Benchmark Tests (Day 10)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

The exterior angle of a regular polygon is 13 1/3 degrees. How many sides does the polygon have?

To solve this problem, we use a theorem from Lesson 13-8 of the U of Chicago text:

Exterior Angles of a Polygon Sum Theorem:
In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360.

Thus we must write and solve the following equation:

360/n = 13 1/3
n = 360/(13 1/3)
n = 27

Therefore the desired polygon is a 27-gon -- and of course, today's date is the 27th.

Today I'm posting the Benchmark Tests. Angles of a polygon do appear on this benchmark, and so today's Pappas question might help us.

Here is the Blaugust prompt for today:

How do you support struggling students?  What intervention strategies have you used?

To respond to today's Blaugust prompt, as usual, I think back to three years ago. Let's look at old some posts from around that time.

January 12th, 2017:
I need to mention my eighth grade class today, especially since they're learning about transformations on this Common Core Geometry blog. In the end, I decided to delay the science lesson to tomorrow and teach transformations today.

This means that this week I had three full days to cover the three transformations. On the first two days, the translations and reflections went well, and most students appeared to understand. But I worried as today's lesson approached, because rotations are probably the most difficult of the three transformations for students to understand.

Now keep in mind that I'm using the Student Journals that are part of the Illinois State text. We know that rotations can be centered either at the origin or away from the origin. Rotations centered at the origin have easier formulas -- for example, the rotation of 180 degrees centered at the origin maps the point (x, y) to the point (-x, -y).

But none of the rotations mentioned in the Illinois State text are centered at the origin. Most of the questions direct a student to rotate a line segment around one of its endpoints. This at least makes it a little easier, since every rotation maps its center to itself.

And so here's what I did today -- on the first page, the students are asked to rotate AB 90 degrees clockwise about point A, The coordinates are A(2, 3) and B(7, 3). I had the students change A to "the origin," and then I show them the 90-degree rotation about the origin. To do this, I had the students the paper 90 degrees counterclockwise -- that is, the opposite direction from the rotation. Then they drew the image A' by going 2 units on the new x-axis and 3 units along the new y-axis. They did the same to find B', and then they restored the paper to its original position. The new segment A'B' now appears to be the clockwise rotation image of AB.

Of course the students are confused by this at first, but in the end, I believe that they're starting to get the hang of this. I like teaching rotations this way because it sets them up nicely to learn the slopes of perpendicular lines later on. By the end of class, I think the most confusion came from changing all the questions in the Illinois State text, which were geared towards the rotation centered at A rather than the origin.

So that's one possible answer to the Blaugust prompt -- I help struggling students by making the questions in the text easier.

January 23rd, 2017:
Meanwhile, today is a coding Monday. In case you're curious, sixth graders create logos for an imaginary company, while seventh graders learn about spreadsheets. The students learn about various Excel functions, including mean, median, and mode. I don't normally have music break on coding Mondays, but I couldn't help singing the Measures of Center song from last month to jog the students' memory.

I notice that often when I wrote that my class "struggled" on something, that something was actually the Monday coding assignment, not math! But in this case, the coding lesson was math-related -- and my solution was to sing a song to remind them of what they had learned.

I admit that in both of these cases, the students simply shut down. They didn't realize why learning math was worth the effort. I used the field trip to inspire them -- I reminded them (especially the girls) that if they studied hard in math, they could become the next Katherine Johnson.

Today is a test day, which makes this our first traditionalists' post of the new school year. Actually, it's been about a month since our main traditionalists (Barry Garelick and SteveH) have posted. So let's seek out some older traditionalist posts.

In fact, I'm still thinking about yesterday's Benjamin Leis post about quadratics and cubics. One question that came up yesterday is, if we're not going to teach the Cubic Formula to high school students, then why are we still teaching the Quadratic Formula? Well, let's see what the traditionalists have to say about the Quadratic Formula.

We might as well start with Barry Garelick again. His most recent mention of the Quadratic Formula was back in January 2018:

https://traditionalmath.wordpress.com/2018/01/15/count-the-tropes-dept-4/

Here Garelick was actually quoting a non-traditionalist (reform math) supporter, Brett Berry:

Brett Berry:
We forget. All the time. Especially when we don’t understand why the algorithm works or if it has been a long time since we last used it. I can guarantee if I took a survey at Starbucks right now, and asked people how to perform long division, convert a mixed number to a fraction, recite the quadratic formula, factor a binomial, and complete the square — most would fail. Not because math is too hard or people are bad at it, but because memorization and algorithms are not the best ways to retain information. We remember through context, understanding and application.

Interestingly enough, the comments read:

BL:

I never did memorize the quadratic formula. But my math teacher showed us how it’s derived, and I quickly did that before each test in the margin of the test paper.

Problem solved. (And, as a result, I never forgot about “completing the square”!)

Hmm, do the initials "BL" stand for "Benjamin Leis"? Indeed, Leis does seem to echo what BL wrote last year:

http://mymathclub.blogspot.com/2019/08/cardanos-method.html

The quadratic formula song troubles me too because I worry it hides a lack of conceptual understanding. I have other memories of dragging kids through a problem that required completing the square where it was clear their mastery was incomplete.

And by extension, even memorizing the Quadratic Formula can be seen as requiring less "conceptual understanding" (those words much reviled by traditionalists) than completing the square. So it makes sense for BL to be Benjamin Leis (as both appear to find completing the square to be superior to just memorizing the Quadratic Formula).

But then Garelick mentioned the Quadratic Formula in an earlier post, dated June 2016:

https://traditionalmath.wordpress.com/2016/06/21/ps-youre-an-idiot-dept/

My grandmother didn’t take Algebra 1, but I took it in the 60’s and I suppose it’s people like me that Ryan is referring to. I have a bunch of textbooks from that era. I’ll be teaching 8th grade algebra at a school in California, in which the school district doesn’t sit on the high horse that SFUSD likes to occupy. In looking through the Common Core-aligned algebra book I’m forced to teach from, I’m aghast at the dearth of good solid word problems, the short shrift given to exponentials, to rational expressions, not to mention the omission of solving quadratic equations by factoring–I guess the quadratic formula saves a lot of time and there’s no value in teaching that approach. There is a chapter on statistics (as if that’s needed in an algebra class), and a superficial look at exponential functions, which I suppose allows people like Ryan to say “Look how deep this course is. Not your grandmother’s algebra 1”.

Here Jim Ryan is also anti-traditionalist. It all goes back to the district in San Francisco which no longer offers eighth grade Algebra I -- the traditionalists' preferred eighth grade math class. Ryan was trying to argue that Common Core Algebra I is more rigorous that traditional pre-Core Algebra I, so it makes sense to wait until freshman year to take it. Garelick countered that Common Core Algebra I isn't more rigorous, since while it contains stats and exponentials, it lacks factoring.

We know that Common Core emphasizes understanding -- and so we must conclude that, unlike completing the square, factoring does not lead to greater understanding of quadratic equations (or otherwise the Core would emphasize it).

Here's a May 2017 post from our other traditionalist website, Joanne Jacobs, where quadratic equations are mentioned:

https://www.joannejacobs.com/2017/05/how-your-brain-beats-google/

Jonathan Rochelle, the director of Google’s education apps group,  “cannot answer” why his children should learn the quadratic equation, according to a recent New York Times story. After all, they can just “ask Google.”

(By context, I interpret "the quadratic equation" to mean the Quadratic Formula.) Of course, there's clearly a conflict of interest here -- it's the director of Google's ed app groups who advocates that Google makes learning the Quadratic Formula archaic.

Tying this back to Leis and the Cubic Formula, though, we see that anyone who needs to solve a cubic formula (that can't easily be factored) just turns to Google or technology to solve it. If we rely on Google to solve cubic equations, then why not quadratic equations?

Then again, I don't care what the employee of Google thinks about Google. I care about what actual students sitting in our classes think about whether technology makes learning obsolete. For that, we scroll down to one of the commenters in the thread, SC Math Teacher:

SC Math Teacher:
I once had a student incorrectly graph a slope-intercept equation on a quiz. She protested and showed me the graph on her TI-83 calculator. She sure knew how to enter the equation and press the graph button, but she knew nothing about slope and intercept. She copied the graph incorrectly onto her quiz sheet. Knowledge of the subject matter is necessary before one uses a calculator. Or Google.

This is a tricky one. I assume that the girl here was arguing that since she copied the answer from the calculator -- and calculators are a priori always correct -- then she must be correct. But it's possible she could be making the larger argument that the existence of graphing calculators make the need to learn how to graph linear equations obsolete.

SC Math Teacher stated that the girl knew how to enter the equation correctly. I suspect her error was in not understanding how to interpret the calculator window -- so that she was unable to convert the line on the screen to the correct coordinates on her paper.

Another commenter asked of SC Math Teacher:

KateC:
[W]hy do you let your students use a calculator?

Indeed, why would they -- presumably in Algebra I -- be allowed to use a graphing calculator?

SC Math Teacher:
That ship has sailed long before the students reach me. I can stand against the tide, I suppose, but with low-level students struggling to multiply single-digit numbers, the benefits would be little, if any.

Of course, graphing calculators are excellent tools in the hands of the adept.

There are several issues here. By "long before the students reach me," does this imply that the students were using graphing calculators in middle school? If not, then the graphing calculator "ship" hasn't really sailed long before Algebra I.

Of course, this comment is all about calculators in general -- especially since drens (that is, "students struggling to multiply") are mentioned later on. The idea being made in this post is that these students (the graphing calculator girl and the drens) all believe that actually knowing math is obsolete. Once again, that's the belief of the students, not just Google execs.

So here's an interesting question -- suppose I were the teacher in this Algebra I class. So I see the girl who graphs lines on her calculator and the drens who multiply on the calculator. Suppose these students make it clear that they believe that actually learning math without a calculator is archaic. So what should I do?

I'm still searching for that proverbial line between what students really need to know and what they can just use technology for. We agree that the Cubic Formula is on the technology side. And I agree with the traditionalists that multiplication is on the "need to know" side -- hence my use of the word "dren" to refer to such students.

Well, I'd start by calling such students "drens," of course. Recall that the use of the word "dren" is not to insult or embarrass a student, but to make a point. So if I see a student try to use a calculator to solve a simple multiplication problem, then I subtly approach him (while the others are working independently) and remind him that what he's doing is "drenny." (Does "drennish" sound better as an adjective than "drenny"?)

If the student continues to argue that he shouldn't have to learn how to multiply because of modern technology, I remind him that most modern technology was developed by students who worked hard and earned A's in their math classes. (This includes the Google exec mentioned above -- he may be arguing that no one needs to learn the Quadratic Formula, but he needed to earn A's in his math classes in order to be hired by Google.)

Thus a "dren" is someone who is more afraid of learning too much ("How dare you try to teach me something I don't need to know?") than too little. And if this is because he's unwilling to work hard and make the sacrifices necessary because he'd rather spend that time doing other things, we must ask whether those other things involve playing video games (developed by A-students), such as Fortnite (developed by A-students), on a cell phone (developed by A-students). Thus a "dren" is also someone who wishes that math class would disappear so that more time can be spent using technology developed by people who earned A's in math. (If math were to disappear, so would that technology!)

OK, so that's what I'd tell the drens, but what about the graphing calculator girl? I typically don't call someone who can't graph lines a "dren." On the other hand, linear equations probably approach the proverbial line between "need to know" and "just use technology."

In previous posts, I argue that the end of Common Core 8 (or Integrated Math I) is a good place to draw the line. This includes some algebra and geometry on the "need to know" side. Quadratic equations would fall on the "just use technology" side, but graphing linear equations would be one of the last things on the "need to know" side. (The SAT's "Heart of Algebra" also represents the approximate position of my line, with "Heart of Algebra" on the "need to know" side.)

But as we see with this girl, it's often one of the first Algebra I lessons where students struggle. So what would I tell her?

Well, I might begin by comparing her test to someone who earned full credit for this question and ask her for the differences between the graphs, then show her why the other student's graph better matches the graph on her calculator. Then I can point out that the students who earned full credit generally did so without a graphing calculator.

All the while, I remind her why it's important for her to learn how to graph lines. Once again, if she enjoys playing games on her phone, I point out that the graphics in the game don't work unless someone codes in where to draw the lines -- and that requires knowing how to graph lines as studied in Algebra I.

Here's one more thing about SC Math Teacher's proverbial "ship" that has sailed -- where all "sailors" on this ship want to use calculators for everything. When I taught at the old charter school, very few sixth graders reached for a calculator, while most eighth graders did. This suggests that the "ship" sails in between -- in seventh grade.

(Let me stop mixing metaphors -- my "line" with SC Math Teacher's "ship." Instead, call my line "a river" for that ship to cross. On one side of the river is simple math that students should know without a calculator, and the other side is advanced math for which a calculator is OK.)

Throughout K-6, students are learning arithmetic, and the teachers show them how to do arithmetic one step at a time. They might use methods that traditionalists disapprove of (such as the lattice method), but they don't reach for a calculator. In seventh grade, the study of arithmetic is complete, and instead the four operations are used to solve problems. Teachers no longer show step-by-step arithmetic, and so they just reach for a calculator. This habit becomes ingrained in their minds as they progress through Grades 8-12.

This is why, in some earlier posts, I wondered whether it's a good idea to delay teaching some operations of arithmetic into seventh and eighth grade -- such as complex multiplication and division of decimals or fractions -- without a calculator. This forces the students to avoid calculators for an extra year or two, so that they can reach Algebra I before SC Math Teacher's ship has sailed.

Today, only one Blaugust participant has posted. So we return to the blog of Sue Jones:

https://resourceroomblog.wordpress.com/
https://resourceroomblog.wordpress.com/2019/08/27/geogebra-anybody-want-to-try/

Whereas, right now I don’t have any official projects with deadlines that keep me motivated to keep learning and doing when I’m not helping students, and

Whereas, I really REALLY want to know Geogebra better, and

Whereas, I work a whole lot better when it’s with other folks,

Therefore I hereby invite anybody out there to spend … hmmm…. a week … dabbling in Geogebra with a *specific* kind of activity to learn to build.

Here Jones is writing about the math software called Geogebra. She's creating a challenge for her readers -- creating an actual quiz in Geogebra. She considers her challenge to be part of Blaugust, even though it actually extends into September.

Hmm, so we jump directly from traditionalists complaining about technology to a Blaugust challenge encouraging the use of technology.

Notice that Jones wants to use Geogebra to make a quiz that the software can grade. She even mentions displaying a "thermometer" to show the students' progress so far. Thus this quiz would be similar to the IXL software that I used at the old charter school. In other words, Jones wishes to use Geogebra because it makes things easier for the teacher.

But what effect will Geogebra have on the students? Suppose, for example, SC Math Teacher were to give that linear equations quiz on Geogebra. I've never used Geogebra, but I assume that it graphs lines just as easily as a TI-83. So the entire class would be just like that girl -- graphing lines without any real effort. And of course, it's possible for drens simply to open another tab and ask Google to multiply small numbers. So I can see why traditionalists might oppose Geogebra.

In the end, I won't join the Geogebra challenge because I don't have the software. It might be interesting to see what sorts of quizzes the other participants can come up with.

OK, let's finally post the Benchmark Tests. These are based on old finals posted to the blog. I admit that the tricky thing about Benchmark Tests in Geometry is that the students are coming off of a year of Algebra I, when they've thought little about Geometry at all. This is different from Benchmark Tests in middle school or Integrated Math, where there should be some continuity from year to year.