Activity: Number Patterns (Days 2-3)

Remember that now I'm a full-time teacher, I no longer have time to post every school day. My plan is to skip every third day. So I'm posting today, Day 2, but my next post won't be until Day 4. The result is that I should be posting approximately three times every week during the school year -- this week I'm posting Tuesday, Wednesday, and Friday.

Let's begin with the Blaugust prompt for today:

17. The best teacher I ever had was …. because ...

Hey, I just wrote about my favorite teachers at the end of the previous school year! I usually don't like reblogging to satisfy MTBoS challenge prompts, but then again, the following prompt encourages it:

26. Reblog an old post - reflect how you see/use it now?

So let me fulfill both Prompts #17 and #26 by reblogging my favorite teachers. From June 2016:

-- My favorite elementary teacher was my second grade teacher -- who later became my fifth grade teacher as well. She was one of the first to notice that I was good at math, and so she came up with the idea of having a Pre-Algebra teacher from the high school (which went from Grades 7-12 in my district) send me a textbook. As a second-grader I would work on the assignments independently, then my teacher would send my work to the high school before I worked on the next assignment. By the time I reached the fifth grade and was in her class again, she had convinced the high school teacher to send me the textbook for "APA," or Advanced Pre-Algebra.

-- Incidentally, my favorite math teacher was that teacher who sent me the advanced work. I finally met her when I was placed in her Algebra I class in the seventh grade. I was the only seventh grader in a class full of eighth graders, but she made me feel welcome in her class.

-- Just like Fawn Nguyen, I had my favorite history teacher when I was an eighth grader. He was also in charge of the Thespian Club at our school, and so he decided to teach history in a unique way -- he would dress up as a historical figure and lecture as if he were that character. Therefore his lectures were more memorable to the students. A few years ago, he retired from teaching, and many of my classmates held a big party for him.

-- My favorite science teacher was my junior-year teacher. I was an up-and-down student when it came to science -- the first two years of Integrated Science were more biology-leaning and I struggled a little, but the third year had more emphasis on physical science, which is more closely related to my strongest subject, math (as we spent over a month discussing with Kline's book). And so I did very well in this teacher's class -- indeed, she told me that I would finish the whole test in a few minutes and spend the rest of the time making my writing neat, and of course my answers were correct. She wondered why I wasn't enrolled in the magnet program, and I replied that I had moved to my new district as a freshman, while magnet students are recruited in the eighth grade. And so my science teacher convinced the school to admit me to the magnet program as a junior. Even though I was no longer in her class, she was still my most memorable science teacher for this reason.

-- My favorite English teacher was my senior-year teacher -- or to be precise, one of two English teachers I had that year. You see, the magnet program I'd entered a year earlier was a year ahead in English -- that is, junior-level English for neighborhood students was equivalent to sophomore English within the magnet. This meant that I would have to double up on English my senior year in order to graduate from the magnet -- and I didn't look forward to this, since my strongest subject was math, not English. So even though I was the only senior in a class full of juniors, I enjoyed this English teacher's class the most. This teacher allowed us to be creative in our writing -- I remember that for extra-credit, I wrote parodies of the literature we were reading, except with my friends and me as the characters. There was also an essay contest for seniors in which we were to write about a journey we had taken -- I wasn't going to participate, except that the junior English teacher whose class I had to take decided to assign the same topic for an in-class grade! I was in the unique position of writing an essay for class and submitting the same essay to the contest.  So I wrote about my journey through my education (much of which I just wrote about in this post) -- and won $200.

When I reflect upon my favorite teachers, I notice that they have some traits in common. Two of my teachers taught subjects I didn't enjoy, English and history -- and made them enjoyable by presenting them in a unique way. The other teachers taught my stronger subjects, math and science -- and they recognized that I was talented enough in those subjects to move me up to the next level.

Some traditionalists lament the fact that the Common Core accountability movement encourages teachers to focus on the weaker students at the expense of the stronger students. They say that some strong students want to move ahead in their classes, but the teachers, who claim their hands are tied by Common Core, won't let them.

I'm torn whether I should focus on my stronger or weaker students as I get ready to teach in the middle school classroom this year. On one hand, neglecting the weaker students is why many people spurn tracking, so I want to help my weaker students get ahead. But on the other hand, I myself am the beneficiary of certain teachers noticing my special talents and allowing me to succeed in more challenging classes. Therefore I owe it to my stronger students to support them and celebrate their talents just as my own teachers celebrated my own talents.

This is so important that it bears repeating. I owe it to my stronger students to support them and celebrate their talents just as my own teachers celebrated my own talents.

Recall back on Square Root Day the story I told about teaching my second grade friend the square roots of 0, 1, and 144. I admit that this incident, along with my admiration of my second grade teacher, formed the foundation of my desire to become a teacher. At first I didn't know that Grades 7 and higher even existed -- I knew that my elementary school was K-6, and I'd always believed that students went directly from sixth grade to college. I remember that as a kindergartner, to me the sixth graders looked like grown-ups, and so I expected that they were nearly college students.

Naturally, it was the arrival of my Pre-Algebra text that alerted me to existence of 7th grade. I wasn't sure whether I wanted to be a teacher because I wasn't sure I'd be good enough at any subject other than math, but the benefactor who gave me the Pre-Algebra text was a single-subject teacher who taught math and nothing else. And so I knew at that moment that I wanted to become a single-subject math teacher -- which meant that I'd most likely teach in a high school.

Now I will be working in a K-8 school, just like Nguyen, But while I will have three preps, Nguyen has just two -- interestingly enough she teaches 6th and 8th grade, but not 7th. This means that my new K-8 school will actually be more like what I thought school was like when I was a little kid, with 6th grade as (one of) the highest grades.

The Blaugust prompt directs us to reflect on this old post, so let me do so. First, as it happens, I was checking the website of my old elementary school and -- believe it or not -- my second/fifth grade teacher still teaches there, nearly 30 years after I was a student in her classes! (At least, she taught there last year, when the site was last updated.) And according to the website, she still teaches both second and fifth grade! (The only other teacher I recognize is my kindergarten teacher.)

Now as I wrote in that old post, I want to focus on my stronger as well as my weaker students. I won't truly know who my strongest students are until after the first Benchmark Testing Week. But I am aware of the best way to protect the top students in the class -- with rules.

As I wrote in yesterday's post, the middle school English teacher came up with the idea of having the students write about the rules. And so this is most of what we're doing on Days 2-3 in the classroom.

I begin by giving each group of four students a sheet of poster paper. Students divide the paper into four quadrants, and each quadrant is labeled with the four rules mentioned earlier on the blog:

1. The Teacher Respects You
2. Respect Your Honesty
3. Respect Yourself and Each Other
4. Respect Your Class Equipment

As it happened, I only had the sixth and eighth graders create rules blogs -- this is due to a mix-up in the Wednesday Common Planning Day schedule. On Wednesdays the students take music, but music lessons don't actually begin until next week. So I gave the seventh graders an alternate assignment.

All three middle school teachers have the students create rules posters. Afterwards, the three of us discuss what the students write, and then the English teacher prints up a common list of rules for all three of us to use. The ultimate rules list will be given to the students tomorrow, and from it we create a Behavior Contract for the parents to sign. We haven't discussed the final rules list at the time of this post, since tomorrow is the day I don't post to the blog.

Because of this, let me post my lesson plans for tomorrow. After passing out the Behavior Contract and spending half the period discussing the rules, I move on to the Music Break. Tomorrow I will play the Square One TV song "Count on It," another song about the need to learn math:

Here are the lyrics, courtesy the following link:

http://wordpress.barrycarter.org/index.php/2011/06/07/square-one-tv-more-lyrics/#.V7UYlSgrKUl

Count On It

Lead vocals by Larry Cedar

Sooner or later, you’re gonna see some math
You can count on it
Sooner or later, those numbers cross your path
You can count on it

You may be hoping it will go away
But let me tell you, math is here to stay
You can count on it, hoo, yeah
You can count on it

Everywhere you look, they’re measuring the action
You can count on it
Everywhere you look, they’re even using fractions
You can count on it

They’re keeping time, and they’re keeping the score
They draw the line, and they’re running the store
You can count on it, hoo
Yeah, you can count on it

Look at the dial; look at how far
Look at how much; look where we are
Look at the gauge; look at the graph
Check out the numbers; you’ve got the last laugh

‘Cause it ain’t mystery; there’s nothing tough about it
You can count on it, that’s right
Soon you’re gonna see that you couldn’t live without it
You can count on it, hoo

Don’t take a genius or a great magician
To make a pretty good mathematician
You can count on it, hoo, yeah
Yeah, you can count on it, whoo
Oh, you can count on it, whoo
Baby, you can count on it

(fade out over Larry singing skat)

In the video the song is played on saxophones, but the only instruments I'll ever play in class are drums and a guitar. I like to play each song over two days, so I'll play the song on the drums on Thursday and the guitar on Friday.

Finally, the main lesson is another Opening Activity, on number patterns. It's time for me to reblog again, since I wrote about it last year:

During the year, I pointed out that many texts begin with a lesson on inductive reasoning, which often entails completing number patterns. I like this sort of lesson at the start of the school year.

In this lesson, students will use inductive reasoning to find patterns in sequences of numbers, letters, and names.I began with some simple sequences where students had to find the next two terms. Notice that Exercise 5 is the Fibonacci sequence -- a nod to Fawn Nguyen, who gives her Geometry students a worksheet on that famous sequence on the first day of school.

Speaking of Fawn Nguyen, yes, many of my opening activities are based on Nguyen's. As it turns out, one of her best known lessons is on patterns -- except these are visual, not numerical, patterns:

http://www.visualpatterns.org/

Exercises 7 and 8 on my worksheet don't come from Nguyen. Instead, they come from the texts used by the students I tutored. Those students enjoyed trying to figure out the patterns in these lists of names. I know that I've spent so recent posts about future presidents, but these lists are all about past presidents. Exercise 7 refers to George Washington, John Adams, Thomas Jefferson, and James Madison, so the correct answer is another James (Monroe), another John (Quincy Adams), and then Andrew (Jackson).

Exercise 8 looks similar, but this time it refers to money -- George Washington ($1), Thomas Jefferson ($2), Abe Lincoln ($5), and Alexander Hamilton ($10). So the correct answer is Andrew (Jackson again, $20), Ulysses (Grant, $50), and then it's all about Benjamin (Franklin, $100). As it turns out, one can actually extend this sequence. I was recently watching old 1960's episodes of the game show Let's Make a Deal on the new BUZZR channel, and often the host Monty Hall would offer contestants $500 bills (William McKinley) and $1000 bills (Grover Cleveland). There was even an episode when Monty showed a contestant an extremely rare $5000 bill that the bank had allowed him to show on that episode only -- had the contestant won it, she would have received a check for $5000 as the bill would have to be returned to the bank. James (Madison, not Monroe) was on the $5000 bill, so the sequence would continue Benjamin, William, Grover, James. It may be a good idea for teachers to give the related number sequence 1, 2, 5, 10, 20, 50, ..., as a hint.

The worksheet was getting long, so I stopped here. but notice that there are still many other types of problems that I could give:

Quatros: 4, 108, 60, 52, 36, 144
Not Quatros: 2, 29, 106, 18, 15, 22, 6
Which are Quatros? 86, 737, 42, 72

Semirps: 2, 13, 11, 23, 53, 97, 71, 47
Not Semirps: 15, 25, 209, 21, 190
Which are Semirps? 123. 67, 51, 27

Notice that "Quatros" are simply multiples of four. The word "Quatro" comes from the Latin word for four -- and we'll see that root later on in Geometry when we cover quadrilaterals. As it turns out, the modern Portuguese word for "four" is quatro [yes -- the Rio Games are still going on now -- dw] A few other Romance languages pronounce the word for "four" identically to the Portuguese, albeit with a slightly different spelling.

As for "Semirps," any nerd -- or even a dren -- can see that "Semirp" is "primes" spelled backwards. I do find it a bit awkward that the text pluralized "prime" to "primes," reversed it as "Semirp," then pluralized it again to "Semirps." Then again, one advantage to calling them "Semirps" rather than "emirps" is that the extra s- may trick readers into thinking about the prefix semi-, which is Latin for one-half -- especially right after seeing the Latin root for "four" in the previous question.

I was also considering including the first two sequences from the "Improving Reasoning Skills" section, which contains some bonus problems:

1. 18, 49, 94, 63, 52, 61, ...
2. O, T, T, F, F, S, S, E, N, ...
3. 4, 8, 61, 221, 244, 884, ...

I was able to figure out the first one, and I'd seen the second one before, but the third question stumped me -- and I suspect that it will stump our students as well. (The second one in the sequence is as easy as One, Two, Three!)

One of my favorite websites when considering number sequences is the On-Line Encyclopedia of Integer Sequences:

http://oeis.org/

The OEIS is one of the oldest sites on the Internet. Notice that it was first created in 1964 -- long before the Internet existed as we know it! Back in the 1960's, users had to submit queries by sending it a primitive form of e-mail. Nowadays, of course, it is web-based like most other sites.

Many famous sequences are entries in the OEIS. Here they are:

1,10,100,1000
180,360,540,720 (Notice the geometrical interpretation -- sum of the angles of an n-gon!)
0,10,21,33,46,60
1,3,6,10,15,21 (triangular numbers)
1,4,9,16,25,36
1,1,2,3,5,8,13
1,3,4,7,11,18 (Lucas numbers, similar to Fibonacci)
1,2,4,8,16,32
1,3,7,15,31,63 (sometimes called Mersenne numbers)
2,6,15,31,56,92 (given by the polynomial that generates these, (n+2)*(2*n^2-n+3)/6)
3,5,11,29,83,245
0,3,8,15,24,35
3,12,48,192,768 (there are some signed sequences in the database, but here the signs were ignored)
1,2,5,14,41,122

In fact, the only sequences I didn't enter were the ones containing letters or fractions, since this is in fact an integer sequence database.

Here's a sequence related to one of the lists of dead presidents that I entered:

1,2,5,10,20,50

Finally, here are the answers to the bonus questions:

18,46,94,63,52,61 (but I really did figure this one out before entering it into the OEIS)
4,8,61,221,244,884 (too hard for me, no problem for OEIS)
2,3,6,1,8,6,8 (too hard for me, no problem for OEIS)

The last sequence did stump the OEIS, however. Neither one of us figured out the sequence:

6,8,5,10,3,14,1, ...

Maybe you can figure it out, then try submitting it to the OEIS! Unfortunately, the OEIS has been swamped with submissions for months. Still, you can see why I enjoy the OEIS as a handy resource for integer sequences.

I hope the students will enjoy tomorrow's activity.


I am not posting tomorrow -- the title of the post includes the notation (Days 2-3) to remind the readers that the next post won't be until Day 4, which is Friday.

Points in Networks (Day 1)

So, what was my first day of school like, you ask? Well, let's first look at today's Blaugust prompt:

16. A  mentor/colleague who impacted your classroom/teaching…

Well, as a first-year math and science teacher, I ought to say that the other math and science teachers at my school are my mentors. But unfortunately, I'm the only math and science teacher here. But at least the other middle school teachers -- one history, the other English -- were somewhat like my mentors, especially on the first day of school.

The history teacher taught at my school last year, so he already knows the school -- and he also knew the names of many of the returning students. The English teacher is also new, but she at least has some teaching experience at another school. Pay attention to the impact my fellow middle school teachers had on my first day of school. I'll post just the basics of what my day was like except for the point where I met my eighth grade class, since this is where I want to focus.

7:45 -- I arrive at my school.

8:00 -- I report to the playground, where many students are beginning to arrive. The students are told to gather in a circle for the flag salute.

8:25 -- My first class, a sixth grade class, begins. There are 18 students in this class. Most of the students behave well. One student is a Spanish speaker who knows very little English, but fortunately another student volunteered to translate for him. As a teacher, I know how important it is to provide alternate material for English learners.

9:45 -- My sixth graders leave and my seventh graders arrive. To our surprise, there are a whopping 25 students in this class. I know -- this is still quite small by the standards of public schools, where there are often 30, 35, or even 40 students in one class. But by our charter school standards, this is large, considering that there are only 24 seats in the classroom. This is the class with which I struggle the most with classroom management -- the history teacher warned me that every 7th grader is either one he remembers as a troublemaker last year, or a newly enrolled student!

11:05 -- My seventh graders leave for nutrition.

11:25 -- My eighth grade class arrives. This is my smallest class, with only 12 students -- but there are only eight students present at the start of class. I begin the class the same way I start all my classes, with a Warm-Up question:

What is 2 * 2 * 2 * 2? (That is, 2 times 2 times 2 times 2.)

Most students answer correctly, although a few tried to add. A student or two is upset that the very first thing we do on the first day of school is multiply. I point out that the answer is 16 -- and that today is the 16th. I always go around to stamp correct papers -- many teachers point out that students enjoy getting stamps, and my students are no exception.

11:35 -- My student support aide arrives -- the English teacher and I are each assigned one. Actually, she arrives with the four missing students, all girls.

We move on to an Opening Activity -- the Konigsberg Bridge Problem. I've written about this problem previously on the blog and even suggested it as a first day of school activity -- well, now I'm finally giving the activity on an actual first day of school. This is a little of what I said about this problem here on the blog:

The Königsberg Bridge Problem is a famous math problem from nearly 300 years ago. Fawn Nguyen, a well-known math blogger and fellow Southern Californian -- she lives in Ventura County -- used this as an activity in her geometry class:

http://fawnnguyen.com/famous-bridge-problem/

As we all know, the Königsberg Bridge Problem is impossible to solve -- it has no solution. But I don't want to start the class with a problem that the students can't solve -- they're already frustrated enough with problems that do have solutions when they just can't find them.

The whole point of this lesson is to point out that students should look for patterns, and that sometimes it's just as important to know why something is impossible as it is to know why something is possible.

Let me complete this with a note on pronunciation. The U of Chicago text points out that the name Euler ends up sounding like "Oiler." But how does one go about pronouncing the name Königsberg? I once read that the o-umlaut ends up sounding like "uh," almost like "ur." A Google search reveals a ten-second video in which this name is pronounced:

By the way. some students believe that they have a solution to the Konigsberg problem, but actually they are crossing one of the bridges twice (they start on island D, cross the bridge towards C, but then head back to D). I start to explain about Euler and why the problem is impossible -- and as I do so, the student who earlier complained about 2 * 2 * 2 * 2 figures out that the impossibility has to do with there being an odd number of bridges from each land! I'm impressed!

12:05 -- Because I know how tough the 80-minute block schedule can be on middle school students, I provide a music break. I get out my guitar and I play the following inspirational song:

The Dren Song -- by Mr. Walker

I don't know why I take math.
I'm all caught up in its wrath.
I'd rather just be a dren.
I would be so happy then.
Tell me what would happen when,
I'm no longer just a dren.
What if I were great at math?
What would be my future path?
Customers won't think it's strange,
When I figure out their change.
Algebra and calculus,
Get me in a cool college.
Once I finish my degree,
Future employers will see,
Of my strong background in STEM.
I know that will impress them.
Reach the moon, be a hero!
I won't just be a zero!
I'll be great, or it may seem,
That this all is just a dream,
'Cause my math skills are so bad.
I can't subtract! I can't add!
I can't multiply by ten.
I will always be a dren.
Now I know why I take math.
Help me find a better path!
I would be so happy then.
But I'm just a dren.

Along the way, I explain that a "dren" is a reverse-nerd -- a nerd is someone who's good at math, and a "dren" is someone who doesn't understand the basics of arithmetic. As it turns out, the student who complained about 2 * 2 * 2 * 2 enjoys this song and looks forward to my next song.

I show my students the September 2015 Boys' Life article about the mathematicians and scientists who work for NASA and the possible future of people traveling to and living on the moon. But as it turns out, eight of the 12 students in my class are girls, so I don't expect Boys' Life to motivate them.

Instead, I tell them about the movie trailer that was released just yesterday -- Hidden Figures, about the scientist Katherine Johnson who worked for NASA and the Apollo projects in the 1960's. For those of you who have read my blog before, it goes without saying that I plan on watching this movie, and I highly recommend that my students watch it in January as well.

12:15 -- I proceed with my next Opening Day activity -- Personality Coordinates. This activity comes from the King of the MTBoS, Dan Meyer:

http://blog.mrmeyer.com/2013/personality-coordinates-icebreaker/

Each person in a group picks a dot and writes her name next to it.

Now the group’s job is to label the axes. Physical attributes don’t require all that much thought and don’t reveal all that much, so don’t allow them.

That’s it. It requires a surprising amount of creativity and conversation. Happy first day of school, teachers.

12:30 -- My support aide leaves, and this is a good time to end the period with an Exit Pass:

If you don't know the answer, ..

The answer is "at least know where to find it," which is posted in a corner of the room. (I mentioned this in an earlier blog post.) Some wrong answers are "ask the teacher" and "you're a dren."

12:45 -- My eighth grade class goes out to lunch.

1:35 -- My sixth grade class returns for a special "Math Intervention" class. There is special software for this class, but I spend the entire period acquainting the students with the laptops, including making sure that the students all have the correct password.

2:35 -- My sixth graders go out to P.E. class. The history teacher, English teacher, and I watch as we discuss our plans for the next day. The English teacher comes up with the idea of having the students come up with rules on the posters.

4:00 -- I go home for the day and head for my computer to type up this blog entry.

If there's anything I could change about the way I ran the class today, it would be to teach the entire class in reverse order. That way, the Exit Pass becomes a Warm-Up, a scavenger hunt to find the rest of the quote, Personality Quotes occur earlier in the class, and the 2 * 2 * 2 * 2 question doesn't turn off students right at the start of the period.

I would also rewrite the Konigsberg worksheet. I'd already changed the worksheet to add more bridge problems, including some trivial ones. But now I'd number those trivial problems #1 and #2 (rather than #3 and #4, as they were numbered today).

Another problem I have has to do with explaining my directions clearly. I was hoping to create a seating chart directly from the Personality Coordinates worksheet (since the students are already seated in groups of four), but I couldn't because some groups randomly labeled the dots rather than place the student sitting northwest in the upper-left corner of the page. Also, some students wrote the Exit Pass on a separate sheet of paper rather than the back of the Warm-Up.

I remember explaining my directions to the students -- but I could be remembering my explanations to the 6th and 7th grade classes, not the 8th grade class. Anyway, I know that I don't always explain instructions clearly to my students from my days as a sub, so I must give the students the benefit of the doubt whenever I see them misinterpreting instructions.

Here is a copy of the Konigsberg worksheet as I actually gave it in class today. Again, I point out that there are still ways that I can improve this worksheet.

And thus ends my first day of school. Today I am a teacher.

Disclaimer

This will be my final post of the summer before school begins next week. There are a few things that I want to clear up here first.

Table of Contents
1. Disclaimer
2. Blaugust Prompt #12
3. A PD Day: More on the Illinois State Text
4. More on Grading and Pacing
5. A Tentative Testing Schedule
6. Conclusion

Disclaimer

I speak for myself only. This blog is not affiliated with any school, district, company, or organization.

Notice that I don't mention the name of my charter school in any post in order to avoid any semblance of conflict of interest. I do mention the LAUSD (Los Angeles Unified School District) in some of my posts, but officially, my sole employer is my charter school, not LAUSD. In any case, this blog is not affiliated with LAUSD.

Blaugust Prompt #12

Since today is August 12th, I will discuss Prompt #12 from the Blaugust list:

12. Something I read/learned this summer that intrigued me…

Earlier, I mentioned The First Days of School, by Harry and Rosemary Wong. Here is a key chapter that I read:

-- Chapter 3: How You Can Be a Happy First-Year Teacher. The Wongs write:

The first day of school or a class -- even the first few minutes -- will make or break a teacher. It is those first few minutes and first few days of school that are the subject of this book.

A PD Day: More on the Illinois State Text

Earlier this week, there was a special training session. I met Dr. Brad Christensen, a STEM specialist from Illinois State -- and he was obviously at our school to discuss the Illinois State text during a Professional Development session.

As I explained earlier on the blog, the Illinois State text is a strong project-based curriculum. During the session, Dr. Christensen demonstrated two math projects that are similar to those that can be found in the text. The first involved using dowel rods and rubber bands to construct as tall a structure as possible. The other was a method of estimating square roots using colored squares.

Back in my Square Root Day post (dated 4/4/16), I mentioned various activities that we can give students in order to learn square roots. This activity would have been perfect for Square Root Day, because it allows students to find rational approximations of irrational square roots. It would have fit in well with some of the activities that I mentioned the first week in April.

Notice that this activity actually directs students to perform a linear interpolation. It's possible to perform this interpolation without using colored squares -- to find the square root of a whole number n, use the formula:

sqrt(n) =approx. floor(sqrt(n)) + (n - floor(sqrt(n))^2) / (2n + 1)

Here floor(sqrt(n)) means the square root of n rounded down to the next whole number, which can be found using inspection or trial and error. We can avoid using the floor function by letting a^2 be the largest perfect square that doesn't exceed n, to obtain:

sqrt(n) =approx. a + (n - a^2) / (2n + 1)

But this demonstrates the whole purpose of the colored squares activity -- those formulas are especially hard to remember! The activity takes a little effort to understand -- but once the students get it, it becomes a nice method for interpolating square roots.

According to Dr. Christensen, it's important when beginning a learning module to have the students just jump in and begin the project. There's no point in having the teacher explain anything to the students first, because there's no reason to believe that the students are even listening -- and so all that time spent explaining is wasted.

This philosophy, of course, is in direct opposition to traditionalism. Traditionalists assume that students will listen to the teacher's explanation just because he or she is the "sage on the stage." That is, they'll listen to the teacher because it's proper to do so, and students do only proper things. I believe that students are less likely to listen to a teacher in middle school than in early elementary school, which is why I agree with traditionalists in the early, but not the middle grades.

Dr. Christensen told us that there is room for traditional lessons within a learning module, especially if some students need them. The important note, though, is that the projects should come first. The projects are the cornerstone of the Illinois State curriculum.

Oh, and by the way, Dr. Christensen showed us the science curriculum. I wasn't sure whether or not it would be divided into Earth Science for sixth graders, Life Science for seventh graders, and Physical Science for eighth graders -- considering that California's "preferred model" for the Next Generation Science Standards is integrated science. Well, the science texts he gave us follow the traditional order of Earth in 6th, Life in 7th, and Physical in 8th. Nevertheless, the texts are definitely NGSS-aligned.

More on Grading and Pacing

Earlier this summer, I wrote that the Illinois State text provides a four-point rubric for us to evaluate the students' projects. Well, we can convert these to letter grades, as follows:

4 = A
3 = B
2 = C
1 = F

You may notice that there is no D grade here. As it turns out, many schools chartered with the LAUSD, including mine, have adopted a no-D policy. The only grades that can appear on a report card are A, B, C, and F.

I've written a little about the no-D grading system earlier on the blog. In particular, I mentioned it back on the last day of the first quarter in 2014 -- second week of October. (I actually reblogged that post at the end of the first quarter in 2015 -- last week of October -- but then I added a discussion of other grading systems that have nothing to what I'm teaching, so I recommend reading the less distracting 2014 post.)

I know that I've changed my mind several times over the summer. But regarding my pacing plan, I want to use the pacing plan that I mentioned back at the end of the first semester in 2016 -- third week of January. According to this plan, we determine which learning module we should be in by looking at the first digit of the day number of the blog calendar. So Module 1 will begin on Day 10 and end on Day 19.

My testing dates will change a little from what I posted earlier. But I did write that there will be a test the second week of school -- and that isn't going to change. And I have an answer ready for why there will be a test so early in the year -- it is a benchmark test.

You see, the second week of school is Benchmark Testing Week. At this time, all students take diagnostic exams in English and math. There is a Benchmark Testing Week every trimester -- the second and third Benchmark Testing Weeks occur at the midpoint of the respective trimesters. This also explains what I'll be doing the first nine days of school if Module 1 doesn't begin until Day 10 -- opening week activities followed by the first Benchmark Testing Week.

The purpose of Benchmark Testing is to measure growth -- indeed, all teachers have a special Data Wall in order to measure growth. Hey, didn't Fawn Nguyen just say something about growth in her post linked above? Exactly -- Benchmark Testing is all about growth, while the SBAC and NGSS tests are all about proficiency.

Using digit pacing and including the Benchmark Testing Week produces the following calendar:

Opening Week Activities: August 16th-19th
Benchmark Testing Week 1: August 22nd-26th: Benchmark Testing Week 1
Module 1: August 29th-September 13th
Module 2: September 14th-27th
Module 3: September 28th-October 13th
Module 4: October 14th-27th
Module 5: October 28th-November 10th
Module 6: November 14th-December 2nd
Module 7: December 3rd-16th
Module 8: January 9th-20th
Benchmark Testing Week 2: January 23rd-27th
Module 9: January 30th-February 6th
Module 10: February 7th-21st
Module 11: February 22nd-March 7th
Module 12: March 8th-21st
Module 13: March 22nd-April 5th
Module 14: April 6th-28th
Benchmark Testing Week 3: May 1st-5th

A Tentative Testing Schedule

At this point the schedule gets murky as we reach the SBAC and NGSS testing block. My plan is to teach Module 15 during the testing block and count the project as a test, so that students can focus on the SBAC and NGSS tests rather than classroom tests. (Note: I may sneak in a little extra science in the eighth grade, since that grade has a NGSS science test to take.)

This plan applies to all three grades -- sixth, seventh, and eighth. Notice that the Illinois State text is identical in all three grades for the first four modules (Unit 0, Tools for Learning), but the traditional workbook that supplements the text does give different problems in each grade.

My plan is still to rotate among the three grades among a Dren Quiz, Grade-Level Quiz, and test, with eighth grade taking a test first -- except now that first graded test will be the third week. Here is a schedule of tests for the eighth grade -- the grade which I'll be blogging the most about:

1. September 1st
2. September 23rd
3. October 14th
4. November 4th
5. December 2nd
6. January 13th
7. February 10th
8. March 3rd
9. March 24th
10. April 21st
11. May 19th (SBAC week -- may be replaced with a project)

Notice that the tests don't necessarily line up with modules -- again, this schedule is designed to spread out the number of tests I have to grade each weekend. On other other hand, Dr. Christensen told us that Illinois State provides tests to give to the students based on the modules. I may end up giving those tests according to the module schedule even if it means I must grade tests for all three grades at the same time -- the extra time it takes to grade is made up for by the time I save not having to write the tests (and besides, my charter school is tiny, with small class sizes, so it's not as if I have that many tests to grade even if I do all three grades together).

But if I keep the rotation pattern, the final week at the end of the year before grades are due will be a Grade-Level Quiz for eighth graders, a Dren Quiz for seventh graders, and a test for sixth graders. So I'll have to teach Module 16 to 6th and 8th grades so that there's something to assess that week. This is convenient because Module 16 is at the end of a unit in the 6th and 8th grade texts, while Module 15 ends a unit in the 7th grade text.

I know I ought to be finishing the entire text, which goes up to the mid-20's in all three grades. But there are two conflicting philosophies here. On one hand, I want there to be ample time for all the projects -- and I'd much rather do the projects right than fast. On the other hand, there's a certain point we need to reach in each text before the January Benchmark Testing Week -- and unfortunately, no one seems to know exactly what that point is. (Recall that I'm the only middle school math teacher on my campus, so there's literally no one I can ask.)

I want to make sure that I at least introduce every topic that appears on the Benchmark to my students prior to the test -- otherwise they may become disillusioned and blow off the entire test, even the questions that they do understand.  But I don't want to follow the pacing plan above and finish Module 7 just before winter break, find out that the Benchmark Test goes up to Module 12, and then suddenly I'm forced to cover Modules 8-12 over two weeks in January. (Obviously, if this were to happen, I'd skip all the projects and use traditional lessons only for Modules 8-12.)

Well, there's nothing I can do about that until I get there.

Conclusion

I usually don't like going back to edit posts (other than my most recent post when I forget to add a link or worksheet), but I'll be going back to edit my July rules posts.

Attention -- Edited Posts:

Here are the posts I edited. Yes, I admit that I change my mind a lot, and it can be annoying to blog readers when I keep changing what I've written. But then again, we as teachers often have to change and adjust to meet our students' needs.

-- First, the list of rules now looks like this:

1. The Teacher Respects You
2. Respect Your Honesty
3. Respect Yourself and Each Other
4. Respect Your Class Equipment

In today's post I explained why the fourth rule now mentions equipment -- as in the equipment that appears in a science classroom. But now I also changed my first rule. Before I can expect my students to show respect, I as a teacher must show them respect. (There was also a story I told in my Rule #4 post that is all about respecting students, so I moved that story to the Rule #1 post.)

-- Second, I quickly finished reading Ron Clark's The Essential 55. Now that I've read it, some of the rules do make sense after all -- and they are all about showing respect. Rule #19 is "When I assign homework, there is to be no moaning or complaining. This will result in a doubled assignment." I'm still not sure whether I would use that or any of his rules in my class, but they do give me something to think about. And no, I won't use his Rule #47, "Do not bring Doritos into the school building."

-- Finally, I've changed my mind about traditionalists. I've decided that the one traditionalist I'll be quoting in my classroom will no longer be Bill. Instead, I will link to and quote Barry Garelick.

Why have I chosen Garelick instead of Bill? And besides, why should I be quoting any traditionalist at all -- shouldn't I be only reading the blogs of actual math teachers?

Well, here's a link to Barry Garelick's blog and one of his recent posts:

https://traditionalmath.wordpress.com/
https://traditionalmath.wordpress.com/2016/07/23/youre-not-the-exception/

I will be starting a teaching assignment in a middle school this August.  A few weeks ago, I met with the woman who will be mentoring me for two years.  (This is part of California’s licensing requirements for new teachers–sort of like being out on parole from ed school.)

So actually, Garelick is a math teacher! And moreover, he's a fellow first-year California middle school math teacher to boot! Recall that the only other active California middle school math blog of which I'm aware belongs to Fawn Nguyen.

But don't expect Garelick to join MTBoS or Blaugust anytime soon. Most MTBoS bloggers like to write about exactly the sort of activities that the traditionalist Garelick despises. I do notice that one frequent commenter on Garelick's site is Dan Meyer, the King of the MTBoS and a strong advocate of activities in the classroom. The two of them often exchange comments about why their respective preferred pedagogy is superior.

As he writes at the above link, Garelick definitely plans to use a traditional pedagogy -- or at least as traditional as he can get away with -- in his classroom. I look forward to reading about his classroom, and especially how he'll convince his middle school students to go along with traditional lessons.

Not only will I be able to quote Garelick himself, but one of his frequent commenters is his fellow traditionalist SteveH, who, like Bill, writes about why students should learn higher math. To Garelick's post with the same date as my post you're reading now, SteveH left this comment:

https://traditionalmath.wordpress.com/2016/08/12/if-you-say-so-dept/

SteveH says:

No, the real world of getting an engineering degree requires at least a differential equations course. Are those institutions and professors wrong – overruled by educational pedagogues? What would happen to those “engineers” who didn’t get to that level? They would be called technicians and end up with a lower salary. If you’re getting paid engineering wages and not using college-level skills, then you better keep quiet about it. You won’t help those coming up the salary pipeline and you will be setting yourself up to be pushed down or laid off.

One of the first engineering courses that all have to take is statics and dynamics. If you can’t (at least) do calculus, then you will flunk out. You may not end up doing calculus day to day in your job, but you darn well need it for a proper engineering degree and understanding of what you’re learning. You can dumb down an engineering degree, but that would catch up to many in the form of dumbed down salaries. BTW, I can’t imagine hiring and paying engineering wages to those who can’t analyze a technical paper with all of those, you know, squiggly lines, in it.

Other than that, this ends my blogging for the summer. My next post will be on the first day of school, which is Tuesday, August 16th. Remember, my plan is for me to write three blog posts during every week of the school year.

I hope my first year of teaching will be a successful one.

My Rule List

In today's post, I will continue to describe the classroom preparations for my very first year of teaching, which will start next week.

Table of Contents
1. Blaugust: Another MTBoS Challenge
2. My First Blaugust Prompt
3. Links to Other Blaugust Participants
4. A Science Teacher Blog
5. My Rule List

Blaugust: Another MTBoS Challenge

Yes, you read that right! The members of the Math Twitter Blogosphere, or MTBoS, certainly enjoy their blog challenges! As the name implies, teachers fulfill the Blaugust challenge by blogging as possible during the month of August.

Here's a link to the creator of the Blaugust challenge:

http://statteacher.blogspot.com/
http://statteacher.blogspot.com/2016/07/mtbosblaugust-is-back.html

The owner of this blog goes only by the username "druin" -- I don't know her full name. (She does list her gender as "female," so I can safely use feminine pronouns.) Based on the name of the blog, druin is a high school statistics teacher.

This is what druin writes about the challenge:

Rules of Blaugust

The rules are pretty simple... all you need to do is blog!  :)

Seriously though, my goal will be to blog daily in August, but I know that's not always possible, so set your own goal!  Maybe it's a goal to blog every other day or to write at least 10 posts this month... whatever it is, you can do it!  This will be my 4th attempt to blog every day for a month and I have YET to achieve that goal, but it's okay - blogging once a week is better than not blogging at all!

Please take a minute to sign-up so I can link to your site as a #MTBoSBlaugust participant and cheer you on!

So apparently this Blaugust thing has been going on for some time, yet this is the first time that I've heard of it! According to druin, participants should post everyday during August. Well, it's too late for that now -- as usual, I'm late to the party.

Still, I'm going to participate in Blaugust anyway, regardless of the sign-up list. I already had plans to post thrice a week during the school year, so I should reach the "at least 10 posts this month" goal, even though it's too late for me to post everyday this month. (Hey, at least I found about Blaugust faster than MTBoS30 -- May was already half over by the time I read about that challenge!)

Oh, and by the way, I will be using the MTBoS label for all of my Blaugust posts. For MTBoS30 I only labeled some of the posts as MTBoS, but this time I want to make sure that all of my challenge posts are marked.

Prompt #9: Start, Stop, Continue

According to druin, we can write about anything we want in our Blaugust posts. But she provided us with some prompts anyway -- just as for the 2016 Blogger Initiative challenge in January. When I first saw her list, I thought that the first prompt was for August 1st, the second prompt for August 2nd, and so on -- until I saw that there were 50 prompts there, not 31. Nonetheless, I think I'll use Prompt #9 today anyway, since today is August 9th.

Notice that many of these prompts are related to the First Day of School. This is probably why druin set up Blaugust in the first place -- she wants teachers to write about the upcoming school year. (And again, I still must get used to the Early Start Calendar and associate "back to school" with August rather than September!)

By the way, I'm retroactively considering my last post, dated August 5th, to be my first official Blaugust post. Of course, I didn't follow Prompt #5 in that post. Let's change that #5 to #45 instead:

45. What’s the toughest challenge you face as a teacher today?

And my toughest challenge will certainly be to teach science in addition to math, as I explained in that August 5th post.

Now for today's prompt:

9. What would you like to Start doing this school year?  What would you like to Stop doing?  What would you like to Continue doing?

And here are my responses:

Start: The one thing I want to start doing this school year is, well, teaching! This is my very first year as a teacher. As we all know, the first year of teaching can be challenging -- and especially sicne I am working at a small charter middle school where there are no other math teachers! But I believe that I can be successful by reading about how more experienced teachers are running their classes -- and that includes other MTBoS participants.

Also, what I need to do is start demanding respect from my students. As a sub, I often found it difficult to demand respect, but now it's time for me to stop thinking of myself as a sub and start acting like a teacher, since that's what I'm going to be. Another way to think of it is this: if I had a bad day as a sub, the regular teacher could fix it the next day, but if I have a bad day as a regular teacher, there's no one to fix it. So I'm the one who's responsible for the students.

Stop: Here's one thing I want to stop from my days as a sub -- stop letting those who want to disrupt the class take over. One of the most important things to have in the classroom is respect, and I want to make sure that students respect their school and community. I must remind the students that part of his or her job is to respect. To do so, I will reward students who contribute to order and punish those who wish to disrespect.

As a sub, a problem I commonly saw was when students didn't sit in their assigned seats. Sometimes a student would sit in adjacent seat because the owner of that seat was absent and the student didn't want to be isolated from the other students, while at other times the seat-changer was doing so for more malicious reasons. But in either case, a seat-changer isn't contributing to order -- and indeed, if I see a student changing seats, I start wondering what other rules that student will break!

And so I must be strict in following my seating chart to the letter. The students in my class are divided into groups of four, which will be convenient for group activities. Any student who sits in a different seat -- even if it's just one seat away from the correct seat -- and refuses to return to the correct seat at the count of five will be immediately given a five-minute detention. This will be separate from my participation points system.

Also, I must stop tolerating cell phones. Again, in the past I usually had the sub mentality -- I can't do anything about the phones because students will never surrender their phones to a sub. I've seen that I must especially be strict with phones with my sixth graders -- eighth graders (and above) usually have phones out to listen to music (on headphones so as not to disturb others), but less mature sixth graders are more likely to use the phones to disrupt the classroom. Just as with seating changes, cell phone violations will be separate from my point system. (Again, Pokemon Go worries me!)

Continue: Just as I want to stop doing anything that takes away from order in the classroom, I wish to keep doing things that contribute to order. To me, the best way to maintain order is to guarantee that the students have something to do at all times. Again, my best days as a sub occurred when students knew exactly what they were supposed to be doing at every moment of the class and knew how to do it.

I will continue to use some sort of point system as much as possible. As a sub, I often used group points because I didn't know the names of all the students. Now I will be leaning more towards individual participation points as I get the opportunity to learn more about all the kids. Again, I'll be giving out rewards and consequences mainly based on how many points a student gains or loses during the lessons.

Also, of course I will continue helping students. I consider one of my greatest strengths as a teacher is to look at a student's paper and figure out what problems he or she is having. I know what the most common errors are, and so I am able to correct these errors and help the students successfully finish the assignment. I must keep in mind, though, that I'm no longer a sub, but a regular teacher -- so I can't help the students forever, but make sure that they can do the work on their own.

Links to Other Blaugust Participants

Let's take a peek at some of the other teachers who are posting for Blaugust. And who's better to start with than druin, the originator of this challenge?

Now druin's first Blaugust posts definitely adhered to the following prompt:

12. Something I read/learned this summer that intrigued me…

And druin definitely read something this summer -- Ron Ritchhart's Making Things Visible. I'll link to her first post about the book, even though this was back in June. She wrote about one chapter in each post and finished the book for Blaugust:

http://statteacher.blogspot.com/2016/06/making-thinking-visible-chapter-1.html

After finishing Ritchhart's book, druin went on to talk about her classroom. Here's a link to the first post she wrote after completing Ritchhart:

http://statteacher.blogspot.com/2016/08/new-bulletin-boards-for-2016.html

druin writes:

New Bulletin Boards

On Wednesday, I went up to work in my classroom for a bit.  My main goal was to finish up the bulletin boards and I was able to get several done. 

First board isn't quite done, but I did get the fabric hung up.  I team teach a Forensic Science class with the science teacher next door and this summer on Etsy, I found this fabric:

Hey, what's this? Apparently druin, just like me, teaches science in addition to math! The main difference is that this "Forensic Science" class sounds like an elective, not a student's main science class -- and that's there's a separate science teacher. Recall that I am the only person teaching both math and science to my middle schools.

Nonetheless, I find some of her bulletin boards inspiring -- and who knows, maybe I'll consider using some of her ideas in my own classroom!

Oh, and by the way, you may be wondering whether there is a "Science Twitter Blogosphere" similar to the MTBoS, but for science teachers. Apparently, there is no close-knit group of science teacher bloggers -- of course, some science teachers do have blogs, but there's nothing analogous to the MTBoS, Blaugust challenges, etc., for science.

Here are a few more Blaugust participants whose blogs I wish to highlight. First, let's look at the list of participants:

http://statteacher.blogspot.com/2016/07/mtbosblaugust-participating-blogs-2016.html

-- Sherrie Nackel:
http://7thgrademathteacherextraordinaire.blogspot.com/

Unlike the Blogging Initiative, Blaugust doesn't have a requirement where we must specifically look at the three blogs right above our own name in the list. But still, I couldn't help but notice the name of the blog right above mine -- "Middle School Math Rules"!

That's right -- Nackel is a middle school teacher from Wisconsin (most likely 7th grade, based on the URL of her blog)! I've said before that most MTBoS participants are high school teachers, so it's refreshing to see a fellow middle school teacher here.

But unfortunately as of today, Nackel has no Blaugust posts. She did put her name on the list, so I hope she'll be posting soon!

-- Author unknown (Mrs. Gamb?)
https://mmgamb.wordpress.com/

The author of this blog doesn't give her full name, but according to her first Blaugust post, she is a 7th and 8th grade math teacher in Texas. In the following post, she writes about classroom materials:

https://mmgamb.wordpress.com/2016/08/06/back-to-school-shopping/

Oh, and notice that so far, my favorite middle school math teacher blogger, Fawn Nguyen, does not appear to be a Blaugust participant. She hasn't posted to her blog at all this month, nor did she add her name to the list. Well, at least I have these two blogs to enjoy.

A Science Teacher Blog

I do want to link to at least one science teacher -- particularly a middle school science teacher, since this is the grade span I'll be teaching. This was a bit difficult since it's hard to find middle school math blogs -- and that's with the existence of MTBoS. Imagine how to find a middle school science teacher without the benefit of an MTBoS for science!

Anyway, I did find the following blog:

http://katesclassroomcafe.blogspot.com/

The author of this blog appears to be a 6th grade science teacher named Kate. Her most recent posts also describe how to set up a classroom for the first day of school -- and these ideas can apply to any middle school class, not just science.

With this, I'm ready to announce my fifth and final rule for my classroom. This rule is based on what I wrote for Start/Stop/Continue:

My Rule List

I keep saying that I want to write my final rule list and then putting it off. Actually, I'll write the list by using one of the other Blaugust prompts:

23. Using your school mascot, create an acrostic of character traits you wish to instill in your students

Actually, my new school doesn't use a mascot, but we really do have an acrostic of character traits based on the word SCHOLAR:

Speak with Integrity
Commit to Success
Honor yourself and others
Own your own future
Lead the way to Excellence
Aspire to learn from everything
Respect your school and community

And so these are the seven rules that I will be enforcing in the classroom this year. Notice that my previous rules posts in July are all aligned with the seventh -- and perhaps most important -- rule:

1. The Teacher Respects You
2. Respect Your Honesty
3. Respect Yourself and Others
4. Respect Your Class Equipment

I guess that the singer Aretha Franklin said it best -- the underlying idea is respect. The rules are all about getting the students to respect the education process.

I will be posting later again this week -- the final post of the summer. And of course I'll be continuing to participate in the Blaugust challenge. Again, I must emphasize that the MTBoS is the only opportunity I have to communicate with my fellow math and science teachers, because I'm working at a small charter school where I'm the only such teacher. Because of this, I highly appreciate the MTBoS and its members who write such thoughtful posts.

The Next Generation Science Standards

I've decided to devote today's post to the Next Generation Science Standards, which are often known as "Common Core" for science.

Table of Contents

1. I am a Science Teacher!
2. Integrated Middle School Science
3. Examples of New Science Standards
4. Fractions Yet Again!
5. Another Bill Comment
6. Fractions in the Illinois State Text

I am a Science Teacher!

Now I've alluded to the NGSS several times here on the blog -- mostly in a special post I posted nine months ago in November, my Black Friday post. That post was part of the Common Core Debate, in which I discussed the pros and cons of national standards by looking a little at the science standards. But in November I took only a cursory look at NGSS, because this is a math blog and I was working on becoming a math teacher.

But now things have changed. Recall that now I am getting ready to teach using of the Illinois State text -- a text that uses project-based learning to integrate math and science together. In other words, strictly speaking I am a science teacher! And so I want to devote an entire post to the NGSS because, again, I am a science teacher, so I am directly affected by the NGSS!

Keep in mind that I still consider myself to be primarily a math teacher. And as I've been saying all summer, the blog will focus on the eighth grade class. Nonetheless, I did already warn you that this eighth grade class will contain some science projects, and I'll be writing about everything that happens in that eighth grade class -- including geometry, algebra, statistics, and science. As of now, it appears that the eighth grade class will be the largest of my three classes, but that may change as time goes on.

So let's look at the NGSS and see how it will affect my teaching. One thing to keep in mind is that currently under No Child Left Behind, all students must be tested in science once during each of elementary, middle, and high school.

Here in California, the chosen grades for the NCLB tests are fifth, eighth, and tenth grades. But this test will not be the official NGSS test, but merely a "pilot test" for the new standards. My current seventh graders will take the "field test" for NGSS next year, and my current sixth graders will take the operational NGSS test the following year.

Here is a link to the NGSS as implemented in California:

http://www.cde.ca.gov/pd/ca/sc/ngssstandards.asp

Just as with the Common Core Math Standards, the NGSS for the elementary Grades K-5 are divided up by grade level, while for high school, the standards are divided by topics (as in Biology and Chemistry) rather than grade level.

But right now, I want to look at the middle school standards, since this is what I'll be teaching. As it turns out, the middle school standards are somewhat divided into 6th, 7th, and 8th grade standards, but how they are divided is very different from what you might expect.

Integrated Middle School Science

Back in my Black Friday post, I wrote that I wasn't sure whether California would follow the integrated pathway or the traditional pathway for middle school science. In many ways this is similar to the Integrated Math I, II, III vs. Algebra I, Geometry, Algebra II debate for high school math -- under the old pre-NGSS California standards, middle school science focused on Earth Science in the sixth grade, Life Science in the seventh grade, and Physical Science in the eighth grade.

As it turns out, the integrated pathway is now the preferred pathway for middle school science. Still, the traditional pathway exists for those districts that prefer the old way. (Notice that the word "preferred" is not used to describe either of the high school mathematics pathways.)

Here's a link to an old EdSource thread from last year, which debates new middle school standards for science:

https://edsource.org/2015/california-adopting-integrated-model-for-science-in-middle-grades/74814

We already know that traditionalists oppose integrated math, and so it goes without saying that traditionalists are the ones who oppose integrated middle school science.

In my new class I will be teaching science from the Illinois State STEM text. This text was written before the NGSS standards, and so we don't expect the three grade-level texts to line up perfectly with either the integrated or traditional pathways. Still, it may be helpful to look up some of the projects in the sixth, seventh, and eighth text to see which pathway the projects correspond to.

For example, suppose the Illinois State eighth grade text corresponded to the old California eighth grade class -- physical science. Then we would expect most of the NGSS physical science topics to be covered in the eighth -- not sixth or seventh -- grade text.

Let's look at the first physical science strand according to California:

MS-PS1 Matter and its Interactions

This appears in both the sixth and eighth grade. Of course, you could argue that those units are all about earth science -- but then that just goes to show that there's a fine line between earth and physical science that's marked only because the topics are taught two years apart (in California).

And the next physical science strand is:

MS-PS2 Motion and Stability: Forces and Interactions

And this is clearly found in all three grade texts because this is covered with the first project of the year (the race cars) in the opening unit, Tools for Learning (which is identical in all three texts).

So it's evident that the Illinois State text is using an integrated model. Keep in mind that we've been using "integrated" to refer to both integration within a subject (Algebra/Geometry, Earth/Physical Science) as well as integration between subjects (Math/Science). Integration in one respect doesn't entail integration in the other -- though clearly the Illinois State text is both. (Then again, we should not be surprised that the Illinois State text doesn't follow the old California traditional pathway.)

I've said before that Integrated Math I teachers would do well to follow my Common Core Math 8 class here on the blog, even though they might be annoyed by the science projects that don't necessarily correspond to freshman year science.

Examples of New Science Standards

So far, I keep talking about the new NGSS, but I have yet to post any actual standards. Well, let's look at some of the standards under this physical science strand, Forces and Interactions. All of these are included in the eighth grade standards (both traditional and integrated models):

MS Forces and Interactions
MS Forces and Interactions  Students who demonstrate understanding can:
MS-PS2-1. Apply Newton’s Third Law to design a solution to a problem involving the motion of two colliding objects.* [Clarification Statement: Examples of practical problems could include the impact of collisions between two cars, between a car and stationary objects, and between a meteor and a space vehicle.] [Assessment Boundary: Assessment is limited to vertical or horizontal interactions in one dimension.]
MS-PS2-2. Plan an investigation to provide evidence that the change in an object’s motion depends on the sum of the forces on the object and the mass of the object. [Clarification Statement: Emphasis is on balanced (Newton’s First Law) and unbalanced forces in a system, qualitative comparisons of forces, mass and changes in motion (Newton’s Second Law), frame of reference, and specification of units.] [Assessment Boundary: Assessment is limited to forces and changes in motion in one-dimension in an inertial reference frame and to change in one variable at a time. Assessment does not include the use of trigonometry.]
MS-PS2-3. Ask questions about data to determine the factors that affect the strength of electric and magnetic forces. [Clarification Statement: Examples of devices that use electric and magnetic forces could include electromagnets, electric motors, or generators. Examples of data could include the effect of the number of turns of wire on the strength of an electromagnet, or the effect of increasing the number or strength of magnets on the speed of an electric motor.] [Assessment Boundary: Assessment about questions that require quantitative answers is limited to proportional reasoning and algebraic thinking.]
MS-PS2-4. Construct and present arguments using evidence to support the claim that gravitational interactions are attractive and depend on the masses of interacting objects. [Clarification Statement: Examples of evidence for arguments could include data generated from simulations or digital tools; and charts displaying mass, strength of interaction, distance from the Sun, and orbital periods of objects within the solar system.] [Assessment Boundary: Assessment does not include Newton’s Law of Gravitation or Kepler’s Laws.]
MS-PS2-5. Conduct an investigation and evaluate the experimental design to provide evidence that fields exist between objects exerting forces on each other even though the objects are not in contact. [Clarification Statement: Examples of this phenomenon could include the interactions of magnets, electrically-charged strips of tape, and electrically-charged pith balls. Examples of investigations could include first-hand experiences or simulations.] [Assessment Boundary: Assessment is limited to electric and magnetic fields, and is limited to qualitative evidence for the existence of fields.]

I admit that this is a bit more advanced than the class I had back when I was an eighth-grader. Even though eighth grade was always devoted to physical science, the exact content of the class was often fuzzy until the introduction of the California Science Standards (by which point I was already past the eighth grade) and later on, the NGSS.

I've discussed many of these ideas as we read Morris Kline's book in the spring. Indeed, I pointed out that I might read the first few chapters of Kline in my classes.

Another Bill Comment

Earlier today, the traditionalist Bill commented in another thread, this time about free college:

http://www.cnn.com/2016/08/03/opinions/free-college-opinion-freeland/index.html
http://www.joannejacobs.com/2016/08/cheap-college-is-better-than-free/

Here Bill is replying to another poster, Rob, so let's give the latter's comment first.

Rob says:

I was agreeing up to the point of the “coding bootcamps”. I suppose you can turn someone into a basic web monkey in a few intensive weeks, but you can’t turn out a journeyman coder in that time. This is a distraction, however.

College isn’t for everyone, because not everyone wants a cerebral job (like coding). Lots of people prefer working with their hands and have a job that just ends at quitting time.

In response to Rob, Bill says:

I would agree with Rob, as being able to write decent software (which isn’t web pages, btw) requires a lot of training and skill which a 8-12 week boot camp isn’t going to cut it at all.

Writing secure software is a very difficult task, as I probably found more than 1000 bugs which are actual software errors across 40 or so software programs (some of which support critical internet functions, btw), and correcting those errors can be very time consuming and frustrating.

Coding Bootcamps seem to be the snake oil of certification bootcamps in the early 2000’s, where anyone could simply book study the material and pass the examinations, without ever having touched a physical piece of equipment.

In STEM fields, real learning takes a great deal of time and patience, and not everyone can handle the required amount of coursework…

There are two issues here -- the first is, do all students need to go to college? Here Rob and Bill agree that the answer is no -- the solution isn't free college or cheap college, but no college for students who don't need it.

The other is that if students really want a STEM jobs, Bill suggests that such students prepare themselves for the required coursework -- and that begins in middle school, by getting A's in their math and science classes and remembering the material long after the class is over.

Fractions Yet Again!

Let's get back to math, since math is not going away. Yesterday, the traditionalist Bill posted in a thread at the Joanne Jacobs website. As usual, I link first to the article, and then the Jacobs website where Bill commented:

http://www.chalkbeat.org/posts/ny/2016/07/29/how-do-you-get-students-fired-up-about-fractions-reinvent-math-class/
http://www.joannejacobs.com/2016/08/teachers-study-how-to-teach-fractions/

Actually, let's look at a comment written by another poster, Tressa.

Tressa says:

I suppose it isn’t revolutionary, but I have found that when I talk to people about math, especially those who say they were never good at it, they always were lost when the school taught fractions. Unfortunately, they tell me that they never caught up again.

I am not sure what it is. Do the schools rush through fractions? Do the kids get lost, but the schools keep going? Higher math is, to put it mildly, frustrating if you don’t have a solid grasp on fractions.

So while this probably isn’t rocket science, I will take it as good news that they understand fractions are important and their current method of instruction is lacking. I also think understanding fractions is a developmental issue. Some kids will grasp it later than others, but by that time the class has moved on. The whole thing frustrates me. It is one of the main reasons why I chose to homeschool my kids.

In response to Tressa, Bill says:

Teaching fractions isn’t all that hard…the old plastic pizza/pie/cake usually did the trick in my day, though without a solid knowledge of fractions, a student will probably not master algebra and will never complete algebra II/trig (unless it’s so watered down that they don’t do fractions in that class anymore).

SIGH

Of course, we all know too well what Tressa is saying here. More often than not, the first topic with which students struggle in math is fractions. Tressa writes that her solution to the fraction problem is to homeschool, so that she can tailor her fraction instruction to the pace at which each of her children individually understands. But not everyone homeschools, and it's impossible to provide such 1-on-1 instruction to every student in a classroom.

Bill responds by claiming that teaching fractions is easy. Yet if teaching it were so easy -- why do so many students struggle with fractions? And kids had trouble with fractions well before Common Core, so we can't blame it on the new standards.

Bill continues his post by saying that he learned fractions by modelling them with pizza. But this reminds me of what Dr. Hung-Hsi Wu -- whom I mentioned in my Pi Approximation Post -- says about fractions:

https://math.berkeley.edu/~wu/Schoolmathematics1.pdf

Wu: If a fraction is just a piece of pizza, then how to multiply two pieces of pizza?

Later on in the same document, he writes:

For example, a teacher familiar with a logical development of fractions would recognize the futility of relying exclusively on cutting pizzas in order to teach fractions. Such a teacher would be more likely to introduce the number line as early as possible.

But note the importance in this context of having precise definitions of both 3/8, a fraction, and 0.375, a decimal, as numbers. Anything less (such as only knowing a fraction as a piece of pizza) and this equality wouldn’t even make sense.

So we see that Wu definitely disagrees with Bill's approach.

By the way, notice what's in the article itself:

But when Kris got to the board, his understanding seemed to vanish. He was supposed to be placing the fraction five-thirds on a number line, but instead he marked off five and one-third, almost four points too far.

For his part, Cipparone never pointed out Kris’ error that morning. He threw the question back to the class for a quick debate. And even though some students argued their classmate was wrong, Cipparone stood back and facilitated.

Now this sounds eerily similar to something posted by Fawn Nguyen, a fellow California middle school math teacher (whom I quoted a few months ago here on the blog):

http://fawnnguyen.com/wbt/

Mrs. Quiggle: Timmy, where on the number line is four-fifths?

Timmy: Ummm. Between four and five?

Entire [...] Class: It’s cool!

T: Yeah? It’s cool? I got it right?

Mrs. Q: Oh, four-fifths is right here. Right about here. If you can just imagine this space spliced up… Yes, that’s right, it’d be sitting right here, Timmy.

T: So I did not get it right.

[Entire Class -- dw]: It’s cool that you said four-fifths is between four and five, Timmy. We wanted you to be merry.

Now Nguyen writes that this "Whole Brain Teaching" link was merely satire -- a commentary on how strange some progressive teaching methods are. And yet the Live Science link above described a math class that taught fractions using almost exactly the method Nguyen considers to be too satirical to be real -- a student gives an incorrect answer, and the teacher would rather start a class discussion than directly tell the student that he's wrong! (This is often referred to as Poe's Law -- it's often difficult to distinguish between a parody of an extreme view and the real thing!)

As usual, I like to take the middle path between Bill's traditionalism and Quiggle's progressivism. I point out that a traditionalist would have emphatically told Kris and Timmy that they were wrong -- and the students might have resented being told that they were wrong. Traditionalists assume that the students would just meekly accepted that they were wrong, even though it's human nature to defend oneself instead of admitting the truth.

I admit that there is a fine balance here -- how to get a student to accept that he is wrong without wanting to resent it. According to the article, Kris does have some understanding of fractions, so it might be preferable to ask Kris to locate 1/3, 2/3, and 3/3 on the number line, or even to have him convert 5/3 to a mixed number before locating it on the number line -- and meanwhile, that class discussion can still take place. This way, Kris can see for himself that he is wrong -- he is neither led to believe that his answer is "cool," nor does he resent being corrected by the teacher.

Here's another example:

Teacher: What's 2+2?
Student: Five.
Teacher: Hmmmm -- when you add 2+1, you get three, right?
Student: Yes.
Teacher: And if you add another 1 to that, you get four, right?
Student: Yes.
Teacher: So when you add 2 things to 2, what's the answer?
Student: Oh, it's four. (Holding up fingers throughout the conversation would help!)

Fractions in the Illinois State Text

The big question, of course, is how will I teach fractions in my classroom? Recall that most fraction instruction occurs in Grades 5-7 (and the article linked above focused on fifth grade), so again, this is not directly related to the eighth grade class that I plan on posting here on the blog.

A big sixth grade topic is fraction division. In the sixth grade module "Playing the Nails," there is an example of the division 3/4 divided by 1/2 using number lines. There's a very similar example of this, dividing 5/6 by 3/4, right on Fawn Nguyen's site:

http://fawnnguyen.com/dividing-fractions/

And so my fraction lessons will be a combination of ideas -- number lines, some class discussion, activating prior knowledge -- possibly even some pizza models, if the students happen to understand these best, as Bill once did.

My next post will be early next week.