**Table of Contents**

**1. Theoni Pappas and MTBoS 2017 Blogging Initiative**

**2. Movie Review:**

*Hidden Figures***3.**

*Hidden Figures*and My Classroom**4. Upcoming Plans for My Algebra I Student**

**5. Upcoming Plans for Other Eighth Graders**

**6. Provability vs. Ease of Understanding**

**7. Upcoming Plans for Grades 6-7**

**8. More on Traditionalists**

**9. Checking in at Fawn Nguyen's Blog**

**10. Today's "Day in the Life Poster"**

**Theoni Pappas and MTBoS 2017 Blogging Initiative**

**As I mentioned in my last post, Theoni Pappas isn't doing her**

*Mathematical Calendar*this year. So here is today's Pappas-inspired question:

25 is which term of the sequence -10, -3, 4, 11, ...

As it happens, 25 is the sixth term of that arithmetic sequence -- and today's date is the sixth.

Also, in my last post, I wondered whether there will be a MTBoS 2017 Blogging Initiative. Since then, they just made the announcement:

https://exploremtbos.wordpress.com/2017/01/05/new-year-new-blog/

Yes, there's a 2017 Blogging Initiative. No, "Day in the Life" isn't the first week's topic -- instead, it's "My Favorite," which last year was the second topic.

Then again, it's just as well because the deadline for this week's post is tomorrow, so I wouldn't be able to double dip "Day in the Life" for two blogging challenges. Hopefully "Day in the Life" will appear during Week 2 or 3 of the challenge. I just hope I'll be able to finish the "My Favorite" post by tomorrow's deadline!

**Movie Review:**

*Hidden Figures***Back in my First Day of School post (August 16th), I wrote that I was looking forward to the new movie**

*Hidden Figures*. Well, that movie is finally out, and today I watched it. So let me give a full review of the movie right here. Of course,

**spoilers**will abound in this post, so those who haven't watched the movie yet should just skip today's post altogether.

The protagonist is Katherine Johnson (nee Goble), a mathematician and scientist. She is a real person, and in fact she's still alive -- she turned 98 just after the trailer was first released. We first meet the young Katherine as she is growing up in West Virginia. She is very smart, especially in math, but she can't attend her local high school because she is black. So a high school for African-Americans contacts her family to invite the girl to attend. Her parents are shocked, because she's just getting ready to complete the sixth grade. But the administrators are impressed when they see Katherine solve a complicated algebra problem on the board. As a math teacher, I can tell you that all the math in the movie appears to be genuine. Katherine solves a quartic, or fourth-degree, equation that has already been factored into two quadratics. The girl explains how she used the Zero Product Property to find all four solutions.

The scene jumps to the early 1960's. The now middle-aged Katherine is riding in a car with her two companions, Dorothy and Mary, when the car breaks down. A police car arrives on the scene, and the cop is impressed when he finds out that the three women work for NASA. This is right after the Soviet launch of Sputnik, and thus the Americans are now working hard on their own launch of a space capsule.

Throughout the film, Katherine uses her knowledge to assist NASA with the launch. She's working as a human computer -- someone who spends all day calculating figures. Her boss, Mr. Harrison, needs someone skilled in Analytic Geometry to assist with determining the flight path -- and of course, he choose Katherine.

Naturally, Katherine faces several challenges due to both her gender and her race. She's assigned to assist her white coworker Paul, who resents her so much that doesn't even want to let her drink from the coffeepot. My own students often ask to go to the restroom during class, and so does Katherine during her work -- but the nearest colored bathroom require her to walk a full mile round trip, in high heels! Unlike my students, though, Katherine carries her work with her. Mr. Harrison is annoyed when she has to leave for forty minutes at a time.

With the subject matter so serious,

*Hidden Figures*provides moments of comic relief. I laughed at the scene where Katherine is asked to help Paul with a calculation, but he has to black out some of the numbers because she lacks a security clearance. She figures out the solution anyway, and both Paul and Mt. Harrison want to know how she was able to see the numbers. Her response is that she just held up the paper to the light! Mr. Harrison admonishes Paul and asks him to use a darker marker.

We also learn a little more about Katherine's family. She is a widow who has to raise three daughters with only her own mother to help out. After NASA learns that a Russian, Yuri Gagarin, has orbited the earth, Katherine and the others must spend long hours working away from their families. Despite this, she meets a new guy, Colonel Jim Johnson, whom she eventually marries. It's revealed that the colonel is also still alive, and the couple has just celebrated their 56th wedding anniversary.

Meanwhile, Katherine's companions Dorothy and Mary are dealing with their own issues. Sometimes it's difficult to tell whether Mary's problems are due more to gender or race. In order to advance at NASA, Mary must take night courses at the all-white high school. After she finally convinces a judge to let her attend the class, she finds out that she's the only woman in the class. Dorothy, on the other hand, finds her job as a human computer threatened by a mechanical computer -- IBM. She wants to learn the computer language FORTRAN, but the book she needs is in the white library. In the end, she learns FORTRAN and becomes the supervisor in charge of coding.

The climax of the movie is when astronaut John Glenn is set to orbit the earth. The engineers must calculate the "go/no go" point where the space capsule would reenter earth's atmosphere. Glenn is worried that the calculations are incorrect, and so NASA calls in the only mathematician whom he trusts to find the exact point of reentry -- Katherine. Glenn is launched into space, and he's supposed to orbit the earth seven times, but instead orbits it only thrice. He's afraid that he will burn up upon reentry, but with the help of Katherine and the other engineers, his capsule safely lands in the water near the Bahamas. By the way, the real John Glenn fell short of seeing his depiction in the movie, as he died about a month ago.

As a math teacher, I enjoyed this movie greatly! I recognized more actual math in the movie. For example, to calculate the "go/no go" point, I see Katherine multiply a certain number of degrees by pi/180 -- that is, she converted the degrees to radians. And I also liked seeing actual clips from the 1960's of the

*Friendship 7*capsule, President Kennedy, and Martin Luther King, Jr.

Of course, in a movie all about calculation, I did see one point where dramatic license was taken. At the end of the movie, it's mentioned that Katherine celebrated her 56th anniversary in 2016 -- so she got married in 1960. So she was already married by the time of the launches depicted in the movie, which were in 1961 and 1962. But I know that the producers include the wedding in between the launches anyway, in order to break up the serious tone of the movie.

Overall, I loved the movie as much as the two science movies from two years ago,

*Imitation Game*(Turing) and

*Theory of Everything*(Hawking). I highly recommend that math teachers -- and anyone else interested -- watch

*Hidden Figures*.

*Hidden Figures*and My Classroom**I try to avoid politics on this blog -- and I especially try to avoid discussing race on the blog. When I do write about politics or race, I always attempt to bury these issues in a post during a vacation period and save school-year posts for content in mathematics (or science, or computers) only.**

(Then again, in my most-viewed post of calendar year 2016, I mentioned Donald Trump -- so much for burying political posts. It was back in January, before the primaries, and I was mulling on a Trump third-party run. Oh, how little I knew then of what was coming...)

But race is one of the central themes of

*Hidden Figures*, and it's impossible for me to write about the movie without mentioning racial issues. Fortunately, this is still technically winter break, and so I can safely write about controversial topics, as I've done before in these vacation posts.

I work at a charter school in Los Angeles. For those of you familiar with my area, you're aware that most charters in L.A. are predominantly minority, and mine is no exception. Here are the approximate racial demographics of my three classes:

Sixth grade: 60% black, 40% Latino

Seventh grade: 60% Latino, 40% black

Eighth grade: 80% black, 20% Latino

I admit it's striking how different the racial ratios are in comparing the different grades, especially in comparing the seventh and eighth grades! In a previous post, I've stated that the large majority of my eighth graders are girls. So we see that most of my eighth grade class consists of black females -- just like Katherine Johnson.

And so it's obvious what message I want to give when I mention the movie

*Hidden Figures*to my students --

*if you work hard in my math and science class, you can grow up to be an influential mathematician or scientist like Katherine Johnson!*And I especially want the eighth graders --

*the majority of whom are African-American girls*-- to receive this message.

And indeed, the dean at our school often warns the students about gentrification. He fears that many of our students will no longer be able to live in their neighborhoods because they'll be pushed out by higher-income white residents. The safeguard against this, he says, is to find a high-paying job, in particular a STEM position, such as Katherine Johnson's, or a tech job in Silicon Beach.

My top eighth grader -- you know, the one who transferred from a school offering Algebra I -- is a black girl. But many of the other black girls are struggling -- indeed, they're often distracted. They spend class time talking about anything other than math and science. I fear that they may think they're not supposed to be good at math and science because they're black, female, or both.

A central theme of my class is that those who excel in math and science are heroes, and those who can't do basic math are "drens." I repeat the word "dren" every time I give the Dren Quiz, but I don't think I talk about the heroes of math and science nearly enough.

I admit that until I heard about the movie, I didn't know who Katherine Johnson was. But now that I know, I definitely consider Johnson to be a hero of mine. She used math and science in the real world to make things work -- and without her, John Glenn may have never gone into orbit, and we may never have landed on the moon.

Someday, we'd like to go to Mars. I'd like to believe that one of my students will be able to contribute to such a mission. In order for that to happen, they will have to do well in my math and science class, especially with eighth grade being such a critical year. Success in my class will set them up to do well in high school classes -- failure will doom them to struggle throughout high school and eventually wind up with a low-paying job.

Here's another way to think about it -- we can drag through the necessary calculations first, or we can just jump into outer space and hope things work out because we hate math so much. We see what Glenn preferred -- and I assume that if my students were in a spaceship headed to Mars, they'll hope that someone has diligently calculated how they can land safely as well.

And so I

*strongly*encourage my students to watch the movie. My plan is to offer a significant amount of extra credit to anyone who watches it and answers a few basic questions about the plot. I haven't mentioned it in class yet, simply because the last time they were in school was over three weeks before the release of the movie, and so they'd have forgotten it by today.

I'm still not sure how many of my students will watch the movie though. I'll give them until the end of the trimester to claim the extra credit. As

*Hidden Figures*is considered to be an Oscar contender, it should be playing in theaters at least until the end of February. Maybe one of my eighth grade girls who desperately needs a grade boost might watch the movie -- and be inspired to keep her grades up in my class from that point on.

Oh, and of course I'll definitely be rooting for

*Hidden Figures*to win an Oscar! I think the Academy will announce its nominations next week.

**Upcoming Plans for My Algebra I Student**

**Who among the eighth grade girls in my class is most likely to pursue a career path similar to Katherine Johnson's? The answer is obvious -- my transfer student. I promised her that I'll get her through Algebra I this year so that she can take freshman Geometry -- the class she would have taken had she not transferred to our charter school.**

The key to her independent study of Algebra I is our online software, IXL. But first things first -- the school (not I, but the school) needs to assign her an IXL account. I'm hoping that she'll be set up by Thursday -- the first time eighth graders have IXL time after the break. If her account isn't ready by Thursday, she'll have to wait an entire week until the following Thursday, since the other day for eighth grade IXL, Monday, is the Martin Luther King Jr. Day holiday.

We're nearly halfway into the year, so I won't start at the beginning of the Algebra I curriculum. Here I link to the Algebra I curriculum provided on the LAUSD website, since she most likely transferred in from a district school:

http://achieve.lausd.net/Page/6079

There are five units listed here, so most likely her old class would be reaching Unit 3 now. But we notice that this unit is called "Descriptive Statistics." At this point, we must revisit the Common Core Standards and California's implementation of them.

Statistics and probability are both included in the Common Core Standards for high school. But recall that the Core doesn't completely divide the high school standards into courses. In particular, it's uncertain what year stats and probability are supposed to be taught.

In California, at least for the traditional (as opposed to integrated) path, statistics is included in the Algebra I standards, while probability is included in Geometry. Notice that I could have included a unit on probability in my Geometry posts during the first two years of this blog, but I didn't. Indeed, far from posting probability units just to suit Californians, I did the exact opposite and ended each year with PARCC questions, when my state doesn't even take the PARCC.

So Algebra I in California includes stats, but the state doesn't dictate when to teach it. That's up to the districts -- and apparently, LAUSD teaches stats during Unit 3.

So far, I haven't written about traditionalists during winter break, but that changes right now. Many traditionalists oppose the teaching of stats in Algebra I -- they say that this time would be better spent covering polynomials and the Quadratic Formula. I even posted last summer that I'd offer some of to teach some of my eighth graders some Algebra I during the Common Core 8 stats units in order to see whether they'll understand the Algebra I lesson better than stats.

This was all before I knew that I'd have an actual Algebra I student in my class. And so to be consistent with what I wrote earlier, I should skip Unit 3 and go straight to Unit 4, which is on Expressions (including polynomial expressions) and Equations. Still, I might offer some of the other stronger students the opportunity to accelerate with a few Algebra I units -- but unfortunately, this will probably end up being mostly the guys.

The best thing for me to do is ask the new girl herself. Let's see whether she'll prefer starting Unit 3 or going straight into Unit 4, as soon as she's given that IXL account.

**Upcoming Plans for Other Eighth Graders**

**Now in Common Core 8, we're in the geometry units. And the lessons to be covered include translations, reflections, rotations, and dilations -- in other words, transformations.**

The name of this blog is "Common Core Geometry." Geometry is my specialty -- and so I hold myself to high expectations when it comes to teaching anything related to geometry. And I am completely

*disappointed*in myself for the way that I taught -- or more accurately, didn't teach -- volumes of cylinders, cones, and spheres, which are clearly classified as geometry.

But

*within*geometry, my specialty is transformations. I've devoted

*two years*to blogging about transformations and how they relate to other geometry topics. Therefore, it is

*unthinkable*that I would wind up failing to teach transformations properly to my eighth graders. I want to make sure that I'm not just teaching transformations, but teaching them

*right*. Anything less is unacceptable.

So right now, while I still have the luxury of time during winter break, let's look at the upcoming Learning Modules in the Illinois State text with the utmost detail. Each of these modules will be listed with the corresponding Common Core Standards:

7. Shapes, Angles, and Structures

CCSS.MATH.CONTENT.8.G.A.1

Verify experimentally the properties of rotations, reflections, and translations:

CCSS.MATH.CONTENT.8.G.A.2

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

8. Tessellate a Structural Design

CCSS.MATH.CONTENT.8.G.A.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

CCSS.MATH.CONTENT.8.G.A.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

CCSS.MATH.CONTENT.8.G.A.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

9. SimilarityUse informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

*For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so*.
CCSS.MATH.CONTENT.8.G.B.6

Explain a proof of the Pythagorean Theorem and its converse.

Explain a proof of the Pythagorean Theorem and its converse.

CCSS.MATH.CONTENT.8.G.B.7

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

CCSS.MATH.CONTENT.8.G.B.8

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

The remaining eighth grade geometry standard is volume -- and I already covered volume (not adequately, though) in Learning Module 6.Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Now recall that I assign ten school days to each learning module. So let's fill the dates in:

7. Shapes, Angles, and Structures (Days 70-79, December 6th-January 12th)

8. Tessellate a Structural Design (Days 80-89, January 13th-27th)

9. Similarity (Days 90-99, January 30th-February 10th)

So as you can see, we're already seven days into Learning Module 7. There are only three days left -- Tuesday, Wednesday, and Thursday of next week -- before I need to begin Module 8. The fact that a three-week break comes right in the middle of the module certainly doesn't help.

Also, we must remember that even though there are ten days assigned to each module, it doesn't mean that we have ten actual days of instruction available. Let's look back at Module 7 again:

Days 70, 71, 72 -- "Shapes, Angles, and Structures" STEM project

Day 73 -- 4's Dren Quiz

Day 74 -- Monday Coding

Day 75 -- Green Team Pretest

Day 76 -- Last Day Frosty Activity

Two IXL days for eighth grade occurred during this module so far (Days 72 and 74). On those days, I assigned an IXL assignment that

*introduced*translations, reflections, and rotations, but of course the students can't completely fully grasp what they are yet. The Frosty activity didn't help -- ironically, the turkey activity before Thanksgiving mentioned dilations, but I ended up giving that graph to only sixth and seventh grade. The Frosty activity had no transformations at all.

There are three days left in this module, and there are three transformations to teach. This seems like a no-brainer -- one transformation each of the three days. But there's another problem -- Thursdays are supposed to be for science with my Bruin Corps member -- and remember that the eighth graders need to have science, too! I'm not sure whether he'll actually be present in my class this Thursday, as he's just coming back from his winter break, too. If he's here, then I really have only

*two*days to cover the three transformations.

(On the other hand, I won't lose any time to music any time soon. Today I received an email that the music teacher injured himself during winter break and will have surgery on Tuesday, January 10th, the day that students return. He'll require all of January, and perhaps part of February, to recover.)

Anyway, which transformation should I teach first? In the U of Chicago text that I used to teach transformations for the first two years of the blog, we covered reflections first, since translations and rotations are each just the composite of two reflections. But is that approach, which I used for a high school Geometry course, necessarily appropriate for eighth graders?

I briefly alluded to the following, but let's repeat in more detail the relationship among Common Core 8, Integrated Math I, and traditional Algebra I and Geometry classes.

1. Question: How much algebra content should be taught in the following classes?

a. Common Core 8

b. the first semester of Algebra I

c. Integrated Math I

Answer: I've always stated that all three classes should have the

*same*algebra content. Using the LAUSD curriculum as a guide, by "first semester of Algebra I" I mean roughly Units 1-2, while the second semester includes Units 4-5. (I've leaving Unit 3 out, since I asked about

*algebra content*, not statistics content.) Put in another way, the first semester of Algebra I emphasizes linear equations, while the second semester emphasizes quadratic equations.

So my opinion is that both Common Core 8 and Integrated Math I should also emphasize linear equations during the algebra portion of these respective classes, Because of this, I often wrote that Common Core 8 and Integrated Math I are essentially identical classes -- and I use that to argue against traditionalists who claim that integrated math doesn't lead to senior-year Calculus. If Common Core 8 and Math I were identical, then we could accelerate students by having them go from Common Core 8 to Math II as freshmen, and freshman Math II readily leads to senior Calculus.

But here's the problem -- we haven't looked at the geometry content yet:

2. Question: How much geometry content should be taught in the following classes?

a. Common Core 8

b. the first semester of high school Geometry

c. Integrated Math I

It would be great if the answer to question 2 was just like question 1 -- all three courses could have the same geometry content. Then this would strengthen the argument that Common Core 8 and Math I are identical, and so Calculus-bound students can take freshman Math II.

But the truth is, the geometry content in a. and b. are decided

*not identical*. A blaring difference between Common Core 8 geometry and high school Geometry is that

*proofs*are required in the latter but not the former. On the other hand, volume is a Common Core 8 topic, but doesn't appear until second semester Geometry (again using the LAUSD curriculum to define semesters of Geometry).

**Provability vs. Ease of Understanding**

During the first two years of this blog, I wrote about the dependence of high school Geometry on proof, and linked to various mathematicians (Joyce, Wu, etc.) who described extensively how to prove various theorems.

In a proof-based course, an easily proved yet difficult to understand theorem must be taught before a difficult to prove yet easily understood theorem -- especially if the former result is used in the proof of the latter. In a course not based on proofs (think Serra), we are free to present results in the order most easily understood by actual students, without our hands being tied by knowing which results are needed to prove other results.

For example, we consider the three main transformations. Of translations, reflections, and rotations, which are the easiest for students to understand? The answer is obvious --

*translations*. We know that the whole purpose of the lesson is to show that any figure mapped to another by any isometry -- including reflections and rotations -- are congruent. Still, two figures that are translation images of each other are more easily seen to be identical than reflection or rotation images.

If you think about it,

*reading*is dependent on translation equivalence. The translation image of the letter

*d*is another

*d*written elsewhere on the page. On the other hand, when we reflect the letter

*d*, we no longer have the letter

*d*, but a new letter,

*b*. Likewise, when we rotate the letter

*d*by 180 degrees, we have a new letter, namely

*p*. The letters

*d*

*q*, written in this order, are glide reflection images of each other. So only when we translate

*d*is the letter still

*d*.

Likewise, let's take two congruent triangles and ask which of the transformations maps one to the other -- since by definition of

*congruent*there must be such a transformation. If it's a reflection or rotation, we can't tell unless we look carefully at the triangles to determine the orientation. But if it's a translation, we can tell at a glance. The conclusion is obvious -- of all the transformations, the translation is the easiest to understand, by far. And so we should begin with translations.

But in a proof-based course, translations can't come first. For example, in the U of Chicago text, a translation is

*defined*to be the composite of reflections in parallel lines. So at least reflections must be studied before translations in order to avoid circularity -- indeed reflections appear in Chapter 4 while translations don't appear until Chapter 6.

And Hung-Hsi Wu, a mathematician at Berkeley, recommends teaching rotations -- at least those of 180 degrees -- before translations as well. This is because translations, by their close relationship to parallel lines, require a Parallel Postulate or the equivalent Playfair, while rotations don't. In the Euclidean tradition, as many results should be proved using neutral geometry as possible before invoking a Parallel Postulate -- this is parsimony of postulates. Therefore translations must be taught after both reflections and rotations, since the first appears after Playfair while the other two appear before Playfair.

So we'd like to cover translations first, yet in high school Geometry we can't, for two reasons that are strongly related to proofs (non-circularity and parsimony of postulates). But in an

*eighth grade course*, we're not tied down by what can and can't be proved. And so I can -- and will -- start with translations, since these are the easiest for students to understand.

The other big issue regarding transformations is the

*coordinate plane*. We remember that the U of Chicago text emphasizes transformations without the plane. Indeed, the simple rule that the images of (

*x*,

*y*) reflected in the

*x*- and

*y*-axes are (

*x*, -

*y*) and (-

*x*,

*y*) respectively appear nowhere in the text! We were also concerned with provability once again. The reflection images in the coordinate axes are easy to prove, but the statement that a translation maps (

*x*,

*y*) to (

*x*+

*h*,

*y*+

*k*) is difficult to prove.

We know that on actual Common Core tests such as the SBAC, the coordinate plane is strongly emphasized during the transformation lessons. Indeed, almost every single question involving transformations shows a coordinate plane. And so rather than hide the coordinate plane, I'm going to dive straight into coordinates when I begin teaching these lessons.

Once again, translations are the easiest to work with. Every single translation maps the point (

*x*,

*y*) to (

*x*+

*h*,

*y*+

*k*) for some

*h*and

*k*, and every single mapping of this form is a translation. On the other hand, eighth graders are only expected to reflect across the coordinate axes, and possibly the lines with equations

*y*=

*x*or

*y*= -

*x*. That's four lines in the entire plane. And eighth graders -- indeed, even high school Geometry students -- only rotate multiples of 90 degrees. A general formula for the point (

*x*,

*y*) rotated by a general angle

*theta*requires trigonometry!

And so my decision has been made. On Tuesday, the first day back for my students, I will teach students about translations on the coordinate plane. I won't try to squeeze in a second transformation that day, because I want to make sure that my students truly understand translations. I'll cover both reflections and rotations on Wednesday, in anticipation of a science day on Thursday. Friday will be the first day of the STEM project for Learning Module 8.

But what does all of this mean for Calculus-bound students at Integrated Math schools? A student who jumps from Common Core 8 to Math II will get all of the algebra content of a traditional student, but will miss some Geometry, especially proofs. It all depends on how important proofs are to those traditionalists who like to push students into Calculus. An argument can be made that Algebra is more important than Geometry from the perspective of preparation for AP Calculus -- but then again, Analytic Geometry is what helped Katherine determine the space capsule flight path.

**Upcoming Plans for Grades 6-7**

**I know that I emphasize eighth grade on the blog, but let's not forget grades six and seven. The seventh graders are learning about angles -- especially supplementary, complementary, vertical, and adjacent angles.**

Let's tie this back to the message of

*Hidden Figures*. In seventh grade, most of the students are Latino, but nonetheless, all students still need to know how important it is to learn math.

Returning to the

*Hidden Figures*target demographic of black females, I point out that one thing that many of the black girls in my classes enjoy is dancing. Indeed, during winter break, one eighth grade girl participated in a dance performance for Kwanzaa, an African-American holiday celebrated from December 26th to January 1st. And while there are fewer black girls in seventh grade than eighth grade, I've seen those in my class dance from time to time as well.

Recall that the first song after winter break that I plan on singing during music break is from

*Square One TV*, and it's called "Angle Dance." That's right -- the song features a

*dance*. And so I have an opportunity to engage the black girls in my class right away. Again, as I wrote in my December 18th post, many of these seventh graders will just watch me raise my hands alone for "Human Protractor," laugh at me, and possibly even record me on their cell phone so others can make fun of me. But I'm hoping that students will want to participate if it's an "Angle Dance" instead.

Meanwhile, in sixth grade, the students will be learning about GCF, greatest common factor. But for the first project on Friday, I'm jumping up to Learning Module 24. That's because this is the first module of Unit 6: Mathematics in Water -- and water and energy conservation are strongly related to the Green Team science unit that's coming up. The Common Core Standards associated with this unit are in statistics.

By the way, our software company IXL just announced that they're adding middle school science and history to their curriculum. But unfortunately, when I tried to click on the science units, I can only participate in a 30-day trial. If that's the case, then I'll just wait until 30 days before the NGSS Science Test, and just use the other science software instead.

**More on the Traditionalists**

**Wow, I've almost gone through an entire winter break without discussing the traditionalists -- and even in this post, I wrote only a little about them. So what exactly have they been up to lately?**

Again, let's tie this back to

*Hidden Figures.*Traditionalists like to point out that it's "traditional math," not "Common Core math," that put a man on the moon. In particular, the math that Katherine Johnson performs in the movie isn't "Common Core math." There's even a scene where Katherine says that she's using not new math, but some rather old math indeed -- Euler's method for estimating the solution of a differential equation. (I've mentioned the seventeenth century Swiss mathematician Euler several times on the blog, especially in connection with the Bridges of Konisgberg puzzle that I gave back on the first day of school.)

Well we know that Johnson skipped two grades so she certainly took math far above what Common Core would prescribe at each age. While it's not necessary that students solve quartic equations as Katherine did in the sixth grade, but the important thing for traditionalists is to take Algebra I in eighth grade to be well-prepared for a STEM-based job with, say, NASA.

It's said that the Sputnik crisis, depicted at the beginning of the movie, is what drove reformers to change the way math is taught in the first place -- nearly a half-century before Common Core. The New Math of the 1960's, like Common Core, is said to focus too much on abstract math rather than teach students the basics. And this, according to traditionalists, is why the New Math had the opposite effect of what was intended -- instead of keeping up with the Soviets, the U.S. finds itself behind many countries in mathematical ability, especially the East Asians.

With all this talk about "Katherine," you may wonder what happened to one of the traditionalists I used to write about all the time, Katharine Beals. Well, she joined forces with yet another Katherine -- Catherine Johnson, of Kitchen Table Math. (Well, she spells it with a "C," but at least her last name is also Johnson.) Their new blog, unfortunately, has nothing to do with math.Johnson no longer updates her old math blog, but Beals has gone back to post her annual "Favorite Comments" written by, who else, her fellow traditionalists.

MTBoS blogger Sarah Carter is not a traditionalist. and neither is Susan Hewett, who recently wrote a guest post on Carter's website. In fact, Hewett teaches middle school in Vietnam:

http://mathequalslove.blogspot.com/2016/12/guest-post-inequality-story.html

Now here's the thing -- in Vietnam, not only do they avoid integrated math (which traditionalists hate), but the eighth grade class is algebra:

https://supermathteacher.wordpress.com/2016/08/26/two-weeks-finished/

Those that dislike algebra are i for it this year, because the 8th grade math class is algebra!

And this is exactly what traditionalists want to see. Defenders of integrated math often say that the countries that score higher than the U.S. teach integrated math rather than the traditional pathway. But here is Vietnam, an East Asian country that scores above the U.S., and avoids integrated math. And furthermore, the eighth grade class is Algebra I -- the class traditionalists like to see in eighth grade!

Because of this evidence, I must admit that a high-scoring traditional pathway nation exists. But still, if this were Vietnam,

*all*of my eighth graders would be in Algebra I, and I'm not sure how many of the actual eighth graders could handle it. Obviously, the new girl can handle it, and perhaps some of the guys can, but this only amounts to about a third of the class. I'd be forced to give two-thirds of my class automatic F's if I had to follow the traditionalist/Vietnamese curriculum

Meanwhile the traditionalist and current middle school teacher Barry Garelick has been actively posting lately. He begins with a link to a HuffPo article by Stanford professor Keith Devlin:

http://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248

Devlin criticizes traditional math as obsolete, so of course Garelick disagrees:

https://traditionalmath.wordpress.com/2017/01/03/ignore-these-messages-dept/

Let me highlight one comment from this thread, from mm:

*The people who are focusing on “understanding” are of several types. There are some good things about understanding. But what SOME of them are attempting to do is to GET RID OF REPETITIVE PRACTICE because THEY DIDN’T LIKE IT. Well, none of us liked it. But it was how we MASTERED problems.*

Now mm is a third grade teacher. The third graders in mm's class might not like drill and practice, but they'll do it anyway because third graders are generally compliant -- at least when compared to middle school students. Middle schoolers don't like drill and practice -- and when middle schoolers don't like something,

*they don't do it*. They'll just throw my packets away -- and I've even seen some just leave them on their desk with no intention of doing it. Yes,

*practice*leads to mastery -- but how much mastery will students who throw their work away or leave it blank on my desk achieve? I want an alternative to traditionalism -- something that middle schoolers will actually

*do*.

**Checking in at Fawn Nguyen's Blog**

Here's a link to Fawn Nguyen's website:

http://fawnnguyen.com/these-twenty-things/

Nguyen provides a list of twenty things teachers should do. Let's look at #19:

19. Let’s not make a list of New Year’s resolutions. It’s like the goddamn pacing guide, sets us up for failure every time. Just repeat #15 above — minus the psycho screaming part, do that just once. Okay, twice. Definitely not more than three times.

Oops, that's too late for me. I already made a New Year's Resolution in my last post. By the way, here's the aforementioned #15:

15. Be kind to yourself. Buy that item you didn’t get for Christmas from your favorite person who is now no longer your favorite. If you sleep next to this person, scream, “I hate you!” in the middle of the night like you are dreaming, except you aren’t.

I'll conclude discussion of this Nguyen post with #14, which is something that I'll like to remember:

14. Remind students that kindness

**trumps**everything you do in your classroom.

[emphasis mine]

Okay, that's the last you'll hear of politics on this blog for a while.

**Today's "Day in the Life" Poster**

**The "Day in the Life" participant with a monthly posting date of the sixth is Dawneen Zabinske:**

http://mszmathmess.blogspot.com/2017/01/day-in-life-expanded-version-december.html

Recall that Ms. Z is a South Carolina middle school teacher, so I'm glad I caught her post! She begins her January 6th post by writing what she did over winter break:

5. Saw "Star Wars: Rogue One" opening night. I'm an old school Star Wars geek (being 46 and having seen the original trilogy first run and in its original format not with the later enhancements). It was amazing and had many call-backs to the original Star Wars movie.

So I guess both of us enjoyed movies during our winter break. Then Ms. Z proceeds to write about a week of school:

We came back to school on Monday, 2 January 2017. Over the past 5 days, we've had two different schedules. Monday, Tuesday, and Wednesday we were on our normal operating schedule; each class I had was 50 minutes plus a lunch period of 35 minutes and a 45 minute planning. The last two days we have had an Semester 1 Exam Schedule so the high school semester classes can have extended classes for exams.

I find two things strange about this schedule:

-- Ms. Z actually had school on Monday, January 2nd. This is odd because January 2nd is supposed to be a legal federal holiday, New Year's Day Observed. Post offices and banks were closed, and so having school on Monday, January 2nd would be just like having school on the 1st itself! I don't think that any school in California held classes that day, and even New York, with its notoriously short winter break, didn't return until the 3rd. But apparently, in South Carolina there can be school on Monday, January 2nd. (I believe in New York, there can be school on the 2nd if it's a Tuesday, Wednesday, or Thursday, but not Monday.)

-- A high school would actually schedule finals for the first week after winter break! (Ms. Z teaches at a 6-12 school.) I've written about the Early Start Calendar and how school must start the first week of August in order for there to be 90 days, or half a year, before winter break. If a school starts later in August, there might be only 85, or nearly 80 days before winter break, but schools would still have finals before winter break (as in the LAUSD) rather than the mathematical halfway point. I've seen schools schedule finals for the second week after break, but never the first until now. Apparently, South Carolina doesn't care if its high school students forget everything over winter break.

(See my December 21st post for more discussion of the school calendar and the madness that ensues when Christmas and New Year's Day are Sundays.)

Well, that's enough about high school -- how about Ms. Z's own middle school classes? She writes about this in her reflection responses. Since she missed her December 6th post, she responds to the questions for both December and January in today's post:

**Whenever I need to try something out on the iPads, I use my second period all boys 7th grade class. They have a mix of ability levels and I can use their feedback to tweak the activity for other class periods.**

__December 2016:__**The whole day was not ideal. I spent most of the day dealing with behavior issues in every class period including those that are not normally an issue. And, yes, I did raise my voice more than normal. We even stopped class to review class rules and where to find them (on the front wall of the room) and what rewards/consequences for various actions. I even stated I'd move our test days next week from Wednesday/Thursday to Monday/Tuesday (for two periods this actually did succeed in calming them down enough to get through the material). I really do hate doing that and only use it as a last resort when all other persuasive measures fail.**

__Today 2017:__In my last post, I wrote about how I don't want to yell to my students. Of course, it's middle school -- and even experienced teachers like Ms. Z have to yell from time to time. Still, I yell almost everyday in my class, and I know that I can reduce this. This is the first time I've heard of moving a test up earlier in order to calm the class down. The closest I've done in my current class is turn a classwork assignment into a pop quiz.

Ms. Z writes that her seventh graders are solving equations and inequalities in her class. In the Illinois State text, equations appear late -- so I fear that we won't reach it before the last day of school. Maybe I'll need to jump around in the seventh grade text, just as I'm about to in the sixth grade text!

**Goal:**

**Working towards a mathematical mindset.**I feel like I am not making progress on this goal. However, that should change in a few weeks. On the 17th of January, we have a teacher workday for district professional development. I have signed up for a half day course on "Creating a Growth Mindset in Middle School Math Classrooms (7th Grade)." Next time I should have more to put here

And I look forward to reading all about it, as this may help me in my own classroom! Notice that her PD day is set up to create a four-day Martin Luther King weekend for the students, just as my school's PD days extend winter break and give a long President's Day weekend.

Tying this back to

*Hidden Figures*, Ms. Z wrote about some of the challenges she has with race back in her September 6th post:

A decision I worry about is the behavior intervention I am currently using with one of my all male classes. It is a class of 23 boys, the majority of whom are African-American and developmentally are below grade level in mathematics. The intervention centers around a achieving a goal for the week. The class started with zero points and points are added every 5 minutes they are compliant with the rules: 40 minutes of class = 8 points. They can also earn additional points for asking a relevant question or answering another student's question or coming to the board to work a problem. However, if they begin to break from the rules, they can lose a point for each 5 minutes they are off task. This worked for about a week and they got close to their goal but didn't quite make it. I have tried rearranging seat assignment. I have tried having students write a discipline essay about their behavior and ways to correct it. I have called or texted parents; I have submitted teacher-managed incident referrals to the office. It's only a few that are causing the disruptions every day; and it's not just my class - it's every class. Usually this point system has worked at least for a few months and a few rewards. I'm seeing with this group - I'm probably going to have to go with individual points/rewards or split the group into two and offer a competition between groups. I might eliminate the taking away of points so to focus more on the positive and less on the negative. Any suggestions??

Notice that Ms. Z's problem is with her black boys, while my problem is with the black girls. She hasn't written about this class in the past four months so I'm hoping that her system is working. You may recall that I came up with something similar in my last post -- both the individual participation point system and a desire to focus on the positive. If it works for Ms. Z, maybe it'll work for me too.

My next post will be tomorrow, for Week 1 of the MTBoS 2017 Blogging Initiative!

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