**Table of Contents**

**1. Pappas Page of the Day**

**2. The Best Laid Plans**

**3. Revisiting My "Rules"**

**4. Why? Because I said so!**

**5. Teacher Tone**

**6. Teacher Look**

**7. Traditionalists**

**8. Valedictorians**

**9. Participation Points**

**10. Plans for a Future Post**

**Pappas Page of the Day**

This is what Theoni Pappas writes on page 160 of her

*Magic of Mathematics*:

"Mathematics certainly offers an abundance of problems. In fact, mathematics and problems are inseparable."

Here Pappas is about to discuss problems that

*haven't*been solved yet. This is the start of the new section "Unsolved Mathematical Mysteries."

On this page, she mentions three specific problems:

-- Is there a formula or a test to determine whether or not a given number is prime?

-- Is there an infinite number of prime pairs? ("Prime pairs" are commonly called "twin primes.")

-- Is there an odd perfect number?

Pappas defines a perfect number as one which is equal to the sum of its proper divisors. There are 49 known perfect numbers, and all of them are even: 6, 28, 496, 8128, and so on. No one knows whether an odd perfect number can exist.

**The Best Laid Plans**

**Yesterday I visited my old school, since I needed the director (principal) to fill out a form confirming employment for my new school. It was after school, but there are always a few students who stay in the after school program. The first student who saw me was -- you guessed it! -- that same seventh grader who's always spotting my car from the street.**

He and a sixth grader wanted me to sing to them, so of course I obliged them. The sixth grader wanted me to sing the Ratio Song, and the most popular song among seventh graders was always Fraction Fever.

While waiting for the director to fill out my paperwork, I spoke to the other administrators who were in the office. They told me about last Friday's eighth grade graduation -- all of the students were dressed up for the occasion. My predecessor teacher (who is now a tutor) had come for a visit -- he was present for the eighth grade ceremony. I was also told about the valedictorian -- he had given a wonderful speech that day.

I noticed a sign near the director's desk that listed the SBAC testing schedule. As it turned out, the tests were given later than expected. In fact, the day I submitted my May "Day in the Life" post (and called it the "last day of testing") was actually the

*first day*of testing -- yes, the science tests I'd fretted about for so long were administered that day. The final day of testing was actually the day I found out I was hired -- the day before the eighth grade graduation!

I've always stated in Common Core debate blog posts that this is how it ought to be -- why should there be wasted time after the SBAC with nothing to do? The state tests should be given just before the last day of school -- and indeed they were at my old school.

But this does contradict something I wrote last summer. I said that the last thing I would do with my eighth graders, between the SBAC and graduation, is give them a "final quiz," a comprehensive exam to prepare them for the real finals that they would take in high school. We now see that even if I hadn't left, I wouldn't have given them a final quiz because there was no extra time between the end of the SBAC and graduation.

It just goes to show me that I shouldn't have spent so many blog posts last summer about academic plans for the upcoming year, since they were based on schedules that could

*change*. Instead, I should have focused (and blogged) more on classroom management, as I'd need to have good management skills no matter what.

The director marked down that I had taught for 117 days at my old school. As I left the school, the seventh grader told me a little about my successor teacher -- recall that she had been teaching kindergarten and first grade at the time of my departure. My former student tells me that she was a little mean, but nonetheless she taught well.

This, of course, is to be expected -- she obviously has better management skills than I, which is why she was chosen to replace me. The students who had neutered all of my rules find out that they can't get away with neutering the rules with my successor. This is something I must keep in mind as I move forward.

Meanwhile, this was the last week before summer break at my old school, It is a week of service -- instead of lessons, the students spend time beautifying the classroom and campus. I actually did know (and write) about the week of service on my blog last year.

**Revisiting My "Rules"**

In this post, I'll continue to write about why I was unsuccessful my first year in the classroom. (So again, experienced teachers who already know all of this may want to skip this entire post.) In my last post, I listed the "rules" that I came up with last summer:

1. The Teacher Respects You

2. Respect Your Honesty

3. Respect Yourself and Others

4. Respect Your Class Equipment

In fact, my original plans were to have two of the rules be the following:

1. Strive to earn an "A" in every class.

2. Strive to earn an "E" in every class.

In my July 22nd post, I wrote that the form of this the rule as "Respect Yourself and Others" came originally from the MTBoS blogger Fawn Nguyen. But still, the idea behind earning A's and E's (excellent conduct grades) underlay this rule, as I mentioned in that post:

*Still, I know that some members of our generation can't go seven waking minutes without using a cell phone, and yet teachers expect them to go seven hours without using one -- that is, 60 times as long as they naturally would. Indeed, I know that for some students, the only effective incentive that will motivate them to work is a few minutes' free time on their cell phone. Again, the idea is for me, after telling the students my library story, to remind them that the people who matter (employers) will criticize -- that is, make fun of -- those who can't go a few minutes without reaching for a phone, and those who do have self-control will get more A's and E's, and ultimately jobs and promotions.*

Notice that this would be in response to a student question, "Why can't we use cell phones?" And so this would be the answer -- the students who avoid cell phones are more likely to earn A's and E's.

But the problem was that

*every time*the students asked

*why*they have to do something or are not allowed to do something, I mentioned A's and E's as the answer:

-- Why can't I talk?

-- Why must I do these problems in the book?

-- Why do I have to sit in my assigned seat?

-- Why can't I sit in the teacher's chair?

First of all, mentioning E's was a bad idea, because our report cards didn't actually have conduct grades (unlike the LAUSD report cards). So mentioning E's was meaningless to the students.

Of course, our reports do have A's and other academic grades. But as it turned out, this didn't help convince the students to change their behavior. One girl told me that she was tired of my mention of A's and grades over and over again.

To me, A's for a student are like W's (wins) for an athlete. Just as every single athlete wants to earn as many W's as possible, every single student should

*want*as many A's as possible. Just as every single athlete wants to be a champion, every single student should

*want*to be a valedictorian. Just as a coach can't stress the importance of getting W's too much, I as a teacher can't stress the importance of getting A's too much. And in case you think that there are some players or teams who aren't trying to win ("tankers"), notice that even tankers are trying to win championships -- in this case, they're trying to earn

*future*rings by drafting top players, rather than present rings.

But a sports analogy didn't work in my classroom -- especially in my eighth grade class, where the majority of students were

*girls*who didn't care about sports either. When I was a teenager, I'd much rather be bored in class with an A than entertained in class with an F, or even a C -- indeed, it wasn't even close. I highly valued the opinion of the people who mattered -- and those people wanted me to earn as many A's as possible. Praise from them was worth gold to me, and I wanted to get that gold by getting lots and lots of A's. On the other hand, the people who would make fun of me for being smart didn't matter to me at all. Their opinion was worth mud to me.

On the other hand, I see that to these eighth grade girls -- and several other students of all grades and genders -- they'd rather be entertained with a low grade than bored with a high grade. And so answering their "Why?" questions with "So you can get an A!" didn't change their behavior.

I remember on my final day in the classroom, the day began with eighth grade SBAC Prep. The administration had been getting on my case for not allowing the students to take the SBAC Practice Test online. And so I distributed the laptops and directed the students to take the test.

The test frustrated many students. The first question was on the laws of exponents -- which I'd taught back in October, but by now many students have forgotten them. (Think back to those few SBAC Prep worksheets that I posted last week.) The second question was on slope -- which I was scheduled to reach later in March or April. And so naturally, many students would ask, "Why? Why do we have to take this test?" And when I wouldn't help them on the test, another "Why?" question came up -- "Why can't you help us on the test?"

I replied, "So you can do better on the SBAC in May." But this didn't placate the students. One girl said, "Just help me. No one will ever know," and another added, "I promise I'll learn all of this by May, but help me now." In other words, I told them that the purpose of the test was to prepare them for May, and their response was that it wasn't May yet.

**Why? Because I said so!**

When I was a young student growing up, I was generally a compliant, obedient student -- especially as I completed eighth grade and headed for high school. Yet I always wanted to know the reasons that teachers were telling me to do something. I hated the phrase, "Because I said so!"

And so when I began teaching, I would do whatever it took to avoid saying "Because I said so!" So students would ask "Why?" and I wanted to give them a good reason -- a stupendous reason that will make them see "Oh, that's why!" and get them back to work. In other words, I wanted to convince them to follow the rules using logic. But unfortunately, this almost never happened. Students would continue to ask "Why" and insist that the reasons I was giving them were invalid.

The problem is that a teacher can

*never*convince a student to obey using logic. After all, I've tried telling students to stop talking, and they claim "I wasn't talking!" though they

*knew*they were. If a student knows that he's talking, then

*no*amount of logic can

*ever*convince him that he was -- since he already knew he was! And yet I always tried in vain to "prove" that he was talking -- which would degenerate into "Yes you were!" and "No I wasn't!"

Likewise, no amount of logic can convince students to take the SBAC Prep seriously. After all, if they were interested in doing well on the test, they'd

*already*be working hard on the practice -- the fact that two girls weren't meant that they didn't care about the test, no matter what I said. (By the way, I knew better than to answer "We're taking the practice because the administrators are on my case to make sure that you take it!" -- after all, why would they care about that?)

The ideal classroom manager already knows how to handle the "Why?" questions. The correct answer to a "Why?" question is that phrase I wanted to avoid so much -- "Because I said so." I've learned that students who ask "Why?" are really looking for

*arguments*, not reasons. By trying to give them reasons I'm legitimizing their argumentation, but by answering "Because I said so," the ideal manager is telling them that it's not okay to argue. And once the ideal manager gains a reputation for answering "Because I said so," the students might even stop asking "Why?" That's the only true way of getting rid of the "Why?" questions, as opposed to trying to answer them.

Yes, students may hate it when I reply "Because I said so!" -- but it's possible that they might hate my alternatives

*even more*! I've mentioned that those eighth grade girls didn't like it when I answered "So you can get an A" and lecture them on the importance of earning high grades. They much would have preferred "Because I said so," if only since this phrase contains only four words.

I remember one day back when I was student teaching. My master teacher told me that a certain special ed student needed to be seated closer to me -- and she wanted me to be the one to tell the girl that she'd have a new seat. Of course, the girl wanted to know "Why?" she had to move -- and so she refused to budge. I started telling the other students affected by the move to pick up their belongings so that I could sit a special ed student near the front of the room. At this, the girl agreed to move -- but she was very offended that I'd told everyone she was a special ed student. (Naturally, I don't post any more details here on the blog -- if she didn't want me telling the whole class about her, of course she wouldn't want me telling the whole world about her. If I write about special ed students on the blog, I keep silent about the details, and of course the students remain anonymous.)

In this case, we see how the girl would have

*strongly*preferred hearing "Because I said so" to hearing the actual reasons for the move. If I'd simply said "Because I said so," she might have changed the seat without being offended. And if I'd had a reputation for answering "Because I said so" -- say that I'd used that phrase to answer the previous ten "Why?" questions asked of me -- most likely she would have moved within ten seconds of my first telling her to move without even asking "Why?" In fact, this is most likely what she would have done if my master teacher had been the one asking her to change seats.

Another situation when "Because I said so" might be useful is when a student implies that a rule is difficult or impossible to follow. This most often occurs with the "no restroom" rule. I wrote back in my July 29th post about the tomb guards who can go up to 24 hours without going to the restroom, and my students complain about having to go 80 minutes without using the restroom. Only a few times did I tell my students about the tomb guards. Instead, I give the story about my own years as a young student, which I wrote on the blog last year:

*I'm proud to say that I completed three full years of school -- Grades 10-12 -- without asking for a restroom pass even once. How did I do it? Each day, I would use the restroom:*

-- Before school

-- During nutrition

-- During lunch

-- After changing from street clothes to P.E. clothes (as I was on the track team)

-- After changing from P.E. clothes back to street clothes

That's five times per day that I went to the restroom. Now 5 * 180 = 900, so that's nine hundred times I used the restroom per year, and that was for three years (Grades 10-12) . So that's a grand total of 2,700 times that I used the restroom -- without missing even one second of class time! Again, I might have been bored and didn't enjoy certain classes (like history), yet I didn't ask for restroom passes. I would much, much rather preserve a reputation as one of the brightest and best-behaved students in the class and earn as many A's, B's, and E's as I could.

-- Before school

-- During nutrition

-- During lunch

-- After changing from street clothes to P.E. clothes (as I was on the track team)

-- After changing from P.E. clothes back to street clothes

That's five times per day that I went to the restroom. Now 5 * 180 = 900, so that's nine hundred times I used the restroom per year, and that was for three years (Grades 10-12) . So that's a grand total of 2,700 times that I used the restroom -- without missing even one second of class time! Again, I might have been bored and didn't enjoy certain classes (like history), yet I didn't ask for restroom passes. I would much, much rather preserve a reputation as one of the brightest and best-behaved students in the class and earn as many A's, B's, and E's as I could.

During snack break, I'd drink no liquids at all. It's possible that I wouldn't have taken any liquids at lunch either, save for the fact that I ran cross country and for runners, it's good to hydrate. But if I drank at lunch, I'd make it all the way to practice time before having to use the restroom.

I'd tell my students this story whenever they asked for a restroom pass. I also told them that on the weekend, they'd probably gone three or more hours between restroom visits. They only go during class because they'd rather miss

*class time*than

*friend time*. I wasn't like that -- to me, I'd rather spend all of my break time in the restroom than miss even one second of class. To me, class time was more precious than not friend time. Even if I hated the class or found it boring, I didn't want to miss a second, because I wanted to maintain a reputation of being smart, compliant, and obedient.

Of course, the point is that this long story hardly convinced anyone to change restroom habits. And again, I point out that I as a young student hadn't avoided the restroom because some teacher told me a long story about tomb soldiers or anything like that. I avoided it because nearly every teacher gave some penalty to those who went during class, and I didn't want that penalty. In other words, it was because the teachers said so.

By the way, my original intention wasn't to have a "no restroom" rule. In July, I wrote about how I would wanted to give restroom passes to students who earned A's on quizzes. I warned myself that this might not work -- and it didn't. The restroom pass system was neutered simply because many students

*lost*their passes (and often it was either a student I knew had earned an A, or it was right after a Dren Quiz when nearly everyone earned an A).

**Teacher Tone**

**Notice that my new school is Grades 6-12 -- that is, students attend it for seven years, and they start at age 11. In the world of J.K. Rowling's**

*Harry Potter*, Hogwarts is also a school that students attend for seven years starting at age 11. Of course, my new school isn't Hogwarts and I'll be teaching the students math, not magic. Still, I often like to compare myself as a teacher to the professors of magic who appear in the series.

One teacher from the series is very laid back -- Professor Flitwick, Charms teacher. Another teacher is more strict -- Professor Snape, Potions teacher. A third teacher is firm yet fair -- Professor McGonagall, Transfiguration teacher. It's obvious which chararacter Rowling considers to be the ideal teacher -- McGonagall, whom the author admits is based on her favorite English teacher.

When I left my old school, the instructional aide suggested that I read Lee Canter. As it turns out, Canter also divides teachers into three categories -- non-assertive, assertive, and hostile. These three categories correspond directly to Flitwick, Snape, and McGonagall. Sometimes I also think of the first three teachers I had in the eighth grade (science, P.E., and history) as corresponding to the three types of teacher.

The ideal teacher is the assertive teacher -- the McGonagall category. I, unfortunately, leaned too much towards the other two categories. According to Canter, teachers who are not assertive may find themselves acting like Flitwick one day and Snape the next.

The three teacher categories also correspond to tones of voice. My normal voice is weak and fits in the Flitwick category. When I speak in a normal voice, students often respond by mocking it. Even the eighth grade guy who became the valedictorian mocked my voice once or twice -- and the girls mocked it all the time. When I raise my voice, it becomes a yell -- the Snape category. When I yelled, students often asked, "Why are you yelling?"

The McGonagall category corresponds to what I call a

*teacher tone*. All ideal classroom managers use the teacher tone to show the students that they mean business. But unfortunately, I simply do not have a teacher tone. I remember back when I was student teaching, sometimes one student wouldn't pay attention to me. Another student suggested that I "yell" at the first student to get his attention -- but here "yell" really means "teacher tone." A true yell (Snape) would only anger the student rather than gain compliance.

Sometimes I hoped that what I lacked in teacher tone, I could make up in

*musical tone*. During the music break I would sing songs to grab their attention. But I couldn't sing all the time -- and besides, when I yelled (Snape) too much, it would hurt my voice and I couldn't sing as well afterward. In the end, I really need a teacher tone.

In my last post, I wrote about the discipline hierarchy that the instructional aide suggested for me:

1st Infraction: Warning

2nd Infraction: Write a 150 word essay telling what did and how you will improve your behavior.

3rd Infraction: Call parents

4th Infraction: Send to Principal

I've said before that the ideal classroom manager only needs the first step in the hierarchy to get the students to comply. This is because the ideal classroom manager uses a

*teacher tone*to issue the warning for the first infraction. In my class, my warnings were ineffective because I lacked the teacher tone to show the students that I meant business. (Actually, the instructional aide told me that it's okay to reach the second level -- it's the third and fourth levels, where other adults are involved, that the ideal manager seeks to avoid.)

So my warnings didn't work, and combined with the second level (already neutered when students didn't put much effort into their 150 words), I found myself at the third and fourth levels too often.

**Teacher Look**

No more pencils!

No more books!

No more teachers' dirty looks!

Tonight marks the start of summer vacation in the LAUSD and at my old school, and so this rhyme is appropriate today. It refers to another tool at the ideal classroom manager's disposal --

*teacher look*.

I already know that I don't have a strong teacher tone. But I don't even know whether I have a strong teacher look or not, because I've never tried to give one! As far as I know, I could have an excellent teacher look that will be very successful in getting the students to comply.

I've stated above that with a strong teacher tone, it's not necessary to go beyond the warning step of the discipline hierarchy. In fact, with a strong teacher

*look*, even a warning it's necessary. Students know from the way the teacher is looking at them that the time for playing around is over.

And so when I tell a student to stop talking and the student replies, "I wasn't talking," this is a great time for me to try out the teacher look. I just stare coldly at the student and make sure that I'm not smiling or laughing. This might be enough to get the student to stop talking -- certainly more than trying to "prove" that the student was talking.

I feel bad for wanting to use the teacher look in class. After all, students don't like it -- which is why they would sing songs about "no more teachers' dirty looks," so I'd love to be the teacher who can be effective without teacher looks. But then again, students don't necessarily like pencils or books either, and I wouldn't consider trying to teach without those! The song is intended to be a celebration of summer,

*not*a suggestion to eliminate pencils, books, or teacher looks in school.

And so now I see that if I want to be a better teacher for my new job in the fall, I need to develop my teacher tone -- or if that's impossible, my teacher look. The only real ways to win arguments with students are teacher tone and teacher look,

*not*logic.

**Traditionalists**

**Let's check out the latest from our main traditionalist, Barry Garelick:**

https://traditionalmath.wordpress.com/2017/06/08/thinking-like-a-mathematician-or-something-that-looks-like-one-anyway/

*This article makes the point that the emphasis on getting students to “understand” by using alternatives to standard algorithms is a subterfuge. The purpose, the article contends, is to make students look smarter than they are. They reason as follows:*

At this point Garelick quotes the article to which he links above, so I'll do the same:

**The problem is this: “number bonds” is a counterfeit of the way kids who are genuinely good at math act by the time they get into elementary. While the other kids are counting on their fingers, kids who’ve been playing with numbers in their heads since they were two or three have figured out all the relationships and will take numbers apart to make it easier to solve. Not something stupid like seven plus seven, of course. More something like 115 + 115.**

**Having figured out that number-gifted children will do this as 100+100=200, 15+15=30, so 115+115= 230. This is quite nifty for a first-grader, but the left thinks it can skip all the work getting there. If they just teach perfectly normal, average children to think in terms of taking numbers apart, voila! Everyone will be a math genius!**Notice that this complaint is about first grade math -- and I've said before that I'm sympathetic to traditionalism in the early grades. Instead, Garelick and Hoyt are criticizing the non-traditional method of "number bonds."

Hoyt mentions 7 + 7 as an example of a question which is to be solved using "number bonds." And without even clicking on the article, we can guess what's go on here. The "number bond" method probably asks students to reduce 7 + 7 to an easier problem, such as 7 + 3, whose sum is 10. And so instead of 7 + 7, students must solve 7 + 3 + 4 = 10 + 4 = 14. I click on the article and, sure enough, that's exactly what "number bonds" means.

We already know how traditionalists like Garelick and Hoyt want students to solve 7 + 7 -- they want them to give the answer 14 in one second or less. The sum 7 + 7 shouldn't reduced to an easier sum -- instead 7 + 7 itself should be a basic sum to which

*other*sums are reduced (and of course, this is via the standard algorithm, as in 77 + 77).

The addition table from 0 + 0 to 9 + 9 contains 100 different sums. To traditionalists, a student should be able to answer any of those 100 sums in one second or less. Remember that I agree with the traditionalists regarding early arithmetic, and so I do like the idea that students should be able to add any two one-digit numbers within one second.

But no one is going to memorize 100 different things at once. Some of the entries in the table are easier to memorize than others, so at some point a young student will know some, but not all, of the entries in the table. To tackle 100 table entries, it's good to look for

*patterns*in the addition table.

An advocate of number bonding would argue that the tens -- 9 + 1, 8 + 2, 7 + 3, and so on -- are easier to add than the others. I have no problem with highlighting the sums that are easier

*en route*to memorizing the harder ones. So a student who doesn't know 7 + 7

*yet*can add four more to 7 + 3, and then after doing this enough times, the student will memorize 7 + 7 as well.

Number bonders don't only emphasize tens. They also like to stress the

*doubles*-- 1 + 1, 2 + 2, 3 + 3, 4 + 4, and so on. On Hoyt's page, the commenter "cirby" mentions doubles:

Instead of "4 + 5 = 9" you get "4 - 1 = 3, and 5 - 2 = 3, and 1 + 2 = 3, so you add 3 + 3 = 6, and 6 + 3 = 9. See how smart I am? Screw the kids, they're all stupid, and I'm going to make sure they stay that way."

Here cirby reduces 4 + 5 to the double 3 + 3. Actually, cirby is exaggerating a little -- in reality, a number bonder would reduce 4 + 5 to 4 + 4 (that is, by writing 4 + 5 = 4 + 4 + 1), not 3 + 3 as cirby does here. The point is that a child who knows 4 + 4 but hasn't memorized 4 + 5

*yet*(that's the key word --

*yet*) can reduce what's unknown to what's known,

*en route*to memorizing the larger sum. Oh, and as a bonus, learning doubles is the key to learning the two's times tables.

But unfortunately, the last part of cirby's comment is

*not*an exaggeration and represents the traditionalists' biggest fears. "I'm going to make sure that they stay that way" -- in this sentence "I" refers to a hypothetical progressive pedagogue. The fear is that the students "will stay that way" -- that is, they'll

*always*add 4 + 5 as 4 + 4 + 1 and

*never*learns 4 + 5 as its own fact.

Correct use of number bonding:

-- No student memorizes all 100 one-digit sums at once.

-- Student so far has memorized 4 + 4 = 8, but not 4 + 5

*yet*.

-- Student uses number bonds to write 4 + 5 as 4 + 4 + 1 = 8 + 1 = 9.

-- A day or two later, student uses this fact to

*memorize*4 + 5 = 9.

-- Student

*never*needs number bonds to add 4 + 5 again.

Incorrect use of number bonding:

-- No student memorizes all 100 one-digit sums at once.

-- Student so far has memorized 4 + 4 = 8, but not 4 + 5

*yet*.

-- Student uses number bonds to write 4 + 5 as 4 + 4 + 1 = 8 + 1 = 9.

-- Student is

*never*made to memorize 4 + 5 = 9

-- Student

*always*needs number bonds to add 4 + 5 from that point on.

The traditionalists argue that too much number bonding is of the incorrect use above. Regarding this, I agree with the traditionalists.

Number bonding also appears in the multiplication table. No student memorizes all 100 products of one-digit numbers at once. Number bonding allows students to use the products they've memorized so far to find the products they haven't.

Tens don't appear as often in multiplication as in addition. In theory, students can use 2 * 5 = 10 to find the products 4 * 5, 6 * 5, and 8 * 5. But doubles -- or their multiplicative equivalent,

*squares*-- can be used, and indeed they are.

I remember once reading an old math text, "Quick Arithmetic" (which I mentioned in an old blog post from October 2014). This text lists the "sexy six" -- the six most difficult products to learn -- these were 6 * 7, 6 * 8, 6 * 9, 7 * 8, 7 * 9, and 8 * 9. Notice that these include all of the products of two numbers from six to nine --

*except*the squares 6 * 6, 7 * 7, 8 * 8, and 9 * 9. The implication is that it's

*easier*to find squares like 7 * 7 and 8 * 8 than 7 * 8,

I began this post with Pappas and primes. Her first unsolved mystery was a primality test -- a method of determining whether a number is prime. It's often said that the smallest composite number that "appears" to be prime is 91. This is because it's clearly not a multiple of two or five. Neither is 91 a multiple of three, due to the "omega" rule -- 9 + 1 = 10, which isn't a multiple of three. The omega rule explains why 91 is the smallest "apparent" prime rather than 87 -- since 8 + 7 is 15, a multiple of three, we conclude that 87 is itself a multiple of three.

Indeed, we see that 91 = 7 * 13, and both 7 and 13 are "opaque" primes -- that is, there is no simple divisibility rule for 7 or 13. But 91 isn't the smallest product of two opaque primes -- that honor goes to 49, which is 7 * 7. So why is 91 the smallest "apparent" prime rather than 49? It's because

*squares are easy*-- 49 is obviously 7 * 7, but 91's factors aren't as obvious.

The whole point of this is that squares are easy -- and so we can number bonding can be used to convert non-squares into squares. A number bonder can multiply 7 * 8 by notice that it is seven more than 7 * 7 = 49, so the product is 49 + 7 = 56. Of course, this is good only if the student proceeds to memorize 7 * 8 itself within the next few days.

Before I leave this post, let me quote the co-author SteveH:

*K-6 pedagogues claim that they love the balance of understanding and skills, but CCSS does not effectively isolate and test the skills portion. K-6 math classes are set up to trust the process and assume that, like thematic learning, facts and skills will be learned automatically, but they only check skills in the context of fuzzy problems or at very low NON-STEM levels. I don’t have any hope that we can find some way to make them see the understanding embedded in mastery of basic skills. I learned all about splitting and combining numbers in different ways in my traditional math classes.*

**Valedictorians**

**In this post, I congratulated my old school's eighth grade valedictorian. Later on, I mentioned that just as every athlete's goal is to be a champion, every student's goal is to be a valedictorian.**

But not everyone agrees with this. From time to time, people write books about how it's

*not*good to be a valedictorian.

I used to link to the Joanne Jacobs site in order to quote the traditionalist Bill. Actually, Bill doesn't comment in the following thread, but I do link to this site to mention an anti-valedictorian book:

http://www.joannejacobs.com/2017/06/valedictorians-but-not-visionaries/

As usual, Jacobs herself provides another link:

https://www.the74million.org/article/top-of-their-class-not-top-of-the-world-valedictorians-less-likely-than-disruptors-to-become-millionaires

According to the author, Eric Barker, a student is more likely to be a millionaire with a 2.9 GPA than a GPA of 3.6. This isn't the first anti-valedictorian book -- the most well-known is probably Robert Kiyosaki's

*Why "A" Students Work for "C" Students*.

I believe that it's good to earn as many A's as possible, just as it's good for an athlete to earn as many W's as possible. But if I were to write a book about how "A" students rule the world, I'd expect the book to sell zero copies.

People don't like to read about how "A" students are successful -- they'd rather read about the "C" students who dropped out of school and started their own companies. I believe that these are exceptions to the general rule that "A" students are the most successful.

One commenter, Walter Underwood, writes:

*That article points out that the key skill for getting straight As is to excel in things that do not interest you. That same skill may be useful in practicing law or medicine, but probably not in extending the frontiers of anything.*

I think back to my own high school days. Of course, I earned straight A's in math all the way through high school, but not every student is interested in math. This is why when I see such a student, I try to compare that student's struggles in math to my own struggles

*in other subjects*. For example, I didn't care for history, just as my students don't care for math.

I remember when I was a junior, I was enrolled in AP US History. But before we took the AP exam, we were to take the "Golden State Exams." The "Golden State Exams" were not state tests like the current PARCC or SBAC. In fact, to this day I don't know the purpose of the Golden States. The Golden States disappeared a few years after the old state tests, the CST's, started appearing (which was also my junior year).

The AP US History exam covers all of American history, but the Golden State History exam covered only the 11th grade curriculum -- basically the 20th century. (All earlier US History was considered to be an eighth grade subject.) In our class we had to rush through more recent history (to the extent that the following year, the teacher started giving summer homework). So I knew that I'd struggle on the essay section of the Golden States, which was on the Vietnam War.

When I received my Golden State score, I found out that I was in the Honors level -- reserved for the top 14% of juniors in the state. (In another subject, Economics, I was in the top 7%, High Honors.) I realized that somehow, in a subject I didn't care for and didn't think I was good at, I had scored in the top 14% range! At the time, I didn't know this, but hearing now about juniors who tank the SBAC, I wouldn't be surprised if 86% of my cohort just wrote a random essay or left it blank on a test that didn't truly count for anything. So simply taking the test seriously is enough for a top 14% score.

I also remember another Golden State Exam in my top subject, math. I was a seventh grade student taking eighth grade Algebra I -- the only such student at my school. As it turns out, the seventh grade Renaissance Fair for history was at the same time as the eighth grade Golden State Exam -- again, I was the only student affected by that scheduling conflict. And I chose to take the math test -- even though this was one time when the history class was actually fun!

At the end of my freshman year, I read about CSF, the California Scholarship Federation. To join this academic club, students must earn points based on grades -- three points for an A, one point for a B (unless it's an honors B for two points), and none for a C. On my report card, I had two A's and three B's, with no C's or lower. This worked out to be nine points (since none of the classes were honors) -- but 10 points are needed to join! So I couldn't join CSF my sophomore year. My grades rose enough for me to join CSF as a junior, but by then I'd lost interest.

And so it surprised me when one day I subbed at a high school on report card day, and when I passed out the report cards, not one student had only A's and B's. Every single student had at least one grade of C or lower. Here I was, thinking myself a failure for not having enough points for CSF, and yet the entire class that day had lower grades. (And of course becoming valedictorian of my high school was out of the question -- our school used unweighted grades, and so straight A's were necessary for valedictorian status.)

The whole point of this story is to show that most of my students aren't as motivated to earn top grades as I was. And so any classroom management style based on motivating students to earn top grades is doomed to failure.

**Participation Points**

**Back in July, I explained how I would use participation points as part of my management. But by January, the participation point system failed. This is what I wrote in my January 3rd post:**

*Moreover, in my current class, I started an individual participation points system where students can gain or lose points -- where most points were gained for "participating" (i.e., giving right answers) and points were lost for "not participating" (i.e. off-task behavior). I would begin giving out consequences when a student has lost all his or her points.*

And there is the problem -- those smart yet loud students would rack up points for answering my questions, and when I took them away for talking, they'd never lose enough points for me to start giving consequences. So again, the students would talk and talk without any punishment (until I started yelling, of course) -- all because of the doublethink of conflating academics with behavior.

Eventually, I did change my participation points system. Students can now gain points whenever they participate, but now consequences are given separately from the point system. The problem is that it's far too late to introduce a change -- the students have already seen the loudest kids get away with being loud for too long, and so they're never quiet.

And there is the problem -- those smart yet loud students would rack up points for answering my questions, and when I took them away for talking, they'd never lose enough points for me to start giving consequences. So again, the students would talk and talk without any punishment (until I started yelling, of course) -- all because of the doublethink of conflating academics with behavior.

Eventually, I did change my participation points system. Students can now gain points whenever they participate, but now consequences are given separately from the point system. The problem is that it's far too late to introduce a change -- the students have already seen the loudest kids get away with being loud for too long, and so they're never quiet.

I still remember the first time my participation point system failed. It was on the third day of school -- and I mentioned it in my monthly "Day in the Life" post for August:

**8:25**-- My first class, a seventh grade class, arrives. Today there is a confrontation with one of the seventh graders. She refuses to do her work, then argues with my student support aide, who asks her to leave the room. I am the teacher, so I should have tried to intervene sooner, though it still might not have made much difference. It is only Day 3, but I already know there's one girl I'll need to watch out for this year.

Well, as I reflect back on this third day of school, the reason I didn't intervene was my flawed participation points system. The girl had gained a few cheap points for turning in her emergency information on time and answering a few questions. So when she confronted my support staff member, I just kept deducting points instead of doing something more substantive.

Ironically, the phrase "I'm taking away a participation point" doesn't scream "I mean business" the same way that a silent

*teacher look*does -- especially if I don't utter that phrase in

*teacher tone*. By this time, the girl had already confronted my assistant, and so it was probably too late for teacher tone or teacher look to make a difference. The best thing for me to have done in that situation was back my assistant out and send the student out of the room, participation points be damned.

As I wrote in January, I'll continue to use participation points in my classes, but I'll only

*give*students points, not

*deduct*them. Participation points can't replace a real classroom management plan, which begins with teacher look and teacher tone, not fooling around with points.

**Plans for a Future Post**

**In a future post, I plan to write about my interactions with one student in particular -- one of my eighth grade girls. I believe that the way I interacted with her is representative of the reasons that I was not successful in my old class, and so I'll write about them in more detail. I haven't decided what day I'll make this post.**

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