she decided to post a Geometry problem.
And for those of you who notice that I keep going back to PARCC questions even though I live in California, which doesn't use PARCC, this week's Beals problem is for California students:
Jose and Tina are studying geometric transformations. Jose is able to move triangle A to triangle A' using the following sequence of basic transformations:
1. Reflection across the x-axis
2. Reflection across the y-axis
3. Translation two units to the right
Tina claims that the same three transformations, done in any order, will always produce the same result. Explain why Tina's claim is incorrect.
As usual, Beals adds an "extra credit question":
For which 21st century careers and college courses are 9th graders preparing for in learning to reason verbally about reflections and translations?
Again, Beals writes a sarcastic remark about the stated goal of the Common Core. The intended answer to her question is that reflections and translations don't make students college or career ready, so they'd be much better off if we just drop transformations from the curriculum entirely.
But for once, someone actually answered Beals's extra credit question. The poster lgm writes:
EC A: science. Many need to develop their visualization skills, and their ability to understand someone else's reasoning on the spot.
To see how to answer the CAASPP question listed above, notice how the question is asking us when can transformations can be done in any order -- that is, when do they commute? I've mentioned on the blog that reflections and transformations commute exactly when the mirror of the reflection is parallel to the direction of the translation. The mirror in step 2 is vertical (the y-axis), while the direction in step 3 is horizontal (to the right). Therefore steps 2 and 3 do not commute, and Tina is mistaken to believe that they do commute.
Indeed, when the mirror of the reflection is perpendicular to the direction of the translation, as they are in this problem, the two transformations are anticommutative. If we started at A and were to perform Step 1, then translate two units to the left before performing Step 2, only then would we obtain figure A'. Simply performing Steps 1, 3, and 2 doesn't map A to A' -- instead, the image of A is four units to the left of A'.
So why should high school students have to learn this? Recall my main reason -- if we didn't use transformations, statements like SSS, SAS, ASA, and SAS Similarity (or at least one of the other similarity statements) are all postulates, but with transformations, they are all proved theorems. Also, by asking questions about when transformations commute, students start to realize that not everything commutes -- and this does come up in Algebra II and later classes.
Of course, Beals and the other traditionalists do make valid points. It's awkward (and circular) to say that we need transformations (dilations) to make SAS Similarity a Theorem, then turn around and ask students to use SAS~ to prove the properties of dilations. Since this week's question comes from CAASPP (part of Smarter Balanced), perhaps the SBAC is more consistent than the PARCC regarding whether statements about transformations are to be proved or assumed.
Also, Beals has a valid point about the need to explain answers -- something which we know is all over the Common Core. Perhaps for this week's question -- since it asks for a counterexample to Tina's claim that the three transformations commute -- it's better to ask the students just to give the three transformations in one order that doesn't map A to A'. In this case, 1, 3, 2 is a valid answer, as are 3, 1, 2 and 3, 2, 1. The other two permutations both map A to A' -- the trick is that since 2 and 3 don't commute, any permutation in which 3 comes before 2 is a counterexample.
The whole point of using the U of Chicago text is to show that of all the pre-Common Core texts, this one actually follows Common Core the best -- but of course, it doesn't follow the Core exactly. Yet I undermine my claim every time I cover the chapters in a different order, skip chapters, or add material from different texts -- especially when the reason is to match PARCC (which may die soon anyway).
I'm considering doing something much simpler next year. I will simply follow the U of Chicago text in order, and follow a simple pattern to determine which lesson to cover on a particular day. Since I already number the school days on the blog anyway:
Day 32: Lesson 3-2, Types of Angles
Day 47: Lesson 4-7, Reflection-Symmetric Figures
Day 52: Lesson 5-2, Types of Quadrilaterals
and so on. This way, I don't have to have to think about the pacing -- all I have to do is look at day number and I instantly know what lesson to teach that day. This pacing plan gets us to the last section of the text, Lesson 15-9, on Day 159 -- and I believe that the last lesson that matters for the Common Core is Lesson 15-3, on Day 153. This is right around the day that the PARCC/SBAC tests begin, so this is a good pacing plan.
I was considering following this plan during the current year. But I didn't, because it means that Lesson 1-1 wouldn't begin until Day 11, since there is no Chapter 0 (which is what would logically fit on the first ten days of school). Consider that the first lesson that I actually posted is Lesson 1-4, which wouldn't be taught until Day 14. This means that the first three weeks of school are empty, and I felt guilty about not really teaching anything for three weeks.
But that was until I subbed in several classrooms in September and saw how late many teachers began and ended the first chapter of their respective texts. One reason for this is obvious -- it may take a week or so before the class is even assigned textbooks! So now I feel less guilty about starting Chapter 1 so late. On the blog, I might actually begin with Chapter 0 -- but from Michael Serra's text, Discovering Geometry, since its Chapter 0 lends itself to opening day/week activities. Then on Day 11, I can begin Lesson 1-1 of the U of Chicago text and then cover the text in order.
Notice that some chapters are short -- Chapter 3 has only six sections. So after covering Lesson 3-6 on Day 36, I spend the next four days adding in extra material that might appear on Common Core tests -- for example, since there are constructions in Chapter 3, I might add in the angle bisector construction that appears in the Common Core but isn't emphasized in the U of Chicago. Then on Day 40 is the Chapter 3 Test, and then Lesson 4-1 begins on Day 41.
In an actual classroom, I probably wouldn't do my pacing so rigidly. I'd probably use the dates as a guideline -- schedule the Chapter 3 Test for Day 40, and then I may decide to spend an extra day on some of the lessons in Chapter 3 or constructions, depending on student need. On the blog though, I'll stick to the digit pattern. I like to schedule activities on Fridays due to the Unexpected Weekend Paradox, but this may be tricky when following the digit pattern.
The first semester will end with Chapter 7. Notice that finals week usually occurs in the Day 80's, right when Chapter 8 should be taught. Since Chapter 8 (on area) is important, I can start Chapter 8 in the second semester, in the Day 90's, and skip Chapter 9, since this chapter usually doesn't appear on any Common Core tests. Chapter 9 introduces 3D figures, but of course the emphasis is on their surface areas and volumes, which is in Chapter 10.
Of course, you may be asking, why wait until next year to start this? I could easily begin Chapter 8 next week, then jump up to Chapter 10 and follow the chapters in order. Well, I already committed to a chapter order for the year and I want to stick to it. Besides, I already covered much of Chapter 13 this year, and I don't wish to repeat it during the Day 130's.
OK, it's time for me to report on the three MTBoS blogs that I mentioned yesterday:
Elayne Miller (yes, this time she posted her full name) wrote about her favorite apps. This is what I wrote in reply:
It was interesting to read about all the apps that you use to maintain classroom management. I don't have a smartphone nor access to computers in the class, so I've never been able to take advantage of any of these.
The best I'm able to do is write a short TI-83/84 program which chooses the name of every student in the class (which I must add by hand), and then the calculator chooses a name at random. But the Plickers app that you mention in this post and your July link is obviously much easier!
Julia Finney-Frock is a North Carolina Algebra I/Geometry teacher who wrote about Instagram. I did express a fear about Instagram in my response:
And this was Finney-Frock's response:
Hi David! That was a fear of mine as well. I have my instagram public, so my students follow me, but I do not follow them. Students can message me (just like email) and the messages are saved.
And thank-you!!! If you do I’d love to hear all about it!
Stacey Strong teaches at a Massachusetts Public School. Here's what I wrote to her:
Sometimes I feel that we teachers are the biggest hypocrites in the world. I posted my Week 1 challenge at the last minute and you skipped yours completely. We doze off during faculty meetings -- then we turn around and yell at our students for doing the same thing!
Don't worry -- I'm not laughing. But you and I seem to be the only two MTBoS participants whose "My Favorites" post has nothing to do with computers/technology!
Of course, I bet traditionalists like Katharine Beals wouldn't like all of this reliance on technology in our classrooms. In fact, here's what Beals wrote recently:
And they overlook the reality that today's social pressures stem not just from the high-powered pressures of social media and social media addiction, but from the toxic social pressures that creep into schools that water down academics with socio-emotional processing activities and under-supervised group activities.
By the way, with all of this discussion about reading and responding to blog posts during MTBoS month, you may wonder why I don't just respond to Beals directly on her blog -- I have no excuse since she uses Blogspot, the same as my blog. Well, Beals is definitely not a member of MTBoS -- but then again, since she mentioned California in her most recent post, I'm very tempted to reply to that post.
Thus ends this post. Since Friday is the aforementioned student free day, my next post will be on Monday, January 25th, Day 89 and the start of the second semester. (There should be no sudden Saturday posts as I've already posted this week's traditionalists and MTBoS posts.) Also next week, Kung's lectures will focus on paradoxes involving physical objects, so stay tuned!