Even though I am not subbing today, I still get emails from Google Classroom from the classes I subbed for last week -- apparently it's easier to join classes than to drop them. I should figure out how to drop classes soon, lest my Classroom page be cluttered with dozens of classes I cover for just a day each.
But I like still having the ability to see that seventh grade math class from last week. Today the regular teacher added a new assignment, and this one has three letters that are very familiar to us -- IXL.
That's right -- just as we did at the old charter school, this teacher uses IXL. Their assignment is to do IXL Skill C.23 (but unfortunately I can't see what exactly that skill is) with a minimum score of 81.
Recall that IXL caused a big problem at my old school, especially with my sixth graders. (I didn't bother to use IXL with my seventh graders, since I didn't see them as often as Grades 6 and 8). There weren't even enough laptops for all the sixth graders, and so there were many arguments regarding who got to use a computer and who didn't -- and the ones without a computer didn't really have anything to do.
It's worth repeating how I should have handled sixth grade IXL. I should have had some sort of pattern where the laptops rotate among the students. Students with laptops are required to answer the questions on IXL, and students without them must answer an equivalent set of questions on paper. All students are given an IXL accountability form where they show the work. (In a way, this is almost like a hybrid schedule where some are working on computer and others aren't -- the difference is that everyone is in the classroom, and there is no pandemic.)
How many questions should they answer? Recall that it takes 28 questions to get to a score of 100 (assuming that all are answered correctly). If we use this teacher's minimum of 81, then it takes fourteen questions to reach 81. (That's right -- it takes just as many questions to get from 0 to 81 as it does to get from 81 to 100.) I might have set a minimum score of 70 instead (since I'm thinking of these as letter grades -- our school had a no-D policy, so the lowest passing grade was C at 70%). Then it takes eleven questions to reach 70. Thus this is how many questions I assign to the students who are working on paper.
Special ed students who need extra help would only have to reach half the score of the other students -- it only takes four questions to reach 35 on IXL. Only one eighth grader (the "special scholar") would have received the lower threshold, but several of my sixth graders would have needed it. And recall that one sixth grader also had trouble typing on the keyboard, so he would always be among the students who are working on paper.
Notice that so far, this is all about how I should have used IXL four years ago at my old school -- back before there was a pandemic. How I would have taught at the old charter school during the pandemic is an interesting question, but it's a counterfactual -- the school was shut down about nine months before the coronavirus outbreak.
Trying to imagine what a hybrid schedule at the old charter school would have looked if there had been a pandemic is tricky because it was a K-8 school. Thus it's likely that we would have implemented an elementary hybrid schedule, with two cohorts (AM and PM) meeting daily -- and thus middle school would also be forced onto this schedule. If an outbreak had occurred during the year I was there, we'd also have to deal with the district elementary school we were co-located with -- that school might control what sort of schedule we could have.
For the sake of argument, assume that at least the middle school would be able to have a secondary hybrid schedule, with each cohort meeting in-person twice a week. Since Wednesdays were our regular early-out days, I'd not be surprised if Wednesdays became the day for all distance learning (similar to Mondays in my subbing districts).
Then Cohort A attends Mondays and Thursdays, and Cohort B attends Tuesdays and Fridays (thus following our old bell schedule pattern). But the classes should meet in the same order everyday to avoid confusion -- the Tuesday-Friday pattern makes the most sense as it would allow me to start with the sixth grade class, which was sort of like my homeroom.
Notice that there are so few eighth graders (during the year I was there) that it might have been possible to place them all on the same cohort. This might reduce some of the need for concurrent hybrid -- if all eighth graders are on Cohort A, then the time that eighth grade Cohort B would have met could instead be an extra Zoom session for students in all grades who opt out of hybrid completely (which shouldn't be that many, since most K-8 students don't opt out). Then asynchronous activities can fill up the days left on the students' schedule to meet minimum time requirements
Middle school students would be released to go home at 12:45 for lunch. After lunch would be something like academic support (in my subbing districts), rather than explicit IXL time. But IXL can still be in my repertoire of online assignments to give during asynchronous time. In fact, in this scenario I would have actually used the Illinois State website more, including the science curriculum and online math homework. Thus ironically, I would have used these resources more (since during distance learning, it would have become obvious that I needed to use them), and so the administration wouldn't have need to yell at me over my failure to use them.
OK, that's enough about this counterfactual. There was no virus while the old charter school was open, and so there's no need to discuss how distance learning or hybrid would have worked there. Instead, let's think about hybrid in my new district -- specifically, how to give math tests during hybrid.
Yes, ever since that seventh grade teacher gave the online test, I've been wondering what math tests might look like when some students are in person and others are at home. And for the sake of this blog, I should figure it out before I post the Chapter 2 Test next week.
We recently looked at the blog of math teacher Jessica Strom. But while Strom's website is helpful in describing an ordinary lesson under concurrent hybrid, she doesn't say much about tests. So instead, I found another blog for us to look at:
http://www.hoffmath.com/2020/09/MathTestDistance.html
The author of this blog is Rebecca Hoff, a Georgia high school math teacher. (Actually, she introduces herself as "Rebecca" on her "About Me" page -- I'm just assuming that "Hoff" is her last name because her blog name is Hoff Math.)
Hoff tells us that she's on a concurrent hybrid plan, which she describes in a earlier post:
My school does a six day rotating schedule. During four of the six days, 5 classes meet per day. On the other two days only 4 classes meet per day and the extra hour is for chapel or assemblies.
They opted *not* to change this rotating schedule during the hybrid learning time.
During hybrid, only students with last names beginning with A-K came to campus on the first day of school; last names L-Z were home doing virtual learning. The next day they switched, and the cycle continues.
Since Hoff mentions "chapel," I deduce that she works at a private religious school. Notice that even though she doesn't fully give her schedule, we can figure much of it out ourselves. She tells us that five classes meet 4/6 days and four classes on 2/6, that's a total of 28 periods over six days. Since 28 is a multiple of seven, it's logical to assume that her school has seven periods, with each class meeting four of the six days. (Seven-period schedules are quite common at religious schools, since students have an extra class for religious instruction. So her schedule might look like this:
Day 1: 1, 2, 3, 4, 5
Day 2: 6, 7, 1, 2, 3
Day 3: 4, 5, 6, 7, Chapel
But notice that this is only a three-day cycle, not a six-day cycle. It could be that chapel is on a different day during the next trio of days, or it could be that it was a pure three-day cycle until hybrid forced it to become a six-day cycle:
Day 1: A-K 1, 2, 3, 4, 5
Day 2: L-Z 6, 7, 1, 2, 3
Day 3: A-K 4, 5, 6, 7, Chapel
Day 4: L-Z 1, 2, 3, 4, 5
Day 5: A-K 6, 7, 1, 2, 3
Day 6: L-Z 4, 5, 6, 7, Chapel
Each class meets four out of six days. Of these, two are in person and two are at home. Notice that the six-day cycle does not correspond to the days of the week. Also, the six-day cycle eliminates the need for an all-distance learning day, like Mondays in my districts.
(Then again, this schedule has the extra complication that classes meet at different times on the clock on different days. In my district, middle schools had a period rotation pre-virus, but not during distance learning or hybrid.)
OK, so now let's figure out how to give tests under the hybrid schedule. Some people might point out that if giving tests on concurrent days (when students are in two different locations, home and school) is tricky, then we could just give all tests on Mondays. Then all students have distance learning, and so we can simply give the test online, just as we did last week.
But Hoff can't do that, because her school doesn't have all-distance Mondays. Since everyday at her school is concurrent, she has no choice but to give tests on concurrent days. Let's see what she says.
First, she describes what to do about the temptation for students at home to cheat:
Then she lists these types of questions, such as open-ended questions, word problems, and so on. I find "sandwich style" questions to be interesting:
A "sandwich style" question is one in which you give the students both the question and the answer, and ask the students to "fill in the fixin's" (a.k.a. show the work to get from the question to the answer). You could have the answer next to the question or have an answer bank.
Recall that the original intent of my Rebecca Rapoport-style Warm-Ups/Exit Passes at my old chartr school was for them to be "sandwich style" as Hoff describes here. In this case there's no need for an "answer bank" -- the answer is just the date. But then I never properly enforced "fill in the fixin's," and so this idea fell apart.
Hoff describes four websites through which these tests can be delivered. One of them is Go Formative, which is what the regular teacher actually used that week. Here's what Hoff says about Formative:
What I like about Formative:
What I don’t like about Formative:
Now Hoff tells us how both her in-class and at-home students submit these tests:
So this is similar to my IXL accountability form idea from earlier -- all students, whether using a home computer or not, must submit their own paper.
Hoff mentions that she's worried about touching student papers during the pandemic, which is why she waits "a couple of days" to touch the students' papers. It turns out that if I naively follow the digit pattern on the blog, then most of my tests just so happen to land on Wednesdays. Teachers can then follow Hoff's suggestion -- we don't have to touch those tests from Wednesday until it's time to take them home over the weekend to be graded. This is much better than giving all tests on Mondays, which is counting on the students actually studying for them over the weekend.
But in my new district, Cohort A is always in-person on Wednesdays and Cohort B is always online, and so the same students would always be using one method or the other to test. It would be better if we could be more balanced -- students could take some of their tests at home and others in person. (Hoff doesn't have this problem because her schedule isn't tied to day of the week -- if she gave all of her tests on Wednesdays, it would automatically result in students taking some, but not all, tests at home.)
Before reading Hoff's post, I was considering rearranging my posts during testing weeks so that the only the cohort in class is taking the test (where it's harder to cheat), while the cohort at home gets some other activity that day. In other words, during the week of the Chapter 2 Test, Cohort A takes the test on Wednesday and has an activity on Friday, while Cohort B does vice versa. (Weeks that don't contain a test are the same for everyone.)
But now that I've read it, I'll just stick to the simple digit pattern. The assumption is that the students in class take it on paper while those at home take it on computer, just as Hoff describes. Again, the only problem is that it will result in most tests being on Wednesdays, and so Cohort A gets more in-person tests while Cohort B gets more at-home tests.
(The reason for this is simple -- the digit pattern is based on decimal or base 10, and ten is a multiple of five -- the number of days in the school week. So all tests land on the same day of the week unless there is a holiday to disrupt the day count. It just so happens that the days are numbered such that the test days during long stretches without holidays are Wednesdays.)
OK, so that's enough about next week's test -- let's get to this week's lesson. [2020 update: Last year's post on this lesson was full of political examples. Even though I've removed many political references from 2019, some of them still remain.]
Lesson 2-4 of the U of Chicago text is called "Converses." (It appears as Lesson 2-3 in the modern edition of the text.)
This is what I wrote last year about today's lesson.
There is a little bit of politics near the end of this post, because I'd perceived one of the examples of fallacious reasoning (assuming that a statement and its converse are equivalent) as one often committed by Republicans. So I added a similar fallacy made by Democrats in order for this post to remain politically balanced -- the point being made that both parties are prone to making logical fallacies. But only the example from the text actually appears on the worksheet:
Lesson 2-4 of the U of Chicago text deals with an important concept in mathematical logic -- converses. We know that every conditional statement has a converse, found by switching the hypothesis and conclusion of that conditional.
The conditional "if a pencil is in my right hand, then it is yellow" has a converse, namely -- "if a pencil is yellow, then it is in my right hand." The original conditional may be true -- suppose all the pencils in my right hand happen to be yellow -- but the converse is false, unless every single yellow pencil in the world happens to be in my right hand. A counterexample to the converse would be a yellow pencil that's on the teacher's desk, or in a student's backpack, or even in my left hand -- anywhere other than my right hand.
If converting statements into if-then form can be confusing for English learners, then writing their converses is even more so. Here's an example from the text:
- Every one of my [Mrs. Wilson's] children shall receive ten percent of my estate.
Converting this into if-then form, it becomes:
- If someone is Mrs. Wilson's child, then he or she shall receive ten percent of the estate.
Now if a student -- especially an English learner -- blindly switches the hypothesis and the conclusion, then the following sentence will result:
- If he or she shall receive ten percent of the estate, then someone is Mrs. Wilson's child.
But this is how the book actually writes the converse:
- If someone receives ten percent of the estate, then that person is Mrs. Wilson's child.
In particular, we must consider the grammatical use of nouns and pronouns. In English, we ordinarily give a noun first, and only then can we have a pronoun referring to that noun. Therefore the hypothesis usually contains a noun, and the conclusion usually contains a pronoun. (Notice that grammarians sometimes refer to the noun to which a pronoun refers as its antecedent -- and of course, the text refers to the hypothesis of a conditional as its antecedent. So the rule is, the antecedent must contain the antecedent.)
And so when we write the converse of a statement, the hypothesis must still contain the noun -- even though the new hypothesis may be the old conclusion that contained a pronoun. So the converse of another conditional from the book:
- If a man has blue eyes, then he weighs over 150 lb.
is:
- If a man weighs over 150 lb., then he has blue eyes.
Saying the converse so that it's grammatical may be natural to a native English speaker, but may be confusing to an English learner.
Now let's look at the questions to see which ones are viable exercises for my image upload. An interesting one is Question 9:
A, B, and C are collinear points.
p: AB + BC = AC
q: B is between A and C
and the students are directed to determine whether p=>q and q=>p are true or false. Now notice that the conditional q=>p is what this text calls the Betweenness Theorem (and what other books call the Segment Addition Postulate). But p=>q is the converse of the Betweenness Theorem -- and the whole point of this chapter is that just because a conditional is true, the converse need not be true. (Notice that many texts that call this the Segment Addition Postulate simply add another postulate stating that the converse is true.)
Now the U of Chicago text does present the converse of the Betweenness Theorem as a theorem, but the problem is that it appears in Lesson 1-9, which we skipped because we wanted to delay the Triangle Inequality until it can be proved.
Here, I'll discuss the proof of this converse. I think that it's important to show the proof on this blog because, as it turns out, many proofs of converses will following the same pattern. As it turns out, one way to prove the converse of a theorem is to combine the forward theorem with a uniqueness statement. After all, the truth of both a statement and its converse often imply uniqueness. Consider the following true statement:
- Donald Trump is currently the President of the United States.
We can write this as a true conditional:
- If a person is Donald Trump, then he is currently the President of the United States.
The converse of this conditional:
- If a person is currently the President of the United States, then he is Donald Trump.
This converse is clearly true as well. By claiming the truth of both the conditional and its converse, we're making a uniqueness statement -- namely that Trump is the only person who is currently the President of the United States.
So let's prove the converse of the Betweenness Theorem. The converse is written as:
- If A, B, and C are distinct points and AB + BC = AC, then B is on
AC.
(I explained why segment AC has a strikethrough back in Lesson 1-8.)
Proof:
Let's let AB = x and BC = y, so that AC = x + y. To do this, we begin with a line and mark off three points on it so that B is between A and C, with AB = x and BC = y. This is possible by the Point-Line-Plane Postulate (the Ruler Postulate). By the forward Betweenness Theorem, AC = x + y.
Now we want to show uniqueness -- that is, that B is the only point that is exactly x units from A and y units from C. We let D be another point that is x units from A, other than B. By the Ruler Postulate again, D can't lie on AC -- it can only be on the same line but the opposite side of A (so that A is between D and C), or else off the line entirely (so that ACD is a triangle).
In the former case, the forward Betweenness Theorem gives us AC + AD = DC. Then the Substitution and Property of Equality give us DC = (x + y) + x or 2x + y, which isn't equal to y (unless 2x = 0 or x = 0, making A and D the same point when they're supposed to be distinct).
In the latter case, with ACD a triangle, we use the Triangle Inequality to derive AD + DC >AC. Then the Substitution Property gives us x + CD > x + y, and then the Subtraction Property of Inequality gives us the statement CD > y, so DC still is not equal to y.
So B is the only point that makes BC equal to y -- and it lies on AC. QED
(The explanation in the book is similar, but this is more formal.) Later on, we're able to use this trick to prove converses of other theorems. So the converse of the Pythagorean Theorem will be proved using the forward Pythagorean Theorem plus a uniqueness statement -- and that statement is SSS, which tells us that there is at most one unique triangle with side lengths a, b, and c up to isometry. And, more importantly, the converse to the what the text calls the Corresponding Angles Postulate (the forward postulate is a Parallel Test, while the converse is a Parallel Consequence) can be proved using forward postulate plus a uniqueness statement. The uniqueness statement turns out to be the uniqueness of a line through a point parallel to a line -- in other words, Playfair's Parallel Postulate. This explains why the forward statement can be proved without Playfair, yet the converse requires it.
In the end, I decided to throw out Question 9 -- I don't want to confuse students by having a statement whose converse requires the Triangle Inequality that we skipped (and the students would likely just assume that the converse is true without proving it) -- and used Question 10 instead.
But now we get to Question 13 -- and here's where the controversy begins. I like to include examples in this lesson that aren't mathematical -- doing so might engage students who are turned off by doing nothing but lifeless math the entire period. This chapter on mathematical logic naturally lends itself to using example outside of mathematics. But the problem is that Question 13 is highly political. Written in conditional form, Question 13 is:
- If a country has communist, then it has socialized medicine.
and its converse is:
- If a country has socialized medicine, then it is communist.
In the end, I will include Question 13 on the worksheet. But I left a space so that if a teacher feels that the question is politically slanted, then he or she could add another question to balance it. For example, that teacher can add a fallacious argument often made by Democrats, for example:
- If a white person is racist, then he or she opposes Obama.
- If a white person opposes Obama, then he or she is racist.
Of course, this question adds a new layer of controversy (race) to the mix. Teachers who want to add a balancing question should just write in their own question, or just throw out the question about socialized medicine altogether.
In the review section, I'd have loved to include Question 15, a review of the last lesson on programming (and of course changed it to TI-BASIC). But in deference to those classes that skipped the lesson because not every student has a graphing calculator, I've thrown it out and included only questions from the fully covered Lessons 2-2, 1-8, and 1-6.
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