Chapter 8 Test (Day 90)

Today is the Chapter 8 Test. It is also Day 90, the mathematical midpoint of the year. As we already know, most Early Start schools don't actually begin early enough in August to have a true semester of 90 days before winter break.

Last year, the mathematical second semester was when SBAC Prep began. Indeed, the bell schedule was changed so that SBAC Prep would replace most of P.E. time. Since we'd lacked a real conference period anyway, things truly became tough for me once SBAC Prep began.

This is what I wrote two years ago about today's test:

Let's worry about the Chapter 8 Test that I'm posting today on the blog. Here are the answers:

1. This is a triangle tessellation. The triangle is isosceles, so it shouldn't be too hard to tessellate.
2. One can find the area by drawing a square grid and estimating how many squares are taken up on the grid. Since the shape happens to be an ellipse, one can also find its area using the formula for the area of an ellipse -- pi * a * b, where a and b are the major and minor axes. That's right -- I had to slip in a reference to pi this week.
3. 5/2 or 2.5.
4. 12 square units.
5. 40 square units.
6. 6 mm. This is a trick if one forgets the 1/2 -- especially with 3-4-5 right triangles featuring in the last few questions, but here the legs are 6 and 4, not 3 and 4.
7. 10 feet.
8. 4.5s.
9. One can find the area by drawing a diagonal to triangulate the trapezoid. Then one adds up the area of the two triangles.
10. Answers may vary. The simplest such rectangle is long and skinny -- 1 foot by 49 feet.
11. Choice (a). The triangles have the same base and height, therefore the same area.
12. 3s^2.
13. 13 feet.
14. 6 minutes.
15. 780,000 square feet.
16. 133 square feet.
17. 1/4 or .25. Probability is a tricky topic -- the U of Chicago assumes that the students already know something about probability. Then again, it should be obvious that the smaller square is 1/4 of the larger square.
18. 4,000 square units.
19. 13.5 square units.
20. a(b - c) or ab - ac square units.

But here's a 2018 update: again, the test I wrote that year was based on Lessons 8-1 to 8-6 (which is what we covered that year), rather than Lessons 8-4 to 8-9. It's awkward to make the students answer so many questions about the first three lessons of the chapter but not the last three. And so if teachers wish, they may replace the first two questions on the test with the following:

1. Give the circumference and area of a circle with radius 10: a. exactly; b. estimated to the nearest hundredth.
2. Could the numbers 1, 2, sqrt(3) be the lengths of sides of a right triangle?

Answers:
1. a. 20pi units, 100pi square units
    b. 62.83 units, 314.16 square units
2. Yes. (Notice that c here needs to be 2, not sqrt(3).)

Today is a test day, and so as usual I'm labeling this post "traditionalists." Let's see what our favorite traditionalist, Barry Garelick, has to say:

https://traditionalmath.wordpress.com/2018/01/16/everyones-happy-in-happy-land-dept-2/

Chicago Tribune carries this story about a controversial approach to education being tried at various high schools in Illinois :

They’re abandoning most aspects of traditional classroom instruction and reshaping the way kids learn.

 Well, that sounds hopeful, now doesn’t it?

Of course, we already know the answer to Garelick's question. No, it doesn't sound helpful -- at least not to traditionalists like him.

So far, this post has drawn five comments. Surprisingly, only one of these was posted by our favorite commenter, SteveH:

“[“I’m scared,” Hardy admitted.
He said the program is not a perfect solution and not every student will graduate prepared for college, work and life after competency-based learning.
Even so, there’s no turning back.
“We had a school on the brink of failure,” he said.]”

Perhaps not prepared for “… work or life after competency-based learning”?

So let’s just lower expectations and make it seem official. Never mind what the requirements are for college, vocational school, work, or even life.

OK, so evidently SteveH's first problem with this "competency-based learning" program is that he sees it as lowering expectations.

Actually, let's back up to Garelick's description of the program itself. We see that it's being taught in Chicago and several other schools throughout Illinois. We already know that Chicago (as in the U of Chicago text and its elementary counterpart Everyday Math) and Illinois (as in the Illinois State text) are hotbeds of anti-traditionalist pedagogy. I wouldn't be surprised if Garelick's opposition to this program is due at least in part to the city and state in which it is being piloted.

The original Chicago Tribune article describes a high school Geometry class, and since this is a Geometry blog, let me post this part in full:

At Ridgewood High School, math teacher Tristan Kumor started off geometry class on a recent day by asking his students, “How do you want to learn today?”

His 9th and 10th graders sat in groups, with one student using Popsicle sticks to build a bridge. The lesson related to triangles. Kumor traversed the room guiding students individually on their progress, which is part of the way competency-based education plays out.

Garelick notes that so far, this sounds just like Project-Based Learning or PBL. Indeed, there might have been a project similar to this in the Illinois State text, but we never reached it last year. Of course, we already know what he would prefer these Geometry students to be doing in class -- answering individual problem sets (or p-sets).

Notice the title of Garelick's post -- "Everyone's Happy in HappyLand." We know that students in a traditional Geometry class might give complaints such as "Why do we have to do proofs?" and "When will we ever use these proofs?" But students in Kumor's class are less likely to complain, because they enjoy building a bridge more than building a proof. The question "When will we ever use this?" never arises. In other words, Kumor's students are happy -- and Garelick acknowledges this by referring to this classroom as "HappyLand."

But Garelick's concern, of course, is that these happy students aren't learning anything. He'd much prefer that students unhappily learn things than happily learn nothing. As usual, there's something that Garelick leaves out -- students who aren't happy doing an assignment are likely to leave the assignment blank.

Presumably, by building a bridge, students learn that triangles are "stable" in that a triangle with three Popsicle sticks doesn't collapse. The reason for this is SSS Congruence -- there is essentially one triangle given three side lengths. Garelick would argue that students would learn this better by writing a traditional proof rather than building a bridge. But I would counter that students learn more by building a bridge than leaving a proof blank -- which is what many students would end up doing.

There are other parts of "competency-based learning" (CBL) that Garelick criticizes. He writes about a "three-before-me" approach, where students are to ask each other for help, not the teacher:

We don’t want teachers teaching  handing it to the student, now do we?

First of all, consider the Latin phrase docendo discimus -- "We learn by teaching." Traditionalists ought to appreciate that this phrase goes back 2000 years, as it's attributed to Seneca the Younger. So you can't get much more traditionalist than that.

Meanwhile, we must also consider the fact that many students, especially teenagers, avoid listening to the teacher at all costs -- especially when the teacher is telling them that they're wrong. Teens are more likely to accept corrections from other students than from the teacher. Garelick just assumes that students will accept the teacher's corrections just because he or she is the teacher.

The next two Garelick comments are about the grading scale. First, CBL doesn't use traditional letter grades -- in particular, there are no F's:

Right, the Jo Boaler approach which holds that mistakes grow your brain. And if mistakes are that powerful, then failing is even better!

No, the idea is that some students think that getting an F is the end of the world. They believe that they are no good at math and simply leave all subsequent assignments blank. Remember, students learn nothing from the assignments they leave blank, no matter how traditional they are. Under the "Jo Boaler approach," students are less likely to leave assignments blank.

Apparently, CBL grades are on a scale of 1 to 4. Garelick responds:

And let me guess; no one gets a 4. At least that’s how it’s played out in other schools that have tried similar things.

Ah -- so now he criticizes the lack of high grades, not the lack of low grades. This strongly reminds me of the French grading system (see my November 15th post), where 20/20 is impossible. But I suppose he goes have a point here -- the impossibility of a 4/4 grade is much different from the impossibility of 20/20, where there are still so many good grades (19/20, 16/20, etc.) possible.

I'm not quite sure what exactly Garelick believes about the lack of 4/4. I do now that by limiting both high and low grades, the grade gap between the top and bottom student is reduced. This is the opposite of tracking, where differences between students are emphasized. Many traditionalists favor tracking, but it is generally avoided for political reasons.

Let's return now to SteveH. He writes:

It appears to be an opt-in pilot program, and I see that Proviso West still offers the traditional AP Calculus track.

This is a response to the first two commenters in the thread. They are concerned that this program is an "experiment" on children -- and indeed, previous traditionalists have accused progressive programs of treating students like "guinea pigs." But that's the problem -- it's possible that there could be a curriculum that is truly more effective at teaching students, but logically, some cohort must be the first to use it, and the parents of that first cohort may think that their kids are being experimented on like guinea pigs.

Of course, traditionalists have a solution -- traditionalism is the most effective pedagogy possible, and so there's no need ever to try anything else. Traditionalism should remain until the end of time. Hence there are no experiments and no guinea pigs needed.

Recall that the U of Chicago has a "Lab School." (If I recall, a former president sent his children to this school.) Parents who send their children there are already aware that new curricula are being tested there, so they won't feel that the kids are being mistreated like guinea pigs. This is another solution to the new curriculum problem -- test the new curriculum only at schools where the parents choose to send their children.

And indeed, this is what SteveH writes. Apparently, at Proviso East High School the new CBL curriculum is being tested, but at Proviso West High traditionalism remains. (There are freshmen in the East High Geometry class and so officially those students are on an AP Calculus track, but SteveH does specify traditional AP Calc track, and CBL isn't traditional.) He allays the fears of the first two commenters that students are being forced to take CBL Geometry.

You end up with a chasm that you can’t Pre-AP away and you can’t claim a program of self-competency.

SteveH explains the problem with "Pre-AP" classes in an earlier post. According to him, the best way to prepare students for AP classes is to teach them traditional math beginning in kindergarten, rather than teach progressive math up through middle school and then place them in "Pre-AP" classes.

That’s the fundamental fuzzies flaw – lower expectations are always less, not more. Engagement plus student-driven PBL is not even a proper vocational education let alone a path to STEM.

I don't know what SteveH does consider to be a proper vocational education if not PBL. I hope it's not just assigning p-sets and assuming that teens will actually complete them simply because the teacher says so.

There is one thing that I agree with Garelick in this post. At the end, we see:

Educators say the grades and transcripts have been a source of concern, with high schools reluctant to change because that data is used in college admissions.

Well yeah, there’s that, but let’s not let that stop such a promising program.

Like Garelick, I'm also concerned that a new grading system can confuse colleges. In past posts, I've written that almost no high school uses a trimester calendar, since college admissions officers expect to see semester grades on the transcripts. Indeed, it also explains why even though schools are split regarding whether sixth grade (or even fifth) is an elementary or a middle grade, nearly all agree that high school starts with ninth grade -- because colleges want to see four years of high school English.

So what should high schools do if they want to pilot a new grading system? This is tricky. Perhaps college applicants -- and only college applicants -- could have their 1-4 scores converted to traditional letter grades. But how should we do this? Should we use the traditional GPA conversion where 4 is an A, 3 is a B, and so on? But then if Garelick is right that there are no 4's, then there are no A's -- and those students would be at a disadvantage compared to seniors from schools that still give out traditional A grades.

Furthermore, such conversion defeats the purpose of having alternate grades if "3" really just means "B" and "2" just means "C," and so on. The conversion between 1-4 grades and letter grades probably shouldn't be a one-to-one conversion.

Otherwise, perhaps 1-4 scores should be reserved for middle school and below only. (Elementary schools often already use 1-4 or other grading systems.) Then when the students reach high school, the expected letter grades can appear on the transcript.

OK, here is the Chapter 8 Test: